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2800	https://www.teacherspayteachers.com/browse/free?search=calculating%20density%20worksheets	Calculating Density Worksheets \| TPT Log InSign Up Cart is empty Total: $0.00 View Wish ListView Cart Grade Elementary Preschool Kindergarten 1st grade 2nd grade 3rd grade 4th grade 5th grade Middle school 6th grade 7th grade 8th grade High school 9th grade 10th grade 11th grade 12th grade Adult education Resource type Student practice Independent work packet Worksheets Assessment Graphic organizers Task cards Flash cards Teacher tools Classroom management Teacher manuals Outlines Rubrics Syllabi Unit plans Lessons Activities Games Centers Projects Laboratory Songs Clip art Classroom decor Bulletin board ideas Posters Word walls Printables Seasonal Holiday Black History Month Christmas-Chanukah-Kwanzaa Earth Day Easter Halloween Hispanic Heritage Month Martin Luther King Day Presidents' Day St. Patrick's Day Thanksgiving New Year Valentine's Day Women's History Month Seasonal Autumn Winter Spring Summer Back to school End of year ELA ELA by grade PreK ELA Kindergarten ELA 1st grade ELA 2nd grade ELA 3rd grade ELA 4th grade ELA 5th grade ELA 6th grade ELA 7th grade ELA 8th grade ELA High school ELA Elementary ELA Reading Writing Phonics Vocabulary Grammar Spelling Poetry ELA test prep Middle school ELA Literature Informational text Writing Creative writing Writing-essays ELA test prep High school ELA Literature Informational text Writing Creative writing Writing-essays ELA test prep Math Math by grade PreK math Kindergarten math 1st grade math 2nd grade math 3rd grade math 4th grade math 5th grade math 6th grade math 7th grade math 8th grade math High school math Elementary math Basic operations Numbers Geometry Measurement Mental math Place value Arithmetic Fractions Decimals Math test prep Middle school math Algebra Basic operations Decimals Fractions Geometry Math test prep High school math Algebra Algebra 2 Geometry Math test prep Statistics Precalculus Calculus Science Science by grade PreK science Kindergarten science 1st grade science 2nd grade science 3rd grade science 4th grade science 5th grade science 6th grade science 7th grade science 8th grade science High school science By topic Astronomy Biology Chemistry Earth sciences Physics Physical science Social studies Social studies by grade PreK social studies Kindergarten social studies 1st grade social studies 2nd grade social studies 3rd grade social studies 4th grade social studies 5th grade social studies 6th grade social studies 7th grade social studies 8th grade social studies High school social studies Social studies by topic Ancient history Economics European history Government Geography Native Americans Middle ages Psychology U.S. History World history Languages Languages American sign language Arabic Chinese French German Italian Japanese Latin Portuguese Spanish Arts Arts Art history Graphic arts Visual arts Other (arts) Performing arts Dance Drama Instrumental music Music Music composition Vocal music Special education Speech therapy Social emotional Social emotional Character education Classroom community School counseling School psychology Social emotional learning Specialty Specialty Career and technical education Child care Coaching Cooking Health Life skills Occupational therapy Physical education Physical therapy Professional development Service learning Vocational education Other (specialty) Calculating Density Worksheets 33+results Sort by: Relevance Relevance Rating Rating Count Price (Ascending) Price (Descending) Most Recent Sort by: Relevance Relevance Rating Rating Count Price (Ascending) Price (Descending) Most Recent Search Grade Subject Free Format All filters (1) Filters Free Clear all Grade Elementary 5th grade Middle school 6th grade 7th grade 8th grade High school 9th grade 10th grade 11th grade 12th grade Higher education Not grade specific More Subject Math Algebra Applied math Decimals Geometry Measurement Science Basic principles Biology Chemistry Earth sciences Forensics General science Physical science Physics Social studies Criminal justice - law Geography For all subjects More Price Free Under $5 $5-10 $10 and up Format Google Apps Microsoft Microsoft Word PDF More Resource type Teacher tools Lessons Teacher manuals Printables Hands-on activities Activities Centers Internet activities Games Laboratory Instruction Handouts Interactive notebooks Student assessment Assessment Critical thinking and problem solving More Student practice Independent work packet Worksheets Homework Task cards Workbooks Standard Theme Seasonal Autumn More Audience Homeschool More Calculating Density Worksheet Created by Kayla B's Classroom In this calculating density worksheet, students will practice finding the mass, volume, and density. They will complete a chart with missing pieces. There is also a graphic included to help students calculate the missing information. A calculator for this activity is helpful. Students will be rounding to the nearest hundredth. 6 th Chemistry, Measurement, Physical Science FREE Rated 4.67 out of 5, based on 12 reviews 4.7(12) Log in to Download Wish List Density Column Worksheet De-6 Created by Bluebird Teaching Materials This one-page student worksheet teaches and reviews the concept of density. The worksheet defines density and gives the mass and volume of four different liquids and two solids. Students calculate the densities of the six substances and complete the density table for mass, volume, and density. They then answer three questions about the results and color the layers in the density column according to density. The one-page answer key is full size and is in color. This worksheet is based on the Cali 7 th - 11 th Physical Science FREE Rated 4.78 out of 5, based on 169 reviews 4.8(169) Log in to Download Wish List Get more with resources under $5 See all Calculating Density Practice Problems Worksheets - Mass Volume Density Amy Brown Science $2.50 Original Price $2.50 Rated 4.85 out of 5, based on 47 reviews 4.9(47) Calculating Density Practice Problems Color by Number Worksheets Amy Brown Science $3.75 Original Price $3.75 Rated 4.82 out of 5, based on 11 reviews 4.8(11) Density Notes and Slides \| Calculating Density Worksheet \| Density Practice The Science Bee $4.00 Original Price $4.00 Rated 4.5 out of 5, based on 2 reviews 4.5(2) Calculating Density Worksheet Mr Wagners Science Store $0.95 Original Price $0.95 Rated 4.84 out of 5, based on 26 reviews 4.8(26) Calculating Density Winter Color By Number \| Science Color By Number Created by The Morehouse Magic This activity consists of 10 questions covering the concepts of mass, volume, and density and calculating either mass, volume, or density given a word problem containing the two other variables. With what standard does this resource align?This resource will aid students in moving towards mastery of NGSS MS-PS1-2.What do I have to do?These science color by codes are no prep! Print the answer sheet on one side and the coloring on the other or print both on one side to save ink and copier counts. 6 th - 8 th Chemistry, Physical Science, Science NGSS MS-PS1-2 FREE Rated 4.72 out of 5, based on 39 reviews 4.7(39) Log in to Download Wish List Free Mapping &Calculating Density-Blank Map Coloring Book Series Created by The Human Imprint Geography Coloring Book Series This coloring book map-page covers three prominent types of density discussed in AP Human Geography. I would use this when discussing the three types of distribution (arithmetic, physiologic, and agricultural). Step 1) Lecture on how to calculate these density types. Step 2) Students use QR codes to find current information needed to calculate densities. Step 3) Discuss comparisons/contrasts between the countries and help explain why. Step 4) These three blank Not Grade Specific For All Subjects, Geography FREE Rated 5 out of 5, based on 30 reviews 5.0(30) Log in to Download Wish List Density Column Worksheet Created by Tillies Science Library Students will: Calculate densityPredict how different fluids interact based on their densityCheck their predictions by analyzing a class demonstrationExtensions: Add different small objects (ex: marble, nail, paperclip - whatever fits!) and have students predict where the object will settle based on its relative densityHave students explore a digital density simulation like this one: it a lab! Have students create their own d 5 th - 8 th Applied Math, General Science, Physical Science CCSS, NGSS RST.6-8.3 , 5-PS1-3 , 5-PS1-4 FREE Rated 4.5 out of 5, based on 2 reviews 4.5(2) Add to Google Drive Wish List Calculating Density Practice Created by Math FUN-damentals This is a practice page designed to get students working with using volume and mass measurements to calculate density of an object. This should be done prior to the use of "Identifying Substances By Density" lab included in the store. There are two versions with identical problems that can be used. The first page is to be used if it is going to be assigned as individual practice following class level work doing these types of density, mass, and volume problems. The second page is if you watch to 6 th - 8 th General Science, Physical Science, Science FREE Rated 4.45 out of 5, based on 11 reviews 4.5(11) Log in to Download Wish List Worksheet - Prologue &Density FREEBIE (Editable) Created by NYS Earth Science and Living Environment Regents This EDITABLE worksheet has 9 multiple choice and constructed response Earth Science Regents questions about density, observations, inferences, and classification. This makes a great homework sheet or in-class review. Topics addressed: Observations, Inferences, Classification, Comparing densities of solids and liquids, Factors affecting density, Calculating density. Note to Customer: Click on the ★ above to follow my store. Leave feedback to earn credit points to save money on future prod 8 th - 12 th Basic Principles, Earth Sciences, General Science FREE Rated 4 out of 5, based on 2 reviews 4.0(2) Log in to Download Wish List Calculating Density Practice Problems Created by Support and Survive A worksheet for students to practice calculating the density with word problems. Answer key included! 5 th - 6 th Math, Science FREE Rated 4.75 out of 5, based on 4 reviews 4.8(4) Log in to Download Wish List Density \| Guided Learning Pack\| Digital Worksheets Created by DewWool Welcome to Density - Learning Track!An engaging and interactive resource designed to help students explore the concept of density and its real-world applications. Ideal for Grades 5–8, this learning track includes interactive worksheets, MCQs, drag-and-drop activities, and educational games to make learning both fun and practical. Students will investigate topics such as mass, volume, calculating density, buoyancy, and how density affects different materials and substances. Note: An active inte 5 th - 8 th Biology, Science NGSS 5-PS1-3 , MS-PS1-4 , MS-PS1-2 FREE Log in to Download Wish List Calculating Density Created by Alyse Consiglio Calculating density, rearranging equation to also solve for mass. 6 th - 8 th Science FREE Rated 5 out of 5, based on 1 reviews 5.0(1) Log in to Download Wish List Density Created by Tonya Raglin This can be used for density practice or quiz. It includes a couple questions regarding safety. Calculate desity, read triple beam balance, etc. Ideal for 6th grade, or pre-test/review in upper grades. 6 th - 8 th Science FREE Rated 4.5 out of 5, based on 2 reviews 4.5(2) Log in to Download Wish List Density& Percent Error Quick Labs Created by Terrier-ific Science Enjoy using these 2 quick labs to practice measurement skills, density calculations, and percent error any time throughout the year. In the density lab, students are asked to determine the density of aluminum by measuring the dimensions and mass of sheets of aluminum foil. In the index card lab, students calculate the area and perimeter of an index card, and determine their percent error based on the provided dimensions. Color and B&W versions included.These are considered quick labs becau 7 th - 10 th Algebra, Chemistry, Physical Science FREE Rated 5 out of 5, based on 1 reviews 5.0(1) Log in to Download Wish List Calculating Density Practice Created by Science by Meredith This worksheet offers students practice calculating density of a cube. This is a great way to get students using a formula - as it gives them guidance on the next steps to take. Each problem is broken down into 4 additional boxes that has students determine what is given, develop a formula from the density triangle, complete work and then finally give an answer. 6 th - 8 th Chemistry, General Science, Science FREE Rated 4.73 out of 5, based on 22 reviews 4.7(22) Log in to Download Wish List Density Practice Problems Created by Schmidt Special Ed Use the density formula to calculate density and identify whether the object will float or sink. 6 th - 10 th Chemistry FREE Rated 4.5 out of 5, based on 4 reviews 4.5(4) Log in to Download Wish List Density of Unknown Glass Samples Forensics Lab Created by Texas Criminal Justice Curriculum Fun and easy glass density lab that has students calculate the density of six unknown glass samples. The equipment needed are graduated cylinders, beakers, balances and 6 different glass samples. I recommend broken Pyrex lab equipment, marbles, large glass beads or even broken glass from car windows or the windshield. The lab can be quickly altered for use in crash scene investigations for Law Enforcement or even chemistry! Included are a copy of the lab in word and pdf as well as teacher instr 10 th - 12 th Chemistry, Criminal Justice - Law, Forensics FREE Rated 4.61 out of 5, based on 8 reviews 4.6(8) Log in to Download Wish List Density Practice Questions Created by Michael Golder Five simple questions around calculating density. 6 th - 8 th Science FREE Rated 4.33 out of 5, based on 3 reviews 4.3(3) Log in to Download Wish List Population Density Created by Ryan Mudry I created this to practice calculating population density and estimating population size. I hope it is useful. 6 th - 12 th Biology, General Science FREE Rated 4.83 out of 5, based on 12 reviews 4.8(12) Log in to Download Wish List Tic-Tac-Toe Four in a Row-Density Created by Retro Solutions Instead of having your students complete a worksheet calculating density, why not make a game of it? Print a Density card for each pair of students, place in a page protector, provide each student with an erasable marker and a paper clip or a centimeter cube and let them play "Tic-Tac-Toe Four in a Row". The complete instructions are included along with a game card. As students play the game and become familiar with the density formula they will discover they need to work from the density in 8 th - 11 th Chemistry, Physical Science NGSS HS-PS1-3 FREE Log in to Download Wish List Mass, Volume, Density Lab Created by Samantha Olson This lab can be modified and used for calculating Mass, Volume, and Density. 5 th - 7 th General Science FREE Log in to Download Wish List Density Calculations: Quick Practice, Warm-up or Formative Assessment Created by Melissa Jaeger Use this assignment to check your students understanding of calculating density and how the density determines which liquids will float and sink in a column. Select "print 4 per page" to have a quarter page handout for your students to do their work on. After students can find the density and understand the connection between density and sinking and floating, they will be able to use density in the determination if a new substance has formed or not. 6 th - 8 th Chemistry, Physical Science, Science NGSS MS-PS1-2 FREE Rated 5 out of 5, based on 1 reviews 5.0(1) Log in to Download Wish List Will it Float? Science Density Experiment Created by Jesse Roe In this experiment, students will develop an understanding of density and use mass and volume to calculate the density of film canisters filled with different objects. 5 th - 9 th Physical Science, Physics FREE Log in to Download Wish List 4 Density Handouts (with keys) Created by Siegfried Howell These handouts are a quick review of density and allow students a chance to practice using the density formula to calculate various unknowns. 9 th - 12 th, Higher Education Chemistry, Physical Science, Physics FREE Rated 4 out of 5, based on 1 reviews 4.0(1) Log in to Download Wish List Density of American Coin Alloys Created by Bill Lewis This homework or class assignment is suitable for physical science, general science, or chemistry. This shows a simple way to calculate the densities of alloys. To increase interest it uses familiar objects, American pennies, nickels, dimes, quarters, half dollars, Susan B. Anthony dollars, and the Sacagawea gold-colored dollar. It is in Word format and is structured for easy grading (all answers are aligned on the left margin). The assignment can be extended experimentally by conducting a 8 th - 12 th Physical Science FREE Rated 3.83 out of 5, based on 3 reviews 3.8(3) Log in to Download Wish List P24 Density Mark Scheme Created by sciencecover Access our full selection of workbooks, presentations and mark-schemes at www.science-cover.com. This download contains a lesson workbook mark scheme. This lesson focuses on density. Students are shown how to calculate the density using mass and volume. Students complete practice questions to calculate density using the equation. Students are shown how to use the equation triangle to rearrange the equation. Students complete practice questions to calculate the mass and volume. Students explore 8 th - 11 th Physics FREE Log in to Download Wish List 1 2 Showing 1-24 of 33+results TPT is the largest marketplace for PreK-12 resources, powered by a community of educators. Facebook Instagram Pinterest Twitter About Who we are We're hiring Press Blog Gift Cards Support Help & FAQ Security Privacy policy Student privacy Terms of service Tell us what you think Updates Get our weekly newsletter with free resources, updates, and special offers. 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2801	https://synonyms-antonyms.fandom.com/wiki/Austere_Synonyms	Austere Synonyms \| Synonyms & Antonyms Wiki \| Fandom Sign In Register Synonyms & Antonyms Wiki Explore Main Page All Pages Community Interactive Maps Recent Blog Posts Wiki Content Recently Changed Pages Winsomeness Synonyms Disturbance Synonyms Beneficial Antonyms Powerful Antonyms Benefic Antonyms Benefic Synonyms Powerful Synonyms Synonyms Friendly Synonyms Double Whammy Synonyms Cute Synonyms Adorable Synonyms Sullenly Synonyms Insignificant Synonyms Merciless Synonyms Nouns Irony Sarcasm. Tease Heaven Hell Heck Bitch Community Recent blog posts Forum Sign In Don't have an account? Register Sign In Menu Explore More History Advertisement Skip to content Synonyms & Antonyms Wiki 2,840 pages Explore Main Page All Pages Community Interactive Maps Recent Blog Posts Wiki Content Recently Changed Pages Winsomeness Synonyms Disturbance Synonyms Beneficial Antonyms Powerful Antonyms Benefic Antonyms Benefic Synonyms Powerful Synonyms Synonyms Friendly Synonyms Double Whammy Synonyms Cute Synonyms Adorable Synonyms Sullenly Synonyms Insignificant Synonyms Merciless Synonyms Nouns Irony Sarcasm. Tease Heaven Hell Heck Bitch Community Recent blog posts Forum Contents 1 Definition 2 Synonyms for Austere 3 Sentences for Austere 4 Examples for Austere 5 See Also in:Synonyms Austere Synonyms Sign in to edit History Purge Talk (0) austere [x] Contents 1 Definition 2 Synonyms for Austere 3 Sentences for Austere 4 Examples for Austere 5 See Also Definition[] Severe or strict in manner, attitude, or appearance (Of living conditions or a way of life) having no comforts or luxuries; harsh or ascetic Having an extremely plain and simple style or appearance; unadorned Synonyms for Austere[] "aloof, chilly, cold, distant, dour, flinty, forbidding, formal, frigid, frosty, grave, grim, hard, hard-nosed, harsh, icy, illiberal, mean-looking, no-nonsense, remote, reserved, rigorous, serious, solemn, steely, stern, stiff, stony, strict, stringent, stuffy, unbending, uncharitable, unfeeling, unforgiving, unfriendly, unsmiling, unsympathetic, unyielding" "abstemious, abstinent, celibate, chaste, continent, frugal, hair-shirt, moderate, moral, nonindulgent, puritanical, restrained, self-abnegating, self-denying, self-disciplined, simple, sober, spartan, strict, temperate, upright" "ascetic, bald, bare, basic, bleak, clinical, functional, modest, muted, no frills, restrained, severe, spartan, stark, subdued, unadorned, undecorated, unornamented, unostentatious" Sentences for Austere[] "An austere man, with a rigidly puritanical outlook." "The house was very cold and austere." "The compound loomed in front of him, the cement walls austere and forbidding." "Yes, well, it's almost a transformation of these serious and austere people." "Though he decided against monastic life, he found the austere lifestyle and important model." "Members of the commune live an austere, simple life." "Ishmael was struck by the man's austere expression." "The cathedral is impressive in its austere simplicity." Examples for Austere[] See Also[] Austere Antonyms Categories Categories: Synonyms Community content is available under CC-BY-SA unless otherwise noted. Comments Start a conversation Sign in to share your thoughts and get the conversation going. SIGN IN Don't have account? 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2802	https://www.chegg.com/homework-help/questions-and-answers/draw-structure-arginine-protonation-state-would-predominate-ph-95-draw-molecule-canvas-cho-q73945537	Solved Draw the structure of arginine in the protonation \| Chegg.com Skip to main content Books Rent/Buy Read Return Sell Study Tasks Homework help Understand a topic Writing & citations Tools Expert Q&A Math Solver Citations Plagiarism checker Grammar checker Expert proofreading Career For educators Help Sign in Paste Copy Cut Options Upload Image Math Mode ÷ ≤ ≥ o π ∞ ∩ ∪           √  ∫              Math Math Geometry Physics Greek Alphabet Science Chemistry Chemistry questions and answers Draw the structure of arginine in the protonation state that would predominate at pH 9.5. Draw the molecule on the canvas by choosing buttons from the Tools (for bonds and charges), Atoms, and Templates toolbars, including charges where needed. H: 120 Exp" cont H HAN с ENH N HN O 2 S 2 CI Br 0 HAN Р F Submit Previous Answers Request Answer X Incorrect; Try Your solution’s ready to go! Our expert help has broken down your problem into an easy-to-learn solution you can count on. See Answer See Answer See Answer done loading Question: Draw the structure of arginine in the protonation state that would predominate at pH 9.5. Draw the molecule on the canvas by choosing buttons from the Tools (for bonds and charges), Atoms, and Templates toolbars, including charges where needed. H: 120 Exp" cont H HAN с ENH N HN O 2 S 2 CI Br 0 HAN Р F Submit Previous Answers Request Answer X Incorrect; Try Show transcribed image text There are 3 steps to solve this one.Solution Share Share Share done loading Copy link Step 1 The aim of this question is to show the structure of given amino acid at given pH value. It gives the ... View the full answer Step 2 UnlockStep 3 UnlockAnswer Unlock Previous questionNext question Transcribed image text: Draw the structure of arginine in the protonation state that would predominate at pH 9.5. Draw the molecule on the canvas by choosing buttons from the Tools (for bonds and charges), Atoms, and Templates toolbars, including charges where needed. H: 120 Exp" cont H HAN с ENH N HN O 2 S 2 CI Br 0 HAN Р F Submit Previous Answers Request Answer X Incorrect; Try Again; 4 attempts remaining Not the question you’re looking for? Post any question and get expert help quickly. 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2803	https://zh.wikipedia.org/zh-hans/%E9%81%8B%E7%AE%97%E6%AC%A1%E5%BA%8F	跳转到内容搜索目录序言 1 基本规则 1.1 负号（一元减号） 1.2 乘除混合 2 记忆术 3 特殊情况 3.1 连续乘幂 3.2 连除 4 参见 5 资料来源运算次序 العربية Беларуская Català Čeština Dansk Deutsch Ελληνικά English Esperanto Español Euskara فارسی Suomi Français Galego ગુજરાતી עברית हिन्दी Bahasa Indonesia Íslenska Italiano 日本語 한국어 मराठी Bahasa Melayu Nederlands Norsk bokmål Polski Português Romnă Русский Simple English Slovenčina Shqip Српски / srpski Svenska Kiswahili தமிழ் ไทย Tagalog Türkçe Українська Tiếng Việt 閩南語 / Bn-lm-gí 编辑链接条目讨论不转换简体繁體大陆简体香港繁體澳門繁體大马简体新加坡简体臺灣正體阅读编辑查看历史工具操作阅读编辑查看历史常规链入页面相关更改上传文件固定链接页面信息引用此页获取短链接下载二维码打印/导出下载为PDF 打印版本在其他项目中维基共享资源维基数据项目外观维基百科，自由的百科全书在数学和计算机科学中，运算次序（也称为运算顺序、运算子优先顺序）是指决定在表示式中的哪一运算子首先被执行的规则。比如，在四则运算中，一般有先乘除后加减的规定。就是说在这样的式子中，按规定会先对3和4作乘法，得出12，然后再把2和12加起来，最后就得出14。自引入现代的代数标记法，乘法总是拥有比加法更高的优先次序。而16和17世纪引入了指数（乘幂）以后，指数总是拥有比加法和乘法更高的优先次序。这个规则的定立消除了混淆并允许了标记更加简洁。当需要越过这规则，甚至简单地强调一下，括号 ( ) 会用以标示另一次序或者是强调预定次序来避免混乱。例如 (2 + 3) × 4 = 20 需要加优先于乘，而 (3 + 5)2 = 64 需要加优先于乘幂。为了方便阅读，有时括弧会换为方括弧，例如 [2 × (3 + 4)] - 5 = 9。基本规则 [编辑] 一般用于算术和编程语言的运算次序如下：乘幂和方根乘除加减当算式中同时存在多层的运算子，最高层级的运算子就优先计算。因为加法与乘法的交换律和结合律，加法可以按任何左右次序计算，乘法亦然。但运算子混合起来时需要依从运算次序。在某些场合，把除换成倒数相乘或者把减当成加的相反对于计算更佳。例如在计算机科学中，这允许用少一点二元运算并让简化了的表示式更容易利用交换律和结合律。转换后，3 ÷ 4 = 3 × 1/4，即3和4的商等于3和1/4的积；3 − 4 = 3 + (−4)，3和4的差等于3与−4的和。于是，1 − 3 + 7 可以想像成 1 + (−3) + 7，三个项用任何次序相加都会得出答案是5。根号√传统上是透过上面的横线（线括号）框下被开方数（这避免了框下被开方数时需要括号）。其他函数用括号框著输入来避免混淆。有时候，当输入只是单项式，括号可被省略。所以 sin 3x = sin(3x)，而 sin x + y = sin(x) + y，因为 x + y 不是单项式。不少计算机和编程语言都要求函数加上括号。组合符号可以凌驾这个一般的运算次序。组合符号可以视为单独表示式。组合符号可以用结合律和分配律去除，而组合符号内的表示式简单得不会在符号移除后造成混淆的时候，就应该移除。例子分数线同样是组合符号：为了方便阅读，其他组合符号例如大括弧 { } 或中括弧 [ ] 会与括弧 ( ) 并用。例如：负号（一元减号） [编辑] 一元运算子 − （通常读“负”）有不同的约定。在手写和印刷数学，表达 −32 解读为 0 − (32) = −9。但部分程式和编程语言，典型例子是Microsoft Excel（和其他试算表程式）和编程语言Bc，一元运算子比二元运算子优先，换言之，负号优先于指数，所以在这些语言中 −32 会解读为 (−3)2 = 9。这不适用于二元减号 − ：例如在Microsoft Excel，=-2^2、=-(2)^2和=0+-2^2会得出4，不过=0-2^2和=-(2^2)会得出−4。乘除混合 [编辑] 同样，使用斜杠符号 / 在诸如 1 / 2x 之类的表达式中可能会有歧义。如果改写成1 ÷ 2x并把除法解读为倒数乘法，就变成： : 1 ÷ 2 × x = 1 × 1/2 × x = 1/2 × x 以此解读 1 ÷ 2x 就等于 (1 ÷ 2)x。不过在部分学术文书中，省略了乘号的乘法被视为优先于除，1 ÷ 2x 等于 1 ÷ (2x) 而不是 (1 ÷ 2)x。举一例子，学术期刊物理评论的交稿说明中讲明乘优先于斜杠符号标示的除，而其他著名物理课本也有此规定，包括朗道和利夫希茨编纂的《理论物理学教程》以及《费曼物理学讲义》。记忆术 [编辑] 基本上，各国都有不同的口诀让学生记著运算次序。英文的口诀会以运算子的字首构成。中文地区一般以“先乘除后加减”作为口诀，但是当中没有涵盖指数美国普遍使用“PEMDAS”，代表“Parentheses（括号）, Exponents（指数）, Multiplication（乘）/Division（除）, Addition（加）/Subtraction（减）” 加拿大与纽西兰使用“BEMDAS”，代表“Brackets（括号）, Exponents（指数）, Multiplication（乘）/Division（除）, Addition（加）/Subtraction（减）” 英国使用“BOMDAS”或“BIMDAS”，与前者相比，这两个缩写纯粹改变了指数的词汇（Orders/Indices）其他英语地区普遍使用“BOMDAS” 这些口诀都有机会导致误导，例如误以为“先加后减”，就会错误地运算。 : 10 - 3 + 2 正确答案是 9（但如果先加后减就会变成 5）。特殊情况 [编辑] 连续乘幂 [编辑] 如果乘幂以堆叠上标的方式呈现，一般规则是从上到下： : abc = a(bc) 不是等同于 (ab)c。但是，以符号 ^ 或箭头 (↑) 作表其运算子的时候，并没有普遍标准。例如，Microsoft Excel把a^b^c理解成 (ab)c，而Google Search和Wolfram Alpha就理解成 a(bc)。所以4^3^2在前者会计成4096，在后者会计成262144。连除 [编辑] 连除也存在同样的含糊，例如 10 ÷ 5 ÷ 2 既可解读为 : (10 ÷ 5) ÷ 2 : 10 ÷ (5 ÷ 2) 参见 [编辑] 运算 6÷2(1+2) 资料来源 [编辑] ^ 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Bronstein, Ilja Nikolaevič; Semendjajew, Konstantin Adolfovič. 2.4.1.1.. Grosche, Günter; Ziegler, Viktor; Ziegler, Dorothea (编). Taschenbuch der Mathematik 1. 由Ziegler, Viktor翻译. Weiß, Jürgen 23. Thun, Switzerland / Frankfurt am Main, Germany: Verlag Harri Deutsch (and B. G. Teubner Verlagsgesellschaft, Leipzig). 1987: 115–120 . ISBN 3-87144-492-8 （德语）. ^ Ask Dr. Math. Math Forum. 2000-11-22 [2012-03-05]. （原始内容存档于2021-04-21）. ^ Angel, Allen R. Elementary Algebra for College Students 8. . Chapter 1, Section 9, Objective 3. ^ Formula Returns Unexpected Positive Value. Microsoft. 2005-08-15 [2012-03-05]. （原始内容存档于2015-04-19）. ^ 5.0 5.1 Ball, John A. Algorithms for RPN calculators 1. Cambridge, Massachusetts, USA: Wiley-Interscience, John Wiley & Sons, Inc. 1978: 31. ISBN 0-471-03070-8. ^ Rules of arithmetic (PDF). Mathcentre.ac.uk. [2019-08-02]. （原始内容存档 (PDF)于2021-02-24）. ^ Physical Review Style and Notation Guide (PDF). American Physical Society. Section IV–E–2–e. [2012-08-05]. （原始内容存档 (PDF)于2013-04-20）. ^ 8.0 8.1 Vanderbeek, Greg. Order of Operations and RPN (Expository paper). Master of Arts in Teaching (MAT) Exam Expository Papers. Lincoln, Nebraska, USA: University of Nebraska. June 2007 [2020-06-14]. Paper 46. （原始内容存档于2020-06-14）. ^ Order of operations (DOC). Syllabus.bos.nsw.edu.au. [2019-08-02]. （原始内容存档于2021-02-24）. ^ Robinson, Raphael Mitchel. A report on primes of the form k · 2n + 1 and on factors of Fermat numbers (PDF). Proceedings of the American Mathematical Society (University of California, Berkeley, California, USA). October 1958, 9 (5): 673–681 [1958-04-07] [2020-06-28]. doi:10.1090/s0002-9939-1958-0096614-7. （原始内容 (PDF)存档于2020-06-28）. ^ Olver, Frank W. J.; Lozier, Daniel W.; Boisvert, Ronald F.; Clark, Charles W. (编). NIST Handbook of Mathematical Functions. National Institute of Standards and Technology (NIST), U.S. Department of Commerce, Cambridge University Press. 2010. ISBN 978-0-521-19225-5. MR 2723248. Archive.today的存档，存档日期2013-07-03 ^ Zeidler, Eberhard; Schwarz, Hans Rudolf; Hackbusch, Wolfgang; Luderer, Bernd; Blath, Jochen; Schied, Alexander; Dempe, Stephan; Wanka, Gert; Hromkovič, Juraj; Gottwald, Siegfried. Zeidler, Eberhard , 编. Springer-Handbuch der Mathematik I I 1. Berlin / Heidelberg, Germany: Springer Spektrum, Springer Fachmedien Wiesbaden. 2013: 590 . ISBN 978-3-658-00284-8. doi:10.1007/978-3-658-00285-5. ISBN 3-658-00284-0 （德语）. (xii+635 pages) ^ Van Winkle, Lewis. Exponentiation Associativity and Standard Math Notation. Codeplea - Random thoughts on programming. 2016-08-23 [2016-09-20]. （原始内容存档于2020-06-28）. 检索自“ 分类：计算机科学代数隐藏分类： CS1德语来源 (de) Webarchive模板archiveis链接添加话题
2804	https://math.montana.edu/jobo/st446/documents/ho7b.pdf	7.4 Systematic Sampling • Systematic sampling is a sampling plan in which the population units are collected systematically throughout the population. More speciﬁcally, a single primary sampling unit consists of secondary sampling units that are relatively spaced with each other in some systematic pattern throughout the population. • Suppose the study area is partitioned into a 20 × 20 grid of 400 population units. A primary sampling unit in a systematic sample could consist of all population units that form a lattice which are 5 units apart horizontally and vertically. In Figure 9a, N = 25 and M = 16. In Figure 9b, each of the N = 50 primary sampling units contains M = 8 secondary sampling units. • Initially, systematic sampling and cluster sampling appear to be opposites because system-atic samples contain secondary sampling units that are spread throughout the population (good global coverage of the study area) while cluster samples are collected in groups of close proximity (good coverage locally within the study area). • Systematic and cluster sampling are similar, however, because whenever a primary sam-pling unit is selected from the sampling frame, all secondary sampling units of that primary sampling unit will be included in the sample. Thus, random selection occurs at the primary sampling unit level and not the secondary sampling unit level. • For estimation purposes, you could ignore the secondary sampling unit yij-values and only retain the primary sampling units ti-values. This is what we did with one-stage cluster sampling. • The systematic and cluster sampling principle: To obtain estimators of low variance, the population must be partitioned into primary sampling unit clusters in such a way that 157 7.4 Systematic Sampling • Systematic sampling is a sampling plan in which the population units are collected systematically throughout the population. More speciﬁcally, a single primary sampling unit consists of secondary sampling units that are relatively spaced with each other in some systematic pattern throughout the population. • Suppose the study area is partitioned into a 20 × 20 grid of 400 population units. A primary sampling unit in a systematic sample could consist of all population units that form a lattice which are 5 units apart horizontally and vertically. In Figure 9a, N = 25 and M = 16. In Figure 9b, each of the N = 50 primary sampling units contains M = 8 secondary sampling units. • Initially, systematic sampling and cluster sampling appear to be opposites because system-atic samples contain secondary sampling units that are spread throughout the population (good global coverage of the study area) while cluster samples are collected in groups of close proximity (good coverage locally within the study area). • Systematic and cluster sampling are similar, however, because whenever a primary sam-pling unit is selected from the sampling frame, all secondary sampling units of that primary sampling unit will be included in the sample. Thus, random selection occurs at the primary sampling unit level and not the secondary sampling unit level. • For estimation purposes, you could ignore the secondary sampling unit yij-values and only retain the primary sampling units ti-values. This is what we did with one-stage cluster sampling. • The systematic and cluster sampling principle: To obtain estimators of low variance, the population must be partitioned into primary sampling unit clusters in such a way that the clusters are similar to each other with respect to the ti-values (small cluster-to-cluster variability). 161 • This is equivalent to saying that the within-cluster variability should be as large as pos-sible to obtain the most precise estimators. Thus, the ideal primary sampling unit is representative of the full diversity of yij-values within the population. • With natural populations of spatially distributed plants, animals, minerals, etc., these con-ditions are typically satisﬁed by systematic primary sampling units (and are not satisﬁed by primary sampling units with spatially clustered secondary sampling units). 7.4.1 Estimation of yU and t • If a SRS is used to select the systematic primary sampling units, we can apply the estima-tion results for cluster sampling to deﬁne (i) estimators, (ii) the variance of each estimator, and (iii) the estimated variance of each estimator. • The following formulas will be the same as those used for one-stage cluster sampling. The subscript sys denotes the fact that data were collected under systematic sampling. • The unbiased estimators of t and yU are: b tsys = N n n X i=1 ti = c yU sys = 1 nM n X i=1 ti = y M = (85) with variance V (b tsys) = V (c yU sys) = N(N −n) M 2 0 S2 t n (86) where S2 t = PN i=1(ti −ti)2 N −1 . • Recall that y = 1 n n X i=1 ti is the sample mean and that s2 t = Pn i=1(ti −y)2 n −1 is the sample variance of the primary sampling units. • Because S2 t is unknown, we use s2 t to get unbiased estimators of the variances: b V (b tsys) = b V (c yU sys) = N(N −n) M 2 0 s2 t n (87) 7.4.2 Conﬁdence Intervals for yU and t • For a relatively small number n of sampled primary sampling units, the following conﬁ-dence intervals are recommended: c yU sys ± t∗ q b V (c yU sys) b tsys ± t∗ q b V (b tsys) (88) where t∗is the upper α/2 critical value from the t(n −1) distribution. Note that the degrees of freedom are based on n, the number of sampled primary sampling units, and not on the total number of secondary sampling units nM. 162 Systematic Sampling Examples In Figure 9a, each of the N = 25 primary sampling units contains M = 16 secondary sampling units corresponding to the same location within the 16 5x5 subregions. n = 3 primary sampling units were sampled. The SSUs sampled are in ( ) Figure 9a 1 1 (1) 1 1 2 1 (0) 0 0 4 5 (0) 1 0 1 2 (1) 0 1 3 2 1 0 1 0 0 0 1 2 2 2 0 2 2 2 0 2 0 1 7 (4) 1 1 1 1 (0) 0 0 2 2 (0) 4 3 2 4 (2) 1 2 2 0 1 2 0 0 0 0 0 4 6 5 1 5 0 0 0 2 1 2 0 1 1 0 (2) 3 2 0 0 (2) 1 3 1 4 (1) 1 1 2 2 (1) 1 2 0 (0) 0 4 3 3 (0) 1 16 5 0 (1) 3 8 0 0 (1) 3 3 0 0 1 14 3 3 1 2 0 8 0 2 0 3 9 0 4 2 1 0 0 (0) 5 1 8 7 (6) 6 6 1 0 (4) 0 0 1 2 (2) 0 1 2 0 0 2 2 3 2 2 3 1 1 1 3 0 0 2 2 0 3 4 0 0 0 0 (0) 1 0 3 1 (1) 1 2 0 2 (0) 2 0 2 1 (1) 0 1 8 (7) 7 8 0 5 (0) 1 0 1 2 (0) 0 2 4 2 (2) 2 4 0 9 1 0 0 1 1 1 0 0 0 1 2 4 0 2 1 3 3 1 0 (0) 0 1 0 2 (4) 3 1 2 2 (0) 0 1 1 2 (2) 0 2 4 0 1 0 0 1 2 0 2 3 5 2 0 0 2 1 1 2 0 1 3 1 0 0 (1) 1 0 0 0 (2) 2 2 1 1 (1) 0 0 2 0 (0) 0 0 2 (0) 2 2 0 1 (1) 0 2 0 0 (1) 0 0 1 1 (1) 5 3 0 0 0 3 2 1 0 0 0 0 0 2 1 0 1 1 1 3 1 2 1 (0) 0 1 0 3 (0) 1 0 0 2 (1) 2 0 0 0 (1) 1 1 0 0 0 0 0 0 0 0 1 1 1 0 1 0 3 0 2 0 1 1 0 2 0 0 (0) 0 0 0 0 (1) 2 0 1 3 (0) 0 1 0 1 (2) 4 The following are the 25 systematic PSU (cluster) totals (ti for i = 1, 2, . . . , 25). The sample contains n = 3 PSU (3 starting locations). The PSUs sampled are in ( ) 25 33 (16) 26 54 15 26 19 32 32 35 (26) 24 21 26 17 13 20 24 23 15 13 15 (15) 19 163 In Figure 9b, each of the N = 50 primary sampling units contains M = 8 secondary sampling units corresponding to the same location within the 8 10x5 subregions. n = 6 primary sampling units were sampled. The SSUs sampled are in ( ) Figure 9b 18 (20) 15 20 20 15 (19) 18 24 23 20 (26) 29 28 28 31 (31) 34 28 32 13 20 16 20 15 23 19 26 21 21 24 30 23 26 25 33 31 28 32 38 (16) 18 20 24 (25) (26) 22 23 26 (26) (22) 27 25 25 (34) (28) 37 36 38 (31) 17 17 16 22 21 23 22 27 27 24 28 32 29 33 27 37 37 38 35 33 15 19 23 17 21 23 21 23 24 25 31 26 32 34 32 33 31 31 36 37 21 (24) 20 21 28 26 (30) 22 31 25 29 (29) 27 30 29 37 (35) 32 38 43 23 17 24 25 24 27 31 29 31 34 27 36 29 29 34 39 37 37 40 36 (18) 24 21 25 27 (22) 32 32 31 26 (28) 34 34 37 35 (34) 38 38 37 40 22 26 28 (26) 24 29 33 26 (27) 27 34 31 39 (32) 36 38 37 40 (44) 43 23 27 28 29 26 32 25 31 35 34 32 33 37 32 42 40 40 37 42 44 23 (21) 31 23 30 27 (31) 30 32 35 30 (40) 32 37 37 36 (40) 44 44 40 26 29 31 26 30 31 34 36 30 38 36 32 38 38 37 42 42 41 40 49 (28) 24 28 27 (26) (31) 32 29 32 (33) (38) 34 39 38 (40) (37) 41 43 42 (43) 32 25 31 32 29 29 35 38 38 32 36 35 39 42 39 40 44 42 41 45 27 29 35 28 35 35 31 40 35 37 38 44 40 40 47 39 49 48 51 49 30 (29) 32 32 33 30 (36) 38 42 36 35 (38) 44 47 45 49 (41) 43 44 51 28 35 35 34 34 33 41 33 34 35 39 44 44 48 44 50 49 48 53 54 (29) 33 32 36 39 (33) 33 34 35 42 (46) 47 48 47 46 (45) 44 52 54 55 28 37 38 (37) 33 33 34 37 (45) 40 39 42 42 (46) 47 48 52 47 (46) 53 38 39 39 37 34 38 39 45 39 42 45 41 44 51 46 50 52 51 51 53 The following are the 50 systematic PSU (cluster) totals (ti for i = 1, 2, . . . , 50). The sample contains n = 6 PSU (6 starting locations). The PSUs sampled are in ( ) 200 (228) 233 236 245 228 237 239 233 253 (226) 235 243 252 (258) 242 247 260 270 250 241 250 272 265 283 257 (262) 258 285 290 266 290 279 294 295 (255) 285 291 302 310 271 292 297 (303) 303 298 296 312 316 321 164 7.4.3 Using R and SAS for Systematic Sampling R code for Systematic Sample in Figure 9a library(survey) source("c:/courses/st446/rcode/confintt.r") # Systematic sample of 3 PSUs from Figure 9a N = 25 n = 3 M = 16 wgt = N/n y <- c(1,0,0,1,0,0,1,1,7,0,0,2,0,1,1,1, 4,0,0,2,0,6,4,2,0,4,0,2,0,0,1,1, 2,2,1,1,0,1,0,1,1,2,1,0,0,1,0,2) clusterid <- c(rep(c(1),M),rep(c(2),M),rep(c(3),M)) fpc <- c(rep(N,nM)) Fig9a <- data.frame(cbind(clusterid,y,fpc)) dsgn9a <-svydesign(ids=~clusterid,weights=c(rep(wgt,nM)),fpc=~fpc,data=Fig9a) esttotal <- svytotal(~trees,design=dsgn9a) print(esttotal,digits=15) confint.t(esttotal,level=.95,tdf=n-1) estmean <- svymean(~trees,design=dsgn9a) print(estmean,digits=15) confint.t(estmean,level=.95,tdf=n-1) R output for Systematic Sample in Figure 9a total SE y 475 82.361 -------------------------------------------------------------------mean( y ) = 475.00000 SE( y ) = 82.36099 Two-Tailed CI for y where alpha = 0.05 with 2 df 2.5 % 97.5 % 120.62924 829.37076 -------------------------------------------------------------------mean SE y 1.1875 0.2059 -------------------------------------------------------------------mean( y ) = 1.18750 SE( y ) = 0.20590 Two-Tailed CI for y where alpha = 0.05 with 2 df 2.5 % 97.5 % 0.30157 2.07343 -------------------------------------------------------------------165 R code for Systematic Sample in Figure 9b # Systematic sample of 6 PSUs from Figure 9b N = 50 n = 6 M = 8 wgt = N/n y <- c(20,19,26,31,21,31,40,40,16,26,22,28,28,31,38,37, 25,26,34,31,26,33,40,43,24,30,29,35,29,36,38,41, 18,22,28,34,29,33,46,45,26,27,32,44,37,45,46,46) clusterid <- c(1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3, 4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6) (The remainder of the code is the same as the previous example) R output for Systematic Sample in Figure 9b -------------------------------------------------------------------mean( y ) = 12766.66667 SE( y ) = 536.93368 Two-Tailed CI for y where alpha = 0.05 with 5 df 2.5 % 97.5 % 11386.43470 14146.89863 --------------------------------------------------------------------------------------------------------------------------------------mean( y ) = 31.91667 SE( y ) = 1.34233 Two-Tailed CI for y where alpha = 0.05 with 5 df 2.5 % 97.5 % 28.46609 35.36725 -------------------------------------------------------------------SAS code for Systematic Sample in Figure 9a (Supplemental) DATA systmtc1; M0 = 400; number of secondary sampling units (SSUs) in population; M = 16; number of SSUs in a PSU; n = 3; number of primary sampling units (PSUs) sampled; wgt = M0/(nM); DO psu = 1 to n; DO ssu = 1 to M; INPUT trees @@; OUTPUT; END; END; DATALINES; 1 0 0 1 0 0 1 1 7 0 0 2 0 1 1 1 4 0 0 2 0 6 4 2 0 4 0 2 0 0 1 1 2 2 1 1 0 1 0 1 1 2 1 0 0 1 0 2 ; TOTAL = number of PSUs in the population ; PROC SURVEYMEANS DATA=systmtc1 TOTAL=25 MEAN CLM SUM CLSUM; VAR trees; CLUSTER psu; WEIGHT wgt; TITLE ’Systematic Sample from Figure 9a’; RUN; 166 SAS output for Systematic Sample in Figure 9a The SURVEYMEANS Procedure Data Summary Number of Clusters 3 Number of Observations 48 Sum of Weights 400 Statistics Std Error Variable Mean of Mean 95% CL for Mean -----------------------------------------------------------------trees 1.187500 0.205902 0.30157311 2.07342689 -----------------------------------------------------------------Statistics Variable Sum Std Dev 95% CL for Sum -----------------------------------------------------------------trees 475.000000 82.360994 120.629244 829.370756 -----------------------------------------------------------------SAS code for Systematic Sample in Figure 9b (Supplemental) DATA systmtc2; M0 = 400; number of secondary sampling units (SSUs) in population; n = 6; number of primary sampling units (PSUs) sampled; m = 8; number of SSUs in a PSU; wgt = M0/(nm); DO psu = 1 to n; DO ssu = 1 to m; INPUT y @@; OUTPUT; END; END; DATALINES; 20 19 26 31 21 31 40 40 16 26 22 28 28 31 38 37 25 26 34 31 26 33 40 43 24 30 29 35 29 36 38 41 18 22 28 34 29 33 46 45 26 27 32 44 37 45 46 46 ; TOTAL = number of PSUs in the population ; PROC SURVEYMEANS DATA=systmtc2 TOTAL=50 MEAN CLM SUM CLSUM; VAR y; CLUSTER psu; WEIGHT wgt; TITLE ’Systematic Sample from Figure 9b’; RUN; SAS output for Systematic Sample in Figure 9b Data Summary Number of Clusters 6 Number of Observations 48 Sum of Weights 400 Statistics Std Error Variable Mean of Mean 95% CL for Mean -----------------------------------------------------------------y 31.916667 1.342334 28.4660867 35.3672466 -----------------------------------------------------------------Variable Sum Std Dev 95% CL for Sum -----------------------------------------------------------------y 12767 536.933681 11386.4347 14146.8986 -----------------------------------------------------------------167 7.4.4 Comments from W.G. Cochran • Cochran (from Sampling Techniques (1953)) makes the following comments about advan-tages of systematic sampling: Intuitively, systematic sampling seems likely to be more precise than simple random sampling. In eﬀect, it stratiﬁes the population into [N] strata, which consist of the ﬁrst [M] units, the second [M] units, and so on. We might therefore expect the systematic sample to be about as precise as the corresponding stratiﬁed random sample with one unit per stratum. The diﬀerence is that with the systematic sample the units all occur at the same relative position in the stratum, whereas with the stratiﬁed random sample the position in the stratum is determined separately by randomization within each stratum. The systematic sample is spread more evenly over the population, and this fact has sometimes made systematic sampling considerably more precise than stratiﬁed random sampling. • Cochran also warns us that: The performance of systematic sampling relative to that of stratiﬁed or simple random sampling is greatly dependent on the properties of the population. There are popu-lations for which systematic sampling is extremely precise and others for which it is less precise that simple random sampling. For some populations and values of [M], [ var(c yU sys)] may even increase when a larger sample is taken — a startling departure from good behavior. Thus it is diﬃcult to give general advice about the situation in which systematic sampling is to recommended. A knowledge of the structure of the population is necessary for its most eﬀective use. • If a population contains a linear trend: 1. The variances of the estimators from systematic and stratiﬁed sampling will be smaller than the variance of the estimator from simple random sampling. 2. The variance of the estimator from systematic sampling will be larger than the vari-ance of the estimator from stratiﬁed sampling. Why? If the starting point of the systematic sample is selected too low or too high, it will be too low or too high across the population of units. Whereas, stratiﬁed sampling gives an opportunity for within-stratum errors to cancel. • Suppose a population has 16 secondary sampling units (t = 130) and is ordered as follows: Sampling unit 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 y-value 1 2 2 3 3 4 5 6 8 9 12 13 14 15 16 17 Note there is a linearly increasing trend in the y-values with the order of the sampling units. Suppose we take a 1-in-4 systematic sample. The following table summarizes the four possible 1-in-4 systematic samples. Sampling unit 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 y-values 1 2 2 3 3 4 5 6 8 9 12 13 14 15 16 17 ti b tsys Sample 1 1 3 8 14 Sample 2 2 4 9 15 Sample 3 2 5 12 16 Sample 4 3 6 13 17 168 • If a population has periodic trends, the eﬀectiveness of the systematic sample depends on the relationship between the periodic interval and the systematic sampling interval or pattern. The following idealized curve was given by Cochran to show this. The height of the curve represents the population y-value. – The A sample points represent the least favorable systematic sample because when-ever M is equal to the period, every observation in the systematic sample will be similar so the sample is no more precise than a single observation taken at random from the population. – The B sample points represent the most favorable systematic sample because M is equal to a half-period. Every systematic sample has mean equal to the true population mean because successive y-value deviations above and below the mean cancel. Thus, the variance of the estimator is zero. – For other values of M, the sample has varying degrees of eﬀectiveness that depends on the relation between M and the period. population has periodic trends, the eﬀectiveness of the systematic sample depends he relationship between the periodic interval and the systematic sampling interval or ern. The following idealized curve was given by Cochran to show this. The height of curve represents the population y-value. The A sample points represent the least favorable systematic sample because when-ever M is equal to the period, every observation in the systematic sample will be similar so the sample is no more precise than a single observation taken at random from the population. The B sample points represent the most favorable systematic sample because M is equal to a half-period. Every systematic sample has mean equal to the true population mean because successive y-value deviations above and below the mean cancel. Thus, the variance of the estimator is zero. For other values of M, the sample has varying degrees of eﬀectiveness that depends on the relation between M and the period. sing a Single Systematic Sample ny studies generate data from a systematic sample based on a single randomly selected ting unit (i.e., there is only one randomly selected primary sampling unit). en there is only one primary sampling unit, it is possible to get unbiased estimators s and b tsys of yU and t. It is not possible, however, to get an unbiased estimator of the ances b V ( c yU sys) and b V (b tsys). e can ignore the fact that the yij-values were collected systematically and treat the econdary sampling units in the single primary sampling unit as a SRS, then the SRS ance estimator would be a reasonable substitute only if the units of the population can onably be conceived as being randomly ordered (i.e., there is no systematic pattern in population such as a linear trend or a periodic pattern). If this assumption is reasonable, then b V ( c yU sys) ≈b V ( c yU) = µN −n N ¶ s2 n h natural populations in which nearby units are similar to each other (spatial correla-), this procedure tends to provide overestimates of the variances of c yU sys and b tsys. 7.5 Using a Single Systematic Sample • Many studies generate data from a systematic sample based on a single randomly selected starting unit (i.e., there is only one randomly selected primary sampling unit). • When there is only one primary sampling unit, it is possible to get unbiased estimators c yU sys and b tsys of yU and t. It is not possible, however, to get an unbiased estimator of the variances b V (c yU sys) and b V (b tsys). • If we can ignore the fact that the yij-values were collected systematically and treat the M secondary sampling units in the single primary sampling unit as a SRS, then the SRS variance estimator would be a reasonable substitute only if the units of the population can reasonably be conceived as being randomly ordered (i.e., there is no systematic pattern in the population such as a linear trend or a periodic pattern). – If this assumption is reasonable, then b V (c yU sys) ≈b V (c yU) = N −n N s2 n • With natural populations in which nearby units are similar to each other (spatial correla-tion), this procedure tends to provide overestimates of the variances of c yU sys and b tsys. • Procedures for estimating variances from a single systematic sample are discussed in Bell-house (1988), Murthy and Rao (1988), and Wolter (1984). 169
2805	https://www.matematica.pt/aulas-exercicios.php?id=190	Aulas do 7º ano sobre Resolução de problemas usando equações. Login Inicio Contactar Toggle navigation Exames Matemática (A) 2º ano (Prova de Aferição) 4º ano (Exame Nacional) 5º ano (Prova de Aferição) 6º ano (Exame Nacional) 8º ano (Prova de Aferição) 9º ano (Exame Nacional) 10º ano (Teste Intermédio) 11º ano (Teste Intermédio) 12º ano (Exame Nacional) Matemática B M.A.C.S. 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V ê com atenção o vídeo que contém a explicação da matéria. De seguida, imprime a ficha de trabalho e tenta resolver o máximo de exercícios que conseguires sobre este tema. Se tiveres alguma dúvida nos exercícios que disponibilizamos, consulta a resolução proposta ou coloca uma questão no fórum. Bom estudo! Equações Algébricas Lição nº: 4 / Total: 4 ant.voltar Concluir o 7º ano 42% Introdução O matemático húngaro George Pólya ficou conhecido na história da matemática pela grande importância que os seus escritos deram na resolução de problemas. Este matemático publicou muitos livros e alguns artigos científicos onde apresenta alguns exemplos de problemas e os métodos para a sua resolução. De acordo com aquilo que ele escreveu, os passos para resolver um problema matemático são os seguintes: 1º) Compreender o enunciado. 2º) Identificar a incógnita. 3º) Escrever a equação. 4º) Resolver a equação. 5º) Verificar a solução. Uma célebre frase sua é: "Aprende-se a resolver problemas, resolvendo problemas." Explicação da matéria Sumário: Resolução de problemas usando equações. Duração: 11:26 Foi interessante? Então partilha! Exercícios resolvidos Exercício nº 1 Nível: Feito Ver Vídeo Votar Dificuldade Guardar Exercício nº 2 Nível: Feito Ver Vídeo Votar Dificuldade Guardar Exercício nº 3 Nível: Feito Ver Vídeo Votar Dificuldade Guardar Exercício nº 4 Nível: Feito Ver Vídeo Votar Dificuldade Guardar Exercício nº 5 Nível: Feito Ver Vídeo Votar Dificuldade Guardar Exercício nº 6 Nível: Feito Ver Vídeo Votar Dificuldade Guardar Exercício nº 7 Nível: Feito Ver Vídeo Votar Dificuldade Guardar Exercício nº 8 Nível: Feito Ver Vídeo Votar Dificuldade Guardar Ficha de Trabalho × Resolução do Exercício Fechar × Conteúdo Restrito ATENÇÃO: Para poder utilizar esta opção terá que fazer login. Se ainda não for um utilizador registado, deverá primeiro criar uma conta. Só depois da conta ser validada pelos nossos serviços é que poderá aceder a todos os conteúdos. Fechar Utiliza este espaço para comentários ou dúvidas N este local poderás colocar os teus comentários e as tuas dúvidas. Todas as mensagens que não estiverem diretamente relacionadas com este tema, ou que eventualmente contenham linguagem considerada imprópria serão removidas. × Foram feitos 10 comentários/dúvidas. 08 de Maio de 2021, 11h47 Mensagem de Sofia O Pedro por um eletrodoméstico pagou 307,50€ incluindo 23% de IVA. Qual é o preço do eletrodoméstico sem o IVA? Eu não estou a conseguir fazer a equação. Pode ajudar-me ? 08 de Maio de 2021, 16h34 Mensagem de Vitor Nunes Olá Sofia, Vamos supor que representa o preço antes de aplicarmos o IVA. Repara que o Pedro vai pagar os do preço (sem IVA) mais os do IVA, ou seja, ele vai pagar . Como equivale a podemos escrever a seguinte equação: . Para resolver a equação, basta fazer e assim vamos obter o preço do eletrodoméstico sem o valor do IVA. 15 de Março de 2022, 08h55 Mensagem de Carla Olá, Consegue ajudar-me a resolver o problema: "a Rita diz que daqui a 18 anos a terça parte da sua idade será metade da sua idade atual. Qual a idade da Rita?" 16 de Março de 2022, 08h59 Mensagem de Vitor Nunes Olá Carla, Não vou dar a resposta, mas posso ajudar a tentar obter a equação. Vamos supor que a variável representa a idade da Rita. Daqui a 18 anos a Rita irá ter anos. A terça parte desta idade, ou seja, vai ser igual a metade da idade atual, isto é, . Portanto, a equação que permite resolver o problema é a seguinte: . Agora é só resolver. Pode começar por aplicar a propriedade distributiva no primeiro membro e de seguida desembaraçar de denominadores. Por fim, basta isolar o e assim descobrir a idade da Rita. 20 de Abril de 2023, 10h47 Mensagem de Artur Consegue ajudar na descoberta da equação? "André comprou 2 cadernos e 3 canetas por 4.90€. O preço de um caderno é o dobro do preço de uma caneta. Determine quanto custa cada caderno e cada caneta." 20 de Abril de 2023, 13h51 Mensagem de Vitor Nunes Olá Artur, Vamos começar por designar por o preço de uma caneta. Tendo em conta que um caderno custa o dobro de uma caneta, então o preço do caderno var ser . No próximo passo, o enunciado refere que ele comprou dois cadernos e 3 canetas, logo o total das compras foi de . Como sabemos que o valor gasto foi de 4.90€, resta-nos construir a equação que vai ser a seguinte: . Agora é só resolver! 13 de Outubro de 2024, 18h20 Mensagem de Maria da Conceição Antunes Será que me podiam ajuda a resolver o seguinte problema? 3 amigas viajaram. A 1ª gastou 61,31 euros em combustível, a 2ª gastou 47,9 euros em portagens e a 3ª não gastou nada. Como fazer as contas, deve e haver no final? 14 de Outubro de 2024, 08h12 Mensagem de Vitor Nunes Olá Maria, A forma mais fácil será começar por somar as despesas. De seguida, como são 3 amigas, vamos dividir o total das despesas por 3. Finalmente, depois de saber a comparticipação de cada uma, vamos subtrair esse valor, ao valor já pago. Se der um valor negativo, é porque essa pessoa tem dinheiro a receber. Se der um valor positivo, então essa pessoa terá de entregar esse dinheiro. Espero ter ajudado! 29 de Agosto de 2025, 16h21 Mensagem de Daniel Realmente a matematica é deslumbrante e pura magia às vezes. No exercicio 5, cheguei ao mesmo resultado com uma abordagem completamente diferente, resolvi recorrendo a um sistema de equaçoes, a primeira 15a+10c = 1485€ (a = adulto e c = criança) e outra a + c = 126, depois multipliquei por -10 a segunda equação e adicionei as duas de forma a anular a variavel c e o resultado foi a = 45, ou seja 45 bilhetes de adulto, a partir de ai também consegui facilmente apurar a variavel c. 30 de Agosto de 2025, 15h56 Mensagem de Vitor Nunes Olá Daniel, Sim, a matemática permite chegar ao resultado final através de diferentes abordagens. Neste caso, utilizaste uma técnica de resolução de sistemas com o nome de adição ordenada. Por vezes, esta técnica permite chegar ao resultado de forma mais rápida do que pela substituição de variáveis. Enviar Comentário/Dúvida Nome (mínimo: 3) Email (o email não será mostrado) Mensagem (mínimo: 10, máximo: 500, restantes: 500) Enviar T odos os vídeos aqui presentes têm por objetivo fornecer ao aluno explicações de matemática online na forma mais intuitiva possível. Todos eles estão disponíveis para consulta através dos canais do YouTube: matematica.PT, Os números da Inês, ExplicaMat, Academia Aberta, Matemática Simples e Matemática no Mocho. Caso encontres algum erro, (por exemplo um vídeo que não funcione ou que não corresponda à explicação do exercício proposto) ou caso queiras dar alguma sugestão de melhoramento, não hesites em nos enviar um email através da página Contactar. Tentaremos dar resposta tão brevemente quanto possível. MATEMATICA.PT © 2025 - Vitor Nunes Se eliminar a causa, o efeito cessa. Receber Novidades × Receber Novidades Subscreve a nossa newsletter para estar informado sobre as mais recentes novidades! Subscrever Fechar
2806	https://people.ucsc.edu/~fmonard/Math216/recap1.pdf	Math 216 partial recap - 01/27/216 - Fran¸ cois Monard This summary covers some of the topics we have been covering in class, in a somewhat compact form. These notes are BY NO MEANS reﬂective of what may or may not appear in the midterm and should not be considered as a replacement for the book or lecture notes. However, as today was a “review” lecture, this might be a good time to provide some brief checkpoint notes. Methods for solving ODEs The general setting is an initial value problem of the form: dy dx = f(x, y), y(x0) = y0. In certain cases1, we are able to work out an explicit expression for the solution, depending on the form of f. Case 1: f does not depend on y. Then we are looking at a problem of the form dy dx = f(x), in which case ﬁnding y consists of ﬁnding an antiderivative for f. Case 2: f is separated, that is, f(x, y) = g(x)h(y). In this case, we may rewrite the ODE as 1 h(y) dy dx = g(x). The task now is to ﬁnd an antiderivative for 1 h (call it H) and an antiderivative for g (call it G), so that the previous equation reads: d dx(H(y(x))) = dH dy dy dx = 1 h(y) dy dx = g(x) = d dxG(x). Then, direct integration gives H(y(x)) = G(x) + C, after which one might be able to solve for y(x), and the integration constant C using the initial condition. Case 3: f is linear in y, of the form f(x, y) = −P(x)y + Q(x). In this case, the ODE takes the form dy dx + P(x)y = Q(x). The recipe to solve the problem goes as follows: 1Note that in real life, most cases cannot be solved explicitely ! 1 1. Compute an integrating factor ρ(x) = exp R P(x) dx , which consists of ﬁnding an an-tiderivative for P, then taking its exponential. 2. Notice, using the product rule, that the ODE can be turned into d dx(ρ(x)y(x)) = ρ(x)Q(x). 3. Integrate this equation once. This will create an integration constant. 4. solve for y and for the integration constant using the initial condition. Tricks for ﬁnding antiderivatives. In all examples above, the main technical step is to ﬁnd antiderivatives. While many of them are easy or can be found in tables, some others may not be so obvious. One trick to be known, is to recognize when an expression takes the form of the derivative of a compound, for instance: • If an expression looks like 2f(t)f′(t) for some function f, then it is nothing but d dt(f2(t)), in which case an antiderivative is f2(t). • If an expression looks like f′(t)/f(t) for some function f, then it is nothing but d dt(ln f(t)), in which case an antiderivative is ln(f(t)). • If an expression takes the form −f′(t)/(f(t))2 for some function f, then it is nothing but d dt(1/f(t)). In all these cases, we recognized that the expression took the form (g′(f(t))f′(t) = d dt(g(f(t))) (with g(x) respectively x2, ln x, 1/x), for which an antiderivative is given by g(f(t)). To practice recognizing such forms, you want to try other cases such as g(x) = xp for any power. . . As examples, try looking for antiderivatives of cos t sin t , 1 t ln t, ln t t , 3(ln t)2 t , 3t2 sin(t3), using the methods above. Parameter-dependent ODEs and bifurcations Autonomous ODEs, equilibrium solutions and stability. An ODE dy dx = f(x, y) is called autonomous if f does not depend on the independent variable x, that is, the ODE looks like dy dx = f(y). In what follows, we will restrict to ODEs of this form. For such an ODE, we deﬁne y⋆to be a critical point if f(y⋆) = 0. For any critical point, the constant function y(x) = y⋆ is a solution curve to the problem, so-called an “equilibrium curve”, hence their importance. Intuitively2, a critical point y⋆is called 2See the book for more precise statements. 2 • stable if nearby solution curves seem to remain close to y⋆. • semi-stable if nearby solution curves converge to that point on one side of it, diverge away on the other side (example: the critical point y⋆= 0 in the ODE dy/dx = y2). • unstable otherwise. As you may notice by plotting examples of velocity ﬁelds, • y⋆is unstable if f(y) > 0 for y near and above y⋆, and f(y) < 0 for y near and below y⋆. • y⋆is stable if f(y) > 0 for y near and below y⋆, and f(y) < 0 for y near and above y⋆. • y⋆is semi-stable if f keeps the same sign over an interval containing y⋆. Bifurcations. Some models may be made parameter-dependent, in the sense that the ODE may involve additional variables which are part of the model. We covered for instance the case of a logistic population with harvesting dy dx = y(4 −y) −λ, where the parameter λ ≥0 quantiﬁes a constant harvesting rate of the population y. Despite the fact that everything in the model looks continuous in term of λ, one may behold drastic (or discontinuous) changes in the number of critical points and/or in the stability properties of these critical points, as λ passes through certain values. We call such phenomena bifurcations. Bifurcations are best observed on a bifurcation diagram, obtained as follows: given a parameter-dependent, autonomous ODE, of the form dy dx = fλ(y), 1. For each value of λ, ﬁnd the critical points y⋆(λ) by solving for y the equation fλ(y) = 0, and determine the sign of fλ in-between the critical points to decide which way trajectories go. From this you are able to decide the type of each critical point (i.e., stable, semi-stable or unstable). 2. Draw the functions y⋆(λ) on a (λ, y)-plane. This is the bifurcation diagram. Summary of numerical methods We have seen three kinds of numerical schemes in order to approximate a solution of an initial value problem of the form dy dx = f(x, y), y(x0) = y0. 3 We suppose that there exists a unique solution y(x) to the problem above, deﬁned on some interval containing x0. The three methods start from a stepsize h > 0, and aim at constructing a sequence of approximations y0, y1, y2, . . . of y(x0), y(x1), y(x2) where we deﬁne xn := x0 + nh for a ﬁnite number of steps 0 ≤n ≤N. We say that a method is accurate at order p (an integer) if there exists a constant C such that \|y(xn) −yn\| ≤Chp, 0 ≤n ≤N. The higher the p, the faster the approximations converge to the true solution as one reﬁnes h. These methods go as follow. Euler: (ﬁrst-order accurate) yn+1 = yn + hf(xn, yn), n = 0, 1, 2, . . . Improved Euler: (second-order accurate) k1 = f(xn, yn), k2 = f(xn+1, yn + k1h), yn+1 = yn + hk1 + k2 2 , n = 0, 1, 2, . . . Runge-Kutta 4: (fourth-order accurate) k1 = f(xn, yn), k2 = f(xn + h/2, yn + (h/2)k1), k3 = f(xn + h/2, yn + (h/2)k2), k4 = f(xn+1, yn + hk3), yn+1 = yn + hk1 + 2k2 + 2k3 + k4 6 , n = 0, 1, 2, . . . As seen in class, these methods are motivated, in the abstract, by more and more accurate rules for approximating a given function over a small interval: indeed, integrating the ODE between xn and xn+1, we see that the true solution satisﬁes y(xn+1) = y(xn) + Z xn+1 xn f(t, y(t)) dt, and, while we cannot compute the rightmost term exactly (we don’t know y(t) for xn ≤t ≤xn+1 !), all the schemes above are based on increasingly accurate rules for computing this integral (respectively, the rectangle rule, the trapezoidal rule, and Simpson’s rule). 4
2807	https://math.stackexchange.com/questions/188107/interval-notation-what-is-the-purpose	Interval Notation - What is the purpose? - Mathematics Stack Exchange Join Mathematics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR == Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Loading… 1. 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You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Interval Notation - What is the purpose? Ask Question Asked 12 years, 11 months ago Modified4 years, 7 months ago Viewed 4k times This question shows research effort; it is useful and clear 3 Save this question. Show activity on this post. I understand that you can write an inequality: x≤3 x≤3 as x∈(−∞,3]x∈(−∞,3] in interval notation, but, I don't understand the practicality of doing this. When is this used? How does it help? notation Share Share a link to this question Copy linkCC BY-SA 3.0 Cite Follow Follow this question to receive notifications edited Aug 28, 2012 at 21:55 Kartik Audhkhasi 1,576 9 9 silver badges 12 12 bronze badges asked Aug 28, 2012 at 21:32 user38392 user38392 5 3 That's not true. x≤3 x≤3 is a claim about x x, while (−∞,3](−∞,3] is just a set. It doesn't _claim_ anything, certainly not about x x. –Harald Hanche-Olsen Commented Aug 28, 2012 at 21:37 Yes, it's interval notation. x = −∞−∞ to 3. –user38392 Commented Aug 28, 2012 at 21:40 3 @AndreOseguera: No! The notation (−∞,3](−∞,3]_does not tell anything about x x_! It is the name for a _set_, and that set does not have anything to do with x x or any other variable. You can combine that set with x x and write x∈(−∞,3]x∈(−∞,3] (which is true exactly when x≤3 x≤3), but the point of intervals is that you can use them in situations where there is _not_ any ∈∈ immediately in front of them. –hmakholm left over Monica Commented Aug 28, 2012 at 21:43 x≤3 x≤3 will be written as x∈(−∞,3]x∈(−∞,3], and not simply as the set (−∞,3](−∞,3]. Can you clarify the question? –Kartik Audhkhasi Commented Aug 28, 2012 at 21:44 Could someone else edit it? I don't want to go about asking incorrectly. –user38392 Commented Aug 28, 2012 at 21:51 Add a comment\| 3 Answers 3 Sorted by: Reset to default This answer is useful 5 Save this answer. Show activity on this post. In general, the interval notation is cleaner to use. E.g. x∈[2,3]∪[5,10]x∈[2,3]∪[5,10] is slightly more concise than the statement: "2≤x≤3 or 5≤x≤10 2≤x≤3 or 5≤x≤10". As other answers will most likely point out, the interval notation seems closer to set theoretic notation. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered Aug 28, 2012 at 22:00 Kartik AudhkhasiKartik Audhkhasi 1,576 9 9 silver badges 12 12 bronze badges 1 Perfect. Thank you very much. –user38392 Commented Aug 28, 2012 at 22:07 Add a comment\| This answer is useful 2 Save this answer. Show activity on this post. As the commenters have mentioned, (−∞,3](−∞,3] is a set; it says nothing about any variable until you declare x∈(−∞,3]x∈(−∞,3] or perhaps x∉(−∞,3]x∉(−∞,3]. This notation can be much more easy to read and write sometimes, but more importantly the notation stresses that (−∞,3](−∞,3] is, indeed, a set, and that its nature as a set may matter. For instance, the domain of a function may be (−∞,3](−∞,3], and when you construct topological arguments about that function, you often want to be dealing with a subset of the extended real numbers. Similarly, the [a,b][a,b] means the set of all numbers between a a and b b inclusive; if you have a function acting on this set, it is much better to write f:[a,b]→R f:[a,b]→R rather than f:R→R,a≤x≤b f:R→R,a≤x≤b (the latter notation is actually awful; there is nothing about x x in the function statement, and so it is unclear when it need not be so). This notation also puts useful topological properties on the outside of the notation, rather than somewhere in between. (a,b)(a,b) is an open set, [a,b][a,b] is closed. This helps reinforce the notions of openness or closedness. The other way to write this set would be {x∣a
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2809	https://math.stackexchange.com/questions/1930468/prove-that-if-gcda-b-gcdb-c-gcda-c-1-then-gcdbc-ac-ab-1	elementary number theory - Prove that if $\gcd(a,b)=\gcd(b,c)=\gcd(a,c)=1$ then $\gcd(bc,ac,ab)=1$? - Mathematics Stack Exchange Join Mathematics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Loading… Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about Stack Overflow the company, and our products current community Mathematics helpchat Mathematics Meta your communities Sign up or log in to customize your list. more stack exchange communities company blog Log in Sign up Home Questions Unanswered AI Assist Labs Tags Chat Users Teams Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Try Teams for freeExplore Teams 3. Teams 4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams Hang on, you can't upvote just yet. You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Prove that if gcd(a,b)=gcd(b,c)=gcd(a,c)=1 gcd(a,b)=gcd(b,c)=gcd(a,c)=1 then gcd(b c,a c,a b)=1 gcd(b c,a c,a b)=1? Ask Question Asked 9 years ago Modified9 years ago Viewed 2k times This question shows research effort; it is useful and clear 5 Save this question. Show activity on this post. So I write 3 Bézout's identities : a u+b v=1 a u+b v=1 b v+c w=1 b v+c w=1 a u+c w=1 a u+c w=1 We could have taken different u,v,w u,v,w... By multiplying I obtain : 1=a b(v 2 u b+u 2 v a+u v w c)+a c(w 2 u c+u 2 w a+u v w b)+b c(v 2 w b+w 2 v c)1=a b(v 2 u b+u 2 v a+u v w c)+a c(w 2 u c+u 2 w a+u v w b)+b c(v 2 w b+w 2 v c) By taking : v 2 u b+u 2 v a+u v w c=x∈Z,w 2 u c+u 2 w a+u v w b=y∈Z,v 2 w b+w 2 v c=z∈Z v 2 u b+u 2 v a+u v w c=x∈Z,w 2 u c+u 2 w a+u v w b=y∈Z,v 2 w b+w 2 v c=z∈Z then gcd(b c,a c,a b)=1 gcd(b c,a c,a b)=1. Now I want to generalize it to a product of n n terms without the i t h i t h term. Are there any faster methods than developing ? PS : This statement is a key to start the proof of the CRT ! Thanks in advance ! elementary-number-theory divisibility alternative-proof gcd-and-lcm Share Share a link to this question Copy linkCC BY-SA 3.0 Cite Follow Follow this question to receive notifications edited Sep 17, 2016 at 23:58 MamanMaman asked Sep 17, 2016 at 16:32 MamanMaman 3,498 1 1 gold badge 19 19 silver badges 31 31 bronze badges 0 Add a comment\| 4 Answers 4 Sorted by: Reset to default This answer is useful 4 Save this answer. Show activity on this post. Suppose a prime p p divides gcd(b c,a c,a b)gcd(b c,a c,a b); then p∣(b c)p∣(b c), so it either divides b b or it divides c c, but not both because gcd(b,c)=1 gcd(b,c)=1. Suppose p∣b p∣b and p∤c p∤c. Since p∣a c p∣a c, we conclude p∣a p∣a, contradicting gcd(a,b)=1 gcd(a,b)=1. Similarly, assuming p∤b p∤b and p∣c p∣c leads to a contradiction, by considering that p∣a b p∣a b. Can we generalize it to more than three numbers? Suppose a 1,a 2,…,a n a 1,a 2,…,a n are pairwise coprime. Set c i=a 1 a 2…a n a i c i=a 1 a 2…a n a i Then gcd(c 1,c 2,…,c n)=1 gcd(c 1,c 2,…,c n)=1. Indeed, if p p is a prime divisor of gcd(c 1,c 2,…,c n)=1 gcd(c 1,c 2,…,c n)=1. Then p∣c n p∣c n, so p p divides exactly one of a 1,…,a n−1 a 1,…,a n−1. Without loss of generality we can say that p∣a 1 p∣a 1; since also p∣c 1 p∣c 1, we get a contradiction, because by assumption gcd(a 1,a j)gcd(a 1,a j) for j>1 j>1. Actually, this is a distributive lattice property. Let's consider (b∨c)∧(a∨c)∧(a∨b)=(b∨c)∧(a∨(b∧c))=((b∨c)∧a)∨(b∧c)=(a∧b)∨(a∧c)∨(b∧c)(b∨c)∧(a∨c)∧(a∨b)=(b∨c)∧(a∨(b∧c))=((b∨c)∧a)∨(b∧c)=(a∧b)∨(a∧c)∨(b∧c) and, if a∧b=a∧c=b∧c=0 a∧b=a∧c=b∧c=0, then also (b∨c)∧(a∨c)∧(a∨b)=0(b∨c)∧(a∨c)∧(a∨b)=0 (where 0 0 denotes the minimum element of the lattice). Here the distributive lattice is the natural numbers under divisibility, where the minimum is 1 1, ∧∧ is the gcd and ∨∨ is the lcm. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications edited Sep 17, 2016 at 17:35 answered Sep 17, 2016 at 16:51 egregegreg 246k 21 21 gold badges 155 155 silver badges 353 353 bronze badges 2 Ok so you use Euclid's lemma. I was confused with this lemma because there were too many terms. Does my proof with Bézout in the case of 3 numbers is right ?Maman –Maman 2016-09-17 17:09:09 +00:00 Commented Sep 17, 2016 at 17:09 @Maman Yes, but it's too much work! ;-)egreg –egreg 2016-09-17 17:12:27 +00:00 Commented Sep 17, 2016 at 17:12 Add a comment\| This answer is useful 1 Save this answer. Show activity on this post. g c d(a,b).=g c d(b,c)=g c d(c,a)=1 g c d(a,b).=g c d(b,c)=g c d(c,a)=1 means a,b,c do not have any common prime factors among them. So g c d(a b,b c)=b g c d(a b,b c)=b and since a,c do not have any factors of b, their g c d(b,c a)=1 g c d(b,c a)=1 Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered Sep 17, 2016 at 16:59 jnyanjnyan 2,525 19 19 silver badges 29 29 bronze badges Add a comment\| This answer is useful 1 Save this answer. Show activity on this post. This is straightforward if you use the properties of g c d()g c d(). I have written out the detailed steps. 1≤g c d(b c,a c,a b)=g c d(g c d(b c,a c),a b)=g c d(c∗g c d(b,a),a b)=g c d(c,a b)≤g c d(c,a)∗g c d(c,b)=1∗1=1,1≤g c d(b c,a c,a b)=g c d(g c d(b c,a c),a b)=g c d(c∗g c d(b,a),a b)=g c d(c,a b)≤g c d(c,a)∗g c d(c,b)=1∗1=1, So g c d(b c,a c,a b)=1 g c d(b c,a c,a b)=1. To extend this to n≥3 n≥3 variables x 1 x 1, ..., x n x n, the idea is the same plus induction. Let s=x 1...x n+1 s=x 1...x n+1. 1≤g c d(s x 1,...,s x n+1)=g c d(x n+1 g c d(...),s x n+1)=g c d(x n+1,x 1...x n)≤g c d(x n+1,x 1)...g c d(x n+1,x n)=1∗...∗1=1 1≤g c d(s x 1,...,s x n+1)=g c d(x n+1 g c d(...),s x n+1)=g c d(x n+1,x 1...x n)≤g c d(x n+1,x 1)...g c d(x n+1,x n)=1∗...∗1=1 Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications edited Sep 17, 2016 at 17:20 answered Sep 17, 2016 at 17:05 S. YS. Y 1,193 6 6 silver badges 5 5 bronze badges Add a comment\| This answer is useful 0 Save this answer. Show activity on this post. a u+b v=1⇒a c u+b c u=c⇒g c d(a c,b c)\|c⇒g c d(a c,b c,a b)\|c a u+b v=1⇒a c u+b c u=c⇒g c d(a c,b c)\|c⇒g c d(a c,b c,a b)\|c Same way g c d(a c,b c,a b)\|b g c d(a c,b c,a b)\|b. Then g c d(a c,b c,a b)g c d(a c,b c,a b) is a common divisor of b b and c c. As g c d(b,c)=1 g c d(b,c)=1 you get g c d(a c,b c,a b)=1 g c d(a c,b c,a b)=1 Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered Sep 18, 2016 at 17:42 N. S.N. S. 135k 12 12 gold badges 150 150 silver badges 271 271 bronze badges Add a comment\| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions elementary-number-theory divisibility alternative-proof gcd-and-lcm See similar questions with these tags. 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2810	https://www.cancer.gov/publications/dictionaries/cancer-terms/def/platelet	Definition of platelet - NCI Dictionary of Cancer Terms - NCI Skip to main content An official website of the United States government Español Menu Search Search About Cancer Cancer Types Research Grants & Training News & Events About NCI Home Publications NCI Dictionaries NCI Dictionary of Cancer Terms Publications Patient Education Publications PDQ® Fact Sheets NCI Dictionaries Dictionary of Cancer Terms Drug Dictionary Dictionary of Genetics Terms Blogs and Newsletters Health Communications Publications Reports platelet Listen to pronunciation (PLAYT-let) A tiny, disc-shaped piece of cell that is found in the blood and spleen. Platelets are pieces of very large cells in the bone marrow called megakaryocytes. They help form blood clots to slow or stop bleeding and to help wounds heal. Having too many or too few platelets or having platelets that don’t work as they should can cause problems. Checking the number of platelets in the blood may help diagnose certain diseases or conditions. Also called thrombocyte. Enlarge this image in new window Blood cells. Blood contains many types of cells: white blood cells (monocytes, lymphocytes, neutrophils, eosinophils, basophils, and macrophages), red blood cells (erythrocytes), and platelets. Blood circulates through the body in the arteries and veins. Search NCI's Dictionary of Cancer Terms Starts with Contains Browse: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z # About About This Website en Español Reuse & Copyright Social Media Resources Contact Us Publications Dictionary of Cancer Terms Find a Clinical Trial Policies Accessibility FOIA Privacy & Security Disclaimers Vulnerability Disclosure Sign up for email updates Enter your email address Enter a valid email address Sign up National Cancer Institute at the National Institutes of Health Contact Us Live Chat 1-800-4-CANCER NCIinfo@nih.gov Site Feedback Follow us U.S. Department of Health and Human ServicesNational Institutes of HealthNational Cancer InstituteUSA.gov
2811	http://www2.hawaii.edu/~plam/ph274/Lecture_outlines/Lecture_35_notes.pdf	Ch43. Nuclear Physics 43.3 – Nuclear Stability and Decay processes 43.4 – Ac<vity and Half-‐Lives Ph274 – Modern Physics 4/20/2016 Prof. Pui Lam Learning Outcomes (1) Understand the alpha decay, beta decay, gamma decay process. (2) Understand decay rate and half-‐life. (3) Understand how to apply carbon da proton + e + antineutrino 0 1n →1 1p + −1 0e+ 0 0νe Note : The A numbers are only approximate (round numbers) both e and νe have mass although << 1 atomic mass unit. Note : If the anti-neutrino were not invloved, then the electron kinetic energy would be a definite value (see the alpha-decay example). However, experiments show that the electron has a range of kinetic energy. In order to account for conservation of momentum and energy, Wolfgang Pauli postulated the existence of neutrino to explain the experiments. A free neutron (i.e. outside a nucleus) will decay naturally (in 15 minutes) A neutron inside a nucleus will decay if the nuclide is "neutron-rich" (left of the Segre stability curve). Example (43.6 on P.1452) 27 60Co →28 60Ni + −1 0e+ 0 0νe Actually this equation does NOT conserve charge because the Co atom is neutral (it has 27 protons and 27 electrons) and hence the RHS of the equation must also be neutral ( 28 protons and 28 electrons) ⇒the Ni atom has only 27 electrons. 27 60Co →28 60Ni+1 + −1 0e+ 0 0νe Note : The mass of 28 60Ni+1 + −1 0e ≈ mass of 28 60Ni and mass of νe ≈0. That is why the text states that "β − decay can occur when the mass of original neural atom is larger than that of the final (neutral) atom." Beta-‐plus Decay β + is a positron (anti-particle to an electron), i.e. β + = +1 0e Basic transformation: 1 1p →0 1n + +1 0e + 0 0νe Note : This transformation does not occur for a proton outside a nucleus (Why?) A proton inside a nucleus will decay if the nucleus is "proton-rich" (right of the Segre stability curve) Example: 12 23Mg →11 23Na + +1 0e + 0 0νe Again this equation does NOT conserve charge because if the parent atom is neutral then the RHS of the equation must be neutral. 12 23Mg →11 23Na−+ +1 0e + 0 0νe →11 23Na + −1 0e+ +1 0e + 0 0νe Hence the text states, "β + decay can occur when the mass of the original atom is at least two electron masses larger than the mass of the final (neutral) atom." Electron Capture Basic transformation: 1 1p + −1 0e →0 1n + 0 0νe Example 12 23Mg →11 23Ni + 0 0νe Note : The orignal atom capture one of its own electrons into the nucleus! Hence the text states, "Electron capture can occur when the mass of the original atom is larger than the mass of the final (neutral) atom." Gamma Decay • A nucleus, similar to an atom, has a ground state and excited states. • The nuclear excita<ons are of order of MeV. • A nucleus can be excited from the ground state by an incident photon (“gamma ray photon”) or by bombarding with high energy par<cles • An excited nucleus can return to its ground state by emi^ng a gamma ray photon. • Gamma ray emission does not change the nucleus mass or charge. Alpha + gamma emission Series of decays from U to Pb
2812	https://www.makeitfrom.com/material-properties/Full-Hard-H04-C36000-Brass	Full-Hard (H04) C36000 Brass :: MakeItFrom.com MakeItFrom.com Menu (ESC) Home>Copper Alloy Up Three>Wrought Brass Up Two>C36000 Brass Up One Full-Hard (H04) C36000 Brass H04 C36000 brass is C36000 brass in the H04 (full hard) temper. It has the highest strength and lowest ductility compared to the other variants of C36000 brass. The graph bars on the material properties cards below compare H04 C36000 brass to: wrought brasses (top), all copper alloys (middle), and the entire database (bottom). A full bar means this is the highest value in the relevant set. A half-full bar means it's 50% of the highest, and so on. Mechanical Properties Elastic (Young's, Tensile) Modulus 100 GPa 15 x 10 6 psi Elongation at Break 5.8 % Poisson's Ratio 0.31 Shear Modulus 39 GPa 5.7 x 10 6 psi Shear Strength 310 MPa 45 x 10 3 psi Tensile Strength: Ultimate (UTS) 530 MPa 77 x 10 3 psi Tensile Strength: Yield (Proof) 260 MPa 38 x 10 3 psi Thermal Properties Latent Heat of Fusion 170 J/g Maximum Temperature: Mechanical 120 °C 250 °F Melting Completion (Liquidus) 900 °C 1650 °F Melting Onset (Solidus) 890 °C 1630 °F Specific Heat Capacity 380 J/kg-K 0.090 BTU/lb-°F Thermal Conductivity 120 W/m-K 67 BTU/h-ft-°F Thermal Expansion 21 µm/m-K Electrical Properties Electrical Conductivity: Equal Volume 26 % IACS Electrical Conductivity: Equal Weight (Specific) 29 % IACS Otherwise Unclassified Properties Base Metal Price 23 % relative Density 8.2 g/cm 3 510 lb/ft 3 Embodied Carbon 2.6 kg CO 2/kg material Embodied Energy 45 MJ/kg 19 x 10 3 BTU/lb Embodied Water 320 L/kg 38 gal/lb Common Calculations Resilience: Ultimate (Unit Rupture Work) 25 MJ/m 3 Resilience: Unit (Modulus of Resilience) 340 kJ/m 3 Stiffness to Weight: Axial 7.0 points Stiffness to Weight: Bending 19 points Strength to Weight: Axial 18 points Strength to Weight: Bending 18 points Thermal Diffusivity 37 mm 2/s Thermal Shock Resistance 18 points Alloy Composition Copper (Cu)Cu 60 to 63 Zinc (Zn)Zn 32.5 to 37.5 Lead (Pb)Pb 2.5 to 3.7 Iron (Fe)Fe 0 to 0.35 Residuals res.0 to 0.5 All values are % weight. Ranges represent what is permitted under applicable standards. Followup Questions How are the material properties defined?How do I compare this exact variant to another material? Further Reading ASTM B16: Standard Specification for Free-Cutting Brass Rod, Bar and Shapes for Use in Screw Machines Copper: Its Trade, Manufacture, Use, and Environmental Status, Gunter Joseph, 2001 Properties and Selection: Nonferrous Alloys and Special-Purpose Materials, ASM Handbook vol. 2, ASM International, 1993 Not BookmarkedBookmarked Member Area Login Get Membership Find Other Materials Search by Property Value Browse from Home Page Table of Contents IntroductionMechanical PropertiesThermal PropertiesElectrical PropertiesOtherwise Unclassified PropertiesCommon CalculationsAlloy CompositionFollowup QuestionsFurther Reading How to Cite Below is a simple format for citing this page as a source. If you belong to an institution that sets its own citation guidelines, use those instead. "Full-Hard (H04) C36000 Brass", www.makeitfrom.com/material-properties/Full-Hard-H04-C36000-Brass, retrieved 2025-09-29. Copyright 2009-25: Disclaimer and Terms. Current page last modified on 2020-05-30.
2813	https://doodlelearning.com/maths/skills/multiplication/powers-of-10	Skip to content What are the powers of ten? Powers of ten are the superheroes of math, saving time and lots of space on every page! Author Amber Watkins Published March 2025 What are the powers of ten? Powers of ten are the superheroes of math, saving time and lots of space on every page! Author Amber Watkins Published March 2025 What are the powers of ten? Powers of ten are the superheroes of math, saving time and lots of space on every page! Author Amber Watkins Published March 2025 Key takeaways Powers of ten are a way to express 10 multiplied by itself a number of times shown as 10n Powers of ten can be written in expanded form, standard form, & exponential form There are many advanced uses of powers of 10 like using them to write numbers in scientific notation and decimals Table of contents Key takeaways Powers of 10 Explained Powers of 10 Chart How to Express Powers of 10 Working with Powers of 10 Practice Did you know the sun is 93,000,000 miles away from earth? Or that the earth is thought to be 4,500,000,000 years old? Just by seeing all those zeros, you know that’s got to be old. In math, the powers of 10 come in handy when expressing big numbers like the age of the earth or the distance to the sun. Let’s learn more about the powers of ten and why they are important in math. Powers of 10, explained By definition, a ‘power’ in math is a number that is multiplied by itself a certain number of times. Therefore, the powers of 10 are how we express 10 multiplied by itself over and over. In fifth grade math, your child will learn the powers of ten along with scientific notation. The powers of ten are most commonly written in exponential form where 10 is the base (the number being multiplied) and the exponent is how many times the base (in our case, 10) is multiplied by itself. When we write powers of ten with exponents, we read them as “10 to the power of ____” and fill in the blank with the exponent. For example, 10 to the power of 2 is written as 102 in exponential form. The powers of ten can also be written in expanded and standard form. Powers of 10 Chart Here are the first ten powers of 10: \| \| \| --- \| \| 101 = 10 \| 101 = 10 \| \| 102 = 100 \| 102 = 100 \| \| 103 = 1,000 \| 103 = 1,000 \| \| 104 = 10,000 \| 104 = 10,000 \| \| 105 = 100,000 \| 105 = 100,000 \| \| 106 = 1,000,000 \| 106 = 1,000,000 \| \| 107 = 10,000,000 \| 107 = 10,000,000 \| \| 108 = 100,000,000 \| 108 = 100,000,000 \| \| 109 = 1,000,000,000 \| 109 = 1,000,000,000 \| \| 1010 = 10,000,000,000 \| 1010 = 10,000,000,000 \| By looking at the table, you’ll see a pattern arise—the number of zeros in the answer is the same as the number of tens you multiplied by. Isn’t that neat? Here are two examples to explain these patterns to your child: Take 10 to the power of 3 (shown as 103). When you multiply three tens (10 x 10 x 10), the answer will have three zeros (1,000). Now look at 10 to the power of 5 (shown as105). When you multiply five tens (10 x 10 x 10 x 10 x 10), the answer will have five zeros (100,000).. Unlock unlimited maths questions Put your skills to the test with fun exercises + maths games that are proven to boost ability! Create Doodle account Try DoodleMaths for free! Select a year group Number Explore the number 10 in a variety of different ways using counting frames and more! ### Shape, space and measure Discover everyday language used to talk about time. ### Patterns ### Number and place value ### Addition and subtraction ### Multiplication and division ### Operations (ASMD) ### Fractions ### Measure ### Shape/geometry ### Statistics ### Ratio and proportion ### Algebra ### Probability How to express powers of 10 In the examples above, we expressed the powers of ten in 3 ways:—expanded form, standard form, & exponential form. Practice each form with your child. Powers of 10 in expanded form is when you write the powers of 10 multiplied out 10 x 10 x 10. Powers of 10 in standard form is when you write the product of 10’s being multiplied as a single number. The number 100,000 is a number in standard form. Powers of 10 in exponential form is the shortcut way of writing powers of 10. 1,000 would be written as 103. Remember, in exponential form, the number 10 is always the base and the number of times you are multiplying 10 is used as the exponent. As we learned earlier, the exponent number is also the number of zeros in the answer. Working with the Powers of 10 Multiplying numbers by the powers of 10 Help your child see that since exponents tell us how many zeros are in each answer, we can easily multiply other numbers by the power of ten. Multiply 4 x 104 Since 104 has 4 zeros, 104 is 10,000. So 4 x 104 is the same as 4 x 10,000 or 40,000. Writing numbers in scientific notation Scientific notation is a way to write large numbers using the powers of 10. Let’s take look with the following example: Write 5,000,000 with the powers of 10. First, count the number of zeros. Since there are six zeros, 5 is being multiplied by 106. So you can write 5,000,000 as 5 x 106. Writing decimals with the powers of 10 Sometimes you will see negative powers of 10. Negative exponents can be written as decimals. Write 10-3 as a decimal. When a negative exponent is used, you must place the 10-3 on the bottom of a fraction to make it positive. 10-3 would become 1 / 103 Since 103 is the same as 10 x 10 x 10, you can rewrite the fraction as 1/(10 x 10 x 10) or 1/1000. 1/1000 can be written as the decimal .001. If your child is struggling with this section, please see our converting fractions to decimals guide. Practise times table with DoodleMaths! DoodleMaths is an award-winning app that’s filled with thousands of questions and games exploring multiplication, division and more! Plus, get free access to DoodleTables with any DoodleMaths subscription! Designed by teachers, it creates each child a unique work programme tailored to their needs, doubling their progression with just 10 minutes of use a day. Try it for free! Create a free Doodle account Practice the powers of ten Here are some beginner problems for your child to practice the powers of ten. What is 10 to the power of 4? What is 10 to the power of 8? What is 2 times 10 to the power of 3? Answers: 104 – The exponent tells us how many times to multiply 10 (4 times). 10 x 10 x 10 x 10 = 10,000. 106 – The exponent tells us how many times to multiply 10 (6 times). It also tells us how many zeros will be in the answer (six). 10 x 10 x 10 x 10 x 10 x 10 = 1,000,000. 2 x 103 is the same as 2 x 10 x 10 x 10 = 2 x 1,000 = 2,000. For more practice with powers of ten, scientific notation, and other math topics for fifth graders, try the DoodleMath. Ready to give it a go? Put your knowledge to the test with these no-risk practice problems to get you ready for the classroom! Additional Counting in Powers of 10 Practice If you are counting in 10s, the tens place changes. If you are counting in 100s, the hundreds place changes. If you are counting in 1000s, the thousands place changes. If you are counting in 10000s, the ten thousands place changes. Now try a few practice questions! Welcome to DoodleMaths! Ready to try some questions? Download the DoodleMaths app to practice questions for free today! Want more practice? Our math help app is a great resource for times table practice! Lesson credits Amber Watkins Amber is an education specialist with a degree in Early Childhood Education. She has over 12 years of experience teaching and tutoring primary through university level maths. "Knowing that my work in maths education makes such an impact leaves me with an indescribable feeling of pride and joy!" Amber Watkins Amber is an education specialist with a degree in Early Childhood Education. She has over 12 years of experience teaching and tutoring primary through university level maths. "Knowing that my work in maths education makes such an impact leaves me with an indescribable feeling of pride and joy!" 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2814	https://asmedigitalcollection.asme.org/appliedmechanics/article/68/1/87/461375/Shear-Coefficients-for-Timoshenko-Beam-Theory	Shear Coefficients for Timoshenko Beam Theory \| J. Appl. 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R. Hutchinson, Life Mem. ASME, Professor Emeritus, J. R. Hutchinson, Life Mem. ASME, Professor Emeritus, Department of Civil and Environmental Engineering, University of California, Davis, CA 95616 Search for other works by this author on: This Site PubMed Google Scholar Author and Article Information J. R. Hutchinson, Life Mem. ASME, Professor Emeritus, Department of Civil and Environmental Engineering, University of California, Davis, CA 95616 Contributed by the Applied Mechanics Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF APPLIED MECHANICS. Manuscript received by the ASME Applied Mechanics Division, June 7, 2000; final revision, August 15, 2000. Associate Editor: R. C. Benson. Discussion on the paper should be addressed to the Editor, Professor Lewis T. Wheeler, Department of Mechanical Engineering, University of Houston, Houston, TX 77204-4792, and will be accepted until four months after final publication of the paper itself in the ASME JOURNAL OF APPLIED MECHANICS. J. Appl. Mech. Jan 2001, 68(1): 87-92 (6 pages) Published Online: August 15, 2000 Article history Received: June 7, 2000 Revised: August 15, 2000 Views Icon Views Open Menu Article contents Share Icon Share Facebook X LinkedIn Email Cite Icon Cite Permissions Search Site Citation Hutchinson, J. R. (August 15, 2000). "Shear Coefficients for Timoshenko Beam Theory ." ASME. J. Appl. Mech. January 2001; 68(1): 87–92. Download citation file: Ris (Zotero) Reference Manager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex ProCite Medlars toolbar search Search Dropdown Menu toolbar search search input Search input auto suggest filter your search Search Advanced Search The Timoshenko beam theory includes the effects of shear deformation and rotary inertia on the vibrations of slender beams. The theory contains a shear coefficient which has been the subject of much previous research. In this paper a new formula for the shear coefficient is derived. For a circular cross section, the resulting shear coefficient that is derived is in full agreement with the value most authors have considered “best.” Shear coefficients for a number of different cross sections are found. Issue Section: Technical Papers Keywords: shear deformation,vibrations,rotation,stress analysis Topics: Cross section (Physics),Shear (Mechanics),Timoshenko beam theory Timoshenko , S. P. , 1921 , “ On the Correction for Shear of the Differential Equation for Transverse Vibrations of Bars of Prismatic Bars ,” Philos. Mag. , 41 , pp. 744 – 746 . Kaneko , T. , 1975 , “ On Timoshenko’s Correction for Shear in Vibrating Beams ,” J. Phys. D 8 , pp. 1927 – 1936 . Timoshenko , S. P. , 1922 , “ On the Transverse Vibrations of Bars of Uniform Cross Section ,” Philos. Mag. , 43 , pp. 125 – 131 . Cowper , G. R. , 1966 , “ The Shear Coefficient in Timoshenko’s Beam Theory ,” ASME J. Appl. Mech. , 33 , pp. 335 – 340 . Hutchinson , J. R. , 1981 , “ Transverse Vibrations of Beams, Exact Versus Approximate Solutions ,” ASME J. Appl. Mech. , 48 , pp. 923 – 928 . Leissa , A. W. , and So , J. , 1995 , “ Comparisons of Vibration Frequencies for Rods and Beams From One-Dimensional and Three-Dimensional Analyses ,” J. Acoust. Soc. Am. , 98 , pp. 2122 – 2135 . Hutchinson , J. R. , 1996 , comments on “ Comparisons of Vibration Frequencies for Rods and Beams From One-Dimensional and Three-Dimensional Analyses ,” J. Acoust. Soc. Am. , 98 , pp. 2122 – 2135 100 , pp. 1890 – 1893 . Spence , G. B. , and Seldin , E. J. , 1970 , “ Sonic Resonances of a Bar and Compound Torsional Oscillator ,” J. Appl. Phys. , 41 , pp. 3383 – 3389 . Spinner , S. , Reichard , T. W. , and Tefft , W. E. , 1960 , “ A Comparison of Experimental and Theoretical Relations Between Young’s Modulus and the Flexural and Longitudinal Resonance Frequencies of Uniform Bars ,” J. Res. Natl. Bur. Stand., Sect. A , 64A , pp. 147 – 155 . Hutchinson , J. R. , and Zillmer , S. D. , 1986 , “ On the Transverse Vibration of Beams of Rectangular Cross-Section ,” ASME J. Appl. Mech. , 53 , pp. 39 – 44 . Hutchinson , J. R. , and El-Azhari , S. A. , 1986 , “ Vibrations of Free Hollow Circular Cylinders ,” ASME J. Appl. Mech. , 53 , pp. 641 – 646 . Armena`kas, A. E., Gazis, D. C., and Herrmann G., 1969, Free Vibrations of Circular Cylindrical Shells, Pergamon Press, Oxford, UK. Leissa , A. W. , and So , J. , 1997 , “ Free Vibrations of Thick Hollow Circular Cylinders From Three-Dimensional Analysis ,” ASME J. Vibr. Acoust. , 119 , pp. 89 – 95 . Reissner , E. , 1950 , “ On a Variational Theorem in Elasticity ,” J. Math. Phys. , 29 , pp. 90 – 95 . Love, A. E. H., 1944, A Treatise on the Mathematical Theory of Elasticity, Dover, New York. Copyright©2001 by ASME You do not currently have access to this content. Sign In Sign In Or Register For Account Sign in via your Institution Purchase this Content $25.00 Purchase Learn about subscription and purchase options Product added to cart. Check OutContinue BrowsingClose Modal 5,762 Views 466Web of Science 486Crossref 508 CITATIONS View Metrics ×Close Modal Get Email Alerts Article Activity Alert Accepted Manuscript Alert New Issue Alert Close Modal Cited By Web Of Science (466) Google Scholar CrossRef (486) Latest Most Read Most Cited Wave Propagation in a Strongly Nonlinear Mass-in-Mass Chain With Soft Springs and Stability and Bifurcation Analyses of Periodic Solutions J. Appl. Mech (December 2025) Closely Piling up of Multiple Adhesive Fronts in Adhesive Friction Due to Re-Attachment J. Appl. Mech (December 2025) A centerline model for bending-dominated configurations of elastic strips J. Appl. Mech Renormalization of mechanical properties in fluctuating active membranes J. Appl. Mech Related Articles Discussion: “Shear Coefficients for Timoshenko Beam Theory” (Hutchinson, J. R., 2001, ASME J. Appl. Mech., 68 , pp. 87–92) J. Appl. 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2815	https://brainly.com/question/18719429	[FREE] Convert 200 km/h to m/s. - brainly.com 7 Search Learning Mode Cancel Log in / Join for free Browser ExtensionTest PrepBrainly App Brainly TutorFor StudentsFor TeachersFor ParentsHonor CodeTextbook Solutions Log in Join for free Tutoring Session +45,2k Smart guidance, rooted in what you’re studying Get Guidance Test Prep +15,1k Ace exams faster, with practice that adapts to you Practice Worksheets +8,4k Guided help for every grade, topic or textbook Complete See more / Physics Textbook & Expert-Verified Textbook & Expert-Verified Convert 200 km/h to m/s. 2 See answers Explain with Learning Companion NEW Asked by obengdavid312 • 10/28/2020 Read More Community by Students Brainly by Experts ChatGPT by OpenAI Gemini Google AI Community Answer 0.0 0 Upload your school material for a more relevant answer 200km × 1000m/1km = 200000 m 1h × 60 mins/h × 60sec/min = 3600s now you divide 200000m/3600s = 55.55m/s Answered by bryanluke403 •2 answers Thanks 0 0.0 (0 votes) Textbook &Expert-Verified⬈(opens in a new tab) 0.0 0 Physics for AP® Courses 2e - Kenneth Podolak, Henry Smith The AP Physics Collection - Gregg Wolfe, Erika Gasper, John Stoke, Julie Kretchman, David Anderson, Nathan Czuba, Sudhi Oberoi, Liza Pujji, Irina Lyublinskaya, Douglas Ingram Introduction to General Chemistry - Muhammad Arif Malik Upload your school material for a more relevant answer To convert 200 km/h to m/s, multiply by 1,000 to convert kilometers to meters and divide by 3,600 to convert hours to seconds. The final result is approximately 55.56 m/s. This conversion is useful for understanding speeds in terms of the Metric System's standard units. Explanation To convert 200 km/h (kilometers per hour) to m/s (meters per second), we can use the conversion factors for distance and time. Understand the conversion factors: 1 kilometer is equal to 1,000 meters, and 1 hour is equal to 3,600 seconds. Set up the conversion: We want to convert kilometers to meters and hours to seconds. 200 km/h=200×1 km 1000 m×3600 s 1 h Calculate the conversion: 200×3600 1000=3600 200000≈55.56 m/s Final result: Therefore, 200 km/h is approximately equal to 55.56 m/s. Examples & Evidence For example, if a car travels at a speed of 100 km/h, to convert it to m/s, you would calculate 100 x 1000 / 3600, which equals approximately 27.78 m/s. Another example would be converting the speed of an athlete running at 15 km/h, which would be 15 x 1000 / 3600, about 4.17 m/s. The conversions from kilometers to meters and hours to seconds are based on internationally recognized measurements, where 1 kilometer = 1,000 meters and 1 hour = 3,600 seconds. Thanks 0 0.0 (0 votes) Advertisement Community Answer This answer helped 10256 people 10K 1.0 0 200 kilometers per hour= 55.556 meters per second Explanation have a great day Answered by hschool613 •22 answers•10.3K people helped Thanks 0 1.0 (1 vote) Advertisement ### Free Physics solutions and answers Community Answer 47 Whats the usefulness or inconvenience of frictional force by turning a door knob? Community Answer 5 A cart is pushed and undergoes a certain acceleration. Consider how the acceleration would compare if it were pushed with twice the net force while its mass increased by four. Then its acceleration would be? Community Answer 4.8 2 define density and give its SI unit Community Answer 9 To prevent collisions and violations at intersections that have traffic signals, use the _____ to ensure the intersection is clear before you enter it. Community Answer 4.0 29 Activity: Lab safety and Equipment Puzzle Community Answer 4.7 5 If an instalment plan quotes a monthly interest rate of 4%, the effective annual/yearly interest rate would be _______. 4% Between 4% and 48% 48% More than 48% Community Answer When a constant force acts upon an object, the acceleration of the object varies inversely with its mass 2kg. When a certain constant force acts upon an object with mass , the acceleration of the object is 26m/s^2 . If the same force acts upon another object whose mass is 13 , what is this object's acceleration Community Answer 20 [4 A wheel starts from rest and has an angular acceleration of 4.0 rad/s2. When it has made 10 rev determine its angular velocity.]]( "4 A wheel starts from rest and has an angular acceleration of 4.0 rad/s2. When it has made 10 rev determine its angular velocity.]") Community Answer 4.1 5 Lucy and Zaki each throw a ball at a target. What is the probability that both Lucy and Zaki hit the target? Community Answer 22 The two non-parallel sides of an isosceles trapezoid are each 7 feet long. The longer of the two bases measures 22 feet long. The sum of the base angles is 140°. a. Find the length of the diagonal. b. Find the length of the shorter base. New questions in Physics Match each discovery about atomic structure to the scientist who made the discovery. Discoveries: • Used cathode ray tubes to discover electrons • Used the gold foil experiment to discover the atomic nucleus • Suggested that electrons orbit the nucleus in fixed energy levels La Luna hace una revolución completa en 28 días. Si la distancia promedio entre la Luna y la Tierra es de 38,4×1 0 7 m, calcular: a. La velocidad tangencial de la Luna con respecto a la Tierra. Which happens during the night at the beach? A. The water is cooler than the land, and wind blows toward the beach. B. The water is cooler than the land, and wind blows toward the water. C. The water is warmer than the land, and wind blows toward the water. D. The water is warmer than the land, and wind blows toward the beach. Fill in the missing part of this equation. (0.030 c m 3 g)⋅□=?m 3 g Noise produced in your workplace is measured in units of A) tonnes, tns. B) sound flux, sf. C) decibels, db. D) wave force, wf. 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2816	https://pubmed.ncbi.nlm.nih.gov/1351281/	Prolonged beta-agonist infusion does not induce desensitization or down-regulation of beta-adrenergic receptors in newborn sheep - PubMed Clipboard, Search History, and several other advanced features are temporarily unavailable. Skip to main page content An official website of the United States government Here's how you know The .gov means it’s official. Federal government websites often end in .gov or .mil. Before sharing sensitive information, make sure you’re on a federal government site. The site is secure. The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely. 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Prolonged beta-agonist infusion does not induce desensitization or down-regulation of beta-adrenergic receptors in newborn sheep H M Stein1,K Oyama,R Sapien,B A Chappell,J F Padbury Affiliations Expand Affiliation 1 Department of Pediatrics, UCLA School of Medicine, Torrance 90502. PMID: 1351281 DOI: 10.1203/00006450-199205000-00009 Item in Clipboard Prolonged beta-agonist infusion does not induce desensitization or down-regulation of beta-adrenergic receptors in newborn sheep H M Stein et al. Pediatr Res.1992 May. Show details Display options Display options Format Pediatr Res Actions Search in PubMed Search in NLM Catalog Add to Search . 1992 May;31(5):462-7. doi: 10.1203/00006450-199205000-00009. Authors H M Stein1,K Oyama,R Sapien,B A Chappell,J F Padbury Affiliation 1 Department of Pediatrics, UCLA School of Medicine, Torrance 90502. PMID: 1351281 DOI: 10.1203/00006450-199205000-00009 Item in Clipboard Cite Display options Display options Format Abstract In adult animals, prolonged beta-agonist exposure leads to down-regulation of beta-adrenergic receptors and desensitization. Prior evidence from our lab suggests that this may not occur in developing animals. To study this, we measured the response to graded epinephrine infusion [2.7, 5.5, 13.6, 27.3 mumol/(kg.min), (0.5, 1.0, 2.5, 5.0 micrograms/(kg.min)], myocardial beta-agonist receptor density, and components of the receptor-cyclase system in newborn lambs before (n = 6) and after (n = 5) 3 d of continuous isoproterenol administration (2 micrograms/kg/min). beta-Adrenergic receptors were measured by radioligand binding. Epinephrine dose-response curves were analyzed for the threshold and slope for changes in mean blood pressure, systolic blood pressure, and heart rate versus plasma epinephrine levels. Despite 3 d of continuous isoproterenol infusion, we observed no desensitization of the hemodynamic response to epinephrine. There was a reduction in receptor density when expressed per membrane protein [155.3 +/- 19.5 (controls) versus 73.2 +/- 3.8 fmol/mg protein (agonist exposed), p less than 0.05], but no alteration in receptor density when expressed per g cardiac wet weight [258.8 +/- 39.9 (controls) versus 406.8 +/- 74.0 fmol/g wet weight (agonist exposed)]. There was no alteration in agonist affinity or in adenylyl cyclase activity after adjustment for membrane protein recovery. Prolonged beta-agonist infusion in newborn lambs does not desensitize hemodynamic responses to infused epinephrine. We propose that receptor regulation in developing animals is fundamentally different than in adult animals. PubMed Disclaimer Similar articles Atypical regulation of hepatic adenylyl cyclase and adrenergic receptors during a critical developmental period: agonists evoke supersensitivity accompanied by failure of receptor down-regulation.Thai L, Galluzzo JM, McCook EC, Seidler FJ, Slotkin TA.Thai L, et al.Pediatr Res. 1996 Apr;39(4 Pt 1):697-707. doi: 10.1203/00006450-199604000-00023.Pediatr Res. 1996.PMID: 8848348 Ontogeny of regulatory mechanisms for beta-adrenoceptor control of rat cardiac adenylyl cyclase: targeting of G-proteins and the cyclase catalytic subunit.Zeiders JL, Seidler FJ, Slotkin TA.Zeiders JL, et al.J Mol Cell Cardiol. 1997 Feb;29(2):603-15. doi: 10.1006/jmcc.1996.0303.J Mol Cell Cardiol. 1997.PMID: 9140819 Desensitization of beta-adrenergic receptor-mediated vascular smooth muscle relaxation.Tsujimoto G, Hoffman BB.Tsujimoto G, et al.Mol Pharmacol. 1985 Feb;27(2):210-7.Mol Pharmacol. 1985.PMID: 2578603 Beta-receptor regulation: dynamics of density and function throughout the cardiac cycle.Gibson RK.Gibson RK.J Cardiovasc Nurs. 1991 Jul;5(4):49-56. doi: 10.1097/00005082-199107000-00006.J Cardiovasc Nurs. 1991.PMID: 1646865 Review. Downregulation of myocardial beta-adrenergic receptors. Receptor subtype selectivity.Muntz KH, Zhao M, Miller JC.Muntz KH, et al.Circ Res. 1994 Mar;74(3):369-75. doi: 10.1161/01.res.74.3.369.Circ Res. 1994.PMID: 8118945 Review.No abstract available. See all similar articles Cited by Association between Beta-Sympathomimetic Tocolysis and Risk of Autistic Spectrum Disorders, Behavioural and Developmental Outcome in Toddlers.Altay MA, Görker I, Aslanova R, Bozatlı L, Turan N, Kaplan PB.Altay MA, et al.Open Access Maced J Med Sci. 2017 Sep 10;5(6):730-735. doi: 10.3889/oamjms.2017.153. eCollection 2017 Oct 15.Open Access Maced J Med Sci. 2017.PMID: 29104681 Free PMC article. Publication types Research Support, Non-U.S. Gov't Actions Search in PubMed Search in MeSH Add to Search Research Support, U.S. Gov't, P.H.S. Actions Search in PubMed Search in MeSH Add to Search MeSH terms Adrenergic beta-Agonists / administration & dosage Actions Search in PubMed Search in MeSH Add to Search Animals Actions Search in PubMed Search in MeSH Add to Search Animals, Newborn Actions Search in PubMed Search in MeSH Add to Search Down-Regulation Actions Search in PubMed Search in MeSH Add to Search Epinephrine / administration & dosage Actions Search in PubMed Search in MeSH Add to Search Hemodynamics / drug effects Actions Search in PubMed Search in MeSH Add to Search Infusions, Intravenous Actions Search in PubMed Search in MeSH Add to Search Isoproterenol / administration & dosage Actions Search in PubMed Search in MeSH Add to Search Myocardium / metabolism Actions Search in PubMed Search in MeSH Add to Search Receptors, Adrenergic, beta / drug effects Actions Search in PubMed Search in MeSH Add to Search Receptors, Adrenergic, beta / metabolism Actions Search in PubMed Search in MeSH Add to Search Sheep Actions Search in PubMed Search in MeSH Add to Search Substances Adrenergic beta-Agonists Actions Search in PubMed Search in MeSH Add to Search Receptors, Adrenergic, beta Actions Search in PubMed Search in MeSH Add to Search Isoproterenol Actions Search in PubMed Search in MeSH Add to Search Epinephrine Actions Search in PubMed Search in MeSH Add to Search Related information MedGen PubChem Compound (MeSH Keyword) Grants and funding HD 07013/HD/NICHD NIH HHS/United States HD 18014/HD/NICHD NIH HHS/United States [x] Cite Copy Download .nbib.nbib Format: Send To Clipboard Email Save My Bibliography Collections Citation Manager [x] NCBI Literature Resources MeSHPMCBookshelfDisclaimer The PubMed wordmark and PubMed logo are registered trademarks of the U.S. Department of Health and Human Services (HHS). 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2817	https://www.motioncontroltips.com/what-are-eddy-currents-and-how-do-they-affect-motor-performance/	Skip to primary navigation Skip to primary sidebar Skip to footer Motion Control Tips Automation • Motion Control • Power Transmission You are here: Home / FAQs + basics / What are eddy currents and how do they affect motor performance? What are eddy currents and how do they affect motor performance? By Danielle Collins Leave a Comment One of the electromagnetic principles that governs the operation of an electric motor is Faraday’s law, which states that when the magnetic environment of a coil of wire is changed — whether by moving the magnet and coil relative to each other or by changing the magnetic field — a voltage, or electromotive force (EMF), will be induced. Another fundamental law of electromagnetism — Lenz’ law — builds on Faraday’s law to ensure that the magnetic flux remains constant. Lenz’ law explains that the polarity of the induced EMF is such that the current it generates (according to Ohm’s law) will have a magnetic field whose direction opposes the change that produced it. In other words, the induced magnetic field opposes the original magnetic field. (Notice the negative sign in the equation for EMF, below.) E = induced EMF (V) N = number of turns in the coil Φ = magnetic flux (weber, Wb) t = time (s) The currents generated according to Faraday’s law are known as eddy currents, because they flow, or circulate, in the conductor — similar to eddies in a body of water. The magnitude of eddy current is directly related to the strength of the magnetic field, the rate of change of the magnetic flux, and the area of the coil, while having an inverse relationship to the resistivity of the conductor. As eddy currents flow through the conductor, heat is generated, referred to as Joule heating. The amount of Joule heating is proportional to the square of the current, so even a small reduction in eddy currents can have a significant effect on the amount of heat produced. One of the most effective ways to reduce eddy currents in motors is to use laminations (thin sheets of metal that are electrically insulated from one another), which have a smaller surface area and higher resistance that a solid core. This reduces the magnitude of the eddy currents that can form, and, in turn, the amount of Joule heating that occurs. The power lost as a result of Joule heating is referred to as eddy current loss, which can be found with the equation: Pe= eddy current loss (W) ke= eddy current coefficient B = flux density (Wb/m2) f = frequency of magnetic reversals per second (Hz) t = thickness of the conductor (m) this is why thin laminations are used rather than solid cores V = volume of the conductor (m3) Notice that the amount of eddy current loss is proportional to the square of the frequency of magnetic reversals, and therefore, to the square of motor speed. This is why the torque produced by an electric motor decreases as speed increases — because of heating due to eddy current losses (as well as other losses, such as hysteresis losses). Motors with high pole counts, such as stepper motors, experience high eddy current losses due to their high frequency of magnetic reversals. Eddy current losses and hysteresis losses are categorized as core losses (also referred to as iron losses or magnetic losses) because they depend on the magnetic paths through the iron core of the motor. Although eddy currents are an undesirable occurrence in electric motors, they are useful in other applications, such as eddy current brakes, non-destructive testing devices, and inductive proximity sensors. You Might Also Like Reader Interactions Leave a Reply You must be logged in to post a comment. Footer DESIGN WORLD NETWORK Design World Online The Robot Report Coupling Tips Linear Motion Tips Bearing Tips Fastener Engineering. Wire and Cable Tips MOTION CONTROL TIPS Subscribe to our newsletter Advertise with us Contact us About us
2818	https://www.reddit.com/r/learnmath/comments/1gxb3pr/subtracting_fractions_please_explain_this_to_me/	Subtracting Fractions - Please explain this to me : r/learnmath Skip to main contentSubtracting Fractions - Please explain this to me : r/learnmath Open menu Open navigationGo to Reddit Home r/learnmath A chip A close button Log InLog in to Reddit Expand user menu Open settings menu Go to learnmath r/learnmath r/learnmath Post all of your math-learning resources here. Questions, no matter how basic, will be answered (to the best ability of the online subscribers). 403K Members Online •10 mo. ago Nervous_Bee8805 Subtracting Fractions - Please explain this to me Hi there, I was wondering if someone could explain this to me: If I try to subtract fractions, ie. a/b - c/d, then I first look for a common multiple of both of the denominators. When I then multiply the second fraction, why am I not multiplying "-"c/d times the common multiple and instead ignore the (-)-sign in the multiplication? I have been studying with two different programs and in neither of the two this is explained. After working through a bunch of problems my results were always wrong because of this. Read more Share Related Answers Section Related Answers How to subtract fractions step by step How to find common denominators for fractions Effective strategies for mastering algebra Best online tools for geometry practice Tips for improving mental math speed New to Reddit? Create your account and connect with a world of communities. Continue with Email Continue With Phone Number By continuing, you agree to ourUser Agreementand acknowledge that you understand thePrivacy Policy. Public Anyone can view, post, and comment to this community 0 0 Top Posts Reddit reReddit: Top posts of November 22, 2024 Reddit reReddit: Top posts of November 2024 Reddit reReddit: Top posts of 2024 Reddit RulesPrivacy PolicyUser AgreementAccessibilityReddit, Inc. © 2025. All rights reserved. Expand Navigation Collapse Navigation
2819	https://www.scribd.com/document/818346084/Matthew-Arnold-the-Study-of-Poetry	Matthew Arnold The Study of Poetry \| PDF \| Poetry \| French Poetry Opens in a new window Opens an external website Opens an external website in a new window This website utilizes technologies such as cookies to enable essential site functionality, as well as for analytics, personalization, and targeted advertising. To learn more, view the following link: Privacy Policy Open navigation menu Close suggestions Search Search en Change Language Upload Sign in Sign in Download free for 30 days 100%(1)100% found this document useful (1 vote) 424 views 10 pages Matthew Arnold The Study of Poetry The document discusses Matthew Arnold's essay 'The Study of Poetry,' which critiques the role of poetry in modern industrial society and its superiority over science and religion. Arnold emp… Full description Uploaded by mk5583482 AI-enhanced title and description Go to previous items Go to next items Download Save Save Matthew Arnold the Study of Poetry For Later Share 100%100% found this document useful, undefined 0%, undefined Print Embed Ask AI Report Download Save Matthew Arnold the Study of Poetry For Later You are on page 1/ 10 Search Fullscreen Pape r 10: Mo dule N o 1 1: E T e x t MHRD-UGC ePG Pathshala - English Principal Inv es tigato r& Affiliation : Prof. T utu n M ukhe rjee, Uni v er si ty of Hydera bad Pape r No & T itl e: L i tera ry Cri ti ci sm an d Theory (Paper 10) Pap e r C oo rdinato r &A ffi liation : Dr. Anita Bhela, Delhi College of Arts and Commerce, University of Delhi Module N um ber &T itl e: Matthew Arnold: “The Study of Poetry” (11) Conte nt Writer's N am e &Af fil iation Dr. Jyotsna Pathak, Associate Professor, University of Delhi Nam e & A ff il iation of C onte nt Reviewer : Dr. Vinita Gupta Chaturvedi, Associate Professor, University of Delhi Nam e & A ff ili ation o f C on ten t Edi tor : Dr. Anita Bhela, Delhi College of Arts and Commerce, University of Delhi adDownload to read ad-free Matth e w Ar no ld: “ T he Study of Poetry ” Ar n o ld is o n e o f t h e fo r e m o s t c r it ic s o f t h e 1 9 t h c e n t u r y. I n h is w r it in g s h e c r it iq u e d a n d c o m-me nte d e x ten si vel y on cultural and soci al i ssue s, reli gion and edu cation. He was th e fi rst cr i ti c to pose questions within the context of the modern industrial society. He was a humanist for whom ma n i n the indu strial soci ety was condem ne d to a mech an i zed exi sten ce with a f ra cture d spiri tua l and moral sensibility. He criticized the narrow mercantile concerns of the 19th century bourgeois an d their obsession wi th util i tar i anism an d rea son. He r ejected the gr owi ng scienti f i c tem per a nd positivism of the age. The central concern for Arnold was the problem of living fulfilled lives in an i ndu strial so ci ety. I n h i s criti ci sm he atte mp ts to m ov e from the exteriori ty of bourg eois exi st-en ce to a n inter i orit y of the self. I t is i n li ne with this that cr i ti ci sm, cul tur e a nd poetry be come mod es of i nte riori ty i n or der to nu l l i f y the exterio rity of bourg eois exi sten ce. It must be noted tha t thi s essay i s not di rect ed at the professi onal men of l ette rs bu t rath er the g ene ra l middle-cl ass rea der w i th an i ntere st i n poetry. C onfli ct betwee n Scienc e and Religion “ The Study of Poetry, ” written a s Gener al I ntrodu cti on to The Eng l i sh Poets, edited by T. H. Ward, is one of the most influential texts of literary humanism. This essay contains some of his best-known pronouncem ents a bout poetry and poets. I t i s pree minentl y an essay about j udgm ent and evaluation. It insists on the social and cultural functions of literature, it ability to civilize and to culti vate m orali ty, as well assist provi ding bu l war k aga i nst th e m ech anisti c excesse s of mod-ern civilization. In the essay, Arnold claims an elevated status for poetry over science, religion, theology and philosophy. He postulates that the fields of science, religion, philosophy and poli-ti cs are awa sh with Charlatan i sm. These ideol ogi es de l i ber ate l y o bf usca te f acts a nd cre ate con f u- adDownload to read ad-free si on be twee n tha t which i s good an d de si ra ble and th at wh i ch is f ake a nd h arm f ul. Rel i gion f ail s to address fundam ental questi ons f aci ng m an since i ts status ha s been threa tened by sc i ence which i n tur n f alsel y presen ts i tself as the ne w arb i ter of k nowledge. Moreover, was he p oi nts out i n the essa y, re l i gion pl ace s me an i ngs on facts which are b eing proved to be incorrect a nd fal se. In contra st to this poetry rests m ea ning in i dea s and thes e a re i nf all i ble. Phil osophy i s i nca pab l e of providing moral and spiritual sustenance to man since it is itself grappling with entrenched an d un res ol ved que sti ons an d pr obl em s. I n vi ew of thi s f act, accord i ng t o the criti c, o nly poetry is in a position to offer any kind of spiritual and emotional succor to man. Poetry is also, accord-ing to him, the only viable method of interpreting life. To interact with poetry it is imperative tha t the re ade r vi ews the poetic obj ect a s it reall y i s by avoi ding the histori cal and p ers onal f all a-ci es. B y re j ecting an ab stra ct sy stem and foregr ounding his to uchs tone m eth od A rnold chall eng-es the re ade r to acce pt hi s cri ti cal taste an d j udg me nt. His assu mp ti on i s that rea sonab l e people, wi thout absolute sta nda rds, can a gre e on th e qu ali ty not onl y of a poe t's artistry but of his 'criti-cism of life'. To his credit, Arnold ’ s surviving notebooks, filled with short quotations from the cl assics, sugg est th at he re all y practiced the m ethod h e ad vo cate d. In thi s essay A rn ol d i s con-cern ed with ran ki ng Eng l i sh poets a nd de ci ding as to whi ch ones m ay be si ngled out as being truly cl assic. In this en dea vo r som e of his statem en ts wer e contr ov ers i al when fi rst sta ted. In to-d a y ’ s ag e of shif ti ng ca nons th ese are provi ng to be extrem ely controversial. The Impo rtance o f Poe try or Poetry as a Spirit ual Force The critic apprises his readers of the fact that if poetry were to play such a central role in the lives of men, then it is imperative that it be of a “ higher order of excellence. ” T hi s mea ns that n ot onl y shoul d poetry maintai n a h i gher standar d but a l so that i t be judged by more str i ngen t para m-ete rs th an any othe r fi eld of study. T her efo re the disti nctions of ‘ excellent and inferior, ’ ‘ sound adDownload to read ad-free and unsound, ’ and true and untrue ’ gain significance in the case of poetry, considering the fact that i t has a “ higher destiny. ” Ac c o r d in g t o Ar n o ld it is n e c e s s a r y t o h o ld p o e t r y t o s u c h h ig h e r stan dar ds since in the increa si ngly me chan i zed world i t wil l prove to be t he on l y source of succor and peace to man. It is only poetry that gives a criticism of life; however the value and credibility of such a criti ci sm is i n direct pr oportio n to the d egr ee with which poem app roach es the i dea l s of trut h an d be au ty. A rn ol d ’ s hum anism i mpli ed tha t he imparted to poetry the power to sustain and deli gh t ma n in the dre ary conf i nes of m oder n exi sten ce. I t i s f or this reason th at he was insi sten t in the creation of “ the best ” poetry. He further elaborates that it i s becau se poetry sustai ns m an in ti m es of troub l es th at he s hould be extre me l y cri ti cal and conscious of wha t he is read i ng. Read-ing is not passive exercise but rather a collaborative endeavor. Since the act of reading poetry i nfl ue nce s the m i nd a nd the s piri t, A rn ol d i nsists that the r ea der b e consta ntl y awa re of wha t he is re ading a nd j udg e a s to whe the r it i s f or his bene f i t or n ot. He insi sts tha t every act of reading poetr y should gi ve a se nse of the excel l en t an d a sen se of j oy. If one f ee l s the se wh i l e r ea ding poem then i t i s the tru e estimate of the worth of t he te x t being read. He goes on to sugge st that it i s only a ca re f ul rea ding of poe try tha t all ows u s to i de ntif y the ca l i be r of poe ts an d to ident i f y them as good or bad. It i s onl y af ter this has been done that the r eade r can choose to accept or re j ect the art i st and this work. The study of poetr y i s an e x er ci se, he says i n the essa y, tha t re-quires consistent scr utiny: the r ea der sh ould be able to i den ti f y when a wor k f all s short in term s of l angu age or me aning and g i v e it the correct rating I t i s onl y when the read er does th i s that he will be in a position to identify good poetry and enjoy it. Thus “ negative criticism ” i n the study of poetry is essen ti al to i den ti f y good l i ter atur e a nd e njo y i t. I n fact he str esse s the f act th at m ere-l y knowl edg e of the eff orts m ade by the ar ti st i n cr ea ti ng th e work, or i nf orm ation rega rding its wea knes ses; o r knowledge of the b i ogra phical deta i l s of the p oet ar e m ean i ng l ess if they do not adDownload to read ad-free adDownload to read ad-free adDownload to read ad-free adDownload to read ad-free adDownload to read ad-free adDownload to read ad-free Share this document Share on Facebook, opens a new window Share on LinkedIn, opens a new window Share with Email, opens mail client Copy link Millions of documents at your fingertips, ad-free Subscribe with a free trial You might also like Medieval and Early Modern English Literature Notes No ratings yet Medieval and Early Modern English Literature Notes 41 pages Honest Criticism and Sensitive Appreciation Is Directed Not Upon The Poet But Upon The Poetry. 100% (1) Honest Criticism and Sensitive Appreciation Is Directed Not Upon The Poet But Upon The Poetry. 6 pages Mathew Arnold No ratings yet Mathew Arnold 13 pages Matthew Arnold As A Critic 100% (1) Matthew Arnold As A Critic 2 pages The Study of Poetry 100% (1) The Study of Poetry 4 pages Unit - VIII: Literary Criticism Aristotle 100% (1) Unit - VIII: Literary Criticism Aristotle 4 pages How Do I Love Thee 100% (1) How Do I Love Thee 11 pages Hazlitt On Going A Journey Summary No ratings yet Hazlitt On Going A Journey Summary 2 pages Question - How Does Eliot Distinguish Between - Unitication of Sensibility and Dissociation of Sensibility - in The Metaphysical Poets 100% (1) Question - How Does Eliot Distinguish Between - Unitication of Sensibility and Dissociation of Sensibility - in The Metaphysical Poets 7 pages Novel, Its Evolution and Features of Modern Novel No ratings yet Novel, Its Evolution and Features of Modern Novel 40 pages Mac Flecknoe 100% (1) Mac Flecknoe 7 pages Ben Jonson 100% (1) Ben Jonson 3 pages Dissociation of Sensibility 50% (2) Dissociation of Sensibility 7 pages Sir Philip Sidney's: Apology Poetry No ratings yet Sir Philip Sidney's: Apology Poetry 19 pages The Faerie Queene LitChart 1 100% (1) The Faerie Queene LitChart 1 17 pages The Romantic Period Was Largely A Reaction Against The Ideology of The Enlightenment Period That Dominated Much of European Philosophy No ratings yet The Romantic Period Was Largely A Reaction Against The Ideology of The Enlightenment Period That Dominated Much of European Philosophy 4 pages Poetics by Aristotle No ratings yet Poetics by Aristotle 8 pages Absolam and Achitophel Notes M.A.II Sem. 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2820	https://www.sciencing.com/use-exponents-scientific-calculator-7337442/	Math Algebra How To Use Exponents On A Scientific Calculator By Chris Deziel Updated SARINYAPINNGAM/iStock/GettyImages Scientific calculators have more functionality that business calculators, and one thing they can do that is especially useful for scientists is to calculate exponents. On most calculators, you access this function by typing the base, the exponent key and finally the exponent. Although this is the convention, it's always good to do a test, because some calculators may require you to enter the numbers in reverse order. Scientific Vs. Business Calculators Scientific Vs. Business Calculators Scientific calculators are easy to distinguish from business calculators because of their many extra function keys. If you aren't sure if you have a scientific calculator, try this calculation: Enter (3+25 =) in that order. A scientific calculator will automatically do the multiplication first and give 13 as the answer. A business calculator will do the operations in the order you enter them and give 25. Here are just a few of the functions on a scientific calculator that you won't find on a business calculator: Negation: This key, denoted by NEG or (-) turns a positive number into a negative one. It is different from the subtraction key. Square Root: Denoted by the square root sign, it automatically displays the square root of the number you enter. Natural Logarithm: Denoted by LN, this key displays loge of the number you enter. Angle Functions: Scientific calculators have six keys the display the sine, cosine, tangent and the inverse of each for the number you enter. In addition to these keys, scientific calculators usually have two keys for exponential functions: Exponent: The key denoted by ^ or by capital E raises ay number to any exponent. Natural Exponent: The key, denoted by ex, raises e to the power you enter. Using the Exponent Key Using the Exponent Key Suppose you want the value yx. On most calculators, you enter the base, press the exponent key and enter the exponent. Here's an example: Enter 10, press the exponent key, then press 5 and enter. (10^5=) The calculator should display the number 100,000, because that's equal to 105. Before you start making a list of calculations, however, you should do a simple test to make sure your calculator isn't one of those that requires you to input the exponent first. Enter the number 2, press the exponent key, then enter 3. The display should read 8. If it reads 9, that's because the calculator interpreted the input as 32 instead of 23. That means you need to enter the exponent before the base. Some calculators have a key marked yx. This is the same as the ^ key. To find 105, enter 10, then the yx key, then 5 and hit the Enter or = key. Reading Exponents Reading Exponents Some numbers, such as 265 billion, have too many digits to display on a calculator. When this happens, the calculator displays the number in scientific notation, using the letter E to denote 10 to the power of whatever number comes after it. For example, 265 billion appears on a scientific calculator as 2.65 E 11. You can add, subtract, multiply and divide large numbers just as you would small ones, and the results will continue to appear in scientific notation a long as they continue to have too many digits to display. Examples: 2.65 E 8 + 5.78 E 7 = 3.23 E 8. 2.65 E 8 / 5.78 E 7 = 4.58 References University of North Carolina: Using Your Calculator zThe Math Forum: Exponents on a Scientific Calculator Cite This Article MLA Deziel, Chris. "How To Use Exponents On A Scientific Calculator" sciencing.com, 13 March 2018. APA Deziel, Chris. (2018, March 13). How To Use Exponents On A Scientific Calculator. sciencing.com. Retrieved from Chicago Deziel, Chris. How To Use Exponents On A Scientific Calculator last modified August 30, 2022.
2821	https://pmc.ncbi.nlm.nih.gov/articles/PMC3093013/	How to Interpret Gastric Emptying Scintigraphy - PMC Skip to main content An official website of the United States government Here's how you know Here's how you know Official websites use .gov A .gov website belongs to an official government organization in the United States. Secure .gov websites use HTTPS A lock ( ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites. Search Log in Dashboard Publications Account settings Log out Search… Search NCBI Primary site navigation Search Logged in as: Dashboard Publications Account settings Log in Search PMC Full-Text Archive Search in PMC Journal List User Guide View on publisher site Download PDF Add to Collections Cite Permalink PERMALINK Copy As a library, NLM provides access to scientific literature. Inclusion in an NLM database does not imply endorsement of, or agreement with, the contents by NLM or the National Institutes of Health. Learn more: PMC Disclaimer \| PMC Copyright Notice J Neurogastroenterol Motil . 2011 Apr 27;17(2):189–191. doi: 10.5056/jnm.2011.17.2.189 Search in PMC Search in PubMed View in NLM Catalog Add to search How to Interpret Gastric Emptying Scintigraphy Ju Won Seok Ju Won Seok 1 Department of Nuclear Medicine, College of Medicine, Chung-Ang University, Seoul, Korea. Find articles by Ju Won Seok 1,✉ Author information Article notes Copyright and License information 1 Department of Nuclear Medicine, College of Medicine, Chung-Ang University, Seoul, Korea. ✉ Correspondence: Ju Won Seok, MD. Department of Nuclear Medicine, College of Medicine, Chung-Ang University, 224-1 Heukseok-dong, Dongjak-Gu, Seoul 156-755, Korea. Tel: +82-2-6299-2896, Fax: +82-2-6299-2899, ethmoid@hanmail.net ✉ Corresponding author. Received 2011 Feb 16; Revised 2011 Feb 21; Accepted 2011 Feb 23; Issue date 2011 Apr. © 2011 The Korean Society of Neurogastroenterology and Motility This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License ( which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited. PMC Copyright notice PMCID: PMC3093013 PMID: 21602998 Abstract Gastric emptying scintigraphy has long been the standard method for measuring gastric motility. Various methodologies for this study have been used including meal composition, patient positioning, instrumentation, frequency of data acquisition, study length and quantitative method. For accurate quantification, it is not important which method is in use, except that the methodology should always be the same and normal values need to be derived and validated for that methodology. Keywords: Gastric emptying, Methods, Stomach Introduction Scintigraphic analysis of gastric motility is noninvasive, reproducible, simple to perform, accurate and quantitative. Ever since the introduction of more than 30 years ago, gastric emptying scintigraphy has been significantly refined and optimized over the years. Radionuclide gastric emptying studies are now well established as the standard method to evaluate gastric emptying. Standard Gastric Emptying Procedure Various methodologies have been used.1,2 Meal composition, patient positioning, instrumentation, frequency of data acquisition, study length and quantitative methods all vary between institutions. In general, the patient is asked to fast overnight or for at least 4 hours prior to the study. Diabetic patients should be studied in the morning, 20-30 minutes after their normal insulin dosage. A wide variety of drugs can affect gastric emptying and therefore most medication should be discontinued prior to a scintigraphic evaluation. Smoking delays gastric emptying and should be avoided. To accurately quantify solid emptying, the radiotracer must be tightly bound to the solid meal. There is currently no consensus on the optimal test meal to study gastric emptying. Generally, radiolabel egg albumen with Tc-99m sulfur colloid is used in most nuclear medicine laboratories. The eggs are mixed with 37 MBq Tc-99m sulfur colloid by frying or scrambling and the prepared egg product is then administered orally, often with toasted bread as an egg sandwich. Labeling efficiency is approximately 85%. Measuring simultaneous solid and liquid emptying is feasible using a dual isotope study by radiolabeling a liquid phase with In-111 DTPA (Indium-111 diethylenetriamine pentaacetic acid). A clear liquid study may be useful in a patient intolerant of solids but able to retain liquids.3,4 To evaluate liquid emptying, water is labeled with 3.7 MBq In-111 DTPA. The test meal should be ingested within 5-10 minutes, after which imaging commences. Data Acquisition and Analysis Immediately after the completion of the meal, the patient is positioned in front of the camera. The study can be acquired on standing, sitting and supine. For accurate quantification, it is not important which method is used, except that the methodology should always be the same and normal values need to be derived and validated for that methodology. The stomach lies obliquely within the abdomen and as food moves from the relatively posterior fundus to the relatively anterior antrum, there is an apparent increase in anterior counts due to lesser depth. For accurate quantitation, images in both anterior and posterior projections are obtained with averaging while attenuation correction should be performed routinely. The standard method for attenuation correction is the geometric mean method (square root of the product of anterior and posterior views). Other work suggests that the use of a single left anterior oblique projection also minimizes geometric effects.5 In that projection, the stomach contents move roughly parallel to the head of gamma camera, thus minimizing the effect of attenuation. The left anterior oblique method requires no mathematical correction of attenuation. Data acquisition is performed for 60-120 minutes either as continuous or intermittent imaging (Fig. 1). In cases where there is prolonged retention of material within the stomach, even more delayed images up to 3 hours may be required. Measurement of the half emptying time, or time required by the stomach to empty 50% of the ingested meal, is the simplest way to assess gastric transit. It is routinely and commonly used for clinical evaluation. Figure 1. Open in a new tab Data acquisition was performed for 120 minutes with intermittent imaging. The region of interest (ROI) is drawn around the activity in the entire stomach in anterior and posterior views (or the left anterior oblique view). The ROI should include any visualized activity in the fundic and antral regions of the stomach, with care to adjust the ROI to avoid activity from adjacent small bowel as in right large figure. Other protocols call for less frequent image acquisition at every 20-30 minutes for 2-3 hours (Fig. 2). In the some study using this acquisition protocol, the normal values were 10%, 65% and 90%, at 1, 2 and 4 hours, respectively.6 Some data suggest that the longer imaging period of 4 hours provides a higher sensitivity for detection of abnormal emptying, compared to 2 hours.7 Figure 2. Open in a new tab Image acquisition was performed at every 30 minutes for 2 hours. Remnant activity at 1 and 2 hours were 88.2% and 67.3%, respectively. The half emptying time (T 1/2) was 163.3 minutes. Gastric emptying time was markedly delayed compared to normal value. Conclusion Gastric emptying scintigraphy is the only satisfactory method of quantitatively measuring the rate of gastric emptying. But this method has not been standardized unfortunately and many variations exist. It is very important to establish normal ranges for the technique employed in each individual laboratory, as the results are highly dependent on the acquisition of parameters and the test meal used. Footnotes Financial support: None. Conflicts of interest: None. References 1.Donohoe KJ, Maurer AH, Ziessman HA, et al. Procedure guideline for adult solid-meal gastric-emptying study 3.0. J Nucl Med Technol. 2009;37:196–200. doi: 10.2967/jnmt.109.067843. [DOI] [PubMed] [Google Scholar] 2.Maurer A. Consensus report on gastric emptying: what's needed to prevent tarnishing a gold standard? J Nucl Med. 2008;49:339. doi: 10.2967/jnumed.106.037507. [DOI] [PubMed] [Google Scholar] 3.Lin E, Connolly LP, Drubach L, et al. Effect of early emptying on quantitation and interpretation of liquid gastric emptying studies of infants and young children. J Nucl Med. 2000;41:596–599. [PubMed] [Google Scholar] 4.Ziessman HA, Chander A, Clarke JO, Ramos A, Wahl RL. The added diagnostic value of liquid gastric emptying compared with solid emptying alone. J Nucl Med. 2009;50:726–731. doi: 10.2967/jnumed.108.059790. [DOI] [PubMed] [Google Scholar] 5.Ford PV, Kennedy RL, Vogel JM. Comparison of left anterior oblique, anterior and geometric mean methods for determining gastric emptying times. J Nucl Med. 1992;33:127–130. [PubMed] [Google Scholar] 6.Tougas G, Eaker EY, Abell TL, et al. Assessment of gastric emptying using a low fat meal: establishment of international control values. Am J Gastroenterol. 2000;95:1456–1462. doi: 10.1111/j.1572-0241.2000.02076.x. [DOI] [PubMed] [Google Scholar] 7.Ziessman HA, Bonta DV, Goetze S, Ravich WJ. Experience with a simplified standardized 4-hour gastric-emptying protocol. J Nucl Med. 2007;48:568–572. doi: 10.2967/jnumed.106.036616. [DOI] [PubMed] [Google Scholar] Articles from Journal of Neurogastroenterology and Motility are provided here courtesy of The Korean Society of Neurogastroenterology and Motility ACTIONS View on publisher site PDF (604.6 KB) Cite Collections Permalink PERMALINK Copy RESOURCES Similar articles Cited by other articles Links to NCBI Databases On this page Abstract Introduction Standard Gastric Emptying Procedure Data Acquisition and Analysis Conclusion Footnotes References Cite Copy Download .nbib.nbib Format: Add to Collections Create a new collection Add to an existing collection Name your collection Choose a collection Unable to load your collection due to an error Please try again Add Cancel Follow NCBI NCBI on X (formerly known as Twitter)NCBI on FacebookNCBI on LinkedInNCBI on GitHubNCBI RSS feed Connect with NLM NLM on X (formerly known as Twitter)NLM on FacebookNLM on YouTube National Library of Medicine 8600 Rockville Pike Bethesda, MD 20894 Web Policies FOIA HHS Vulnerability Disclosure Help Accessibility Careers NLM NIH HHS USA.gov Back to Top
2822	https://calculat.io/en/number/bin-to-dec/101101	Send You can also email us on infocalculat.io 101101 binary to decimal Binary Number to convert Convert What is binary 101101 in decimal? Binary number 101101 to Decimal Conversion Explanation Binary to Decimal Conversion Formula: (Decimal Number)10 = (d0 × 20) + (d1 × 21) + (d2 × 22) + ... + (dn−1 × 2n-1) According to Binary to Decimal Conversion Formula if you want to convert Binary 101101 to its Decimal form you have to multiply each digit of the binary number by the corresponding power of two which depends on the digit position in the number. There are 6 digits in 101101 so there are 6 positions. So you need to write down the powers of two from right to left according to its position starting from index 0 and ending with 5 and multiply it by the corresponding binary digit. \| \| \| \| \| \| \| \| --- --- --- \| Digit \| 1 \| 0 \| 1 \| 1 \| 0 \| 1 \| \| Pow of 2 \| 25 \| 24 \| 23 \| 22 \| 21 \| 20 \| Binary 101101 to Decimal Calculation Steps: (1 × 25) + (0 × 24) + (1 × 23) + (1 × 22) + (0 × 21) + (1 × 20) = 32 + 0 + 8 + 4 + 0 + 1 = 45 (101101)2 = (45)10 Related Calculations See Also Bin to Dec Conversion Table \| Binary Number \| Number \| --- \| \| 11110 \| 30 \| \| 11111 \| 31 \| \| 100000 \| 32 \| \| 100001 \| 33 \| \| 100010 \| 34 \| \| 100011 \| 35 \| \| 100100 \| 36 \| \| 100101 \| 37 \| \| 100110 \| 38 \| \| 100111 \| 39 \| \| 101000 \| 40 \| \| 101001 \| 41 \| \| 101010 \| 42 \| \| 101011 \| 43 \| \| 101100 \| 44 \| \| 101101 \| 45 \| \| 101110 \| 46 \| \| 101111 \| 47 \| \| 110000 \| 48 \| \| 110001 \| 49 \| \| 110010 \| 50 \| \| 110011 \| 51 \| \| 110100 \| 52 \| \| 110101 \| 53 \| \| 110110 \| 54 \| \| 110111 \| 55 \| \| 111000 \| 56 \| \| 111001 \| 57 \| \| 111010 \| 58 \| \| 111011 \| 59 \| About "Binary Number Converter" Calculator "Binary Number Converter" Calculator Binary Number to convert Convert FAQ What is binary 101101 in decimal? See Also
2823	https://www.quora.com/What-are-some-creative-ways-of-proving-Fermat-s-Little-Theorem	Something went wrong. Wait a moment and try again. What are some creative ways of proving Fermat’s Little Theorem? Phil Scovis I play guitar. And sing in the car. · Upvoted by Dan Grubb , Ph. D. Mathematics, Kansas State University (1986) and Daniel Roebuck , Integrated Masters Mathematics, University of St Andrews (2024) · Author has 6.8K answers and 12.8M answer views · Updated 6y A clever proof that I once saw, asks us to imagine making beaded bracelets. How many possible bracelets can be made with [math]p[/math] beads in [math]a[/math] colors? Obviously, [math]a^p[/math] different bracelets can be made, if we allow cyclic permutations to be considered distinct. But many of these bracelets are the same, if we don’t consider cyclic permutations to be different. (For example, ROYGB is the same as OYGBR). Let’s collect equivalent bracelets into piles, so they can be counted properly. It might seem that each pile now contains [math]p[/math] bracelets — one for each cyclical permutation. But this is not necessarily true. It’s c A clever proof that I once saw, asks us to imagine making beaded bracelets. How many possible bracelets can be made with [math]p[/math] beads in [math]a[/math] colors? Obviously, [math]a^p[/math] different bracelets can be made, if we allow cyclic permutations to be considered distinct. But many of these bracelets are the same, if we don’t consider cyclic permutations to be different. (For example, ROYGB is the same as OYGBR). Let’s collect equivalent bracelets into piles, so they can be counted properly. It might seem that each pile now contains [math]p[/math] bracelets — one for each cyclical permutation. But this is not necessarily true. It’s certainly not true for bracelets of one color (which are identical to all their cyclical permutations, and so are in a pile by themselves). Note that there are [math]a[/math] of these. It’s also not true for bracelets like RYGRYG, which only has three distinct equivalents, not six. But if [math]p[/math] is prime, then we can’t have any smaller equivalent segments, and the number of bracelets in the pile will be [math]p[/math]. So, assuming [math]p[/math] is prime, and excluding the single-color bracelets, the rest of the bracelets can be grouped into piles of size [math]p[/math]. That is, if [math]p[/math] is prime, [math]a^p-a[/math] is divisible by [math]p[/math]. The theorem follows easily from there. Mohammad Afzaal Butt B.Sc in Mathematics & Physics, Islamia College Gujranwala (Graduated 1977) · Upvoted by Aditya Garg , M.Sc. Mathematics, Indian Institute of Technology, Delhi (2013) · Author has 24.6K answers and 22.7M answer views · 6y [math]\mathbf{\text{Fermat’s Little Theorem}}[/math] [math]\text{It states that if p is a prime number then for any integer a relatively prime}[/math] [math]\text{to p, the number}\,\,a^p - a\,\,\text{is divisible by p. The statement can }[/math] [math]\text{be expressed using modular arithmetic notation as}[/math] [math]a^p\equiv a\pmod{p} \quad \text{or}\quad a^{p-1}\equiv 1\pmod{p}[/math] [math]\text{The theorem is named after Pierre de Fermat, who stated it in 1640. It}[/math] [math]\text{is called ‘little theorem’ to distinguish it from Fermat’s Last Theorem}[/math] [math]\mathbf{\text{Proof by using Modular Arithmetic}}[/math] [math]\text{Let a is any positive integer relatively prime to a prime p.}[/math] [math]\tex[/math] [math]\mathbf{\text{Fermat’s Little Theorem}}[/math] [math]\text{It states that if p is a prime number then for any integer a relatively prime}[/math] [math]\text{to p, the number}\,\,a^p - a\,\,\text{is divisible by p. The statement can }[/math] [math]\text{be expressed using modular arithmetic notation as}[/math] [math]a^p\equiv a\pmod{p} \quad \text{or}\quad a^{p-1}\equiv 1\pmod{p}[/math] [math]\text{The theorem is named after Pierre de Fermat, who stated it in 1640. It}[/math] [math]\text{is called ‘little theorem’ to distinguish it from Fermat’s Last Theorem}[/math] [math]\mathbf{\text{Proof by using Modular Arithmetic}}[/math] [math]\text{Let a is any positive integer relatively prime to a prime p.}[/math] [math]\text{Consider the sequence of numbers}[/math] [math]a (1), a (2), a (3) ,\cdots, a (p-1)[/math] [math]\text{If we divide each number by p. Every number will leave a unique remainder}[/math] [math]\text{that will belong to the set S} = {1, 2, 3, \cdots, p-1}[/math] [math]\text{by multiplying all these numbers and re-arrangement we get}[/math] [math]a^{p-1} (1\times 2\times 3\times\cdots\times (p-1)) \equiv ( (1\times 2\times 3\times\cdots\times (p-1)\pmod{p}[/math] [math]\implies a^{p-1}\equiv 1\pmod{p}\quad [1, 2, 3, \cdots, p-1\,\, \text{are relatively prime to p}][/math] [math]\implies a^p\equiv a\pmod{p}[/math] [math]\mathbf{\text{Euler's Proof by Mathematical Induction}}[/math] [math]a^P\equiv a\pmod{p}[/math] [math]\text{The result is true for a = 0}\quad \because\quad 0\equiv 0\pmod{p}[/math] [math]\text{The result is also true for a = 1}\quad \because\quad 1\equiv 1\pmod{p}[/math] [math]\text{Let the result be true for a = k. That is}[/math] [math]k^p\equiv k\pmod{p}[/math] [math]\text{Now we need to prove that the result is also true for a = k + 1. That is}[/math] math^p\equiv (k + 1)\pmod{p}[/math] [math]\text{by binomial theorem}[/math] math^n = \displaystyle\sum_{m = 0}^{n}\displaystyle\binom{n}{m}x^{n - m}a^m[/math] [math]\text{Replacing x by k, a by 1 and n by p}[/math] math^p = \displaystyle\sum_{m = 0}^{p}\displaystyle\binom{p}{m}k^{p - m}[/math] [math]\displaystyle\binom{p}{m} = \dfrac{p!}{m! (p - m)!}\text{is an integer and divisible by p for all }\,\,0\lt m\lt p[/math] [math]\displaystyle\binom{p}{m} = \dfrac{p!}{m! (p - m)!} = 1 \quad \text{for m = 0 and m = p}[/math] [math]\therefore (k + 1)^p\equiv (k^p + 1)\pmod{p}[/math] [math]\implies (k + 1)^p\equiv (k^p - k + k + 1)\pmod{p}[/math] [math]\implies (k + 1)^p\equiv( k + 1)\pmod{p}\quad \because\,\,\text{by our assumption }\,\,p \,\|\,k^p - k[/math] [math]\therefore\,\,\text{The result is also true for n = k + 1. Hence by the principle}[/math] [math]\text{of mathematical induction the result is true for all n}\,\,\in\Z^{+}[/math] [math]\text{For a good understanding of the theorem please follow the link.}[/math] Fermat’s Little Theorem Assistant Bot · 10mo Fermat’s Little Theorem states that if [math]p[/math] is a prime number and [math]a[/math] is an integer not divisible by [math]p[/math], then: [math]a^{p-1} \equiv 1 \mod p[/math] Here are some creative proofs and approaches to understanding this theorem: Counting with Groups: Group Theory Approach: Consider the multiplicative group of integers modulo [math]p[/math], denoted math^[/math]. This group has [math]p-1[/math] elements (the integers from [math]1[/math] to [math]p-1[/math]). Since [math]a[/math] is not divisible by [math]p[/math], we can consider the subgroup generated by [math]a[/math]. The order of this subgroup divides [math]p-1[/math], leading to the conclusion that [math]a^{p-1} \equiv 1 \mod p[/math]. Induction: Mathematical Indu Fermat’s Little Theorem states that if [math]p[/math] is a prime number and [math]a[/math] is an integer not divisible by [math]p[/math], then: [math]a^{p-1} \equiv 1 \mod p[/math] Here are some creative proofs and approaches to understanding this theorem: Counting with Groups: Group Theory Approach: Consider the multiplicative group of integers modulo [math]p[/math], denoted math^[/math]. This group has [math]p-1[/math] elements (the integers from [math]1[/math] to [math]p-1[/math]). Since [math]a[/math] is not divisible by [math]p[/math], we can consider the subgroup generated by [math]a[/math]. The order of this subgroup divides [math]p-1[/math], leading to the conclusion that [math]a^{p-1} \equiv 1 \mod p[/math]. Induction: Mathematical Induction: We can prove the theorem using induction on [math]n[/math]. Base case [math]n = 1[/math] is trivial. For the induction step, assume it holds for [math]n[/math]. Then consider [math]n+1[/math], and use properties of binomial coefficients and modular arithmetic to show that [math]a^{p-1} \equiv 1 \mod p[/math]. Using Binomial Theorem: Binomial Expansion: Consider math^{p-1}[/math] and expand it using the binomial theorem. The terms involving [math]p[/math] will vanish modulo [math]p[/math]. This leads to the conclusion that [math]a^{p-1} \equiv 1 \mod p[/math]. Combinatorial Proof: Counting Residues: Consider the set of integers [math]{ a, 2a, 3a, \ldots, (p-1)a }[/math]. Each of these integers is distinct modulo [math]p[/math] (since [math]a[/math] is not divisible by [math]p[/math]). This set is a permutation of the integers [math]{ 1, 2, \ldots, p-1 }[/math]. The product of the integers in both sets can be shown to be congruent modulo [math]p[/math], leading to [math]a^{p-1} \equiv 1 \mod p[/math]. Using Lifting the Exponent Lemma: Lifting Exponents: This approach leverages properties of prime factorization and modular arithmetic. It shows that if [math]a^{p-1} \equiv 1 \mod p[/math] holds for some integers, it will hold for all integers not divisible by [math]p[/math]. Field Theory: Finite Fields: Consider the field [math]\mathbb{F}_p[/math]. The non-zero elements of [math]\mathbb{F}_p[/math] form a cyclic group under multiplication. If [math]g[/math] is a generator of this group, then every non-zero element can be expressed as [math]g^k[/math] for some [math]k[/math]. Therefore, [math]a^{p-1} = (g^k)^{p-1} = g^{k(p-1)} \equiv 1 \mod p[/math]. Using Euler's Theorem: Connection to Euler's Theorem: Since Fermat’s Little Theorem is a special case of Euler's theorem (which states that [math]a^{\phi(n)} \equiv 1 \mod n[/math] for [math]a[/math] coprime to [math]n[/math]), you can derive it from Euler’s theorem by considering [math]n = p[/math] (where [math]\phi(p) = p-1[/math]). These proofs showcase the richness of mathematical reasoning behind Fermat's Little Theorem and highlight connections to various fields such as group theory, combinatorics, and number theory. Each proof provides a different perspective on why the theorem holds true, offering deeper insights into its implications and applications. Related questions What are the real life applications of Fermat's little theorem? Are mathematicians who wrote highly complex proofs (say Fermat’s last theorem) able to re-do the whole proof in their mind? What is the easiest way to prove Fermat's Last Theorem for n = 3? What are some clever applications of Wilson's Theorem and Fermat's Little Theorem? What are the uses of Fermat's little theorem? Steve Morris interested in Number Theory · Author has 190 answers and 178.4K answer views · 5y Originally Answered: Number Theory: How do you prove Fermat's little theorem? · Fermat's little theorem states that “If a and p are two integers such that p is a prime then [math]a^p-a[/math] is divisible by [math]p[/math]. We will use a slightly different statement ie. If a and p are co-prime and p is a prime then, [math]a^{p-1}\equiv 1 (mod p)[/math] Now, Consider the following numbers, [math]a,2a,3a,...,(p-1)a[/math] When divided by [math]p[/math] They all leave the remainders, [math]1,2,3,...,(p-1)[/math] Not necessary in the same order. ie. What is important is that every multiple of [math]a[/math] leaves a unique and distinct remainder which is less than [math]p[/math]. Obviously they don't leave the remainder zero since p is a prime. Let us prove the facts given above, Suppose Fermat's little theorem states that “If a and p are two integers such that p is a prime then [math]a^p-a[/math] is divisible by [math]p[/math]. We will use a slightly different statement ie. If a and p are co-prime and p is a prime then, [math]a^{p-1}\equiv 1 (mod p)[/math] Now, Consider the following numbers, [math]a,2a,3a,...,(p-1)a[/math] When divided by [math]p[/math] They all leave the remainders, [math]1,2,3,...,(p-1)[/math] Not necessary in the same order. ie. What is important is that every multiple of [math]a[/math] leaves a unique and distinct remainder which is less than [math]p[/math]. Obviously they don't leave the remainder zero since p is a prime. Let us prove the facts given above, Suppose two distinct multiples of [math]a[/math] (say [math]x_1 [/math] & [math]x_2 [/math] ) gives same remainder r. Thus, [math]x_1a \equiv r (mod p)[/math] And, [math]x_2a \equiv r (mod p)[/math] Thus, [math]x_1a \equiv x_2a (mod p)[/math] Since, a is co-prime to p, we can ‘cancel' a from both sides. Thus, [math]x_1 \equiv x_2 (mod p)[/math] Since the coefficient of multiples we considered are less than p and positive, we deduce, [math]x_1=x_2[/math] But we had considered two distinct multiples. Thus their coefficients must be different. Thus we have a contradiction. Therefore we conclude that no two distinct multiples we considered above will give same remainder when divided by p. Thus, we have a general modular equation. [math]na \equiv r (mod p)[/math] Where [math]0<n<p [/math] and [math]0<r<p[/math] We can multiply all such congruences to get, math!.a^{p-1} \equiv (p-1)! (mod p)[/math] Since math! [/math] is co-prime to [math]p[/math] thus we can cancel out math![/math] from both sides. Therefore we get, [math]a^{p-1} \equiv 1 (mod p)[/math] We can easily prove the original statement by multiplying [math]a[/math] on both sides. If a and p are not co-prime there is no need to prove the theorem because it obviously holds true. Check it yourself Damien Jiang Math major at MIT. · Upvoted by Igor Markov , MA in Mathematics, PhD in CS and Vladimir Novakovski , silver medals, IOI 2001 and IPhO 2001 · Updated 9y Originally Answered: Number Theory: How do you prove Fermat's little theorem? · There are a number of proofs at Proofs of Fermat's little theorem on Wikipedia. I'll restate some here, and give some analysis. Sridhar Ramesh's answer gives a "higher-level" (i.e. more general) view on proofs 2 and 4. For reference, Fermat's Little Theorem asserts that for integer [math]a[/math] and prime [math]p[/math], [math]a^p \equiv a \pmod{p}[/math]. Equivalently, for integer [math]a[/math] not divisible by [math]p[/math], [math]a^{p-1} \equiv 1 \pmod{p}[/math]. The set (actually ring) of residues modulo an integer [math]n[/math] is usually denoted by [math]\mathbb{Z}/n\mathbb{Z}[/math]. We're concerned about the multiplicative structure here, and we know [math]n[/math] is prime, so we'll abbreviate t There are a number of proofs at Proofs of Fermat's little theorem on Wikipedia. I'll restate some here, and give some analysis. Sridhar Ramesh's answer gives a "higher-level" (i.e. more general) view on proofs 2 and 4. For reference, Fermat's Little Theorem asserts that for integer [math]a[/math] and prime [math]p[/math], [math]a^p \equiv a \pmod{p}[/math]. Equivalently, for integer [math]a[/math] not divisible by [math]p[/math], [math]a^{p-1} \equiv 1 \pmod{p}[/math]. The set (actually ring) of residues modulo an integer [math]n[/math] is usually denoted by [math]\mathbb{Z}/n\mathbb{Z}[/math]. We're concerned about the multiplicative structure here, and we know [math]n[/math] is prime, so we'll abbreviate the group of (non-zero) residues mod a prime [math]p[/math] under multiplication as [math]\mathbb{Z}_p^{\times}[/math]. (If you don't know what a group is, pretend it's a set where you can multiply and divide. Recall that non-zero residues mod [math]p[/math] have inverses, so "division" is a legal operation mod [math]p[/math].) Proof 1. We use the fact that multiplication by [math]a[/math] modulo [math]p[/math] permutes the elements of [math]\mathbb{Z}_p^{\times}[/math]. That is, we claim that the set [math]S = {1, 2, 3, \ldots, p-1}[/math] is equivalent to the set [math]T = {a, 2a, 3a, \ldots, (p-1)a}[/math] when viewed modulo [math]p[/math]. (So each element of [math]S[/math] is congruent to exactly one element of [math]T[/math] mod [math]p[/math].) Why is this? First, we can't have two distinct elements [math]ma, na[/math] of [math]T[/math] that are congruent [math]\pmod{p}[/math], since you could then "divide" by [math]a[/math] (really, multiply by [math]a^{-1}[/math]) and get [math]m \equiv n \pmod{p}[/math]. Furthermore, none of the elements of [math]T[/math] are 0 [math]\pmod{p}[/math], so the [math]p-1[/math] distinct elements of [math]T[/math] must be exactly the [math]p-1[/math] non-zero residues modulo [math]p[/math]! But now, the product of the elements of [math]S[/math] is congruent to the product of the elements of [math]T[/math] mod [math]p[/math], so [math]1\cdot2\cdot\ldots\cdot(p-1)[/math] [math]\equiv a \cdot 2a \cdot \ldots \cdot (p-1)a \pmod{p}[/math], and multiplying by [math] ( (p-1)! )^{-1} [/math] (which happens to be -1 by Wilson's Theorem!) on both sides gives [math] 1 \equiv a^{p-1} \pmod{p}, [/math] as desired. This approach seems somewhat roundabout (it seems more like the byproduct of playing around with the first fact I noted--that multiplication by [math]a[/math] permutes [math]\mathbb{Z}_p^{\times}[/math]). But it's simple and good for beginning number theory students--it doesn't use anything other than unique inverses modulo [math]p[/math]. Proof 2. This is a "counting proof," and certainly very cute. The goal here is to show that [math] p \mid a^p-a [/math]. That is, we want to find a set of size [math]a^p-a[/math] which has [math]p[/math] symmetries. Say you have access to [math]a[/math] colors of beads, and you want to create a necklace (closed, so it forms a loop) of [math]p[/math] of these beads. How many such necklaces are there? If you fix a certain spot as the "top" of the necklace, there are [math]a[/math] possibilities for each of [math]p[/math] spots, for a total of [math]a^p[/math] necklaces. That's nice, but we're really only interested in counting distinct necklaces; if you can rotate one necklace to get another, they're not any different. There are [math]a[/math] necklaces which only have one color of bead, and for the remaining [math]a^p - a[/math] of these necklaces, we claim that each one has [math]p[/math] distinct rotations. (You can check this yourself.) But there are an integer number of distinct necklaces, so [math]\frac{a^p-a}{p}[/math] has to be an integer, and we're done. This looks like a really magical proof, but there's something really deep going on here. As Sridhar Ramesh's answer mentioned, this proof is (not-so?) secretly about group actions and their orbits. We exploit the fact that you can rotate (the "action") each of the non-monochromatic necklaces [math]p[/math] times, and equivalently, only one of these rotations preserves the orientation of the original necklace. In particular, the number of distinct necklaces you get by rotating (the size of the action's "orbit") and the number of ways you can fix the original necklace (the "stabilizer") multiply to the total number of rotations one can perform (the order of the group.) You can check this for monochromatic necklaces too. This fact gives rise to powerful ways of counting things with symmetries, such as Burnside's lemma and Pólya enumeration theorem. Proof 3. This proof is by induction. Unfortunately, it reveals little of the intuition behind Fermat's Little Theorem, but it's easy to remember. As in the necklace proof, we'll show that [math] p \mid a^p-a [/math]. We'll repeatedly use the fact that [math]p \mid \binom{p}{a}[/math] for [math] 1 \le a \le p-1. [/math] You can show this by noticing that in [math] \binom{p}{a} = \frac{p!}{a!(p-a)!} [/math], the numerator is divisible by [math]p[/math], but the denominator isn't. The base case is the assertion that [math]p \mid 0^p - 0 = 0,[/math] which is obvious. Now, say [math]p \mid k^p-k.[/math] Then by the binomial theorem and the above fact, [math] (k+1)^p - (k+1) = [/math] [math]k^p + \binom{p}{1}k^{p-1} + \ldots + \binom{p}{p-1}k + 1 - (k+1) [/math] [math] \equiv k^p-k \equiv 0 \pmod{p}, [/math] completing the induction. The induction is actually unnecessary if you use the Multinomial theorem (which is the way I usually remember it; in fact, this proof is the easiest one for me to remember, since I'm more algebraically inclined.) On the other hand, I don't feel like I learn anything from this proof, other than the result; I'd love it if someone else could point out some sort of takeaway from here :) It is worth noting, though, that identities like math^p \equiv a^p + b^p \pmod{p}[/math] are examples of the Frobenius map, which has some pretty deep applications. Proof 4. We basically use Lagrange's theorem; I'll only give a short summary because this is more technical. A non-zero residue [math]a[/math] generates the subgroup [math]{a, a^2, \ldots }[/math] of [math]\mathbb{Z}_p^{\times}[/math]. This subgroup is finite because [math]\mathbb{Z}_p^{\times}[/math] is finite, and its order is the order of [math]a[/math] (the smallest integer [math]k[/math] such that [math]a^k \equiv 1 \pmod{p}[/math]. The order of the subgroup divides the order of the whole group, so the order of [math]a[/math] divides [math]p-1[/math]; thus [math]a^{p-1} \equiv (a^k)^{(p-1)/k} \equiv 1 \pmod{p}[/math] as desired. This proof can be restated in terms of cosets of the subgroup [math]{a, a^2, \ldots }[/math], and you can even get rid of any notion of "group." On the other hand, this proof seems to be most useful as an example of Lagrange's theorem in action. Sponsored by Sheryl Duncan Closing Sale - Up to 75% Off! I am closing my boutique shop — this is my last sale. Thank you for being part of the journey. Jiten Ahuja I love math, art and psychology · Author has 99 answers and 283.6K answer views · 10y Originally Answered: Number Theory: How do you prove Fermat's little theorem? · The proof that I am about to give has already been stated by Damien Jiang(proof 3). I will be trying to make it more simplistic. The multinomial theorem is as follows: (a1 + a2 + a3 + .... + an )^p = a1^p + a2^p + .... + an^p + kp ; where p and k are integers, Now substitute a1=a2=a3=...=an=1, you get (1 + 1 + 1 +... n times)^p = 1^p + 1^p + .... n times + kp; Hence, n^p = n + kp; Therefore, n^p - n = kp; Hence, n(n^(p-1) - 1) = kp; Therefore, p divides n(n^(p-1) - 1) as k is an integer. If n is not divisible by a prime p, then p divides (n^(p-1) - 1) which is Fermat's little theorem. Related questions What are some creative solutions for handling cooking when your partner dislikes your food? What are the most surprising uses of Fermat's Last Theorem? What are some creative ways to handle in-game chaos when one player's actions lead to a full-blown disaster for the party? What books do I need to read and what branches of mathematics do I need to learn to understand the proof for Fermat's last theorem? Are there any counterintuitive implications from Fermat's little theorem that beginners should be aware of? Peter Aouad Director at Cybernet Associates Inc. (2006–present) · Author has 1.8K answers and 1.6M answer views · 6y I suggest you watch this simple video on you Tube: Select: Fermat's Little Theorem Example. But before you start, you need to be familiar with the concept of : Dividing 2 numbers m/n = Q + remainder not divisible by n; => m / n = Q + r/n; Q is the quotient , r the remainder;=>m / n = r (modulo n)=r((mod n); Example: 22 / 7 = 3 + 1/7; now add 3 + 1/7 = 21/7 + 1/7 = 22/7; the math is correct; Using the modulo terminology, we say: 22 = 1 (modulo 7) = 1 (mod7); note also: 43 = 1 (mod 7); Q does not matter. 44 = 222=2 1 (mod7); =2(mod7); A whole new math, ver I suggest you watch this simple video on you Tube: Select: Fermat's Little Theorem Example. But before you start, you need to be familiar with the concept of : Dividing 2 numbers m/n = Q + remainder not divisible by n; => m / n = Q + r/n; Q is the quotient , r the remainder;=>m / n = r (modulo n)=r((mod n); Example: 22 / 7 = 3 + 1/7; now add 3 + 1/7 = 21/7 + 1/7 = 22/7; the math is correct; Using the modulo terminology, we say: 22 = 1 (modulo 7) = 1 (mod7); note also: 43 = 1 (mod 7); Q does not matter. 44 = 222=2 1 (mod7); =2(mod7); A whole new math, very useful opens up. Now go watch the YouTube link for bigger numbers beyond what a calculator can do. Promoted by The Penny Hoarder Lisa Dawson Finance Writer at The Penny Hoarder · Updated Jun 26 What's some brutally honest advice that everyone should know? Here’s the thing: I wish I had known these money secrets sooner. They’ve helped so many people save hundreds, secure their family’s future, and grow their bank accounts—myself included. And honestly? Putting them to use was way easier than I expected. 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Right now, a direct deposit of at least $1K not only sets you up for higher returns but also brings you closer to earning up to a $300 welcome bonus (terms apply). You can easily deposit checks via your phone’s camera, transfer funds, and get customer service via chat or phone call. There are no account fees, no monthly fees and no overdraft fees. And your money is FDIC insured (up to $3M of additional FDIC insurance through the SoFi Insured Deposit Program). It’s quick and easy toopen an account with SoFi Checking and Savings (member FDIC) and watch your money grow faster than ever. Read Disclaimer Luboš Motl MSc in Mathematics and Physics & Theoretical Physics, Charles University in Prague (Graduated 1997) · Author has 7.3K answers and 17M answer views · 2y Originally Answered: How did Pierre de Fermat prove his "little" theorem? · Fermat published the statement as a conjecture (justified by “mathematical experiments” that worked) in 1640 and he had no proof. Euler published the first proof in 1736, almost a century later! There exist many proofs now. Alon Amit PhD in Mathematics; Mathcircler. · Upvoted by Tom Davis , PhD in mathematics from Stanford University. and Robert Martin , Ph.D. Mathematics, Columbia University (1976) · Author has 8.7K answers and 171.7M answer views · 6y Related Has Fermat’s Last Theorem been used to prove anything else yet? No, and it’s extremely unlikely that it ever will. In the tree of ideas of mathematics there is a trunk (“math”), big branches, small branches, twigs, tiny twigs, and leaves. Branches are lemmas and theorems which are useful for a variety of applications, and are repeatedly used to prove other results. Leaves are “end results” which may be important, or cool, or historically significant, or applicable outside of math, but are not themselves useful ingredients for proving other things. The Classification of Finite Simple Groups is a huge, massive branch. So are the Fundamental Theorem of Arithmeti No, and it’s extremely unlikely that it ever will. In the tree of ideas of mathematics there is a trunk (“math”), big branches, small branches, twigs, tiny twigs, and leaves. Branches are lemmas and theorems which are useful for a variety of applications, and are repeatedly used to prove other results. Leaves are “end results” which may be important, or cool, or historically significant, or applicable outside of math, but are not themselves useful ingredients for proving other things. The Classification of Finite Simple Groups is a huge, massive branch. So are the Fundamental Theorem of Arithmetic, the main theorems of Fourier analysis, Zorn’s Lemma, and many others. The modularity theorem , which was the main final ingredient in the proof of FLT, is a branch. The statement of FLT is a tiny, tiny leaf. It has great historical significance due to the way its legend motivated research in algebraic number theory, but never stood a chance of becoming a lemma. Footnotes Modularity theorem - Wikipedia Sponsored by Vrindavan Chandrodaya Mandir Aid In Building a Spiritual Icon dedicated to Lord Krishna. The world's tallest temple is shaping up in Vrindavan. Saral Vidyarthi Author has 124 answers and 62.9K answer views · 4y Originally Answered: How do you prove Fermat's little theorem? · There are several proofs, but this one is sort of one liner. The nonzero numbers (integers) modulo a prime number form a group under multiplication. As there are [math]p-1 [/math] elements in the group, the order of any element [math]a[/math] is [math]p-1, [/math] which immediately implies that [math]a^{p-1}\equiv1\pmod p\forall a\neq0. [/math] For zero, it is elementarily true. Tony Nixon Mavely Computer verification of Goldbach conjecture isn't enough · 7y Fermat’s Little Theorem states that for any prime [math]p[/math] and any number [math]a[/math], [math]a^p-a[/math] is divisible by [math]p[/math]. We shall try to solve it using modular arithmetic. If [math]p \mid a[/math] , then the result is trivial (Try it yourself). If [math]p \nmid a[/math], consider the set [math]S = {a,2a,3a, \ldots ,(p-1)a}[/math] None of the elements in [math]S[/math] is congruent to [math]0 \: mod p[/math], as math=1[/math] and [math]p \nmid i[/math] where [math]i={1,2, \ldots ,(p-1)}.[/math] We also claim that each element is distinct If [math]ma \equiv na \: mod p[/math] for some [math]m,n \in {1,2, \ldots (p-1)}[/math] [math]\implies (m-n)a \equiv 0 \: mod p[/math] [math]\implies p\|(m-n) \: (\because[/math][math] (p,a)=1[/math] and Euclid’s Lemma[math] )[/math] But [math]m,n \in {1,2, \ldots [/math] Fermat’s Little Theorem states that for any prime [math]p[/math] and any number [math]a[/math], [math]a^p-a[/math] is divisible by [math]p[/math]. We shall try to solve it using modular arithmetic. If [math]p \mid a[/math] , then the result is trivial (Try it yourself). If [math]p \nmid a[/math], consider the set [math]S = {a,2a,3a, \ldots ,(p-1)a}[/math] None of the elements in [math]S[/math] is congruent to [math]0 \: mod p[/math], as math=1[/math] and [math]p \nmid i[/math] where [math]i={1,2, \ldots ,(p-1)}.[/math] We also claim that each element is distinct If [math]ma \equiv na \: mod p[/math] for some [math]m,n \in {1,2, \ldots (p-1)}[/math] [math]\implies (m-n)a \equiv 0 \: mod p[/math] [math]\implies p\|(m-n) \: (\because[/math][math] (p,a)=1[/math] and Euclid’s Lemma[math] )[/math] But [math]m,n \in {1,2, \ldots (p-1)}[/math], [math]\therefore m=n[/math] Thus, when we reduce them under [math]modulo \: p[/math], we get [math]{1,2, \ldots ,p-1}[/math][math].[/math] Therefore multiplying all the elements in set [math]S[/math] gives us, [math]a2a3a \ldots (p-1)a \equiv 123 \ldots (p-1) \: mod p[/math] [math]\therefore (p-1)!a^{p-1} \equiv (p-1)! \: mod p[/math] Therefore [math]p \mid ([/math][math]p-1)!(a^{p-1}-1)[/math] By Euclid’s lemma ,as p is a prime, we get [math]p \mid a^{p-1}-1[/math] Which means [math]a^{p-1} \equiv 1 mod p[/math] Multiplying [math]a [/math] on both sides gives us Fermat’s little theorem Studied Advanced Mathematics at University of New South Wales · 8mo Originally Answered: Number Theory: How do you prove Fermat's little theorem? · Observe that Fp is the initial object of Field with Char F equal to p. Hence the only endomorphism of Fp is identity. Hence the Frobenius automorphism equal to identity, that is the Fermat’s little theorem. Sridhar Ramesh PhD in Logic (mathematics), University of California, Berkeley · Upvoted by Anurag Bishnoi , Ph.D. Mathematics, Ghent University (2016) and Alon Amit , Lover of math. Also, Ph.D. · Author has 954 answers and 6.6M answer views · 11y Originally Answered: Number Theory: How do you prove Fermat's little theorem? · See What is a layman's proof of Fermat's Little Theorem? (or, better yet, the Wikipedia article it links to: Proofs of Fermat's little theorem). There are essentially two proofs: one by considering the orbits of the action on [math]\mathbb{Z}_p^{\times}[/math] of the subgroup generated by any element A (this is the Lagrange's theorem approach); the other by considering the orbits of the action of [math]\mathbb{Z}_p^{+}[/math] on the space [math]A^p[/math] for any set [math]A[/math] (this is the "necklace-counting" approach). Every proof I've ever seen has been a variant of one of these two (I'd be very curious to know if somehow these two proofs See What is a layman's proof of Fermat's Little Theorem? (or, better yet, the Wikipedia article it links to: Proofs of Fermat's little theorem). There are essentially two proofs: one by considering the orbits of the action on [math]\mathbb{Z}_p^{\times}[/math] of the subgroup generated by any element A (this is the Lagrange's theorem approach); the other by considering the orbits of the action of [math]\mathbb{Z}_p^{+}[/math] on the space [math]A^p[/math] for any set [math]A[/math] (this is the "necklace-counting" approach). Every proof I've ever seen has been a variant of one of these two (I'd be very curious to know if somehow these two proofs could be seen as reframings of the same core idea as each other as well, but I have not been able to see this to be the case, despite the common idea of breaking a set into its orbits under a group action for counting purposes). For what it's worth, both of these approaches can in fact establish the Euler-Fermat theorem more generally. Related questions What are the real life applications of Fermat's little theorem? Are mathematicians who wrote highly complex proofs (say Fermat’s last theorem) able to re-do the whole proof in their mind? What is the easiest way to prove Fermat's Last Theorem for n = 3? What are some clever applications of Wilson's Theorem and Fermat's Little Theorem? What are the uses of Fermat's little theorem? What are some creative solutions for handling cooking when your partner dislikes your food? What are the most surprising uses of Fermat's Last Theorem? What are some creative ways to handle in-game chaos when one player's actions lead to a full-blown disaster for the party? What books do I need to read and what branches of mathematics do I need to learn to understand the proof for Fermat's last theorem? Are there any counterintuitive implications from Fermat's little theorem that beginners should be aware of? Which is the easiest way to explain Fermat's Little Theorem? What was Fermat's first theorem, and how many theorems Fermat's had introduced? Is there a proof for Fermat's Last Theorem that can be easily understood by those without formal training in mathematics? How long was Fermat's Last Theorem unsolved? How can engineering students with no background in art/design or creative thinking techniques apply creative problem solving principles to industrial/product design? About · Careers · Privacy · Terms · Contact · Languages · Your Ad Choices · Press · © Quora, Inc. 2025
2824	https://avishek.net/2021/10/25/functional-analysis-exercises-6-completion-metric-spaces.html	Total Internal Reflection : Musings on Machine Learning, Technology, and Art Functional Analysis Exercises 6 : Completion of Metric Spaces \| Total Internal Reflection Total Internal Reflection Technology and Art Enter your email to receive updates Home Machine Learning Mathematics Engineering References Art Archives About Me Resume Code Cobol REKT Tape/Z Plenoxels Transformer Basis-Processing Cataract COMRADE Duck-Angular Exo IRIS MuchHeap Snail-MapReduce Underline Lambda-Queuer jQuery-Jenkins Radiator Contact Github Twitter LinkedIn Site Feed Functional Analysis Exercises 6 : Completion of Metric Spaces Avishek Sen Gupta on 25 October 2021 This post lists solutions to the exercises in the Completion of Metric Spaces section 1.6 of Erwin Kreyszig’sIntroductory Functional Analysis with Applications. This is a work in progress, and proofs may be refined over time. 1.6.1 Show that if a subspace Y Y of a metric space consists of finitely manypoints, then Y Y is complete. Proof: By definition, a limit point L L of the set Y Y has at least one point x≠L x≠L within every neightbourhood ϵ ϵ. Since Y Y has a finite number of points, then it has no limit points, and thus (vacuously) contains all its limit points. Thus, Y Y is a complete metric subspace. ■◼ 1.6.2 What is the completion of (X,d)(X,d), where X X is the set of all rational numbers and d(x,y)=\|x−y\|d(x,y)=\|x−y\|? Proof: The completion of (X,d)(X,d), where X X is the set of all rational numbers and d(x,y)=\|x−y\|d(x,y)=\|x−y\|, is R R, since every real number is the limit of a sequence of rational numbers. ■◼ 1.6.3. What is the completion of a discrete metric space X X? (Cf. 1.1-8.). Proof: A discrete metric space X X has no limit points, since no point in it has at least one point in every neighbourhood ϵ ϵ. Thus, it vacuously contains all its limit points. Thus, the completion of the discrete metric space X X is itself. ■◼ 1.6.4 If X 1 X 1 and X 2 X 2 are isometric and X 1 X 1 is complete, show that X 2 X 2 is complete. Proof: Assume x,y∈X 1 x,y∈X 1, and let T:X 1→X 2 T:X 1→X 2. Since X 1 X 1 and X 2 X 2 are isometric, we have: d(x,y)=¯d(T x,T y)d(x,y)=d¯(T x,T y) We know that X 1 X 1 is complete: let us assume, for an arbitrary ϵ ϵ, a point x 1∈X 1 x 1∈X 1 lying in the ϵ ϵ-neighbourhood of a limit point x∈X 1 x∈X 1. Then, we have: d(x,x 1)=d(T x,T x 1)<ϵ d(x,x 1)=d(T x,T x 1)<ϵ Thus, for an arbitrary ϵ ϵ, there is a point T x∈X 2 T x∈X 2 which has a point T x 1 T x 1 in its ϵ ϵ-neighbourhood as well. Thus, T x T x is a limit point of X 2 X 2 as well. Since T x∈X 2 T x∈X 2 for all x∈X 1 x∈X 1, X 2 X 2 contains all its limit points as well. Thus, X 2 X 2 is complete. ■◼ 1.6.5 (Homeomorphism) A homeomorphism is a continuous bijective mapping T:X→Y T:X→Y whose inverse is continuous; the metric spaces X X and Y Y are then said to be homeomorphic. (a) Show that if X X and Y Y are isometric, they are homeomorphic. (b) Illustrate with an example that a complete and an incomplete metric space may be homeomorphic. Proof: Consider a Cauchy sequence (x n)(x n) in X X. Then, we have, ∀ϵ>0∀ϵ>0, ∃N∃N, such that d(x m,x n)<ϵ d(x m,x n)<ϵ for all m,n>N m,n>N. Let T:X→Y T:X→Y. Since X X is isometric to Y Y, we have: d(x m,x n)=d(T x m,T x n)<ϵ d(x m,x n)=d(T x m,T x n)<ϵ This implies that for every ϵ>0 ϵ>0, there exists a δ>0 δ>0, such that d(x m,x n)<δ⇒d(T x m,T x n)<ϵ d(x m,x n)<δ⇒d(T x m,T x n)<ϵ. In this case δ=ϵ δ=ϵ. Thus, T T is continuous at x n x n. The above argument can be used for T−1 T−1 to prove that it is also continuous. To prove injectivity, we note that x≠y⇒d(x,y)≠0⇒⇒d(T x,T y)≠0⇒T x≠T y x≠y⇒d(x,y)≠0⇒⇒d(T x,T y)≠0⇒T x≠T y. To prove surjectivity, we pick a point y∈Y y∈Y. Assume x 1∈X x 1∈X. Then, by isometry we must have: d(y,Tx_1)=d(x, x_1), where x∈X x∈X. Thus, there is a corresponding preimage for every y∈Y y∈Y. ■◼ Consider the f:(0,1)→R f:(0,1)→R defined as f(x)=x f(x)=x. Then f(x)f(x) and its inverse are continuous and bijective. (0,1)(0,1) is an incomplete metric space and R R is complete. 1.6.6 Show that C[0,1]C[0,1] and C[a,b]C[a,b] are isometric. Proof: We note that f(t)=t−a b−a,a≠b f(t)=t−a b−a,a≠b is a mapping f:[a,b]→[0,1]f:[a,b]→[0,1], and that f−1(t)=a+(b−a)t f−1(t)=a+(b−a)t is a mapping f−1:[0,1]→[a,b]f−1:[0,1]→[a,b]. We note that f f and f−1 f−1 are bijections. The distance metric in C C is defined as d(x,y)=sup\|x(t)−y(t)\|d(x,y)=sup\|x(t)−y(t)\|. Define a mapping T:C t∈0,1→C t∈a,bT:C t∈0,1→C t∈a,b Think of C(f(t)C(f(t) as the original function applied to [0,1][0,1] even though the input t∈[a,b]t∈[a,b]. Then, practically we have C t∈0,1=C t∈a,bC t∈0,1=C t∈a,b. Then: d(T x,T y)=sup[a,b]\|x(f(t))−y(f(t))\|=sup[0,1]\|x(t)−y(t)\|=d(x,y)d(T x,T y)=sup[a,b]\|x(f(t))−y(f(t))\|=sup[0,1]\|x(t)−y(t)\|=d(x,y) Thus, T T preserves distances. To prove injectivity, suppose T x=T y T x=T y, then we have: d(T x,T y)=sup[a,b]\|x(f(t))−y(f(t))\|=0⇒sup[0,1]\|x(t)−y(t)\|=d(x,y)=0⇒x=y d(T x,T y)=sup[a,b]\|x(f(t))−y(f(t))\|=0⇒sup[0,1]\|x(t)−y(t)\|=d(x,y)=0⇒x=y For surjectivity, we note that for an arbitrary function y(f(t))∈C[a,b]y(f(t))∈C[a,b], we always have x(t)∈C[0,1]x(t)∈C[0,1], since T−1 x=x(f−1(f(t)))=x(t)T−1 x=x(f−1(f(t)))=x(t). ■◼ 1.6.7 If (X,d)(X,d) is complete, show that (X,~d)(X,d~), where ~d=d/(l+d)d~=d/(l+d), is complete. Proof: Let (x n)(x n) be a Cauchy sequence in (X,¯d)(X,d¯), so that we have, ∀ϵ>0,∃N∀ϵ>0,∃N, such that d(x m,x n)<ϵ d(x m,x n)<ϵ for all m,n>N m,n>N. Then, we have: d(x m,x n)1+d(x m,x n)<ϵ d(x m,x n)<ϵ+ϵ d(x m,x n)d(x m,x n)(1−ϵ)<ϵ d(x m,x n)<ϵ 1−ϵ d(x m,x n)1+d(x m,x n)<ϵ d(x m,x n)<ϵ+ϵ d(x m,x n)d(x m,x n)(1−ϵ)<ϵ d(x m,x n)<ϵ 1−ϵ Set ϵ=1 k ϵ=1 k, so that we get: d(x m,x n)<1 k−1 d(x m,x n)<1 k−1 k k can be made as large as needed to make ϵ ϵ as small as needed. Thus, the sequence (x n)(x n) is Cauchy in (X,d)(X,d), and thus has a limit x x, i.e., x n→x x n→x. Then, d(x n,x)<ϵ d(x n,x)<ϵ. ¯d(x n,x)<d(x n,x)<ϵ d¯(x n,x)<d(x n,x)<ϵ■◼ 1.6.8 Show that in Prob. 7, completeness of (X,~d)(X,d~) implies completeness of (X,d)(X,d). Proof: Suppose (X,~d)(X,d~) is complete. Then, we have: ■◼ 1.6.9 If (x n)(x n) and (x′n)(x n′) in (X,d)(X,d) are such that (1) holds and x n→l x n→l, show that (x′n)(x n′) converges and has the limit l l. Proof: ■◼ 1.6.10 If (x n)(x n) and (x′n)(x n′) are convergent sequences in a metric space (X,d)(X,d) and have the same limit l l, show that they satisfy (1). (1) defines equivalence of two sequences as (x n)~(x′n)⇒lim n→∞d(x n,x′n)=0(x n)(~x n′)⇒lim n→∞d(x n,x n′)=0. Proof: Since (x n)(x n) and (x′n)(x n′) are convergent, we have, ∀ϵ>0,∃N 1,N 2∀ϵ>0,∃N 1,N 2 such that d(x m,l)<ϵ d(x m,l)<ϵ and d(x′n,l)<ϵ d(x n′,l)<ϵ, for m>N 1,n>N 2 m>N 1,n>N 2. Choose N=max(N 1,N 2)N=max(N 1,N 2), so that we have d(x n,l)<ϵ d(x n,l)<ϵ and d(x′n,l)<ϵ d(x n′,l)<ϵ for all n>N n>N. d(x n,x′n)≤d(x n,l)+d(l,x′n)<ϵ+ϵ=2 ϵ⇒lim n→∞d(x n,x′n)=0 d(x n,x n′)≤d(x n,l)+d(l,x n′)<ϵ+ϵ=2 ϵ⇒lim n→∞d(x n,x n′)=0■◼ 1.6.11 Show that (1) defines an equivalence relation on the set of all Cauchy sequences of elements of X X. (1) defines equivalence of two sequences as (x n)~(x′n)⇒lim n→∞d(x n,x′n)=0(x n)~(x n′)⇒lim n→∞d(x n,x n′)=0. Proof: We will check for the following properties: Reflexive Symmetric Transitive We know that d(x n,x n)=0 d(x n,x n)=0 always because of the Principle of Indiscernibles. Thus, we get: lim n→∞d(x n,x n)=0 lim n→∞d(x n,x n)=0 By the Symmetry Property of a distance metric, we know that d(x n,x′n)=d(x′n,x n)d(x n,x n′)=d(x n′,x n). Thus if we have lim n→∞d(x n,x′n)=0 lim n→∞d(x n,x n′)=0, then we also have: lim n→∞d(x′n,x n)=0 lim n→∞d(x n′,x n)=0 By the Triangle Inequality, we have: d(x n,z n)<d(x n,y n)+d(y n,z n)d(x n,z n)<d(x n,y n)+d(y n,z n) Taking limits, we get: lim n→∞d(x n,z n)≤lim n→∞d(x n,y n)+lim n→∞d(y n,z n)lim n→∞d(x n,z n)≤lim n→∞d(x n,y n)+lim n→∞d(y n,z n) If we have lim n→∞d(x n,y n)=0 lim n→∞d(x n,y n)=0 and lim n→∞d(y n,z n)=0 lim n→∞d(y n,z n)=0, we get: lim n→∞d(x n,z n)≤0 lim n→∞d(x n,z n)≤0 Since distances are always nonnegative, we have: lim n→∞d(x n,z n)=0 lim n→∞d(x n,z n)=0. ■◼ 1.6.12 If (x n)(x n) is Cauchy in (X,d)(X,d) and (x′n)(x n′) in X X satisfies (1), show that (x′n)(x n′) is Cauchy in X X. Proof: Since (x n)(x n) is Cauchy, we have, ∀ϵ>0,∃N∀ϵ>0,∃N such that d(x m,x n)<ϵ d(x m,x n)<ϵ for m,n>N m,n>N. d(x m,x n)<ϵ d(x m,x n)<ϵ We also have (x n)(x n) and (x′n)(x n′) being equivalent, so we can write: lim n→∞d(x n,x′n)=0 lim n→∞d(x n,x n′)=0 By the Triangle Inequality, we have: d(x′m,x′n)≤d(x′m,x m)+d(x m,x n)+d(x n,x′n)d(x m′,x n′)≤d(x m′,x m)+d(x m,x n)+d(x n,x n′) Taking limits on both sides, we get: lim n→∞d(x′m,x′n)≤lim n→∞d(x′m,x m)0 because equivalent+lim n→∞d(x m,x n)0 because Cauchy+lim n→∞d(x n,x′n)0 because equivalent lim n→∞d(x m′,x n′)≤lim n→∞d(x m′,x m)⏟0 because equivalent+lim n→∞d(x m,x n)⏟0 because Cauchy+lim n→∞d(x n,x n′)⏟0 because equivalent Since distance metric has to be nonnegative, we conclude that: lim n→∞d(x′m,x′n)=0 lim n→∞d(x m′,x n′)=0■◼ 1.6.13 (Pseudometric) A finite pseudometric on a set X X is a function d:X×X→R d:X×X→R satisfying (M1), (M3), (M4), Sec. 1.1, and (M2) d(x,x)=0 d(x,x)=0. What is the difference between a metric and a pseudometric? Show that d(x,y)=\|ξ 1−η 1\|d(x,y)=\|ξ 1−η 1\| defines a pseudometric on the set of all ordered pairs of real numbers, where x=(ξ 1,ξ 2),y=(η 1,η 2)x=(ξ 1,ξ 2),y=(η 1,η 2). (We mention that some authors use the term semimetric instead of pseudometric.) Proof: (M1) We know that d(x,y)=\|ξ 1−η 1\|d(x,y)=\|ξ 1−η 1\| is always nonnegative, real-valued, and finite. (M3) Because d(x,y)=\|ξ 1−η 1\|d(x,y)=\|ξ 1−η 1\| has a modulus sign, we always have: \|ξ 1−η 1\|=\|η 1−ξ 1\|\|ξ 1−η 1\|=\|η 1−ξ 1\|, and thus we have symmetry. (M4) We have: d(x,y)=\|ξ 1−η 1\|=\|ξ 1−κ 1+κ 1−η 1\|≤\|ξ 1−κ 1\|+\|κ 1−η 1\|d(x,y)=\|ξ 1−η 1\|=\|ξ 1−κ 1+κ 1−η 1\|≤\|ξ 1−κ 1\|+\|κ 1−η 1\|. Thus, the Triangle Inequaity is shown. (Modified M2) An example pair which satisfies this condition is (1,2)(1,2) and (1,3)(1,3). We see that any pair (κ,ξ)(κ,ξ) and (κ,η)(κ,η) will satisfy (Modified M2). We see that if x 1=(κ,ξ)x 1=(κ,ξ) and x 2=(κ,η)x 2=(κ,η), then d(x,y)=\|κ−κ\|=0 d(x,y)=\|κ−κ\|=0. ■◼ 1.6.14 Does d(x,y)=b∫a\|x(t)−y(t)\|d t d(x,y)=∫a b\|x(t)−y(t)\|d t define a metric or pseudometric on X X if X X is (i) the set of all real-valued continuous functions on [a,b][a,b], (ii) the set of all real-value Riemann integrable functions on [a,b][a,b]? Proof: [TODO] ■◼ 1.6.15 If (X,d)(X,d) is a pseudometric space, we call a set B(x 0;r)=x∈X:d(x,x 0)O)B(x 0;r)=x∈X:d(x,x 0)O) an open ball in X X with center x 0 x 0 and radius r r. (Note that this is analogous to 1.3-1.) What are open balls of radius 1 1 in Prob. 13? Answer: The open ball in this case is a vertical rectangles with open width 2 centered at x 0 x 0. tags: Mathematics - Proof - Functional Analysis - Pure Mathematics - Kreyszig
2825	https://www.it.psu.edu/services/canvas/	Services All Services Services For Students Faculty Staff Researchers Service Logins Canvas Office 365 Google Workspace Kaltura LinkedIn Learning LionPATH SIMBA Sites at Penn State Top Hat Voicethread WorkLion Zoom Get Connected Change my Penn State Account Password Connect to Wi-Fi Set up Multifactor Authentication Software Security Support Get Support Service Status Accessibility IT Open a Support Ticket Knowledge Base Get Training Projects Learning Analytics Course Insights Elevate Telephony Transformation Project Wireless University IT Strategic Plan About Our Leadership Student Employment News Search Canvas Training & Support Integrations Updates Log in to Canvas Canvas is the world’s leader in learning and teaching software. Log in Service Status Help Desk About Use Canvas to communicate with students. Design courses with a student-centered approach. Grade your students and create assignments and quizzes. Starting the fall 2025 semester: Reminder to Publish Courses in Canvas Aug 15, 2025 This is a reminder to instructors to publish your courses in Canvas at the beginning of the semester. Some students may want to view their courses in Canvas prior to the start of the semester, however,... read more View More News Training & Support We can help you with your training needs, questions, and resources. Choose your own pace and method with our learning resources, or get one-on-one assistance. Penn State Training & Resources View Penn State’s online, cloud-based system offering features and integrations for teaching and learning. Penn State IT Support Get 24-hour assistance with our IT consultants in person, by phone, or online. Canvas Advisory Group Canvas Advisory Group (CAG) informs the Canvas improvement program (enhancements, third-party learning tools, and known issues). In turn, the LMS Operations Team coordinates and implements technical solutions, documentation, communication, and support as appropriate. Integrations Learning Tools Integrations enhances writing assessments, video communications, and more with Canvas. Learn more about Learning Tool Integrations.
2826	https://philosophy.institute/metaphysics/plato-ideals-forms-eternal-realities/	Plato: Ideals and Forms as Eternal Realities When we think of philosophy, it’s hard to avoid the name of Plato. His ideas continue to influence how we understand reality, knowledge, and existence itself. Central to Plato’s philosophy is his theory of Forms, which introduces a radically different view of the world. According to Plato, the world we experience with our senses is just a pale imitation of a higher, more perfect realm of abstract entities—the Forms. These Forms are eternal, unchanging, and the truest form of reality, as opposed to the fleeting, material world we perceive. In this blog, we’ll dive deep into Plato’s concept of Forms, exploring how these eternal realities shape his metaphysical ideas and how they still resonate with us today. Table of Contents Understanding Plato’s Theory of Forms The realm of the Forms: The world of eternal realities The difference between the material world and the world of Forms Key Characteristics of the Forms 1. Eternal and unchanging 2. Perfect and immutable 3. Independent of human perception 4. Accessible only through reason The Allegory of the Cave: A Visual Metaphor for the Theory of Forms The Influence of Plato’s Theory of Forms Criticism of the Theory of Forms Conclusion Understanding Plato’s Theory of Forms To understand Plato’s philosophy, it’s crucial to first grasp the core of his metaphysical system—the theory of Forms, or Ideas. Plato believed that the physical world we experience is not the true reality but merely a reflection or shadow of a higher, non-material world. In other words, what we see, touch, or hear with our senses is not the “real” world, but only a poor imitation of the perfect and eternal Forms that exist beyond sensory perception. The realm of the Forms: The world of eternal realities According to Plato, the material world is constantly changing. Everything we perceive through our senses is in a state of flux—objects grow, decay, and eventually disappear. For example, the tree you see today may one day wither and die, but the essence of “tree-ness” is not affected by this change. In Plato’s view, this “tree-ness” exists in the realm of Forms as an eternal, unchanging reality. Imagine the perfect Form of a tree, a perfect abstraction of what it means to be a tree—without any flaws, changes, or imperfections. This ideal tree exists independently of any physical tree you might see in the world around you. To Plato, these Forms are the only true objects of knowledge. They are not products of sensory perception but can only be apprehended through reason and intellect. The difference between the material world and the world of Forms Plato’s distinction between the material world and the realm of Forms can be better understood through a few examples. Let’s take the concept of a chair. When you sit in a chair, you’re interacting with a physical object that has a certain shape, material, and function. But, according to Plato, no physical chair is perfect. Some might be wobbly, others might be uncomfortable, or their design might not appeal to you. In the realm of Forms, however, there exists the “perfect chair”—a flawless ideal that is not subject to imperfections or change. This perfect chair is not a physical object, nor can it be perceived with the senses. It exists only as a perfect abstraction, accessible only to the intellect. The physical chairs we encounter in our daily lives are mere imitations of this perfect chair, which exists in the higher, intelligible realm of Forms. In this way, the physical world is seen as an imperfect copy of the true reality that the Forms represent. Key Characteristics of the Forms Plato’s theory of Forms is not just about abstract ideas or ideals; it’s a deep metaphysical claim about the nature of reality. The Forms possess several key characteristics that set them apart from the material world and shape Plato’s understanding of existence itself. 1. Eternal and unchanging The most important feature of the Forms is their permanence. Unlike the material world, which is constantly changing and subject to decay, the Forms are eternal and unchanging. They exist outside of time and space, unaffected by the fluctuations of the physical world. For example, while specific instances of beauty might fade over time (a beautiful sunset, a blooming flower), the Form of Beauty itself is eternal, always existing in its perfect, unchanging state. 2. Perfect and immutable The Forms are also perfect. Each Form is the ultimate expression of a concept or quality. For instance, the Form of Beauty represents beauty in its purest, most ideal form—free from any flaws or imperfection. In the material world, no object can fully capture the perfection of its Form. We might say that a work of art is beautiful, but it can never embody the absolute, flawless beauty that the Form of Beauty represents. 3. Independent of human perception Another important characteristic of the Forms is that they exist independently of human perception or opinion. The Forms are objective and exist whether or not we recognize them or understand them. For Plato, the Form of a tree, a chair, or beauty is not defined by our individual experiences or interpretations. It exists independently, waiting to be discovered through the intellect and reason. 4. Accessible only through reason Because the Forms exist outside of the material world, they cannot be apprehended through our senses. Instead, Plato argued that we can only access the Forms through the exercise of reason and intellect. This is why knowledge of the Forms is a higher form of knowledge—unlike empirical knowledge, which is based on sensory perception, knowledge of the Forms requires philosophical contemplation and intellectual insight. The Allegory of the Cave: A Visual Metaphor for the Theory of Forms Plato’s most famous metaphor for his theory of Forms is the Allegory of the Cave, which appears in his work The Republic. This allegory offers a vivid picture of how human beings are trapped in a world of illusion and how they can escape into the realm of true knowledge and reality. In this allegory, Plato compares human existence to prisoners chained inside a dark cave. These prisoners can only see the shadows cast on the wall by objects passing in front of a fire behind them. The prisoners mistakenly believe these shadows are the only reality, because they have never seen the objects that cast them. However, one prisoner is freed and brought outside of the cave into the light of the sun. At first, the sunlight is blinding, and the prisoner is reluctant to accept it, but eventually, he comes to understand that the sunlight represents true reality. The shadows on the wall were mere illusions, and the objects outside the cave are the true Forms, which can be known only through intellectual enlightenment. The Allegory of the Cave encapsulates Plato’s view that most people are trapped in a world of sensory perception, mistaking the illusions of the material world for true reality. Only through philosophical reasoning and the pursuit of wisdom can one transcend the limitations of the physical world and apprehend the higher, eternal truths of the Forms. The Influence of Plato’s Theory of Forms Plato’s theory of Forms has had a profound influence on Western thought and continues to shape our metaphysical views today. His distinction between the world of appearances and the world of true reality has influenced various philosophical movements, including Neoplatonism, medieval scholasticism, and even modern debates in metaphysics and epistemology. In the Middle Ages, Christian philosophers like Augustine adopted aspects of Plato’s philosophy, using the idea of eternal Forms to explain the nature of God and the creation of the world. The notion that there are perfect, timeless ideals that transcend the material world resonated with religious views about divine perfection and the existence of an ultimate truth. In the modern era, thinkers like Immanuel Kant, George Berkeley, and even some forms of existentialism wrestled with the legacy of Plato’s thought. Kant, for example, argued that we can never know the “thing-in-itself” (the ultimate reality behind appearances), which mirrors Plato’s claim that we can never fully access the realm of the Forms through sensory experience alone. Criticism of the Theory of Forms While Plato’s theory of Forms has been hugely influential, it has also faced significant criticism. One of the major objections comes from Plato’s own student, Aristotle, who argued that Plato’s distinction between the material world and the realm of the Forms was problematic. Aristotle believed that the Forms were not separate from the material world but were instead inherent in the objects themselves. He criticized Plato for creating an unnecessary dualism between the world of the senses and the world of the intellect. Another criticism is the challenge of how we can know these Forms if they are entirely separate from the material world. If the Forms are beyond sensory perception, how can we come to know them with certainty? This question has led to debates about the relationship between perception, reality, and knowledge that continue to this day. Conclusion Plato’s theory of Forms offers a radically different view of reality—one in which the material world is seen as a mere shadow of a higher, more perfect realm. The Forms, eternal and unchanging, represent the true reality, while the physical world is only an imperfect copy. Although Plato’s theory has been critiqued and developed by later philosophers, its central ideas continue to influence metaphysical thought and shape how we understand the nature of reality, knowledge, and existence. What do you think? Do you believe that the material world is just a shadow of a higher reality, or do you think the physical world is all there is? How do you think Plato’s ideas influence modern philosophy and science today? How useful was this post? Click on a star to rate it! Average rating / 5. Vote count: No votes so far! Be the first to rate this post. We are sorry that this post was not useful for you! Let us improve this post! Tell us how we can improve this post? PDF 📄 Comments Leave a Reply Cancel reply Metaphysics 1 Etymology (Definition and Scope) Etymology of Metaphysics Definition of Metaphysics Scope of Metaphysics 2 Fundamental Notions and Principles Fundamental Notions and Principles (Western) Fundamental Notions and Principles (Indian) 3 Brief History of Western Metaphysics Survey of Major Metaphysicians Heraclitus (C. 500 B.C.E.E) Parmenides (C. 550 B.C.E.) Plato (428-348 B.C.E.) Aristotle (384-322 B.C.E.) Augustine (353-430 C.E.) Aquinas (1225-1274) Francis Bacon (1561-1626) Christian Wolff (1679-1754) Immanuel Kant (1724-1804) 4 Brief History of Indian Metaphysics The Vedas The Upanisads The Bhagavad Gita Metaphysical Systems Vedanta Buddhism Jainism Saivism Sikhism: Guru Nanak Contemporary Indian Metaphysics 5 Being and Essence Characteristics of Being Characteristics of Essence 6 Substance and Accidents Etymological Meaning Development of the Doctrine Kinds of Substance Characteristics of Substance Accidents Characteristics of the Accident 7 Matter and Form Classical Views Scientific Views Prime Matter Substantial Form 8 Act and Potency Act Potency Potency and Possibility Potency and Change Potency and Evolution 9 Being and Relation Ontology of Relation Epistemology of Relation 10 Being and Causality Main Kinds of Cause Intrinsic Causes: Material Cause and Formal Cause Extrinsic Causes: Efficient Cause Final Cause and Exemplary Cause Allied Concepts: Condition, Occasion, Sufficient Reason, and Chance 11 Being and Analogy Analogy of Being from Western Perspectives Analogy of Being from Indian Perspectives 12 Being and the Problem of One and Many Early Western Perspectives Early Indian Perspectives Being as One in Many Finite Being as One in Many 13 Categorial Notion of Being (Being and Categories) Early Western Perspective: Aristotle Early Indian Perspective: Vaiseshika School 14 Agapeic Notion of Being (Being and Agape) Notion of Being Being as Agape (love) Freedom as Condition of Love Agapeic Love 15 Transcendental Notion of Being (Being and Transcendentals) Being as One Being as True Being as Good Being as Beautiful 16 Absolute Notion of Being (Being and Absolute) Starting With Heidegger The Notion of Absolute God the Unlimited Existence The Be-ing of Beings is Personal Be-ing is Bliss Responding to Be-ing Share on Mastodon
2827	https://math.stackexchange.com/questions/3692909/proving-that-caratheodorys-criterion-holds-but-how-to-make-it-precise	lebesgue measure - Proving that Caratheodory's Criterion holds..... but How to Make it Precise? - Mathematics Stack Exchange Join Mathematics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Loading… Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about Stack Overflow the company, and our products current community Mathematics helpchat Mathematics Meta your communities Sign up or log in to customize your list. more stack exchange communities company blog Log in Sign up Home Questions Unanswered AI Assist Labs Tags Chat Users Teams Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Try Teams for freeExplore Teams 3. Teams 4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams Hang on, you can't upvote just yet. You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Proving that Caratheodory's Criterion holds..... but How to Make it Precise? Ask Question Asked 5 years, 4 months ago Modified4 years, 11 months ago Viewed 400 times This question shows research effort; it is useful and clear 1 Save this question. Show activity on this post. I'm really sorry for the obscenely long post. This question can be thought of as an exercise in being able to articulate carefully why something is true. If that's not what your in the mood for right now... I understand and I'll catch you on the flip side. Problem Let A A and B B be open boxes in R d R d. Let m∗m∗ denote the Lebesgue outer measure (defined in terms of coverings by open elementary sets). Prove that m∗(B)=m∗(B∩A)+(B∖A).m∗(B)=m∗(B∩A)+(B∖A). I know it might seem tedious, but this result is pivotal in the development of the theory of Lebesgue measurable sets (well, at least for the strategy in Tao's Analysis II). So, I'd really like to be able to articulate precisely why it is true and I'm mildly concerned that I actually don't know how to do so. I would love to hear how other people would attack this problem! Definitions (nothing surprising I expect): An open box in R d R d is defined to be either the empty set, or a set of the form (a 1,b 1)×⋯×(a d,b d)(a 1,b 1)×⋯×(a d,b d) where a j<b j a j<b j for each j∈{1,…,d}j∈{1,…,d}. The volume of an open boxB B in R d R d is defined to be zero if B=∅B=∅, and (b 1−a 1)⋯(b d−a d)(b 1−a 1)⋯(b d−a d) otherwise. In either case it is denoted by vol d(B)vol d(B). In abbreviation of the statement that B B is a finite or countably infinite collection of open boxes in R d R d, let us write B∈B d B∈B d. The outer measure of a subset Ω Ω of R d R d is defined to be the extended real number m∗d(Ω):=inf{a∈R∪{±∞}:there exists a B∈B d such that both a=∑B∈B vol d(B)and Ω⊆⋃B∈B B}.m d∗(Ω):=inf{a∈R∪{±∞}:there exists a B∈B d such that both a=∑B∈B vol d(B)and Ω⊆⋃B∈B B}. For brevity, the d d subscript may be dropped when this does not cause ambiguity. Facts that can be used Basic properties of the outer measure: sub-additive, monotone, m∗d(∅)=0 m d∗(∅)=0, translation invariant, etc. If B B is an non-empty open box in R d R d and β β is any subset of the boundary of B B, then m∗d(B∪β)=vol d(B)m d∗(B∪β)=vol d(B). Any open set is equal to a union of countably many open boxes. If Ω Ω is an arbitrary subset of R d R d, then there exists a sequence of open sets (G n)(G n) such that Ω⊆⋂n G n⊆⋯⊆G 2⊆G 1 Ω⊆⋂n G n⊆⋯⊆G 2⊆G 1 and m∗(Ω)=m∗(⋂n G n)m∗(Ω)=m∗(⋂n G n). Further, if m∗(Ω)<+∞m∗(Ω)<+∞, then (G n)(G n) can be chosen so that m∗(G n)<+∞m∗(G n)<+∞ for every n n and lim n→∞m∗(G n)=m∗(Ω).lim n→∞m∗(G n)=m∗(Ω). Stuff that is off limits Basically anything having to do with the theory of Lebesgue measurable sets, as the development of that theory would (in my case) depend on the result that we're currently trying to prove. Any open set is equal to the union of countably many closed boxes, the interiors of which are mutually disjoint (this is at least as difficult as the problem at hand). My attempt so far Fix ε>0 ε>0. Fix d∈Z+d∈Z+. Let A A and B B be open boxes. For what it's worth, the fact that m∗(B)=m∗(cl(B))m∗(B)=m∗(cl(B)) gives a little more room, at least in principle. By definition of the outer measure, we have a finite or infinite sequence {B n}{B n} of open boxes which cover cl(B)cl(B) and satisfy ∑n vol(B n)≤m∗(cl(B))+ε.∑n vol(B n)≤m∗(cl(B))+ε. In fact, since cl(B)cl(B) is compact, the same can be said of a finite sub-collection {B 1,…,B N}{B 1,…,B N} of {B n}{B n}. That is, for some finite number of open boxes {B 1,…,B N}{B 1,…,B N} ∑n=1 N vol(B n)≤m∗(cl(B))+ε.∑n=1 N vol(B n)≤m∗(cl(B))+ε. On the other hand, Q 1:={B n:n≤N,B n∩A≠∅}Q 1:={B n:n≤N,B n∩A≠∅} covers cl(B)∩A cl(B)∩A (does it cover cl(B)∩cl(A)cl(B)∩cl(A)? what about cl(B∩A)cl(B∩A)?) and Q 2:={B n:n≤N,B n∖A≠∅}Q 2:={B n:n≤N,B n∖A≠∅} covers cl(B)∖A cl(B)∖A. So, using monotonicity and then sub-additivity of the outer measure, m∗(cl(B)∩A)+m∗(cl(B)∖A)≤∑B n∈Q 1 m∗(B n)+∑B n∈Q 2 m∗(B n).m∗(cl(B)∩A)+m∗(cl(B)∖A)≤∑B n∈Q 1 m∗(B n)+∑B n∈Q 2 m∗(B n). But ∑B n∈Q 1 m∗(B n)+∑B n∈Q 2 m∗(B n)−∑B n∈Q 1∩Q 2 m∗(B n)=∑n=1 N m∗(B n).∑B n∈Q 1 m∗(B n)+∑B n∈Q 2 m∗(B n)−∑B n∈Q 1∩Q 2 m∗(B n)=∑n=1 N m∗(B n). Therefore, m∗(cl(B)∩A)+m∗(cl(B)∖A)≤m∗(B)+ε+∑B n∈Q 1∩Q 2 m∗(B n)m∗(cl(B)∩A)+m∗(cl(B)∖A)≤m∗(B)+ε+∑B n∈Q 1∩Q 2 m∗(B n) where Q 1∩Q 2 Q 1∩Q 2 are precisely those boxes among our cover for cl(B)cl(B) that intersect both A A and A c A c. Since this reasoning applies to any collection of boxes {B n}{B n} that cover cl(B)(B), it seems like we should be able to refine the given {B n}{B n} so that each box in B n∈Q 1∩Q 2 B n∈Q 1∩Q 2 is made smaller and smaller, eventually either being shrunk out of the class Q 1∩Q 2 Q 1∩Q 2, or being made so small that vol(B n)<ε/2 n vol(B n)<ε/2 n. But, I really can't grasp how to make this precise and it's bothering me... Will you join me in this indulgence of detail? Miscellaneous other notes Breaking things into cases, it is tractable to prove by brute force that the claim is true if d=1 d=1 (I've already written this proof, actually). Is it then possible to induct on d d? (unrelated) We can assume WLOG that B∩A≠∅B∩A≠∅, but I've yet to find this helpful. (unrelated) The following is why I have used the "alternative proof" tag. The approach taken by Tao is to prove two lemmas which lead to an even stronger result in the end: (i) Prove that an open half space H H (e.g., H=R×(0,∞)×R H=R×(0,∞)×R) satisfies m∗(B)=m∗(B∩H)+m∗(B∖H)m∗(B)=m∗(B∩H)+m∗(B∖H) for any open box B B. Consequently (pixelating a given set into open boxes plus a tolerance of ε ε), any open half space H H is Lebesgue measurable (LM) by Caratheodory's criterion (which we would take as the definition). (ii) The translate of a LM set remains LM, as does the intersection of any finite number of LM sets (these are both actually quite easy to prove). Then, one argues that an open box B B can be expressed as an intersection of a finite number of translates of open half spaces. Thus, an open box is LM and the claim is true. This strategy is pretty, and I cannot disrespect Tao... but getting down to brass tacks, when one goes to prove (i), I don't see how you can treat all O(d 2)O(d 2)-many cases without just asserting that they're mutually similar. (fyi Tao left both lemmas as an exercise; so the lack of insight is mine alone, not his). Again, I feel that if something is so clear, we should be able to articulate why it's true, and we should be capable of coming up with a more precise explanation than just "do a couple of cases by hand and figure that we're good", which is currently all I've got. Edit(s): just cosmetic changes lebesgue-measure alternative-proof Share Share a link to this question Copy linkCC BY-SA 4.0 Cite Follow Follow this question to receive notifications edited May 27, 2020 at 19:08 Thomas WinckelmanThomas Winckelman asked May 26, 2020 at 19:14 Thomas WinckelmanThomas Winckelman 869 7 7 silver badges 16 16 bronze badges 3 1 I agree with you that this is the crux of the matter. I followed Rudin, who did this for d=1 d=1. Then d>1 d>1 follows only after introducing product measure, and invoking the complete machinery. I recommend you look in a variety of sources.Stephen Montgomery-Smith –Stephen Montgomery-Smith 2020-06-02 21:05:13 +00:00 Commented Jun 2, 2020 at 21:05 Could I ask which of Rudin's books?Thomas Winckelman –Thomas Winckelman 2020-06-02 23:18:46 +00:00 Commented Jun 2, 2020 at 23:18 1 Sorry. I meant Royden. amazon.com/Real-Analysis-4th-Halsey-Royden/dp/013143747XStephen Montgomery-Smith –Stephen Montgomery-Smith 2020-06-03 00:59:43 +00:00 Commented Jun 3, 2020 at 0:59 Add a comment\| 1 Answer 1 Sorted by: Reset to default This answer is useful 1 Save this answer. Show activity on this post. Here is an update: Now, I'm taking a class on real analysis, and it has been pointed out to me (thanks Alan!): in order to cover B B with a fininte number of open boxes with an error of at most ε ε, we don't need to merely rely on the definition of the outer measure to tell us that such a collection exists. Rather, we can actually write down the explicit form of such a collection {B 1,…,B N}{B 1,…,B N}, and the additional information provided by this explicit form gives us the extra information that we needed. Let A A and B B be open boxes in R d R d. Say, B=∏j=1 d(a j,b j).B=∏j=1 d(a j,b j). Fix m m a positive integer. For each (ℓ 1,…,ℓ d)∈{1,…,m}d(ℓ 1,…,ℓ d)∈{1,…,m}d, let δ(ℓ 1,…,ℓ d)>0 δ(ℓ 1,…,ℓ d)>0 be arbitrary. Define the open box B((ℓ 1,…,ℓ d)):=∏j=1 d(a j+ℓ j−1 m(b j−a j)−δ(ℓ 1,…,ℓ d)2,a j+ℓ j m(b j−a j)+δ(ℓ 1,…,ℓ d)2).B((ℓ 1,…,ℓ d)):=∏j=1 d(a j+ℓ j−1 m(b j−a j)−δ(ℓ 1,…,ℓ d)2,a j+ℓ j m(b j−a j)+δ(ℓ 1,…,ℓ d)2). Then, using the formula for volume, we have vol(B((ℓ 1,…,ℓ d)))=(b 1−a 1 m+δ(ℓ 1,…,ℓ d))⋯(b d−a d m+δ(ℓ 1,…,ℓ d)).vol(B((ℓ 1,…,ℓ d)))=(b 1−a 1 m+δ(ℓ 1,…,ℓ d))⋯(b d−a d m+δ(ℓ 1,…,ℓ d)). Now, for each given ℓ∈{1,…,m}d ℓ∈{1,…,m}d, we can make this bigger than vol(B)/m d(B)/m d by as small an error as we d well please. In particular, since the preceding is values of δ(ℓ)δ(ℓ) were all arbitrary, we can now, for each ℓ∈{1,…,m}d ℓ∈{1,…,m}d, choose δ(ℓ)>0 δ(ℓ)>0 sufficiently small that vol(B(ℓ))<vol(B)m d+ε 2 ι(ℓ).vol(B(ℓ))<vol(B)m d+ε 2 ι(ℓ). where ι:{1,…,m}d→{1,…,m d}ι:{1,…,m}d→{1,…,m d} is any enumeration (a bijection). Finally, as alluded to previously, we now have that the collection of open boxes {B(ℓ)}ℓ{B(ℓ)}ℓ covers B B, covers B∩A B∩A, and covers B∖A B∖A. Hence, {B(ℓ):B(ℓ)∩A≠∅}ℓ covers B∩A{B(ℓ):B(ℓ)∩A≠∅}ℓ covers B∩A and, likewise, {B(ℓ):B(ℓ)∖A≠∅}ℓ covers B∖A.{B(ℓ):B(ℓ)∖A≠∅}ℓ covers B∖A. Next, using the definition of the outer measure m∗(B∩A)+m∗(B∖A)≤∑ℓ B(ℓ)∩A≠∅vol(B(ℓ))+∑ℓ B(ℓ)∖A≠∅vol(B(ℓ)),m∗(B∩A)+m∗(B∖A)≤∑ℓ B(ℓ)∩A≠∅vol(B(ℓ))+∑ℓ B(ℓ)∖A≠∅vol(B(ℓ)), Second, we have the following elementary (though, perhaps, non-trivial) fact about partitioning a finite sum based on a logical predicate: ∑ℓ vol(B(ℓ))=∑ℓ B(ℓ)∩A≠∅vol(B(ℓ))+∑ℓ B(ℓ)∖A≠∅vol(B(ℓ))−∑ℓ B(ℓ)∩A≠∅B(ℓ)∖A≠∅vol(B(ℓ)).∑ℓ vol(B(ℓ))=∑ℓ B(ℓ)∩A≠∅vol(B(ℓ))+∑ℓ B(ℓ)∖A≠∅vol(B(ℓ))−∑ℓ B(ℓ)∩A≠∅B(ℓ)∖A≠∅vol(B(ℓ)). So, to summarize the last two facts, m∗(B∩A)+m∗(B∖A)≤∑ℓ vol(B(ℓ))+∑ℓ B(ℓ)∩A≠∅B(ℓ)∖A≠∅vol(B(ℓ)).m∗(B∩A)+m∗(B∖A)≤∑ℓ vol(B(ℓ))+∑ℓ B(ℓ)∩A≠∅B(ℓ)∖A≠∅vol(B(ℓ)). Third, using our previously established upper bound on vol(B(ℓ))(B(ℓ)), we have ∑ℓ vol(B(ℓ))+∑ℓ B(ℓ)∩A≠∅B(ℓ)∖A≠∅vol(B(ℓ))<∑ℓ(vol(B)m d+ε 2 ι(ℓ))+∑ℓ B(ℓ)∩A≠∅B(ℓ)∖A≠∅(vol(B)m d+ε 2 ι(ℓ)).∑ℓ vol(B(ℓ))+∑ℓ B(ℓ)∩A≠∅B(ℓ)∖A≠∅vol(B(ℓ))<∑ℓ(vol(B)m d+ε 2 ι(ℓ))+∑ℓ B(ℓ)∩A≠∅B(ℓ)∖A≠∅(vol(B)m d+ε 2 ι(ℓ)). In other words, m∗(B∩A)+m∗(B∖A)≤2 ε+vol(B)+vol(B)m d∑ℓ B(ℓ)∩A≠∅B(ℓ)∖A≠∅1.m∗(B∩A)+m∗(B∖A)≤2 ε+vol(B)+vol(B)m d∑ℓ B(ℓ)∩A≠∅B(ℓ)∖A≠∅1. As noted in the original post, we know by this point in our devlopment of the theory that vol(B)=m∗(B)vol(B)=m∗(B). Finally, since m m is an arbitrary positive integer, the argument is reduced to estimating the number of boxes B(ℓ)B(ℓ) which can intersect both A A and A c A c. This is still hard to do!!! Edit: Notation and stuff. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications edited Oct 6, 2020 at 22:37 answered Oct 6, 2020 at 20:55 Thomas WinckelmanThomas Winckelman 869 7 7 silver badges 16 16 bronze badges Add a comment\| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions lebesgue-measure alternative-proof See similar questions with these tags. Featured on Meta Introducing a new proactive anti-spam measure Spevacus has joined us as a Community Manager stackoverflow.ai - rebuilt for attribution Community Asks Sprint Announcement - September 2025 Report this ad Related 0Alternative proof for Lebesgue outer measure of elementary sets 0A confusion over the lebesgue outer measure of (0,1)∩Q c(0,1)∩Q c 3Help save this proof about the regularity of the lebesgue measure on R d R d 1Show: ∀A∈L(R)∀A∈L(R) the set A×R⊂R 2∈L(R 2)A×R⊂R 2∈L(R 2) 0Can a Non-Measurable Set be Approximated by Closed Sets from Below? 5A A Lebesgue measurable iff ∀ε>0∃G=⋃N n=1 I n∀ε>0∃G=⋃n=1 N I n bounded open intervals: \|A∖G\|+\|G∖A\|<ε\|A∖G\|+\|G∖A\|<ε 6Proving a Volume Identity for Open Sets in R n R n Hot Network Questions Are there any world leaders who are/were good at chess? For every second-order formula, is there a first-order formula equivalent to it by reification? Can a cleric gain the intended benefit from the Extra Spell feat? 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2828	https://journals.lww.com/otology-neurotology/fulltext/2012/07000/cochlear_implantation_in_patients_with.30.aspx?generateEpub=Article%7Cotology-neurotology:2012:07000:00030%7C10.1097/mao.0b013e318254fba5%7C	Otology & Neurotology Log in or Register Subscribe to journal Subscribe Get new issue alerts Get alerts;;) Subscribe to eTOC;;) ### Secondary Logo Enter your Email address: Privacy Policy ### Journal Logo Articles Advanced Search Toggle navigation SubscribeRegisterLogin Browsing History Articles & Issues Current Issue Previous Issues Latest Articles Collections For Authors Submit a Manuscript Information for Authors Language Editing Services Author Permissions Journal Info About the Journal Editorial Board Affiliated Societies Advertising Open Access Subscription Services Reprints Rights and Permissions Articles Advanced Search July 2012 - Volume 33 - Issue 5 Previous Abstract Next Abstract Cite Copy Export to RIS Export to EndNote Share Email Facebook X LinkedIn Favorites Permissions More Cite Permissions Article as EPUB Export All Images to PowerPoint FileAdd to My Favorites Email to Colleague Colleague's E-mail is Invalid Your Name: Colleague's Email: Separate multiple e-mails with a (;). Message: Your message has been successfully sent to your colleague. Some error has occurred while processing your request. Please try after some time. Export to End Note Procite Reference Manager [x] Save my selection Tumors of the Ear and Cranial Base Cochlear Implantation in Patients With Neurofibromatosis Type 2 Variables Affecting Auditory Performance Carlson, Matthew L.; Breen, Joseph T.; Driscoll, Colin L.†; Link, Michael J.†; Neff, Brian A.; Gifford, René H.‡; Beatty, Charles W. Author Information Departments of Otolaryngology – Head and Neck Surgery and †Neurologic Surgery, Mayo Clinic School of Medicine, Rochester, Minnesota; and ‡Vanderbilt Bill Wilkerson Center, Department of Hearing and Speech Sciences, Vanderbilt University, Nashville, Tennessee, U.S.A. Address correspondence and reprint requests to Colin L. Driscoll, M.D., Department of Otolaryngology – Head and Neck Surgery, Mayo Clinic, 200 First Street S.W., Rochester, MN 55905; E-mail: driscoll.colin@mayo.edu No funding or other support was required for this study. Dr Driscoll is a consultant for Cochlear Corporation, Advanced Bionics Corporation, and Med-El GmbH. Presented at the Combined Otolaryngology Spring Meeting 2012, American Neurotology Society section. Otology & Neurotology 33(5):p 853-862, July 2012. \| DOI: 10.1097/MAO.0b013e318254fba5 Buy Abstract Objective To investigate cochlear implant performance outcomes among patients with Neurofibromatosis type 2 (NF2). Study Design Retrospective case series, patient questionnaire, and systematic review of the literature. Setting Tertiary academic referral center. Patients All patients with NF2 having an anatomically intact ipsilateral cochlear nerve who underwent cochlear implantation (CI). Intervention(s) Cochlear implantation. Main outcome measures Postimplantation audiometric performance and patient perceived benefit. Results Ten patients met study criteria. The median duration of follow-up after CI was 42 months (mean, 46.9 mo; range, 12–97 mo). Five patients received previous microsurgical resection of their ipsilateral vestibular schwannoma, 4 underwent previous stereotactic radiosurgery, and 1 patient had no tumor treatment before CI. Nine subjects achieved sound awareness, 6 attained open-set speech recognition and 7 are daily users. Variables including prolonged auditory deprivation, cochlear ossification, unfavorable electrical promontory stimulation testing, and useful contralateral hearing were associated with poor cochlear implant performance. No statistical associations were found between open-set recognition capacity and previous tumor management strategy, surgical approach, or ipsilateral tumor size. Conclusion Cochlear implantation is an attractive alternative to auditory brainstem implantation for hearing rehabilitation in patients with NF2. Approximately 70% of patients achieve open-set speech discrimination, many scoring at the ceiling of audiometric testing. Given a favorable risk profile and superior audiometric outcomes, CI should be strongly considered in patients with nonserviceable hearing who have an anatomically intact cochlear nerve, whereas auditory brainstem implantation should be reserved for patients with evidence of cochlear nerve loss. Akin to conventional cochlear implant recipients, prolonged hearing loss, unfavorable electrophysiological testing, and cochlear ossification may predict poor subject performance. Finally, useful hearing in the contralateral ear may present a barrier to daily device use. © 2012 Otology & Neurotology, Inc. Full Text Access for Subscribers: ##### Individual Subscribers Log in for access ##### Institutional Users Access through Ovid® Not a Subscriber? Buy Subscribe Request Permissions Become a Society Member You can read the full text of this article if you: Log InAccess through Ovid Source Cochlear Implantation in Patients With Neurofibromatosis Type 2: Variables Affecting Auditory Performance Otology & Neurotology33(5):853-862, July 2012. Full-Size Email Favorites Export View in Gallery Email to Colleague Colleague's E-mail is Invalid Your Name: Colleague's Email: Separate multiple e-mails with a (;). Message: Your message has been successfully sent to your colleague. Some error has occurred while processing your request. Please try after some time. Most Popular Gene Therapy for Hearing Loss: Which Genes Next? Hearing Preservation of Slim Modiolar and Slim Straight Electrodes: A Systematic Review and Meta-Analysis Sensitivity and Specificity of Intraoperative TransImpedance Matrix Recordings Compared With X-ray Imaging in Detecting Perimodiolar Cochlear Implant Tip Foldovers: A Multicenter Study Assessment of Safety, Tolerability, Pharmacokinetics, and Volume-Dependent Conductive Hearing Loss in Healthy Volunteers: First-in-Human, Open-Label, Placebo-Controlled Study of a Single Intratympanic Injection of AC102 Long-Term Risk of Progression From Unilateral to Bilateral Méniere's Disease: A Systematic Review and Meta-Analysis Back to Top Never Miss an Issue Get new journal Tables of Contents sent right to your email inbox Get New Issue Alerts Browse Journal Content Most Popular For Authors About the Journal Past Issues Current Issue Register on the website Subscribe Get eTOC Alerts;;) For Journal Authors Submit an article How to publish with us Customer Service Live Chat Chat Offline Activate your journal subscription Activate Journal Subscription Browse the help center Help Contact us at: Support: Submit a Service Request TEL: (USA): TEL: (Int’l): 800-638-3030 (within USA) 301-223-2300 (international) Manage Cookie Preferences Privacy Policy Legal Disclaimer Terms of Use Open Access Policy Feedback Sitemap RSS Feeds LWW Journals Your California Privacy Choices Copyright©2025 Otology & Neurotology, Inc. \| Content use for text and data mining and artificial intelligence training is not permitted. 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Students must unlock 5 locks by solving for x in 20 equations. Questions are grouped 4 per puzzle, resulting in five 4-letter codes that will unlock all 5 locks. In all equations, the variable is on one side. The entire activity is housed in one GOOGLE Form. There are no links to outside websites. The 4-letter codes are set with answer validation so that students cannot move to the next puzzle until they enter the correct code. Includes 7 th - 9 th Algebra, Algebra 2 CCSS 8.EE.C.7 Also included in:Middle School Digital Math Escape Room Activity Bundle 6th 7th 8th Grade $3.00 Original Price $3.00 Rated 4.79 out of 5, based on 321 reviews 4.8(321) Add to cart Wish List 8th Grade Math Spiral Review \| Warm Ups, Math Homework, Progress Monitoring Created by One Stop Teacher Shop 100% EDITABLE } This 8th Grade math spiral review resource can easily be used as math HOMEWORK, WARM UPS, or a DAILY MATH REVIEW! This resource was designed to keep math concepts fresh all year and to help you easily track student progress. All pages are 100% EDITABLE and easy to differentiate to fit your students' needs. ⭐ UPGRADE for just $10 and get both PRINT & DIGITAL versions } CLICK HERE⭐ CLICK HERE to see more SUBJECTS & GRADE LEVELS! ⭐ SAVE $$ with these BUNDLES★★ 8th Grade Math BU 7 th - 9 th Algebra, Geometry, Math CCSS 8.G.A.1 , 8.G.A.1a , 8.G.A.1b +33 Bundle (4 products) $39.96 Original Price $39.96 $29.99 Price $29.99 Rated 4.85 out of 5, based on 1229 reviews 4.9(1.2K) Add to cart Wish List 8th Grade Math Word Wall \| 8th Grade Math Classroom Vocabulary Created by Scaffolded Math and Science This 8th grade math word wall shows vocabulary and concepts through bright visuals and examples. The word wall comes in printable color, printable B & W, digital in Google Slides, and includes printable Spanish vocabulary. With references for scatter plots, correlation, function vs not a function, combining like terms, one/zero/infinite solutions when solving equations, exponent rules, transformations, triangle sums, Pythagorean Theorem, systems of equations, squares, cubes, D=RT, rational and i 8 th Math, Other (Math), Vocabulary CCSS 8.G.A.1 , 8.G.A.2 , 8.G.A.3 +12 Also included in:Math Word Wall Bundle \| Math Bulletin Board Vocabulary Posters $12.00 Original Price $12.00 Rated 4.91 out of 5, based on 785 reviews 4.9(785) Add to cart Wish List Linear Equations Unit \| Two-Step Equations with Distributive Property 8.EE.7 Created by Maneuvering the Middle An 11 day CCSS-Aligned Linear Equations Unit - including simplifying expressions, solving multi-step equations, solving equations with variables on both sides, writing equations, and equations with special cases. Students will practice with both skill-based problems, real-world application questions, and error analysis to support higher level thinking skills. You can reach your students and teach the standards without all of the prep and stress of creating materials!Standards: 8.EE.7a, 8.EE.7b 8 th Algebra, Math, Other (Math) CCSS 8.EE.C.7 , 8.EE.C.7a , 8.EE.C.7b $14.00 Original Price $14.00 Rated 4.89 out of 5, based on 526 reviews 4.9(526) Add to cart Wish List 8th Grade Math Warm-Ups (CCSS-Aligned Math Bell Ringers) \| Daily Warm-Ups Created by Maneuvering the Middle These 120 daily math warm-ups can be used as bell work to immediately engage your students. They are designed to be used as spiral review bell ringers throughout the year! However, eight additional ideas for use have been included. These warm-ups/bell ringers allow you to reinforce each of the 8th Grade Math Common Core State Standards throughout the year and refresh your student's memory by continually spiraling the content. What is included?Printable PDF (120 Warm-Ups)16 spiral review be 7 th - 9 th Math, Math Test Prep CCSS 8.G.A.1 , 8.G.A.1a , 8.G.A.1b +33 $16.00 Original Price $16.00 Rated 4.85 out of 5, based on 803 reviews 4.9(803) Add to cart Wish List Solving One Step, Two Step and Multi Step Equations Guided Notes Created by 4 the Love of Math Make solving equations clear and approachable with this comprehensive Solving Equations Guided Notes and Practice Set! Whether you're introducing one-step equations or guiding students through multi-step equations with variables on both sides, these structured notes break down each skill with step-by-step clarity. These solving equation guided notes walk students through solving equations with strategies, examples, and built-in practice problems Included in this solving equations guided notes 7 th - 12 th, Higher Education Algebra, Math CCSS 8.EE.C.7 , HSA-REI.A.1 , HSA-REI.B.3 Also included in:Solving Equations Bundle - One-Step, Two-Step, Multi-Step Notes and Activities $3.00 Original Price $3.00 Rated 4.75 out of 5, based on 337 reviews 4.8(337) Add to cart Wish List Multi-Step Equations Digital Math Escape Room Variables on Both Sides Activity Created by Scaffolded Math and Science An engaging algebra escape room activity for solving equations with variables on both sides. Students must unlock 5 locks by solving 20 equations, most with x on both sides. Some equations involve distribution and some include fractions. Questions are grouped 4 per puzzle, resulting in five 4-letter codes that will unlock all 5 locks. The entire activity is housed in one GOOGLE Form. There are no links to outside websites. The 4-letter codes are set with answer validation so that students canno 8 th - 10 th Algebra, Algebra 2, Math CCSS 8.EE.C.7 , HSA-REI.B.3 Also included in:Digital Math Escape Room Activity Bundle for Algebra $3.00 Original Price $3.00 Rated 4.84 out of 5, based on 196 reviews 4.8(196) Add to cart Wish List Two-Step Equations Algebra Bundle \| 10 Activities + Best-Selling Whodunnit Created by Clark Creative Math Master two-step equations with this Algebra bundle of 10 engaging activities! Perfect for 6th–8th grade math classes, these print and digital resources include puzzles, games, practice sheets, and the best-selling Whodunnit activity. Use them for review, test prep, homework, or sub plans to keep students motivated and confident in solving two-step equations. In this activity pack you will find these resources: Adventure - Two Step Equations - India Basic Training - Two-Step Cards Basic Trainin 6 th - 12 th Algebra, Algebra 2, Math CCSS 8.EE.C.7 , 8.EE.C.7a , HSA-REI.B.3 Also included in:Equations Unit Bundle - Algebra Curriculum - Notes, Homework & Activities $24.00 Original Price $24.00 $8.00 Price $8.00 Rated 4.85 out of 5, based on 772 reviews 4.9(772) Add to cart Wish List 8th Grade Math Pre Algebra Guided Notes Lessons Mega Bundle Entire Year Created by mrscasiasmath Looking for ready-to-use 8th Grade Math lessons that will last you the entire year? This full-year 8th Grade Math guided notes bundle includes 95 no-prep lessons, all aligned to the 8th Grade Math TEKS and compatible with CCSS and other standards—and all for less than $1 per lesson! These concise, student-friendly guided notes are designed to get straight to the point—perfect for the attention span of busy teenagers. Whether you’re introducing a concept, reviewing, or planning for a sub da 8 th Algebra, Math, Other (Math) CCSS 8.G.A.1 , 8.G.A.1a , 8.G.A.1b +24 Bundle (96 products) $135.00 Original Price $135.00 $92.99 Price $92.99 Rated 4.91 out of 5, based on 77 reviews 4.9(77) Add to cart Wish List Solving One and Two Step Equations Activity: Algebra Escape Room Math Game Created by Escape Room EDU This breakout escape room is a fun way for students to test their skills with one and two step equations. Most of my Escape Rooms can be used for distance learning, this one; however, is not recommended for distance learning. I have recently opened a new store that sells 100% digital math escape rooms that are specifically designed for distance learning. That store's name is Digital Escape Rooms and contains this topic. Here is the link to the One and Two Step Equations Escape Room for d 6 th - 9 th Algebra, Math Test Prep, Other (Math) CCSS 8.EE.C.7 $6.50 Original Price $6.50 Rated 4.83 out of 5, based on 366 reviews 4.8(366) Add to cart Wish List 8th Grade Math Review \| CCSS Test Prep \| End of Year Math Review \| Exam Prep Created by Maneuvering the Middle This CCSS-Aligned 8th Grade Math Review and Test Prep Unit includes teacher guides, warm ups, cheat sheets, class activities, and 8-10 question mini assessments for each of the 10 topics covered. There is more than enough for a 10 day review, so think of it like a buffet with many options to pick from and multiple ideas for use. Teachers use this COMPLETE review unit to organize your 8th Grade Math Common Core Review and Test Prep to best suit your students’ needs. This resource is aligned to 8 th Math, Math Test Prep CCSS 8.EE.A.1 , 8.EE.A.2 , 8.EE.A.3 +6 $35.00 Original Price $35.00 Rated 4.88 out of 5, based on 299 reviews 4.9(299) Add to cart Wish List 8th Grade Math Mini Assessments - Test Prep and Skills Review for 8th Grade Math Created by Beyond the Worksheet with Lindsay Gould ⭐️⭐️⭐️ This collection of 28 8th Grade Math Mini Assessments is designed to be a quick and easy way to assess critical 8th grade math skills. Each assessment includes 4 to 8 targeted questions, ensuring a thorough evaluation of each standard. Not just limited to assessments, these versatile pages double as effective homework, quizzes, or review material before tests. ❤️ What if I don't teach Common Core?These assessments will still work for you! Also included is a version that does NOT have t 8 th Math, Math Test Prep CCSS 8.G.A.1 , 8.G.A.2 , 8.G.A.3 +19 Also included in:8th Grade Math Curriculum Supplemental Activities Bundle $12.00 Original Price $12.00 Rated 4.96 out of 5, based on 468 reviews 5.0(468) Add to cart Wish List 8th Grade Math Pre Algebra Foldables and Activities Bundle Created by Lisa Davenport This editable foldable bundle includes all my 8th grade math pre-algebra foldables and notes for interactive notebooks, Task Cards, Scavenger Hunts, MATHO (Math Bingo) games, Puzzles, worksheets, and more! Take a look at the individual contents listed below to get a better idea of what is included! Many of these activities were updated as of May 2025. Please re-download the bundle if you made the purchase before then! Download the preview to take a closer look at what's included! This bundl 8 th Algebra, Geometry, Math CCSS 8.G.A.1 , 8.G.A.1a , 8.G.A.1b +32 Bundle (190 products) $426.50 Original Price $426.50 $60.00 Price $60.00 Rated 4.94 out of 5, based on 209 reviews 4.9(209) Add to cart Wish List 8th Grade Math Interactive Notebook Guided Notes Created by Math in Demand This is a whole years worth of an interactive notebook that is very organized and engaging. CHECK OUT THE PREVIEW TO SEE HOW AMAZING THIS NOTEBOOK IS! The notebook has 5 units (6 including notebook starter). The units include: Unit 0 - Notebook Starter Unit 1 - The Number System Unit 2 - Expressions & Equations Unit 3 - Functions Unit 4 - Geometry Unit 5 - Statistics & Probability A breakdown of each unit is below: Unit 0 - Notebook Starter-Notebook Cover for Students -Classroom Rules -Noteb 7 th - 9 th Algebra, Algebra 2, Math CCSS 8.G.A.1 , 8.G.A.1a , 8.G.A.1b +33 Also included in:Custom Bundle for Math Interactive Notebooks $59.99 Original Price $59.99 Rated 4.85 out of 5, based on 221 reviews 4.9(221) Add to cart Wish List Fun Solving Multi Step Equations with Variables on Both Sides Activity Mystery Created by Lauren Fulton Looking for a LOW PREP, HIGH ENGAGEMENT Solving Equations with Variables on Both Sides Activity? Your students will love this MYSTERY! In this multi-step equations mystery, students are given unique suspect cards and assume the identity of an amusement park guest. When one of their fellow park guests is robbed, it's up to your class to figure out who the thief is and solve the mystery!Students must be able to write & solve multi-step equations, including the use of distributive property, with 7 th - 9 th Algebra, Basic Operations, Math CCSS 8.EE.C.7 , 8.EE.C.7b , HSA-CED.A.1 +1 Also included in:Writing and Solving Equations & Inequalities Bundle! $3.75 Original Price $3.75 Rated 4.86 out of 5, based on 132 reviews 4.9(132) Add to cart Wish List Solving Equations \| Self-Checking, Digital Pixel Art-Two-Step, Multi-Step & VOBS Created by Math with Mrs Markowski Make solving equations fun and engaging with this self-checking Google Sheets™ activity! Students will solve equations and watch as mystery images are revealed, keeping them motivated and providing instant feedback with every answer. This no-prep resource includes three differentiated Google Sheets™, perfect for independent practice, review, or homework. ✅ What’s Included: 3 separate Google Sheets™ mystery picture activities: 15 two-step equations 12 multi-step equations 10 equations with va 7 th - 9 th Algebra, Math CCSS 8.EE.C.7 , 8.EE.C.7b , HSA-REI.B.3 $6.00 Original Price $6.00 Rated 4.89 out of 5, based on 110 reviews 4.9(110) Add to cart Wish List Integer Operations Rules Student Handout or Anchor Chart Created by Math With Ms Murphy You will receive a PDF document with integer operations rules for addition, subtraction, multiplication and division. These can be printed and given to students to keep in their notebooks, you can upload them to your Canvas and Google Classroom courses for students to reference or even use an anchor chart within your classroom! Brief examples and descriptions are provided for each operation. 5 th - 9 th Algebra, Math, Other (Math) CCSS 6.NS.C.5 , 6.NS.C.6a , 7.NS.A.1a +9 $1.50 Original Price $1.50 Rated 4.76 out of 5, based on 93 reviews 4.8(93) Add to cart Wish List Distributive Property & Combining Like Terms Partner Activity Created by The Math Cafe This resource contains: 1) Distributive Property Partner Practice 2) Combining Like Terms Partner Practice 3) Answer Keys This activity is meant to be used as a "self-check" activity between partners. Each partner receives a different expression to simplify but they should both end up with the same simplified expression. If they produce different expressions, they must try to prove to the other why they think their answer is correct. Great for checking integer rules, fraction distribution, etc. 7 th - 8 th Algebra, Math CCSS 7.EE.A.1 , 7.EE.A.2 , 7.EE.B.4a +1 Also included in:Linear Expressions MINI-BUNDLE $1.00 Original Price $1.00 Rated 4.82 out of 5, based on 109 reviews 4.8(109) Add to cart Wish List Solving Multi-Step Equations Digital and Printable Flipbook Notes Created by 2ndary Math Engage your Math students by using this set of interactive guided notes to practice multistep equations. Each page of the scaffolded notes includes the previous step for solving multistep equations for easy to follow notes. This Flipbook helps students remember the steps for solving a multi-step equation using the acronym, Don't (Distribute) Call (Combine Like Terms) Me (Move the Variable to One Side of the Equation) After (undo any Addition or Subtraction) Midnight (undo any Multiplication or 7 th - 9 th Algebra, Math CCSS 8.EE.C.7 , 8.EE.C.7b Also included in:BUNDLE of Solving Equations Activities $4.00 Original Price $4.00 Rated 4.89 out of 5, based on 136 reviews 4.9(136) Add to cart Wish List Pre Algebra 8th Grade Math Printable Homework Worksheet Bundle and Google Forms Created by Lisa Davenport This 8th Grade Math Pre-Algebra Homework and Assessment Bundle with Google Forms includes 92 (and growing!) auto-grading digital assignments that can be easily shared with your students through Google Classroom. Each assignment also includes a corresponding PDF to use alongside or instead of the digital version. Assign digitally or print + go! Students will practice working with real numbers, laws of exponents, scientific notation, solve multi-step equations, linear relationships, linear s 8 th Geometry, Math CCSS 8.G.A.1 , 8.G.A.1a , 8.G.A.1b +33 Bundle (92 products) $164.75 Original Price $164.75 $60.00 Price $60.00 Rated 4.86 out of 5, based on 125 reviews 4.9(125) Add to cart Wish List 6th, 7th, 8th Grade SPIRAL MATH REVIEW BUNDLE \| Google BACK TO SCHOOL Created by Exceeding the CORE Need a SPIRAL REVIEW of the 6th, 7th and 8th Grade Common Core standards? This BUNDLE contains 300 QUESTIONS of review specifically written for the common core math standards for 6th, 7th and 8th grade. The review sheets are organized by grade level into 5 boxes. Each box contains problems from the 5 domains of that grades CCSS standards. This BUNDLE RESOURCE includes:✔ 6th Grade Math WEEKLY/DAILY REVIEW WARM-UPS (20 weeks / 100 questions)✔ 7th Grade Math WEEKLY/DAILY REVIEW WARM-UPS (2 6 th - 8 th Math, Math Test Prep, Other (Math) CCSS 6.G.A.1 , 6.G.A.2 , 6.G.A.3 +121 Bundle (3 products) $18.00 Original Price $18.00 $14.40 Price $14.40 Rated 4.82 out of 5, based on 328 reviews 4.8(328) Add to cart Wish List 8th Grade Math Escape Rooms Bundle ★ Digital and Printable Created by Math in the Midwest Looking for no-prep, fun, and engaging activities for middle school students? Look for further! This bundle is a great way to review all the 8th Grade Math Common Core Standards with your students in a fun and engaging way. Your students will be asking for more math!These escape rooms will have your students using mathematical skills from the 8th Grade math standards to find codes that will unlock each puzzle. Plus there is NO PREP all you need is an internet browser as well as either a lapt 8 th Math, Math Test Prep CCSS 8.G.A.1 , 8.G.A.1a , 8.G.A.1b +33 Bundle (10 products) $53.94 Original Price $53.94 $31.24 Price $31.24 Rated 4.9 out of 5, based on 176 reviews 4.9(176) Add to cart Wish List Solving Equations with Variables on Both Sides Multi Step Equations Guided Notes Created by The Sassy Math Teacher Want to breeze through teaching multi step equations with ease? These guided notes are a great way to teach your students to solve multi-step equations that require using the distributive property and combining like terms along with variables on both sides. This resource includes:3 Scaffolded Guided Notes PagesSpace for Page Numbers to help with notebook organizationLearning Target at the top of each page for easy reference at testing time3 Practice Pages for Classwork or HomeworkAnswer KeysTop 7 th - 8 th Algebra, Math, Numbers CCSS 8.EE.A.4 , 8.EE.C.7 , 8.EE.C.7a +1 Also included in:8th Grade Math Curriculum - Notes Activities Warm Ups Spiral Review Exit Tickets $5.50 Original Price $5.50 Rated 4.86 out of 5, based on 14 reviews 4.9(14) Add to cart Wish List Solving Multi Step Equations with Variables on Both Sides Digital Pixel Art Created by Math with Ms. Rivera Could your student use more practice solving multi step equations with variables on both sides? Of course! With this digital self-checking activity, your students will solve 10 problems to practice solving multi step equations with variables on both sides to reveal a pixel art picture. If they answer correctly a portion of the mystery picture will appear. If they get it wrong, the boxes will not be colored in. Students will love guessing what the picture will be. Here is what's included: ➡️ 1 8 th - 10 th Algebra, Math CCSS 8.EE.C.7 , HSA-REI.B.3 Also included in:Solving Multi Step Equations Variables on Both Sides Activity Pack for Algebra 1 $3.50 Original Price $3.50 Rated 4.87 out of 5, based on 87 reviews 4.9(87) Add to cart Wish List 1 2 3 4 5 Showing 1-24 of 8,069 results TPT is the largest marketplace for PreK-12 resources, powered by a community of educators. Facebook Instagram Pinterest Twitter About Who we are We're hiring Press Blog Gift Cards Support Help & FAQ Security Privacy policy Student privacy Terms of service Tell us what you think Updates Get our weekly newsletter with free resources, updates, and special offers. 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2830	https://www.bhf.org.uk/informationsupport/heart-matters-magazine/medical/ask-the-experts/atrial-flutter	The difference between atrial flutter and atrial fibrillation - BHF Skip to main content Who we are Learn CPR Store finder Search Close Popular across the BHF How to do CPR Leave a gift in your Will Cost of living Recipe finder Information and support Information and support Conditions Risk factors Support Tests Treatments CPR and defibrillators: How to save a life How your heart works For professionals Health at work Heart Matters How you can help How you can help Volunteer Donate Fundraise Take part in an event Partner with us ### London to Brighton Bike Ride Sign up now for our next London to Brighton Bike Ride on 21st June 2026! ### Volunteer in our shops A great way to meet new people and share your skills – we'd love for you to join our team Shop Shop Buy furniture and items for your home Our eBay shop Buy BHF gifts and merchandise Donate your items Find your local shop Buy health monitoring devices Buy defibrillators Book a house clearance Volunteer in our shops What we do What we do Who we are Our research Our strategy News from BHF Jobs at BHF ### Find BHF near you Find your nearest BHF shop, and book and clothing bank ### Our research: Heart statistics Read the most comprehensive statistics on the effects, prevention, treatment, and costs of heart and circulatory diseases in the UK Donate Donate Donate items Donate money Leave a gift in your Will Lottery ### Donate items We pick up furniture and electrical items from your home for free. The items you donate are sold in our stores to help fund our lifesaving research ### Donate money BHF funds the science that saves lives. Your donation helps us discover new ways to beat heart diseases Search Sign in/Register MenuClose Information and support How you can help Shop What we do Back Information and support Conditions Risk factors Support Tests Treatments CPR and defibrillators: How to save a life How your heart works For professionals Health at work Heart Matters Back How you can help Volunteer Donate Fundraise Take part in an event Partner with us ### London to Brighton Bike Ride Sign up now for our next London to Brighton Bike Ride on 21st June 2026! ### Volunteer in our shops A great way to meet new people and share your skills – we'd love for you to join our team Back Shop Buy furniture and items for your home Our eBay shop Buy BHF gifts and merchandise Donate your items Find your local shop Buy health monitoring devices Buy defibrillators Book a house clearance Volunteer in our shops Back What we do Who we are Our research Our strategy News from BHF Jobs at BHF ### Find BHF near you Find your nearest BHF shop, and book and clothing bank ### Our research: Heart statistics Read the most comprehensive statistics on the effects, prevention, treatment, and costs of heart and circulatory diseases in the UK Back Donate Donate items Donate money Leave a gift in your Will Lottery ### Donate items We pick up furniture and electrical items from your home for free. The items you donate are sold in our stores to help fund our lifesaving research ### Donate money BHF funds the science that saves lives. Your donation helps us discover new ways to beat heart diseases Atrial flutter and atrial fibrillation Heart Matters magazine Understanding health Ask the experts Understand health What's the difference between atrial flutter and atrial fibrillation? I have been diagnosed with atrial flutter, although I only have it occasionally. I have never heard of this and can only find information on atrial fibrillation. What’s the difference? byChristopher Allen, Cardiac Nurse Published: 5 January 2016 byChristopher Allen, Cardiac Nurse Published: 5 January 2016 Senior Cardiac Nurse Christopher Allen says: Atrial flutter andatrial fibrillationare both abnormal heart rhythms. They occur when there is an issue with the electrical signals and pathways in your heart, which usually help it beat in an organised, effective way. Normally, the top chambers (atria) contract and push blood into the bottom chambers (ventricles). In atrial fibrillation, the atria beat irregularly. In atrial flutter, the atria beat regularly, but faster than usual and more often than the ventricles, so you may have four atrial beats to every one ventricular beat. Atrial flutter is less common than atrial fibrillation Atrial flutter is less common, but has similar symptoms (feeling faint, tiredness, palpitations, shortness of breath or dizziness). Some people have mild symptoms, others have none at all. About a third of people with atrial flutter also have atrial fibrillation. Both conditions carry increased risk of stroke, usually managed by drugs (such as warfarin or a newer anticoagulant). This is why, whether you have atrial fibrillation or atrial flutter, it is vital to be diagnosed early so you can get the right treatment and reduce your stroke risk. Either condition may require medications to prevent your heart rate becoming too rapid. Catheter ablation is usually considered the best treatment for atrial flutter, whereas medication is often the first treatment for atrial fibrillation. Catheter ablation is a procedure that is done under local anaesthetic, where radiofrequency energy is used to destroy the area inside your heart that’s causing the abnormal heart rhythm. Find out what to expect during an ablationin our exclusive video. Read more about treatments for abnormal heart rhythms. Meet the expert Christopher Allen helps manage the BHF’s genetic information service and has extensive specialist experience of working in coronary care. Related Links What is atrial fibrillation? Expert tips on living with atrial fibrillation What to expect in a cath lab: 360 degree video Sign up to Heart Matters Related Publications Atrial fibrillation - your quick guide British Heart Foundation Get in touch Contact us Customer services:0300 330 3322 Phone lines are open Mon - Fri 9am-5pm Nurse Helpline Work for us Jobs Volunteer Your BHF Online community Sign in/register Get our email newsletter Find the BHF near you Our policies Privacy Cookies Terms & conditions Accessibility Executive pay Modern slavery statement All policies Cookie settings Follow us British Heart Foundation is a registered charity in England and Wales (225971), Scotland (SC039426) and the Isle of Man (1295). A company limited by guarantee, registered in England and Wales (699547) and the Isle of Man (6201F). Registered office: Greater London House, 4th Floor, 180 Hampstead Road, London NW1 7AW To find out more, or to support British Heart Foundation’s work, please visit www.bhf.org.uk. You can speak to one of our cardiac nurses by calling our helpline on 0808 802 1234 (freephone), Monday to Friday, 9am to 5pm. For general customer service enquiries, please call 0300 330 3322, Monday to Friday, 9am to 5pm. British Heart Foundation is a registered Charity No. 225971. Registered as a Company limited by guarantee in England & Wales No. 699547. Registered office at Greater London House, 180 Hampstead Road, London NW1 7AW. Registered as a Charity in Scotland No. SC039426 We use cookies. Some are necessary to make our site work and others are optional, allowing us to analyse site usage, personalise content and to tailor advertising. These are stored on your device and are placed by us and trusted partners. Find out more in our cookie policy. Manage cookies OK Cookie Preference Centre Cookie Preference Centre Your Privacy Your Privacy When you visit any website, it may store or retrieve information on your browser in the form of cookies. 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2831	https://it.wikipedia.org/wiki/Viaggio_nel_tempo	Vai al contenuto Ricerca Indice 1 La "macchina del tempo" 2 Il viaggio nel tempo per la fisica attuale 2.1 Campi gravitazionali 2.2 Sistemi cinematici a velocità differenti 2.2.1 Alla velocità della luce 2.2.2 Sopra la velocità della luce 2.3 Meccanica quantistica 3 Speculazioni teoriche 4 Sperimentazioni 5 Viaggi nel tempo e paradossi 5.1 Paradosso di "coerenza" (o del nonno) 5.2 Paradosso di "conoscenza" (o del pittore o della Monna Lisa) 5.3 Paradosso di predestinazione 5.4 Paradosso di "co-esistenza" 5.5 Paradosso dell'infattibilità 5.6 Risoluzioni possibili dei paradossi 5.6.1 Protezione cronologica 5.6.2 Esistenza di mondi paralleli 6 Il viaggio nel tempo nella fantasia 7 Viaggi nel tempo e leggende metropolitane 8 Note 9 Bibliografia 10 Voci correlate 11 Altri progetti 12 Collegamenti esterni Viaggio nel tempo العربية Asturianu Azərbaycanca Български বাংলা Català Cebuano کوردی Čeština Dansk Deutsch Ελληνικά English Esperanto Español Eesti Euskara فارسی Suomi Français Frysk Galego עברית हिन्दी Hrvatski Magyar Հայերեն Interlingua Bahasa Indonesia Ido 日本語 ქართული 한국어 Latina Lietuvių Latviešu Македонски മലയാളം Bahasa Melayu Nederlands Norsk bokmål ਪੰਜਾਬੀ Polski پنجابی Português Romnă Русский سنڌي සිංහල Simple English Slovenčina Slovenščina Soomaaliga Shqip Српски / srpski Sunda Svenska தமிழ் Тоҷикӣ ไทย Tagalog Türkçe Українська اردو Oʻzbekcha / ўзбекча Tiếng Việt 吴语中文粵語 Modifica collegamenti Voce Discussione Leggi Modifica Modifica wikitesto Cronologia Strumenti Azioni Leggi Modifica Modifica wikitesto Cronologia Generale Puntano qui Modifiche correlate Link permanente Informazioni pagina Cita questa voce Ottieni URL breve Scarica codice QR Stampa/esporta Crea un libro Scarica come PDF Versione stampabile In altri progetti Wikimedia Commons Wikiquote Elemento Wikidata Aspetto Da Wikipedia, l'enciclopedia libera. Disambiguazione – Se stai cercando altri significati, vedi Viaggio nel tempo (disambigua). Il viaggio nel tempo è il concetto del viaggio tra diverse epoche o momenti temporali, inteso in una maniera analoga al viaggio tra diversi punti dello spazio, sia verso il passato sia verso il futuro, senza che il soggetto debba far esperienza di tutto l'intervallo temporale presente tra l'epoca di partenza e quella di arrivo. Per la fisica attuale, questa esperienza sarebbe possibile solo per quanto riguarda il "viaggio" nel futuro, seppur non in modo proprio istantaneo. L'idea di viaggiare istantaneamente nel tempo, sia nel passato che nel futuro, ha da sempre affascinato l'umanità, diffondendosi soprattutto nella fantascienza, alcune volte utilizzata come espediente narrativo per storie ambientate nel passato, altre volte con storie o viaggi ambientati o simulati. La "macchina del tempo" [modifica \| modifica wikitesto] Lo stesso argomento in dettaglio: Cronovisore. Nell'immaginario collettivo, la "macchina del tempo" è il nome dato all'ipotetico mezzo di trasporto per viaggiare nel tempo, in grado di spostare oggetti ed esseri viventi da un'epoca temporale all'altra, sia nel passato sia nel futuro. La fantascienza, in genere, ha abituato a vedere tale macchina come una sorta di vero e proprio "veicolo" o "apparecchio" nel quale si entra, si configurano i parametri di viaggio, si dà inizio al viaggio e poco dopo si può uscire ritrovandosi nell'epoca temporale programmata. Con le attuali conoscenze, tale macchina dovrebbe compiere anche enormi balzi spaziali oltre che temporali, poiché il pianeta Terra occupa, secondo per secondo, una posizione diversa lungo l'orbita di rivoluzione intorno al Sole, così come il Sole occupa uno spazio ben preciso durante il suo moto intorno al centro della Via Lattea, e così via. In conclusione, un viaggio nel tempo così concepito dovrebbe necessariamente comportare anche un movimento spaziale, altrimenti l'ipotetico crononauta si ritroverebbe sperduto nel vuoto cosmico al momento dell'arrivo. Per ora, le uniche "macchine" tecnologiche in grado di far soltanto "vedere" il passato – o il futuro – pur rimanendo nel presente, sono quelle attraverso la simulazione in realtà virtuale come, ad esempio, i cronovisori, dispositivi con i quali, con le tecnologie di oggi, si possono solamente generare immagini di paesaggi o scenari di interi mondi, attraverso dettagliate ricostruzioni grafiche al computer, proiettate su schermi ad alta definizione, monitor interattivi e vari dispositivi multimediali. Il viaggio nel tempo per la fisica attuale [modifica \| modifica wikitesto] Lo stesso argomento in dettaglio: Freccia del tempo, Entropia e Secondo principio della termodinamica. La fisica classica ha esaminato per secoli e con attenzione la possibilità di viaggiare nel tempo; in particolare, le difficoltà ravvisate emergono soprattutto per il viaggio nel passato. Tali difficoltà sono legate al concetto di "tempo" secondo l'esperienza e la conoscenza classica del mondo, ovvero un "tempo" naturalmente percepito soltanto come "lo scorrere degli eventi", come visione classica del "divenire" di Eraclito. Secondo questa visione, il tempo risulta quindi un parametro immutabile e unidirezionale, come lo scorrere dell'acqua di un fiume, e tutti gli eventi dell'Universo si susseguono seguendo le leggi del modello di causalità (causa → effetto). Tutte le leggi della natura infatti, seguono una cosiddetta "freccia del tempo", che è strettamente legata al concetto di entropia, e ciò che è compiuto risulta irreversibile. Esempi ne sono l'impossibilità di ricomporre i cocci di un vaso rotto fino al vaso intero originario, di rimettere un inchiostro disciolto in acqua dentro il flacone che lo conteneva all'inizio, il fumo della combustione di una sigaretta dentro la precedente sigaretta nuova e integra, e riportare in vita un essere morto. Tutti i fenomeni naturali sono assoggettati da un aumento di entropia, sintetizzata, in altre parole, come "disordine", "caos". In qualsiasi fenomeno, la natura ha una direzione per la quale "preferisce" scegliere il maggior numero di stati possibili successivi allo stato iniziale. Se ne conclude che il processo inverso sia (non viene detto "impossibile", ma) altamente improbabile. L'esempio classico di questo concetto, è quello di un neonato al quale proviamo a mettere dei piccoli guanti alle mani: il neonato, dopo vari tentativi goffi e disordinati, aiutandosi con la bocca, o le braccia, i piedi, ecc. riuscirà comunque a togliersi i guanti dalle mani. Ma è altamente improbabile che egli riuscirà a ricalzarli di nuovo perfettamente, perché esiste un solo e unico modo ordinato per metterli: la natura si comporta esattamente come quel neonato. Sul finire del XIX secolo, il fisico Boltzmann studiò a fondo tali principi, soprattutto nel ramo della termodinamica, aprendo così la strada alla cosiddetta "fisica statistica": sulla tomba gli fu dedicata la formula dell'entropia (), sebbene questa fosse stata perfezionata da Planck. All'inizio del XX secolo poi, il concetto di "freccia del tempo", il concetto filosofico di causalità (causa → effetto) e dello stesso "scorrere del tempo", all'epoca considerato immutabile, furono rivoluzionati dalla nascente teoria della relatività ristretta e dalla relatività generale di Albert Einstein. Il tempo non resta più una costante, un parametro fisso, immutabile e universale, così come concetto di fenomeno esperienziale, bensì una componente variabile. Un primo modello fu dato dalle teorie della relatività einsteiniane, che identificarono nei fenomeni una struttura quadridimensionale, dove non esiste più lo spazio in sé, e non più un tempo in sé, ma una dimensione plastica detta di spaziotempo. Risulta chiaro che, viste le grandezze in gioco, lo sviluppo di tali teorie avvenne in costante evoluzione con il progresso dell'astrofisica dell'epoca. Ci si rese conto che in natura esistono due sistemi dove lo spaziotempo è variabile: in un sistema di campi gravitazionali (ovvero la Teoria della Relatività generale) in un sistema cinematico di corpi a velocità differenti (ovvero la Teoria della Relatività ristretta) Campi gravitazionali [modifica \| modifica wikitesto] La relatività generale postula che lo spaziotempo si curva quando un corpo – o anche la luce stessa – attraversa un qualsiasi campo gravitazionale. Le osservazioni, eseguite soprattutto durante le eclissi solari del 1912 e 1919, scoprirono che anche la luce (o un qualsiasi flusso di onde elettromagnetiche), quando attraversa una massa (meglio quella il cui campo gravitazionale è particolarmente significativo) subisce una curvatura – e quindi una variazione – dello stesso spaziotempo, fenomeno successivamente battezzato col nome di "lente gravitazionale". Per capire meglio il concetto di tempo influenzato dalla gravità, dobbiamo raffigurarci lo spaziotempo, o "cronotopo", mutuando il termine dalla geometria, proprio come un telo uniforme disteso in tutto l'Universo, perfettamente elastico, ben tirato, ma tuttavia increspato, in un qualche punto, da alcune zone occupate da corpi celesti, in questo caso chiamati "gravi", perché dotati, appunto, di gravità. Ogni increspatura – o avvallamento – è detta "curvatura spaziotemporale", ed è influenzata proporzionalmente dal campo gravitazionale generato dal corpo stesso immerso in esso. Grazie a questo fenomeno della dilatazione temporale gravitazionale, il tempo quindi scorre a differenti velocità in regioni di diverso potenziale gravitazionale, ovvero più veloce se si trova lontano dal centro di gravità, più lentamente se si trova vicino. Sul nostro pianeta, dotato di una certa massa e di una certa gravità, il tempo scorre leggermente più veloce in alto, ad esempio in montagna, rispetto alla pianura, anche se questo, naturalmente, in modo del tutto impercettibile e trascurabile. Nell'ambito dell'astrofisica, dove troviamo spesso campi gravitazionali molto elevati, come in prossimità di un buco nero o di una stella di neutroni, la cosa si fa ancora più interessante. Se un ipotetico equipaggio di un viaggio interstellare riuscisse a recarsi in prossimità di tali masse enormi, il tempo scorrerebbe molto più lentamente rispetto a tutto il resto dell'Universo e quindi, una volta allontanatosi dal buco nero, esso si troverebbe, a tutti gli effetti, nel futuro. Il tempo tenderebbe addirittura a fermarsi, in taluni casi estremi, come, nel caso del buco nero, sul suo bordo, ovvero in prossimità dell'orizzonte degli eventi. Non a caso, i buchi neri, che sono gli oggetti fisici dove sono massime sia la densità di materia, sia il campo gravitazionale, rientrano nella possibilità di creare dei "ponti" spaziotemporali. Questo ipotetico passaggio nello spaziotempo viene chiamato Ponte di Einstein-Rosen, o altrimenti detto wormhole, letteralmente "buco di verme": si tratta di ipotetiche "porte" spaziotemporali, collocate nel vuoto cosmico, chiamate metaforicamente così perché, proprio come un verme che scava dentro una mela, attraverseremmo la stessa mela dall'interno, ovvero percorrendo una "scorciatoia" spaziotemporale, piuttosto che prendere invece la strada convenzionale sull'esterno, ovvero sulla buccia. Sistemi cinematici a velocità differenti [modifica \| modifica wikitesto] Quando la differenza di velocità di due sistemi cinematici, uno di riferimento e l'altro di misurazione risulta apprezzabile, allora anche lo spaziotempo (e quindi anche il tempo) tra i due sistemi risulterà diverso. Questo fenomeno venne chiamato dilatazione del tempo, ed è dimostrabile con la teoria della relatività ristretta di Einstein e le relative trasformazioni matematiche di Lorenz. Soltanto la velocità della luce nel vuoto rimane un parametro fisso, costante e invalicabile, e che viene chiamato c (= 299 792,458 km/s), e questo in tutti i sistemi di riferimento. Quando la materia percorre delle velocità apprezzabilmente elevate, meglio quindi se vicine a quella della luce, poiché la funzione matematica è esponenziale (le formule matematiche ammettono che deve essere almeno il 10% di essa), il suo tempo subisce un rilevabile rallentamento rispetto all'altro sistema di riferimento, che si trova, ad esempio, in stato di quiete. In queste condizioni, essendo lo stesso spaziotempo a deformarsi, ne consegue che si ha anche un aumento della massa del corpo in movimento, con una conseguente riduzione del suo volume – ovvero dello spazio occupato, dando luogo a quello che, in fisica relativistica, è conosciuto come il "paradosso dell'auto" o "del garage". Tipici esempi per comprendere la dilatazione del tempo, sono il paradosso dei gemelli, o l'esperimento mentale del treno di Einstein. Un esempio pratico per osservare in natura il limite della velocità della luce è il brillare che ci arriva da corpi celesti lontani, come stelle e pianeti, dove, in un certo senso, noi viaggiamo già nel passato; la luce della Luna è di un secondo fa, la luce del Sole che ci scalda è di circa 8 minuti fa, mentre la luce di Sirio è di circa 8 anni fa, e così via. Parimenti, ipotetici alieni lontani da noi vedrebbero la vita sul nostro pianeta Terra di migliaia di anni fa. Questo vale lo stesso, sebbene impercettibilmente, anche per piccole distanze: quando noi guardiamo un nostro interlocutore, la sua immagine, a causa della velocità della luce, risulta infatti quella di qualche milionesimo di miliardesimo di secondo fa. La dilatazione del tempo fu dimostrata anche con un esperimento cinematico, e aggiungendo anche il fatto che fu esclusa dalla sola parte relativa al sistema cinematico anche il valore relativo alla dilatazione temporale gravitazionale dovuta alle differenti altezze dal centro di gravità: ponendo un orologio di precisione su un velivolo, si riscontrarono delle discrepanze tra esso e l'orologio di riferimento con cui era stato precedentemente sincronizzato, posto in un sistema in quiete rispetto al velivolo (per esempio sulla pista), dimostrando che l'orologio del velivolo, spostandosi ad alta velocità dal suo riferimento, viaggiò qualche frazione di secondo indietro rispetto all'orologio posto a terra. In sintesi, la "velocità" con cui scorre localmente il tempo in un sistema in quiete (cioè la rapidità con cui si muovono le lancette di un orologio in tale sistema di riferimento) è di un secondo al secondo, se si prende come sistema di riferimento lo stesso sistema (in quiete) in cui ci si trova. Nel precedente esempio sul velivolo il tempo scorre a meno di un secondo (tempo locale, sistema del velivolo) al secondo (tempo del sistema di riferimento, in quiete, sulla pista). Nella pratica, il ritardo dell'orologio sul velivolo risulterà lievissimo: la velocità del velivolo è assai minore della velocità della luce nel vuoto, sicché gli effetti della relatività speciale non sono facilmente percepibili. Questo esperimento fu condotto per la prima volta nel 1971 dai fisici Joseph C. Hafele e Richard E. Keating, ed oggi noto come Esperimento di Hafele-Keating – o H-K –, calcolando il piccolissimo scarto temporale (decine-centinaia di nanosecondi) tra i precisissimi orologi atomici al cesio portati a bordo di un Boeing 747 che viaggiava a circa 800 km/h e gli orologi a terra, ovviamente considerando e quindi scartando il calcolo della differenza di tempo dovuta all'effetto di dilatazione temporale gravitazionale, questa invece influenzata dalla differenza di gravità tra le due altezze. Alla velocità della luce [modifica \| modifica wikitesto] La stessa relazione matematica tra velocità e tempo conferma inoltre che, per corpi che si muovessero alla velocità della luce, il tempo t' tenderebbe a infinito, mentre il tempo t di riferimento terrestre smetterebbe praticamente di scorrere. Purtroppo, sappiamo che un qualsiasi corpo dotato di massa non può raggiungere la velocità della luce, ma avvicinarsi solo al suo valore. Esperimenti eseguiti su una particella subatomica detta muone, dimostrarono che essa vive più a lungo man mano che si avvicina a velocità prossime alla luce. Sopra la velocità della luce [modifica \| modifica wikitesto] Lo stesso argomento in dettaglio: Velocità superluminale. Un capitolo interessante è quello di un ipotetico sistema di particelle di materia che si muovano a velocità superluminali, dove le equazioni prevedono che lo scorrere del tempo diventi addirittura negativo: il loro "futuro" quindi, sarebbe il passato di tutti gli altri corpi. Ipotizzando un corpo a velocità sopraluminale, questo potrebbe possedere soltanto una massa immaginaria, sia a riposo sia accelerata. A questa ipotetica particella subatomica sopraluminale fu attribuito il nome di "tachione", ma la sua esistenza non fu mai dimostrata. Per la fisica attuale infatti, dal punto di vista dell'ipotetico tachione, anche il principio dell'entropia sarebbe invalidato: cocci di vetro si ricomporrebbero per generare un bicchiere infranto, mentre un cadavere potrebbe riprendere vita e tornare al momento del concepimento. Dal punto di vista filosofico inoltre, in un "mondo sopraluminale", le conseguenze precederebbero la causa generante, quindi si entrerebbe in contraddizione del principio di causalità (es. paradosso detto del cacciatore e della tigre). Queste ipotesi suggestive furono gradualmente abbandonate, anche se le teorie einsteiniane non proibiscono velocità superluminali; il raggiungimento di tali velocità è vietato solo a corpi con massa reale e positiva, ovvero tutti i corpi costituiti dalla materia al momento conosciuta. Non si sa se nell'universo esistano oggetti per cui tale divieto non sia valido. Infatti fu dimostrata l'esistenza della cosiddetta "materia oscura", chiamata così poiché non direttamente osservabile ma per la quale sono comprovati i suoi effetti, oltre che alla teoria per la quale nei viaggi nel tempo potrebbe essere implicata anche la cosiddetta "materia esotica", teoria avanzata dal fisico Kip Thorne. In ogni caso, tutte le formule della teoria della relatività contengono un termine temporale elevato alla seconda potenza, per cui la definizione di un tempo negativo non crea particolari problemi al modello matematico. Meccanica quantistica [modifica \| modifica wikitesto] Nelle stesso periodo di inizio XX secolo inoltre, furono sviluppate delle interessanti teorie fisiche della nascente meccanica quantistica, che analizzò i fenomeni della natura principalmente nei sistemi microcosmici. In particolare, fu osservato il comportamento di particelle subatomiche presunte "gemelle" in differenti e lontane località dello spazio, osservandone il loro cambiamento di stato fisico in modo praticamente istantaneo. La meccanica quantistica nacque infatti per dimostrare il comportamento della stessa materia in interazione con particolari "particelle" che compongono la stessa luce, dette fotoni o quanti, poiché non costituite da materia in sé ma da differenze, o "quanti", di energia, sotto forma di radiazione elettromagnetica e che quindi non solo non sono dotate di massa a riposo, ma per loro il tempo risulterebbe uguale a zero. Tali teorie della fisica quantistica vennero successivamente prese in considerazione con il nome di Paradosso di Einstein-Podolsky-Rosen (E.P.R.) e fenomeno di entanglement quantistico. Queste teorie furono avanzate grazie all'ipotesi di definizione del tempo come un semplice "cambiamento di stato" delle particelle subatomiche, strettamente legato anche alla dimensione dello spazio – in relazione all'osservatore stesso del fenomeno – oltre che alla materia, all'energia, alle fluttuazioni quantistiche e ai campi gravitazionali. I due sistemi di "particelle", molto distanti tra di loro, sarebbero in un qualche modo correlate da delle fluttuazioni quantistiche, reagendo in "sincrono" e contraddicendo quindi l'affermazione che nessuna informazione può viaggiare oltre la velocità della luce. A partire dalla metà del XX secolo quindi, la cosiddetta "filosofia del tempo" risultò totalmente rivoluzionata. Vi furono parecchie divergenze e opinioni varie soprattutto nell'astrofisica, come, ad esempio, l'introduzione delle teorie degli "universi a blocchi", dove lo stesso avanzare dello spaziotempo sarebbe suddiviso in una sorta di "blocchi" di spaziotempo nei quali passato-presente-futuro coincidono, ma non dal punto di vista degli osservatori immersi nel blocco, pertanto il mio esatto "istante presente" non è uguale a quello di un altro, il mio tempo passato e il mio tempo futuro pure, poiché tutto relativo. Un altro interessante modello matematico simile dello spaziotempo è quello dell'universo a "eventi", altrimenti detto universo di Minkowski, per il quale gli eventi si sviluppano nello spaziotempo attraverso coni di luce o, più genericamente, coni detti "degli eventi". Grazie a queste affermate teorie quindi, la possibilità di viaggiare nel tempo in modo apprezzabile sarebbe quindi ammessa, ma soltanto in condizioni estreme, attualmente impossibili da realizzare con le più recenti tecnologie e, tendenzialmente, soltanto in avanti, ovvero in un futuro rispetto all'istante temporale per il quale si decidesse di iniziare il viaggio.Il celebre fisico Stephen Hawking ad esempio, fu un forte sostenitore dell'impossibilità di viaggi nel passato, perché se questi fossero possibili per la sola relatività generale, avrebbero comunque effetti significativi anche sulla natura quantistica. Finché non si riusciranno ad unificare le due teorie nella cosiddetta grande teoria della gravità quantistica (quantum gravity), le due attuali descrizioni dell'universo resteranno incompatibili. Hawking sostenne l'impossibilità anche per un altro motivo, ovvero la conservazione della materia-energia nel continuum spaziotempo, affermando che, se fosse veramente possibile viaggiare nel tempo l'Universo sarebbe pieno di crononauti "cloni di sé stessi", portando quindi in saturazione tutto il sistema. Tuttavia, alcuni esperimenti del 2011 eseguiti dallo scienziato russo Igor Smolyaninov, partendo dalle teorie sulla materia esotica e con l'impiego di metamateriali plasmonici iperbolici, dimostrarono la possibilità di "modellare" il flusso del tempo, rendendo quindi accettabile l'ipotesi di un viaggio soltanto nel tempo futuro. Speculazioni teoriche [modifica \| modifica wikitesto] Se per la fisica attuale è ipotizzabile soltanto un viaggio nel futuro, anche tale spostamento sarebbe comunque vincolato dallo stesso principio di causalità che regola gli eventi dal passato verso l'istante presente, scelto come riferimento per l'osservazione del fenomeno. Nel 1949, un matematico amico di Einstein, Kurt Gödel, ammise la possibilità dei viaggi nel tempo partendo da alcune equazioni relativistiche, seppur con alcune limitazioni, che egli chiamò teoremi di incompletezza matematica. Se le teorie einsteiniane ponevano un limite teorico alle velocità, che non può superare quella della luce, non vi sono limiti teorici all'intensità di un campo gravitazionale e, quindi, alla deformazione dello spaziotempo. Le speculazioni teoriche sulla creazione di "macchine per il viaggio nel tempo" sono quindi incentrate sull'ipotesi di deformazioni spaziotemporali di varia natura (oltre che su alcune soluzioni particolari delle equazioni presenti nelle teorie di Einstein, come ad esempio la Curva spaziotemporale chiusa di tipo tempo). La realizzazione di tali deformazioni, sempre estreme, necessita però di quantità immense di energia, che eccedono di gran lunga persino quelle prodotte nel Sole. Le strade attuali per un ipotetico viaggio nel tempo quindi, resterebbero quelle sullo studio sui buchi spaziotemporali e l'analisi dei buchi neri, ma confrontati sempre con lo studio sulla legge di conservazione dell'energia. I fisici Paul Davies, Kurt Gödel, Frank Tipler e John Richard Gott III (vedi Bibliografia) proposero delle metodologie necessariamente ideali per costruire una macchina del tempo. In particolare, il modello di Gott III si basò sul fatto che la gravità dei corpi massivi influenza lo scorrere del tempo. In breve, il modello prevede di usare un corpo di massa paragonabile a quella di Giove per creare una sfera cava, all'interno della quale porre il cosiddetto "crononauta". Da calcoli fatti, il campo gravitazionale della sfera cava (generata dalla massa del corpo fortemente compressa) rallenterebbe il tempo di un numero variabile di volte (massimo quattro) a seconda della densità della sfera, che deve essere sempre inferiore a quella necessaria per la contrazione in un buco nero. Da citare tra le più classiche e moderne teorie sui viaggi nel tempo del fisico britannico e pioniere del computer quantistico David Deutsch, che riprese quelle degli anni cinquanta di Hugh Everett III e Bryce Seligman DeWitt sull'ipotesi dell'"interpretazione a molti mondi" e che furono di ispirazione a Bob Gale nella saga cinematografica di "Ritorno al futuro". Di contro, la teoria di Gödel sarebbe invece valida solo in un universo chiuso in rotazione dove, muovendosi a velocità prossime a quella della luce, si potrebbe raggiungere ogni istante di tempo semplicemente viaggiando sempre in una stessa direzione. Ipotizzando tale universo costituito da dei coni di luce - o altrimenti detti "coni di eventi" -, si può saltare da un "cono" all'altro attraverso linee immaginarie chiuse dette CTC (Closed Timelike Curves, ovvero curva spaziotemporale chiusa di tipo tempo). La teoria di Tipler invece, risulta una variante di quella di Gödel che si basa sull'esistenza di un corpo materiale, e non utilizza dunque l'intero universo come nel precedente esempio; si tratterebbe di un ipotetico cilindro rotante di massa esorbitante (si parla di miliardi di masse solari), ma di densità inferiore a quella necessaria perché si trasformi in un buco nero, creerebbe un'attrazione gravitazionale tale da far sì che un corpo che si muova intorno a esso a velocità elevatissime anche se non necessariamente prossime a quella della luce si sposti nel passato o nel futuro, a seconda che si muova nel verso opposto o uguale a quello della rotazione del cilindro. Questo modello pone però due importanti limitazioni: non si può andare in un passato precedente la creazione del cilindro, e non si può andare in futuro successivo la sua distruzione. Il modello matematico, inoltre, presuppone un cilindro infinitamente lungo, e non è ancora chiaro se questa condizione sia necessaria per il viaggio nel tempo. I principali ipotetici mezzi per un viaggio nel tempo resterebbero quindi: un wormhole (ponte di Einstein-Rosen), o altri metodi molto simili, ovvero che utilizzino le deformazioni spaziotemporali il raggiungimento di velocità elevate, meglio se prossime alla velocità della luce, soprattutto per i viaggi nel futuro, come effetto già provato della dilatazione temporale di Einstein L'utilizzo di intensi campi gravitazionali, sempre sfruttando l'effetto della dilatazione temporale di Einstein Sperimentazioni [modifica \| modifica wikitesto] Vari esperimenti realizzati danno l'impressione di un effetto retrogrado, ossia di un viaggio nel tempo verso il passato, ma sono interpretati in modo diverso dalla comunità scientifica. Un esempio fu l'esperimento di cancellazione quantistica a scelta ritardata del 1999 (che è ispirato al paradosso EPR e richiede l'utilizzo di fessure di Young) che lascia supporre che su scala quantica una particella nel futuro determini il suo passato. Secondo alcuni, questo mette semplicemente in evidenza le difficoltà di qualificare la nozione di tempo all'interno della scala quantica; in ogni caso, quest'esperimento non costituisce una violazione della causalità. Infine, il programma "Effetto STL" effettuato dal fisico Ronald Mallett ha lo scopo ufficiale di osservare una violazione della causalità mediante il passaggio di un neutrone attraverso un cristallo fotonico che rallenta la luce. Si è potuto constatare che il neutrone riappare nel dispositivo prima di essere disintegrato. La relazione è uscita nel novembre 2006 e beneficia del sostegno di molte università degli Stati Uniti. Il teletrasporto e il viaggio temporale sono quindi temi collegati, che presuppongono la copertura di enormi distanze nello spazio o nel tempo. Le tematiche del viaggio nel tempo e nello spazio vengono a essere in stretta relazione, per almeno due ragioni: secondo la relatività generale, spazio e tempo sono parte di un continuum a quattro dimensioni; il paradosso dei gemelli ammette la possibilità teorica di un viaggio nel futuro; il ponte di Einstein-Rosen è una costruzione fisica e matematica che ammette la possibilità teorica di un viaggio nel passato e nel futuro. I ponti di Einstein-Rosen descrivono sia un collegamento fra due punti arbitrariamente distanti nello stesso universo, oppure che possono distare arbitrariamente nel tempo. I punti possono appartenere allo stesso universo o a due universi paralleli. La massa che è oggetto del teletrasporto può comparire nel punto di arrivo in un tempo superiore a quello che impiegherebbe muovendosi alla velocità della luce, rispettando il limite teorico imposto dalla relatività generale. Esiste però una variante del teletrasporto che presuppone di collegare due punti a velocità inferiori a quella della luce, riproducendo l'informazione della massa nel punto di arrivo. La realizzazione di un viaggio nel passato o nel futuro, oltre ai problemi teorici, presenterebbe notevoli difficoltà tecniche. Secondo le teorie che ammettono la possibilità di un viaggio nel tempo, come quella dei ponti di Einstein-Rosen, sarebbe necessaria una quantità enorme di energia, pari alla potenza elettrica mondiale. Le potenze in gioco sono simili a quelle che un'esplosione nucleare produce in pochi minuti. Onda d'urto e radiazioni di una bomba atomica, tuttavia, si disperdono a distanza di migliaia di chilometri e di anni. In base alla formula , 600 grammi di massa d'uranio possono infatti produrre un'energia pari a Joule, per un tempo di 10 minuti (assumendo una velocità della luce pari a 300.000 km/s). Un ulteriore modalità di viaggio nel tempo è l'attraversamento di dimensioni esterne allo spaziotempo; la teoria delle stringhe ad esempio, ipotizza l'esistenza di dieci dimensioni. Le dimensioni aumentano a seconda della lente, della scala di misura con la quale si osserva l'universo. Sei di queste dimensioni sono in più rispetto a quelle note dello spazio tempo, "arrotolate" e compresse in un piccolissimo raggio, per cui punti diversi dello spazio-tempo potrebbero essere collegati da una di queste dimensioni. Viaggiando attraverso di esse, si otterrebbe una "scorciatoia" per collegare due punti, nello spazio e/o nel tempo, senza superare il limite teorico della velocità della luce. Viaggi nel tempo e paradossi [modifica \| modifica wikitesto] Lo stesso argomento in dettaglio: Paradosso temporale. Oltre al noto paradosso dei gemelli, che riguarderebbe comunque viaggi nel futuro, furono avanzati anche paradossi su ipotetici viaggi nel tempo passato. I paradossi che contengano vere e proprie contraddizioni logiche sarebbero da evitare nei calcoli della fisica e nella matematica. Paradosso di "coerenza" (o del nonno) [modifica \| modifica wikitesto] Lo stesso argomento in dettaglio: Paradosso del nonno. È utilizzato nelle tematiche relative al continuum spaziotempo, ed è più comunemente noto come paradosso del nonno. L'esempio più classico è viaggiare nel passato per tornare a far visita a vostro nonno. Il viaggio riesce, e vi trovate finalmente a tu per tu con lui, che però è giovane e non si è ancora sposato con quella che diventerà, in seguito, la vostra nonna. Se uccidete vostro nonno, oppure lo distraete dalla sua vita normale, egli potrebbe non presentarsi mai all'appuntamento con la ragazza che diventerà la vostra futura nonna. Di conseguenza, sia i vostri genitori che voi stessi non nascereste; ma se non foste mai nati, come avreste potuto impedire ai nonni di incontrarsi? Un esempio di questo problema è altresì rappresentato nel film-trilogia di fantascienza Ritorno al futuro: il viaggiatore nel tempo, impedendo ai suoi genitori d'incontrarsi, sarebbe dovuto scomparire dalla realtà in quanto mai nato. Questo tipo di paradosso è detto di "coerenza". Il paradosso fu ripreso anche in una puntata del cartone animato Futurama, creato da Matt Groening, quando il protagonista, Fry, viaggiando indietro nel tempo, uccide involontariamente suo nonno, ma continua a vivere in quanto ha messo incinta sua nonna, scoprendo così di essere sempre stato il nonno di se stesso. Una situazione d'incoerenza analoga a questo paradosso si verificherebbe qualora l'ipotetico viaggiatore nel tempo incontrasse se stesso in un momento in cui aveva un'età minore, così come viene citato anche nella trilogia di Ritorno al futuro. Paradosso di "conoscenza" (o del pittore o della Monna Lisa) [modifica \| modifica wikitesto] Una variante del paradosso di coerenza è quella proposta dal filosofo Michael Dummett; un critico d'arte torna nel passato al fine di conoscere quel che diventerà il più famoso pittore del futuro. Il viaggio riesce, il critico incontra il pittore, che però è molto giovane, e dipinge quadri in verità molto mediocri, ben lontani dai capolavori che realizzerà nel futuro. Il critico allora gli mostra le stampe, portate con sé nel viaggio, dei suoi futuri capolavori. Il pittore ne è talmente entusiasta che li copia. Nel frattempo, il critico d'arte si reimbarca nella macchina del tempo per tornare alla sua epoca e lascia le copie nel passato. La domanda è: considerando l'intera vicenda, da dove arriva, in definitiva, la conoscenza necessaria a creare i capolavori? Non può venire dal pittore perché la conoscenza non l'ha elaborata lui, ma l'ha appresa dal futuro. Non può venire dal critico d'arte perché egli a sua volta l'aveva semplicemente appresa dalle opere che il pittore avrebbe realizzato nel futuro, come conseguenza di quanto appreso dal critico. La profondità del paradosso è che, a tutti gli effetti, questa conoscenza sembra nascere dal nulla e senza una reale causa. Nella fantascienza il problema è ripreso nel film Terminator, con i suoi seguiti: il microchip che sta alla base tecnica degli androidi che vengono sviluppati è copiato da un androide che ha viaggiato nel tempo. Anche qui lo stesso problema del pittore: la conoscenza complessa e sofisticata presente nel chip innovativo nasce dal nulla, non prodotta da niente e nessuno. Il problema è riproposto nel racconto La scoperta di Morniel Mathaway di William Tenn e affrontato marginalmente anche nella trilogia di Ritorno al futuro: quando Marty (Michael J. Fox) alla fine del primo film suona la canzone Johnny B. Goode, un membro della band che assiste alla sua esibizione fa sentire al telefono la canzone al suo parente Chuck Berry, che diventerà l'autore del futuro brano. Il quarto episodio della nona stagione di Doctor Who si basa su questo paradosso, che viene anche citato come Paradosso di Dummet. Paradosso di predestinazione [modifica \| modifica wikitesto] Lo stesso argomento in dettaglio: Paradosso della predestinazione. A causa di una sorta di "legge naturale" legata alla predestinazione degli eventi stessi, un viaggio indietro nel tempo o le azioni del viaggiatore non potrebbero cambiare gli eventi. Paradosso di "co-esistenza" [modifica \| modifica wikitesto] Supponiamo, di nuovo, che il viaggio nel tempo sia possibile, e che un oggetto qualsiasi torni indietro nel tempo. Limitiamo l'infinita gamma di momenti passati in cui si potrebbe tornare soltanto a quelli in cui l'oggetto già esisteva. Dal punto di vista dell'universo, al momento di arrivo nel passato, la massa costituente l'oggetto comparirebbe praticamente dal nulla; in altre parole, la sua "copia ridondante" sarebbe dunque priva di passato. Ciò sembra inconcepibile, in quanto violerebbe molte delle leggi fisiche (oltre che logiche) esistenti. Bisogna tuttavia osservare che, se un corpo viaggia nel tempo, viene meno una quantità di massa e energia nel punto di partenza che, però, ricompare nel punto di arrivo. La massa non viene creata, c'è una trasformazione nello spaziotempo in cui si trova, ovvero un "semplice" cambio di coordinate. In questo caso, le leggi di conservazione di massa e la conservazione dell'energia sono rispettate, purché siano estese a quattro dimensioni, includendo quella temporale: non sono rispettate nelle tre dimensioni dello spazio di arrivo dove una massa sembra comparire dal nulla, mentre lo sono se si prendono lo spaziotempo di partenza e di arrivo. Un esempio di questo problema è rappresentato sempre nel film di fantascienza Ritorno al futuro Parte II: il 12 novembre 1955 si trovano contemporaneamente ben quattro macchine del tempo: la DeLorean al plutonio che riporta Marty nel 1985 la DeLorean volante guidata da Doc che, colpita da un fulmine, lo porta nel 1885, nel vecchio West la DeLorean danneggiata che Doc del 1985, intrappolato nel 1885, ha lasciato nel vecchio cimitero abbandonato dei pistoleri, la quale apparirà solo dopo che la DeLorean volante verrà colpita dal fulmine – e quindi mandata nel 1885 – a causa di un errore nei circuiti spaziotemporali la DeLorean volante guidata dal Biff del futuro che è tornato indietro nel tempo per dare al "se stesso" del 1955 un almanacco. Questo paradosso si fa ancora più intricato se coinvolge persone viventi. Ad esempio, in Ritorno al Futuro, Marty, nel tentativo di salvare Doc, anticipa il momento del suo rientro nel futuro. Riesce quindi a vedersi salire sulla DeLorean e dare quindi inizio al ciclo di eventi che egli conclude col suo ritorno. Se il Marty ritornato al futuro avesse impedito la partenza del Marty del presente, l'intera linea temporale non sarebbe mai esistita. Il paradosso di co-esistenza non è relativo al viaggio nel futuro: supponiamo che un uomo voglia vedersi nel futuro, e parte per il viaggio. La linea temporale di tutti gli eventi continua senza di lui, e quindi lui non si incontrerà mai, perché partito nel passato. A meno che non riesca perfettamente un viaggio di ritorno eseguito sulla stessa linea del tempo, che però implica un viaggio nel passato dopo averne effettuato uno nel futuro: dal presente si va ad un futuro remoto e da questo si torna indietro ad un tempo futuro meno remoto rispetto al presente, ma pur sempre passato dal punto di vista del tempo di partenza (il futuro remoto) del secondo viaggio. Questo tema viene affrontato nel film L'uomo che visse nel futuro (The Time Machine, 1960) di George Pal quando George, il viaggiatore del tempo, torna per un breve momento nella sua vecchia casa, alcune decine di anni dopo la sua partenza. Qui incontra James, il figlio del suo vecchio amico Filby, che racconta dell'amico del padre, partito tanti anni prima e mai più tornato. Anche qui la linea degli eventi è continuata senza il viaggiatore del tempo, del quale si ha solo il ricordo. Paradosso dell'infattibilità [modifica \| modifica wikitesto] In tale paradosso la situazione prevede che il viaggio nel tempo sia già stato effettuato, ma a causa della sua pericolosità, le persone che lo hanno effettuato hanno fatto in modo di non permettere in alcun modo la sua attuazione, a causa dei cambiamenti troppo pericolosi della realtà. L'ipotesi prevede che vi siano già stati fatti molteplici viaggi da tali soggetti. Tali viaggi sono stati fatti per non permettere la realizzazione di uno strumento che attui il viaggio temporale. Ovviamente in questo paradosso si ipotizza che il flusso del tempo sia univocamente unidirezionale e che non vi siano deviazioni in universi paralleli. Si ritorna dunque all'ipotesi dello scorrere dell'acqua del fiume. Questo paradosso non è risolvibile, a meno che non vi siano dei soggetti che potrebbero inventare uno strumento che permetta i viaggi nel tempo, diversi da chi lo ha inventato in "precedenza". Si può ancora ipotizzare che i nuovi inventori però abbiano fatto le stesse azioni dei precedenti nell'impedire i viaggi nel tempo oppure che gli sia stato impedito dai predecessori di attuare viaggi temporali e così via. Risoluzioni possibili dei paradossi [modifica \| modifica wikitesto] Protezione cronologica [modifica \| modifica wikitesto] Lo stesso argomento in dettaglio: Congettura di protezione cronologica. Alcuni scienziati, come i celebri Stephen Hawking e Roger Penrose, ritengono che, qualora tentassimo in qualche modo di fare qualcosa in grado di mutare significativamente il passato, ad impedirlo interverrebbe una sorta di censura cosmica. Si tratta di un'ipotesi strettamente correlata alla congettura di protezione cronologica, secondo cui le leggi della fisica sono tali da impedire la nascita di curve temporali chiuse, almeno su scale non sub-microscopiche. La stessa ipotesi fu avanzata dai fisici Kip Thorne e il premio Nobel 2004 David Politzer, i quali lasciarono aperta la possibilità di viaggi nel tempo in linee temporali chiuse, una dove il crononauta può modificare il passato, l'altra invece no. Ad esempio, nel "paradosso del nonno", potrebbe intervenire qualche meccanismo fisico ancora ignoto che, a protezione della catena degli eventi, impedirebbe l'intervento del nipote nel negare l'incontro con la nonna, affinché il nipote continui sempre ad esistere. Un esempio di questo problema è rappresentato dal film di fantascienza L'esercito delle 12 scimmie: nonostante fossero possibili i viaggi indietro nel tempo, non era possibile modificare il presente, in quanto tutto ciò che faceva il viaggiatore era già accaduto e documentato nella storia. Egli poteva soltanto raccogliere informazioni nel passato come un mero spettatore, e modificare il futuro agendo soltanto dal presente da cui proveniva. Tuttavia, le domande che sorgono partendo dalla censura cosmica sono: che ne sarebbe del potere di decisione di ognuno, del suo libero arbitrio? E poi in che modo questa cosiddetta "censura" agirebbe? Come farebbe l'universo, in modo del tutto razionale, ad "accorgersi" che qualcosa non va, che ci sarebbe il rischio che un piccolo crono-vandalo provochi seri guai alla storia futura? L'argomento è ulteriormente trattato nella serie televisiva Lost. In essa, i personaggi riescono a tornare indietro nel tempo, e Jack, uno di essi, cerca di cambiare il futuro facendo esplodere una bomba a idrogeno. Non ci è dato di sapere se egli riesce a cambiare lo scorrere degli eventi. È assumibile però, che lui sia già parte integrante del passato, considerato che altri personaggi hanno tentato di cambiare il passato, ma hanno constatato che il fatto di tornare nel passato era già contemplato nel passato stesso. Questo, comunque, comporta un gravoso paradosso che è riassumibile nella domanda: "qual è stato il primo Jack che ha deciso di tornare nel passato?" Infatti, dato che nel suo passato il suo io-futuro è già presente, non si riesce a discriminare il primo Jack che decide di cambiare lo scorrere degli eventi. Un ulteriore esempio lo si ha nel videogioco picchiaduro Tekken 5. Nel suo video conclusivo, la protagonista Ling Xiaoyu utilizza una macchina del tempo con l'intento di impedire a Heihachi Mishima di gettare il figlio Kazuya nel cratere di un vulcano; l'unico risultato che ottiene è, tuttavia, quello di restare nel suo tempo, mentre la macchina del tempo "parte" senza di lei e colpisce Heihachi, facendogli cadere di mano Kazuya proprio nel vulcano. Esistenza di mondi paralleli [modifica \| modifica wikitesto] Lo stesso argomento in dettaglio: Dimensione parallela. Relativamente opposta all'ipotesi della censura cosmica, fu avanzata la teoria quantistica nota come "teoria a molti mondi", proposta nel 1956 da Hugh Everett III e successivamente riadattata da David Deutsch nel 1998. La teoria ipotizza tante copie del nostro mondo quante sono le possibili variazioni quantistiche delle particelle che lo compongono. Nell'ipotetico paradosso del nonno, ci saranno mondi in cui il nonno si sposa con la nonna, e mondi in cui questo fatto non avviene più, pertanto, se un ipotetico viaggio nel passato si impedisse a nostro nonno di incontrare nostra nonna, si approderebbe in un mondo parallelo nel quale noi stessi non siamo mai nati. Limitazioni a questa teoria è che, in questo caso, ci si sposterà soltanto tra dimensioni parallele, e non nel tempo come lo si concepisce. Inoltre, rimane da spiegare quale sia il principio di carattere generale che ci permetta di scegliere "questo universo"; in questo caso, però, sia il libero arbitrio che il principio di causalità sono salvi, anche se le varianti possibili sarebbero potenzialmente infinite. Il viaggio nel tempo nella fantasia [modifica \| modifica wikitesto] Lo stesso argomento in dettaglio: Viaggio nel tempo nella fantascienza. Il viaggio temporale ha da sempre affascinato l'umanità, presentandosi in molti miti, come ad esempio in quello di mago Merlino, che sperimenta delle regressioni temporali. Il tema, benché presente già in precedenza in varie opere fantastiche, venne reso popolare dal romanzo La macchina del tempo di H. G. Wells del 1895, riconosciuto come un classico, in cui il protagonista viaggia nel remoto futuro alla scoperta del destino dell'umanità. Altri racconti simili furono proposti da Dickens, George Pal, Mark Twain, Audrey Niffenegger, Isaac Asimov. Il viaggio nel tempo rimane un tema tipico della fantascienza, tanto che alcuni lo considerano un vero e proprio sottogenere, ma è presente anche nel fantasy e nei racconti fantastici. Un meccanismo narrativo spesso utilizzato è quello di portare un personaggio in un particolare tempo a cui non appartiene, ed esplorare le possibili ramificazioni dell'interazione del personaggio con le persone e la tecnologia dell'epoca (una derivazione del campagnolo che va nella grande città, o viceversa). Questo espediente narrativo si è evoluto per esplorare le idee di cambiamento e le reazioni a esso, e anche per esplorare le idee di universi paralleli o ucronia dove alcuni piccoli eventi avvengono, o non avvengono, ma causano massicci cambiamenti nel futuro (a causa tipicamente dell'effetto farfalla). Il concetto di viaggio nel tempo applicato alla letteratura e alla sceneggiatura consente di sviluppare trame particolarmente elaborate e avvincenti, con elementi ricorsivi, possibilità di analizzare evoluzioni parallele di un evento e colpi di scena estremi, come la riapparizione di personaggi scomparsi. Viaggi nel tempo e leggende metropolitane [modifica \| modifica wikitesto] Lo stesso argomento in dettaglio: Viaggi nel tempo e leggende metropolitane. Note [modifica \| modifica wikitesto] ^ Il legame tra il tempo e lo spazio: lo spaziotempo e la relatività generale, su manuelmarangoni.it. URL consultato il 24 aprile 2019. ^ ^ a b ^ ^ Mi spiegate l'esperimento dei muoni scoperti e la dilatazione dei tempi?, su scienzapertutti.lnf.infn.it. URL consultato il 24 aprile 2019 (archiviato dall'url originale il 4 maggio 2017). ^ ^ Il tempo? Ora sappiamo che non esiste, su l'Espresso, 23 ottobre 2014. URL consultato il 24 aprile 2019. ^ Chi dice che il tempo scorre?, su Il Sole 24 ORE. URL consultato il 24 aprile 2019. ^ l'Universo e l'Uomo, su luniversoeluomo.blogspot.com. URL consultato il 24 aprile 2019. ^ I viaggi nel tempo e il paradosso del nonno, su Le Scienze. URL consultato il 24 aprile 2019. ^ I metamateriali dimostrano che non è possibile viaggiare nel tempo passato, su manuelmarangoni.it. URL consultato il 24 aprile 2019. ^ (EN) Yu-Ju Hung e Igor I. Smolyaninov, Modeling of Time with Metamaterials, 4 aprile 2011, DOI:10.1364/JOSAB.28.001591, ISSN 0740-3224 (WC · ACNP). URL consultato il 24 aprile 2019. ^ John Richard Gott III, Viaggiare nel tempo: la possibilità fisica di spostarsi nel passato e nel futuro, Mondadori, Milano 2002, traduzione di Tullio Cannillo. ^ Il Pesa-Nervi. Ipotesi sulla manipolazione dello spazio-tempo Archiviato il 26 ottobre 2011 in Internet Archive. ^ In generale, quale concetto filosofico e culturale e nella prospettiva vetero- e neo-testamentaria, cristiana e protestante: predestinazione, su treccani.it. URL consultato il 14 dicembre 2023.; predestinazione, su treccani.it. URL consultato il 14 dicembre 2023.; Predestinazione, su it.cathopedia.org. URL consultato il 14 dicembre 2023.. Bibliografia [modifica \| modifica wikitesto] Bibliografia scientifica Marcus Chown, The Universe Next Door, Londra, 2003. Paul Davies, About Time, 1995. Paul Davies, Come costruire una macchina del tempo(How to Build a Time Machine), Milano, Mondadori, 2003. David Deutsch e Franck Lockwood, La fisica quantistica del viaggio nel tempo, in Le Scienze, n. 309, maggio 1994. J. Richard Gott, Time Travel in Einstein's Universe: The Physical Possibilities of Travel Through Time, 2002. Viaggiare nel tempo: La possibilità fisica di spostarsi nel passato e nel futuro, Milano, Arnoldo Mondadori Editore, 2002. Ronald Mallett, Time Traveler: A Scientist's Personal Mission to Make Time Travel a Reality, Thunder's Mouth Press, 2006. Paul J. Nahin, Time Machines: Time Travel in Physics, Metaphysics, and Science Fiction, 1993. Clifford A. Pickover, Time: A Traveler's Guide, 1999. Frank J. Tipler, Rotating Cylinders and the Possibility of Global Causality Violation, in Physical Review D 9 (1974), 2003. Vatinno G., Storia naturale del Tempo. L' "Effetto Einstein" e la Teoria della Relatività, Armando editore, 2014. Bibliografia letteraria Renato Giovannoli, Capitoli VI-VII, in La scienza della fantascienza, Bompiani, 1991. Voci correlate [modifica \| modifica wikitesto] Macchina del tempo Curva chiusa di tipo tempo Dilatazione del tempo Dimensione parallela Anacronismo Freccia del tempo Geometria iperbolica Paradosso temporale Pseudoscienza John Titor OOPArt Vadim Aleksandrovič Černobrov Viaggiatori extratemporali Viaggio nel tempo nella fantascienza Viaggi nel tempo e leggende metropolitane Altri progetti [modifica \| modifica wikitesto] Altri progetti Wikiquote Wikimedia Commons Wikiquote contiene citazioni di o su viaggio nel tempo Wikimedia Commons contiene immagini o altri file su viaggio nel tempo Collegamenti esterni [modifica \| modifica wikitesto] (EN) time travel, su Enciclopedia Britannica, Encyclopædia Britannica, Inc. (EN) Viaggio nel tempo, su Internet Encyclopedia of Philosophy. (EN) Viaggio nel tempo / Viaggio nel tempo (altra versione), su Stanford Encyclopedia of Philosophy. (EN) Opere riguardanti Viaggio nel tempo, su Open Library, Internet Archive. (EN) Time Travel, su The Visual Novel Database. (EN) Viaggio nel tempo, su Comic Vine, Fandom. Siti accademici (EN) John Gribbin, La teoria del viaggio nel tempo, su Università del Sussex. URL consultato il 13 maggio 2006 (archiviato dall'url originale il 17 giugno 2006). Siti divulgativi (EN) Time, Time Travel & Traversable Wormholes Archiviato il 5 febbraio 2024 in Internet Archive. and other time travel related; science & technology Archiviato il 2 dicembre 2024 in Internet Archive. topics (EN) On the Net: Time Travel by James Patrick Kelly in Asimov's Science Fiction (EN) Howstuffworks' article on "How Time Travel Will Work", su science.howstuffworks.com. (EN) NOVA Online: Time Travel, su pbs.org. (EN) Time Travel in Flatland?, su theory.caltech.edu. URL consultato il 22 marzo 2005 (archiviato dall'url originale il 19 marzo 2005). Buchi neri e Macchine del Tempo su Intercom Pseudoscienza Chernobrov, un ricercatore russo che asserisce di avere costruito una macchina del tempo, su fisicamente.it. URL consultato il 23 gennaio 2006 (archiviato dall'url originale l'8 maggio 2006). \| V · D · M Tempo \| \| Concetti principali \| Tempo· Eternità· Immortalità Tempo profondo· Storia· Passato· Presente· Futuro· Futurologia \| \| \| Misure e standard \| Cronometria· UTC· UT· TAI· Secondo· Minuto· ora· Tempo siderale· Tempo solare· Fuso orario Orologio· Orologeria· Storia della misurazione del tempo· Astrario· Cronometro marino· Meridiana· Clessidra (ad acqua)· Orologio a incenso· Cronometro Calendario· Giorno· Settimana· Mese· Anno· Anno tropico· Giuliano· Gregoriano· Islamico Intercalazione· Secondo intercalare· Anno bisestile \| \| Cronologia \| Cronologia astronomica· Tempo geologico· Storia geologica· Geocronologia· Datazione archeologica Era del calendario· Anno di regno· Cronaca· Periodizzazione \| \| Religione e mitologia \| Chronos· Decani· Eone· Kairós· Kāla· Kālacakratantra· Padre Tempo· Profezia· Ruota del tempo· Tempo del Sogno· Zeitgeist \| \| Filosofia \| Causalità· Eterno ritorno· Evento· Kalpa· Presentismo· Sincronicità· Tempo ciclico· Tempo storico· Yuga \| \| Scienze fisiche \| Tempo in fisica· Spaziotempo· Tempo assoluto· Simmetria temporale Freccia del tempo· Chronon· Quarta dimensione· Epoca di Planck· Tempo di Planck· Dominio del tempo· Dimensioni temporali multiple Teoria della relatività· Dilatazione del tempo· Dilatazione gravitazionale del tempo· Tempo coordinato· Tempo proprio \| \| Biologia \| Cronobiologia· Ritmi circadiani· Tempo di reazione \| \| Sociologia e antropologia \| Cronemica· Futurologia· Long Now Foundation· \| \| Scienze economiche \| Tempo newtoniano nelle scienze economiche· Valore tempo del denaro· Tempo bancario· Banca del tempo· Ora legale \| \| Argomenti correlati \| Spazio· Durata· Capsula del tempo· Viaggio nel tempo· Misura (musica)· Tempo di sistema· Tempo metrico· Tempo esadecimale· Carpe diem· Tempus fugit \| \| \| \| --- \| \| Controllo di autorità \| LCCN (EN) sh88005014 · GND (DE) 4190617-2 · J9U (EN, HE) 987007529956905171 \| Portale Fantascienza Portale Fisica Estratto da " Categoria: Viaggio nel tempo Categorie nascoste: Template Webarchive - collegamenti all'Internet Archive P1417 letta da Wikidata P5088 letta da Wikidata P3123 multipla letta da Wikidata senza qualificatore P3847 letta da Wikidata Voci con template Collegamenti esterni e qualificatori sconosciuti P3180 letta da Wikidata P5905 letta da Wikidata Voci con codice LCCN Voci con codice GND Voci con codice J9U Voci non biografiche con codici di controllo di autorità Viaggio nel tempo Aggiungi argomento
2832	https://mathworld.wolfram.com/UnitVector.html	Unit Vector -- from Wolfram MathWorld TOPICS AlgebraApplied MathematicsCalculus and AnalysisDiscrete MathematicsFoundations of MathematicsGeometryHistory and TerminologyNumber TheoryProbability and StatisticsRecreational MathematicsTopologyAlphabetical IndexNew in MathWorld Algebra Vector Algebra Unit Vector A unit vector is a vector of length 1, sometimes also called a direction vector (Jeffreys and Jeffreys 1988). The unit vector having the same direction as a given (nonzero) vector is defined by where denotes the norm of , is the unit vector in the same direction as the (finite) vector. A unit vector in the direction is given by where is the radius vector. When considered as the th basis vector of a vector space, a unit vector may be written (or ). See also Direction, Norm, Radius Vector, Vector, Zero Vector Explore with Wolfram\|Alpha More things to try: vector algebra normalize (1,1) normalize ((10.5, 3., 0.1) cross (0.02, 20.3, 0.1)) References Jeffreys, H. and Jeffreys, B.S. "Direction Vectors." §2.034 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, p.64, 1988.Stephens, M.A. "The Testing of Unit Vectors for Randomness." J. Amer. Stat. Assoc.59, 160-167, 1964. Referenced on Wolfram\|Alpha Unit Vector Cite this as: Weisstein, Eric W. "Unit Vector." From MathWorld--A Wolfram Resource. Subject classifications Algebra Vector Algebra About MathWorld MathWorld Classroom Contribute MathWorld Book wolfram.com 13,278 Entries Last Updated: Sun Sep 28 2025 ©1999–2025 Wolfram Research, Inc. Terms of Use wolfram.com Wolfram for Education Created, developed and nurtured by Eric Weisstein at Wolfram Research Created, developed and nurtured by Eric Weisstein at Wolfram Research
2833	https://www.quora.com/What-is-the-difference-between-quartile-exc-and-quartile-inc	What is the difference between quartile.exc() and quartile.inc()? - Quora Something went wrong. Wait a moment and try again. Try again Skip to content Skip to search Sign In Data Analysis Common Excel Functions Quartile.inc () Data Science Microsoft Excel Percentile and Quartile Worksheet-function Statistical Analysis Excel Formulas 5 What is the difference between quartile.exc() and quartile.inc()? All related (36) Sort Recommended Assistant Bot · 1y The functions QUARTILE.EXC() and QUARTILE.INC() in Excel (or similar spreadsheet applications) are used to calculate quartiles, which are values that divide a dataset into four equal parts. The main difference between these two functions lies in how they handle the calculation of quartiles: : Definition: Calculates the quartile based on exclusive methods, meaning it excludes the lowest and highest values in the dataset when determining quartiles. Quartile Values: quart = 1: First quartile (25th percentile) quart = 2: Second quartile (50th percentile, or median) quart = 3: Third quartile Continue Reading The functions QUARTILE.EXC() and QUARTILE.INC() in Excel (or similar spreadsheet applications) are used to calculate quartiles, which are values that divide a dataset into four equal parts. The main difference between these two functions lies in how they handle the calculation of quartiles: : Definition: Calculates the quartile based on exclusive methods, meaning it excludes the lowest and highest values in the dataset when determining quartiles. Quartile Values: quart = 1: First quartile (25th percentile) quart = 2: Second quartile (50th percentile, or median) quart = 3: Third quartile (75th percentile) Usage: This function is used when you want a more statistical approach that does not include the extremes of the dataset in the calculation. : Definition: Calculates the quartile based on inclusive methods, meaning it includes all values in the dataset to determine quartiles. Quartile Values: quart = 1: First quartile (25th percentile) quart = 2: Second quartile (50th percentile, or median) quart = 3: Third quartile (75th percentile) Usage: This function is used when you want to include all data points, providing a more comprehensive view of the dataset. Summary Use for exclusive quartile calculations (excludes extremes). Use for inclusive quartile calculations (includes all data points). Example If you have the dataset {1, 2, 3, 4, 5, 6, 7, 8, 9}: QUARTILE.EXC() would calculate quartiles excluding the lowest (1) and highest (9). QUARTILE.INC() would calculate quartiles including all values from 1 to 9. This distinction can lead to different quartile values depending on the dataset and the method used. Your response is private Was this worth your time? This helps us sort answers on the page. Absolutely not Definitely yes Upvote · Related questions More answers below What is the difference between quartile EXC and quartile Inc. in Excel? How do I find quartile 1 in Excel? What is quartile retirement? What is the difference between quartile vs percentile? What is the right way to calculate quartiles? Ly Tran Lives in Hanoi, Vietnam (1994–present) ·7y For discrete distributions, there is no universal agreement on selecting the quartile values. Method 1(inclusive) Use the median to divide the ordered data set into two halves. If there are an odd number of data points in the original ordered data set, do not include the median (the central value in the ordered list) in either half. If there are an even number of data points in the original ordered data set, split this data set exactly in half. The lower quartile value is the median of the lower half of the data. The upper quartile value is the median of the upper half of the data. This rule is empl Continue Reading For discrete distributions, there is no universal agreement on selecting the quartile values. Method 1(inclusive) Use the median to divide the ordered data set into two halves. If there are an odd number of data points in the original ordered data set, do not include the median (the central value in the ordered list) in either half. If there are an even number of data points in the original ordered data set, split this data set exactly in half. The lower quartile value is the median of the lower half of the data. The upper quartile value is the median of the upper half of the data. This rule is employed by the TI-83 calculator boxplot and "1-Var Stats" functions. In excel, this method was implemented by QUARTILE.INC function. Syntax: QUARTILE.INC(array, quart) The QUARTILE.INC function syntax has the following arguments: Array: The array or cell range of numeric values for which you want the quartile value. Quart: Indicates which value to return. Method 2(exclusive) Use the median to divide the ordered data set into two halves. If there are an odd number of data points in the original ordered data set, include the median (the central value in the ordered list) in both halves. If there are an even number of data points in the original ordered data set, split this data set exactly in half. The lower quartile value is the median of the lower half of the data. The upper quartile value is the median of the upper half of the data. The values found by this method are also known as "Tukey's hinges". In excel, this method was implemented by QUARTILE.EXC function. Syntax: QUARTILE.EXC(array, quart) The QUARTILE.EXC function syntax has the following arguments: Array: The array or cell range of numeric values for which you want the quartile value. Quart: Indicates which value to return. Upvote · 99 16 9 5 Sponsored by Grammarly 92% of professionals who use Grammarly say it has saved them time Work faster with AI, while ensuring your writing always makes the right impression. Download 999 210 Ly Tran Lives in Hanoi, Vietnam (1994–present) ·7y Originally Answered: What is the difference between the percentile.inc() and percentile.exc() function in Excel and which one should I use? · The syntax is PERCENTILE.EXC(array, k) and PERCENTILE.INC(array, k) Array: The array or range of data that defines relative standing. K: The percentile value in the range 0..1, exclusive. PERCENTILE.EXC function returns the k-th percentile of values in a range, where k is in the range 0..1, exclusive. PERCENTILE.EXC only works if k is between 1/n and 1-1/n, where n is the number of elements in an array. PERCENTILE.INC function returns the k-th percentile of values in a range, where k is in the range 0..1, inclusive. PERCENTILE.INC uses a slightly less accurate algorithm, but it works for any value Continue Reading The syntax is PERCENTILE.EXC(array, k) and PERCENTILE.INC(array, k) Array: The array or range of data that defines relative standing. K: The percentile value in the range 0..1, exclusive. PERCENTILE.EXC function returns the k-th percentile of values in a range, where k is in the range 0..1, exclusive. PERCENTILE.EXC only works if k is between 1/n and 1-1/n, where n is the number of elements in an array. PERCENTILE.INC function returns the k-th percentile of values in a range, where k is in the range 0..1, inclusive. PERCENTILE.INC uses a slightly less accurate algorithm, but it works for any value of k between 0 and 1. The traditional statistical definition of the k-th percentile of values in a range: it is the (interpolated) data point below which k% of the data lie. With that definition, we cannot ask "where is the 100th percentile?" because there is no data point below which 100% of the data lie. (Alternatively, there are infinite data points below which 100% of the data lie, namely any value above the highest value.). PERCENTILE.EXC tries to fit that traditional statistical definition. Upvote · 99 11 Ranjan Choudhary Learner. Optimist. Fighter. Believer. Comic · Author has 55 answers and 459K answer views ·8y Originally Answered: What is the difference between the percentile.inc() and percentile.exc() function in Excel and which one should I use? · From name it appears as inc means inclusive and exc means exclusive but that is not the case. It is completely different from inclusive and exclusive because both of these formula exclude one number from your series. inc includes first number of the series and excludes last number. Exc excludes first number of series and includes last number. Eg you have series of 0 to 11 numbers i.e. total 12 numbers then inc will use 0 to 10 ie total 11 numbers and similarly exc will also use 11 numbers but those will be 1 to 11. I hope this clarifies your doubt in simple manner. If you require more explanation vi Continue Reading From name it appears as inc means inclusive and exc means exclusive but that is not the case. It is completely different from inclusive and exclusive because both of these formula exclude one number from your series. inc includes first number of the series and excludes last number. Exc excludes first number of series and includes last number. Eg you have series of 0 to 11 numbers i.e. total 12 numbers then inc will use 0 to 10 ie total 11 numbers and similarly exc will also use 11 numbers but those will be 1 to 11. I hope this clarifies your doubt in simple manner. If you require more explanation visit Microsoft official page. Take care! Upvote · 9 9 Related questions More answers below How do you find quartiles in grouped data? In Excel, what is the difference between quart.Inc and quart.ext? What is the difference between the lower quartile and the upper quartile? What does the lower and upper quartile mean? How can I calculate quartiles in Excel? Prashant Rai Studied at National Institute of Technology, Rourkela ·9y Originally Answered: What is the difference between the percentile.inc() and percentile.exc() function in Excel and which one should I use? · Both PERCENTILE.EXC and PERCENTILE.INC (same as the original PERCENTILE) first rank the N values of the data Array argument from 1 (lowest value) to N (highest), then determine the possibly-non-integer calculated rank for the specified percentage argument K (a decimal number between 0.00 and 1.00), and finally use linear interpolation between the closest integer-rank values of the data array. PERCENTILE.EXC and PERCENTILE.INC differ only in the way the possibly-non-integer rank is calculated. For PERCENTILE.INC (and PERCENTILE) the calculated rank is K(N-1)+1. For PERCENTILE.EXC the calculated Continue Reading Both PERCENTILE.EXC and PERCENTILE.INC (same as the original PERCENTILE) first rank the N values of the data Array argument from 1 (lowest value) to N (highest), then determine the possibly-non-integer calculated rank for the specified percentage argument K (a decimal number between 0.00 and 1.00), and finally use linear interpolation between the closest integer-rank values of the data array. PERCENTILE.EXC and PERCENTILE.INC differ only in the way the possibly-non-integer rank is calculated. For PERCENTILE.INC (and PERCENTILE) the calculated rank is K(N-1)+1. For PERCENTILE.EXC the calculated rank is K(N+1). (For more details about the calculated rank, refer to the "Alternative methods" section of the Wikipedia entry.) The functions then use linear interpolation to determine the value of the percentile function. (For more details about the interpolation, refer to the "Linear interpolation between closest ranks" section of the Wikipedia entry.) For a very small data set you might choose to use PERCENTILE.INC (or PERCENTILE) instead of PERCENTILE.EXC because PERCENTILE.EXC returns the #NUM! error value for values of K <= 1/(N+1) and for values of K >= N/(N+1). Upvote · 99 17 9 1 Sponsored by JetBrains JetBrains Startup Program. Save 50% on the software development tools including IntelliJ IDEA, WebStorm, PhpStorm and more! Apply Now 999 422 Jim Kenyon I live in Ann Arbor · Author has 547 answers and 917.9K answer views ·11y Originally Answered: What is the difference between the percentile.inc() and percentile.exc() function in Excel and which one should I use? · I'd summarize the answer by re-typing it but it would only be a full cut-n-paste of PERCENTILE.INC and PERCENTILE.EXC - difference -- this link is short and to the point. Upvote · 9 1 Brad Yundt Mechanical engineer and Excel aficionado · Author has 3.6K answers and 18.4M answer views ·4y Related In Excel, what is the difference between quart.Inc and quart.ext? The old QUARTILE and new QUARTILE.INC functions are supposed to produce the same answers. They differ from QUARTILE.EXC in how many “segments” the data are divided into before interpolating to find the 25%, 50% and 75% percentile breakpoints. QUARTILE.EXC returns #NUM! error value when asked for the 0% and 100% quartiles, while QUARTILE.INC returns the MIN and MAX respectively. Assuming n data points arranged in ascending order, QUARTILE.INC uses n-1 segments while QUARTILE.EXC uses n+1. You may find the number line illustrations in Microsoft Excel MVP Jon Peltier webpage on the QUARTILE functio Continue Reading The old QUARTILE and new QUARTILE.INC functions are supposed to produce the same answers. They differ from QUARTILE.EXC in how many “segments” the data are divided into before interpolating to find the 25%, 50% and 75% percentile breakpoints. QUARTILE.EXC returns #NUM! error value when asked for the 0% and 100% quartiles, while QUARTILE.INC returns the MIN and MAX respectively. Assuming n data points arranged in ascending order, QUARTILE.INC uses n-1 segments while QUARTILE.EXC uses n+1. You may find the number line illustrations in Microsoft Excel MVP Jon Peltier webpage on the QUARTILE function to be helpful in explaining the difference. Excel’s QUARTILE.EXC function should give you the same answers as statistics software packages like JPM, Minitab and SAS. Upvote · 9 1 9 2 Sponsored by EGNITION Automatically sort and organize large inventory on Shopify. This Shopify app will organize products in collections automatically based on flexible sorting rules. Get the App 2.5K 2.5K Murali Krishna Former Retired Senior Lecturer in DIET at Government (General) (1990–2004) · Author has 6.5K answers and 6.7M answer views ·5y Related How do you find the 1st and 3rd quartile of a data set? Q1 is the MEDIAN (the middle) of the lower half Of the data And Q3 is the median(middle) of the upper half of the data X (3 5 7 8 9 )( 11 15 16 20 21 ) Q1=7 Q3=16 Q3-Q1 =16-7=9 INTER QUARTILE RANGE is Q3-Q1 =9 Upvote · 99 10 9 1 Harshal SG Lived in Mumbai, Maharashtra, India ·5y Related In statistics, what is the difference between a quartile and a quantile? Quantile is something which divides the dataset into equal parts. A quantile which divides the dataset into 4 parts is a quartile ie at 0.25, 0.5 , 0.75 . Quartile is a type of Quantile. There are diff types of quantiles. Please look for the image given below: Image Source : Wikipedia Thanks! Continue Reading Quantile is something which divides the dataset into equal parts. A quantile which divides the dataset into 4 parts is a quartile ie at 0.25, 0.5 , 0.75 . Quartile is a type of Quantile. There are diff types of quantiles. Please look for the image given below: Image Source : Wikipedia Thanks! Upvote · 9 9 Sponsored by H&R Block Late on Expat taxes? Avail Amnesty Now. US expats: Amnesty ending soon! Catch up on taxes and avoid heavy penalties now. Sign Up 99 26 Shoumi Author has 112 answers and 713.3K answer views ·8y Related What is quartile deviation? Quartile deviation, also known as semi-interquartile range, is a measure of scatteredness (dispersion) of your data points - that is, whether the observations are densely clustered around a central value or are far apart from each other on an average. It is half the difference between the third and first quartiles. Quartile Deviation = Q 3−Q 1 2 Q 3−Q 1 2 Quartile deviation prevents outliers from overly inflating the actual scatteredness of your data points. Say you ask someone to make a note of the daily average temperature (in °C) of a given city, over a certain week. Suppose they give you this Continue Reading Quartile deviation, also known as semi-interquartile range, is a measure of scatteredness (dispersion) of your data points - that is, whether the observations are densely clustered around a central value or are far apart from each other on an average. It is half the difference between the third and first quartiles. Quartile Deviation = Q 3−Q 1 2 Q 3−Q 1 2 Quartile deviation prevents outliers from overly inflating the actual scatteredness of your data points. Say you ask someone to make a note of the daily average temperature (in °C) of a given city, over a certain week. Suppose they give you this: 23.4 21.5 22.4 23.6 21.5 23.7 32.4 All the readings are in the 21 - 23 range, except for the absurd 32.4. Something just doesn't look right in the data, does it? The temperature isn't suppose to fluctuate so widely in a span of a single week. Perhaps the person typed in the digits of those observations in a wrong order (that is, maybe he typed 32.4 instead of 23.4). Or maybe he didn't make any copying errors on his part, perhaps his source of information has misquoted the temperature. Or maybe the temperature actually was like that? You have no idea what happened, but you suspect that 32.4 is a suspicious value. This is where the quartiles come in. Instead of considering the highest and lowest values in your dataset to arrive at a measure of dispersion, you consider the bulk of the observations sandwiched somewhere in between them (because the highest value can be abnormally high if wrongly reported; similarly, the lowest value can be abnormally low if it were wrongly reported in the first place). You arrange your observations in increasing (or decreasing) order: 21.5 21.5 22.4 23.4 23.6 23.7 32.4 You identify the quartiles. 21.5 21.5 22.4 23.4 23.6 23.7 32.4 Here Q 1 Q 1 = 21.5, Q 2 Q 2=23.4, Q 3 Q 3=23.7 You work with only the data within Q 1 Q 1 and Q 3 Q 3 (both inclusive). 21.5 22.4 23.4 23.6 23.7 Now the data taken from the middle doesn't contain any blatantly absurd observations! This the essence behind using such a measure for dispersion based on the difference of the third and first quantiles of the original data. The Quartile Deviation this case is 23.7−21.5 2=1.1 23.7−21.5 2=1.1 Upvote · 99 23 9 2 Roger Asdale Author has 2.3K answers and 327.9K answer views ·2y Related How do you calculate the interquartile range (IQR), first quartile (Q1), and third quartile (Q3) in a set of data? The interquartile range indicates spread of data in the middle half of a statistical distribution. Quartiles segmenting any distribution are ordered from low to high in 4 equal parts. The interquartile range (IQR) contains the 2nd and 3rd quartiles, or the middle half, of a data set. Formulas for Quartiles (N items in data set): Lower Quartile (Q1) = (N+1) / 4. Middle Quartile (Q2) = (N+1) / 2. Upper Quartile (Q3 )= (N+1) 3 / 4. Interquartile Range = Q3 – Q1. Upvote · 9 2 Gopal Menon B Tech in Chemical Engineering, Indian Institute of Technology, Bombay (IITB) (Graduated 1975) · Author has 10.2K answers and 15.2M answer views ·7y Related What is meant by quartile and formula? Quartiles divide the population, or data set into four equal parts, when the elements of the population or data set are kept in an ascending order. The first (or lower) quartile has 25%25% of the population below it. The second quartile has 50%50% of the population below it. It is also called as the median. The third (or upper) quartile has 75%75% of the population below it. The value of the i t h i t h quartile is Q i=value of \,(i(n+1)4)t h Q i=value of \,(i(n+1)4)t h observation. Upvote · 9 2 Leela Subramanian Phd in Statistics (academic discipline), University of Mumbai (Graduated 2018) · Author has 408 answers and 847.2K answer views ·7y Related What is the difference between percentile quartile and median? Percentiles are 99 points that divide any data set or distribution into 100 equal parts. These are denoted as P1, P2, …P99. Thus 1% of the observations, 2% of the observations,….99% of the observations are less than or equal to P1, P2,…P99 respectively. Quartiles Q1, Q2, Q3 are 3 points that divide the distribution into 4 equal parts. Thus 25%, 50% and 75% of the observations are less than or equal to Q1, Q2 and Q3 respectively. It can be seen that the first quartile is same as the 25th percentile, second quartile is same as the 50th percentile while the third quartile is same as the 75th perce Continue Reading Percentiles are 99 points that divide any data set or distribution into 100 equal parts. These are denoted as P1, P2, …P99. Thus 1% of the observations, 2% of the observations,….99% of the observations are less than or equal to P1, P2,…P99 respectively. Quartiles Q1, Q2, Q3 are 3 points that divide the distribution into 4 equal parts. Thus 25%, 50% and 75% of the observations are less than or equal to Q1, Q2 and Q3 respectively. It can be seen that the first quartile is same as the 25th percentile, second quartile is same as the 50th percentile while the third quartile is same as the 75th percentile. Median on the other hand divides the distribution into two equal parts. Thus it coincides with Q2 and P50. It may be noted that the percentiles, median and quartiles are positional measures. Median in particular is also a measure of location and is used instead of the arithmetic mean when there are extreme values in the data or in the case of open ended classes. Upvote · 99 15 Related questions What is the difference between quartile EXC and quartile Inc. in Excel? How do I find quartile 1 in Excel? What is quartile retirement? What is the difference between quartile vs percentile? What is the right way to calculate quartiles? How do you find quartiles in grouped data? In Excel, what is the difference between quart.Inc and quart.ext? What is the difference between the lower quartile and the upper quartile? What does the lower and upper quartile mean? How can I calculate quartiles in Excel? What is the difference between a quartile and a quadrillion? What does quartiles mean? What is the 1st quartile, median and 3rd quartile of the following? What is an upper and lower quartile? How do you break data into quartiles? Related questions What is the difference between quartile EXC and quartile Inc. in Excel? How do I find quartile 1 in Excel? What is quartile retirement? What is the difference between quartile vs percentile? What is the right way to calculate quartiles? How do you find quartiles in grouped data? 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2834	https://www.rarediseaseadvisor.com/disease-info-pages/systemic-sclerosis-guidelines/	Systemic Sclerosis (SSc) Guidelines Systemic sclerosis (SSc) is a heterogeneous disorder whose pathogenesis is characterized by 3 features: small-vessel vasculopathy, autoantibody production, and dysfunction of fibroblasts resulting in increased deposition of extracellular matrix. The clinical signs and symptoms and the prognosis of SSc are variable, with skin thickening and internal organ involvement noted in many patients. Various types of SSc include limited cutaneous SSc, diffuse cutaneous SSc, and SSc without skin involvement.1 ACR-EULAR Classification Criteria To improve the sensitivity and specificity of the 1980 criteria and reflect advancements in the knowledge of SSc, the American College of Rheumatology and European League Against Rheumatism (ACR-EULAR) created a joint proposal for new classification criteria in 2013. The objectives were to create criteria that (1) encompass a wider range of cases of SSc, including patients in both early and late stages of the disease process; (2) include vascular, immunologic, and fibrotic manifestations; (3) are practical to use in routine clinical practice; and (4) are consistent with the criteria currently used for diagnosing SSc in clinical practice. Rheumatologists, researchers, national and international drug authorities, pharmaceutical companies, and others interested in SSc studies can use these criteria for diagnosis.1 The ACR-EULAR classification criteria established that SSc can be diagnosed if thickening of the skin of the fingers extends proximal to the metacarpophalangeal joints. Absent this finding, the presence of the following 7 features should be noted and scored: thickening of the skin of the fingers, lesions on the fingertips, telangiectasia, abnormal nail fold capillaries, interstitial lung disease or pulmonary arterial hypertension, Raynaud phenomenon, and SSc-related autoantibodies. Patients with a minimum score of 9 are classified with definite SSc.1,2 ACR-EULAR Criteria for the Classification of Systemic Sclerosis3 \| Classify a patient as having systemic sclerosis if the sum of points for the criteria below is ≥9. \| Criterion \| Points \| Skin thickened on the fingers of both hands, extending proximal to the metacarpophalangeal joints \| 9 1 \| Skin on fingers thickened (count only the highest score)Puffy fingersSclerodactyly \| 24 2 \| Lesions on fingertips (count only the highest score)Ulcers on tip of digitsPitting scars on fingertips \| 23 3 \| Telangiectasia \| 2 4 \| Abnormal nail fold capillaries \| 2 5 \| Pulmonary arterial hypertension and/or interstitial lung disease (maximum score is 2)Pulmonary arterial hypertensionInterstitial lung disease \| 22 6 \| Raynaud phenomenon \| 3 7 \| Presence of ≥1 of the following:Centromere antibodyScl-70 antibodyRNA polymerase III antibody \| 3 Read more about SSc diagnosis Treatment Guidelines No conclusive therapy or widely recognized disease-modifying agent can change the course of SSc. However, outcomes can be improved with an early diagnosis. The effectiveness of treatment depends on the clinical assessment, identification of the affected organ(s), and progression of the disease. Additionally, it is critical that the objectives of treatment be comprehensive and designed to improve the patient’s quality of life and halt further organ damage. Every person with SSc should be educated about the condition, engage in regular exercise, maintain a healthy diet and lifestyle, and receive emotional support.4 Some commonly used immunosuppressive agents in the treatment of SSc include: methotrexate for skin disease, inflammatory arthritis, and myositis; CellCept® (mycophenolate mofetil) for both skin and lung disease; Cytoxan® (cyclophosphamide) for both skin and lung disease; Imuran® (azathioprine) for skin disease, lung disease, and myositis; and Plaquenil® (hydroxychloroquine) for skin disease.4 Corticosteroids should generally be avoided in SSc. The use of high doses of corticosteroids as well as the prolonged use of low or moderate doses has been linked to scleroderma renal crisis (SRC). If absolutely necessary, such as in cases of active inflammatory alveolitis, refractory inflammatory arthritis, or inflammatory myositis, corticosteroids should be administered in the smallest possible dose for the shortest possible time.4 Read more about SSc treatment Organ-Based Guidelines for Therapies Raynaud Phenomenon and Digital Ulcers Raynaud phenomenon is an exaggerated response of blood vessels to emotional stress or cold temperatures that manifests as clearly demarcated changes in the color of the skin of digits. Ischemia and ulceration further worsen the condition. Therapy should be aimed at improving quality of life and preventing tissue loss due to ulceration and gangrene.5 Recommendations for Raynaud Phenomenon Recommendations for Digital Ulcers Read more about SSc therapies SSc-Related Lung Disease Interstitial lung disease can develop in up to 80% of patients with SSc. It may be mild and stable, but if it is extensive or progressive, immunosuppression should be considered. Lung involvement can take the form of interstitial pneumonitis, bronchiolitis, or pulmonary vascular disease. Lung involvement can be a significant cause of morbidity and mortality in SSc.5 Recommendations for Lung Fibrosis Recommendations for Pulmonary Arterial Hypertension Read more about SSc complications Scleroderma Renal Crisis SRC occurs following an abrupt elevation of blood pressure; severe hypertension then drives the development of progressive renal failure, hypertensive encephalopathy, congestive heart failure, and/or microangiopathic hemolytic anemia. Close monitoring of a patient with SSc is required to detect any progression to SRC during the first 4 to 5 years. The optimal anthypertensive treatment recommended for SRC is an angiotensin-converting enzyme (ACE) inhibitor such as Capoten® (captopril), which is preferred over enalapril or ramipril.5 Recommendations for Renal Crisis Read more about SSc risk factors SSc-Related Gastrointestinal Disease The gastrointestinal (GI) tract is frequently affected in SSc, and any or all parts, from mouth to anus, may be involved. GI manifestations in SSc include gastrointestinal reflux disease (GERD), delayed gastric emptying, delayed motility leading to postprandial bloating and small-intestinal bacterial overgrowth, chronic constipation, and vascular complications such as gastric antral vascular ectasia. The goal of treatment, which is primarily supportive, is to reduce symptoms.5 Recommendations for Gastrointestinal Manifestations Read more about SSc signs and symptoms Use of Hematopoietic Stem Cell Transplant in SSc Hematopoietic stem cell transplant (HSCT) should be a treatment option only for carefully selected patients with rapidly progressing SSc who are at risk for organ failure. The careful identification of patients with SSc for whom this type of treatment is appropriate and a skilled medical team are of the utmost importance because HSCT carries a significant risk of treatment-related side effects and early mortality.7 Read more about SSc prognosis References Reviewed by Debjyoti Talukdar, MD, on 5/4/2023. Harshi Dhingra is a licensed medical doctor with specialization in Pathology. She is currently employed as faculty in a medical school with a tertiary care hospital and research center in India. Dr. Dhingra has over a decade of experience in diagnostic, clinical, research, and teaching work, and has written several publications and citations in indexed peer reviewed journals. She holds medical degrees for MBBS and an MD in Pathology. READ MORE ON Rare Disease Advisor, a trusted source of medical news and feature content for healthcare providers, offers clinicians insight into the latest research to inform clinical practice and improve patient outcomes. Copyright © 2025 Haymarket Media, Inc. All Rights Reserved. This material may not be published, broadcast, rewritten or redistributed in any form without prior authorization. Your use of this website constitutes acceptance of Haymarket Media’s Privacy Policy and Terms & Conditions.
2835	https://www.invertekdrives.com/support/iknow/selection-and-installation/braking-and-regeneration-14	Braking and Regeneration \| iKnow Knowledge Base \| Invertek Drives \| Invertek Drives iSource Sign in Select Language English Deutsch Español Italiano Português Polski Global Contacts Variable Frequency Drives All Variable Frequency Drives General Purpose Variable Frequency Drives Optidrive E3 Optidrive E3 for Single Phase Motors High Performance Variable Frequency Drives Optidrive P2 Application Specific Variable Frequency Drives Optidrive Eco HVAC HVAC Building Services Optidrive Eco Pump Advanced Pump Control Optidrive Elevator Core Advanced Elevator Control Optidrive Elevator Lift/Elevator Drive OEM Variable Frequency Drives Optidrive Compact 2 Compact OEM Drive Optidrive Coolvert Compressor & Heat Pump Drive Options & Accessories Software Ecodesign Energy Savings Calculator View our Case Studies ON ALL OPTIDRIVES What is a Variable Frequency Drive?iKnow VFD Knowledge Base Industries Agriculture Cranes & Hoists Food & Beverage General Automation HVAC Leisure Lifts/Elevators Marine Material Handling Mining Printing Refrigeration Solar Pumping Textiles & Paper Water & Wastewater Support Sustainability Sustainability Cooler Planet Ecodesign Energy Savings Calculator News Careers About About Invertek Drives InvertekTV Community Policy Documents Invertek Drives in Sumitomo Heavy Industries Contact Contact Worldwide Contacts Become a Distributor Braking and Regeneration Article 14 Regeneration and torque control are important for cranes and hoists Most of the time, in most applications, a variable frequency drive controls the motor by supplying it with energy which then powers the load. However, occasionally the energy flow will be in reverse, that is, from the load, through the motor, back to the drive. This will occur if the load is giving up energy, such as when a crane or elevator is lowering a load, or maybe when a conveyer is transporting material downhill. Regeneration, as it is called, will also take place if a high inertia load is decelerated; in this case, the energy stored in the rotating mass flows back through the motor to the drive. Fans often regenerate when slowed quickly. Of course, if there is significant friction in the system, or if there are other braking effects (such as the airflow through a fan) then the energy returned to the drive may be greatly reduced. However, if regeneration is significant, the energy will get back to the drive. If this happens, the continued operation of the drive will maintain a voltage on the motor, so a magnetic flux will be present, but the phase of the currents will change, so energy – that is current – will flow into the drive from the motor. The IGBTs and commutation diodes work as normal – the diodes don’t act as rectifiers or anything. As a result, current flows into the DC link and onto the DC link capacitor. Here it charges the capacitor, so the voltage rises. The current can’t go back to the supply (the rectifiers block this), so if regeneration continues, the voltage on the capacitor will continue to rise. To prevent damage, the drive will detect this and turn off the switching of the output IGBTs. Now there is no output voltage and therefore no magnetising current, so the flux in the motor collapses, leaving the rotor and load spinning freely with no more energy returning to the drive. The drive has tripped on overvoltage; this is a symptom of too much regeneration. We can prevent over voltage trips in several ways. When the system regenerates during deceleration, the easiest solution is to reduce the deceleration; that is, increase the ramp down time. Now the regenerated energy is less over a longer period, and this can maybe be absorbed by the losses in the system or by the drive itself. However, with high inertia loads you’ll probably end up with a very long deceleration time. Another approach is to simply turn off the output of the variable frequency drive and allow the motor and load to coast to a stop. You can do this by changing the Stop Mode parameter P-05 (P1-05 on P2 and Eco) from 0 to 1. Now there is no ramp down, the drive shuts off and the motor coasts. The disadvantage with this solution is that there is no control of the load and motor during stopping, so you don’t know if and when it will all stop. This isn’t convenient in process industries, but may be ok for a cooling fan. Neither of these options help if the load regenerates as part of its normal operation, rather than when stopping. If you are continuously accelerating and decelerating, or if your crane or lift is lifting and lowering all the time, a controlled solution is needed. Some applications, such as the downhill conveyer mentioned earlier, or an unwinder, regenerate all the time. The energy which flows to the DC link of the drive must therefore be dissipated. The solution is to connect a resistor across the DC to burn off the energy, The resistor is switched on and off by an IGBT, built into most industrial drives, which is controlled by the drive software which monitors the DC voltage and switches the IGBT on and off accordingly. This ‘chops’ the voltage, so it is sometimes called a braking chopper. Perhaps a better description is dynamic, or resistive braking. The resistor is not normally included in the drive package, and must be selected and bought separately. This arrangement is shown in Figure 1. Fig. 1 Variable frequency Drive showing Braking Chopper Operation With the correct braking resistor, the drive will now allow typically full load current to be returned to the drive and the power dissipated in the resistor. This allows for controlled lowering of loads and unwinding, as well as rapid deceleration of high inertia loads if required. As mentioned before, the ‘chopper’ IGBT is usually built into the drive, but the resistor must be selected by the customer. Selection needs a little care. Firstly, any resistor selected must be capable of working at high DC voltage, and must be protected (i.e. fused) accordingly. Secondly, the resistor must have a minimum Ohmic value to limit the current in the IGBT. Then it must be selected to absorb the expected power over the duty cycle of the machine. Finally, consideration should be given to the protection of the resistor. Because of the heat they dissipate, braking resistors are often mounted outside the cubicle, and should be out of harm’s way and protected against liquid, dirt and fingers. This solution is excellent for low and medium powers, and for control of braking that occurs occasionally. However, if you are operating a container crane that is continuously lifting and lowering large containers, it is wasteful to be burning all that energy in a resistor. The solution here is a fully regenerative drive that will feed energy back into the supply. DC drives were quite good at this, but AC drives need to replace the input rectifier will a fully functioning inverter as shown in Figure 2. Fig. 2 Fully Regenerative Variable frequency Drive We’ve already established that power can flow both ways through an inverter, so now the regenerative power flows back along the DC link, through the inverter at the front, and back to the mains. An inverter rectifier like this also has the advantage that it can control the input harmonics under normal operation, which can be quite important at these high powers. The extra cost of the second inverter, its control electronics, and the associated inductors required for the system is justified in drives that regenerate above 200kW or so. They are also used extensively in locomotive variable frequency drives as well, so slowing the train pumps power back into the supply rather than wearing out the brakes. Another trick to recover braking energy in a system with several drives is to connect the DC links of the drives together, so that when one drive regenerates it simply supplies power to another drive that’s motoring. This needs a bit of care; there’s an application note that helps. E3 Invertek drives include braking IGBTs as standard, except for the smallest unit. P2 drives also have built in braking IGBTs (optional on Frame sizes 6 and above). The Elevator variant of P2 includes a braking IGBT. 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Global Contacts Select Language English Deutsch Español Italiano Português Polski Home Variable Frequency Drives All Variable Frequency Drives General Purpose Drives Optidrive E3 Optidrive E3 for Single Phase Motors High Performance Drives Optidrive P2 Application Specific Drives Optidrive Eco HVAC Optidrive Eco Pump Optidrive Elevator Core Optidrive Elevator OEM Drives Optidrive Compact 2 Optidrive Coolvert Options & Accessories Software Ecodesign Energy Savings Calculator Datasheets Case Studies iKnow VFD Knowledge Base Industries Agriculture Cranes & Hoists Food & Beverage General Automation HVAC Leisure Lifts/Elevators Marine Material Handling Mining Printing Refrigeration Solar Pumping Textiles & Paper Water & Wastewater Support Support iKnow VFD Knowledge Base Invertek News & Media Global Service Centres User Guides Optidrive E3 Optidrive E3 for Single Phase Motors Optidrive P2 Optidrive Eco Optidrive Elevator Optidrive Compact 2 Optidrive Coolvert Software OptiTools Studio OptiTools Mobile Communication Interface Configuration Files BLE & NFC Drivers Sustainability Sustainability Cooler Planet Ecodesign Energy Savings Calculator News Careers About About Invertek Drives InvertekTV Community Policy Documents Invertek Drives in Sumitomo Heavy Industries Contact Contact Worldwide Contacts Become a Distributor Privacy Policy Sitemap iSource Sign in
2836	https://www.uotbih.ba/images/pdf-knjige/netters-concise-atlas-of-orthopaedic-anatomy.pdf	COVER FRONTMATTER PREFACE ABOUT THE AUTHOR INTRODUCTION CHAPTER 1 - SPINE CHAPTER 2 - SHOULDER CHAPTER 3 - ARM CHAPTER 4 - FOREARM CHAPTER 5 - HAND CHAPTER 6 - PELVIS CHAPTER 7 - THIGH/HIP CHAPTER 8 - LEG/KNEE CHAPTER 9 - FOOT/ANKLE CHAPTER 10 - BASIC SCIENCE ABBREVIATIONS USED IN THIS BOOK Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier Netter's Concise Atlas of Orthopaedic Anatomy Jon C. Thompson, M.D. Dedication T o my parents, for their unwavering faith in me. T o my in-laws, for their continual support. T o my daughters, who make it meaningful and fun. Especially to my wife Tiffany, who inspires me in every aspect of my life. SAUNDERS ELSEVIER Elsevier Inc. 1600 John F. Kennedy Boulevard Suite 1800 Philadelphia, PA 19103-2899 Netter's Concise Atlas of Orthopaedic Anatomy ISBN-13: 978-0-914168-94-2 ISBN-10: 0-914168-94-0 Published by Icon Learning Systems LLC, a subsidiary of Elsevier, Inc. Copyright © 2002 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Permissions may be sought directly from Elsevier's Health Sciences Rights Department in Philadelphia, PA,USA: phone: (+1) 215 239 3804, fax: (+1) 215 239 3805, e-mail: healthpermissions@elsevier.com . Y ou may also complete your request on-line via the Elsevier homepage ( ), by selecting ‘Customer Support’ and then ‘Obtaining Permissions’. NOTICE Medicine is an ever-changing field. Standard safety precautions must be followed, but as new research and clinical experience broaden our knowledge, changes in treatment and drug therapy may become necessary or appropriate. Readers are advised to check the most current information provided by the manufacturer of each drug to be administered to verify the recommended dose, the method and duration of administration, and contraindications. It is the responsibility of the licensed health care provider, relying on experience and knowledge of the patient, to determine dosages and the best treatment for each individual patient. Neither the publisher nor the editor assumes any liability for any injury and/or damage to persons or property arising from this publication. The Publisher Library of Congress Catalog No: 00-130477 Printed in U.S.A. Last digit is the print number: 9 8 7 6 5 4 Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier PREF ACE While working on the Orthopedic Service as a medical student I found myself in need of a quick, but comprehensive reference to help me get through my busy clinics and morning rounds. Having had success with pocket references, I searched the bookstores for something similar for orthopedics. Several were available, but none of them had the quick and easy-to-read format I wanted. As a result, I made pocket-sized note cards for my own use. These cards started with basic anatomy such as diagrams of the Brachial plexus or fascial compartments of the leg. I then added cards for various conditions including notes on pertinent History and Physical Exam findings and treatment options. Many years later, when the growing stack of note cards was too big, unwieldy and tattered to use any longer, I converted the information into a more usable book format. That original hand-assembled book is the foundation of the atlas you are now holding. One w ell-draw n anatomic picture often explains far more than several pages of detailed text. This concise, quick-reference atlas covers the spine and extremities as well as diagnosis and treatment of orthopedic conditions with primary emphasis on illustrations that educate, oftentimes without the need for explanatory text. T ext, when necessary, is presented in tabular form to allow for fast review of essential information. The first nine chapters are divided anatomically. Because I believe quite strongly that the treatment of orthopedic problems is based in anatomy, I have incorporated an extensive review of the anatomy of both the spine and extremities. There are also subsections within each chapter to help in the clinical diagnosis and treatment of the orthopedic patient. For example, the History table offers help in developing a differential diagnosis while the Trauma and Disorder tables assist in the work-up and treatment options of many orthopedic conditions. Chapter T en is a brief introduction to orthopedic-related basic science. From the first time I opened Frank Netter's Atlas of Human Anatomy, I was impressed, and even inspired, by the clarity and the incredible amount of information contained within each of his illustrations. I consider his work incomparable. As the basis for this text is also deeply rooted in its extensive use of illustrations, you can imagine how pleased I was when Icon Learning Systems asked me to combine our efforts to create this new publication. I thank them for their diligence, expertise,and patience with this project. I would also like to thank Dr. Jim Heckman for lending his wisdom and years of publishing experience to this effort. This book is the result of several years of accumulating and condensing Orthopedic-related data. Indeed, as it stands now, this is truly the reference I had searched for as a medical student, but was never able to find. The information inside these covers served to help me synthesize and retain a large body of information when I was a student and young physician. I trust its readers will be as equally well served. Jon C. Thompson, MD Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier ABOUT THE AUTHOR Jon Thompson, MD, received his medical degree from the Uniformed Services University of the Health Sciences in Bethesda, Maryland. He received his undergraduate degree from Dartmouth College. Dr. Thompson has worked as both an emergency room physician and a research assistant in the Extremity Trauma Branch of the Institute of Surgical Research. Currently, he is a resident in orthopedic surgery in the San Antonio Uniformed Services Health Education Consortium at Brooke Army Medical Center and is a corresponding member of the Department of Surgery at the Uniformed Services University of the Health Sciences. Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier INTRODUCTION Netter's Concise Atlas of Orthopedic Anatomy is an easy-to-use reference and compact atlas of orthopedic anatomy for students and clinicians. Using images from both the Atlas of Human Anatomy and the 13-Volume Netter Collection of Medical Illustrations, this book brings together over 450 Netter images together for the first time in one book. T ables are used to highlight the Netter images and offer key information on bones, joints, muscles and nerves, and surgical approaches. Clinical material is presented in a clear and straightforward manner with emphasis on trauma, minor procedures, history and physical exam, and disorders. Users will appreciate the unique color-coding system that makes information look-up even easier. Key material is highlighted in black, red, and green to provide quick access to clinically relevant information. BLACK for standard text RED highlights key information that if missed could result in morbidity or mortality GREEN highlights “must know” clinical information. Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com CHAPTER 1 - SPINE TOPOGRAPHIC ANATOMY OSTEOLOGY TRAUMA SPINAL CORD TRAUMA JOINTS LIGAMENTS HISTORY PHYSICAL EXAM MUSCLES: ANTERIOR NECK MUSCLES: POSTERIOR NECK SUPERFICIAL MUSCLES: POSTERIOR NECK AND BACK DEEP MUSCLES: POSTERIOR NECK AND BACK NERVES OF THE UPPER EXTREMITY: CERVICAL PLEXUS NERVES: BRACHIAL PLEXUS NERVES: LUMBAR PLEXUS NERVES: SACRAL PLEXUS ARTERIES DISORDERS PEDIATRIC DISORDERS SURGICAL APPROACHES Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier CHAPTER 1 – SPINE TOPOGRAPHIC ANATOMY Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier OSTEOLOGY CHARACTERISTICS OSSIFY FUSE COMMENT C1 ATLAS • Ring shaped • Two lateral masses with facets on them • No body, no spinous process • Post. Arch has a sulcus/groove Anterior arch (1) Posterior arch (2) (1 for each half) 6 yrs Birth • Superior facet articulates with occiput, anterior arch articulates with dens • Fractures: most have 2 sites • Vertebral artery runs in groove on posterior arch C2 AXIS • Dens/odontoid articulates w/atlas at median atlantoaxial joint Lower body (2) Dens (2) Arch (2) Body Tip 6yrs Birth 12yrs Birth • Odontoid has precarious vascular supply watershed area): increased incidence of nonunion with fractures • Rotation in neck mostly occurs between C1 and C2 CERVICAL (C3-7) • Foramina in transverse process • Facets: “semi-coronal” allow flex/extension, no rotation • Narrow intervertebral foramina • Bifid spinous processes Primary Arch Body Secondary 7-8wk (fetal) 11-14 yr 1-2 yr 7-10 yr 18-25 yr • Vertebral artery runs through transverse foramina • Nerve roots at risk of compression • No foramina in transverse process of C7 • C7 is vertebral prominens, nonbifid spinous process • Klippel-Feil syndrome: congenital fusion of cervical vertebrae THORACIC • Facets: form semi-circle: allow rotation Costal facets (for 7-1-2 yr • T1 spinous process is as • Costal facets (for ribs) T1-9: on the transverse process T10-12: on the pedicle Primary Arch Body Secondary 7-8wk (fetal) 11-14 yr 1-2 yr 7-10 yr 18-25 yr • prominent as that of C7 • Rotation of spine occurs within the thoracic region • Spinous processes overlap the next lower vertebrae CHARACTERISTICS OSSIFY FUSE COMMENT LUMBAR • Large vertebral bodies • Short lamina and pedicles Primary Arch 7-8 1-2 • L5 is the largest vertebrae pedicles • Mamillary and accessory processes • Facets: sagittal: good for flexion/extension, not rotation • No costal facets Body Secondary Mamillary process 7-8 wk (fetal) 11-14 yrs yrs 7-10 yrs 18-25 yrs vertebrae • Large vertebral bodies capable of bearing weight • L5 has a ligamentous attachment to the ilium SACRAL • 5 vertebrae are fused • 4 pairs of sacral foramina • Sacral canal opens to hiatus Body Arches Cpstal elements Secondary 8 wk (fetal) 11-14 yrs 2-8 yrs 2-8 yrs 2-8 yrs 20 yrs • Transmits weight of body to the pelvis • Nerves exit through the sacral foraminae • Segments fuse to each other at puberty COCCYGEAL • 4 vertebrae are fused • Lacks most of the features of typical vertebrae Primary Arch Body 7-8 wk (fetal) 1-2 yrs 7-10 yrs • Is attached to Gluteus maximus and coccygeal muscle Ossification: Typically 3 primary (body each arch), 5 secondary ossification centers (spinous process, transverse process (2), upper and lower plates of the body (2)) The arches fuse dorsally; spina bifida occurs when it does not fuse The arches unite with the bodies (6-10years old) in order: thoracic, cervical, lumbar, sacral (7 years). Neurocentral joint (fusion of arch and body) is in the body GENERAL INFORMATION • 33 Vertebrae: 7 cervical, 12 thoracic, 5 lumbar, 5 sacral (fused), 4 coccygeal • Cancellous bone in cortical shell • Vertebral canal between body and lamina: houses the spinal cord. • Spinal Curves: Cervical: lordosis Thoracic: kyphosis (increase in Scheuermann's disease) Lumbar: lordosis • Vertebrae: 1. Body (centrum): have articular cartilage on superior/inferior aspects; get larger inferiorly 2. Arch (pedicles lamina) [no arch develops in spina bifida] 3. Processes: spinous, transverse, costal, mamillary 4. Foramina: vertebral, intervertebral, transverse • 3 Columns Anterior ALL, anterior half of body annulus Middle PLL, posterior half of body annulus Posterior Ligamentum flavum, lamina, pedicles, facets LEVEL CORRESPONDING STRUCTURE C2-3 Mandible C3 Hyoid cartilage C4-5 Thyroid cartilage C6 Cricoid cartilage C7 Vertebral prominens T3 Spine of scapula T7 Xiphoid, tip of scapula T10 Umbilicus L1 End of spinal cord L3 Aorta bifurcation L4 Iliac crest Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier TRAUMA DESCRIPTION EVALUATION CLASSIFICATION TREATMENT CERVICAL FRACTURE • High energy injury: Y oung - MVA, old - fall • Axial compression (most common mech.-anism) results in burst fracture HX: Trauma. Pain, worse with movement, +/-numbness weakness. PE: T ender to palpation, +/-“step off” neurologic or myelopathic Based on level location: C1-Jefferson fracture: both arches fractured C1-Lateral mass fracture C2-Hangman's (isthmus): Immobilize all fractures, traction on unstable, lower c-spine fractures C1 and 2: Stable: Collar or halo Unstable: Halo for 3 months • Flexion/distraction injury results in dislocation • Neurologic injury rare (esp. with C12 fracture) seen • Often have associated injuries • 9 criteria checklist predicts instability myelopathic signs. Do rectal genital exams. XR: AP, lateral, odontoid: note anterior soft tissue CT : Shows canal (fragments may compress canal) MR: Evaluate soft tissues Levine classification C2-Odontoid: Type 1,2,3 C3-7 Fracture Spinous process (Clay shoveler's fracture): C6, 7, T1 (C7 most common) months and/or fusion Odontoid type 2: ORIF (worse with traction) C3-7: Stable: Collar or halo Unstable: Fusion Spinous process: Symptomatic COMPLICATIONS: Neurologic injury (e.g., CN VIII with C1 fracture, etc.); Residual pain; Osteoarthritis; Nonunion (especially odontoid type 2 fracture) Three-Column Concept of Spinal Stability DESCRIPTION EVALUATION CLASSIFICATION TREATMENT THORACOLUMBAR FRACTURE • Mechanism: MVA, fall • 1 column fracture: stable • 2 column fracture: unstable • Anterior column (Wedge) fracture 50% height loss is HX: Trauma. Pain, +/-numbness weakness PE: T ender to palpation, +/- “step off” neurologic or myelopathic signs. Do rectal genital exams Mechanism: Compression/wedge: anterior column Burst: fragments displace posteriorly; anterior middle columns (unstable) Stable fractures: bed rest, orthosis (TLSO) Unstable (or with • 50% height loss is considered 2 columns • Compression/wedge fracture: (most common) • Chance fracture: rare • Neurologic deficits rare, but seen with Burst fractures exams XR: AP, lateral T-L spine: body height, splaying pedicle CT : Shows any canal impingement MR: Evaluate soft tissues Flexion/distraction (Chance/seatbelt fracture): 2 (or 3) columns: posterior middle (anterior). Fracture/dislocation: all 3 columns involved. neurologic symptoms/compressed canal): Spinal canal decompression and spinal fusion COMPLICATIONS: Neurologic injury; Osteoarthritis; Associated injuries. Stable Fracture Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier SPINAL CORD TRAUMA Cervical Spine Injury: Incomplete Spinal Syndromes DESCRIPTION EVALUATION CLASSIFICATION TREATMENT • Y oung males most common • Complete cord injury: no function AND bulbocavernosus reflex has returned. (spinal shock over) • Incomplete cord injury: 4 types • Anterior cord: #2. Flexion injury; worst prognosis • Central cord: most common. Hyperextension injury, seen in elderly (who fall), associated with spondylosis • Posterior: very rare (may not exist) • Brown-Sequard: rare, best prognosis HX: Trauma. Symptoms depend on injury/lesion. PE: Depends on injury Complete: no motor or sensory function below injury level. Anterior: LEUE paralysis, pain temperature sensory loss, vibratory proprioception intact. Central: Weakness UELE, sacral sensation spared. Posterior: Loss of vibratory sensation and proprioception. B-S: Ipsilateral motor, vibratory, proprioception loss; contralateral pain temperature loss. XR: C-spine series, +/- TL spine CT : if evidence of fracture Complete cord injury: cord severed, no function (spinal shock must be resolved to diagnose it) Incomplete: Anterior: Spinothalamic corticospinal tracts out, posterior columns spared. Central: gray matter injury Posterior: posterior columns disrupted Brown-Séquard (lateral): hemi-section of cord Treat associated injuries: lifethreatening first. Mannitol and early IV steroids may improve neurologic function Immobilization is the key to treatment Stable injures: collar, brace Unstable injuries: Halo vest or internal fixation COMP: Neurogenic shock; Autonomic dysreflexia (requires urinary catheterization and/or fecal disimpaction); Neurologic sequelae Spinal Shock: Physiologic cord injury/dysfunction (often from compression or swelling) including paralysis areflexia. Return of bulbocavernosus reflex (arc reflexes) marks the end of spinal shock. Neurogenic Shock: Hypotension w ith bradycardia. Cord injury results in decreased sympathetic release (unopposed vagal tone) Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier JOINTS LIGAMENT ATTACHMENT COMMENT ATLANTOOCCIPITAL (Ellipsoid) Primarily involved in flexion, extension, lateral bending movements T ectoral membrane Anterior/Posterior capsule Axis body to occiput around facets Extension of the PLL Joint stabilized by attachment to dens; known to be weak in Down's Syndrome MEDIAN ATLANTOAXIAL C1-2 (Plane and Pivot) Primarily involved in rotation; dependent on ligaments for stability; instability in Down's syndrome Transverse Apical Alar Superior Longitudinal Inferior Longitudinal Lateral mass-dens-lateral mass Dens to occiput Dens to occiput condyles Dens to basilar occiput Dens to axis body Strongest ligament: holds dens in place Part of cruciate ligament Prevent excessive head rotation With transverse apical forms cruciate ligament LIGAMENT ATTACHMENT COMMENT ZYGAPOPHYSEAL (Facet Plane) Has articular discs: this joint allows the most mobility in the spine Capsule Around facets Changes orientation at different vertebral levels Orientation dictates plane of motion; C5-6 most mobile (#1 degeneration site) L4-5 most flexion INTERVERTEBRAL Intervertebral disc ALL PLL Inferior superior aspect of bodies Anterior: body to body Posterior: body to body Strongest attachments of bodies Thicker than PLL Thinner, disc herniation usually posterolateral. COSTOVERTEBRAL (Luschka) Capsule Intraarticular Radiate Surrounds rib head joint Head of rib to disc Anterior head to both bodies Holds head to vertebrae Reinforces joint anteriorly Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier LIGAMENTS LIGAMENT LOCATION COMMENT Anterior Longitudinal [ALL] Posterior Longitudinal [PLL] Intertransverse Apophyseal joint capsule Ligamentum Flavum Ligamentum Nuchae Supraspinous Interspinous T ectoral membrane Transverse ligament Alar Iliolumbar Anterior surface of vertebral bodies Posterior surface of bodies (connects discs] Between transverse processes Around facet joint Connects anterior surfaces of laminae C7 to occipital protuberance Along dorsal spinous processes to C7 Between spinous processes Posterior aspect of bodies dens to clivus Lateral mass to dens to lateral mass Dens to occiput tubercles L5 transverse process to ilium Strong; thicker in center of body Weaker thinner [herniation occurs laterally or posterolaterally] Weak, adds little support Weak, adds little support Strong; constantly in tension Extension of supraspinous ligament Unknown contribution to stability Unknown contribution to stability Extension of PLL Part of cruciate ligament, major stabilizer Resists excessive rotation Avulsion fracture can occur in trauma INTERVERTEBRAL DISCS [made of fibrocartilage] Annulus fibrosis Nucleus pulposus Outside, type I collagen, connects to vertebral hyaline cartilage, buffers compression Inside, type II collagen, high water content until old age, derived from notochord, can protrude/herniate through annulus, is avascular Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier HISTORY QUESTION ANSWER CLINICAL APPLICATION 1. AGE Y oung Middle age Elderly Disc injuries, spondylolisthesis Sprain/strain, herniated disc, degenerative disc disease Spinal stenosis, herniated disc, degenerative disc disease, arthritis 2. PAIN a. Character Radiating (shooting) Diffuse, dull, non-radiating Radiculopathy (Herniated disc, spondylosis) Cervical or lumbar strain (soft tissue injury) b. Location Unilateral vs. bilateral Neck Arms (+/-radiating) Lower back Legs (+/-radiation) Unilateral: herniated disc; Bilateral: systemic or metabolic disease;space occupying lesion Cervical spondylosis, neck sprain or muscle strain Cervical spondylosis (+/- myelopathy), herniated disc Degenerative Disc Disease, back sprain or muscle strain, spondylolisthesis, tumor Herniated disc, spinal stenosis c. Occurrence Night pain With activity Tumor Usually mechanical etiology d. Alleviating Arms elevated Sit down Herniated cervical disc Spinal stenosis (stenosis relieved) e. Exacerbating Back extension Spinal stenosis (e.g. going down stairs) 3. TRAUMA MVA (seatbelt?) Cervical strain (whiplash), cervical fractures, ligamentous injury 4. ACTIVITY Sports (stretching injury) “Burners/stingers” (especially in football) 5. NEUROLOGIC SYMPTOMS Pain, numbness, tingling Spasticity, clumsiness Bowel or bladder symptoms Radiculopathy, neuropathy Myelopathy Cauda equina syndrome 6. SYSTEMIC Fever, weight 6. SYSTEMIC COMPLAINTS Fever, weight loss Infection, tumor Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier PHYSICAL EXAM EXAM TECHNIQUE CLINICAL APPLICATION INSPECTION Gait Leaning forward Wide-based Spinal stenosis Myelopathy Alignment Malalignment Dislocation, scoliosis, lordosis, kyphosis Posture Head tilted Pelvis tilted Dislocation, spasm, spondylosis, torticollis Loss of lordosis: spasm Skin Disrobe patient Cafe-au-lait spots, growths: possibly neurofibromatosis Port wine spots, soft masses: possibly spina bifida PALPATION Bony structures Spinous processes Focal/point tenderness: fracture. Step-off: dislocation/spondylolisthesis Soft tissues Cervical facet joints Coccyx-via rectal exam Paraspinal muscles Supraclavicular fossa Skin T enderness: osteoarthritis, dislocation T enderness: fracture or contusion Diffuse tenderness indicates sprain/muscle strain. Trigger point: spasm Swelling suggests clavicle fracture Fatty masses: possibly spina bifida RANGE OF MOTION Flexion/extension: Cervical Lumbar Chin to chest/occiput back T ouch toes with straight legs Normal: Flexion: chin within 3-4cm of chest; Extension 70 degrees Normal: 45-60 degrees in flexion, 20-30 degrees in extension Lateral flexion: Cervical Lumbar Ear to shoulder Bend to each side Normal: 30-40 degrees in each direction Normal: 10-20 degrees in each direction Stabilize Rotation: Cervical Lumbar shoulders: rotate Stabilize hip: rotate Normal: 75 degrees each direction Normal: 5-15 degrees in each direction NEUROVASCULAR A complete neurologic examination should be performed Sensory CERVICAL Supraclavicular (C2-3) Axillary nerve (C5) Musculocutaneous nerve (C6) Radial Nerve (C6) Median Nerve (C7) Ulnar Nerve (C8) Medial Cutaneous nerve forearm(T1) Anterior neck clavicle area Lateral shoulder Lateral forearm Dorsal thumb web space Radial border mid finger Ulnar border small finger Medial forearm Deficit indicates corresponding nerve/root lesion Deficit indicates corresponding nerve/root lesion Deficit indicates corresponding nerve/root lesion Deficit indicates corresponding nerve/root lesion Deficit indicates corresponding nerve/root lesion Deficit indicates corresponding nerve/root lesion Deficit indicates corresponding nerve/root lesion Straight Leg Test EXAM TECHNIQUE CLINICAL APPLICATION LUMBAR Femoral/Saphenous nerve (L4) Superficial/Deep Peroneal Nerve (L5) Tibial/sural nerve (S1) Sacral nerves (S 2, 3, 4) Medial leg ankle Dorsal foot 1st-2ndtoe web space Lateral foot Perianal sensation Deficit indicates corresponding nerve/root lesion Deficit indicates corresponding nerve/root lesion Deficit indicates corresponding nerve/root lesion Deficit indicates corresponding nerve/root lesion Motor CERVICAL Spinal accessory (CN11) Axillary nerve (C5) Musculocutaneous nerve (C5-6) Radial nerve (PIN) (C7) Median nerve (C8) Ulnar nerve (Deep branch) (T1) Neck flexion rotation Resisted shoulder abduction Resisted elbow flexion Finger extension Thumb flexion, opposition, abduction Finger cross (abduct/adduct) Weakness = Sternocleidomastoid or nerve/root lesion Weakness = Deltoid or nerve/root lesion Weakness = Brachialis or nerve/root lesion Weakness = EDC, EIP, EDM or nerve/root lesion Weakness = FPL/thenar muscles or corresponding nerve/root lesion Weakness = DIO/VIO or nerve/root lesion LUMBAR Deep Peroneal nerve (L4) Deep Peroneal nerve (L5) Superficial Peroneal Foot inversion dorsiflexion Great toe extension Foot eversion Weakness = Tibialis anterior or nerve/root lesion Weakness = Extensor hallucis longus or nerve/root lesion Weakness = Peroneus longus/brevis or Superficial Peroneal (S1) Tibial nerve (S1) Great toe flexion nerve/root lesion Weakness = Flexor hallucis longus or nerve/root lesion Reflexes C5 C6 C7 L4 S1 S1, 2, 3 Biceps Brachioradialis Triceps Patellar Achilles reflex Bulbocavernosus Hypoactive/absence indicates C5 radiculopathy Hypoactive/absence indicates C6 radiculopathy Hypoactive/absence indicates C7 radiculopathy Hypoactive/absence indicates L4 radiculopathy Hypoactive/absence indicates S1 radiculopathy Finger in rectum, squeeze/pull penis (Foley), anal sphincter contracts UMN Babinski/clonus Upgoing toe is consistent with upper motor neuron lesion Pulses Upper extremity Lower extremity Brachial, radial, ulnar Femoral, popliteal, dorsalis pedis, posterior tibial Diminished/absent = vascular injury or compromise Diminished/absent = vascular injury or compromise Forward Bending Test EXAM TECHNIQUE CLINICAL APPLICATION SPECIAL TESTS CERVICAL Spurling Axial load, then laterally flex rotate neck Radiating pain indicates nerve root compression Distraction Upward distracting force Relief of symptoms indicates foraminal compression of nerve root LUMBAR Straight leg Flex hip to pain, dorsiflex foot Symptoms reproduced (pain below knee) indicative of radicular etiology Straight leg 90/90 Supine: flex hip knee 90°, extend knee 20° of flexion = tight hamstrings: source of pain Bowstring Raise leg, flex knee, Radicular pain with popliteal pressure indicates sciatic Bowstring apply popliteal pressure nerve etiology Sitting root (flip sign) Sit: distract patient, passively extend knee Patient with sciatic pain will arch or flip backward on knee extension Kernig Supine: flex neck Pain in or radiating to legs indicates meningeal irritation or infection Brudzinski Supine: flex neck, flex hip Pain reduction with knee flexion indicates meningeal irritation. Forward Bending Standing, bend at waist Asymmetry of back (scapula/ribs) is indicative of scoliosis Trendelenburg Stand on one leg Drooping pelvis on elevated leg side: gluteus medius weakness Hoover Supine: hands under heels, patient then raises one leg Pressure should be felt under opposite heel (not being raised). No pressure indicates lack of effort, not true weakness Waddell signs Presence indicates non-organic pathology: 1) exaggerated response or overreaction, 2) pain to light touch, 3) non-anatomic pain localization, 4) negative flip sign with positive straight leg test. Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier MUSCLES: ANTERIOR NECK MUSCLE ORIGIN INSERTION ACTION NERVE ANTERIOR NECK Platysma Fascia: Deltoid/pectoralis major Mandible; skin Depress jaw CN 7 SUPRAHYOID MUSCLES Digastric Anterior: Mandible Posterior: Mastoid notch Hyoid body Elevate hyoid, depress mandible Anterior: Mylohyoid (CN 5) Posterior: Facial (CN 7) Mylohyoid Mandible Raphe on hyoid Same as above Mylohyoid (CN 5) Stylohyoid Styloid process Body of hyoid Elevate hyoid Facial nerve (CN 7) Geniohyoid Genial tubercle of mandible Body of hyoid Elevate hyoid C1 Via CN 12 INFRAHYOID MUSCLES [STRAP MUSCLES INCLUDES THE SCM] SUPERFICIAL Sternohyoid Manubrium clavicle Body of hyoid Depress hyoid Ansa cervicalis (C1-3) Omohyoid Suprascapular notch Body of hyoid Depress hyoid Ansa cervicalis (C1-3) DEEP Thyrohyoid Thyroid cartilage Greater horn of hyoid Depress/retract hyoid/larynx C1 via CN 12 Sternothyroid Manubrium Thyroid cartilage Depress/retract hyoid/larynx Ansa cervicalis (C1-3) Sternocleidomastoid Manubrium clavicle Mastoid process Turn head opposite side CN 11 Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier MUSCLES: POSTERIOR NECK MUSCLE ORIGIN INSERTION ACTION NERVE POSTERIOR NECK: SUBOCCIPITAL TRIANGLE Rectus capitis posterior: major Spine of axis Inferior nuchal line Extend, rotate, laterally flex Suboccipital nerve Rectus capitis posterior: minor Posterior tubercle of atlas Occipital bone Extend, laterally flex Suboccipital nerve Obliquus capitis superior Atlas transverse process Occipital bone Extend, rotate, laterally flex Suboccipital nerve Obliquus capitis inferior Spine of axis Atlas transverse process Extend, laterally rotate Suboccipital nerve Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier SUPERFICIAL MUSCLES: POSTERIOR NECK AND BACK MUSCLE ORIGIN INSERTION ACTION NERVE SUPERFICIAL (EXTRINSIC) Trapezius Spinous process C7-T12 Clavicle; Scapula (AC, SP) Rotate scapula CN 11 Latissimus dorsi Spinous process T6-S5 Humerus Extend, adduct, IR arm Thoracodorsal Levator scapulae Transverse process C1-4 Scapula (medial) Elevate scapula C3, 4, Dorsal scapular Rhomboid minor Spinous process C7-T1 Scapula (spine) Adduct scapula Dorsal scapular Rhomboid major Spinous process T2-T5 Scapula (medial border) Adduct scapula Dorsal scapular Serratus posterior superior Spinous process C7-T3 Ribs 2-5 (upper border) Elevate ribs Intercostal nerve (T1-4) Serratus posterior inferior Spinous process T11-L3 Ribs 9-12 (lower border) Depress ribs Intercostal nerve (T9-12) Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier DEEP MUSCLES: POSTERIOR NECK AND BACK MUSCLE ORIGIN INSERTION ACTION NERVE DEEP (INTRINSIC) SUPERFICIAL LAYER: SPINOTRANSVERSE GROUP Splenius capitis Ligamentum nuchae Mastoid nuchal line Both: laterally flex rotate neck to same side Dorsal rami of inferior cervical nerves Splenius cervicus Spinous process T1-6 Transverse process C1-4 INTERMEDIATE LAYER: SACROSPINALIS GROUP (Erector spinae) All have 3 parts: thoracis, cervicis and capitis Iliocostalis Longissimus Spinalis Common origin: Sacrum, iliac crest, and lumbar spinous process. Ribs TC spinous process, mastoid process T-spine: spinous process Laterally flex, extend, rotate head (to same side) and vertebral column Dorsal rami of spinal nerves MUSCLE ORIGIN INSERTION ACTION NERVE DEEP (INTRINSIC) DEEP LAYERS: TRANSVERSOSPINALIS GROUP Semispinalis (CT) Transverse process Spinous process Extend, rotate opposite side Dorsal primary rami Semispinalis capitis Transverse process T1-6 Nuchal ridge Dorsal primary rami Multifidi [C2-S4] Transverse process Spinous process Flex laterally, rotate opposite Dorsal primary rami Rotatores Transverse process Spinous process +1 Rotate superior vertebrae opposite Dorsal primary rami Interspinales Spinous process Spinous process +1 Extend column Dorsal primary rami Intertransversarii Transverse process Transverse process +1 Laterally flex column Dorsal primary rami Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier NERVES OF THE UPPER EXTREMITY : CERVICAL PLEXUS CERVICAL PLEXUS (C1-C4 ventral rami) Behind IJ and SCM 1. Lesser Occipital Nerve(C2-3): arises from posterior border of SCM Sensory: Superior region behind auricle Motor: NONE 2. Great Auricular Nerve (C2-3): exits inferior to Lesser Occipital nerve, then ascends on SCM Sensory: Over parotid gland and below ear Motor: NONE Transverse Cervical Nerve (C2-3): exits inferior to Greater Auricular nerve, then to anterior neck Sensory: Anterior triangle of the neck Motor: NONE Supraclavicular (C2-3): splits into 3 branches: anterior, middle, posterior Sensory: Over clavicle, outer trapezius deltoid Motor: NONE Ansa Cervicalis (C1-3): superior (C1-2) inferior (C2-3) roots form loop Sensory: NONE 3. 4. 5. Motor: Omohyoid Sternohyoid Sternothyroid 6. Phrenic Nerve (C3-5): On anterior scalene, into thorax between subclavian artery and vein Sensory: Pericardium and mediastinal pleura Motor: Diaphragm Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier NERVES: BRACHIAL PLEXUS BRACHIAL PLEXUS (C5-T1 ventral rami) [variations: C4-T2] (also see Shoulder) SUPRACLAVICULAR [approach through posterior triangle] ROOTS Dorsal Scapular (C5): pierces middle scalene, deep to Levator Scapulae Rhomboids. Sensory: NONE Motor: Levator scapulae Rhomboid Minor and Major Long Thoracic (C5-7): on anterior surface of Serratus Anterior with Lateral Thoracic artery. Sensory: NONE Motor: Serratus Anterior (wing scapula with nerve dysfunction) UPPER TRUNK Suprascapular (C5-6): through scapular notch, under superior transverse scapular ligament. Sensory: Shoulder joint Motor: Supraspinatus Infraspinatus Nerve to Subclavius (C5-6): descends anterior to plexus, posterior to clavicle Sensory: NONE Motor: Subclavius INFRACLAVICULAR [approach through axilla] LATERAL CORD • Lateral root to Median nerve Lateral Pectoral (C5-7): named for lateral cord, is medial to Medial Pectoral nerve runs with pectoral artery. Sensory: NONE Motor: Pectoralis Major Pectoralis Minor (via loop to MPN] Musculocutaneous (C5-7): pierces coracobrachialis, runs between biceps brachialis. Sensory: Lateral forearm [via Lateral cutaneous nerve] Motor: ANTERIOR COMPARTMENT OF ARM Coracobrachialis Biceps brachialis Brachialis INFRACLAVICULAR [approach through axilla] MEDIAL CORD • Medial root to Median nerve Medial Pectoral (C8-T1): named for medial cord, is lateral to Lateral Pectoral nerve Sensory: NONE Motor: Pectoralis Minor Pectoralis Major (overlying muscle] Medial Cutaneous Nerve of Arm (Brachial, C8-T1): joins Intercostalbrachial Sensory: Medial (inner) arm Motor: NONE Medial Cutaneous Nerve of Forearm (Antibrachial, (C8- T1): runs with basilic vein. Sensory: Medial forearm anterior arm Motor: NONE 10. Ulnar (C (7) 8-T1): runs behind medial epicondyle in groove. Multiple sites of possible compr Sensory: Medial palm 1 1/2 digits via: palmar palmar digital branches Medial dorsal hand 1 1/2 digits via: dorsal, dorsal digital, proper palmar digital bra Motor: FOREARM [runs between the two muscles] Flexor carpi ulnaris Flexor digitorum profundus [digits 4,5] HAND [divides at hypothenar eminence] Superficial Branch [lateral to pisiform] Palmaris brevis Deep (Motor) Branch [around hook of hamate] Adductor pollicis THENAR MUSCLES Flexor pollicis brevis[FPB][with median] HYPOTHENAR MUSCLES Abductor digiti minimi [ADM] Flexor digiti minimi brevis [FDMB] Opponens digiti minimi [ODM] INTRINSIC MUSCLES Dorsal interossei [DIO] [abduct DAB] Volar interossei [VIO] [adduct PAD] Lumbricals [medial two (3,4)] 1. 2. 3. 4. 5. 6. 7. 8. 9. Lumbricals [medial two (3,4)] BRACHIAL PLEXUS (C5-T1 ventral rami) [variations: C4-T2] (also see Shoulde INFRACLAVICULAR [approach through axilla] MEDIAL AND LATERAL CORDS 11. Median (C (5) 6-T1): runs anteromedial, no branches in arm Multiple sites of possible com Sensory: Dorsal distal phalanges of lateral 3 1/2 digits via: proper palmar digital branche Volar 3 1/2 digits and lateral palm via: palmar palmar digital branches Motor: ANTERIOR COMPARTMENT OF FOREARM Superficial Flexors Pronator T eres [PT] Flexor Carpi Radialis [FCR] Palmaris longus [PL] Flexor digitorum superficialis [FDS] [sometimes considered a “middle” flexor] Deep Flexors: AIN (Anterior Interosseous Nerve) Flexor digitorum profundus [digits 2,3] Flexor pollicis longus [FPL] Pronator Quadratus [PQ] HAND: Motor Recurrent (Thenar motor) Thenar Abductor pollicis brevis [APB] Opponens pollicis Flexor pollicis brevis [FPB][with ulnar] Intrinsic Lumbricals [lateral two (1,2)] POSTERIOR CORD 12. Upper Subscapular (C5-6) Sensory: NONE Motor: Subscapularis [upper portion] 13. Lower Subscapular (C5-6) Sensory: NONE Motor: Subscapularis [lower portion] T eres major 14. Thoracodorsal (C7-8): runs with Thoracodorsal artery Sensory: NONE Motor: Latissimus dorsi Axillary (C5-6): runs with Posterior Circumflex Humeral ar space Sensory: Lateral upper arm: via Superior lateral cutaneo Motor: Deltoid (Deep branch) T eres minor (Superficial branch) 16. Radial (C5-T1): runs with Deep Artery of Arm in Tr Sensory: Lateral arm: via Inferior lateral cutaneou Posterior arm: via Posterior cutaneous Posterior forearm: via Posterior cutane Dorsal 3 1/2 digits and hand: via super branches) Motor: POSTERIOR COMPARTMENT OF AR Triceps [medial, long, lateral heads] Anaconeus MOBILE WAD: (Radial nerve-Deep bra Brachioradialis [BR] Extensor carpi radialis longus [ECRL] Extensor carpi radialis brevis [ECRB] POSTERIOR COMPARTMENT OF FO PIN Multiple possible compression si (see Forearm) Superficial Extensors Extensor carpi ulnaris [ECU] Extensor digiti minimi [EDM] Extensor digitorum [ED] Deep Extensors Supinator Abductor pollicis longus Extensor pollicis longus Extensor pollicis brevis Extensor indicis proprius 15. Copyright © 2008 Elsevier Inc. All rights reserved. - ww Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier NERVES: LUMBAR PLEXUS LUMBAR PLEXUS (Deep to Psoas muscle) ANTERIOR DIVISION Subcostal (T12): Sensory: Subxiphoid region Motor: NONE Iliohypogastric (L1): Sensory: Above pubis Posterolateral buttocks Motor: Transversus abdominus Internal Oblique Ilioinguinal (L1): Sensory: Inguinal region Motor: NONE Genitofemoral (L1-2): pierces Psoas, lies on anteromedial surface. Sensory: Scrotum/mons Motor: Cremaster Obturator (L2-4): exits via obturator canal, splits into anterior posterior divisions. Can be injured by retractors placed behind the transverse acetabular ligament. Sensory: Inferomedial thigh via cutaneous branch of Obturator nerve External oblique Adductor longus (anterior division) 1. 2. 3. 4. 5. Motor: Adductor brevis (ant post division) Adductor magnus (posterior division) Gracilis (anterior division) Obturator externus (posterior division) 6. Accessory Obturator (L2-4): inconsistent Sensory: NONE Motor: Psoas POSTERIOR DIVISION 7. Lateral Femoral Cutaneous LFCN: crosses ASIS, can be compressed at ASIS Sensory: Lateral thigh Motor: NONE 8. Femoral (L2-4): lies between psoas major and iliacus Sensory: Anteromedial thigh via anterior intermediate cutaneous nerves Medial leg foot via medial cutaneous nerves (Saphenous Nerve) Motor: Psoas Iliacus Pecineus Quadriceps Rectus femoris Vastus lateralis Vastus intermedialis Vastus Medialis Sartorius Articularis genu Copyright © 2008 Elsevier Inc. All rights reserved. -www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier NERVES: SACRAL PLEXUS SACRAL PLEXUS ANTERIOR DIVISION Tibial (L4-S3): descends between heads of Gastrocnemius to medial malleolus Sensory: Posterolateral proximal calf: via Medial sural Posterolateral distal calf: via Sural Medial plantar heel: via Medial calcaneal Medial plantar foot: via Medial plantar Lateral plantar foot: via Lateral plantar Motor: POSTERIOR THIGH Biceps femoris [long head] Semitendinosus Semimembranosus SUPERFICIAL POST . COMPARTMENT OF LEG Soleus: via nerve to Soleus Gastrocnemius Plantaris DEEP POSTERIOR COMPARTMENT OF LEG Popliteus: via nerve to Popliteus Tibialis posterior [TP] (T om) Flexor digitorum longus [FDL] (Dick) Flexor hallucis longus [FHL] (Harry) FIRST PLANTAR LAYER of FOOT Abductor hallucis: Medial plantar Flexor digitorum brevis [FDB]: Medial plantar Abductor digiti minimi: Lateral plantar SECOND PLANTAR LAYER of FOOT Quadratus plantae: Lateral plantar Lumbricals: Medial lateral plantar THIRD PLANTAR LAYER of FOOT Flexor hallucis brevis [FHB]: Medial plantar Adductor hallucis: Lateral plantar Flexor digitorum minimus brevis [FDMB]: Lateral plantar FOURTH PLANTAR LAYER of FOOT Dorsal interosseous: Lateral plantar Plantar interosseous: Lateral plantar Nerve to Quadratus femoris (L4-S1): Sensory: NONE Motor: Quadratus femoris Inferior gemelli Nerve to Obturator internus (L5-S2): exits greater sciatic foramen Sensory: NONE Motor: Obturator internus Superior gemelli Pudendal (S2-4): exit greater then re-enters lesser sciatic foramen Sensory: Perineum: via Perineal (scrotal/labial branches) via Inferior rectal nerve via Dorsal nerve to penis/clitoris Motor: Bulbospongiosus: Perineal nerve Ischiocavernosus: Perineal nerve Urethral sphincter: Perineal nerve Urogenital diaphragm: Perineal nerve Sphincter ani externus: Inferior rectal nerve Nerve to Coccygeus (S3-4) Sensory: NONE Motor: Coccygeus 1. 2. 3. 4. 5. Motor: Levator ani POSTERIOR DIVISION 6. Common Peroneal (L4-S2): in groove between biceps lateral head of Gastrocnemius. Wraps around fibular head, deep to peroneus longus; the divides Sensory: Proximal lateral leg: via Lateral sural nerve Distal lateral leg dorsal foot: via Superficial peroneal Lateral foot: via Sural (lateral calcaneal dorsal cutaneous branches) 1st/2nd interdigital space: Deep peroneal Motor: POSTERIOR THIGH Biceps femoris [short head] ANTERIOR COMPARTMENT of LEG: Deep Peroneal Tibialis anterior [TA] Extensor hallucis longus [EHL] Extensor digitorum longus [EDL] Peroneus tertius LATERAL COMPARTMENT of LEG: Superficial Peroneal Peroneus longus Peroneus brevis FOOT : Deep Peroneal Extensor hallucis brevis [EHB] Extensor digitorum brevis [EDB] Superior Gluteal (L4-S1): Sensory: NONE Motor: Gluteus medius Gluteus minimus T ensor fascia lata Inferior Gluteal (L5-S2): Sensory: NONE Motor: Gluteus maximus Nerve to piriformis (S2): Sensory: NONE Motor: Piriformis 10. Posterior Femoral Cutaneous Nerve [PFCN] (S1-3) Sensory: Posterior thigh Motor: NONE 7. 8. 9. Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier ARTERIES ARTERY COURSE BRANCHES COMMENT Vertebral Major arterial supply of cervical spine and cord. Off both subclavian through transverse foramen of C1-6 Anterior and posterior segmental medullary Feed Anterior Posterior spinal arteries respectively Anterior spinal Forms superiorly from both vertebrals Posterior spinal Each branch superiorly from vertebrals Ascending cervical From Thyrocervical Contributes to Anterior Posterior spinal arteries via segmental medullary arteries Deep cervical From Costocervical Contributes to Anterior Posterior spinal arteries via segmental medullary arteries Segmental/Intercostal Branch from aorta Dorsal branch Dorsal branch Spinal branch Ventral branch Major anterior segmental medullary (Adamkiewicz Artery) Supplies dura, posterior elementsSupplies cord and bodies Supplies vertebral bodies Supplies inferior thoracic superior, L-spine, feeds anterior spinal artery in L-spine Spinal branch Along vertebral bodies Anterior segmental medullary Posterior segmental medullary Radicular arteries (Anterior Posterior) On ventral root; feeds anterior spinal artery Feeds posterior spinal arteries Along nerve roots, do not feed spinals Anterior segmental medullary On Posterior On ventral root; feeds anterior Lumbar arteries Branch from aorta segmental medullary Radicular arteries (Anterior Posterior) spinal artery Feeds Posterior spinal arteries Anterior segmental medullary Along nerve roots Anterior spinal artery Anterior radicular arteries Single artery, runs midline Do not feed spinal arteries Posterior segmental medullary Along nerve roots Posterior spinal artery Posterior radicular arteries Paired arteries (left/right) Do not feed spinal arteries Anterior spinal Midline anterior surface of cord Supplies anterior 2/3 of cord; has multiple contributions from segmental arteries Sulcal branches Pial arterial plexus Supplies center of cord Supplies cord peripheries Posterior spinal Off midline (L R) Supplies post 1/3 of cord; has multiple contributions from segmental arteries Each nerve root has either a segmental medullary or a radicular artery associated with it. Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier DISORDERS DESCRIPTION H P WORK-UP/FINDINGS TREATMENT CAUDA EQUINA SYNDROME • Compression of cauda equina • Etiology: usually a large midline disc herniation • A surgical emergency HxPE: Back, buttock, leg pain. Bladder (#1) and bowel dysfunction. Leg numbness paralysis XR: no emergent need MR (or myelography): to show compression Immediate surgical decompression (when diagnosis is confirmed) CERVICAL SPONDYLOSIS • Disc degeneration with vertebral and facet arthritis • 3 pain sources: disc, ligament, root (HNP) • C5-6 #1 site • PLL ossifies, results in stenosis (most common in Asians) Hx: Older, men. Neck UE pain, stiffness or grinding. PE: Decreased ROM, midline neck TTP. Radicular or myelopathic signs if HNP or cord compressed XR: AP, lateral: 1. Osteophytes 2. Spinal stenosis 3. Disc space narrowed 4. Facet osteoarthritis 5. Instability 1. Discogenic: soft collar, NSAID, Physical therapy, +/-traction 2. Persistent radiculopathy or myelopathy: decompression and fusion (not for discogenic pain) CERVICAL STRAIN/MUSCLE STRAIN (Whiplash) • Not a sprain. Soft tissue (muscle/ligament) strain • Etiology: trauma or some minor movement Hx: Stiffness, pain (dull/nonradiating) in neck traps PE: Paraspinal muscles tender to palpation (+/-spasm). Spurling test XR: if history of trauma or neurologic or persistent symptoms 1. Soft collar immobilization (Philadelphia collar) 2. NSAID, muscle relaxant 3. +/- Ice, heat, massage DEGENERATIVE DISC DISEASE (DDD) • Aging process: disc desicates and tears. Facet degeneration and sclerosis • Associated with tobacco use Hx: Chronic LBP (+/-buttock), stiffness (worse with activity) PE: Back tender to palpation +/-Waddell's signs. XR: AP, lateral: aging, osteophytes, disc space narrowed, “vacuum sign” 1. NSAIDs (no narcotics) 2. Antidepressants if indicated 3. Physical therapy, exercise, weight control HERNIATED CERVICAL DISC (Herniated nucleus pulposus) • Nucleus pulposus protrudes presses on root. • Usually posterolateral at C5-6 or C6-7. Hx: Y oung or middle age. Numbness radiating pain. PE: 1weakness, decreased sensation reflexes, 1 Spurling test XR: AP, lateral: spondylosis MR: bulging nucleus pulposus 1. Soft collar, rest 2. Physical therapy, NSAIDs 3. Surgical decompression DESCRIPTION H P WORK-UP/FINDINGS TREATMENT HERNIATED LUMBAR DISC (HNP) • DDD annulus tear: nucleus herniates, +/-root or cauda compression. • Can be Asymptomatic • L4-5 most common • Most posterolateral (PLL weak) Hx: DDD sx (+/-radicular sx). Increased with sneeze, decreased with hip flexion PE: Root weakness, decreased sensation reflexes, 1straight leg bowstring tests. XR: AP, lateral: age changes EMG/NCS: + after 3 weeks MR: shows herniation 1. Bed rest, NSAIDs 2. Physical therapy, fitness program 3. Discectomy 4. Cauda Equina Syndrome: a surgical emergency DESCRIPTION H P WORK-UP/FINDINGS TREATMENT LUMBAR BACK SPRAIN/MUSCLE STRAIN • Strain or lifting injury • Soft tissue injury (muscle spasm, ligament or tendon injury, disc tear-without bulge) Hx: LBP (+/- radiation to buttock, not leg), paraspinous spasm tenderness PE: Normal neurologic exam XR: if neurologic symptoms present or refractory to treatment 1. Rest (1-2 day bed rest), NSAIDs (no narcotics) 2. Physical therapy 3. Increase fitness SCHEUERMANN'S DISEASE • Increased thoracic kyphosis (Cobb angle 45°) with 3 vertebrae with anterior wedging • Unknown etiology • Schmorl nodes (cartilage) in the vertebral body Hx: Adolescent with poor posture, +/-back pain PE: “rounded back” on examination, usually nontender to palpation XR: AP, lateral T-spine: 1. Increased kyphosis 2. Anterior wedging (3) 3. Schmorl nodes Immature: exercise, brace or orthosis Mature: Anterior release and posterior fusion SCOLIOSIS • Lateral spine curve (+/-rotation) • Multiple etiologies: #1 idiopathic • Girls.boys (needing tx) • Find on school screening • Progression: based on skeletal maturity, curve angle Hx: +/-pain, fatigue, visible physical deformity. PE: Neurologic exam usually normal. 1forward bend test. Determine plumb line (hang string from C7) XR: Full length AP, lateral: Lateral curve on AP. Measure Cobb angle: angle between lines drawn perpendicular to most superior inferior affected vertebrae Curves: 1. 30° observation 2. 30-40° bracing 3. 40° surgery: spinal fusion. SPINAL STENOSIS • Congenital vs. acquired (most common) • Canal narrowing with symptoms • Etiology: DDD or facet osteoarthritis ligament laxity Hx: Neurogenic claudication (fatigue), +/-pain; Back extension reproduces sx. PE: Weakness, decreased pin prick reflexes XR: AP, lateral: age changes CT/MR: better to evaluate canal, shows stenosis 1. Physical Therapy: abdominal strength back flexion exercises 2. NSAIDs (+/-steroids) 3. Laminectomy SPONDYLOLISTHESIS • Forward slipped vertebrae • 6 Types (common sites): 1. Congenital: facet defect (S1) 2. Isthmic (most common): pars defect (L5-S1; Hx: Type: I (peds), II (young), III (elderly). Mechanical back pain, +/-radicular symptoms XR: AP, lateral: measure forward slippage for grade (I-V, 0-100°) Type: 1. Scottie dog: long 1. Activity modification, rest, NSAIDs 2. Flexion exercises Surgical 2. defect (L5-S1; associated with hyperextention); 3. Degenerative: facet arthropathy (L4-5) 4. Traumatic 5. Pathologic 6. Post-surgical PE: +/-palpable step-off spasm. +/-radicular signs (e.g. weakness, decreased sensation reflexes) neck 2. Scottie dog: broken neck 3. Facet arthritis 3. Surgical decompression and fusion for progressive slippage or radicular symptoms SPONDYLOLYSIS • Defect or stress fracture (without slippage) in pars interarticularis • Leads to spondylolisthesis • L5 most common site Hx: Y oung, athlete (football, gymnast). Low back pain, worse with activity (#1 cause in pediatrics) XR: Oblique L-spine “Scottie dog has a collar” 1. Symptomatic treatment 2. Activity restriction, +/-brace 3. Back muscle strengthening TUMORS Metastatic are most common. Most common primary: Multiple Myeloma (malignant) Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier PEDIATRIC DISORDERS DESCRIPTION EVALUATION TREATMENT/COMPLICATIONS MYELODYSPLASIA • Neural tube (closure) defect; • No function below level of lesion; level determines function (L1 paraplegic/S1 near normal) • Associated with increased AFP • Associated with many deformities Hx: Some have family history PE/XR: Depends on type of defect: 1. Spina bifida occulta 2. Meningocele 3. Myelomeningocele 4. Rachischisis Must individualize for each patient: Most need ambulation assistance, orthoses, surgical releases, etc.Common problems requiring treatment: Deformities and/or contractures of spine, hips, knees, ankles, and feet SCOLIOSIS • Lateral spine curve +/- rotation • Multiple etiologies: #1 idiopathic • Cases needing tx: girls boys • Curve progression predicted: 1. Angle of curve 2. Skeletal maturity (Risser stages: iliac Apophysis) Hx: +/- pain fatigue, visible deformity, found in school screening PE: + forward bend test (asymmetric). Neurologic exam usually normal. Determine plumb line from C7 XR: AP full length: measure Cobb angle. (See Disorder T able) Based on curves and Risser stage; 1. 30°: observation (most) 2. 30-40°:bracing (Boston, for apex below T8 vs. Milwaukee brace) 3. 40°: spinal fusion TORTICOLLIS • Contracture of SCM • Associated with other disorders • Associated with intrauterine position Hx: Parents note deformity PE: Head tilted to one side, chin to opposite side, 1/2facial asymmetry XR: Spine hips: rule out 1. Physical therapy/stretching of the sternocleidomastoid 2. Surgical release if persistent Complication: poor eye intrauterine position • Etiology: several theories XR: Spine hips: rule out other anomalies development Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier SURGICAL APPROACHES USES INTERNERVOUS PLANE DANGERS COMMENT ANTERIOR APPROACH 1. Herniated disc removal 2. Vertebral fusion 3. Osteophyte removal 4. Tumor or biopsy Superficial: 1. SCM (CN 11) 2. Strap muscles (C1-3) Deep: Between left and right Longus colli muscles 1. Recurrent laryngeal nerve 2. Sympathetic nerve 3. Carotid artery 4. Internal jugular 5. Vagus nerve 6. Inferior thyroid artery • Access C3 to T1 • Right recurrent laryngeal nerve more susceptible to injury-most choose approach on left side. • Thyroid arteries limit extension of the approach USES INTERNERVOUS PLANE DANGERS COMMENT POSTERIOR APPROACH CERVICAL 1. Posterior fusion 2. Herniated disc 3. Facet dislocation Left and Right paracervical muscles (posterior cervical rami) 1. Spinal cord 2. Nerve roots 3. Posterior rami 4. Vertebral artery 5. Segmental vessels 1. Most common c-spine approach 2. Mark the level of pathology with a radiopaque marker pre-op to assist finding the appropriate level intraoperatively LUMBAR 1. Herniated disc 2. Explore nerve roots Left and Right paraspinal muscles (dorsal rami) Segmental vessels to paraspinals Incision is along the spinous processes. Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com CHAPTER 2 - SHOULDER TOPOGRAPHIC ANATOMY OSTEOLOGY TRAUMA JOINTS MINOR PROCEDURES HISTORY PHYSICAL EXAM MUSCLES: INSERTIONS AND ORIGINS MUSCLES: BACK/SCAPULA REGION MUSCLES: ROTATOR CUFF MUSCLES: DELTOID/PECTORAL REGION NERVES ARTERIES DISORDERS SURGICAL APPROACHES Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier CHAPTER 2 – SHOULDER TOPOGRAPHIC ANATOMY Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier OSTEOLOGY CHARACTERISTICS OSSIFY FUSE COMMENT CLAVICLE • Cylindrical; S shaped • Middle: narrowest, no ligament attachments Primary (2) (medial/lateral) Secondary (sternal/acromial) 7 weeks fetal 18-20 years 9 weeks fetal 25 years (sternal) 19-20 yrs (acromial) • Clavicle is first to ossify, last to fuse • It starts as intramembranous ossification, ends as membranous. SCAPULA • Flat, triangular shape • Only attachments to axial skeleton are muscular. 1. Body 2. Coracoid 3. Coracoid/glenoid 4. Acromion 5. Inferior angle 8 weeks (fetal) 1 year 15 yrs 15 yrs 16 yrs All fuse between 15-20 years Blood supply: 1. Subscapular (and circumflex scapular arteries) 2. Suprascapular artery Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier TRAUMA DESCRIPTION EVALUATION CLASSIFICATION TREATMENT CLAVICLE FRACTURE • Most common fracture • Fall on shoulder or direct blow. • Football, hockey • Rare neurovascular damage (subclavians) HX: Trauma. Cannot raise arm. Pain. PE: Gross deformity at fracture site with ttp. Must do neurological and vascular exams. XR: AP and 45° cephalad Group II: stress views I. Middle 1/3: 80% II Distal 1/3: 15% Type I: minimally displaced; between ligaments. Type II: Displaced, fracture medial to CC ligament. Type IIA: CC ligaments both attached to distal fragment Type IIB: Conoid ruptured Trapezoid ligament attached. Type III: Fracture through AC joint. Ligaments intact. III Proximal 1/3: 5% Closed treatment (no reduction) with figure of eight brace or sling for mid/ proximal 1/3, distal 1/3 (Types I and III) (3-4 weeks; ROM) Open treatment for Type II to prevent nonunion. (also open fracture, vascular injury) COMPLICATIONS: Nonunion: esp. with distal 1/3: type II injury; Brachial plexus (medial cord/ulnar nerve) or subclavian injury; Pneumothorax. SCAPULAR FRACTURE • Relatively uncommon • Males-young • High-energy trauma • 85% w/associated injuries (including HX: Trauma. Pain in back and/or shoulder. PE: Swelling and tenderness to palpation XR: AP/Axillary Anatomic classification: A-G Idleberg (glenoid fracture) Type I: Anterior avulsion fracture Type II: Tranverse/oblique fracture thru glenoid; exits inferiorly Type III: Oblique Closed treatment with a sling for 2 weeks for most fractures. Then early ROM. ORIF for intraarticular fx (including severe) • Dx often delayed due to associated injuries (esp pulmonary great vessels). XR: AP/Axillary lateral/ scapular Y ; CXR CT : intraarticular glenoid fracture through glenoid, exits superiorly Type IV: Transverse fracture exits through the scapula body Type V: Types II + IV ORIF for intraarticular fx and/or large displaced (25%) fragments COMPLICATIONS: Associated injuries: Rib fracture #1, pneumothorax, pulmonary contusion, vascular injury, brachial plexus inury; AC injury (esp w/type III; acromion fx); Suprascapular nerve injury DESCRIPTION EVALUATION CLASSIFICATION TREATMENT ACROMIOCLAVICULAR (AC) SEPARATION • Separation is subluxation or dislocation of AC joint • Fall onto acromion • Contact sports: hockey football, wrestling • Males HX: Trauma. Range of pain: minimal to severe. PE: AC joint TTP, gross deformity with grade III up. XR: AP, stress view: grade II vs. grade III I: normal, II: minimal separation, III and up: clavicle displaced. 6 Grades: (based on ligament tear clavicle position) Grade I: Sprain, AC ligament intact Grade II: AC tear, CC sprain Grade III: AC/CC (both) torn AC joint is dislocated. Grade IV: III with clavicle posterior into/thru trapezius muscle Grade V: III with clavicle elevated 100% superiorly Grade VI: III with clavicle inferior Grade I, II: sling until pain subsides (+/-injection/pain medication) for 1-2 wks, then increase ROM Grade III: nonoperative for most; operative for laborers/athletes Grade IV-VI: Open reduction and repair. COMPLICATIONS: Permanent deformity; Stiffness, early OA; Distal clavicle osteolysis (pain); Associated injuries: Fracture, pneumothorax. GLENOHUMERAL DISLOCATION • Anterior: Abd/ER injury 2 mechanisms 1. TUBS [Traumatic Unilateral, Bankart lesion, Surgery] 2. AMBRI [Atraumatic Multi-directional, Bilat- eral, responds to Rehab, Inferior capsule repair) 20 yo: 80% recur Hill Sachs Bankart lesions predisposed to recurrence • Posterior: after seizure often missed HX: Trauma or hx of shoulder slipping out. Intense pain. PE: Deformity, flattened shoulder silhouette. Exquisitely tender. Do full neurovascular PE XR: AP/axillary lateral (also Stryker notch) Anterior: Hill Sacks Lesion Posterior: Rev Hill Sachs, “empty glenoid” MRI: Bankart lesion (anterior/inferior labral tear) Anatomic Classification: where humeral head is: • Anterior (90%) • Posterior (5%) • Inferior (luxatio erecta) very rare • Superior: very, very rare Reduce dislocation: Pre and Post neurological exam Conscious sedation (IV benzo + narcotic) Methods: 1. Traction/counter-traction 2. Hippocratic 3. Stimson 4. Milch Immobilize (2-6 weeks), rehabilitation Surgery for recurrent/TUBS, posterior dislocation 3 wks COMPLICATIONS: Recurrence rate (young age predicts it, decreases w/increased age); Axillary nerve injury; Rotator cuff tear; Glenoid/Greater tuberosity fracture; Dead arm syndrome Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier JOINTS JOINT TYPE LIGAMENTS COMMENTS Glenohumoral Spheroidal Ball and Socket Highly mobile, decreased stability (needs Rotator cuff); #1 dislocated joint (anterior 90%) Capsule Loose, redundant, with gaps; minimal support Coracohumoral Provides anterior support Glenohumoral Discrete capsular thickenings; 3 ligaments: superior, middle, inferior-strongest Glenoid labrum Increases surface area depth of glenoid. Injuries: SLAP lesion/Bankart lesion Transverse humeral Holds biceps (LH) tendon in groove Sternoclavicular Double sliding Capsule Anterior and Posterior SC ligaments Posterior stronger; Anterior dislocation more common Interclavicular Costoclavicular Strongest SC ligament Acromioclavicular [AC joint] Plane/Gliding Capsule has a disc in joint; Acromioclavicular Horizontal stability; torn in Grade II AC injury Coracoacromial Can cause impingement Coracoclavicular Vertical stability; torn in Grade III AC injury Trapezoid Anterior/lateral position Conoid Posterior/medial position; stronger Scapulothoracic not an articulation Allows scapula to move along the posterior rib cage. Other ligaments Superior transverse Separates Suprascapular Artery Other ligaments transverse scapular and Nerve STRUCTURE FUNCTION MUSCLES ROTATOR CUFF Holds humeral head in glenoid Supraspinatus Most commonly torn tendon Infraspinatus T eres Minor Subscapularis Anterior support LIGAMENTS Capsule Rotator cuff tendons fused to it Glenohumeral Superior: resists inferior translation Middle: resists anterior translation Inferior: resists ant/inf translation Coracohumeral Resists post/inferior translation Labrum Deepens glenoid Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier MINOR PROCEDURES STEPS INJECTION OF THE ACROMIOCLAVICULAR (AC) JOINT 1. Ask patient about allergies 2. Palpate clavicle distally to AC joint (sulcus) 3. Prepare skin over AC joint (iodine/antiseptic soap) 4. Anesthetize skin with local (quarter size spot) 5. Use 21 gauge or smaller, insert needle into joint vertically. Aspirate to ensure not in a vessel, then inject 2ml of 1:1 local/ corticosteroid preparation into AC joint. (Y ou will feel the needle "pop/give" into the joint) 6. Dress injection site INJECTION OF SUBACROMIAL SPACE 1. Ask patient about allergies 2. Palpate the acromion: define it's borders 3. Prepare skin over shoulder (iodine/antiseptic soap) 4. Anesthetize skin with local (quarter size spot) 5. Hold finger (sterile glove) on acromion, insert needle under posterior acromion w/cephalad tilt. Aspirate to ensure not in a vessel, then inject 5-10cc of preparation-will flow easily if in joint). Use: a. diagnostic injection: local only b. therapeutic injection: local/corticosteroid 5:1 6. Dress injection site GLENOHUMERAL ARTHROCENTESIS 1. Palpate the coracoid process/humeral head 2. Prepare skin over shoulder (iodine/antiseptic soap) 3. Anesthetize skin (quarter size spot) 4. Abduct arm/downward traction (by an assistant) 5. Insert needle between humeral head and coracoid process 6. Synovial fluid should aspirate easily 7. Dress insertion site Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier HISTORY QUESTION ANSWER CLINICAL APPLICATION 1. AGE OLD YOUNG Rotator cuff tear/impingement, arthritis (OA), adhesive capsulitis (frozen shoulder), humerus fracture (after trauma) Instability, AC injury, osteolysis, impingement in athletes 2. PAIN a. Onset b. Location c. Occurrence d. Exacerbating /relieving Acute Chronic On top/AC joint Night pain Overhead worse Overhead better Fracture, rotator cuff tear, acromioclavicular injury, dislocation Impingement, arthritis AC joint arthrosis Classic for Rotator Cuff tear, tumor Rotator Cuff tear Cervical radiculopathy 3. STIFFNESS Y es Osteoarthritis, adhesive capsulitis 4. INSTABILITY “Slips in and out” Dislocation: 90% anterior - occurs with abduction external rotation (e.g. throwing motion) 5. TRAUMA Direct blow Fall on outstretched hand Acromioclavicular injury Glenohumeral dislocation Overhead usage Osteolysis (distal clavicle) 6. WORK/ACTIVITY Weight lifting Athlete: throwing type Long term manual labor Osteolysis (distal clavicle) Rotator cuff tear/impingement Arthritis (OA) 7. Neurologic Symptoms Numbness/tingling/ “heavy” Thoracic outlet syndrome, brachial plexus injury 8. PMHx Cardiopulmonary/GI Referred pain to shoulder Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier PHYSICAL EXAM EXAM TECHNIQUE/FINDINGS CLINICAL APPLICATION INSPECTION Symmetry Compare both sides Wasting Loss of contour/muscle mass Rotator Cuff tear Gross deformity Superior displacement Acromioclavicular injury (separation) Gross deformity Anterior displacement Anterior dislocation (glenohumeral joint) Gross deformity "Popeye" arm Biceps tendon rupture (usually proximal end of long head) PALPATION AC joint Feel for end of clavicle Pain indicates Acromioclavicular pathology Subacromial bursa Feel acromion-down to acromiohumeral sulcus Pain: bursitis and/or supraspinatus tendon rupture Coracoclavicular ligament Feel between acromion coracoid Pain indicates impingement Greater tuberosity Prominence on lateral humeral head Pain indicates Rotator Cuff tendinitis Biceps tendon Feel proximal insertion on humerus Pain indicates biceps tendinitis RANGE OF MOTION Forward flexion Arms from sides forward 0-160° normal Abduction Arms from sides outward 0-160/180° normal Internal rotation Reach thumb up back-note level Mid thoracic normal-compare sides External rotation 1. Elbow at side, rotate forearms lateral 2. Abduct arm to 90°, externally rotate up 30-60° normal External rotation decreased in adhesive capsulitis Rotator Cuff tear: AROM decreased, PROM ok, Adhesive Capulitis: both are decreased NEUROVASCULAR Sensory Light touch, pin prick, 2 pt Supraclavicular nerve (C4) Superior shoulder/ clavicular area Deficit indicates corresponding nerve/root lesion Axillary nerve (C5) Lateral shoulder Deficit indicates corresponding nerve/root lesion T2 segmental nerve Axilla Deficit indicates corresponding nerve/root lesion Motor Spinal accessory (CN11) Resisted shoulder shrug Weakness = Trapezius or corresponding nerve lesion. Suprascapular (C5-6) Resisted abduction Weakness = Supraspinatus or corresponding nerve/root lesion. Resisted external rotation Weakness = Infraspinatus or corresponding nerve/root lesion. Axillary nerve (C5) Resisted abduction Weakness = Deltoid or corresponding nerve/root lesion. Resisted external rotation Weakness = T eres minor or corresponding nerve/root lesion. Dorsal scapular Shoulder shrug Weakness = Lev Scap/Rhomboid or nerve (C5) Shoulder shrug nerve/root lesion. Thoracodorsal nerve (C7-8) Resisted adduction Weakness = Latissimus dorsi or nerve/root lesion. Lateral pectoral nerve (C5-7) Resisted adduction Weakness = Pectoralis major or corresponding nerve/root lesion. U/L subscabular nerve (C5-6) Resisted internal rotation Weakness = T eres min or subscapularis or nerve/root lesion. Long thoracic nerve (C5-7) Scapular protraction /reach Weakness = Serratus anterior or nerve/root lesion EXAM TECHNIQUE/FINDINGS CLINICAL APPLICATION SPECIAL TESTS Supraspinatus (empty can) Bilateral:30°add,90°FF,IR,resist down force Weakness indicates Rotator cuff (supraspinatus) tear, impingement Drop Arm Passively abduct 90°, lower slowly Weakness or arm drop indicates rotator cuff tear Liftoff Hand behind back, push posteriorly Weakness or inability indicates subscapularis rupture Speed Resist forward flexion of arm Pain indicates biceps tendinitis Y ergason Hold hand, resist supination Pain indicates biceps tendinitis, biceps tendon subluxation Impingement sign (Neer) Forward flex greater than 90° Pain indicates Impingement Syndrome Hawkins sign Forward flex 90°, elbow @ 90°, then IR Pain indicates Impingement Syndrome Cross Body Adduction 90°Forward flex then adduct arm across body Pain indicates Acromioclavicular pathology, Decreased ROM indicates tight posterior capsule AC Shear Cup hands over clavicle/scapula: then squeeze Pain/movement indicates AC pathology Active Compression (O’Brien's) 90°FF, max IR, then adduct/flex Pain or pop indicates a SLAPlesion Load and shift Push into glenoid, translate ant/post Motion indicates instability in that direction (anterior vs. posterior) Apprehension sign Throwing position- continue to externally rotate Apprehension indicates anterior instability Relocation (Jobe) 90°abd, full ER, posterior force on humeral head Relief of pain/apprehension, or increased externalrotation indicates anterior instability Posterior Apprehension sign FF 90°,internally rotate, posterior force Apprehension indicates posterior instability Inferior instability Abd 90°, downward force on mid- humerus Slippage of humeral head or apprehension: inferior instability or Multidirectional instability Sulcus sign Arm to side, downward traction Increased acromiohumeral sulcus: inferior instability or Multidirectional instability Adson Palpate radial pulse, rotate neck to ipsilateral side Reproduction of symptoms indicates thoracic outlet syndrome Roo (EAST) Bilateral arm: abduct/ER, open and close fist 3 minutes Reproduction of symptoms indicates thoracic outlet syndrome Spurling Lateral flex/axial compression of neck Reproduction of symptoms indicates cervical disc pathology Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier MUSCLES: INSERTIONS AND ORIGINS CORACOID PROCESS GREATER TUBERCLE ANTERIOR PROXIMAL MEDIAL EPICONDYLE LATERAL EPICONDYLE ORIGINS INSERTIONS INSERTIONS ORIGINS ORIGINS Biceps (SH) Supraspinatus Pectoralis major Pronator T eres Anaconeus Corcobrachialis Infraspinatus Latissimus dorsi Common Flexor Common. Extensor INSERTIONS T eres minor T eres major T endon [FCR, PL, T endon [ECRB,ED, Pectoralis minor FCU, FDS] EDM, ECU] Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier MUSCLES: BACK/SCAPULA REGION MUSCLE ORIGIN INSERTION NERVE ACTION COMMENT Trapezius C7-T12 spinous process Clavicle, Acromion spine of scapula Cranial nerve XI Elevate rotate scapula Connect UE to spine Latissimus dorsi T7-T12, iliac crest Humerus (intertubercular groove) Thoracodorsal Adduct, extend arm, IR humerus Connect UE to spine Levator scapulae C1-C4 transverse process Superior medial scapula Dorsal scapular/ C3-4 Elevates scapula Connect UE to spine Rhomboid minor C7-T1 spinous process Medial scapula (at the spine) Dorsal scapular Adduct scapula Connect UE to spine Rhomboid major T2-T5 spinous process Medial scapula Dorsal scapular Adduct scapula Connect UE to spine Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier MUSCLES: ROTATOR CUFF SPACE BORDERS STRUCTURES Triangular Space T eres Minor Circumflex Scapular Artery T eres Major Triceps (Long Head) Quadrangular Space T eres Minor Axillary Nerve T eres Major Posterior Circumflex Artery Triceps (Long Head) Humeral Artery Triceps (Lateral Head) Triangular Interval T eres Major Radial Nerve Triceps (Long Head) Deep Artery of Arm Triceps (Lateral Head) MUSCLE ORIGIN INSERTION NERVE ACTION COMMENT Deltoid Clavicle, Acromion spine of scapula Humerus (Deltoid tuberosity) Axillary Abduct arm Atrophy: Axillary nerve damage T eres major Inferior angle of the scapula Humerus (intertubercular groove) Lower subscapular IR, adduct arm Protects radial nerve in posterior approach Rotator Cuff(4) 1.Supraspinatus Supraspinatus fossa (scapula) Greater tuberosity (superior) Suprascapular Abduct arm (initiate), Trapped in impingement #1 torn tendon (RC tear) 2.Infraspinatus Infraspinatus fossa (scapula) Greater tuberosity (middle) Suprascapular ER arm, stability Weak ER: damage to nerve. lesion in notch Dissection 3.T eres Minor Lateral scapular Greater tuberosity (inferior) Axillary ER arm, stability Dissection can damage circum-flex vessels 4.Subscapularis Subscapular fossa (scapula) Lesser tuberosity Upper Lower Subscapular IR, adduct arm, stability Can rupture in anterior dislocation Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier MUSCLES: DELTOID/PECTORAL REGION MUSCLE ORIGIN INSERTION NERVE ACTION COMMENT Deltoid Clavicle, Acromion, spine of scapula Humerus (Deltoid tuberosity) Axillary Abduct arm Atrophy: Axillary nerve damage Pectoralis major 1.Clavicle 2.Sternum Humerus (intertubercular groove) Lateral/medial pectoral Adducts arm, IR humerus Can rupture during weight lifting Pectoralis minor Ribs 3-5 Coracoid process (scapula) Medial pectoral Stabilizes scapula Divides Axillary artery into 3 parts Serratus anterior Ribs 1-8 (lateral) Scapula (antero-medial border) Long thoracic Holds scapula to chest wall Paralysis indicates wing scapula Subclavius Rib 1 (and costal cartilage) Clavicle (inferior border/mid 3rd) Nerve to subclavius Depresses clavicle Cushions sub- clavian vessels Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier NERVES BRACHIAL PLEXUS • C5-T1 ventral rami Variations: C4 (prefixed) T2 (post-fixed) • Rami (Roots), Trunks, Divisions, Cords, Branches (Rob Taylor Drinks Cold Beer) • Supraclavicular (rami trunks) portion in posterior triangle of neck Rami exit between Anterior Medial Scalene, then travel with Subclavian artery in axillary sheath • Divisions occur under (posterior) to clavicle and subclavius muscle Anterior Divisions: Flexors Posterior Divisions: Extensors • Infraclavicular (cords branches) portion in the axilla 1. Spinal Accessory (CN11,C1-C6): in posterior cervical triangle on levator scapulae Sensory: NONE Motor: Trapezius, Sternocleidomastoid CERVICAL PLEXUS 2. Supraclavicular(C2-3): splits into 3: anterior middle, posterior branches Sensory: over clavicle, outer trap, deltoid Motor: NONE BRACHIAL PLEXUS SUPRACLAVICULAR [approach through posterior triangle] INFRACLAVICULAR [approach through axilla] LATERAL CORD ROOTS •Lateral root to Median nerve 3.Dorsal Scapular (C3, 4, 5): pierces middle scalene, deep to Levator 7. LateralPectoral(C5-7):named for cord,runs with pectoral artery Sensory: NONE Scapulae Motor: Pectoralis Major Sensory: NONE Pectoralis Minor Motor: Levator scapulae MEDIAL CORD Rhomboid Minor and Major •Medial root to Median nerve 4.Long Thoracic(C5-7): on anterior surface of Serratus Anterior. Runs with lateral thoracicartery 8. MedialPectoral(C8-T1): named for cord Sensory: NONE Motor: Pectoralis Minor Sensory: NONE Pectoralis Major (overlying muscle] Motor: Serratus Anterior POSTERIOR CORD UPPER TRUNK 9. UpperSubscapular(C5-6) 5.Suprascapular(C5-6): thru scapular notch, under ligament Sensory: NONE Motor: Subscapularis [upper portion] Sensory: Shoulder joint 10. LowerSubscapular(C5-6) Motor: Supraspinatus Sensory: NONE Infraspinatus Motor: Subscapularis [lower portion] 6.Nerve to Subclavius (C5-6): descends anterior to plexus, posterior to clavicle T eres major 11. Thoracodorsal(C7-8): runs with thoracodorsal artery Sensory: NONE Sensory: NONE Motor: Latissimus dorsi Motor: Subclavius 12. Axillary(C5-6):with posterior circumflex humeral arterythrough Quadranglar space. Injured in Anterior dislocations, or proximal humerus fractures Sensory: Lateral upper arm: via Superior Lateral Cutaneous Nerve of arm Motor: Deltoid: via deep branch T eres minor: via superficial branch Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier ARTERIES TRUNK BRANCH COURSE/COMMENT Thyrocervical Trunk Suprascapular Over superior transverse scapular ligament. Infraspinatous branch Bends around spine of scapula Subclavian artery comes off: Left - aorta, Right - brachiocephalic. Then goes between anterior and middle scalene muscles with brachial plexus Subclavian Artery Dorsal Scapular Splits around levator scapulae; descends medial to scapula Parts determined by pectoralis minor. Part I of the axillaryartery has 1 branch, Part II has 2 branches, Part III has 3 branches Axillary (Part I) Superior thoracic T o serratus anterior and pectoralis muscles Axillary (Part II) Thoracoacromial Clavicular branch Acromial branch Deltoid branch Courses with basilic vein Pectoral branch Lateral thoracic T o serratus anterior with Long Thoracic nerve. Axillary (Part III) Subscapular Circumflex scapular Seen posteriorly in Triangular space Thoracodorsal Follows Thoracodorsalnerve Anterior circumflex Supplies humeral head ( anterior humerus) Posterior circumflex Seen posteriorly in Quadrangular space. Injury in proximal humeral fracture. Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier DISORDERS DESCRIPTION H P WORK-UP/FINDINGS TREATMENT ADHESIVE CAPSULITIS (FROZEN SHOULDER) •Inflammatoryprocess; leads to joint fibrosis Hx: Middle age women, DM Slow onset: pain/stiffness XR: Usually normal 1.NSAIDs Arthrogram: decreased joint volume. 2.Physical therapy and home therapy program (3 month minimum) •3 stages: 1. Pain, 2. Stiffness3. Resolving; PE: Decreased active ROM passive ROM •Associated with old Colles fracture ARTHRITIS:ACROMIOCLAVICULAR (AC) JOINT •Usually osteoarthritis Hx: Pain at AC, esp. with motion XR: Osteophytes, joint narrowing 1.NSAIDs, rest 2.Distal clavicle resection (Mumford) PE: T ender to palpation ARTHRITIS:GLENOHUMORAL JOINT •Multiple etiologies: OA, RA, post-traumatic Hx: Older, pain increases with activity XR: True AP,axillary lateral: joint space narrowed 1. NSAIDs, ice/heat, ROM steroid inject controversial •Often overuse condition PE:+/- wasting, crepitus, decreased AROM 2.Refractory: hemi vs.total joint arthroscopy BICEPS TENDINITIS •Associated with impinge- ment or subluxation/transverse humeral ligament tear Hx : Pain in shoulder XR: Normal views: usually normal 1.Treat the impingement PE: T enderness along groove 2.Biceps strengthening +Speed, + Y ergason 3.T enodesis (rare procedure) BICEPS TENDON RUPTURE •Long Head of biceps rupture Hx: Old, or young weight lifter, sudden pain XR: Normal; rule out fracture 1.Old: conservative treatment Arthrogram: rule out RC tear 2.Y oung/laborer: surgery •Due to impingement, micro- trauma or trauma PE: Proximal arm bulge (Popeye arm) •Associated with RC tear BRACHIAL PLEXUS INJURY •Traction of brachial plexus Hx: Football players, parathesias in XR: Shoulder series: normal Most resolve with rest arm BURSITIS:SUBACROMIAL •Often from impingement Hx/PE: Pain at shoulder Treat the impingement IMPINGEMENT •RC (supraspinatus), Biceps tendon trapped under acromion or coracoacromial ligament Hx: Older, or athlete. Pain/inability to do overhead activity. XR: Normal views +outlet view: type III acromion or subacromial spur 1. Decrease/modify activity 2. NSAID, ROM, strengthen 3. Corticosteroid injection 4. Subacromial decompression •Associated with Type III acromion PE: +Neer,+Hawkins INSTABILITY/DISLOCATION: GLENOHUMORAL JOINT TWO TYPES 1. TUBS [Trauma Unilateral Bankart lesion, Surgery] Hx:Pain, "arm slips out" TUBS history XR: Trauma (+/-Stryker) Bankart/Hill Sachs lesion 1. Reduce (if dislocated): 3 ways. Immobilize in IR for 4 weeks, RC strengthening, then ROM •90% anterior (posterior after seizure) PE: +PE for unilateral instability (e.g. + Axillary nerve injury (esp. with anterior) •Pts 20yrs: 80% recur Apprehension, relocation) 2. Surgical repair for recurrence (notin posterior) 2. AMBRI Atraumatic Multi- directional, Bilateral, Rehab responsive, Inferior capsule repair Hx: Pain, "arms slip out" + AMBRI history XR: Trauma series 1. Reduce if dislocated: 3 ways2. Long term conservative treatment PE: +sulcus, general joint laxity in MDI 3. Life style modifications DESCRIPTION H P WORK-UP/FINDINGS TREATMENT INSTABILITY/DISLOCATION:STERNOCLAVICULAR JOINT •T ear of capsule Hx: Large force: sports/MVA, pain (anterior: ant prominence, posterior: +/- pulm, XR: May not show injury Anterior: sling/closed reduction GI) •Most anterior; Posterior rare, has increased Complications (great vessels) CT : Helpful in diagnosis Posterior: early closed reduction immobilize, PT LABRUM INJURY (SLAP LESION) Bicep tendon attachment injury I. Bicep fraying/anchor intact II.T ear in anchor (labrum) III. Bucket handle tear IV.III 1tear in bicep Hx: Pain, 1/2instability symptoms PE: 1 O’Brien test XR: Shoulder series MR/Arthroscopy to diagnose SLAP lesion By type: I . Debridement II. Reattachment III.Debridement IV.Repair vs. tenodesis LONG THORACIC NERVE INJURY •Nerve injuryresults in serratus anterior dysfunction Hx: Usually trauma PE: Winged scapula NONE Conservative treatment, most resolve within weeks/ months OSTEOLYSIS •Often in weight-lifters Hx: Pain in shoulder XR: Distal clavicle lucency 1.Activity modification. 2.Mumford PECTORALIS MAJOR RUPTURE •Maximal eccentric contraction Hx/PE: Sudden, pain, palpable defect NONE Surgical repair ROTATAR CUFF TEAR •Due to poor vascularity, overuse, micro or macro trauma, degeneration, or abnormal acromion Hx: Older; pain is deep at night, worse with overhead activity XR: Trauma series: high-riding humerus 1.Conservative: NSAID, rest, activity modification, ROM, RC strengthening •Supraspinatus most common PE: Atrophy,decreased AROM, normal PROM, + drop arm/empty can, +lift off (subscapular tear) Arthrogram (or MR/Arthrogram): Gold standard: shows communication with subdeltoid bursa 2.Surgical repair with subacromial decompression for complete tears THORACIC OUTLET SYNDROME •Compression of neuro- vascular structure (vein, artery, or plexus) between first rib and scalene muscle•Also seen with cervical ribs Hx: Women 20-50 yo. Worse with overhead activity Vein: edema, discolor,stiff Artery: cool, claudication Plexus: parathesias XR: Shoulder usually normalC-spine: Rule out massCXR: Rule out mass 1. Activity modification (until symptoms resolve)2. Posture training3. Surgery: especially for a cervical rib PE: +Adson, +Roos tests Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier SURGICAL APPROACHES USES INTERNERVOUS PLANE DANGERS COMMENT ANTERIOR (DELTOPECTORAL) APPROACH (HENRY) 1.Shoulder reconstruction 1.Deltoid [Axillary] 1.Musculocutaneous nerve 1.Keep arm adducted to avoid bringing brachial plexus into the field. 2.Biceps tendon repair. 2.Pectoralis major [lat/med pectoral] 2.Cephalic vein 3.Arthroplasty 3.Axillary nerve 2.Keep dissection to lateral side of coracobrachialis: protect MC nerve. ARTHROSCOPY PORTALS 1.Anterior “Soft spot” between biceps tendon, anterior glenoid, superior edge of subscapular tendon 1.Musculocutaneous nerve 1.Usually placed AFTER the posterior portal 2.Cephalic vein 3.Axillary nerve 2.Posterior “Soft spot”between teres minor and infraspinatus 1.Superior AC ligament 1.Primary portal for shoulder 2.RC tendons 2.Aim to coracoid when placing 3.Lateral Through deltoid 1.Axillary nerve 1.T o access subacromial space Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com CHAPTER 3 - ARM TOPOGRAPHIC ANATOMY OSTEOLOGY TRAUMA ELBOW JOINTS MINOR PROCEDURES HISTORY PHYSICAL EXAM MUSCLES: INSERTIONS AND ORIGINS ANTERIOR MUSCLES POSTERIOR MUSCLES MUSCLES: CROSS SECTION NERVES ARTERIES DISORDERS SURGICAL APPROACHES Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier CHAPTER 3 – ARM TOPOGRAPHIC ANATOMY Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier OSTEOLOGY CHARACTERISTICS OSSIFY FUSE COMMENT HUMERUS • Long bone characteristics Primary: Shaft 8-9th wk (fetal) By birth • Surgical neck: common fracture site • Lateral condyle • Blood supply 1. Epicondyle: non-articular Secondary Proximal (3): Proximal: Anterior/Posterior circumflex 2. Capitellum: articular 1. Head 17-20 yrs Middle: Nutrient artery (from Deep artery) • Medial condyle 2. Tuberosities (2) Birth 1. Epicondyle: non-articular 3-5 yrs Distal: Branches from anastomosis 2. Trochlea: articular • Elbow ossification order: Capitellum, Radial head, Medial epicondyle, Trochlea, Olecranon, Lateral epicondyle (Captain Roy Makes Trouble On Leave) 3. Cubital tunnel: covered with Osbourne's fascia. Distal (4): 1. Capitellum 1 yr 2. Medial epicondyle 4-6 yr 13-14 yrs 3. Trochlea 9-10 yr 4. Lateral epicondyle 12 yr 15-20 yrs Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier TRAUMA DESCRIPTION EVALUATION CLASSIFICATION TREATMENT PROXIMAL HUMERUS FRACTURE • Common fracture HX: Fall/trauma. Pain worse with movement Neer: based on number of fragments(parts) 1-4 1 part: sling, early motion. • Osteoporosis, elderly, female PE: Swelling, ecchymosis, good neurovascular exam Multiple combinations of fractures possible 2 part: closed reduction splint. Irreducible, intraarticular anatomic neck fx: ORIF. Greater tuberosity fx: ORIF and Rotator Cuff repair • Mechanism: 1. Elderly: fall on outstretched hand XR: Trauma series Also fracture dislocation, and intraarticular fx 3 4 part : ORIF or hemiarthroplasty (elderly) CT : shows intraarticular glenoid involvement 2. Y oung: high energy trauma (e.g. MVA, fall) MR: sensitive for AVN 4 parts: head, shaft, greater and lesser tuberosities Fracture/Dislocation: • 80% non or minimally displaced (1 part fx) Each part: 1cm displaced or 45° angulated 2 part: closed treatment except when displaced • Most heal well 3-4 part: ORIF or hemiarthorplasty • Early pendulum motion is key for full ROM Fragment displacement due to attached muscle Intraarticular: ORIF or hemiarthroplasty COMPLICATIONS: Stiffness/adhesive capsulitis; Avascular necrosis (AVN):4 part anatomic neck, axillary nerve and brachial plexus injury; axillary artery injury, nonunion DESCRIPTION EVALUATION CLASSIFICATION TREATMENT HUMERUS SHAFT FRACTURE • Common fracture HX: Trauma, fall. Severe pain, swelling Descriptive: Closed: Most fractures: coaptation splint or fracture brace for 6-8 weeks • Mechanism: direct blow or fall on outstretched arm PE: Swelling, deformity + / - radial nerve findings Location: level of humerus Open Neurovascular injury, multitrauma, pathologic fracture. Severe comminution requires plates/screws or intermedullary (IM) nail • Displacement based on fracture site relation to deltoid pectoralis major insertion XR: AP lateral arm, shoulder and elbow series Pattern: oblique, spiral, transverse • Almost 100% union Displacement or comminution • Site of pathologic fx COMPLICATIONS: Radial nerve injury (esp. Holstein/Lewis fracture, spiral fracture of distal third) most resolve. Malunion is rare. DISTAL HUMERUS FRACTURE • Uncommon HX: Pain, deformity, discoloration, swelling Displaced vs. nondisplaced Early motion important to avoid loss of motion • High morbidity PE: Swelling, ecchymosis crepitus, tenderness, good neurovascular exam Multiple types: Intercondylar: ORIF or total joint arthroplasty (closed treatment if comminuted or elderly) • Often intraarticular XR: AP lateral: posterior fat pad/sail sign Intercondylar Transcondylar: reduce, percutaneous pinning • Mechanism: fall onto hand, ulna forced into humerus CT : Optional: useful in pre-operative planning Transcondylar Others: • Intercondylar most common in adults Supracondylar Nondisplaced: closed treatment; 10-14 days and early motion. • Condylar, capitellum, Trochlea, Epicondylar all rare Condylar Displaced or comminuted (or elderly) require ORIF Capitellum Trochlea Epicondylar (medial or lateral) COMPLICATIONS: Stiffness/arthritis; Compartment syndrome; Median/Ulnar nerve injury; Brachial artery injury; Nonunion DESCRIPTION EVALUATION CLASSIFICATION TREATMENT SUPRACONDYLAR FRACTURE • Common childhood fracture HX: Fall. Pain, swelling, will not use arm. Extension (common): Undisplaced Partially displaced Fully displaced Neurovascularly intact: closed reduction and percutaneous pinning under general anesthesia (fluoroscopy) •Occurs at metaphysis, above growth plate PE: Swelling, point tenderness, + / -neurovascular signs: check distal pulses do neurologic exam Flexion (rare) Pulseless/Perfused: same • Extension type most common(90%): shaft is anterior, distal fragment is posterior XR: AP lateral (note capitellum position to anterior humeral line) Pulseless/Unperfused: open reduction exploration • Associated with signifcant morbidity; prompt treatment essential. Arteriogram: if pulseless COMPLICATIONS: Neurovascular injury: brachial artery; AIN injury; Compartment syndrome can lead to Volkmann's ischemic contracture; Deformity: cubitus varus DESCRIPTION EVALUATION CLASSIFICATION TREATMENT ELBOW DISLOCATION • Common in children and young adults HX: Fall/trauma. Pain, inability to flex elbow Location of ulna (radius) Posterior (common) Posterolateral (90%) Anterior Lateral Medial Closed reduction: + / - local anesthesia and/or conscious sedation • Y ounger, sports related fall on hand PE: Deformity, tenderness, + / -neurovascular signs. Check distal pulses neurologic exam Splint 7days for comfort, then early ROM • Associated with radial head fracture, brachial artery, median XR: AP lateral: rule out fracture Open: if unstable or with entrapped artery, median nerve injury bone or soft tissue • Both collateral ligaments ruptured Divergent (ulna and radius opposite) COMPLICATIONS Neurovascular injury: brachial artery; median or ulnar nerve; Loss of extension; Instability/redislocation; Heterotopic ossification RADIAL HEAD SUBLUXATION (NURSEMAID'S ELBOW) • Common in children Usually ages 2-4, 7 rare Hx: Pulled by hand, child will not use arm. NONE Reduce: with gentle, full supination and flexion (should feel it “pop” in). • Mechanism: child pulled or swung by hand or forearm PE: Arm held pronated/flexed. Radial head supination tender. Immobilize a recurrence • Annular ligament stretches, radial head lodges within it. XR: only if suspect fracture COMPLICATIONS: Recurrence Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier ELBOW JOINTS JOINT TYPE ARTICULATION LIGAMENTS COMMENTS ELBOW Includes 3 joints Capsule (common to all 3) Carrying angle: 10-15°valgus Ulnohumeral “Trochlear joint” Ginglymus [Hinge] Trochlea and trochlear notch Ulnar(medial) collateral: 1. Anterior band 2. Posterior band 3. Transverse band T orn in posterior dislocation Strongest: resists valgus stress Radiohumeral Trochoid [Pivot] Capitellum radial head Radial (lateral) collateral 1. Ulnar part 2. Radial part Weak Gives posterolateral stability Proximal radioulnar Radial head radial notch Annular Keeps head in radial notch Oblique cord Quadrate Supports rotary movements Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier MINOR PROCEDURES STEPS ELBOW ARTHROCENTESIS 1. Extend elbow, palpate lateral condyle, radial head and olecranon laterally; feel triangular sulcus between all three 2. Prepare skin over sulcus (iodine/antiseptic soap) 3. Anesthetize skin locally (quarter size spot) 4. May keep arm in extension or flex it. Insert needle in the “triangle” between bony landmarks 5. Fluid should aspirate easily 6. Dress injection site OLECRANON BURSA ASPIRATION 1. Prepare skin over olecranon (iodine/antiseptic soap) 2. Anesthetize skin locally (quarter size spot) 3. Insert 18 gauge needle into bursa and aspirate fluid. 4. If suspicious of infection, send fluid for Gram stain and culture 5. Dress injection site TENNIS ELBOW INJECTION 1. Ask patient about allergies 2. Flex elbow 90°, palpate ERCB distal to lateral epicondyle. 3. Prepare skin over lateral elbow (iodine/antiseptic soap) 4. Anesthetize skin locally (quarter size spot) 5. Insert 22 gauge or smaller needle into ERCB tendon at its insertion just distal to the lateral epicondyle. Aspirate to ensure needle is not in a vessel, then inject 2-3ml of 1:1 local/corticosteroid preparation. 6. Dress insertion site 7. Annotate improvement in symptoms Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier HISTORY QUESTION ANSWER CLINICAL APPLICATION 1. AGE Y oung Dislocation, fracture Middle age, elderly T ennis elbow (epicondylitis), arthritis 2. PAIN a. Onset Acute Dislocation, fracture, tendon avulsion/rupture, ligament injury Chronic Cervical spine pathology b. Location Anterior Biceps tendon rupture, arthritis Posterior Olecranon bursitis Lateral Lateral epicondylitis, fracture (especially radial head-hard to see on x-ray) Medial Medial epicondylitis, nerve entrapment, fracture, MCL strain c. Occurrence Night pain/at rest Infection, tumor With activity Ligamentous and/or tendinous etiology 3. STIFFNESS Without locking Arthritis, effusions (trauma) With locking Loose body, Lateral collateral ligament injury 4. SWELLING Over olecranon Olecranon bursitis. Other: dislocation, fracture, gout 5. TRAUMA Fall on elbow, hand Dislocation, fracture 6. ACTIVITY Sports, repetitive motion Epicondylitis, ulnar nerve palsy 7. NEUROLOGIC SYMPTOMS Pain, numbness, tingling Nerve entrapments (multiple possible sites), cervical spine pathology, thoracic outlet syndrome 8. HISTORY OF ARTHRITIDES Multiple joints involved Lupus, rheumatoid arthritis, psoriasis Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier PHYSICAL EXAM EXAM/OBSERVATION TECHNIQUE CLINICAL APPLICATION INSPECTION Gross deformity, swelling Compare both sides Dislocation, fracture, bursitis Carrying angle (normal 5-15°) Negative ( 5 degrees) Cubitus varus: physeal damage (e.g. malunion supracondylar fracture) Positive ( 15 degrees) Cubitus valgus: physeal damage (e.g. lateral epicondyle fracture) PALPATION Medial Epicondyle supracondylar line Pain: medial epicondylitis (Golfer's elbow), fracture, MCL rupture Ulnar nerve in ulnar groove Parathesias indicate ulnar nerve entrapment Lateral Epicondyle supracondylar line Pain: lateral epicondylitis (T ennis elbow), fracture Radial head Pain: arthritis, fracture, synovitis Anterior Biceps tendon in antecubital fossa Pain can indicate biceps tendon rupture Posterior Flex elbow: olecranon olecranon fossa Olecranon bursitis, triceps tendon rupture EXAM/OBSERVATION TECHNIQUE CLINICAL APPLICATION RANGE OF MOTION Elbow at side, flex extend Normal: 0-5° to 140-150°; Flex and extend Elbow at side, flex extend at elbow Normal: 0-5° to 140-150°; note if PROM AROM Pronate and supinate Tuck elbows, pencils in fists, rotate wrist Normal: supinate 90 degrees, pronate 80-90 degrees NEUROVASCULAR Sensory (LT , PP, 2 pt) Axillary nerve (C5) Superolateral arm Deficit indicates corresponding nerve/root lesion Radial nerve (C5) Inferolateral and posterior arm Deficit indicates corresponding nerve/root lesion Medial Cutaneous nerve of the Arm (T1) Medial arm Deficit indicates corresponding nerve/root lesion Motor Musculocutaneous n. (C5-6) Resisted elbow flexion Weakness = Brachialis/biceps or corresponding nerve/root lesion. Musculocutaneous n. (C6) Resisted supination Weakness = Biceps or corresponding nerve/root lesion. Median nerve (C6) Resisted pronation Weakness = Pronator T eres or corresponding nerve/root lesion. Median nerve (C7) Resisted wrist flexion Weakness = FCR or corresponding nerve/root lesion. Radial nerve (C7) Resisted elbow extension Weakness = Triceps or corresponding nerve/root lesion. Radial nerve/PIN (C6-7) Resisted wrist extension Weakness = ECRL-B/ECU or corresponding nerve/root lesion. Ulnar nerve (C8) Resisted wrist flexion Weakness = FCU or corresponding nerve/root lesion. Reflexes C5 Biceps Hypoactive/absence indicates corresponding radiculopathy C6 Brachioradialis Hypoactive/absence indicates corresponding radiculopathy C7 Triceps Hypoactive/absence indicates corresponding radiculopathy Pulses Brachial, Radial, Ulnar SPECIAL TESTS T ennis Elbow Make fist, pronate, extend wrist and fingers against resistance Pain at lateral epicondyle suggests lateral epicondylitis Golfer's Elbow Supinate arm, extend wrist Elbow Pain at medial epicondyle suggests medial epicondylitis Ligament Instability 25° flexion, apply varus/valgus stress Pain or laxity indicates LCL/MCL damage Tinel's Sign (at the elbow) T ap on ulnar groove (nerve) Tingling in ulnar distribution indicates entrapment Elbow Flexion Maximal elbow flexion for Tingling in ulnar distribution Elbow Flexion 3-5min indicates entrapment Pinch Grip Pinch tips of thumb and index finger Inability (or pinching of pads, not tips) indicates AIN pathology Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier MUSCLES: INSERTIONS AND ORIGINS CORACOID PROCESS GREATER TUBEROSITY ANTERIOR PROXIMAL HUMERUS MEDIAL EPICONDYLE LATERAL EPICONDYLE ORIGINS INSERTIONS INSERTIONS ORIGINS ORIGINS Biceps (SH) Supraspinatus Pectoralis major Pronator T eres Anconeus Coracobrachialis Infraspinatus Latissimus dorsi Common Flexor T endon Common Extensor T endon INSERTIONS T eres minor T eres major [FCR, PL, FCU, FDS] [ECRB, ED, EDM, ECU] Pectoralis minor Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier ANTERIOR MUSCLES MUSCLE ORIGIN INSERTION NERVE ACTION COMMENT Coracobrachialis Coracoid process Middle humerus Musculocutaneous Flex and adduct arm Brachialis Distal anterior humerus Ulnar tuberosity Musculocutaneous Flex forearm Often split in anterior surgical approach Biceps brachii Long Head Supraglenoid tubercle Radial tuberosity (proximal radius) Musculocutaneous Flex supinate forearm Can rupture proximally-results in Popeye arm Short Head Coracoid process Radial tuberosity (proximal radius) Musculocutaneous Flex supinate forearm Covers brachial artery Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier POSTERIOR MUSCLES MUSCLE ORIGIN INSERTION NERVE ACTION COMMENT Triceps Brachii Long Head Infraglenoid tubercle Olecranon (proximal) Radial n. Extends forearm Border of quadrangular triangular space interval Lateral Head Posterior humerus (proximal) Olecranon (proximal) Radial n. Extends forearm Border in lateral approach Medial Head Posterior humerus (distal) Olecranon (proximal) Radial n. Extends forearm One muscular plane in posterior approach Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier MUSCLES: CROSS SECTION Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier NERVES INFRACLA VICULAR [approach through axilla] LATERAL CORD 1. Musculocutaneous (C5-7): pierces coracobrachialis between bicep and brachialis. At risk for injury during anterior approach to shoulder. Sensory: NONE (in arm) Motor: ANTERIOR COMPARTMENT OF ARM Coracobrachialis Biceps brachii Brachialis MEDIAL CORD 2. Medial Cutaneous Nerve of Arm (C8-T1): joins intercostal-brachial nerve Sensory: Medial (inner) arm Motor: NONE 3.Ulnar (C(7)8-T1): travels from anterior to posterior compartment via arcade of Struthers[], then to cubital tunnel[]. Sensory: NONE (in arm) Motor: NONE (in arm) POSTERIOR CORD 4.Radial (C5-T1): runs with deep artery of arm in triangular interval, then spiral groove 15cm from elbow (injured in shaft fx; at risk in surgery), then it divides at the elbow: 1. PIN (motor), 2. superficial radial nerve (sensory) Sensory: Lateral arm: via Inferior Lateral Cutaneous Nerve of arm Posterior arm: via Posterior Cutaneous Nerve of arm Motor: POSTERIOR COMPARTMENT OF ARM Triceps [medial, long, lateral heads] Anconeus possible compression site Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier ARTERIES ANASTOMOSES AROUND THE ELBOW SUPERIOR INFERIOR Superior Ulnar Collateral Posterior Ulnar Recurrent Inferior Ulnar Collateral Anterior Ulnar Recurrent Middle Collateral (branch of Deep Artery) Interosseous Recurrent Radial Collateral (branch of Deep Artery) Radial Recurrent TRUNK BRANCH COURSE/COMMENT Brachial Artery Continuation of axillary artery Medial to biceps, runs with median nerve 1. Deep artery of arm Runs with radial nerve in radial groove (posterior humerus) 2. Nutrient humeral artery Enters nutrient canal 3. Superior ulnar collateral Branches in middle of arm, runs with ulnar nerve Anastomosis with posterior ulnar collateral at elbow 4. Inferior ulnar collateral Anastomosis with anterior ulnar collateral at elbow Brachial artery can be clamped below this branch: collateral circulation is usually sufficient. 5. Muscular 5. Muscular branches Variable, usually branch laterally 6. Radial artery These are the two terminal branches of Brachial artery, it divides in the cubital fossa. 7. Ulnar artery Deep Artery of arm Radial collateral Anastomosis with Radial recurrent artery at elbow Middle collateral Anastomosis with Recurrent interosseous artery at elbow Radial Artery Radial Recurrent Anastomosis with radial collateral artery at elbow Ulnar Artery Anterior ulnar recurrent Anastomosis with inferior ulnar collateral artery at elbow Posterior ulnar recurrent Anastomosis with superior ulnar collateral artery at elbow Common interosseous artery Recurrent interosseous artery Anastomosis with middle collateral artery at elbow Collateral branches are all superior branches, recurrent branches are all inferior branches of the anastomosis at the elbow Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier DISORDERS DESCRIPTION H P WORK-UP/FINDINGS TREATMENT ARTHRITIS • Uncommon condition Hx: Chronic pain stiffness XR: OA vs. inflammatory 1. Conservative (rest, NSAID) • Osteoarthritis seen in athletes PE: Decreased ROM tenderness Blood: RF, ESR, ANA 2. Debridement • Site for arthritides Joint fluid: crystals, cells, culture 3. Joint replacement BICEPS TENDON RUPTURE • Trauma: forced elbow flexion against resistance Hx: Acute onset of pain XR: usually normal Surgical reattachment • Rare (proximal distal) PE: Decreased or absent elbow flexion CUBITAL TUNNEL SYNDROME • Trauma or stretching of ulnar nerve in cubital tunnel Hx: Numbness/tingling (+ / - pain) in ulnar distribution XR: Usually negative 1. Rest, ice, NSAID • Occurs near FCU origin PE: + / - decreased grip strength, Tinel's and/or elbow flexion test Nerve conduction: gives objective data, but often not necessary 2. Splints (day and/or night) • Can also be trapped at arcade of Struthers 3. Casting 4. Nerve decompression and transposition LATERAL EPICONDYLITIS (T ennis Elbow) • Degeneration of common extensor tendons (esp. ECRB) Hx: Age 30-60, chronic pain at lateral elbow, worse with wrist finger extension XR: Rule out fracture OA. Calcification of tendons can occur (esp. ECRB) 1. Activity modification, ice, NSAIDs • Due to overuse (e.g. tennis) or injury (microtrauma) PE: +T ennis elbow test 2. Use of brace or strap 3. Stretching/strengthening 4. Corticosteroid injection 5. Surgical release of tendon LCL SPRAIN • Rare condition Hx: + / - catching and locking XR: Usually negative Conservative unless recurrent subluxation, then surgical reconstruction PE: + instability with varus stress, + posterolateral (pivot shift) drawer MCL SPRAIN • Due to single traumatic or repetitive valgus stress Hx: Y oung, throwing athletes, chronic pain or acute onset of pain at MCL, + / -“pop” XR: occasional spur; rule out fracture (+ / -stress view) Grade I II: conservative (rest, ice, NSAID) • Usual mechanism: throwing PE: + / - instability with valgus stress MRI: before surgery Grade III (complete tear): surgical repair (use PL) • Anterior Band is affected MEDIAL EPICONDYLITIS (Golfer's Elbow) • Degeneration of pronator/ flexor group (PT FCR) Hx: Medial elbow pain XR: Rule out fracture OA. Calcification of tendons can occur Same as T ennis elbow • Due to injury or overuse PE: Focal medial epicondyle tenderness, + Golfer's elbow test Surgery is less effective than for lateral epicondylitis OLECRANON BURSITIS • Inflammation of bursa (Infection/trauma/other) Hx: Swelling, acute or chronic Aspirate bursa: send purulent fluid for culture and Gram stain 1. Compressive dressing PE: Palpable mass at olecranon 2. Reaspirate if recurs 3. Corticosteroid injection OSTEOCHONDRITIS DISSECANS OF ELBOW: OCD • Repetitive valgus stresses (e.g. throwing or gymnastics) Hx: Y oung, active (thrower or gymnast), lateral elbow pain XR: lucency and/or loose body Type I (fragment stable): Ice, discontinue activity, NSAID • Vascular compromise and microtrauma of capitellum PE: + / - catching and/or locking, crepitus with pronation and supination CT/MRI: determine articular and subchondral involvement Type II-III (loose fragment): Drill or curette fragment TRICEPS TENDON RUPTURE • Trauma: forced elbow extension against resistance Hx: Pain in posterior elbow XR: usually normal Surgical reattachment PE: Loss of active elbow extension Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier SURGICAL APPROACHES USES INTERNERVOUS PLANES DANGERS COMMENT HUMERUS: ANTERIOR APPROACH 1. ORIF of fractures Proximal 1. Deltoid [Axillary] 2. Pectoralis Major [Pectoral] Proximal 1. Axillary nerve 2. Humeral circumflex artery • Anterior humeral circumflex artery may need ligation. 2. Bone biopsy or tumor removal. • The brachialis has a split innervation which can be used for an internervous plane. Distal 1. Brachialis splitting Lateral [Radial] Medial [MC] Distal 1. Radial nerve ELBOW: LATERAL APPROACH (KOCHER) Most radial head procedures 1. Anconeus [Radial] 1. PIN • Protect PIN: stay above annular ligament; keep forearm pronated 2. ECU [PIN] 2. Radial nerve USES INTERNERVOUS PLANES DANGERS COMMENT ELBOW: POSTERIOR APPROACH (BRY AN/MORREY) 1. Arthroplasty 2. Distal humerus and olecranon fractures 3. Loose body removal No planes Ulnar nerve • Triceps is detached from the olecranon. • MCL release may be necessary. Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com CHAPTER 4 - FOREARM TOPOGRAPHIC ANATOMY OSTEOLOGY OF THE FOREARM OSTEOLOGY OF THE WRIST TRAUMA JOINTS: WRIST OTHER WRIST STRUCTURES MINOR PROCEDURES HISTORY PHYSICAL EXAM MUSCLES: ORIGINS & INSERTIONS ANTERIOR COMPARTMENT MUSCLES: SUPERFICIAL FLEXORS POSTERIOR COMPARTMENT MUSCLES: SUPERFICIAL EXTENSORS ANTERIOR COMPARTMENT MUSCLES: DEEP FLEXORS POSTERIOR COMPARTMENT MUSCLES: DEEP EXTENSORS MUSCLES: CROSS SECTIONS NERVES ARTERIES DISORDERS: ARTHRITIS & INSTABILITY DISORDERS: NERVE COMPRESSION OTHER DISORDERS SURGICAL APPROACHES Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier CHAPTER 4 – FOREARM TOPOGRAPHIC ANATOMY Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier OSTEOLOGY OF THE FOREARM CHARACTERISTICS OSSIFY FUSE COMMENT RADIUS • Cylindrical long bone • Head within elbow joint • Tuberosity outside joint • Palpate head laterally • Styloid is distal Primary: Shaft Secondary 1. Proximal epiphysis 2. Distal epiphysis 8-9 weeks (fetal) 1-9 years 14-21 years • Elbow ossification: used to determine bone age in peds • Elbow ossification order: Capitellum, Radial head, Medial epicondyle, Trochlea, Olecranon, Lateral Epicondyle (Captain Roy Makes Trouble On Leave) ULNA • Cylindrical long bone Olecranon Primary: Shaft 8-9 weeks (fetal) Olecranon • Olecranon palpable posteriorly at elbow • Styloid process distally Secondary 1. Olecranon 2. Distal epiphysis (fetal) 10 years 5-6 yrs 16-20 years • Olecranon and coronoid give the elbow bony stabilization. Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier OSTEOLOGY OF THE WRIST CHARACTERISTICS OSSIFY FUSE COMMENT PROXIMAL ROW Scaphoid: boat shaped, 80% of surface is articular (not the waist) 5th 5 years 14-16 yrs • Lies beneath the anatomic snuffbox • Distal (to waist) blood supply (radial artery); proximal pole is susceptible to necrosis if injured Lunate: moon shaped 4th 4 years 14-16 yrs • Dislocations often missed • Blood supply is palmar: palmar fractures need ORIF to protect against osteonecrosis; dorsal fractures treated nonsurgically Triquetrum: pyramid shaped 3rd 3 years 14-16 yrs Pisiform: large sesamoid bone 8th 9-12 years 14-16 yrs • In the FCU tendon; TCL attaches DISTAL ROW Trapezium: most radial 6th 5-6 years 14-16 yrs • Articulates with 1st metacarpal; TCL attaches, FCR Trapezoid: wedge shape 7th 5-6 years 14-16 yrs • Articulates with 2nd metacarpal Capitate: largest carpal bone 1st 1 year 14-16 yrs • First to ossify Hamate: has a hook 2nd 1-2 14-16 • TCL, FCU attach to the hook Hamate: has a hook years 16 yrs • TCL, FCU attach to the hook Ossification: each from a single center: counterclockwise (anatomic position) starting with capitate Carpal tunnel borders: Roof: Transverse carpal ligament; Lateral wall: scaphoid trapezium; Medial wall: pisiform hamate Contents: Median nerve, flexor tendons Guyon's canal: Roof: volar carpal ligament; Floor: TCL; Lateral wall: hamate (hook); Medial wall: pisiform Contents: Ulnar nerve and artery Anatomic snuffbox: Between tendons of EPL and EPB; Contents: Radial artery (scaphoid directly deep to snuffbox) Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier TRAUMA DESCRIPTION EVALUATION CLASSIFICATION TREATMENT OLECRANON FRACTURE • Mechanism: fall directly on elbow; fall on hand • Articular surface always involved • Triceps tendon pulls fragment HX: Fall/trauma. Swelling, pain, +/- numbness. PE: Effusion, tenderness +/- decreased elbow extension. Good neurovascular exam (esp. ulnar nerve) XR: AP/lateral Colton: Undisplaced: 2mm Displaced -avulsion -transverse/oblique -comminuted -fracture/dislocation Undisplaced: Cast at 45-90° for 3 weeks, then gentle ROM Displaced: ORIF with tension band wires or bicortical screw. (comminuted fracture: excise bone then reattach triceps) COMPLICATIONS: Ulnar nerve injury (most resolve); Decreased ROM; Arthritis RADIAL HEAD FRACTURE • Common • Fall on outstretched arm radius pushed into capitellum • Intraarticular fracture • Can be associated with elbow dislocation HX: Fall. Pain, swelling, decreased function. PE: T enderness of radial head, decreased ROM especially pronation/supination. T est MCL stability XR: AP/lateral: +fat pad Mason: 4 Types I: Undisplaced II: Displaced III: Comminuted (head) IV: Fracture with elbow dislocation Type I: Splint for 3 days, then early ROM Type II: If motion intact-splint, then early ROM. If 1/3 of head involved or 3mm displaced-ORIF or excision Type III: Radial head excision COMPLICATIONS: Decreased ROM; Instability BOTH BONE FRACTURE • Mechanism: high energy injuries • Fractures in shaft of single bone shorten, resulting forces cause fracture in other bone • Nightstick fracture: ulnar shaft fracture only HX: Trauma. Pain, swelling. PE: T enderness, deformity. Check compartments and do neurovascular PE XR:AP/lateral: including wrist and elbow Descriptive: • Undisplaced • Displaced • Comminuted ORIF (usually plates and screws) through two separate incisions. Nightstick: Undisplaced-closed treatment; Displaced-ORIF Peds: closed, LAC 6-8wks COMPLICATIONS: Loss of Pronation and supination; Nonunion DESCRIPTION EVALUATION CLASSIFICATION TREATMENT MONTEGGIA FRACTURE • Proximal ulna fracture, shortening forces result in radial head dislocation. HX: Fall. Pain, swelling. PE: T enderness, deformity. Check compartments and Bado (based on radial head location): I: Anterior (common) II: Posterior Ulna: ORIF (plates/screws) Radial head: closed reduction (open if irreducible or dislocation. • Mechanism: direct blow or fall on outstretched hand. neurovascular exam. XR: AP/lateral: including wrist and elbow series. III: Lateral IV: Anterior with associated both bone fracture. irreducible or unstable). Peds: closed reduction cast. COMPLICATIONS: Radial nerve/PIN injury (most resolve); Decreased ROM; Compartment Syndrome; Nonunion GALEAZZI/PIEDMONT FRACTURE • Mechanism: fall on outstretched hand. • Distal radial shaft fracture, shortening forces result in distal radioulnar dislocation. HX: Fall. Pain, swelling. PE T enderness, deformity. Check compartments and do neurovascular exam. XR: AP/lateral: including wrist and elbow By mechanism: Pronation: Galeazzi Supination: Reverse Galeazzi (ulna shaft fracture with DRUJ dislocation) Radius: ORIF (plate/screws) DRUJ: closed reduction, +/-percutaneous pins. (open treatment if unstable) Cast immobilization for 4-6wks. Peds: closed reduction, cast. COMPLICATIONS: Nerve injury; Decreased ROM; Nonunion; Distal radioulnar joint (DRUJ) arthrosis DESCRIPTION EVALUATION CLASSIFICATION TREATMENT DISTAL RADIUS FRACTURE • Very common (Colles#1) • Fall on outstretched arm HX: Fall. Pain, swelling. PE: Swelling, Frykman (for Colles): Type I, II: extraarticular Close reduce, immobilize with WELL molded cast. (volar flexion ulnar • Colles fracture: dorsal displacement (apex volar), radial shortening, dorsal angulation. • Smith fracture: volar displacement (apex dorsal) • Barton fracture: radial rim carpus displace together • Radial styloid (chauffeur fracture) deformity, tenderness to palpation.Good neurovascular exam. XR: AP/lateral: normal radius: 1. 23° radial inclination 2. 13 mm radial height 3. 11° volar tilt Type III, IV: radiocarpal joint. Type V, VI: radioulnar joint Type VII, VIII: radiocarpal and radioulnar joints involved (even numbers also have ulna styloid fx) Barton: 1. Dorsal 2. Volar (most common) flexion ulnar deviation). If unstable add percutaneous pins, ORIF or external fixation. Smith: closed treatment +/-percutaneous pinning (often unstable needs ORIF) Barton fracture: Most need ORIF Styloid fracture: ORIF COMPLICATIONS: Loss of motion; Deformity; Median nerve injury; Malunion; Scapholunate dislocation DESCRIPTION EVALUATION CLASSIFICATION TREATMENT SCAPHOID FRACTURE • Most common carpal fracture • Fall on outstretched arm • High complication rate Proximal pole HX: Fall. Pain worse with gripping, swelling. PE: “Snuffbox” tenderness, swelling on radial wrist XR: AP/lateral: By location: Proximal pole Middle (“waist”) most common Distal pole If clinical symptoms with negative xray: thumb spica for 10-14days then re-evaluate. Nondisplaced: cast 6-12 wks Displaced: • Proximal pole with tenuous blood supply also PA with ulnar deviation/oblique Displaced: ORIF (K-wire or Herbert screw) COMPLICATIONS: Nonunion/malunion; Osteonecrosis: especially of proximal pole; Degenerative Joint Disease (DJD) CARPAL DISLOCATION: PERILUNATE INSTABILITY • Uncommon: hyperextension supination injury • Injury determined by a progression of ligament disruption (see joint chart) • Space of Poirer is weak (Capitate-lunate joint) HX: Fall. Pain. PE: Wrist pain, + Watson sign. XR: AP/lateral: 3mm SL gap is T erry Thomas sign.+/-2 Scaphoid ring sign Cinearthrogram: definitive diagnosis Mayfield (4 stages): I: Scapholunate diastasis II: Perilunate dislocation III: Lunotriquetral diastasis IV: Volar lunate dislocation. Closed reduction and cast simple cases. Open reduction, pin fixation, and primary ligament repair usually required. COMPLICATIONS: Wrist instability and/or pain; SLAC wrist DESCRIPTION EVALUATION CLASSIFICATION TREATMENT INCOMPLETE FRACTURE: TORUS GREENSTICK FRACTURE • Common in children (usually ages 6-12) Mechanism: Hx: Trauma. Pain, inability to use arm. T orus(Buckle):concave cortex compresses T orus: reduction rarely needed, • Mechanism: fall on hand most common • Distal radius most common • Increased flexibility of pediatric bone allows only one cortex to be involved use arm. PE:+/-deformity. Point tenderness swelling. XR: AP and lateral: only one cortex involved. (buckles), convex/tension side: intact Greenstick: concave cortex intact, convex/tension side fracture/plastic deformity needed, splint 2-4 weeks Greenstick: reduce if 10° of angulation. Long arm cast for 6 weeks. COMPLICATIONS: Deformity; Malunion; Neurovascular injury (rare) Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier JOINTS: WRIST LIGAMENTS ATTACHMENTS COMMENTS RADIOCARPAL (Ellipsoid type) Bones: radius, scaphoid, lunate, triquetrum Capsule Surrounds joint Loose, provides little support Volar radiocarpal [VRC] Multiple intracapsular ligaments Strong; space of Poirier (lunocapitate) is weak. Injury leads to instability. Radioscaphocapitate [RCL] Radial styloid to capitate Stabilizes radial wrist, distal row, midcarpal joint. Disrupted in perilunate instability stage II. Radioscapholunate [RSL] Radial styloid to lunate Stabilizes radial wrist, scapholunate joint; Disrupted in DISI, perilunate instability stage I. Radiolunotriquetral [RTL] Radial styloid to triquetrum Largest, volar sling for lunate, lunotriquetral joint stabilizer. Disrupted in perilunate instability stage III. Dorsal radiocarpal [DRC] Radius, scaphoid, lunate, triquetrum Weak; stabilizes proximal row, radiolunate joint. Disrupted in perilunate instability stage IV. Radial collateral Radius, scaphoid, trapezium, TCL Stabilizes proximal row. Radial artery runs adjacent to it. RADIOULNAR (Pivot type) Triangular Fibrocartilage Complex (TFCC): Multiple components stabilize joint, absorbs axial load; any tear or injury results in pain COMPONENT ORIGIN INSERTION Dorsal Volar Radioulnar Ulnar radius Caput ulna Triangular fibrocartilage (disc) Radius/ulna Triquetrum Meniscus homologue Ulna/disc Triquetrum Ulnar collateral/ECU Ulna Fifth metacarpal OTHER LIGAMENTS Ulnocarpal: Often considered part of TFCC; Stabilizes proximal row of carpus Ulnolunate Ulna Lunate Ulnotriquetral Ulna Triquetrum JOINT TYPE LIGAMENTS ATTACHMENTS COMMENTS INTERCARPAL Dorsal stronger Stabilize Proximal Row Gliding 2 Dorsal intercarpal 2 Palmar intercarpal 2 Interosseous Scapholunate, lunotriquetral Scapholunate, lunotriquetral Scapholunate, lunotriquetral. Stabilize SL or LT joints DISI: SL ligament injury VISI: LT ligament injury Pisiform Articulation Capsule Ulnar collateral Volar radiocarpal Pisohamate Pisometacarpal Pisiform triquetrum Ulna to pisiform RCL to pisifrom Pisiform to hamate Pisiform to 5th metacarpal Holds it proximally Holds it proximally Assists FCU; roof of Guyon's canal Assists FCU flexion Distal Row Gliding 3 Dorsal intercarpal 3 Palmar intercarpal 2 interosseous All four bones in distal row All four bones in distal row Trapezoid to capitate to hamate Thicker than proximal MIDCARPAL Ellipsoid Palmar (Volar) intercarpal Carpal collaterals Capitotriquetral (CTL) Proximal distal carpal rows Capitate to triquetrum 1/3 of wrist extension, 2/3 of wrist flexion occurs here Radial stronger than ulnar Stabilizes distal row Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier OTHER WRIST STRUCTURES STRUCTURE FUNCTION COMMENT Extensor Retinaculum Dorsal Compartments Covers dorsum of the wrist I: APL, EPB II: ECRL, ECRB III: EPL IV: EDC, EIP V: EDM VI: ECU Forms six fibroosseous dorsal compartments DeQuervain's tenosynovitis can develop here T endinitis (carpal bossing) Around Lister's tubercle: tendon can rupture T enosynovitis, ganglions Jackson-Vaughn syndrome (rupture from RA) T endon can “snap” over ulnar styloid Transverse Carpal Ligament (TCL, Flexor Retinaculum) Covers volar wrist Attaches to: Medial: pisoform hook of hamate Lateral: scaphoid trapezium Roof of carpal tunnel, floor of Guyon's canal (ulnar nerve can entrap here) Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier MINOR PROCEDURES STEPS WRIST ASPIRATION/INJECTION 1. Ask patient about allergies 2. Palpate radiocarpal joint dorsally for EPL,ECRB, Lister's tubercle and the space ulnar to them 3. Prepare skin over dorsal wrist (iodine/antiseptic soap) 4. Anesthetize skin locally (quarter size spot) 5. Aspiration: Insert 20 gauge needle into space ulnar to Lister's tubercle/ECRB and radial to EDC, aspirate. Injection: Insert 22 gauge needle into same space,aspirate to ensure not in vessel, then inject 1-2ml of local or local/steroid preparation into RC joint. 6. Dress injection site 7. If suspicious for infection, send fluid for Gram stain culture CARPAL TUNNEL INJECTION/MEDIAN NERVE BLOCK 1. Ask patient about allergies 2. Ask patient to pinch thumb and small finger tips, Palmaris longus (PL) tendon will protrude (10-20% do not have one) median nerve is directly beneath PL, just ulnar to FCR 3. Prepare skin over volar wrist (iodine/antiseptic soap) 4. Anesthetize skin locally (quarter size spot) 5. Insert 22 gauge or smaller needle into wrist under PL at flexion crease. Aspirate to ensure needle is not in a vessel. Inject 1-2ml of local or local/steroid preparation. 6. Dress injection site Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier HISTORY QUESTION ANSWER CLINICAL APPLICATION 1. AGE Y oung Middle age-elderly Trauma: fractures and dislocations, ganglions Arthritis, nerve entrapments, overuse 2. PAIN a. Onset b. Location Acute Chronic Dorsal Volar Radial Ulnar Trauma Arthritis Kienbock's disease, ganglion Carpal tunnel syndrome (CTS), ganglion (especially radiovolar) Scaphoid fracture, DeQuervain's tenosynovitis, arthritis Triangular Fibrocartilage Complex(TFCC) tear, tendinitis 3. STIFFNESS with dorsal pain with volar pain (at night) Kienbock's disease Carpal tunnel syndrome 4. SWELLING Joint: after trauma Joint: no trauma Along tendons Fracture or sprain Arthritides, infection, gout Flexor or extensor tendinitis (calcific), DeQuervain's disease 5. INSTABILITY Popping, snapping Scapholunate dissociation 6. MASS Along wrist joint Ganglion 7. TRAUMA Fall on hand Fractures: distal radius, scaphoid; Dislocation: lunate, ulna TFCC tear 8. ACTIVITY Repetitive motion (typing) Carpal Tunnel Syndrome (CTS), DeQuervain's tenosynovitis 9. NEUROLOGIC SYMPTOMS Numbness, tingling Weakness Nerve entrapment, thoracic outlet syndrome, radiculopathy Nerve entrapment (median (e.g. CTS), ulnar, or radial) 10. HISTORY OF ARTHRITIDES Multiple joints involved Arthritides Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier PHYSICAL EXAM EXAMINATION TECHNIQUE CLINICAL APPLICATION INSPECTION Gross deformity Swelling Bones and soft tissues Especially dorsal or radial Diffuse Fractures, dislocations: forearm and wrist Ganglion Trauma, infection PALPATION Skin changes Warm, red Cool, dry Infection, gout Neurovascular compromise Radial and Ulnar styloids Palpate each separately T enderness may indicate fracture Carpal bones Both proximal and distal row Snuffbox tenderness: scaphoid fracture; lunate tenderness: Kienbock's disease. Proximal row Pisiform Scapholunate dissociation T enderness: pisotrequetral arthritis or FCU tendinitis Soft tissues 6 dorsal extensor compartments TFCC: distal to ulnar styloid Compartments T enderness over 1st compartment: DeQuervain's disease T enderness indicates TFCC injury Firm/tense compartments: compartment syndrome RANGE OF MOTION Flex and extend Flex (toward palm), extend opposite Normal: flexion 80°, extension 75° Radial/ulnar deviation Pronate and supinate In same plane as the palm Flex elbow 90°: hold pencil, rotate wrist Normal: radial 15-20°, ulnar 30-40° Normal: supinate 90°, pronate 80-90° (only 10-15° is in the wrist, most motion is in elbow) NEUROVASCULAR Sensory (LT , PP, 2 pt) Musculocutaneous nerve (C6) Lateral forearm Deficit indicates corresponding nerve/root lesion Medial Cutaneous nerve of forearm (T1) Medial forearm Deficit indicates corresponding nerve/root lesion Motor Radial Nerve (C6-7) Resisted wrist extension Weakness=ECRL/B or corresponding nerve/root lesion PIN (C6-7) Resisted ulnar deviation Weakness=ECU or corresponding nerve/root lesion Ulnar Nerve (C8) Resisted wrist flexion Weakness=FCR or corresponding nerve/root lesion Median Nerve (C7) Resisted wrist flexion Weakness=FCR or corresponding nerve/root lesion Median Nerve (C6) Resisted pronation Weakness=Pronator T eres or nerve/root lesion Musculocutaneous (C6) Resisted supination Weakness=Biceps or corresponding nerve/root lesion Reflex C6 Brachioradialis Hypoactive/absence indicates corresponding radiculopathy Pulses Radial, Ulnar Diminished/absent = vascular injury or compromise (perform Allen test) EXAMINATION TECHNIQUE CLINICAL APPLICATION SPECIAL TESTS Phalen Maximal flexion of both wrists for several minutes Reproduction of symptoms (numbness or tingling): Carpal Tunnel Syndrome (CTS) Tinel T ap volar wrist (carpal tunnel/TCL) Pain, numbness suggests Median nerve compression (CTS) Finkelstein Make fist with thumb inside, then ulnar deviation Pain over 1st compartment (APL, EPB) suggests DeQuervain's tenosynovitis Watson Push scaphoid anteroposterior with wrist in radial or ulnar deviation Positive if scaphoid subluxes or reduces: carpal ligament injury Allen Occlude radial ulnar arteries, pump fist then release one artery only Delay or absent of “pinking up” of palm suggest arterial compromise of artery released Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier MUSCLES: ORIGINS INSERTIONS PROXIMAL ULNA PROXIMAL RADIUS ANTERIOR INSERTIONS INSERTIONS Brachialis Biceps Supinator ORIGINS ORIGINS Flexor Digitorum Flexor Digitorum Superficialis [1 head] Superficialis [1 head] Pronator teres Flexor Pollicis longus Supinator PROXIMAL ULNA PROXIMAL RADIUS POSTERIOR INSERTIONS INSERTIONS Triceps Biceps Anaconeus Supinator ORIGINS ORIGINS Flexor carpi ulnaris NONE Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier ANTERIOR COMPARTMENT MUSCLES: SUPERFICIAL FLEXORS MUSCLE ORIGIN INSERTION NERVE ACTION COMMENT Pronator T eres [PT] Medial epicondyle coronoid process Lateral radius-middle 1/3 Median Pronate and flex forearm May trap AIN (AIN syndrome) Flexor carpi radialis [FCR] Medial epicondyle Base of 2nd 3rd metacarpal Median Flex wrist, radial deviation Radial artery is immediately lateral Palmaris Longus [PL] Medial epicondyle Flexor retinaculum palmar aponeurosis Median Flex wrist Used for tendon transfers. 10% congenitally absent Flexor carpi ulnaris [FCU] Medial epicondyle posterior ulna Pisoform, hook of hamate, 5th MC Ulnar Flex wrist, ulnar deviation Most powerful wrist flexor MUSCLE ORIGIN INSERTION NERVE ACTION COMMENT Flexor digitorum superficialis [FDS] 1. Medial epicondyle, coronoid process Middle phalanges of digits (not thumb) Median Flex PIP (also flex digit and hand) Sublimus test will isolate test function 2. Anteroproximal radius Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier POSTERIOR COMPARTMENT MUSCLES: SUPERFICIAL EXTENSORS MUSCLE ORIGIN INSERTION NERVE ACTION COMMENT Flexor digitorum profundus [FDP] Anterior ulna Interosseus membrane Distal phalanx (IF/MF) Median/AIN Flex DIP (also flex digit and hand) Avulsion: Jersey finger. Distal phalanx (RF/SF) Ulnar FDP and FPL are most susceptible to Volkmann's contracture. Flexor pollicis longus [FPL] Anterior radius coronoid process Distal phalanx of thumb Median/AIN Flex thumb (IP) Pronator quadratus [PQ] Medial distal ulna Anterior distal radius Median/AIN Pronate forearm Primary pronator (initiates pronation) Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier ANTERIOR COMPARTMENT MUSCLES: DEEP FLEXORS MUSCLE ORIGIN INSERTION NERVE ACTION COMMENT Anaconeus Posterior-lateral epicondyle Posterior-poximal ulna Radial Forearm extension Must retract on Kocher approach Mobile Wad(3) Brachioradialis [BR] Lateral supra-condylar humerus Lateral distal radius Radial Forearm flexion Is a deforming force in radius fractures. Extensor carpi radialis longus [ECRL] Lateral supra-condylar humerus Base of 2nd MC Radial Wrist extension Used for tendon transfer Extensor carpi radialis brevis [ECRB] Lateral epicondyle Base of 3rd MC Radial Wrist extension Inflamed in T ennis elbow, can compress PIN Extensor digitorum [ED] Lateral epicondyle Sagittal bands, central slip, distal phalanx Radial-PIN Digit extension Distal avulsion is mallet finger injury Extensor digiti minimi [EDM] Lateral epicondyle Sagittal bands, central slip, distal phalanx of SF Radial-PIN SF extension In 5th dorsal compartment. Extensor carpi ulnaris [ECU] Lateral epicondyle Base of 5th MC Radial-PIN Hand extension and Must retract on Kocher approach adduction approach Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier POSTERIOR COMPARTMENT MUSCLES: DEEP EXTENSORS MUSCLE ORIGIN INSERTION NERVE ACTION COMMENT Supinator Posterior medial ulna Proximal lateral radius Radial-PIN Forearm supination Can compress PIN Abductor pollicis longus [APL] Posterior radius/ulna Base of 1st MC Radial-PIN Abduct and extend thumb (CMC) 1st compartment: DeQuervain Disease Extensor pollicis brevis [EPB] Posterior radius Base of proximal phalanx of thumb Radial-PIN Extend thumb (MCP) Same as above, radial border of snuffbox Extensor pollicis longus [EPL] Posterior ulna Base of thumb distal phalanx Radial-PIN Extend thumb (IP) T endon turns 45° on Lister's tubercle Border of snuffbox Extensor indicis proprius [EIP] Posterior ulna Sagittal bands, central slip, distal phalanx of index finger Radial-PIN Index finger extension Used in tendon transfer Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier MUSCLES: CROSS SECTIONS Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier NERVES INFRACLA VICULAR LATERAL CORD Musculocutaneous (C5-7): only sensory in the forearm Sensory: Lateral forearm [via Lateral cutaneous nerve of forearm] Motor: NONE (in forearm) MEDIAL CORD Medial Cutaneous Nerve of Forearm (Antibrachial) (C8-T1): runs with basilic vein Sensory: Medial forearm anterior arm Motor: NONE Ulnar (C(7)8-T1): runs behind medial epicondyle in groove and between 2 heads of ECU[], then under FCU[], then to Guyon's canal[]. 3. Sensory: NONE (in forearm) Motor: Flexor carpi ulnaris Flexor digitorum profundus [digits 4, 5] MEDIAL AND LATERAL CORDS 4. Median(C(5)6-T1): runs between 2 heads of PT[], through ligament of Struthers[] and lacertus fibrosus[], under FDS [] into carpal tunnel[] (Martin Gruber formation: ulnar motor branches run with median nerve then branch to ulnar nerve distally). In wrist, median divides to Motor branch and palmar cutaneous (runs between FCR/PL): at risk in CTS release Sensory: NONE (in forearm) Motor: ANTERIOR COMPARTMENT OF FOREARM Superficial Flexors Pronator T eres [PT]Flexor Carpi Radialis [FCR]Palmaris longus [PL]Flexor digitorum superficialis[FDS][sometimes considered a “middle” flexor] Deep Flexors Anterior Interosseous N. (AIN) AIN compressed by PT in forearm, injuredin supracondylar fractures Flexor digitorum profundus [digits 2, 3] Flexor pollicis longus [FPL] Pronator Quadratus [PQ] Potential nerve compression site 1. 2. INFRACLA VICULAR POSTERIOR CORD 5. Radial (C5-T1): Divides into 2 branches: 1. superficial radial (sensory) and 2. deep (motor)-which then pierces supinator and becomes PIN) Sensory: Posterior forearm: via Posterior CutaneousNerve of forearm Motor: MOBILE WAD(3): Radial Nerve (deep branch): runs around radius into posterior compartment, through radial tunnel[]becomes PIN Superficial Extensors Brachioradialis [BR]Extensor carpi radialis longus [ECRL]Extensor carpi radialis brevis [ECRB] POSTERIOR COMPARTMENT : PIN-PosteriorInterosseous Nerve Multiple sites ofcompression: 1. fibrous tissue of radialhead, 2. leash of Henry, 3. Arcade ofFrohse, 4. distal supinator, 5. ECRB Superficial Extensors Extensor carpi ulnaris [ECU]Extensor digiti minimi [EDM]Extensor digitorum communis [EDC] Deep Extensors SupinatorAbductor pollicis longusExtensor pollicis longusExtensor pollicis brevisExtensor indicis proprius Potential nerve compression site Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier ARTERIES ARTERY COURSE BRANCHES Radial over PronatorT eres, underBrachioradialis. Radial recurrent muscularbranches (leash of Henry) Ulnar on FDP, underFDS Anterior ulnar recurrentPosterior ulnar recurrentCommon interosseousAnterior interosseousPosterior interosseousRecurrent interosseousMuscular branches See Arm chapter for arterial anastomosis around the elbow ARTERY COURSE BRANCHES COMMENT Radial Volar: lateral to FCRDorsal: between EPL APL/EPB 3 branchesPalmar carpal branchDorsal carpal branchSuperficial palmar branchDeep palmar arch Is in anatomic snuffboxDeep to flexor tendonsDeep to extensor tendonsAnastomoses with ulnar artery completes superficial palmar archT erminal branch of radial artery Ulnar on the TCL, lateral to pisoform. 4 branchesPalmar carpal branchDorsal carpal branchDeep palmar branch Deep to FDSDeep to extensor tendonsAnastomoses with radial artery completes deeppalmar arch Superficial palmar arch T erminal branch of ulnar artery Allen test 1. Occlude both radial and ulnar arteries at wrist 2. Patient should squeeze fist several times 3. Release pressure on one artery 4. Repeat releasing other artery Hand perfusion (“pinking up”) after release indicates patent arches collateral circulation. Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier DISORDERS: ARTHRITIS INSTABILITY DESCRIPTION H P WORK-UP/FINDING TREATMENT ARTHRITIS OSTEOARTHRITIS/DEGENERATIVE JOINT DISEASE • “Wear tear”: articular cartilage loss • 1° or 2° (e.g. trauma.) • Seen in SLAC wrist Hx: Older, women, pain (worse with activity) PE: Swelling, decreased ROM XR: OA findings: spurs, joint space loss, sclerosis 1. NSAID, splint, steroid injection 2. Arthrodesis (pain relief) DEQUERVAIN'S DISEASE • Stenosing tenosynovitis of 1st dorsal compartment (APL/EPB) Hx: Often history of tennis or golf. Pain, swelling. PE: 1Finkelstein test XR: Possible calcified tendons Lab: Uric acid (rule out gout) 1. Splint, NSAID, injection 2. Surgical release RHEUMATOID ARTHRITIS • Systemic inflammatory disorder affecting synovium, destroys joint • Wrist common site • Associated with tenosynovitis CTS Hx: Pain, stiffness (worse In AM) PE: Swelling throughout joint. Decreased ROM, ulnar drift at MCPs. XR: Hand series: joint destruction erosion Labs: RF, ANA, WBC, ESR, uric acid 1. Medical management, splint joints 2. Synovectomy (single joint) 3. T endon transfer or repair 4. Arthrodesis or arthroplasty INSTABILITY SLAC: SCAPHOLUNATE ADVANCED COLLAPSE • Degenerative arthritis secondary to instability (SL ligament disruption or scaphoid fracture/injury) Hx/PE: Chronic pain, remote history of trauma. XR: Radioscaphoid OA: (CL joint also involved, RL joint spared) 1. Scaphoid excision, capitolunate fusion 2. Proximal row carpectomy or fusion SCAPHOLUNATE DISSOCIATION: (static/dynamic) • SL/RCL ligament disrupted: lunate displaced dorsally [DISI: Dorsal Intercalated Segment Instability] • LT ligament disrupted: lunate displaced volarly [VISI:Volar ISI] Hx: Fall (extension supination wrist injury). Pain in wrist. PE: 1Watson's test XR: SL space .3mm 5 “T erry Thomas” sign. Closed fist: increases SL gap Early: closed reduction, splint/cast. Repair ligament if full tear Late: STT fusion, carpectomy, or wrist fusion. Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier DISORDERS: NERVE COMPRESSION DESCRIPTION H P WORK-UP/FINDING TREATMENT AIN (Anterior Interosseous Nerve) SYNDROME • AIN trapped under: 1. PT 2. FDS 3. FCR Hx: No sensory findings XR: Rule out other pathology 1. Conservative treatment PE: decreased thumb flexion, no “OK” sign (+ Kiloh-Nevinsign) 2. Surgical release if does not resolve CARPAL TUNNEL SYNDROME (CTS) • Median nerve trapped in carpal tunnel Hx: Repetitive motion, night pain, parathesias, clumbsy XR: Rule out other pathology 1. Activity modification • Most common nerve entrapment PE: Weak thenar muscles, + Tinel Phalen tests EMG/NCS: Localize the lesion 2. Cock-up splint, NSAID, steroid injection • Associated with metabolic disease (DM, EtOH, pregnancy, thyroid disease) 3. Carpal tunnel release [avoid palmar branch] PIN SYNDROME (Saturday Night Palsy) • PIN trapped by: 1. Supinator (proximal border most common) 2. Arcade of Frohse 3. Leash of Henry 4. Fibrous bands 5. ECRB Hx: +/- pain XR: Rule out other pathology 1. Observe. It may resolve PE: No sensory findings. Wrist drop, EMG/NCS: 2. Surgical findings. Wrist drop, decreased wrist digit extension Localize the lesion decompression if symptoms persist PRONATOR SYNDROME • Median nerve trapped by:1. PT , 2. Ligament of Struther, 3. Lacertus fibrosus, 4. FDS Hx: Forearm pain, increases with activity XR: Rule out other pathology 1. NSAID, rest, splint PE: Thenar weakness, Tinel Phalen tests EMG/NCS: Localize the lesion 2. Surgical release after 3-4 months RADIAL TUNNEL SYNDROME • Radial nerve trapped in radial tunnel (1 of 4 places) Hx: Pain in lateral forearm XR: Rule out other pathology 1. Rule out lateral epicondylitis PE: No motor/sensory findings 2. Activity modification, splinting 3. Surgical exploration/release ULNAR TUNNEL SYNDROME • Ulnar nerve trapped in Guyon's canal Hx: Pain, numbness, intrinsic weakness XR: not indicated 1. Activity modification, rest, immobilize • Can be trauma related PE: +Tinel of ulnar nerve at wrist EMG/NCS: will localize lesion 2. Surgical decompression Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier OTHER DISORDERS DESCRIPTION H P WORK-UP/FINDING TREATMENT GANGLION • Cyst with mucinous/joint fluid Hx/PE: Round, large or small transilluminating mass, +/-pain XR: Wrist series, no radiographic evidence of ganglion 1. Asymptomatic: reassurance • Communicates with joint 2. Symptomatic: aspirate or surgically excise (with stalk or it will recur) • Most common mass in wrist1. Dorsal (SL)2. Volar (ST) KIENBÖCK'S DISEASE • Osteonecrosis of lunate Hx: Pain, swelling, stiffness XR: Opacity of lunate I. NSAID, splinting • Wrist trauma or short ulna PE: Grip strength may be reduced. Bone scan/MRI: will confirm diagnosis II/III. Joint leveling procedure/carpal fusion • 4 stages: based on collapse IV. Proximal row carpectomy or fusion Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier SURGICAL APPROACHES USES INTERNERVOUS PLANE DANGERS COMMENT FOREARM: ANTERIOR APPROACH (HENRY) 1. ORIF fractures Distal1. Brachioradialis [Radial]2. FCR [Median] 1. PIN 1. Radial recurrent artery (Leash of Henry) vein need ligation. 2. Osteotomy Proximal1. Brachioradialis [Radial]2. Pronator T eres [Median] 2. Superficial radial nerve 2. If not ligated, hemorrhage could result in Compartment syndrome and/or Volkmann's contracture 3. Biopsy bone tumors 3. Radial artery WRIST: DORSAL APPROACH 1. Fusion 1. 3rd dorsal compartment [EPL] Radial nerve (Superficial) 1. Incise to the extensor retinaculum. This leaves cutaneous nerves intact in the subcutaneous fat. 2. Stabilization 2. 4th dorsal compartment [EDC, EIP] 2. Neuroma can develop from cutting cutaneous nerves. 3. ORIF fractures 4. Carpectomy WRIST: VOLAR APPROACH 1. Carpal tunnel No planes 1. Median nerve• Palmar cutaneous 1. Retract PL/FPL radially decompression No planes branch• Recurrent motor Retract FDS/FDP ulnarly 2. ORIF volar fracture 2. Palmar arch 2. Dissect TCL carefully to avoid nerve damage. 3. Dislocated lunate 4. T endon laceration Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com CHAPTER 5 - HAND TOPOGRAPHIC ANATOMY OSTEOLOGY OF THE HAND TRAUMA JOINTS OTHER STRUCTURES: FLEXOR TENDON SHEATH AND PULLEYS OTHER STRUCTURES: HAND SPACES OTHER STRUCTURES: FINGER FLEXOR TENDON INJURY ZONES MINOR PROCEDURES HISTORY PHYSICAL EXAM MUSCLES INTRINSIC MUSCLES NERVES ARTERIES DISORDERS: ARTHRITIS DISORDERS: LIGAMENT INJURIES DISORDERS: INFECTIONS DISORDERS: MASSES & TUMORS SURGICAL APPROACHES Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier CHAPTER 5 – HAND TOPOGRAPHIC ANATOMY Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier OSTEOLOGY OF THE HAND CHARACTERISTICS OSSIFY FUSE COMMENT METACARPALS • Triangular in cross section: gives 2 volar muscular attachment sites Primary: Body 9 wks (fetal) 18 yrs • Named I-V (thumb to small finger) • Thumb MC has saddle shaped base: increases it mobility Epiphysis 2 yrs • Only one epiphysis per bone in the head. In thumb MC it is in the base. PHALANGES • Palmar surface is almost flat Primary: Body 8 wks (fetal) 14-18 years • 3 phalanges in each digit except thumb • Tubercles and ridges are sites for attachment. Epiphysis 2-3 yr • Only one epiphysis per bone in base. Nomenclature for digits: thumb, index finger, middle finger, ring finger, small finger Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier TRAUMA DESCRIPTION EVALUATION CLASSIFICATION TREATMENT METACARPAL FRACTURES • Common in adults • 5th MC most common (Boxer's fracture at neck) • 1st MC base. Bennett Rolando fracture: displaced, intraarticular. • 4th 5th MC tolerate angulation; 2nd 3rd do not HX: Trauma. Swelling, pain, deformity. PE: Swelling, tenderness, +/- rotational deformity, shortening. Decreased ROM. XR: PA, lateral, oblique By location: • Head • Neck (most common) • Shaft (transverse, spiral, Oblique) • Base (Bennett, Rolando, “Baby Bennett ”-base of 5th MC) Nondisplaced: ulnar gutter splint 4 weeks, then ROM. Severely Angulated or shortened: percutaneous pins or ORIF Displaced or intraarticular: reduce then pin. Unstable: ORIF COMPLICATIONS: Rotational deformity grip abnormalities (malunion) PHALANGEAL FRACTURES HX: Trauma. Descriptive/location: • Intra vs extraarticular •Displaced/undisplaced Extraarticular Undisplaced: buddy tape and/or splint • Childrenadults Swelling, pain, deformity. • Open/closed • Transverse/oblique • Base, shaft, neck, condyle Displaced: reduce, splint Unstable: pin or ORIF • Distal phalanx most common (MF) PE: Swelling, tenderness, +/-rotational deformity, shortening. Decreased • Early ROM important for good results ROM, 2 pt discrimination, capillary refill. • Articular surfaces do not T olerate incongruity. Close follow up is critical for intraarticular fractures XR: AP, lateral, blique Splint must have MCP in flexion, IPs extended Intraarticular: ORIF Repair nail bed if needed COMPLICATIONS: Rotational deformity (malunion); Decreased motion; Degenerative Joint Disease (DJD) Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier JOINTS JOINT TYPE LIGAMENTS ATTACHMENTS COMMENTS CARPOMETACARPAL Thumb Saddle Capsule Highly mobile; common site for arthritis Dorsal, palmar, radial CMC Trapezium to metacarpals Finger Gliding Capsule Dorsal palmar CMC Carpal to metacarpal bones Dorsal strongest Interosseous CMC METACARPOPHALANGEAL Ellipsoid Capsule Metacarpal to proximal phalanx 2 collateral (radial and ulnar) Metacarpal to proximal phalanx Loose in extension, tight in flexion Cast in flexion or ligaments will shorten Thumb ulnar collateral: • stabilizes pinch • injury is Gamekeeper's Palmar [volar plate] Metacarpal to proximal phalanx Deep transverse metacarpal INTERPHALANGEAL Hinge Capsule 2 collateral Adjacent phalanges Obliquely oriented Palmar [volar Adjacent phalanges Prevents hyperextension plate] Adjacent phalanges Prevents hyperextension Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier OTHER STRUCTURES: FLEXOR TENDON SHEATH AND PULLEYS STRUCTURE CHARACTERISTICS COMMENT Flexor tendon sheath Fibroosseous tunnel, lined with tenosynovium Pulleys (5 annular, 3 cruciate) are thickenings of sheath. A2, A4 most important mechanically. A1, 3, 5 cover joints; A1 common cause of triggering. Protect, lubricate, nourish tendons In sheath: vinculae are vascular supply to tendons Site of potential infection: Kanavel signs often present (see Disorders) Intrinsic Apparatus Sagittal bands EDC attaches extends MCP Central Slip EDC attaches extends PIP: injury can result in Boutonniere deformity Lateral bands Lumbricals attach extend PIP Volar plate (transverse fibers) FDS attaches flexes PIP Oblique retinacular ligaments Interossei attach flex MCP EDC attaches extends DIP Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier OTHER STRUCTURES: HAND SPACES STRUCTURE CHARACTERISTICS COMMENT HAND SPACES Thenar Between flexor tendon and Adductor pollicis Potential space: site of possible infection Mid-palmar Between flexor sheath and metacarpal Potential space: site of possible infection Radial bursa Proximal extension of FPL sheath Infection can track proximally Ulnar bursa Communicates with SF, FDS, FDP flexor tendon sheath Flexor sheath infection can track proximally into bursa Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier OTHER STRUCTURES: FINGER STRUCTURE CHARACTERISTICS COMMENT FINGERTIP Nail Cornified epithelium If completely avulsed, replace to keep eponychium and matrix separated until nail can grow back. Nail bed/Matrix Germinal: to lunula, under eponychium Where nail grows (1mm a week), must be intact (repaired) for nail growth Sterile: distal to lunula If injured, does not need repair to function Pulp Multiple septae, nerves, arteries Felon is an infection of the pulp Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier FLEXOR TENDON INJURY ZONES ZONE BOUNDARIES COMMENT I FDS insertion to distal tip Injuries amenable to repair (e.g. Jersey finger) II Midpalm fibroosseous tunnel to FDS insertion Called “No man's land” because high rate of complications. Careful PE is required for diagnosis, the injury may not be at skin laceration site . FDS FDP may both require repair. A2, A4 must be preserved. Repair in zones 3-5 should be immediate III Transverse Carpal ligament to fibro-osseous tunnel Injuries often associated with Median nerve or arterial arch injuries. Explore and repair all. IV Transverse carpal ligament (carpal tunnel) Uncommon site of injury. Repair usually requires carpal tunnel release and repair. Median nerve at risk. V Proximal to the TCL Injuries require end-to-end repair Thumb I Thumb IP to distal tip Similar to finger Thumb II Thumb CMC to IP Similar to finger Thumb III Thenar eminence Repair may require lengthening or graft procedure Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier MINOR PROCEDURES STEPS INJECTION OF THUMB CMC JOINT 1. Ask patient about allergies 2. Palpate thumb CMC joint on volar radial aspect 3. Prepare skin over CMC joint (iodine/antiseptic soap) 4. Anesthetize skin locally (quarter size spot) 5. Palpate base of thumb MC, pull axial distraction on thumb with slight flexion to open joint. Use 22 gauge or smaller needle, and insert into joint. Aspirate to ensure needle is not in a vessel. Inject 2-3ml of 1:1 local (without epinephrine)/corticosterioid preparation into CMC joint. (The fluid should flow easily if needle is in joint) 6. Dress injection site FLEXOR TENDON SHEATH BLOCK 1. Ask patient about allergies 2. Palpate the flexor tendon at the distal palmar crease. 3. Prepare skin over palm (iodine/antiseptic soap) 4. Insert 22 gauge needle into flexor tendon at the level of the distal palmar crease. Withdraw needle so it is just outside tendon, but inside sheath. Inject 2-5ml of local anesthetic without epinephrine. 5. Dress injection site DIGITAL BLOCK 1. Prepare skin over dorsal proximal finger web space (iodine/antiseptic soap) 2. Insert 22 gauge needle between metacarpal heads on both sides of finger. Aspirate to ensure needle is not in a vessel. Inject 2- 5ml of local anesthetic without epinephrine. The dorsum of the proximal digit may also require anesthesia for adequate anesthesia. 3. Care should be taken not to inject too much fluid into the closed space of the proximal digit 4. Dress injection site Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier HISTORY QUESTION ANSWER CLINICAL APPLICATION 1. HAND DOMINANCE Right or left Dominant hand injured more often 2. AGE Y oung Trauma, infection Middle age, elderly Arthritis, nerve entrapments 3. PAIN a. Onset Acute Trauma, infection Chronic Arthritis b. Location CMC (thumb) Arthritis (OA) especially in women Volar (fingers) Purulent tenosynovitis (1 Kanavel signs) 4. STIFFNESS In AM, with “catching” Trigger finger, rheumatoid arthritis 5. SWELLING After trauma Infection (e.g. purulent tenosynovitis, felon, paronychia) No trauma Arthritides, gout, tendinitis 6. MASS Ganglion, Dupuytren's contracture, giant cell tumor 7. TRAUMA Fall, sports injury in dirty environment Fracture, tendon avulsion Infection 8. ACTIVITY Sports, mechanic Trauma (e.g. fracture, dislocation, tendon rupture) 9. NEUROLOGIC SYMPTOMS Pain, numbness, tingling Nerve entrapment (e.g. carpal tunnel), thoracic outlet syndrome, radiculopathy Weakness Nerve entrapment (usually in wrist or more Weakness proximal) 10. HISTORY OF ARTHRITIDES Multiple joints involved Rheumatoid arthritis, Reiter syndrome, etc. Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier PHYSICAL EXAM EXAMINATION TECHNIQUE CLINICAL APPLICATION INSPECTION Gross deformity Ulnar drift or swan neck Rheumatoid arthritis Rotational or angular deformity Fracture Finger position Flexion Dupuytren contracture, purulent tenosynovitis Skin, hair, nail changes Cool, hairless, spoon nails, etc. Neurovascular disorders: Raynaud's, diabetes, nerve injury Swelling DIPs Nodes from osteoarthritis: Heberden's (at DIPs: #1), Bouchard's (at PIPs) PIPs MCP's Rheumatoid arthritis Fusiform shape finger Purulent tenosynovitis Muscle wasting Thenar eminence Median nerve injury, CTS, C8/T1 pathology, CMC arthritis Hypothenar eminence or intrinsics Ulnar nerve injury EXAMINATION TECHNIQUE CLINICAL APPLICATION PALPATION Skin Warm, red Infection Cool, dry Neurovascular compromise Metacarpals Each along its length T enderness may indicate fracture Phalanges finger joints Each separately T enderness: fracture, arthritis; Swelling: arthritis Soft tissues Thenar hypothenar eminences Wasting indicates median ulnar nerve injury respectively Palm (palmar fascia) Nodules: Dupuytren's contracture; Snapping with finger extension: Trigger finger Flexor tendons: along volar finger T enderness suggests purulent tenosynovitis Sides of finger Giant cell tumors All aspects of finger tip T enderness: paronychia or felon RANGE OF MOTION Finger: MCP joint Flex 90°, extend 0°, Add/abd 0-20° Decreased flexion if casted in extension (collateral ligaments shorten) PIP joint Flex 110°, extend 0° Hyperextension leads to swan-neck deformity DIP joint Flex 80°, extend 10° All fingers should point to scaphoid at full flexion Thumb: CMC joint Radial abduction: Flex 50°, extend 50° Motion is in plane of palm Palmar abduction: Abduct 70°, adduct 0° Motion is perpendicular to plane of the palm MCP joint In plane of palm: Flex 50°, extend 0° IP joint In plane of palm: Flex 90°, extend 10° Opposition T ouch thumb to small fingertip Motion is mostly at CMC joint EXAMINATION TECHNIQUE CLINICAL APPLICATION NEUROVASCULAR Sensory Light touch pinprick, 2 point Radial Nerve (C6) Dorsal thumb web space Deficit indicates corresponding nerve/root lesion Median Nerve (C6-7) Radial border middle finger Deficit indicates corresponding nerve/root lesion Ulnar Nerve (C8) Ulnar border small finger Deficit indicates corresponding nerve/root lesion Motor Number in parenthesis indicates compartment Radial nerve/PIN (C7) Finger extension Weakness 5 EDC(4), EIP(4), EDM(5) or nerve lesion Thumb abduction extension Weakness 5 APL(1) / EPL(3) or nerve/root lesion Median nerve/AIN (C8) PIP flexion Weakness 5 FDS or corresponding nerve/root lesion DIP flexion Weakness 5 FDP (1/2 of muscle) or nerve lesion Thumb IP flexion Weakness 5 FPL or corresponding nerve/root lesion Motor Recurrent Branch “OK” sign Weakness 5 APB, OP, 1/2 FPB or nerve lesion; (CTS) MCP flexion (index/middle fingers) Weakness 5 IF, MF lumbricals or c nerve/root lesion Ulnar nerve (Deep branch) (T1) Finger cross (abduct/adduct) Weakness 5 Dorsal/Volar interosseous or nerve lesion Small finger abduction Weakness 5 FDM, ODM, ADM or nerve/root lesion MCP flexion (ring/small fingers) Weakness 5 RF, SF lumbricals or nerve/root lesion Reflex: Hoffmann T ap a finger distal phalanx Only pathologic (1 if different phalanx flexes): UMN syndrome T ests ulnar and radial artery patency Pulses/capillary refill Allen's test Doppler: arches, digital pulses SPECIAL TESTS Stabilize PIP in Profundus Stabilize PIP in extension, flex DIP only Inability to flex DIP alone indicates FDP pathology Sublimis Extend all fingers, flex a single finger at PIP Inability to flex PIP of isolated finger indicates FDS pathology Froment's sign Hold paper with thumb index finger, pull paper Thumb PIP flexion is positive, suggest Adductor Pollicis or Ulnar nerve palsy CMC grind Axial compress rotate CMC joint Pain indicates arthritis at CMC and/or MCP joints of thumb Finger instability Stabilize proximal joint, apply varus valgus stress Laxity indicates collateral ligament damage Thumb instability Stabilize MCP, apply valgus stress Laxity indicates ulnar collateral ligament strain (Gamekeeper's thumb) Murphy sign Make fist, observe height of MCP's If 3rd MC (normally elevated) is flat with 2nd 4th MC, suggests lunate dislocation Bunnel-Littler Extend MCP, passively flex PIP Tight or inability to flex PIP, improved with MCP flexion indicates tight intrinsic muscles Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier MUSCLES MUSCLE ORIGIN INSERTION NERVE ACTION COMMENT THENAR COMPARTMENT Abductor pollicis brevis [APB] Scaphoid, trapezium Lateral proximal phalanx of thumb Median Thumb abduction Palpable in lateral thenar eminence Flexor pollicis brevis [FPB] Trapezium Base of proximal phalanx of thumb Median Thumb MCP flexion Palpable in medial thenar eminence Opponens pollicis Trapezium Lateral thumb MC Median Oppose thumb, rotate medially Opposition is most important action ADDUCTOR COMPARTMENT Adductor pollicis 1. Capitate, 2nd 3rd MC Base of proximal phalanx of thumb Ulnar Thumb adduction Radial artery between its two heads 2. 3rd Metacarpal HYPOTHENAR COMPARTMENT Palmaris brevis [PB] Transverse carpal ligament [TCL] Skin on medial palm Ulnar Wrinkles skin Protects ulnar nerve Abductor digiti minimi [ADM] Pisiform Base of proximal phalanx of SF Ulnar SF abduction Palpable laterally Flexor digiti minimi brevis [FDMB] Hamate, TCL Base of proximal phalanx of SF Ulnar SF MCP flexion Palpable medially Oppose SF, Deep to other Opponens digiti minimi [ODM] Hamate, TCL Medial side 5th MC Ulnar Oppose SF, rotate laterally Deep to other muscles in the group Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier INTRINSIC MUSCLES MUSCLE ORIGIN INSERTION NERVE ACTION COMMENT INTRINSICS Lumbricals 1 2 FDP tendons (lateral 2) Lateral bands Median Extend PIP, flex MCP Only muscles in body to insert on their own antagonist. Lumbricals 3 4 FDP tendons (medial 3) Lateral bands Ulnar Extend PIP, flex MCP Interosseous: Dorsal [DIO] Adjacent metacarpals Proximal phalanx extensor expansion Ulnar Digit abduction DAB: Dorsal ABduct Interosseous: Volar [VIO] Adjacent metacarpals Proximal phalanx extensor expansion Ulnar Digit adduction PAD: Palmar Adduct (volar 5 palmar) Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier NERVES INFRACLAVICULAR MEDIAL CORD 1. Ulnar (C(7)8-T1): through Guyon's canal, past hook of hamate Sensory: Medial palm 1 1/2 digits via: palmar, palmar digital branches Medial dorsal hand 1 1/2 digits via: dorsal, dorsal digital, proper digital branches Nerve divides at hypothenar eminence Motor: Superficial Branch @[lateral to pisiform] Palmaris brevis Deep (Motor) Branch [around hook of hamate] Adductor pollicis THENAR MUSCLES Flexor pollicis brevis [FPB] [with median] HYPOTHENAR MUSCLES Abductor digiti minimi [ADM] Flexor digiti minimi brevis[FDMB] Opponens digiti minimi [ODM] INTRINSIC MUSCLES Dorsal interossei [DIO] [abduct DAB] Volar interossei [VIO] [adduct PAD] Lumbricals [medial two (3,4)] INFRACLAVICULAR MEDIAL AND LATERAL CORDS 2. Median (C(5)6-T1): runs through carpal tunnel, then cutaneous branches off at (risk in Carpal Tunnel release) Sensory: Palmar Cutaneous Branch Dorsal distal phalanges of 3 1/2 digits: via proper palmar digital branches Volar wrist capsule Volar 3 1/2 digits and lateral palm: via palmar palmar digital branches (multiple variations of thumb sensory innervation) Motor: Motor Recurrent (Thenar motor) Branch: Usually branches off median before carpal tunnel THENAR Abductor pollicis brevis [APB] Opponens pollicis Flexor pollicis brevis [FPB] l(Joint innervation with ulnar nerve)/l INTRINSIC Lumbricals [lateral two (1,2)] POSTERIOR CORD 3. Radial (C5-T1): Sensory: Dorsal 3 1/2 digits and hand: via superficial branch (dorsal digit branches) Dorsal wrist capsule Motor: NONE (in hand) Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier ARTERIES COURSE BRANCHES COMMENT DEEP PALMAR ARCH Through heads of the adductor pollicis T erminal branch of radial artery deep branch of the ulnar artery Princeps pollicis Radialis indicis Proper digital artery of thumb Under FPL, along 1st metacarpal May come from deep arch Palmar metacarpal (3) Joins common digital artery SUPERFICIALS PALMAR ARCH Just deep to aponeurosis. T erminal branch of ulnar artery superficial branch of the radial artery Common palmar digital (3) Bifurcates Proper palmar digital Along sides of fingers Proper palmar digital Of small finger only Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier DISORDERS: ARTHRITIS DESCRIPTION HISTORY/PHYSICAL EXAM WORK-UP/FINDINGS TREATMENT ARTHRITIS: OSTEOARTHRITIS/DEGENERATIVE JOINT DISEASE (DJD) • Wear and tear arthritis Hx: Older, women, pain worsewith activity XR: OA findings:osteophytes, joint spaceloss, sclerosis,subchondral cysts 1. NSAID, splint, steroid injection • Loss of articular cartilage PE: + IP (DIP and/or PIP)nodes, + CMC grind test 2. DIP: arthrodesis, CMC/PIP: arthroplasty • DIP #1 [Heberden's nodes] CMC, IP #2 [Bouchard's nodes] ARTHRITIS: RHEUMATOID • Systemic inflammatorydisease affecting synovium:destroys joints. MCP #1 Hx: Painful, stiff (worse in AM) XR: Hand series: joint destruction I. Medical management splinting • Has 4 stages PE: Multiple joint swelling. deformities: ulnar drift (MCP)swan neck, boutonniere Labs: RF, ANA, WBC, ESR, uric acid II. Synovectomy (single joint) • Associated with tenosynovitis,Carpal Tunnel Syndrome III/IV. T endon transfer orrepair, arthrodesis,arthroplasty FLEXOR TENOSYNOVITIS: TRIGGER FINGER/THUMB • Nodule on tendon Hx: Age: 401, tender 1. Steroid injection (+/-catcheson pulley (A1 most common) Hx: Age: 401, tender nodule XR: None needed 1. Steroid injection (+/-splint) • Also seen in Diabetes Mellitus PE: Pain. Locking with flexion extension 2. A1 release [must spare A2] Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier DISORDERS: LIGAMENT INJURIES DESCRIPTION HISTORY/PHYSICAL EXAM WORK-UP/FINDINGS TREATMENT CENTRAL SLIP INJURY: BOUTONNIERE DEFORMITY • Extensor tendon (central slip) at PIP ruptures, lateral bands slip volar and flex PIP. Hx: Hand trauma XR: Hand series: normal 1. Splint PIP in extension, DIP free PE: PIP flexed, no active extension, DIP extended 2. Reconstruct central slip and bands • Associated with RA 3. Severe: fusion or arthroplasty FLEXOR TENDON INJURY: JERSEY FINGER • Flexor tendon avulses from forceful extension Hx: Extension injury, 1/2 pain. XR: Rule out fracture (1/2 avulsion fracture) 1. Primary repair • In football; RF#1; FDPFDS PE: FDS: 1 sublimus test FDP: 1 profundus test 2. Older patient: DIP fusion MALLET FINGER • Extensor tendon rupture atdistal phalanx Hx: Minor trauma XR: 1/2 avulsion fracture 1. CONSTANT splint (DIP only) for 8 weeks PE: Cannot extend DIP, minimal pain swelling • FDP unopposed so DIP flexes . 2. Repair if large bony avulsion fracture SWAN NECK DEFORMITY • FDS rupture/volar plate injury Hx: Trauma, RA, spastic XR: Hand series 1. Early: splint • Lateral bands subluxes dorsally, PIP hyperextends DIP flexes PE: PIP yperextended, DIP flexed 2. Late: surgical repair (individualize flexes flexed each case) ULNAR COLLATERAL OF THUMB: GAMEKEEPER'S THUMB • Ulnar collateral ligament torn Hx: Trauma. Pain swelling. XR: 1/2 avulsion fracture. 1. Incomplete: splint 2-4 weeks • Mechanism: forceful radial deviation PE: Ulnar thumb unstable with radial extension/abduction Stress view shows injury 2. Complete: surgical repair (treat Stener lesion) • Often in ski pole injury Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier DISORDERS: INFECTIONS DESCRIPTION HISTORY/PHYSICAL EXAM WORK-UP/FINDINGS TREATMENT BITES: HUMAN/ANIMAL • Usually dominant hand Hx: Laceration or puncture,dorsal MCP most common location XR: Rule out fracture 1. Thorough ID, Td if necessary • Classic mechanism: fist fight Labs: Aerobic anaerobic cultures, WBC 2. IV antibioticsAnimal: Unasyn Human: Augmentin • Human: poly bacterial including Eikenella corrodens PE: Red, swollen, 1/2 drainage, streaking. Decreased extension if tendon torn [Contact health officials if animal possibly rabid] 3. Do not close wound, dress appropriately • Animal: Pasteurella multocida DEEP SPACE INFECTION • From palm puncture or spread from finger (+/-Horseshoe) Hx/PE: Erythema, fluctuance, and tenderness XR: Usually normal Dorsal volar ID and IV antibiotics FELON • Deep infection or abscess in pulp Hx/PE: Erythematous, swollen, and painful. XR: Usually normal 1. ID, release septae 2. IV antibiotics • Staph Aureus #1 organism PARONYCHIA/EPONYCHIA • Nail bed infection (most common finger infection) Hx/PE: Red, painful, swollen, often purulent drainage XR: Usually normal 1. Soaks and oral antibiotics 2. ID with nail removal if necessary • Staph Aureus #1 organism PURULENT TENOSYNOVITIS • Infection of flexor tendon sheath Hx: Puncture wound XR: Possible foreign body or subcutaneous air 1. Mild (early): IV antibiotics, re-evaluate within 24 hours • Usually from puncture wound PE: KANAVEL SIGNS: 1. Flexed position, 2. Pain on passive extension, 3. Fusiform swelling, 4. T ender flexor sheath 2. Most: I D (1/2 drain) and IV antibiotics • May extend into palm and develop “horseshoe” infection No treatment results in adhesions necrosis SPOROTRICHOSIS • Lymphatic infection (from roses) Hx/PE: Discoloration or rash XR: None Potassium iodine solution Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier DISORDERS: MASSES TUMORS DESCRIPTION HISTORY/PHYSICAL EXAM WORK-UP/FINDINGS TREATMENT DUPUYTREN'S DISEASE • Proliferation of fascia (long bands) Hx: Male, 401 years old XR: None needed 1. No proven conservative treatment • Northern European descent PE: nodule, non-tender, flexed digit (RF#1, SF#2) • Associated with DM, epilepsy 2. Fasciotomy ENCHONDROMA • #1 Primary bone tumor Hx: Pain after pathologic fracture XR: Lytic lesion Curettage and bone graft • Usually proximal phalanx EPIDERMAL INCLUSION CYST • Epidermal cells embedded deep into tissue Hx: Trauma or puncture XR: Normal Excision (get all epidermal cells or it will recur) PE: Painless mass, usually on digits, no transillumination GANGLION RETINACULAR CYST • Cyst (arises from joint or tendon) with mucinous joint fluid Hx: Y oung patient XR: No osteophyte in corresponding area Aspiration of cyst if symptomatic. (may recur) PE: Visible, firm mass (volar MCP flexor tendon #1 site). • Most common mass in hand GIANT CELL TUMOR (FIBROXANTHOMA) • Originates from tendon sheath Hx/PE: Firm, painless mass, usually volar finger (IF,MF) XR: Normal Excise, they do recur • 2nd most common hand mass MALIGNANT TUMORS • #1 Primary: squamous cell Hx/PE: Mass, usually on dorsum of hand XR: Normal Excise • #1 Metastatic: lung MUCOUS CYST • A ganglion of dorsal DIP Hx: Women, older patients XR: OA and/or spur at DIP Excision and osteophyte or joint debridement • Associated with OA at DIP PE: Dorsal DIP mass, 1/2 pain Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier SURGICAL APPROACHES USES INTERNERVOUSPLANE DANGERS COMMENT FINGER: VOLAR APPROACH 1. Flexor tendons (repair/explore) No planes 1. Digital artery 1. Make a “zig-zag” incision with angles of 90° 2. Digital nerve 2. Digital nerve 3. Soft tissue releases 2. Neurovascular bundle is lateral to the tendon sheath 4. Infection drainage FINGER: MID-LATERAL APPROACH Phalangeal fractures No planes 1. Digital nerve Soft tissues are thin, capsule can be incised if care is not taken. 2. Digital artery Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com CHAPTER 6 - PELVIS TOPOGRAPHIC ANATOMY OSTEOLOGY LANDMARKS AND OTHER STRUCTURES TRAUMA JOINTS HISTORY AND PHYSICAL EXAM PHYSICAL EXAM OF THE PELVIS PHYSICAL EXAM MUSCLES: ORIGINS AND INSERTIONS ANTERIOR MUSCLES (also see muscles of the thigh/hip) GLUTEAL MUSCLES (also see muscles of the thigh/hip) NERVES ARTERIES Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier CHAPTER 6 – PELVIS TOPOGRAPHIC ANATOMY Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier OSTEOLOGY CHARACTERISTICS OSSIFY FUSE COMMENT INNOMINATE: COXAL BONE • One bone: started as 3, connected by tri-radiate cartilage at acetabulum Ilium: body ala Ischium: body ramus Pubis: body 2 rami Primary (one in each body) 2-6 mo to acetabulum 15 yrs • Iliac wing and superior pubic ramus are “weak spots” • ASIS: avulsion fracture can result from sartorius Secondary Iliac crest Acetabulum Ischial tuberosity AIIS Pubis 15 yrs All fuse 20 yrs • AIIS: avulsion fracture can result from rectus femoris • Two innominate per pelvis (L R) • Iliac crest ossification used to determine skeletal maturity (Risser stage) • Acetabulum: anteverted and oblique orientation (approx. 45°) • Iliac crest contusion referred to as “hip pointer” SACRUM See spine chapter Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier LANDMARKS AND OTHER STRUCTURES STRUCTURE ATTACHMENTS/ RELATED STRUCTURES COMMENT ASIS Sartorius Inguinal ligament • LFCN crosses the ASIS can be compressed there (Meralgia paresthetica) Transverse internal oblique abdominal muscles • Sartorius can avulse from it (avulsion fracture) AIIS Rectus femoris T ensor fascia lata Iliofemoral ligament (hip capsule) • Rectus femoris can avulse from it (avulsion fracture) PSIS Posterior sacroiliac ligaments • Excellent bone graft site Marked by skin dimple Arcuate line Pectineus muscle • Strong, weight bearing region Gluteal lines 3 lines: anterior, inferior, posterior • Separate origins of gluteal muscles Greater trochanter SEE ORIGINS/INSERTIONS • T ender with trochanteric bursitis Lesser trochanter Iliacus Psoas muscles Ischial tuberosity SEE ORIGINS/INSERTIONS Sacrotuberous ligaments • Excessive friction can cause bursitis (Weaver's bottom) Ischial spine Coccygeus Levator ani attach Sacrospinous ligaments Anterior (iliopubic) column of acetabulum Consists of: 1. Pubic ramus 2. Anterior acetabulum 3. Anterior iliac wing • Involved in several different fracture patterns Posterior (ilioischial) column of acetabulum Consists of: 1. Ischial tuberosity 2. Posterior acetabulum 3. Sciatic notch • Involved in several different fracture patterns Lesser sciatic foramen Short external rotators exit: Obturator externus Obturator internus Greater sciatic foramen Structures that exit: 1. Superior gluteal nerve 2. Superior gluteal artery 3. Piriformis muscle 4. Pudendal nerve 5. Inferior pudendal artery 6. Nerve to the Obturator internus 7. Posterior • Piriformis muscle is the reference point • Superior Gluteal nerve and artery exit superior to the piriformis • POP'S IQ is a mnemonic for the Cutaneous nerve of thigh 8. Sciatic nerve 9. Inferior gluteal nerve 10. Inferior gluteal artery 11. Nerve to Quadratus femoris nerves (structures) that exit inferior to the piriformis (medial to lateral) Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier TRAUMA Classification of Pelvic Fractures (Y oung and Burgess) DESCRIPTION EVALUATION CLASSIFICATION TREATMENT PELVIC FRACTURE • Mechanism 1: High energy force (e.g. MVA). Lateral force more common than AP • Usually associated with other injuries (often life threatening). • Open pelvic fracture with associated GI and/or GU injury: 50% mortality • Posterior SI ligament is key to pelvic stability • Mechanism HX: Trauma. Swelling, pain, deformity. PE: ABC's. Affected LE shortened, +/-blood in rectum/vagina/urethra. Do good neurovascular exam: +/-pulses in groin LE with neurologic deficits including loss of rectal tone bulbocavernosus reflex. XR: AP, Inlet, Outlet Judet views of the pelvis. Y oung and Burgess: • AP compression (APC): I. 2.5cm pubic diastasis fracture of 1-2 rami II. 2.5cm diastasis; SI disruption, but stable III. Complete disruption pubis symphysis SI joint: unstable fracture • Lateral Compression (LC): I. Sacral compression with rami fractures II. Rami Treat life threatening injuries first (ABC's). Treat pelvic hemorrhage with external fixation (+/-2embolization) Diverting colostomy for GI injury (avoid sepsis) Stable fractures: (single ramus, avulsion fx, APC or LC I): conservative treatment; bedrest, decreased Mechanism 2: Minor trauma (e.g. fall on osteopenic bone): stable single ramus fracture • Mechanism 3: Stable avulsion fracture -ASIS (Sartorius) -AIIS (Rectus femoris) -Ischium (hamstring) CT : Scan entire pelvis AGRAM: for hemorrhage II. Rami fracture, posterior SI ligment disrupted, but stable III. LC II, with contralateral APC III (“windswept” ) • Vertical shear: anterior posterior pelvic injury (displacement): vertically unstable. decreased activity Unstable fractures: external fixation with ORIF as needed Early mobilization aids recovery COMPLICATIONS: Associated injuries (especially with APC III): 1. GI, 2. GU, 3. Vascular/hemorrhage, 4. Neurologic; Prolonged hospital stay with associated risks (infection, DVT , etc.); Residual deformity and/or pain (lower back or SI); Leg length discrepancy DESCRIPTION EVALUATION CLASSIFICATION TREATMENT ACETABULAR FRACTURE • Uncommon, younger • High energy or violent injury; femoral head is forced into acetabulum • Dislocation of hip is often associated • Also GI, GU, vascular associated injuries. HX: Trauma (e.g. dashboard injury). Pain, deformity. PE: LE shortened, rotated. Usually neurovascularly intact distally. XR: AP. Internal external obliques (Judet views): many possible fracture sites CT : shows fracture pattern and loose fragments Judet/Letournel: I. Posterior wall II. Posterior column III. Anterior wall IV. Anterior column V. Transverse VI. Posterior column wall VII. Transverse post. wall VIII. T-type IX. Anterior column posterior emi-transverse X. Both columns Traction on affected side Nondisplaced, congruent joint, Displaced, dislocation, unstable fx: ORIF XRT (600 rads) prophylaxis for heterotopic bone. COMPLICATIONS: Need for T otal Hip Arthroplasty; Nerve injury (sciatic); Heterotopic bone formation; Osteonecrosis steoarthritis Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier JOINTS LIGAMENTS ATTACHMENTS COMMENTS SACROILIAC (GLIDING) Posterior SI (short long) Sacrum to ilium: Short are horizontal Long are vertical Strongest SI ligaments: key to stability. Short: resist rotation Long: resist vertical shear Disruption: rotational vertical instability Anterior SI Sacrum to ilium (horizontal) Rotational stability Interosseous Sacral to iliac tuberosities Strong LIGAMENTS ATTACHMENTS COMMENTS SYMPHYSIS PUBIS Superior pubic ligament Both pubic bones superiorly There is a fibrocartilage disc between the two hemipelvi Arcuate pubic ligament Both pubic bones inferiorly OTHER LIGAMENTS Sacrospinous Anterior sacrum to ischial spine Divides greater lesser sciatic foramina; provides rotational stability Sacrotuberous Anterior sacrum to ischial tuberosity Inferior border of lesser sciatic foramina; provides vertical stability Iliolumbar L5 transverse process to crest Can result in avulsion fracture Lumbosacral L5 transverse process to ala Vertical stability Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier HISTORY AND PHYSICAL EXAM QUESTION ANSWER CLINICAL APPLICATION 1. AGE Y oung Middle age, elderly Ankylosing Spondylitis (1HLA-b27) Decreased mobility 2. PAIN a. Onset b. Character c. Occurrence Acute Chronic Deep, non-specific Radiating In out of bed, on stairs Adducting legs Trauma: fracture, sprain Systemic inflammatory disorder Sacroiliac etiology T o thigh or buttock on ipsilateral side: SI joint injury Sacroiliac etiology Symphysis pubis etiology 3. PMHx Pregnancy Laxity of ligaments of SI joint causes pain 4. TRAUMA Fall on buttock, twist injury Sacroiliac joint injury High velocity: MVA, fall Fracture 5. ACTIVITY/WORK Twisting, stand on one leg Sacroiliac etiology 6. NEUROLOGIC SYMPTOMS Pain, numbness, tingling Spine etiology, sacroiliac etiology 7. HISTORY of ARTHRITIDES Multiple joints involved SI involvement of RA, Reiter's syndrome, Ankylosing Spondylitis, etc. Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier PHYSICAL EXAM OF THE PELVIS EXAM/ OBSERVATION TECHNIQUE CLINICAL APPLICATION INSPECTION Skin Discoloration, wounds ASIS's, Iliac crests Both level (same plane) If on different plane: Leg length discrepancy, sacral torsion Lumbar curvature Increased lordosis Flexion contracture Decreased lordosis Paraspinal muscle spasm PALPATION Bony structures Standing: ASIS, Pubic Iliac tubercles, PSIS Unequal side to side 5pelvic obliquity: leg length discrepancy Lying: Iliac crest, Ishial tuberosity Mass: cluneal neuroma Soft tissues Inguinal ligament Protruding mass: hernia Femoral pulse nodes Diminished pulse: vascular injury; palpable nodes: infection Muscle groups Each group should be symmetric bilaterally RANGE OF MOTION Forward flexion Standing: bend forward PSIS's should elevate slightly (equally) Extension Standing: lean backward PSIS's should depress (equally) Hip flexion Standing: knee to chest PSIS should drop but will elevate in hypomobile SI joint Ischial tuberosity should move laterally, will elevate in hypomobile SI joint Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier PHYSICAL EXAM EXAM/ OBSERVATION TECHNIQUE CLINICAL APPLICATION NEUROVASCULAR Sensory Iliohypogastric nerve (L1) Suprapubic, lateral buttocks thigh Deficit indicates corresponding nerve/root lesion Ilioinguinal nerve (L1) Inguinal region Deficit indicates corresponding nerve/root lesion (e.g. abdominal muscle compression) Genitofemoral nerve (L1-2) Scrotum or mons Deficit indicates corresponding nerve/root lesion Lateral femoral cutaneous nerve (L2-3) Lateral hip thigh Deficit indicates corresponding nerve/root lesion (e.g. Meralgia paresthetica) Pudental nerve (S2-4) Perineum Deficit indicates corresponding nerve/root lesion Motor Femoral (L2-4) Hip flexion Weakness 5Iliopsoas or corresponding nerve/root lesion Inferior Gluteal nerve External rotation Weakness 5Gluteus maximus or nerve/root lesion Nerve to Quadratus femoris External rotation Weakness 5Short rotators or corresponding nerve/root lesion Nerve to Obturator internus Nerve to Piriformis Superior Gluteal nerve Abduction Weakness 5Gluteus medius/minimus, TFL or corresponding nerve/root lesion Reflex Bulbocavernosus Finger in rectum, squeeze or pull penis (Foley), anal sphincter should contract Pulses Femoral pulse SPECIAL TESTS Straight leg Supine: extend Pain radiating to LE: HNP with radiculopathy Straight leg knee, flex hip Pain radiating to LE: HNP with radiculopathy SI stress Press ASIS, iliac crest, sacrum Pain in SI could be SI ligament injury Trendelenburg sign Standing: lift one leg (flex hip) Flexed side: pelvis should elevate; if pelvis falls: Abductor or gluteus medius dysfunction Patrick (FABER) Flex, ABduct, ER hip, then abduct more Positive if pain or LE will not continue to abduct below other leg: SI joint pathology Meralgia Pressure medial to ASIS Reproduction to pain, burning, numbness: LFCN entrapment Rectal Vaginal exam Especially after trauma Gross blood indicates trauma communicating with those organ systems Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier MUSCLES: ORIGINS AND INSERTIONS PUBIC RAMI (ASPECT) GREATER TROCHANTER ISCHIAL TUBEROSITY LINEA ASPERA/ POSTERIOR FEMUR Pectineus (pectineal line/superior) Piriformis (anterior) Inferior gemellus Adductor magnus Adductor magnus (inferior) Obturator internus (anterior) Quadratus femoris Adductor longus Adductor longus (anterior) Superior gemellus Semimembranosus Adductor brevis Adductor brevis (inferior) Gluteus medius (posterior) Semitendinosus Biceps femoris Gracilis (inferior) Gluteus minimus (anterior) Biceps femoris (LH) Pectineus Psoas minor (superior) Adductor magnus Gluteus maximus Vastus lateralis Vastus medialis Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier ANTERIOR MUSCLES (also see muscles of the thigh/hip) MUSCLE ORIGIN INSERTION NERVE ACTION COMMENT HIP FLEXORS ANTERIOR Psoas T12-L5 vertebrae Lesser trochanter Femoral Flex hip Covers lumbar plexus Iliacus Iliac fossa Lesser trochanter Femoral Flex hip Covers anterior ilium Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier GLUTEAL MUSCLES (also see muscles of the thigh/hip) MUSCLE ORIGIN INSERTION NERVE ACTION COMMENT HIP ABDUCTORS T ensor fascia latae Iliac crest, ASIS Iliotibial band Superior Gluteal Abducts, flex, IR thigh A plane in anterior approach to hip HIP ABDUCTORS Gluteus medius Ilium between anterior posterior gluteal lines Greater trochanter Superior Gluteal Abduct (IR) thigh Trendelenburg gait if muscle is out. Gluteus minimus Ilium between anterior interior gluteal lines Anterior greater trochanter Superior Gluteal Abduct (IR) thigh Works in conjunction with medius HIP EXTERNAL ROTATORS Gluteus maximus Ilium, dorsal sacrum Gluteal tuberosity (femur), ITB Inferior Gluteal Extend, ER thigh Must detach in post. approach to hip Piriformis Anterior sacrum Superior greater trochanter Piriformis ER thigh Used as landmark Obturator externus Ischiopubic rami, obturator membrane Trochanteric fossa Obturator ER thigh Muscle actually in medial thigh Short Rotators Obturator internus Ischiopubic rami, obturator membrane Medial greater trochanter N. to Obturator internus ER, abduct thigh Muscle makes a right turn Superior gemellus Ischial spine Medial greater trochanter N. to Obturator internus ER thigh Assists obturator internus Inferior gemellus Ischial tuberosity Medial greater trochanter N. to Quadratus femoris ER thigh Assists obturator internus Quadratus femoris Ischial tuberosity Intertrochanteric crest N. to Quadratus femoris ER thigh Runs with ascending branch of medial circumflex artery Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier NERVES LUMBAR PLEXUS ANTERIOR DIVISION 1. Subcostal (T12): Sensory: Subxyphoid region Motor: NONE 2. Iliohypogastric (L1) Sensory: Above pubis Posterolateral buttocks Motor: Transversus abdominus Internal Oblique 3. Ilioinguinal (L1) Sensory: Inguinal region Motor: NONE 4. Genitofemoral(L1-2): pierces Psoas, lies on anteromedial surface. Sensory: Scrotum or mons Motor: Cremaster 5. Obturator (L2-4): exits via obturator canal, splits into ant. post. divisions. Can be injured by retractors placed behind the transverse acetabular ligament. Sensory: Inferomedial thigh via cutaneous branch of Obturator nerve Motor: External oblique Obturator externus (posterior division) 6. Accessory Obturator (L2-4): inconsistent Sensory: NONE Motor: Psoas POSTERIOR DIVISION 7. Lateral Femoral Cutaneous LFCN: crosses, ASIS, can be compressed at ASIS Sensory: NONE (in pelvis) Motor: NONE 8. Femoral (L2-4): lies between psoas major and iliacus Sensory: NONE (in pelvis) Motor: Psoas Iliacus Pectineus SACRAL PLEXUS ANTERIOR DIVISION 9. Nerve to Quadratus femoris (L4-S1): Sensory: NONE Motor: Quadratus femoris Inferior gemelli 10. Nerve to Obturator internus (L5-S2): exits greater sciatic foramen Sensory: NONE Motor: Obturator internus Superior gemelli 11. Pudendal (S2-4): exits greater then re-enters lesser sciatic foramen Sensory: Perineum:via Perineal (scotal/labial branches)via Inferior rectal nervevia Dorsal nerve to penis/clitoris Motor: Bulbospongiosus: Perineal nerve Isiocavernosus: Perineal nerve Urethral sphincter: Perineal nerve Urogenital diaphragm: Perineal nerve Sphincter ani externus: Inf. rectal nerve 12. Nerve to coccygeus (S3-4) Sensory: NONE Motor: Coccygeus Levator ani POSTERIOR DIVISION 13. Superior Gluteal (L4-S1): Sensory: NONE Motor: Gluteus medius Gluteus minimus T ensor fascia lata 14. Inferior Gluteal (L5-S2): Sensory: NONE Motor: Gluteus maximus 15. Nerve to piriformis (S2): Sensory: NONE Motor: Piriformis OTHER NERVES (non-plexus) 16. Cluneal nerves: branches of lumbar and sacral dorsal rami. Can be injured during bone grafts. Sensory: Skin of gluteal region Motor: NONE Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier ARTERIES COURSE BRANCHES COMMENT AORTA Along anterior vertebral bodies ALL Common iliacs at L4 Lumbar arteries (4 sets) Paired: posterior branch supplies cord, meninges paraspinal muscles Median sacral artery 5th Lumbar arteries (2) Unpaired vessel Anastomoses with lat. sacral artery COMMON ILIACS Still on anterior L-spine sacrum Divide into internal external iliacs at S1 INTERNAL ILIAC Under ureter near SI joint, divides into its divisions at edge of greater sciatic foramen Supplies most of pelvis and the pelvic organs ANTERIOR DIVISION Obturator Runs with nerve through foramen Fovea artery (artery of ligamentum teres in hip) Minor contributions to the vascular supply of the femoral head Inferior gluteal Supplies muscles of the Inferior gluteal buttocks Multiple visceral branches[] POSTERIOR DIVISION Superior gluteal Supplies muscles of the buttocks Iliolumbar Supplies iliopsoas and ilium Lateral sacral Supplies sacral roots, meninges, muscles covering sacrum EXTERNAL ILIAC Under inguinal ligament over the pubic rami, on the psoas muscle Does not supply much in the pelvis Deep circumflex iliac artery Inferior epigastric artery Femoral artery (under inguinal ligament) At risk T otal Hip Arthroplasty (THA) Other branches of the Internal iliac include: Umbilical, V aginal/Inferior vesical, Uterine, Middle rectal, Inferior pudendal Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com CHAPTER 7 - THIGH/HIP TOPOGRAPHIC ANATOMY OSTEOLOGY TRAUMA JOINTS MINOR PROCEDURES HISTORY PHYSICAL EXAM MUSCLES: ORIGINS AND INSERTIONS MUSCLES: ANTERIOR MUSCLES: MEDIAL MUSCLES: POSTERIOR (HAMSTRINGS) THIGH MUSCLES: CROSS SECTIONS NERVES ARTERIES ARTERIES OF THE FEMORAL NECK DISORDERS TOTAL HIP ARTHROPLASTY TIPS ON TOTAL HIPS PEDIATRIC DISORDERS SURGICAL APPROACHES Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier CHAPTER 7 – THIGH/HIP TOPOGRAPHIC ANATOMY Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier OSTEOLOGY CHARACTERISTICS OSSIFY FUSE COMMENT FEMUR • Long bone characteristics • Proximally: head, neck, greater lesser trochanters • Neck: bone comprised of tensile compressive groups • Distally: 2 condyles Lateral: more anterior proximal Medial: larger, more posterior distal Primary (Shaft) Secondary 1. Distal physis 2. Head 3. Greater trochanter 4. Lesser trochanter 7-8 wks (fetal) Birth 1 yr 4-5 yr 10 yr 16-18 years 19 years 18 years 16 years 16 years • Blood supply Head neck: branches of the Medial Lateral circumflex artery (from profunda) Shaft: nutrient (from profunda) • Head neck vascularity tenuous: increased risk of ischemia in fracture or dislocation. • Femoral neck weakens with age: susceptable to • Femoral anteversion: 12-14° • Neck/shaft angle: 126° susceptable to fracture • Anatomic axis: along shaft of femur • Mechanical axis: femoral head to intercondylar notch Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier TRAUMA DESCRIPTION EVALUATION CLASSIFICATION TREATMENT HIP DISLOCATION • High energy trauma (esp MVA-dashboard injury or significant fall.) • Orthopaedic emergency • Multiple associated injuries +/- fractures, (e.g. femoral head neck) • Posterior most common (85%) HX: Trauma. Severe pain, Cannot move thigh/hip. PE: Thigh position: Post: adducted, flexed, IR Ant: abducted, flexed, ER. Pain (esp. with motion), good neurovascular exam XR: AP pelvis, frog lateral (Femoral head is different size) Also femur knee series CT : Rule out fracture or bony fragments Posterior. Thompson: I. Simple, no posterior fragment II. Simple, large posterior fragment III. Comminuted posterior fragment IV. Acetabular fracture V. Femoral head fracture Anterior. Epstein: I. (A, B, C): Superior II. (A, B, C): Inferior A: No associated fracture B: Femoral head fracture C: Acetabular fracture Early reduction essential, then repe XR neurologic exam Posterior: I: Closed reduction abduction pillow II-V: 1. Closed Reduction (ope if irreducible) 2. ORIF fracture or excise fragme Anterior: closed reduction, ORIF if necessary. COMPLICATIONS: Osteonecrosis (AVN) reduced risk with early reduction; Sciatic nerve injury (posterior dislocations); Femoral artery nerve injury (anterior dislocations); Instability recurrence; Osteoarthritis; Heterotopic ossification DESCRIPTION EVALUATION CLASSIFICATION TREATMENT FEMORAL NECK FRACTURE • Mechanism: 1. Fall by elderly woman most common; 2. High velocity injury in young adults • Intracapsular fractures • Associated with osteoporosis • Often caused by medical condition (syncope, etc) • High morbidity complication rate (25%) HX: Fall. Pain, inability to bear weight or walk. PE: LE shortened, abducted, externally rotated. Pain with “rolling” of leg. XR: AP pelvis (+/-IR), groin lateral MR: If symptomatic with negative XR Garden (4 types): I. Incomplete fracture; valgus impaction II. Complete fracture; nondisplaced III. Complete fracture, Partial displacement (varus) IV. Complete fracture, total displacement Early reduction essential All fractures: Closed (open) reduction then IF of fracture: Y oung: 3 parallel screws Old: hemi-arthroplasty (Stable fracture, type I, may heal without surgery, ORIF because of displacement risk) COMPLICATIONS: Osteonecrosis (AVN) incidence increases with fracture type (displacement) +/- late segmental collapse; Nonunion; Hardware failure DESCRIPTION EVALUATION CLASSIFICATION TREATMENT SUBTROCHANTERIC FRACTURE • Fall by a more elderly woman most common • Associated with osteoporosis • Occurs along or below the intertrochanteric line • Extracapsular fractures • Stable vascularity • Most heal well with proper fixation HX: Fall. Pain, inability to bear weight or walk PE: LE shortened, ER. Pain with “log rolling” of leg XR: AP pelvis (+/-IR), groin lateral MR: If symptomatic with negative XR Evans (based on post-reduction stability) Type I. Stable Type II. Unstable Nonoperative is very rarely indicated. Operative treatment with sliding compression hip screw and side plate. Early mobilization with partial weight-bearing COMPLICATIONS: Nonunion/Malunion; Hardware failure or loss of reduction; Infection. Mortality rate, first 6 months after fracture, is 15-25% SUBTROCHANTERIC FRACTURE • Mechanism: 1. Fall in elderly 2. Trauma in young Occurs below HX: Trauma or fall. Pain, swelling PE: Swelling, Seinsheimer (5 types): I. Non or minimally displaced II. Displaced: 2 parts Nonoperative treatment: traction hip spica cast for 6-8 wks (not commonly used) • the lesser trochanter (up to 5cm below it). • Pathologic fractures seen here. • Decreased vascularity = tenuous healing Swelling, tenderness +/-shortening of LE XR: AP lateral II. Displaced: 2 parts III. Displaced: 3 parts IV. Comminuted (41parts) V. Subtrochanteric/intertrochanteric fracture. used) Operative treatment: Locked IM nail, compression screw, or Zickel nail, +/-bone graft COMPLICATIONS: Nonunion/Malunion; Hardware failure or loss of reduction; Refracture with hardware removal DESCRIPTION EVALUATION CLASSIFICATION TREATMENT FEMORAL SHAFT FRACTURE • Orthopaedic emergency • High energy injury • Multiple associated injuries (many serious) • Potential source of significant blood loss • Patient should be transported with leg in traction HX: Trauma. Pain, swelling deformity PE: Deformity, +/-open wound soft tissue injury; Check distal pulses XR: AP lateral thigh, knee trauma series. Winquist/Hansen (4 types): Stable I. No/minimal comminution II. Comminuted: 50% of cortices intact Unstable III. Comminuted: 50% of cortices intact IV. Complete comminution, no intact cortex Extensive irrigation of any open fractures Operative: Interlocking intramedullary rods (closed) Early mobilizaton with crutch ambulation COMPLICATIONS: Neurovascular injury and/or hemorrhagic shock; Nonunion/Malunion; Hardware failure or loss of reduction; Knee injury (5%) DISTAL FEMUR FRACTURE • Mechanism: direct blow • Metaphysis or epiphysis • Quadriceps or gastrocnemius often displace fragments • Restoration of articular surface is essential to regain normal knee mobility function HX: Trauma. Cannot bear weight, pain, swelling. PE: Effusion, tenderness, do good neurovascular exam XR: Knee trauma series CT : Better defines fracture AGRAM: if pulseless Extraarticular Supracondylar Intraarticular Intercondylar: T or Y Condylar +/- aspirate hemarthroses Undisplaced/extraarticular: reduce, immobilize (less commonly used method) Displaced/intraarticular: ORIF: plates and screws or intramedullary nails Early mobilization COMPLICATIONS: Osteoarthritis and/or pain; Decreased range of motion; Malunion/nonunion; Instability Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier JOINTS LIGAMENTS ATTACHMENTS COMMENTS HIP JOINT (Spheroidal/Ball and Socket type) Transverse acetabular Anteroinferior to posteroinferior acetabulum Cups the acetabulum Labrum Acetabular rim Deepens stabilizes acetabulum JOINT CAPSULE Acetabular rim to femoral neck Pubofemoral (anterior/inferior) Femoral neck to superior pubic ramus Covers femoral NECK Iliofemoral (anterior) (Y ligament of Bigelow) AIIS to intertrochanteric line Strongest, most support Ishiofemoral (posterior) Posterior rim to intertrochanteric crest Posterior femoral neck only partially covered (weak) Zona orbicularis (posterior) Ligament of T eres Fovea to cotyloid notch Artery runs in ligament Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier MINOR PROCEDURES STEPS HIP INJECTION OR ASPIRATION 1. Ask patient about allergies 2. Place patient supine, palpate the greater trochanter. 3. Prepare skin over insertion site (iodine/antiseptic soap) 4. Anesthetize skin locally (quarter size spot) 5. ANTERIOR: Find the point of intersection between a vertical line below ASIS and horizontal line from Greater trochanter. Insert 20 gauge (3 inch/spinal needle) upward slightly medial direction at that point. LATERAL: Insert a 20 gauge (3 inch/spinal needle) superior and medial to greater trochanter until it hits the bone (the needle should be within the capsule which extends down the femoral neck). Inject (or aspirate) local or local/steroid preparation into joint. (The fluid should flow easily if needle is in joint) 6. Dress injection site TROCHANTERIC BURSA INJECTION 1. Ask patient about allergies 2. Place patient in lateral decubitus position, palpate the greater trochanter. 3. Prepare skin over lateral thigh (iodine/antiseptic soap) 4. Insert 20 gauge needle (at least 1 1/2inches) into thigh to the bone at the point of most tenderness. Withdraw needle (1—2mm) so it is just off the bone and in the bursa. Aspirate to ensure needle is not in a vessel. Inject 10ml of local or 4:1 local/corticosteroid preparation into bursa 5. Dress injection site Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier HISTORY QUESTION ANSWER CLINICAL APPLICATION 1. AGE Y oung Trauma, developmental disorders Middle age, elderly Arthritis (inflammatory conditions), femoral neck fractures 2. PAIN a. Onset b. Location c. Occurrence Acute Chronic Lateral hip or thigh Buttocks/posterior thigh Groin/medial thigh Anterior thigh Ambulation/motion At night Trauma, infection Arthritis (inflammatory conditions) Bursitis, LFCN entrapment, snapping hip Consider spine etiology Hip joint or acetabular etiology (less likely to be from pelvis or spine) Proximal femur Hip joint etiology (i.e. not pelvis or spine) Tumor, infection 3. SNAPPING With ambulation Snapping hip syndrome, loose bodies, arthritis, synovitis 4. ASSISTED AMBULATION Cane, crutch, walker Use (and frequency) indicates severity of pain condition 5. ACTIVITY TOLERANCE Walk distance activity cessation Less distance walked and fewer activities no longer performed = more severe 6. TRAUMA Fall, MVA Fracture, dislocation, bursitis 7. ACTIVITY/WORK Repetitive use Femoral stress fracture 8. NEUROLOGIC SYMPTOMS Pain, numbness, tingling LFCN entrapment, spine etiology 9. HISTORY OF ARTHRITIDES Multiple joints involved Systemic inflammatory disease Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier PHYSICAL EXAM EXAM/OBSERVATION TECHNIQUE CLINICAL APPLICATION INSPECTION Skin Discoloration, wounds Trauma Gross deformity Fracture, dislocation Gait 60%stance, 40%swing Normal gait: 20% double stance (both feet on ground) Antalgic (painful) Decreased stance phase Knee, ankle, heel (spur), midfoot, toe pain Lurch (Trendelenburg) Laterally (on WB side) Gluteus medius weakness, hip disease (OA, AVN) Lurch Posteriorly (hip extended) Gluteus maximus weakness Steppage More hip knee flexion Foot drop, weak anterior leg muscles Flat foot No push off Hallux rigidus, gastrocnemius/soleus weakness Wide Feet 4 inches apart Neurologic/cerebellar disease Decreased step size Less than previous normal Pain, age, other pathology PALPATION Bony structures Greater trochanter/bursa Pain/palpable bursa: infection/bursitis, gluteus medius tendinitis Soft tissues Sciatic nerve (hip Pain: disc herniation, piriformis spasm Soft tissues flexed) Pain: disc herniation, piriformis spasm Muscle groups Each group should be symmetric bilaterally EXAM/OBSERVATION TECHNIQUE CLINICAL APPLICATION RANGE OF MOTION Flexion Supine: knee to chest Normal: 130 degrees Thomas test: see next page Rule out flexion contracture Extension Prone: lift leg off table Normal: 20 degrees Abduction/adduction Supine: leg lateral/medial Normal: Abd: 40 degrees, Add: 30 degrees Internal / External rotation Seated: foot lateral/medial Normal: IR: 30 degrees, ER: 50 degrees Prone: flex knee leg: in out Normal: IR: 30 degrees, ER: 50 degrees NEUROVASCULAR Sensory Genitofemoral nerve (L1-2) Proximal anteromedial thigh Deficit indicates corresponding nerve/root lesion Obturator nerve (L2-4) Inferomedial thigh Deficit indicates corresponding nerve/root lesion Lateral Femoral Cutaneous nerve (L2-3) Lateral thigh Deficit indicates corresponding nerve/root lesion Femoral nerve (L2-4) Anteromedial thigh Deficit indicates corresponding nerve/root lesion Posterior Femoral Cutaneous nerve (S1-3) Posterior thigh Deficit indicates corresponding nerve/root lesion Motor Obturator nerve (L2-4) Thigh adduction Weakness =Adductor muscle group or nerve/root lesion. Superior Gluteal nerve (L5) Thigh abduction Weakness =Gluteus medius or nerve/root lesion. Femoral nerve (L2-4) Hip flexion Weakness =Iliopsoas or corresponding nerve/root lesion. Weakness =Quadriceps or Knee extension Weakness =Quadriceps or corresponding nerve/root lesion. Inferior Gluteal nerve (L5-S2) Hip extension Weakness =Gluteus maximus or nerve/root lesion. Sciatic: Tibial portion (L4-S3) Knee flexion Weakness =Biceps Long Head or nerve/root lesion. Peroneal portion (L4-S2) Knee flexion Weakness =Biceps Short Head or nerve/root lesion Reflex None Pulses Femoral EXAM/OBSERVATION TECHNIQUE CLINICAL APPLICATION SPECIAL TESTS Thomas sign Supine: one knee to chest If opposite thigh elevates off table: flexion contracture of that side Ober On side: flex abduct hip Leg should then adduct, if stays in abduction: ITB contracture Piriformis On side: adduct hip Pain in hip/pelvis indicates tight piriformis (compressing sciatic nerve) Leg length discrepancy ASIS to medial malleolus A measured difference of 1cm is positive 90-90 straight leg Flex hip knee 90°, extend knee 20 degrees of flexion after full knee extension =tight hamstrings Ely's Prone: passively flex knee If hip flexes as knee is flexed: tight rectus femoris muscle Log roll Supine, hip extended: IR/ER Pain in hip is consistent with arthritis Patrick (FABER) Flex, ABduct, ER hip, then abduct more (figure of 4) Positive if pain or LE will not continue to abduct below other leg: Hip or SI joint pathology Meralgia Pressure medial to ASIS Reproduction to pain, burning, numbness: LFCN entrapment Ortolani (Peds) Hips at 90°, abduct hips A clunk indicates the hip(s) was dislocated and now reduced Barlow (Peds) Hips at 90°, posterior force A clunk indicates the hip(s) is now dislocated, should reduce with Ortolani Galeazzi (Peds) Supine:Flex hips knees Any discrepancy in knee height : 1. Dislocated hip, 2. Short femur Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier MUSCLES: ORIGINS AND INSERTIONS PUBIC RAMI (ASPECT) GREATER TROCHANTER ISCHIAL TUBEROSITY LINEA ASPERA/ POSTERIOR FEMUR Pectineus (pectineal line/sup) Piriformis (anterior) Inferior gemellus Adductor magnus Adductor magnus (inferior) Obturator internus (anterior) Quadratus femoris Adductor longus Adductor longus (anterior) Superior gemellus Semimembranosus Adductor brevis Adductor brevis (inferior) Gluteus medius (posterior) Semitendinosus Biceps femoris Gracilis (inferior) Gluteus minimus (anterior) Biceps femoris (LH) Pectineus Psoas minor (superior) Adductor magnus Gluteus maximus Vastus lateralis Vastus medialis Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier MUSCLES: ANTERIOR MUSCLE ORIGIN INSERTION NERVE ACTION COMMENT Articularis genu Distal anterior femoral shaft Synovial capsule Femoral Pulls capsule superiorly in extension May join with vastus intermedius Sartorius ASIS Proximal medial tibia (Pes anserinus) Femoral Flex, ER hip Can avulse from ASIS (fracture) QUADRICEPS Rectus femoris AIIS, superior rim of acetabulum Patella/tibial tubercle Femoral Flex thigh, extend leg Can avulse from AIIS (fracture) LEG EXTENSORS Vastus lateralis Greater trochanter, lateral linea aspera Lateral patella, tibial tubercle Femoral Extend leg Oblique fibers can affect Q angle Vastus intermedius Proximal femoral shaft Patella; tibial tubercle Femoral Extend leg Covers articularis genu Vastus medialis Intertrochanteric line, medial linea aspera Medial patella, tibial tubercle Femoral Extend leg Weak in many patello-femoral disorders. Copyright © 2008 Elsevier Inc. 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Copyright © 2001 Saunders, An Imprint of Elsevier MUSCLES: MEDIAL MUSCLE ORIGIN INSERTION NERVE ACTION COMMENT Obturator externus Ischiopubic rami, obturator membrane Trochanteric fossa Obturator ER thigh T endon posterior to femoral neck HIP ADDUCTORS Adductor longus Body of pubis (inferior) Linea aspera (mid 1/3) Obturator Adducts thigh T endon can ossify Adductor brevis Body and inferior pubic ramus Pectineal line, upper linea aspera Obturator Adducts thigh Deep to pectineus Adductor magnus Ischiopubic ramus ischial tuberosity Linea aspera/adductor tubercle Obturator/ Sciatic Adducts flex/ extend thigh 2 portions: separate insertions innervation Gracilis Body and inferior pubic ramus Proximal medial tibia (Pes anserinus) Obturator Adducts (flex) thigh flex, IR leg Used in ligament reconstruction (ACL) HIP FLEXORS (also iliopsoas) Pectineus Pectineal line of pubis Pectineal line of femur Femoral Flex and adduct thigh Part of femoral triangle floor Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier MUSCLES: POSTERIOR (HAMSTRINGS) MUSCLE ORIGIN INSERTION NERVE ACTION COMMENT Semitendinosus Ischial tuberosity Proximal medial tibia (Pes anserinus) Sciatic (tibial) Extend thigh, flex leg Used in ligament reconstructions (ACL) Semimembranosus Ischial tuberosity Posterior medial tibial condyle Sciatic (tibial) Extend thigh, flex leg A border in medial approach Biceps femoris: Long Head Ischial tuberosity Head of fibula Sciatic (tibial) Extend thigh, flex leg Covers sciatic nerve Biceps femoris: Short Head Linea aspera, supra condylar line Fibula, lateral tibia Sciatic (peroneal) Extend thigh, flex leg Shares insertion tendon with Long Head Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier THIGH MUSCLES: CROSS SECTIONS Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier NERVES LUMBAR PLEXUS ANTERIOR DIVISION Genitofemoral (L1-2): pierces Psoas, lies on anteromedial surface Sensory: Proximal anteromedial thigh Motor: NONE (in thigh) 2. Obturator (L2-4): exits via obturator canal, splits into anterior posterior divisions. Can be injured by retractors placed behind the transverse acetabular ligament. Sensory: Inferomedial thigh: via cutaneous branch of obturator nerve Motor: Gracilis (anterior division) Adductor longus (anterior division) Adductor brevis (ant/post divisions) Adductor magnus (posterior division) 1. LUMBAR PLEXUS POSTERIOR DIVISION Lateral Femoral Cutaneous LFCN: crosses ASIS, can be compressed at ASIS. Sensory: Lateral thigh Motor: NONE 4. Femoral (L2-4): lies between psoas major and iliacus; Saphenous nerve branches in Femoral Triangle runs under sartorius. Sensory: Anteromedial thigh: via anterior/intermediate cutaneous nerves Motor: Psoas Sartorius Articularis genu QUADRICEPS Rectus femoris Vastus lateralis Vastus intermedius Vastus medialis 3. SACRAL PLEXUS ANTERIOR DIVISION 5. Tibial (L4-S3): descends (as sciatic) in posterior thigh Sensory: NONE (in thigh) Motor: POSTERIOR THIGH Biceps femoris [long head] Semitendinosus Semimembranosus POSTERIOR DIVISION 6. Common peroneal (L4-S2): descends(as sciatic) in posterior thigh Sensory: NONE (in thigh) Motor: Biceps femoris [short head] 7. Posterior Femoral Cutaneous Nerve [PFCN] (S1-3) Sensory: Posterior thigh Motor: NONE Copyright © 2008 Elsevier Inc. All rights reserved. -www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier ARTERIES ARTERY BRANCHES COMMENT Obturator Anterior posterior branches Runs through obturator foramen Femoral (Superficial Femoral) [SFA] In femoral triangle, runs in medial thigh between vastus medialis and adductor longus, to obturator canal, through adductor hiatus, then becomes Popliteal Artery behind knee. Superficial circumflex iliac Superficial epigastric Superficial external pudendal Deep external pudendal Deep artery of thigh (Profunda) See below Descending genicular artery Anastomosis at knee to supply knee Articular branch Saphenous branch Deep Artery of the thigh (Profunda) Medial circumflex Supplies femoral neck Lateral circumflex Supplies femoral neck Ascending branch Forms anastomosis at femoral neck Transverse branch Contributes to anastomosis at femoral neck Descending branch Contributes to anastomosis at femoral neck Perforators/muscular branches Supplies femoral shaft and thigh muscles Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier ARTERIES OF THE FEMORAL NECK ARTERY COURSE COMMENT Obturator: Fovea artery (A. of Ligament T eres) Runs through the ligament of femur head Relatively minor contribution to femoral head Deep Artery of thigh Branches from Femoral in Femoral triangle. Supplies anterior medial thigh Medial circumflex Between pectineus iliopsoas to posterior femoral neck Anastomosis: posterior supply Ascending branch Runs on Quadratus femoris Can be injured in posterior approach Lateral circumflex Deep to sartorius and rectus femoris Extracapsular anastomosis at neck Ascending branch T o greater trochanter anteriorly Anastomosis: anterior supply Cervical branches Extracapsular branches of anastomosis Pierce the capsule Retinacular arteries Intracapsular branches: run along neck, enter bone at base of femoral head. Most of femoral head supply is posterior (at risk in injury: AVN) Transverse branch Extends laterally Minor contribution to anastomosis Descending branch Under rectus femoris Minor contribution to anastomosis Inferior Superior Gluteal arteries Branches make small contributions to femoral neck anastomosis Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier DISORDERS DESCRIPTION H P WORK-UP/FINDINGS TREATMENT INFLAMMATORY ARTHRITIS • Host immunologic response results in synovitis. • RA, Lupus, SeroNegative arthropathies, gout, etc. Hx: Pain, stiffness, +/-other joints involved. PE: Antalgic gait, decreased ROM (especially IR) XR: AP, frog leg lateral Labs: RF, ESR, CRP ANA, CBC, uric acid, crystals, culture 1. Physical therapy, NSAIDs 2. Cane or crutch 3. Synovectomy (early) 4. T otal hip Arthroplasty (late) OSTEOARTHRITIS • Loss or damage to articular cartilage • Etiology: developmental, trauma, infection, metabolic, idiopathic Hx: Chronic hip or groin pain, increasing over time with activity PE: Decrease ROM (first IR), + log roll, +/- flexion contracture antalgic gait XR: AP/lateral hip 1. Joint space narrowing 2. Osteophytes 3. Subchondral sclerosis 4. Bony cysts 1. NSAIDs, Physical Therapy 2. Injection, activity modification, cane 3. Osteotomy (young) 4. Arthrodesis (young) 5. T otal Hip Arthroplasty (elderly) LATERAL FEMORAL CUTANEOUS NERVE ENTRAPMENT (Meralgia Paresthetica) • Nerve trapped near ASIS. • Due to activity (hip extension), or clothing (e.g. belt) Hx: Pain/burning in lateral thigh PE: Decreased sensation on lateral thigh, + Meralgia XR: AP/lateral of hip: rule out other pathology 1. Remove compressive entity 2. Surgical release: rare OSTEONECROSIS (Avascular necrosis: AVN) • Necrosis of femoral head (trabecular bone) • Due to vascular disruption • Associated with trauma, Etoh, steroid use, RA • Ficat classification: 4 stages based on sx, XR, bone scan Hx: Insidious onset dull hip ache PE: With collapse: pain with IR ER Without collapse: discomfort with IR ER XR: AP, frog leg lateral: femoral head sclerosis MR: Double line sign (T2) Early: core decompression or vascularized fibular graft Late or collapse: T otal hip arthroplasty SNAPPING HIP (Iliotibial band) • ITB snapping over greater trochanter of iliopsoas tendon over pectineal eminence Hx: Snapping in hip with walking (as hip extends). Pain rare. PE: Adduct flex XR: AP pelvis, AP/latearl of hip: usually normal, rule 1. Reassurance 2. Avoid activity, Physical therapy eminence • Women (wide pelvis) most common PE: Adduct flex hip, then extend: + snap out other pathology 3. Injection for acute bursitis 4. Surgery rare TROCHANTERIC BURSITIS • Inflammation of bursa over greater trochanter or gluteal tendons Hx: Lateral hip pain. Cannot sleep on affected side. PE: Point tenderness at greater trochanter XR: AP pelvis, AP/lateral of hip: rule out spur, OA, calcified tendons 1. NSAIDs 2. Physical therapy (IT Band stretching) 3. Steroid injection Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier TOTAL HIP ARTHROPLASTY TIPS ON TOTAL HIPS GENERAL INFORMATION • Types of implants: cemented, noncemented (press fit porous ingrowth), hybrid - “Supermetals”: cobalt chrome titanium (shaft/head) - Acetabular cup: Ultra high-molecular weight polyethylene - Porous ingrowth: best pore size 200-400 microns - Cemented usually used in elderly patients, noncemented for younger patients • Cement: Polymethylmethacralate • Head size: 26-28mm is optimal INDICATIONS 1. Arthritis of hip: common etiologies: OA, RA, AVN Most patients complain of pain, worsening over time (wakes them from sleep), and decreased ability to ambulate. Patient should have appropriate radiographic evidence of arthritis It is preferable when the patient is elderly (needs only one replacement) OSTEOARTHRITIS RHEUMATOID ARTHRITIS 1. Joint space narrowing 2. Sclerosis 3. Subchondral cysts 4. Osteophyte formation 1. Joint space narrowing 2. Periarticular osteoporosis 3. Joint erosions 4. Ankylosis 2. Failed conservative treatment: activity modification, weight loss, physical therapy/strengthening, NSAIDs, ambulation assistance (cane used on unaffected side, walker, etc.), injections. 3. Other: Fractures, tumors, developmental disorders (DDH, etc.) CONTRAINDICATIONS • Y oung, active patient (will wear out replacement many times) • Medically unstable (e.g. severe cardiopulmonary disease) • Neuropathic joint • Any infection ALTERNATIVES • Considerations: Age, activity level, overall health • Osteotomy: Femoral or pelvic; not common in U.S. • Arthrodesis/Fusion: good for young patients/laborers, unilateral disease, no other joint disease (e.g. spine, knee). Fuse with hip in slight flexion PROCEDURE • Posterior or lateral approach usually used • Femoral component should be in valgus (“Thou shalt not Varus”) • Acetabular cup at 45° COMPLICATIONS • Failure of Implant 1. Loosening (#1 complication in cemented joints) 2. Varus alignment 3. Implant breakage (patients: active, heavy, young, will wear out prosthetic) • Hip thigh pain post-operatively (#1 complication in noncemented joints) • Deep Venous Thrombosis (DVT)/Pulmonary emboli: patients should be anticoagulated (Heparin/warfarin) postoperatively • Infection: often leads to removal of prosthesis (Staph #1 cause) • Dislocation: posterior are most common (abduction pillow can help prevent) • External iliac/Femoral artery and vein injury with anterior/superior quadrant screw • Obturator nerve, artery, vein injury with anterior/inferior quadrant screw. Posterior screw placement is preferable • Nerve injury (sciatic: peroneal portion) by retractors: Foot drop • Heterotopic ossification: one dose prophylactic XRT can help prevent it. • Osteolysis: Macrophage response; due to polyethylene wear debris Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier TIPS ON TOTAL HIPS Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier PEDIATRIC DISORDERS DESCRIPTION EVALUATION TREATMENT/COMPLICATIONS DEVELOPMENTAL DYSPLASIA • 1. Capsule/ligament laxity, or 2. Acetabular roof abnormal: hip does not develop correctly • Associated with: First female, breech delivery, + family health, decreased intrauterine space conditions • Early diagnosis and treatment essential (3mo) • Poor outcomes if diagnosis delayed Hx: Twins, other risk factors. Often unnoticed by parents. PE: + Barlow (dislocation), + Ortalani (relocation), + Galeazzi tests. Decreased abduction XR: In older patients US: if PE not conclusive • Goal: maintain femoral head in the acetabulum (concentric reduction): 1. Pavlik harness (3mo) 2. Closed reduction cast (6-18mo) 3. Osteotomy (18mo) • Post reduction films essential COMPLICATIONS: Osteonecrosis (femoral head) FEMORAL ANTEVERSION • Internal rotation of femur, femoral anteversion does not decrease properly • #1 cause of intoeing Hx: Usually presents 3-6 yrs PE: Femur IR (IR 65°), patella is medial, intoeing gait 1. Most spontaneously resolve 2. Derotational osteotomy if it persists past age 10 (mostly cosmetic) DESCRIPTION EVALUATION TREATMENT/COMPLICATIONS LEGG-CALVE-PERTHES DISEASE • Osteonecrosis of femoral head • Idiopathic, vascular etiology (hypercoaguable/sludging) • Associated with: + family history, breech birth • Catteral classification: 4 stages • Poor prognosis: after age 9 or with large femoral head involvement Hx: Boys(4:1) usually 4-8 yo, unilateral thigh or knee pain limp PE: Decreased abduction, no point tenderness on exam XR: AP pelvis, frog lateral (density of the femoral head is indicative; crescent sign: subchondral fx) The femoral head must revascularize Based on age: 5 yrs: observation NSAIDs 5-8 yrs: concentric containment: abduction brace or osteotomy 9+ yrs: operative treatment often fails (many need THA as adult) SLIPPED CAPITAL FEMORAL EPIPHYSIS (SCFE) • Proximal femoral epiphysis falls off femur (posterior) head in acetabulum • Obese adolescents • Early diagnosis and treatment essential Hx: 11-14 yo, often obese, slow onset hip, thigh, knee pain, +/-limp PE: Decreased ROM (especially IR, Do not attempt reduction 1. Non weight-bearing 2. Percutaneous pinning COMPLICATIONS: Osteonecrosis, chondrolysis, osteoarthritis, decreased ROM abduction) XR: AP pelvis, frog lateral Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier SURGICAL APPROACHES USES INTERNERVOUS PLANE DANGERS COMMENT POSTERIOR (Moore/Southern) APPROACH TO HIP 1. T otal Hip Arthroplasty 2. Arthroplasty 3. ORIF posterior acetabulum 3. Posterior hip dislocations Split gluteus maximus [Inferior gluteal n] 1. Sciatic nerve 2. Inferior gluteal artery 1. Superior and inferior gluteal arteries need to be controlled. 2. The short external rotators must be detached to access the joint. LATERAL (Hardinge) APPROACH TO HIP T otal Hip Arthroplasty (not used for revisions) Split gluteus medius [Superior gluteal n] 1. Superior gluteal artery 2. Femoral nerve 3. Femoral Artery vein 1. No osteotomy of greater trochanter required. Leads to earlier mobilization. 2. Less exposure than posterior approach, thus not used for revision THA. LATERAL APPROACH TO THIGH 1. Fractures 2. Tumors Split vastus lateralis (and intermedius) [Femoral 1. Branch of Lateral femoral circumflex artery 1. Incision can be large or small; it is made along the line between greater trochancter and lateral condyle. 2. Tumors [Femoral nerve] 2. Superior lateral geniculate artery 2. Arteries (#1 2 at left) encountered if incision extended proximally or distally; ligate them. Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com CHAPTER 8 - LEG/KNEE TOPOGRAPHIC ANATOMY OSTEOLOGY TRAUMA KNEE JOINTS MINOR PROCEDURES: KNEE HISTORY PHYSICAL EXAM MUSCLES: ORIGINS AND INSERTIONS MUSCLES: ANTERIOR COMPARTMENT MUSCLES: LATERAL COMPARTMENT MUSCLES: SUPERFICIAL POSTERIOR COMPARTMENT MUSCLES: DEEP POSTERIOR COMPARTMENT MUSCLES: CROSS SECTIONS NERVES ARTERIES DISORDERS DISORDERS: LIGAMENT INJURIES DISORDERS TOTAL KNEE ARTHROPLASTY TOTAL KNEE ARTHROPLASTY PEDIATRIC DISORDERS SURGICAL APPROACHES Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier CHAPTER 8 – LEG/KNEE TOPOGRAPHIC ANATOMY Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier OSTEOLOGY CHARACTERISTICS OSSIFY FUSE COMMENT TIBIA • Long bone characteristics Primary: Body 7 wks (fetal) 18 years • Ossification site at the tibial tuberosity can be confused with a fracture. • Wide proximal end (plateau) articulates with the femoral condyles Secondary 18-20 years • Traction (quadriceps) apophysitis at the tibial tuberosity: Osgood Schlatter disease • Distal end (plafond) cups the talus 1. Proximal epiphysis 9 mo • Primary weight-bearing bone in leg • Medial malleolus is distal end 2. Distal epiphysis 1 yr • IT Band inserts on Gerdy's tubercle 3. Tibial tuberosity FIBULA • Long bone characteristics Primary: Body 8 wks (fetal) 20 years • Common peroneal nerve runs across the neck, injured in fractures (foot drop) • Distal end (lateral malleolus) is lateral wall of ankle mortise. Secondary 18-22 years • Used to determine “lateral” on radiographs 1. Proximal epiphysis 1-3 yr 2. Distal 2. Distal epiphysis 4 yr CHARACTERISTICS OSSIFY FUSE COMMENT PATELLA • Largest sesamoid bone in the body Primary (single center) 3 years 11-13 years • Failure to fuse: Bipartite patella (can be confused with patella fracture). • Two facets (lateral is larger) • Functions: 1. Enhances quadriceps pull 2. Protects knee • Triangular in cross-section • Very thick articular cartilage (bearing heavy loads) Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier TRAUMA DESCRIPTION EVALUATION CLASSIFICATION TREATMENT PATELLA FRACTURE • Mechanism: direct indirect: (e.g. fall, dashboard or kicking injury) • Pull of quadriceps and patella tendons displace most fractures • If intact, retinaculum resists displacement • Do not confuse with bipartite patella HX: Trauma. Pain, cannot extend knee, swelling. PE: ”Dome” effusion. T enderness, +/-palpable defect. Inability to extend knee. XR: Knee trauma series CT : Not usually needed Descriptive location: Nondisplaced Transverse Vertical Stellate Inferior/superior pole Comminuted Nondisplaced or comminuted: cylinder cast for 6 wks Displaced(2-3mm): ORIF (e.g. tension bands) to restore articular surface Severely comminuted: may require patellectomy COMPLICATIONS: Osteoarthritis and/or pain, Decreased motion and/or strength; Osteonecrosis; Refracture TIBIAL PLATEAU FRACTURE • Mechanism: Direct blow (e.g. MVA) • Intraarticular fracture • Restoration of articular surface is important • Most often lateral • Metaphyseal injury: bone compresses, leads to functional bone loss. • Associated with ligament injuries HX: Trauma. Cannot bear weight. Pain, swelling. /PE: Effusion, tenderness, do good neurovascular PE XR: Knee trauma series CT : Better defines fracture. AGRAM: if pulseless Schatzker (6 types): I. Lateral plateau split fx II. Lateral split/depression fx III. Lateral plateau depression IV. Medial plateau split fx V. Bicondylar plateau fx VI. Fx with metaphyseal-diaphyseal separation +/- Aspirate hemarthroses Undisplaced (6 mm): cast, ROM at 6 wks, WB 3mos. Displaced/unstable: ORIF: plates and screws +/- bone graft Mobilize early, weight- bear at 2 months COMPLICATIONS: Compartment syndrome; Hardware failure or loss of reduction; OA; Popliteal artery or nerve injury KNEE DISLOCATION • Rare: Ortho emergency • Usually high energy injury • Ligaments other soft tissue are disrupted • High incidence of associated fracture neurovascular injury • Close follow up is important for good result HX: Trauma. Pain, inability to bear weight. PE: Effusion, deformity, pain, +/-distal pulses peroneal nerve function XR: AP/lateral AGRAM: ID arterial injury MR: Ligament injury By position: Anterior Posterior Lateral Medial Rotatory: Anteromedial or anterolateral. Early reduction essential Post reduction neuro-logic exam and x-rays. Immobilize (cast): 6-8 wks (not if ligaments torn) Open: If irreducible, vascular injury (+/-pro-phylactic fasciotomy), early repair of ligaments if needed. COMPLICATIONS: Neurovascular: Popliteal artery, peroneal nerve injury; Decreased motion; Instability DESCRIPTION EVALUATION CLASSIFICATION TREATMENT TIBIA SHAFT FRACTURE • Common long bone fracture • Y oung adults • Often tibia/fibula fracture or tibia fracture/dislocation combination injuries • T enuous blood supply: union is a problem. • Up to 5% residual angulation is acceptable HX: Trauma. Cannot bear weight, pain, swelling. PE: Swelling, deformity, +/-tense compartments open wound. Palpate pulse XR: AP/lateral leg, + knee and ankle series AGRAM: if pulseless Descriptive: Location Displaced/comminuted Type: transverse, spiral oblique Rotation/angulation Stable, non or minimally displaced, closed injury: Long leg cast 4-6 wks then shorter cast Unstable, displaced, comminuted injury: ORIF Intramedullary nails (external fixation for severe open fractures) COMPLICATIONS: Malunion/nonunion: especially mid-distal 1/3; Compartment syndrome; Decreased motion; Hardware failure; Neurovascular injury; Reflex Sympathetic Dystrophy (RSD) MAISONNEUVE FRACTURE • Complete syndesmosis disruption with diastasis proximal fibula fracture • Variant of ankle fracture deltoid ligament rupture • Unstable fracture HX: Trauma. Ankle pain, +/-knee pain. PE: Ankle pain, swelling, +/- knee signs. XR: Knee series with each ankle fracture Reduce and stabilize syndesmosis with a screw COMPLICATIONS: Ankle instability; Ankle arthritis PILON (DISTAL TIBIA) FRACTURE • Intraarticular: through distal articular/WB surface. • Comminution common • Associated soft HX: Trauma. Cannot bear weight, pain, swelling PE: Effusion, tenderness, do good neurovascular PE Ruedi-Allgower (3 types): I. Non or minimally displaced. II. Displaced: articular Nondisplaced: Long leg cast NWB for 6 wks Displaced/Comminuted: ORIF: plates screws +/- • tissue injuries • Articular surface repair is difficult essential • Healing is often slow PE XR: AP/lateral (obliques) CT : Needed: better image of articular surface surface incongruous. III. Comminuted articular surface. bone grafting Severely comminuted: external fixation COMPLICATIONS: Post-traumatic Osteoarthritis (almost 100% in comminuted fractures); Decreased motion; Malunion/nonunion Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier KNEE JOINTS SUPPORT ATTACHMENTS COMMENTS FEMORAL/TIBIAL: CONDYLOID ANTERIOR Patellofemoral joint See page 212 Anterior cruciate (ACL) Tibial eminence to medial aspect of lateral femoral condyle Prevents anterior translation, tight in flexion, must reconstruct if injured Transverse meniscal ligament Anterior menisci Meniscus support stability MEDIAL Meniscus Between femoral condyle tibial plateau More crescentic than lateral Capsule (III) Surrounds joint Minimal support Medial collateral (MCL) Medial epicondyle to tibia (II) meniscus (III) Superficial (II) and Deep (III) portion Coronary ligament (III) Meniscus to medial tibia Stabilizes meniscus Semimembranous membrane (II) Attach to posterior tibial condyle Pes anserinus tendons (I) Medial tibial condyle T endinitis can occur at insertion LATERAL Meniscus Between femoral condyle tibial plateau More circular than medial Popliteus muscle tendon Proximal tibia Intraarticular tendon Capsule (III) Surrounds joint Minimal support Arcuate ligament (III) Posterolateral femoral condyle to fibular head Covers popliteus tendon Fabellofibular ligament (III) Fabella to fibula Variable Lateral collateral (LCL) Lateral femoral condyle to Prevents varus angulation (III) fibular head Prevents varus angulation Biceps muscle tendon (I) Gerty's tubercle fibular head Iliotibial band (I) Lateral tibial condyle If tight, ITB syndrome can occur POSTERIOR Capsule (III) Surrounds joint Minimal support Ligament of Humphrey Posterior lateral meniscus to medial femoral condyle In front of PCL Posterior cruciate (PCL) Tibial sulcus to anterior medial femoral condyle Prevents posterior translation Ligament of Wrisberg Posterior lateral meniscus to medial femoral Behind the PCL condyle Oblique popliteal ligament Semimembranous to lateral femoral condyle Derived from semimembranous Gastrocnemius/plantaris muscle Origin: posterior medial lateral femoral condyles Two heads originate above knee SUPPORT ATTACHMENTS COMMENTS PATELLOFEMORAL Quadriceps tendon Attach on superior patellar pole Superior extensor mechanism Patellar ligament (tendon) Inferior patella pole to tibial tuberosity Inferior extensor mechanism Medial lateral retinaculum (quadriceps oblique fibers) (II) Quadriceps extensions to patella, then to tibial condyles Stabilizes patella in motion. Can affect Q angle if tight Medial lateral patellofemoral ligaments (II) Patella to femoral condyles Stabilizes patella Medial lateral patellotibial ligaments Patella to tibial condyles Stabilizes patella PROXIMAL TIBIOFIBULAR : Plane Anterior ligament of head of fibula Fibula head to lateral tibia Broader than posterior Posterior ligament of head of fibula Fibula head to lateral tibia Weaker than anterior OTHER STRUCTURES Interosseous membrane Lateral tibia to medial fibula Strong; runs length of leg • Three compartments in the knee: Medial, Lateral, Patellofemoral • Meniscus: Made of fibrocartilage. Function: 1) Protects articular cartilage (increases weight bearing surface area, 2) Stabilizes by deepening facet, 3) Load transmission Peripheral 1/3 vascular (geniculate arteries): can be repaired; Inner 2/3 supplied by synovial fluid: must debride in injured • There are three layers of support in the knee: I, II, III (noted in parentheses next to structure) • Posterolateral corner complex: Arcuate ligament, popliteus, posterolateral capsule • Muscles attaching at the pes anserinus: sartorius, gracilis, semitendinosus Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier MINOR PROCEDURES: KNEE STEPS ARTHOCENTESIS/INJECTION 1. Ask patient about allergies 2. Place patient supine, knee extended, palpate the lateral patella and lateral distal femur. 3. Prepare skin over the knee (iodine/antiseptic soap) 4. Anesthetize skin locally (quarter size spot) 5. Insert an 18 gauge needle laterally into the suprapatella pouch (between the patella and femur) proximal to the joint. Aspirate fluid from joint (or inject 3-5cc of local/steroid preparation). Fluid should flow easily if needle is in joint. 6. If suspicious of infection, send fluid for GS culture. 7. Dress injection site Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier HISTORY QUESTION ANSWER CLINICAL APPLICATION 1. AGE Y oung Trauma: fractures, ligamentous or meniscal injury Middle age, elderly Arthritis 2. PAIN a. Onset Acute Trauma: fracture, dislocation, soft tissue (ligament/meniscus) injury, septic bursitis Chronic Arthritis, infection, tendinitis/bursitis, tumor b. Location Anterior Quadricep or patellar tear or tendinitis, prepatellar bursitis, patellofemoral arthritis Posterior Meniscus tear (posterior horn), Baker's cyst, popliteal aneurysm Lateral Meniscus tear (jointline), collateral ligament injury, arthritis, ITB friction syndrome Medial Meniscus tear (jointline), collateral ligament injury, arthritis, pes bursitis c. Occurrence Night pain Tumor, infection With activity Etiology of pain likely from joint 3. STIFFNESS Without locking Arthritis, effusion (trauma, infection) With locking or catching Loose body, meniscal tear (especially bucket handle), arthritis, synovial plica 4. SWELLING Within joint Infection, trauma Acute (post injury) Acute (hours): ACL injury; Subacute (day): meniscus injury Acute (without injury) Infection: prepatellar bursitis, septic joint 5. INSTABILITY Giving away/collapse Cruciate ligament injury, extensor mechanism injury Giving away,+/-pain Patellar subluxation/dislocation, pathologic plica, osteochondritis dissecans 6. TRAUMA Mechanism: valgus force MCL injury (+/- terrible triad: MCL, ACL, medial meniscus injuries) Varus force LCL injury Flexion/posterior force PCL injury (e.g. dashboard injury) Contact injury Non-contact: ACL injury, Contact: multiple ligaments Popping noise Cruciate ligament injury (especially ACL), osteochondral fracture NONE Degenerative and overuse etiology 7. ACTIVITY Agility sports Cruciate and/or collateral ligament injury Running, cycling, climbing Patellofemoral etiology Squatting Mensicus tear Walking Distance able to ambulate equates with severity of arthritic disease 8. NEUROLOGIC SYMPTOMS Pain, numbness, tingling Neurologic disease, trauma 9. SYSTEMIC COMPLAINTS Fevers, chills Infection, septic joint 10. HISTORY OF ARTHRITIDES Multiple joints involved Rheumatoid Arthritis, gout, etc. Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier PHYSICAL EXAM EXAM TECHNIQUE/FINDINGS CLINICAL APPLICATION INSPECION Gait Observe patella tracking Abnormal patella tracking can lead to patellofemoral problems Flexed knee gait Tight Achilles tendon or hamstrings: patellofemoral problems Anterior Genu valgum (knock knee) Genu varum (bow leg) Normal: 7 degrees valgus; varus or valgus deformity with ligamentous or osseous deficiency Swelling Effusion (arthritis, trauma, infection/inflammation), bursitis (prepatellar, infrapatellar) Posterior Swelling, mass Effusion (arthritis), Baker's cyst Lateral Back knee, high/low riding patella Genu recurvatum (PCL injury), patella alta (patellar instability) Musculature Atrophy Vastus medialis atrophy: can lead to patellofemoral problems PALPATION Bony structures Patella: medial lateral aspects T enderness at distal pole: tendinitis (Jumpers knee) Tibial tubercle T enderness with Osgood Schlatter disease Soft tissues Compress suprapatellar pouch (“milk” knee) Ballotable patella (effusion): arthritis, trauma, infection Prepatellar/infrapatellar bursae Edematous or tender bursae indicate correlating bursitis Pes anserine bursa T enderness indicates bursitis Plica (medial to patella) Thickened, tender plica is pathologic Medial jointline MCL T enderness: medial meniscus tear or MCL injury Lateral jointline LCL T enderness: lateral meniscus tear or LCL injury Iliotibial band (anterolateral knee) Pain or tightness is pathologic Popliteal fossa Mass consistent with Baker's cyst, popliteal aneurysm Compartments of leg (anterior, posterior, lateral) Firm or tense compartment: Compartment syndrome EXAM TECHNIQUE/FINDINGS CLINICAL APPLICATION RANGE OF MOTION Flexion extension Supine: knee to chest, then straight Normal: Flex 0 to 125-135°, Extend 0 to 5-15°; Extensor lag (final 20° difficult): weak quadriceps; Decreased extension with effusion Note patellar tracking, pain, crepitus Abnormal tracking leads to anterior knee pain; pain crepitus: arthritis Tibial IR ER Stabilize femur, rotate tibia Normal: 10-15° IR ER NEUROVASCULAR Sensory Femoral nerve (L4) Medial leg (Medial cutaneous nerves) Deficit indicates corresponding nerve/root lesion Peroneal nerve (L5) Lateral leg (common superficial) Deficit indicates corresponding nerve/root lesion Tibial nerve (S1) Posterior leg (Sural nerves) Deficit indicates corresponding nerve/root lesion Motor Femoral nerve (L2-4) Knee extension Weakness = Quadriceps or nerve/root lesion Sciatic: Tibial (L4-S3) Knee flexion Weakness = Biceps (LH) or nerve/root lesion Peroneal (L4-S2) Knee flexion Weakness = Biceps (SH) or nerve/root lesion Tibial nerve (L4-S3) Foot plantarflexion Weakness = TP, FHL, FDL or nerve/root lesion Peroneal (deep) n. (L4-S2) Foot dorsiflexion Weakness = TA, EHL, EDL or nerve/root lesion Reflex L4 Patellar Hypoactive/absence indicates L4 radiculopathy Pulse Popliteal EXAM TECHNIQUE/FINDINGS CLINICAL APPLICATION SPECIAL TESTS Q (quadriceps) angle ASIS to mid-patella to tibia tubercle Normal: 13° male, 18° female; Increased angle: PF Syndrome, subluxation Patella grind Extend knee: fire quads, compress patella Pain: patellofemoral joint pathology, patella chondromalacia Patella apprehension Relax knee: push patella lateral Pain/apprehension: subluxation; Medial retinaculum injury McMurray Flex/ER leg/valgus force, then extend knee Pop/click on extension indicates medial meniscal tear Flex/IR leg/varus force, then extend knee Pop/click on extension indicates lateral meniscal tear Apley compression Prone: knee 90°, compress rotate tibia Pain/popping: meniscal injury, arthritis Ligament Stability Tests Valgus stress Lateral force: knee at: 1) 30°, 2) 0° Laxity at: 1) 30°: MCL, at 2) 0°: MCL/PCL/posterior capsule injury Varus stress Medial force: knee at 1) 30° 2) 0° Laxity at: 1) 30°: LCL, at 2) 0° LCL/PCL/posterior capsule injury Lachman Flex knee 30°: anterior force on tibia Laxity/displacement: ACL injury (most sensitive exam for ACL) Anterior drawer Flex knee 90°: anterior force on tibia Laxity/displacement: ACL injury Posterior drawer Flex knee 90°: posterior force on tibia Posterior translation: PCL injury Posterior sag Supine: hip 45°/knee 90°: lateral view Posterior translation of tibia on femur: PCL injury Quadriceps active Supine: flex knee 90°, fire quadriceps Posterior translated tibia will translate anterior when quadriceps fire: PCL injury Pivot shift Supine: extend knee, IR, valgus force on proximal tibia, then flex Clunk with flexion: AnteroLateral Rotary Instability (ALRI): ACL and/or posterior capsule injury Reverse pivot shift Supine: knee at 45°, ER, valgus force on proximal tibia, extend Clunk with extension: PosteroLateral Rotary Instability (PLRI): PCL and/or Posterolateral corner injury Slocum Knee 90°, ER foot 15°, anterior force Displacement: AnteroMedial Rotary Instability Knee 90°, IR foot 30°, anterior force Displacement: AnteroLateral Rotary Instability (ALRI): ACL injury Posterior lateral drawer Knee 90°, ER foot 15°, posterior force Displacement: PosteroLateral Rotary Instability (PLRI): PCL/corner Posterior medial drawer Knee 90°, IR foot 30°, posterior force Displacement: PosteroMedial Rotary Instability (PMRI): PCL Prone ER at 30° 90° Prone: ER both knees at: 1)30°, 2)90° Increased ER at: 1) 30: PL corner, 2) 90: PCL PL corner injury Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier MUSCLES: ORIGINS AND INSERTIONS Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier MUSCLES: ANTERIOR COMPARTMENT MUSCLE ORIGIN INSERTION NERVE ACTION COMMENT Tibialis anterior [TA] Lateral tibia, interosseous membrane Medial cuneiform, base of 1st metatarsal Deep peroneal Dorsiflex invert foot T est L4 motor function Extensor hallucis longus [EHL] Medial fibula, interosseous membrane Base of distal phalanx of great toe Deep peroneal Dorsiflex extend great toe T est L5 motor function Extensor digitorum longus [EDL] Lateral tibia condyle proximal fibula Base of middle distal phalanges (4 toes) Deep peroneal Dorsiflex extend lateral 4 toes Single tendon divides into four tendons Peroneus tertius Distal fibula, interosseous membrane Base of 5th metatarsal Deep peroneal Dorsiflex Evert foot Often adjoined to the EDL Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier MUSCLES: LATERAL COMPARTMENT MUSCLE ORIGIN INSERTION NERVE ACTION COMMENT Peroneus longus Proximal lateral fibula Medial cuneiform, base of 1st MT (plantarly) Superficial peroneal Evert, plantar flex foot T est S1 motor function. Runs under the foot Peroneus brevis Distal lateral fibula Base of 5th metatarsal Superficial peroneal Evert foot Can cause avulsion fx at base of 5th MT Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier MUSCLES: SUPERFICIAL POSTERIOR COMPARTMENT MUSCLE ORIGIN INSERTION NERVE ACTION COMMENT Gastrocnemius Lateral and medial femoral condyles Calcaneus (via Achilles tendon) Tibial Plantarflex foot T est S1 motor function Has two heads Soleus Posterior fibular head/soleal line of tibia Calcaneus (via Achilles tendon) Tibial Plantarflex foot Fuses to gastrocnemius at Achilles tendon Plantaris Lateral femoral supracondylar line Calcaneus Tibial Plantarflex foot Short muscle belly is proximal, has a long tendon. Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier MUSCLES: DEEP POSTERIOR COMPARTMENT MUSCLE ORIGIN INSERTION NERVE ACTION COMMENT Popliteus Lateral condyle Proximal posterior tibia Tibial Flex ( IR) knee Anterior distal to LCL on femur Flexor hallucis longus [FHL] Posterior fibula Base of distal phalanx of great toe Tibial Plantarflex great toe T est S1 motor function Flexor digitorum longus [FDL] Posterior tibia Bases of distal phalanges of 4 toes Tibial Plantarflex lateral 4 toes At ankle, tendon is just anterior to tibial artery. Tibialis posterior [TP] Posterior, interosseous membrane, tibia, fibula Navicular tuberosity, cuneiform, MT's Tibial Plantarflex invert foot T endon can degenerate rupture: 2° pes planus Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier MUSCLES: CROSS SECTIONS ANTERIOR LATERAL SUPERFICIAL POSTERIOR DEEP POSTERIOR MUSCLES Tibialis anterior [TA] Peroneus longus Gastrocnemius Popliteus Extensor hallucis longus [EHL] Peroneus brevis Soleus Flexor hallucis longus [FHL] Extensor digitorum longus [EDL] Plantaris Flexor digitorum longus [FDL] Peroneus tertius Tibialis posterior [TP] NEUROVASCULAR Deep peroneal nerve Superficial peroneal nerve NONE Tibial nerve Anterior tibial artery and vein Posterior tibial artery and vein Peroneal artery and vein Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier NERVES LUMBAR PLEXUS POSTERIOR DIVISION 1. Femoral (L2-4): Sensory: Medial leg: via medial cutaneous nerve (Saphenous N) Motor: NONE (in leg) SACRAL PLEXUS ANTERIOR DIVISION 2. Tibial (L4-S3): descends between heads of gastrocnemius to medial malleolus Sensory: Posterolateral proximal calf: via Medial sural Posterolateral distal calf: via Sural Motor: SUPERFICIAL POSTERIOR COMPARTMENT OF LEG Soleus: via nerve to soleus Plantaris Gastrocnemius DEEP POSTERIOR COMPARTMENT OF LEG Popliteus: via nerve to popliteus Tibialis posterior [TP] (T om) Flexor digitorum longus [FDL] (Dick) Flexor hallucis longus [FHL] (Harry) POSTERIOR DIVISION 3. Common peroneal (L4-S2): in groove between biceps lateral head of Gastrocnemius. Wraps around fibular head, deep to peroneus longus, then divides. Can be injured in lateral approach to the knee. Sensory: Proximal lateral leg: via Lateral sural Distal lateral leg: via superficial peroneal Motor: ANTERIOR COMPARTMENT of LEG: Deep Peroneal Nerve Tibialis anterior [TA] Extensor hallucis longus [EHL] Extensor digitorum longus [EDL] Peroneus tertius LATERAL COMPARTMENT of LEG: Superficial Peroneal Nerve Peroneus longus Peroneus brevis Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier ARTERIES COURSE BRANCHES SUPPLY/COMMENT POPLITEAL Through popliteal fossa. T erminates at the popliteus muscle. Superior Inferior Medial Geniculate All four arteries anastomose around knee patella (supply meniscus) Superior Inferior Lateral Geniculate Middle Geniculate Cruciate ligaments synovium Anterior Posterior Tibial T erminal branches ANTERIOR TIBIAL Supplies muscles of the ANTERIOR COMPARTMENT Through 2 heads of Tibialis Posterior interosseous membrane. Then lies on anterior surface of the membrane with deep peroneal nerve, between TA and EHL. Anterior Tibial recurrent Supplies knee Anterior Medial malleolar Supplies ankle Anterior Lateral malleolar Supplies ankle Dorsalis Pedis T erminal branch in foot POSTERIOR TIBIAL Supplies muscles of the POSTERIOR COMPARTMENT From popliteal, through posterior compartment with tibial nerve to behind medial malleolus (between FDL FHL). Posterior Tibial recurrent Supplies the knee Peroneal artery LATERAL COMPARTMENT Posterior medial malleolar Perforating/muscular branches Medial calcaneal Medial Lateral plantar T erminal branches in sole PERONEAL Supplies muscles of the LATERAL COMPARTMENT From posterior tibial between tibialis posterior and FHL. Posterior lateral malleolar T erminal branch Lateral calcaneal Artery Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier DISORDERS DESCRIPTION H P WORK-UP/FINDINGS TREATMENT ANTERIOR FAT PAD SYNDROME (Hoffa disease) • Fat pad (under patellar tendon) is pinched (2° to trauma) Hx: Intermittent anterior knee pain XR: AP/Lateral: possible patella baja 1. RICE, activity modification PE: +/- click with motion 2. Surgical excision (rare) ARTHRITIS: INFLAMMATORY • Synovitis (pannus formation) destroys articular cartilage and joint Hx: Any age (disorder dependent), female male, multiple joints, AM pain. XR: Arthritis series 1. Early: medical management • RA, Gout, SeroNegative arthropathy PE: +/- warm, effusion, crepitus Labs: RF, ESR, CRP, ANA, CBC, crystals, culture 2. Late: a) Conservative: like OA b) Operative: 1. Synovectomy 2. T otal knee ARTHRITIS: OSTEOARTHRITIS • Primary or posttraumatic Hx: Elderly, pain (worse with activity or weight bearing), stiffness, sticking/grinding. XR: Arthritis series 1. NSAIDs, Physical Therapy • Loss or damage to articular cartilage PE: Effusion, jointline tenderness, +/- angular deformity (varus #1) or contracture. 1. joint space narrowing 2. Injection, activity modification (cane) • Knee (Medial compartment) #1 site 2. osteophytes 3. Fusion (young/worker) • All 3 compartments are possible 3. subchondral sclerosis 4. High tibial osteotomy (young, 1 compartment disease) sites 4. bony cysts 5. T otal Knee Arthroplasty (old, 1 compartment) BAKER'S CYST • Posterior knee (popliteal fossal) Hx: Stiffness, +/-knee tenderness XR: AP/lateral: normal 1. Aspiration initially • Arises from MM or hamstring tendon (may communicate) PE: Mass in popliteal fossa MR or aspiration: confirm diagnosis 2. Surgical resection for recurrence or pain BURSITIS: PREPATELLAR (Housemaid's knee) • Continuous irritation of bursa leads to inflammation Hx: Pain with activity XR: AP/lateral: normal rule out infection (common problem) 1. NSAID, knee pads, injection • Most common bursitis in knee PE: “egg” shaped swelling over patella 2. Bursal removal (rare) 3. Treat infection if present BURSITIS: PES ANSERINE • Bursa under tendon insertion inflamed (overuse, runner, etc.) Hx: Pain in medial knee XR: AP/lateral: normal+/- OA, rule out tumor 1. NSAID, activity modification, stretch PE: Pes anserine tenderness 2. Partial excision (rare) DESCRIPTION H P WORK-UP/FINDINGS TREATMENT CHONDROMALACIA: PATELLOFEMORAL SYNDROME [PFS] • Damage or softening of the patellar articular cartilage. Hx: Anterior knee pain, worse with sitting (theater sign), and/or stairs XR: AP/lateral/sunrise to evaluate alignment. Rule out patellofemoral OA 1. Physical therapy: quadricep strengthening stretching • Multiple etiologies: trauma, dislocation, malalignment leads to patellofemoral OA PE: +/- VMO atrophy, valgus deformity, high Q angle, patellar apprehension, + crepitus 2. Orthosis if patella subluxes 3. Lateral release (early) 4. Tibial tuberosity realignment COMPARTMENT SYNDROME • Increased pressure in closed space Hx: 5 P's: pain, parathesias, pulseless, pallor, paralysis. Compartment pressures: 40 mmHg (normal: 0-10 mmHg) 1. Fasciotomy within 4 hours (Usually two incisions) • From: trauma, (e.g. fracture, burn, vascular injury, overexertion) PE: Firm compartments (check all three) 2. Debride nonviable soft tissue. • Results in nerve injuries soft tissue necrosis ILIOTIBIAL BAND FRICTION SYNDROME • ITB rubs on lateral femoral condyle Hx: Pain with activity XR: AP/lateral: normal Rule out tumor 1. NSAID, activity modification, stretching • Common in runners, cyclists PE: Lateral femoral condyle TTP (knee at 30° flexion) 2. Partial excision (rare) DESCRIPTION H P WORK-UP/FINDINGS TREATMENT MENISCUS INJURY: TEAR • Y oung: trauma/twisting injury Hx: Pain, catching/locking (esp. bucket-handle tears) XR: AP (extension 30° flexion)/lateral/sunrise, +/-arthrocentesis 1. Conservative for minor symptoms • Old: Degeneration/squat injury PE: Effusion, jointline tenderness, + McMurray test 2. Debride (inner 2/3 lesion) • Seen with ACL injuries 3. Repair (outer 1/3 or longitudinal lesion) • Medial lateral (cysts develop) Improved results with ACL repair OSTEOCHONDRITIS DISSECANS • Subchondral bone injury Hx: Insidious onset knee pain XR: AP/lateral: shows radiolucency, +/- fragment or loose body 1. Often spontaneously heals in children • Unknown etiology: AVN, repetitive microtrauma PE: Crepitus on flexion extension, femoral condyle tender to palpation 2. Adults: drill lesion vs. bone graft/chondroplasty • Lateral aspect of medial femoral condyle #1 DESCRIPTION H P WORK-UP/FINDINGS TREATMENT PLICA • Synovial tissue (embryonic remnant) thickens rubs medial femoral condyle. Hx: Anteromedial knee pain, catching/popping XR: AP/lateral Arthrography 1. NSAIDs • Medial patellar plica: #1 PE: Palpable plica, jointline tenderness 2. Activity modification 3. Arthroscopic debridement PATELLAR COMPRESSION SYNDROME • Compression of patella due to tight lateral retinaculum Hx: Anterior knee pain XR: AP/lateral: normal 1. Quadriceps strengthening PE: Lateral patella (facet) tender to palpation 2. Lateral release of retinaculum PATELLAR INSTABILITY • Spectrum: malalignment-recurrent subluxation-instability-dislocation Hx: Knee buckles, +/-pain XR: AP/lateral/sunrise: Lateral displacement of the patella. +/-patella alta 1. PT : VMO strengthening • Usually lateral, leads to OA PE: +/- genu valgum, increased Q angle, VMO atrophy, + patellar apprehension 2. Orthosis for subluxation 3. Lateral release, realignment procedures (especially for MMS) Miserable Malalignment Syndrome (MMS): associated with femoral anteversion, increased Q angle, genu valgum DESCRIPTION H P WORK-UP/FINDINGS TREATMENT PATELLAR TENDINITIS: JUMPER'S KNEE • Seen in jumpers (e.g. basketball volleyball players) Hx: Sports, anterior knee pain XR: AP/lateral: normal 1. NSAIDs, strengthen quadriceps [no steroid injection-tendon rupture] PE: Patella: inferior pole tender to palpation MR: Increased signal in inferior pole 2. Debride tendon (rare) PATELLAR TENDON (LIGAMENT) RUPTURE • Direct trauma (also systemic/metabolic disorders) Hx: Y oung, history of trauma XR: AP/lateral: relative patella alta Primary surgical repair • Quadriceps patella tendon rupture PE: Decreased or no active extension, + palpable defect QUADRICEPS TENDON RUPTURE • Result of minor trauma Hx: Older, cannot actively extend knee XR: AP/lateral: relative patella baja Primary surgical repair • Metabolic disorders weaken tendon PE: Palpable defect or sulcus TUMORS #1 in Adolescents: Osteosarcoma; #1 in Adults: Chondrosarcoma; #1 benign (young adult): Giant cell Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier DISORDERS: LIGAMENT INJURIES DESCRIPTION H P WORK-UP/FINDINGS TREATMENT ANTERIOR CRUCIATE (ACL) • Twisting injury, often no contact Hx: “Popping,” swelling XR: AP/lateral/sunrise:+/-capsular avulsion 1. Closed chain exercises • Associated with MCL meniscus tear (all 3 = T errible Triad) PE: Effusion. + Lachman, anterior drawer and pivot shifts tests (Lachman most sensitive) Arthrocentesis (+ /-): 70% have hemarthrosis 2. Reconstruction needed (usually after several weeks of rehabilitation) • Segond fracture: avulsion fx MR: confirms diagnosis POSTERIOR CRUCIATE (PCL) • Anterior force on flexed knee (e.g. dashboard) Hx: Pain, unable to ambulate XR: AP/lat/sunrise: +/- avulsion fracture 1. Non-operative: crutches • Also with other ligament njuries PE: + posterior drawer, posterior sag, quad active tests MR: confirms diagnosis 2. Quadriceps strengthening (Complication: OA) MEDIAL COLLATERAL (MCL) • Valgus force (football clip) Hx: Medial knee pain XR: AP/lateral: possibly an avulsion. 1. Hinged knee brace • Graded 1, 2 (partial), 3 (complete) PE: Laxity and/or pain with valgus stress (at 30° flexion) 2. Physical therapy: early ROM strengthening LATERAL COLLATERAL (LCL) • Varus force (isolated, rare) Hx: Trauma. Pain swelling XR: AP/lateral: possibly an avulsion. 1. Nonoperative: see MCL • Associated with other ligament and peroneal nerve injuries PE: Laxity pain with varus stress (at 30°). T est for foot drop 2. Surgical for grade III (usually combination injury) Isolated PCL, MCL, and LCL injuries are primarily treated non-operatively; operative repair is used when these injuries occur in combination. POSTEROLATERAL CORNER COMPLEX (PLC) • Often with PCL injury Hx: Pain, instability XR: AP/lateral Early surgical repair • LCL torn PE: Increased ER at 30° flexion, + posterolateral drawer test • Popliteofibular ligament torn Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier DISORDERS Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier TOTAL KNEE ARTHROPLASTY KEYS TO TOTAL KNEES GENERAL INFORMATION • Implants: unlike hip, all are cemented (to reduce complications with loosening) Cement: Polymethylmethacralate Femoral condylar and tibia components are metallic Tibial component surface plate: Polyethylene INDICATIONS 1. End stage DJD: results in disabling pain in knee secondary to arthritis in 2 + compartments (medial lateral patellofemoral). • Common etiologies: OA, RA, AVN • Most patients complain of PAIN, worsening over time (wakes them from sleep), and decreased ability to ambulate • Patient should have appropriate radiographic evidence of arthritis OSTEOARTHRITIS RHEUMATOID ARTHRITIS 1. Joint space narrowing 1. Joint space narrowing 2. Sclerosis 2. Periarticular osteoporosis 3. Subchondral cysts 3. Joint erosions 4. Osteophyte formation 4. Ankylosis • It is preferable that the patient is elderly (needs only one replacement) 2. Failed conservative treatment: activity modification, weight loss, orthosis, physical therapy/strengthening, NSAIDs, ambulation assistance (cane, walker, etc.), injections. CONTRAINDICATIONS • Y oung, active patient (will wear out replacement many times) • Knee extensor mechanism dysfunction • Medically unstable (e.g. severe cardiopulmonary disease) • Neuropathic joint • Any infection Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier TOTAL KNEE ARTHROPLASTY KEYS TO TOTAL KNEES ALTERNATIVES • Considerations: Age, activity level, overall health • Osteotomy: for unicompartmental disease, young, active (not in elderly patients) Medial compartment (varus deformity): high tibial osteotomy Lateral compartment (valgus deformity): distal femoral osteotomy • Arthrodesis/Fusion: totally destroyed, neuropathic, or septic joint • Unicompartment arthroplasty: for unicompartment disease. Only in selected patients not eligible for osteotomy. PROCEDURE • Medial parapatellar approach used (lateral parapatellar for severe valgus deformity) • ACL is sacrificed • Using specialized guides, the distal femur and proximal tibia are removed and replaced with metallic/plastic components. Underside of patella also replaced. • Flexion and extension gap should be equal COMPLICATIONS • Infection: often leads to removal of prosthesis (Staph #1) • Loosening of components • Patellofemoral joint pain • Decreased ROM (usually from inadequate postoperative physical therapy) • Patella fracture • Superolateral geniculate artery is at risk • Fat embolism • Peroneal nerve palsy • Deep Venous Thrombosis (DVT)/Pulmonary emboli: patients should be anticoagulated (Heparin/warfarin) postoperatively Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier PEDIATRIC DISORDERS DESCRIPTION EVALUATION TREATMENT/COMPLICATIONS GENU VARUM: BOW LEGS • Normal: neonate to 2 yrs old Hx: Parents observe deformity 1. Most resolve spontaneously with normal development • Etiology: PE: Measure tibiofemoral angle 2. Night bracing rarely required 1. Blount's disease XR: Only large deformity or if concerned about dysplasia. 3. Osteotomy if persistent (15°) 2. Rickets (nutritional) 3. Skeletal dysplasia 4. Trauma GENU VALGUM: KNOCK KNEES • Normal for 2 yrs to 4 yrs Hx: Parents observe deformity 1. Most resolve spontaneously with normal development • Adult: 5-10° valgus is normal PE: Measure tibiofemoral angle 2. Surgery if persists past age 10 • Etiology: XR: Only large deformity or if concerned about dysplasia. 1. Rickets (renal) 2. Skeletal dysplasia 3. Trauma OSGOOD SCHLATTER DISEASE • Osteochondritis/traction apophysitis of tibial tubercle (at 2° ossification center) Hx: Early adolescent. Knee pain worse after activity 1. Activity restriction/modification (at 2° ossification center) • From repetitive extensor (quadriceps) pull on tubercle PE: Pain, swelling at tubercle 2. Most resolve with fusion of apepnysis in midadolesence XR: Knee AP/lateral: may show heterotopic ossification TIBIAL TORSION • Congenital IR of tibia (associated with intrauterine position) Hx: 1-2 yo, often tripping, no pain Will resolve spontaneously (between 24-48 months) • Often bilateral PE: Negative foot to thigh angle (normal 10-30°),with knee/patella pointed forward, intoeing gait observed Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier SURGICAL APPROACHES USES INTERNERVOUS PLANE DANGERS COMMENT KNEE: MEDIAL PARAPATELLAR APPROACH 1. Ligament reconstruction No planes: Capsule is under skin 1. Infrapatellar branch of Saphenous Nerve 1. Most commonly used approach 2. T otal knee arthoplasty 2. Most/best exposure 3. Meniscectomy 3. Neuroma may develop from cutaneous nerves LEG/TIBIA: POSTEROLATERAL APPROACH (Harmon) 1. Fractures 1. Gastrocnemius/soleus/FHL [Tibial] 1. Lesser saphenous vein 1. A technically difficult approach 2. Nonunions 2. Peroneus longus/brevis [Superficial peroneal] 2. Posterior tibial artery 2. Bone grafting of nonunion ARTHROSCOPY PORTALS 1. Anteromedial Just above joint line, Anterior horn of medial menicus Used to view lateral compartment 1 cm inferior to patella 1 cm medial to patellar ligament 2. Anterolateral Just above joint line, Anterior horn of lateral meniscus 1. Used to view medial compartment, ACL, and menisci 1 cm inferior to patella 1 cm lateral to patellar ligament 2. PCL posterior structures hard to see 3. Superolateral 2.5 cm above joint line, lateral to quadricep tendon Used to view patellofemoral articulation, patella tracking, Superolateral lateral to quadricep tendon etc. 4. Posteromedial Flex knee to 90°, 1 cm posterior to femoral condyle Used to view PCL, posterior horns of menisci Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com CHAPTER 9 - FOOT/ANKLE TOPOGRAPHIC ANATOMY OSTEOLOGY TRAUMA ANKLE JOINTS FOOT JOINTS OTHER STRUCTURES MINOR PROCEDURES HISTORY OF THE FOOT/ANKLE PHYSICAL EXAM MUSCLES: DORSUM MUSCLES: FIRST PLANTAR LAYER MUSCLES: SECOND PLANTAR LAYER MUSCLES: THIRD PLANTAR LAYER MUSCLES: FOURTH PLANTAR LAYER NERVES ARTERIES DISORDERS PEDIATRIC DISORDERS SURGICAL APPROACHES TO THE ANKLE Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier CHAPTER 9 – FOOT/ANKLE TOPOGRAPHIC ANATOMY Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier OSTEOLOGY CHARACTERISTICS OSSIFY FUSE COMMENT See leg chapter for Tiba and Fibula TALUS • Head (anterior-navicular) • Neck: susceptible to fracture • Body/trochlea: in ankle mortise • Lateral process • Posterior process: medial lateral tubercles Primary: Body 7mo. (fetal) 13-15 years • T alus is only tarsal bone to articulate with tibia and fibula. No muscular attachments. • AVN a concern due to retrograde blood supply from branches of posterior tibial dorsalis pedis arteries • Weight from tibia is transmitted through the trochlea • FHL runs between medial lateral tubercle of posterior process • Unfused lateral tubercle: Os trigonum, not a fracture CHARACTERISTICS OSSIFY FUSE COMMENT CALCANEUS • Multiple facets: posterior largest • Sustentaculum tali: has the middle facet; supports talar neck Primary: Body Secondary: Tubercle 6 mo. (fetal) 9 year 13-15 years • Largest tarsal bone; posterior support for longitudinal arch • FHL runs under sustentaculum tali; spring ligament attaches to it • Painful spurs can develop on tuberosity NAVICULAR • “Boat-shaped” • Tuberosity (medial) Primary: 4 years 13-15 years • Tibialis posterior inserts on to the tuberosity • Articulates with talus, cuneiforms, cuboid • Shape of tarsals create transverse arch CUNEIFORMS • Three bones • Medial: largest • Intermediate: shorter than others • Lateral Primary: 3 years 4 years 1 year 13-15 years • 2nd MT is in “recess” of short intermediate bone; can lead to fracture of it's base, unstable TMT joint. • Peroneus longus partially inserts on plantar aspect of med. cuneiform CHARACTERISTICS OSSIFY FUSE COMMENT CUBOID • Tuberosity inferiorly • Cuboid groove inferiorly Primary: Birth 13-15 yrs • Most lateral tarsal bone • Peroneus longus tendon passes through groove on inferior surface METATARSALS • Long bone characteristics • Base of 2nd MT in tarsal “recess” • Anterior support of longitudinal arch of the foot Primary: Shaft Secondary: Epiphysis 9 wks (fetal) 5-8 yrs Birth 14-18 years • Numbered medial to lateral: I to V. • Only one epiphysis per bone: in the head except for the 1st MT [in the base] • Peroneus brevis inserts on base of 5th MT (avulsion can occur) PHALANGES • Great toe has only two phalanges Primary: Body 10 wks (fetal) 14-18 years • 14 total phalanges in each foot • Only one epiphysis per bone: in the base • Great toe has two sesamoid bones Secondary: Epiphysis 2-3 yrs years base • Sesamoid bones with other toes can occur as a normal variant Ossification of each tarsal bone occurs from a single center Borders of ankle mortise: Superior: tibia (plafond), medial: medial malleolus (tibia), lateral: lateral malleolus (fibula) T arsal Tunnel: A fibroosseous tunnel formed by the posterior medial malleolus, medial walls of calcaneus and talus, and flexor retinaculum. Contents: T endons (TP, FDL, FHL), Posterior Tibial artery, Tibial nerve (can be compressed in tunnel) Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier TRAUMA Lauge-Hansen Classification of Ankle Fractures DESCRIPTION EVALUATION CLASSIFICATION TREATMENT ANKLE FRACTURE (see Knee Trauma table for Maisonneuve fracture) • Very common in all ages • Malleoli and/or talar dome are involved • 1 malleolus fx: stable; • 2 malleoli and/or ligaments injured: unstable • Perfect symmetrical mortise reduction required • Also must correct fibular length HX: Trauma. Pain, swelling PE: Effusion, intense tenderness at 1 or both malleoli +/- proximal fibula. Check posterior tibial pulse and tibial nerve function XR: Ankle trauma seriesCT : Good for intraarticularfractures needing repair Lauge-Hansen – 4 types with subdivided stages • SA: supination/adduction stage I, II • SER: supination/external rotation: stages I-IV • PA: pronation/abduction stages I, II, III • PER: pronation/external rotation: stages I-IV Dislocation: immediately reduce Stable/nondisplaced: short leg cast 4-6 weeks Unstable/displaced: ORIF, repair articular surface fibular length, +/- need for syndesmosis screw COMPLICATIONS: Post-traumatic osteoarthritis/pain; Decreased motion and/or strength; Instability; Nonunion/malunion; RSD Extraarticular Fracture of Calcaneus Intraarticular Fracture of Calcaneus DESCRIPTION EVALUATION CLASSIFICATION TREATMENT CALCANEUS FRACTURE • Most common tarsal fracture • Mechanism: large axial load (e.g. high fall or jump) • Must rule out spine injury • Subtalar joint affected • Most fractures are intraarticular (worse prognosis) HX: Trauma. Cannot bear weight, pain, swelling. PE: T ender to palpation. Check Tibial nerve function, pulses arch swelling. XR: AP/lateral (+/- Harris) and spine films CT : Needed to better define fx Extraarticular: Body Tuberosity Anterior/medial process Intraarticular: Nondisplaced T ongue-type Joint depression Comminuted Extraarticular: Cast. ORIF if unstable Displaced/intraarticular: ORIF: plates and screws +/- bone graft Severely comminuted: Closed treatment. COMPLICATIONS: Osteoarthritis: subtalar; Decreased motion; Malunion/nonunion; Compartment syndrome; Sural nerve injury Fracture of Talar Neck DESCRIPTION EVALUATION CLASSIFICATION TREATMENT TALUS FRACTURE • MVA, fall from height • Neck most common site, head body rare • T enuous blood supply adds complications • Semi-emergent injury • Hawkins sign (on XR) resorption of subchondral bone indicates healing (no AVN) HX: Trauma. Cannot bear weight, pain, swelling. PE: T ender to palpation. Check Tibial nerve function, pulses, arch swelling XR: AP/lateral (+/-Canale) CT : usually not needed Hawkins types [neck] predicts osteonecrosis: I. Nondisplaced II. Displaced; subtalar subluxation/dislocation III. Displaced; talar body dislocation IV. T alar head (+/-body) dislocation Type I: Cast 2 months. Manyprefer ORIF to reduce risk ofdisplacement Type II, III, IV: ORIF emergentlyto avoid necrosis +/- bonegraft Early ROM COMPLICATIONS: Osteoarthritis: ankle and subtalar joints; Osteonecrosis of body (incidence decreased with ORIF); Delayed union/nonunion Injury to Tarsometatarsal (Lisfranc) Joint Complex DESCRIPTION EVALUATION CLASSIFICATION TREATMENT MIDFOOT FRACTURES • Involves tarsal bones • Usually high energy • Midtarsal joint injuries result from fractures of adjacent bones. • Cuneiform cuboid fractures are rare • 2nd MT in tarsal recess: fracture of its base destabilizes TMT joint, dislocation may result. HX: Trauma. Dorsal pain. PE: Swelling, severe pain atMidtarsal or TMT jointincreases with midfootmotion. XR: AP/lateral/oblique,+/-foot stress filmMed. 2nd MT and middlecuneiform should align CT/MR: if unsure of fracture Midtarsal: Navicular fracture Avulsion Tuberosity Body Cuboid fracture Cuneiform fracture T arsometatarsal -LisfrancFracture (2ndMT) dislocationHomolateral, Isolated,Divergent Midtarsal: Nondisplaced: cast. Other: ORIF Navicular: Reduce, +/-PCP. Many require ORIF Lisfranc injury: Close reduce fracture and/ordislocation (+/- PCP). ORIF: if displaced orirreducible-most COMPLICATIONS: Neurovascular injury: Dorsalis pedis artery; Compartment syndrome; Decreased motion; Post-traumatic osteoarthritis or chronic pain. DESCRIPTION EVALUATION CLASSIFICATION TREATMENT METATARSAL AND PHALANGEAL FRACTURES • Common injuries: most are benign. • Fracture at metaphyseal/diaphyseal junction of 5th MT (Jones fracture) is not benign • Base of 5th MT avulsion fracture [PB]: benign • T oe fx: usually stub injury 5th toe most common HX: Pain with weight bearing, swelling PE: Swelling, ecchymosis, bony pain (increases with motion) XR: MT : AP/lateral/oblique T oe: AP only Metatarsal: Head neck fractureShaft Base (esp. of 5th)Phalanges: Shaft Joint injuries Metatarsal Fractures:Undisplaced: hard soledshoe or walking cast. Displaced/angulated: ORIF5th MT Jones fx: Cast andNWB 6 weeks vs. ORIF Phalange Fractures:Great toe: Reduce. PCP jointinjuries. Others: splint or buddy tape COMPLICATIONS: Neurovascular injury: Dorsalis pedis artery; Osteoarthritis/pain; Decreased motion; Nonunion, especially in 5th Metatarsal (Jones) fracture; Deformity Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier ANKLE JOINTS LIGAMENTS ATTACHMENTS COMMENTS INFERIOR TIBIOFIBULAR SYNDESMOSIS: Distal tibia/fibula support: must be stabilized if disrupted Anterior/inferior tibiofibular [AITFL] Distal anterior tibia fibula Oblique, connects bones anteriorly Posterior/inferior tibiofibular [PITFL] Distal posterior tibia fibula Weaker, posterior support of mortise Inferior transverse ligament Inferior deep to PITFL Strong posterior support of mortise Interosseous ligament Lateral tibia to med. fibula A continuation of interosseous membrane, strong support; torn in Maisonneuve fracture • Syndesmosis widening seen on radiographs if both the AITFL and PITFL are ruptured ANKLE (mortise/talus) (Ginglymus/hinge type) Capsule Tibia to talus Extends to interosseous ligament MEDIAL: Deltoid ligament (4 parts) Medial malleolus to: Strong medial support: fewer sprains. Tibionavicular Navicular tuberosity Overlaps the anterior tibiotalar ligament Tibiocalcaneal Sustentaculum tali Oriented vertically Posterior tibiotalar Medial tubercle of talus Thickest part of deltoid ligament Anterior tibiotalar T alus Minimal support LATERAL: Lateral malleolus to: Weaker lateral support: more sprains Anterior talofibular [ATFL] Neck of talus Weak, most often sprained, positive anterior drawer test when ruptured Calcaneofibular [CFL] Calcaneus Stabilizes subtalar joint Posterior talofibular [PTFL] Posterior process (talus) Strong, seldom torn Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier FOOT JOINTS JOINT LIGAMENTS COMMENTS INTERTARSAL Subtalar (talocalcaneal) Allows inversion/eversion of foot (e.g. walking on uneven surface) Medial talocalcaneal Medial tubercle to sustentaculum tali Lateral talocalcaneal Deep to calcaneofibular ligament Posterior talocalcaneal Short; Posterior process to calcaneus Interosseous talocalcaneal Strong; in sinus tarsus Also supported by the ligaments of the ankle (see ankle joints) Transverse/Midtarsal (Chopart's Joint): assists subtalar joint with inversion eversion T alonavicular Plantar calcaneonavicular (Spring) Sustentaculum tali to navicular: plantar support for head of talus; Strong. Dorsal talonavicular Dorsal support Calcaneonavicular (Bifurcate 1) Lateral support Calcaneocuboid Calcaneocuboid (Bifurcate 2) Stabilizes two rows of tarsus Dorsal calcaneocuboid Dorsal support Plantar calcaneocuboid (short plantar) Strong plantar support Calcaneocuboid MT (long plantar) Additional plantar support Cuboideonavicular Cuneonavicular Intercuneiform Cuneocuboid Each of these four joints have dorsal, plantar, and interosseous ligaments, each bearing the name of the corresponding joint These joints are small, have little motion or clinical significance. Share a common articular capsule. Plantar ligaments are stronger than the dorsal ligaments TARSOMETATARSAL (Lisfranc) Gliding type Dorsal, plantar, interosseous, tarsalmetatarsals (TMT) ligaments Medial cuneiform to 2nd metatarsal: Lisfranc's ligament INTERMETATARSAL Dorsal, plantar, interosseous MT Strengthen transverse arch Deep transverse metatarsal Connect the MT heads METATARSOPHALANGEAL Ellipsoid/condyloid type Plantar plate and Intersesamoid Part of weight bearing surface Collateral Strong Deep transverse metatarsal ligaments add support to this joint INTERPHALANGEAL Ginglymus/hinge type Plantar plate Similar to the IP joints of the hand Collateral Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier OTHER STRUCTURES STRUCTURE FUNCTION COMMENT Superior extensor retinaculum Covers tendons, nerves vessels of anterior compartment at the ankle Distal fibula to medial tibia Inferior extensor retinaculum Surrounds covers tendons, etc. of the anterior compartment in the foot “Y” shaped; calcaneus to medial malleolus and navicular Flexor retinaculum Covers tendons of posterior compartment Medial malleolus to calcaneus. Roof of tarsal tunnel. Superior Inferior peroneal retinaculum Covers tendons sheaths of the lateral compartment at the hindfoot Superior: Lateral malleolus to calcaneus Inferior: Inferior extensor retinaculum to calcaneus Plantar Aponeurosis (Plantar fascia) Supports longitudinal arch Inflammed: plantar fascitis. Can develop nodules Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier MINOR PROCEDURES STEPS ANKLE ARTHROCENTESIS 1. Ask patient about allergies 2. Plantarflex foot, palpate medial malleolus and sulcus between it and the tibialis anterior tendon. Use the visible EHL tendon if TA is not palpable. 3. Prepare skin over ankle joint (iodine/antiseptic soap) 4. Anesthetize skin locally (quarter size spot) 5. Insert 20 gauge needle perpendicularly into the sulcus/ankle joint (medial to the tendon, inferior to distal tibia articular surface, lateral to medial malleolus). Aspirate fluid. If suspicious for infection, send fluid for Gram Stain and culture. The fluid should flow easily if needle is in joint. 6. Dress injection site DIGITAL BLOCK 1. Same as in hand. See Hand chapter. Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier HISTORY OF THE FOOT/ANKLE QUESTION ANSWER CLINICAL APPLICATION 1. AGE Y oung Sprain, fractures Middle age, elderly Overuse injuries, arthritis, gout 2. PAIN a. Onset Acute (less common) Fracture, stress fracture Chronic Most foot ankle disorders are chronic b. Location Ankle Fracture, osteoarthritis, instability, posterior tibial tendinitis Hindfoot Plantar fascitis, fracture, retrocalcaneal bursitis, Achilles tendinitis Midfoot Osteoarthritis of tarsal joints, fracture Forefoot Hallux rigidus, fractures, metatarsalgia, Morton's neuroma, bunions, gout Bilateral Consider systemic illness, RA c. Occurrence Morning pain Plantar fascitis (improves with stretching/walking) With activity Overuse type injuries 3. STIFFNESS Without locking Ankle sprain, RA With locking Loose body 4. SWELLING Y es Fracture, arthritis 5. TRAUMA Mechanism/foot position Inversion: ATFL injury/sprain Bear weight? Y es: less severe injury; No: more severe (rule out fracture) 6. ACTIVITY/OCCUPATION Sports, repetitive motion Achilles tendinitis, overuse injuries Standing all day Overuse injuries 7. SHOE TYPE Tight/narrow toe box Hallux valgus (bunion, overwhelmingly seen in women) 8. NEUROLOGIC SYMPTOMS Pain, numbness, tingling T arsal tunnel syndrome 9. HISTORY OF SYSTEMIC DISEASE Manifestations in foot Diabetes mellitus, gout, peripheral vascular disease, RA, Reiter's syndrome Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier PHYSICAL EXAM EXAM TECHNIQUE CLINICAL APPLICATION INSPECTION Foot (standing/weight-bearing) Anterior view Alignment/rotational deformities, toe deformities, bunions Posterior view Minimal valgus is normal, “pump bump” exostosis Superior view Bunion, bunionette Medial view Flat foot (pes planus); high arch foot (pes cavus) Foot (supine/sitting/ non-WB) Inferior/plantar view Callus, warts, ulcers (especially in diabetic foot) Swelling Foot and ankle Swelling sign of infection, trauma (bilateral): cardiovascular etiology Color Change WB to non-WB If foot changes color: pink to RED: arterial insufficiency Shoes All aspects of the shoe Abnormal wear may indicate disease (e.g. scuffed toe, drop foot) EXAM TECHNIQUE CLINICAL APPLICATION PALPATION Bony structures 1st MTP joint (MT head) Bunion, bursitis, callus; pain: gout, sesamoiditis, tendinitis Other MTP joint (MT head) Pain: metatarsalgia, Freiberg's infraction, fracture, tailor's bunion (5th MT head) T arsal bones (T alus) T enderness suggests fracture, osteonecrosis, osteochondritis Calcaneus Pain: fracture. Posterior: bursitis (pump bump); Plantar: spur, plantar fascitis; Medial pain: nerve entrapment Both malleoli Pain indicates fracture, syndesmosis injury in leg Soft tissue Skin Cool: peripheral vascular disease. Swelling: trauma or infection vs. venous insufficiency Between metatarsal heads Mass pain: neuroma Medial ankle ligaments Pain suggests ankle sprain (Deltoid ligament) T endons at med. malleolus Pain indicates tendinitis, rupture (sprain) Lateral ankle ligaments Pain suggests ankle sprain ATFL, CFL, PTFL (rare) Peroneal tendons (lateral malleolus) Pain indicates tendinitis, rupture/sprain, dislocation Achilles tendon Pain: tendinitis. Defect suggests Achilles rupture RANGE OF MOTION Ankle: dorsiflex/plantarflex Stabilize subtalar joint Normal: Plantarflex 50°, Dorsiflex (extend) 25 ° Subtalar: inversion/eversion Stabilize tibia Normal: Invert 5-10°, Evert 5° Midtarsal: adduction/ abduction Stabilize heel/hindfoot Normal: Adduct 20°, abduct 10° Great toe: MTP: flex/extend Stabilize foot Normal: Flex 75°, extend 75°. Decreased in hallux rigidus IP: flex/extend Stabilize foot Normal: Flex 90, extend 0° Pronation: dorsiflexion, eversion, abduction. Supination: plantarflexion, inversion, adduction EXAM TECHNIQUE CLINICAL APPLICATION NEUROVASCULAR Sensory Saphenous (L4) Med. foot (med. cutaneous) Deficit indicates corresponding nerve/root lesion Tibial nerve (L4) Plantar foot (calcaneal/plantar) Deficit indicates corresponding nerve/root lesion Superficial Peroneal (L5) Dorsal foot Deficit indicates corresponding nerve/root lesion Deep Peroneal (L5) 1st dorsal web space Deficit indicates corresponding nerve/root lesion Sural nerve (S1) Lateral foot Deficit indicates corresponding nerve/root lesion Motor Deep Peroneal nerve (L4) Foot inversion/dorsiflexion Weakness = Tibialis Anterior or nerve/root lesion Deep Peroneal nerve (L5) Great toe extension (dorsiflex) Weakness = EHL or corresponding nerve/root lesion Tibial nerve (S1) Great toe plantarflexion Weakness = FHL or corresponding nerve/root lesion Superficial Peroneal (S1) Foot eversion Weakness = Peroneus muscles or nerve/root lesion Reflex S1 Achilles reflex Hypoactive/absence indicates S1 radiculopathy Upper Motor Neuron Babinski reflex Upgoing toes indicates an Upper Motor Neuron disorder Pulses Dorsalis pedis Decreased pulses: trauma or vascular compromise, peripheral vascular disease Posterior tibial SPECIAL TESTS Anterior drawer Hold tibia, anterior force to calcaneus Anterior translation: AnteriorT aloFibular Ligament (ATFL) rupture (sprain) T alar tilt Hold tibia, invert ankle Increased laxity compared to contralateral: CFL/ATFL sprain Eversion/abduct stress Hold tibia, evert/abduct Ankle Increased laxity compared to contralateral: Deltoid ligament sprain “T oo many toes” sign Standing, view foot posteriorly “T oo many toes” (more seen laterally than other side): acquired flat foot Squeeze Compress distal tibia/fibula Pain indicates a syndesmosis injury Heel lift Standing, raise onto toes Heel into varus is normal. Decreased lift with posterior compartment pathology Tinel's sign at the Ankle T ap nerve posterior to medial malleolus Tingling/parathesia is positive for posterior tibial nerve entrapment Compression Squeeze foot at MT heads Pain, numbness, tingling: interdigital neuroma (Morton's) Thompson Prone: feet hang, squeeze calf Absent plantar flexion indicates Achilles tendon rupture Homans' sign Knee extended: passively dorsiflex foot Pain in calf suggestive of deep venous thrombophlebitis (DVT) Copyright © 2008 Elsevier Inc. All rights reserved. -www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier MUSCLES: DORSUM MUSCLE ORIGIN INSERTION NERVE ACTION COMMENT Extensor hallucis brevis [EHB] Dorsal calcaneus Base of proximal phalanx of Great toe Deep peroneal Extends great toe Assists EHL with its action Extensor digitorum brevis [EDB] Dorsal calcaneus Base of proximal phalanx: 4 lateral toes Deep peroneal Extends toes Injury can result in dorsal hematoma Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier MUSCLES: FIRST PLANTAR LAYER MUSCLE ORIGIN INSERTION NERVE ACTION COMMENT FIRST LAYER Abductor hallucis Calcaneal tuberosity medial process Through med. sesamoid to proximal phalanx of great toe Medial plantar Abducts great toe Supports longitudinal arch medially. Flexor digitorum brevis [FDB] Calcaneal tuberosity medial process Sides of middle phalanges: lateral 4 toes Medial plantar Flex lateral 4 toes Supports longitudinal arch Abductor digiti minimi [ADM] Calcaneal tuberosity medial lateral processes Lateral base of proximal phalanx: 5th toe Lateral plantar Abducts small toe Supports longitudinal arch laterally Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier MUSCLES: SECOND PLANTAR LAYER MUSCLE ORIGIN INSERTION NERVE ACTION COMMENT SECOND LAYER Quadratus plantae Medial and lateral plantar calcaneus Lateral FDL tendon Lateral plantar Assists FDL with toe flexion Two heads/bellies join on FDL tendon Lumbricals Separate FDL tendons Proximal phalanges, extensor expansion 1. Medial plantar 2-4. Lateral plantar Flex MTP joint, extend IP joint 1st lumbrical attaches to 1 FDL tendon T endons of FHL and FDL also pass through in the second layer Medial and lateral plantar nerves are terminal branches of the Tibial nerve: they run in the 2nd layer. Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier MUSCLES: THIRD PLANTAR LAYER MUSCLE ORIGIN INSERTION NERVE ACTION COMMENT THIRD LAYER Flexor hallucis brevis [FHB] Cuboid, lateral cuneiform Through sesamoids to proximal phalanx of great toe Medial plantar Assist great toe flexion Sesamoid bones attach to each tendon Adductor hallucis Oblique: base 2-4 MT Transverse: Lateral 4 MTP Through lateral sesamoid to proximal phalanx of great toe Lateral plantar Adducts great toe Supports transverse arch. 2 heads have different orientations Flexor digiti minimi brevis [FDMB] Base of 5th metatarsal Base of proximal phalanx small toe Lateral plantar Flex small toe Small, relatively insignificant muscle Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier MUSCLES: FOURTH PLANTAR LAYER MUSCLE ORIGIN INSERTION NERVE ACTION COMMENT FOURTH LAYER Plantar interossei (3) Med. 3, 4, 5th MT s Medial proximal phalanges: toes 3-5 Lateral plantar Adduct toes (PAD) Attachment to MT is medial for all 3 Dorsal interossei (4) Adjacent MT shafts Proximal phalanges toes 2-5 Lateral plantar Abduct toes (DAB) Larger than the plantar interossei muscles Peroneus longus and Tibialis posterior tendons pass through the fourth layer Medial and lateral plantar nerves are terminal branches of the Tibial nerve. PAD = 5 Plantar ADduct, DAB 5 = Dorsal ABduct; the second digit is used as the reference point for abduction/adduction in the foot Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier NERVES LUMBAR PLEXUS POSTERIOR DIVISION 1. Femoral (L2-4): Saphenous nerve branches in proximal thigh, descends in superficial medial leg, then anterior to medial malleolus in foot. Sensory: Medial foot: via medial cutaneous nerve (Saphenous nerve) Motor: NONE (in foot or ankle) SACRAL PLEXUS ANTERIOR DIVISION 2. Tibial (L4-S3): behind medial malleolus, splits on plantar surface Sensory: Medial heel: via Medial calcaneal Medial plantar foot: via Medial plantar Lateral plantar foot: via Lateral plantar Motor: FIRST PLANTAR LAYER of FOOT Abductor hallucis: Medial plantar Flexor digitorum brevis[FDB]: Medial plantar Abductor digiti minimi: Lateral plantar SECOND PLANTAR LAYER of FOOT Quadratus plantae: Lateral plantar Lumbricals: Medial Lateral plantar THIRD PLANTAR LAYER of FOOT Flexor hallucis brevis [FHB]: Medial plantar Adductor hallucis: Lateral plantar Flexor digiti minini brevis [FDMB]: Lateral plantar FOURTH PLANTAR LAYER of FOOT Dorsal interosseous: Lateral plantar Plantar interosseous: Lateral plantar POSTERIOR DIVISION 3. Common peroneal (L4-S2): Superficial peroneal divides into intermediate and medial dorsal cutaneous branches in leg. Deep peroneal divides under extensor retinaculum into medial lateral branches. Sensory: Lateral foot: via Sural (lateral calcaneal dorsal cutaneous). Dorsal foot: Superficial peroneal. Dorsal (med.) (Med. dorsal cutaneous branch). 1st/2nd interdigital space: Deep peroneal (med. branch) Motor: FOOT : Deep Peroneal (Lateral branch) Extensor hallucis brevis [EHB] Extensor digitorum brevis [EDB] Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier ARTERIES ARTERY STEM ARTERY/ COMMENT Artery to the T arsal Sinus Dorsalis pedis and Peroneal arteries Artery to the T arsal Canal Posterior tibial artery Deltoid artery Posterior tibial artery; supplies medial body Capsular ligamentous vessels Multiple sources Interosseous anastomosis Extensive, protects against AVN ARTERY COURSE COMMENT (See Leg/Knee chapter for stem arteries) Anterior Medial Malleolar Under TA EHL tendons to medial malleolus From Anterior tibial artery, supplies medial malleolus Anterior Lateral Malleolar Under EDL tendon to lateral malleolus From Anterior tibial artery, supplies lateral malleolus Posterior Medial Malleolar Under tendons of TP and FDL, not FHL, to medial malleolus From Posterior tibial artery, supplies medial malleolus Posterior Lateral Malleolar Under Peroneus longus/brevis tendons to lateral malleolus From Peroneal artery, supplies lateral malleolus Perforating and communicating branches Anastomosis with anterior lateral malleolar and posterior tibial arteries From Peroneal artery, contributes supply to lateral malleolus An anastomosis occurs at each malleolus between the above arteries ARTERY COURSE BRANCHES COMMENT/SUPPLY (see Leg Knee chapter for stem arteries) Lateral Calcaneal with Lateral calcaneal nerve (Sural nerve) NONE From Peroneal artery; supplies heel Medial Calcaneal with Medial calcaneal nerve (Tibial nerve) NONE From Posterior tibial artery; supplies heel Lateral plantar Between quadratus plantae FDB, runs w/ lateral plantar nerve Deep plantar arch Larger terminal branch of Posterior tibial artery Medial plantar Between Abductor hallucis FDB runs with medial plantar nerve Superficial branch 1 proper plantar digital Deep branch Smaller terminal branch of Posterior tibial artery; supplies medial Great toe Anastomose with plantar MT artery Dorsalis Pedis Dorsum of foot with medial branch of deep peroneal nerve Supplies dorsum of foot via: Medial T arsal No branches Lateral T arsal No branches Arcuate artery 3 Dorsal MT arteries branch off Deep Plantar Descends to deep plantar arch 1st dorsal metatarsal T erminal branch of dorsalis pedis 3 dorsal digital arteries Supply dorsal great toe ARTERY COURSE BRANCHES COMMENT/SUPPLY (see Leg Knee chapter for stem arteries) Medial T arsal Across tarsals, under EHL tendon NONE Supplies dorsum of foot (can be 2 or 3 of these arteries). Lateral T arsal Across tarsals with lateral branch of Deep peroneal nerve NONE Supplies EDB, lateral tarsal bones, anastomoses laterally Arcuate Across bases of metatarsals, under extensor tendons 2nd, 3rd, 4th dorsal MT artery 7 dorsal digital arteries Deep plantar Descends between 1st 2nd MT's Deep plantar arch Anastomosis with Lateral calcaneal Deep plantar arch On plantar interosseous muscles in 4th layer of foot. 4 posterior perforating Join dorsal metatarsal arteries 1 Common/proper plantar digital Most lateral artery in foot toes 4 plantar metatarsal 4 anterior perforating Join dorsal metatarsal arteries 4 Common plantar digital 8 Proper plantar digital Supplies the distal tip of phalanx T otal of 4 Dorsal Metatarsal arteries leading to 10 dorsal digital arteries. They do not reach the distal tip of the digit. T otal of 4 Plantar Metatarsal arteries leading to 10 proper plantar digital arteries via common plantar digital arteries. Each digit has 2 dorsal digital and 2 proper plantar digital arteries. Dorsal branch of proper plantar digital artery supply distal tip. Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier DISORDERS DESCRIPTION HISTORY/PHYSICAL EXAM WORK-UP/FINDINGS TREATMENT ACHILLES TENDINITIS • Occurs at or above insertion of Achilles tendon Hx/PE: Heel pain, worse with push off. T ender to palpation XR: Standing lateral: spur at Achilles insertion 1. Rest, NSAID, heel lift 2. Excise bone or bursa (rare) ACHILLES TENDON RUPTURE • “Weekend warriors.” Middle age men in athletics. Hx: “hit with bat” sensation PE: Defect, + Thompson test XR: Standing AP/lateral: usually normal Casting (in equinus) vs. surgical repair ACQUIRED FLAT FOOT (POSTERIOR TIBIALIS DYSFUNCTION) • Tibialis posterior tendon dysfunction: tears or degeneration • No arch support results in valgus foot Hx: Pain and swelling PE: + “too many toes” sign, no heel varus on toe rise XR: Standing AP/lateral: middle foot sag 1. Orthosis 2. Activity modification 3. Calcaneal osteotomy and FDC transfer 4. Arthrodesis ANKLE INSTABILITY • Multiple/recurrent sprains • Also neurologic etiology decreased proprioception Hx: Inversion instability esp. on uneven groundPE: + anterior drawer talar tilt test XR: AP/lateral/stress view: gapping laterally 1. PT : strengthen peroneals 2. Surgical reconstruction if condition persists ANKLE SPRAIN • #1 musculoskeletal injury • Lateral 90% -ATFL alone 60%, with syndesmosis 5% • Inversion most common mechanism Hx: “Pop,”pain, swelling, +/- ability to bear weightPE: + Anterior drawer, +/-talar tilt test XR: only if cannot bear weight or + bony point tenderness 1. RICE, NSAIDs 2. Immobilize grade III 3. PT ROM exercises 4. Surgery: athletes or severe injury ARTHRITIS: OA/DJD • Can occur in any joint • Associated with trauma, obesity, overuse activity Hx/PE: Older, pain at affected joint. XR: Standing AP/lateral: classic OA findings 1. NSAID, activity modifcation, orthosis 2. Fusion/arthroplasty (rare) CHARCOT JOINT : NEUROPATHIC JOINT • Neurologic disease results in decreased sensation • Joint destroyed/deformed by fx undetected by patient Hx/PE: Patient is insensate-no pain. Red, warm, swollen joint XR: Standing AP/lateral: fractures (callus or unhealed), joint destroyed 1. Immobilze (skin checks) 2. Bony excision or fusion CLAW TOE • Deformity: MTP extended, PIP flexed. Usually all toes • Etiology: Neurologic disease Hx: T oe painPE: T oe deformity, +/- callus corn, neurologic exam XR: Standing AP/lateralMR/EMG/lab: to rule out neurologic disease 1. Shoes with extra deep toe box 2. Surgical reconstruction: based on Neurologic disease (e.g. Charcot-Marie-T ooth) disease based on deformity CORN • Two types: 1. Hard 2.Soft 1. Hyperkeratosis: pressure on bones (5th toe #1) 2. Interdigital maceration Hx/PE: Tight shoes. Pain at lesion site. XR: AP/lateral: look for bone spurs 1. Wide toe box shoe, pads 2. Debride callus 3. Excise bony prominence DIABETIC FOOT : NEUROPATHIC FOOT • Neuropathy leads to unperceived injury (ulcer, infection) • Vascular insufficiency leads to decreased healing Hx: Burning tingling, +/- painPE: +/-: skin changes, ulcers, deformity, swelling, warmth XR: Standing AP/lateral: rule out osteomyelitis or Charcot jointDo Ankle Brachial Index 1. Skin care (prevention) 2. Protective shoe 3. Treat ulcers, infections 4. Amputation if necessary GOUT (Podagra) • Purine metabolism defect • Urate crystals create synovitis • Great toe most common site Hx: Men, acute exquisite pain PE: Red, swollen toe. Labs: 1. Elevated uric acid 2. Negatively birefringent crystals 1. NSAIDs, colchicine 2. Rest 3. Allopurinol (prevention) DESCRIPTION HISTORY/PHYSICAL EXAM WORK-UP/FINDINGS TREATMENT HALLUX RIGIDUS • DJD of MTP of Great toe • Often post traumatic Hx: Middle age. Painful, stiff PE: MTP T ender to palpation, decreased ROM XR: Standing AP/lateral OA findings at 1st MTP 1. NSAID, stiff sole shoe 2. Arthroplasty/fusion HALLUX VALGUS (Bunion) • Great toe valgus; MTP bursitis • Multiple etiologies: genetic, flat feet, narrow shoes, RA • 10:1 women (shoes) Hx: Pain, swelling (worse with shoe wear (narrow toe box) PE: Medial 1st MTP TTP, +/- decreased great toe ROM XR: Standing AP: measure: 1. Distal MT Articulation Angle (normal 10°) 2. Inter MT angle (9°) 3. Hallux Valgus angle (15°) 1. Shoes: wide toe box 2. Refractory cases: multiple corrective surgical procedures based on deformity and severity HAMMER TOE • T oe PIP flexion deformity • Associated with trauma, Hallux Valgus (shoes) Hx: T oe pain, worse when wearing shoes PE: T oe deformity, +/-corn XR: Standing AP/lateral: PIP deformity 1. Extra deep shoe toe box 2. Surgery: resect or fuse PIP MALLET TOE • Lesser toe DIP flexion deformity • 2nd toe most common Hx: T oe pain PE: T oe deformity, callus XR: Standing AP/lateral: DIP deformity 1. Shoe modification 2. FDL release METATARSALGIA • Metatarsal head pain • Etiology: flexor tendinitis, ligament rupture, callus (#1) Hx/PE: Pain under MT head (2nd MT most common) XR: Standing AP/lateral: look for short MT 1. Metatarsal pads 2. Modify shoes 3. Treat underlying cause MORTON'S NEUROMA (Interdigital) • Fibrosis of irritated nerve • Usually between 2nd 3rd metatarsals • 5:1 female(shoes) Hx: Plantar MT pain PE: MT TTP, +/-numbness, + compression test XR: Standing AP/lateral: usually normal, not helpful 1. Wide toe shoes, steroid injections, MT pads 2. Nerve excision PLANTAR FASCITIS • Inflammation and/or degeneration of fascia. Female 2:1 • Associated with obesity Hx: AM pain, improves with ambulation or stretching PE: Medial plantar calcaneus tender to palpation XR: Standing lateral: +/-calcaneal bone spur 1. Stretching, NSAID 2. Heel cup 3. Splint (night), casting PLANTAR WARTS • Hyperkeratosis • Due to Papilloma virus Hx/PE: Painful plantar lesions Histopathology if necessary 1. Pads vs. freeze or debride lesion RETROCALCANEAL BURSITIS: HAGLUND'S DISEASE • Bursitis at insertion of Achilles tendon on calcaneus Hx: Pain on posterior heel PE: Red, tender to palpation, “pump bump” XR: Standing lateral: spur at Achilles insertion 1. NSAID, heel lift, casting 2. Excise bone/bursa (rare) RHEUMATOID ARTHRITIS • Synovitis destroys joints • More common in females Hx: Forefoot: pain, swelling PE: Red, tender, +/-deformity (e.g. Hallux XR: AP/lateral: joint destroyed Lab: positive 1. Medical management 2. Custom molded shoes • Associated with HLA-DR4 deformity (e.g. Hallux Valgus) RF, ANA 3. Fusion or resection SERONEGATIVE SPONDYLOARTHROPATHY : REITER'S, AS, PSORIASIS • Multiple manifestations • Associated with HLA-B27 • Most common in males Hx/PE: Y oung, forefoot/toe/ heel: red, swollen, tender XR: AP/lateral: +/- calcification Lab: negative RF, ANA 1. Conservative treatment 2. Rheumatology consult TAILOR'S BUNION: BUNIONETTE • Prominent 5th MT head Laterally • Bony exostosis/bursitis Hx/PE: Difficulty fitting shoes, painful lateral 5th metatarsal prominence XR: Standing AP: 5th toe medially deviated, MT head laterally deviated 1. Pads, stretch toe box 2. Metatarsal osteotomy DESCRIPTION HISTORY/PHYSICAL EXAM WORK-UP/FINDINGS TREATMENT TARSAL TUNNEL SYNDROME • Tibial nerve trapped by flexor retinaculum and/or tendons Hx/PE: Pain, tingling, burning on sole (made worse with activity) XR: AP/lateral: normalEMG: confirms diagnosisMR: to find mass lesion 1. NSAID, steroid injection2. Surgical release (must follow plantar nerves also) Copyright © 2008 Elsevier Inc. All rights reserved. -www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier PEDIATRIC DISORDERS DESCRIPTION EVALUATION TREATMENT/COMPLICATIONS METATARSUS ADDUCTUS • Forefoot adduction (varus) • #1 pediatric foot disorder • Associated with intrauterine position or other disorders Hx: Parent notices deformity PE: ”Kidney bean” deformity, negative thigh/foot angle, + intoeing gait 1. Most spontaneously resolve with normal development 2. Serial casting 3. Rarely, midfoot osteotomies TALIPES EQUINOVARUS: CLUBFOOT • Congenital, boys, 50% bilateral • Genetic environment factors • Idiopathic or associated with other disorders (neuromuscular, etc.) • 4 deformities with soft tissue contractures Hx: Deformity at birth PE: Rigid foot with: 1. plantarflexed ankle (equinus) 2. inverted hindfoot (varus) 3. adducted forefoot 4. cavus midfoot XR: if diagnosis is unclear 1. Manipulation and casting 2-4 mo. 2. Surgical correction (release, lengthening, etc.) with post operative casting COMPLICATION: recurrence of deformity DESCRIPTION EVALUATION TREATMENT/COMPLICATIONS PES PLANUS: CONGENITAL FLATFOOT • Normal in infants (up to 6 yo) • No longitudinal arch • Ankle everted (valgus) • Classified: 1. Rigid (tarsal coalition/vertical talus) 2. Flexible (variant of normal) Hx: Usually adolescent, 1/2 foot pain PE: Rigid: always flat Flexible: only flat when WB XR: AP/lateral: may see coalition/or vertical talus in rigid foot Flexible: 1. Asymptomatic: no treatment 2. Symptomatic: arch supports, stretching Rigid: Treat underlying condition (see tarsal coalition) PES CAVUS: HIGH ARCH FOOT • High arch due to muscle imbalance in immature foot (T . A. and peroneus longus) • Ankle flexed: causes pain • Must rule out neuromuscular disease (e.g. Charcot-Marie-T ooth) Hx: 8-10 yrs, ankle pain PE: T oe walking, tight heel cord decreased ankle dorsiflexion XR: AP/lateral foot and ankle EMG/NCS: test for weakness MR: spine: r/o neuromuscular disease 1. Braces/inserts/AFO as needed (used with mixed results) 2. Various osteotomies 3. T endon transfer balance TARSAL COALITION • Connection (fibrous, cartilage then bony) of two tarsals • #1 Calcaneus/navicular (13-16yo) • #2 T alus/calcaneus (9-13yo) Hx: Foot pain during adolescence PE: Stiff, decreased ROM (subtalar), flatfoot (peroneal spasm) XR: AP/lateral/oblique: coalitions can be 1. Mild: observe 2. Casting 3. Coalition resection 4. Triple arthrodesis • Flatfoot deformity results seen CT : often necessary to confirm PE Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier SURGICAL APPROACHES TO THE ANKLE USES INTERNERVOUS PLANE DANGERS COMMENT ANKLE: ANTERIOR LATERAL APPROACH 1. Fusions/triple arthrodesis 2. T alar procedures 3. Intertarsal joint access 1. Peroneals [Superficial peroneal] 2. EDL [Deep peroneal] 1. Deep peroneal nerve 2. Ant. Tibial artery 1. Can access hindfoot 2. Preserving fat pad (sinus talus) helps wound healing. Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com CHAPTER 10 - BASIC SCIENCE BONES NERVES MUSCLES (SKELETAL) MICROBIOLOGY IMAGING Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier CHAPTER 10 – BASIC SCIENCE BONES STRUCTURE COMMENT Bone function Attachment of muscles Protection of organs Reservoir of minerals for body Hematopoiesis site Bone Forms Long bones Form by enchondral ossification, except clavicle Have a physis at each end (except in hand foot) 4 parts: epiphysis, physis, metaphysis, diaphysis Length is derived from the growing physis Flat bones Form by intramembranous ossification, (e.g., pelvis) Physeal Anatomy Divided into multiple zones Reserve zone Matrix production and storage Proliferative zone Cell proliferation, matrix production Hypertrophic zone Broken into 3 zones, calcification of matrix STRUCTURE COMMENT Microscopic Bone Types Woven Immature bone; normal in infants, also found in callus tumors Lamellar Mature bone; well organized, normal both cortical cancellous after age 4 Structural Bone Types Cortical (compact) 80% of bone, highly organized (osteons), blood supply in haversian canal. Volkmann's canal has vessels connecting osteons. Cancellous (spongy/trabecular) 20% of bone, crossed lattice structure, higher bone turnover STRUCTURE COMMENT Cell Types Osteoblasts Make bone (secrete matrix, collagen, GAG, stimulated by PTH) Osteoclasts Resorb bone (giant cells, mineralized bone found only in Howship's lacunae) Osteocytes Maintain bone (90% of cells, inhibited by PTH) STRUCTURE COMMENT Bone Composition Organic matrix (40%) Produced by osteoblasts—becomes osteocytes when trapped in matrix Collagen (Type I) 90% of matrix, gives strength. Mineralization occurs at gaps at the end of each collagen fiber Proteoglycan Glycosaminoglycans structure (GAGs) Non-collagen protein Osteonectin is most abundant Inorganic (60%) Mineralized portion Calcium Hydroxyapatite Adds strength to bone, found in the collagen gaps Types of Ossification Enchondral Bone replaces a cartilage template in long bones Intramembranous Mesenchymal template in flat bones and clavicle STRUCTURE COMMENT Fracture Types Point tenderness and swelling are common findings Open vs. closed Break in skin is open. Gustilo classification (grade I, II, III A, B, C) Direction Transverse, spiral, oblique, comminuted Displacement Displaced or nondisplaced Other • Salter-Harris—fracture involving an open physis in adults, growth plate in children. • Greenstick—only one cortex disrupted • T orus—one cortex impacted, but intact • Pathologic results—from bone tumor/disease STRUCTURE COMMENT Stages of Bone Healing Inflammation Hematopoietic cells, fibroblasts, osteoprogenitor cells Repair Callus formation (hard or soft), woven bone formation (enchondral) Remodeling Lamellar bone replaces woven, bone assumes normal shape, and repopulation of the marrow STRUCTURE COMMENT Bone Healing Factors Minerals Calcium, Phosphate STRUCTURE COMMENT Main Hormones Parathyroid hormone (PTH), Vitamin D, Calcitonin (see fig.__) Other Hormones Estrogen Inhibits bone resorption Corticosteroids Increases bone loss Thyroid hormone Normal levels promote bone formation, increased levels enhance resorption Growth hormone Promotes bone formation STRUCTURE COMMENT Metabolic Disorders Hypercalcemia Symptoms: constipation, nausea, abdominal pain, confusion, stupor, coma 1° hyperparathyroidism Increased urine calcium, decreased serum phophate, “brown tumors” result 2° hyperparathyroidism Malignancy #1, Multiple Endocrine Neoplasm (MEN) syndromes Hypocalcemia Symptoms: hyperreflexia, tetany +Chvostek's/Trousseau's sign, papilledema, prolonged QT interval 1° hypoparathyroidism Hair loss, vitiligo Renal osteodystrophy Chronic renal failure, “Rugger jersey” spine Rickets/osteomalacia Decreased/failed mineralization, Vitamin D deficiency Osteoporosis Decreased bone mass, elderly Scurvy Vitamin C deficiency results in defective collagen Osteopetrosis Increased bone density due to reduced osteoclast activity Paget's Disease Simultaneous osteoblast osteoclast activity results in dense, but more brittle bones STRUCTURE COMMENT Cartilage Several types: Hyaline Articular surfaces, physeal plates Fibrocartilage Annulus fibrosis, meniscus, pubic symphysis Elastic Nose, ears Articular Cartilage Function Distribute load over large surface, low friction motion surface Components Water, collagen type II, proteoglycans, chondrocytes Water content Decreases with age, increases in osteoarthritis Osteoarthritis #1 form of arthritis, articular cartilage defect/damage. Primary, “wear and tear”; or secondary, (e.g., posttraumatic.) Often found in hands and weight-bearing joints, knees #1 site Classic radiographic findings: 1. Osteophytes 2. Subchondral cysts 3. Subchondral sclerosis 4. Joint space narrowing Inflammatory Arthritis Rheumatoid, SLE, spondyloarthropathy, gout Rheumatoid Arthritis Immune disorder targeting the synovium. Chronic synovitis and pannus ormation lead to articular surface and joint destruction. 3: 1 women, associated with HLA-DR4, +RF, increased ESR/CRP Multiple joints affected: MCPs: ulnar deviation, feet: claw toe common Findings: morning stiffness, nodules, radiographs: 1. Bone erosions (periarticular) 2. Osteopenia 3. Swelling Reiter's Syndrome Triad: Urethritis, conjunctivitis, asymmetric arthritis; + HLA-B27 Gout Mono-sodium urate crystals in the joint induce an inflammatory rxn Old men, great toe #1 site, elevated uric acid levels often seen Crystals: negatively birefringent Ligaments Attach one bone to another Ligament bone attachment 1. Ligament to fibrocartilage 2. Fibrocartilage to calcified fibrocartilage, (most injuries occur here) 3. Calcified fibrocartilage to bone (Sharpey's fibers) Sprain T ear of a ligament. Grade I Stretching of, or minor tear in, ligament; no laxity Grade II Incomplete tear, laxity is evident (usually swelling) Grade III Complete tear, increased laxity (swelling/hematoma) Ligament Strength Relative strength difference between ligament and one predict injury Pediatrics Stronger than physis. Injury will occur at physis first Adult Bone stronger than ligament. Ligament will rupture first Geriatrics Ligament stronger than bone. Bone will fracture first Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier NERVES STRUCTURE COMMENT Cellular Anatomy Neuron Cell body. Dendrites receive signal, axon conveys signal Glial cells Schwann cells produce myelin to cover the axon Microanatomy Peripheral nerve has both afferent and efferent fibers Afferent fibers (axon) Transmits sensory signals from peripheral nerve endings to the CNS Cell bodies are in the dorsal root ganglion (DRG) Efferent fibers (axon) Transmits motor signals from CNS via ventral horn/ventral root to peripheral muscles. Endoneurium Surrounds each individual fiber (axon) Fascicles Group of endoneurium coated fibers Perineurium Surrounds each fascicle Peripheral nerve Groups of fascicles, blood vessels, and connective tissue Epineurium Surrounds the groups of fascicles (nerves) Nerve Injuries Based on microanatomy Neuropraxia Conduction disruption, axon intact; resolves in days to weeks Axonotmesis Axon disrupted, endoneurium intact allows axon regeneration; recovery is slow, growth 1mm/day, but usually full Neurotmesis Nerve transection, recovery requires surgical repair Poliomyelitis Viral destruction of ventral horn (motor) cells resulting in weakness/paralysis, but normal sensation. Vaccine for prevention. Nerve Conductions Facilitated by myelin coating on axon (larger/coated fibers are faster) Resting potential Maintained by a polar difference between intra/extracellular environments Action potential Change in permeability of Na+ ions depolarizes cell. Nodes of Ranvier Gaps between Schwann cells that facilitate conduction Nerve Conduction Evaluates motor and sensory peripheral nerves Studies (NCS) Stimuli is given and followed by surface electrodes. Latency (delay) and amplitude (strength of signal) are measured. Conduction velocities, 50m/s are abnormal Guillain-Barré Syndrome Ascending motor weakness/paralysis. Caused by demyelination of peripheral nerves following viral illness. Most self-limiting. Charcot-Marie-Tooth Autosomal dominant disorder. Demyelinating disorder affecting motorsensory nerves. Onset 5-15yrs, peroneal muscles first, then hand foot intrinsics. Can result in cavus foot, claw toe, intrinsic minus hand. Neuromuscular junction Axon of motor neuron synapses with the muscle (motor end plate) Neurotransmitter Acetylcholine stored in axon crosses synaptic cleft and binds to receptors on sarcoplasmic reticulum and depolarizes Pharmacologic agents Nondepolarizing agents (e.g., vecuronium) competively bind Ach receptor Depolarizing agents (e.g. succinylcholine) bind short term to Ach receptor T oxins/nerve gas: also bind these receptors competively; treat with anticholinesterase agents (increase Ach levels in cleft) Myasthenia gravis Relative shortage of acetylcholine receptors due to competitive binding by thymus derived antibodies. Treat with thymectomy or anti-acetylcholinesterase agents (increase acetylcholine levels in cleft) Motor Unit All the muscles innervated by a single motor neuron Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier MUSCLES (SKELETAL) STRUCTURE COMMENT Types of Muscle Smooth, cardiac, skeletal Skeletal Voluntary control, have an origin and insertion Anatomy Muscles cells have two types of contractile filaments: actin, myosin Muscle Comprised of multiple bundles or fascicles; surrounded by epimysium Bundle/Fascicle Comprised of multiple muscle fibers (cells); surrounded by perimysium Fiber (cell) Comprised of multiple myofibril; surrounded by endomysium Myofibril Comprised of multiple sarcomeres, end to end; no surrounding tissue Sarcomere Comprised of interdigitated thick and thin filaments; organized into bands. Z line to Z line defines the sarcomere A band: length of thick filaments, does not change with contraction I band, H zone, and sarcomere length all shorten with contraction Myosin Thick filament: have “heads” that bind ATP and attach to thin filaments Actin Thin filaments: fixed to Z bands; associated with troponin and tropomyosin Troponin Associated with actin and tropomyosin, binds Ca++ ions Tropomyosin Long molecule, lies in helical groove of actin and blocks myosin binding Contraction Initiated when Acetylcholine binds to receptors on sarcoplasmic reticulum and depolarizes them. Depolarization causes a release of Ca++ which then binds to troponin molecules. This binding causes the tropomyosin to move and the “charged” head (ATP bound) of myosin can bind to actin. Breakdown of ATP causes contraction of filaments, (shortening of sarcomere), and the release of the myosin from the actin filament. Electromyography (EMG) Intramuscular electrodes used to evaluate muscle function. Increased frequency, decreased duration, decreased amplitude indicate myopathy; opposite findings indicative of neuropathy. Types of Contraction Isometric Muscle fires against increasing resistance, muscle length is constant Isotonic Resistance is constant through contraction Isokinetic Muscle contracts at a constant speed Eccentric Muscle lengthens when it fires; can cause injury Concentric Muscle shortens when it fires Strength Related to cross sectional area of muscle Duchene Muscular Dystrophy X-linked recessive disorder affecting boys. Progressive, noninflammatory process affecting proximal muscles (increased CPK). Birth and development to age 3-5 usually normal, then weakness, clumsy walking, + Gower's sign (uses hands to rise from floor) and calf pseudohypertrophy. Most wheelchair bound by 15. Multiple associated deformities, contractures, scoliosis, etc. STRUCTURE COMMENT Compartments Muscles are located within confined fibroosseous/fascial spaces Compartment Syndrome Multiple causes of increased compartment pressures. Increased pressures and decreased perfusion resulting in myonecrosis. 5 P's: Pain, parathesias, paralysis, pallor, pulselessness (not all needed for diagnosis). Firm tense compartments on exam. Fasciotomy within 6 hours needed. Contracture can result. Musculotendinous Weakest portion of muscular attachment to bone (injuries occur here) Junction Muscle strain is a partial tear of this unit Tendon Anatomy Attaches muscles to bones Fibril Type I collagen grouped into microfibrils, then subfibrils, then fibrils, surrounded by endotenon Fascicle Fibroblasts and fibrils surrounded by peritenon T endon Groups of fascicles surrounded by epitenon Vascular Tendon Vascular paratenon surrounds tendon to supply vascularity; no sheath Avascular Tendon These tendons are in a sheath, have a vincula to supply vascularity Tendon bone Junction 1. T endon to fibrocartilage 2. Fibrocartilage to calcified fibrocartilage (Sharpey's fibers) 3. Sharpey's fibers to bone. Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier MICROBIOLOGY INFECTION COMMENT Osteomyelitis Bacterial infection of bone or bone marrow. Staph. aureus #1 organism. Hematogenous spread most common. Classified as acute, subacute, or chronic. Pain, swelling, increased WBC, ESR, positive blood cultures. XR shows radiolucencies, +/-sequestrum (dead cortical bone), involucrum (periosteal new bone). Bone scan helps diagnosis. I D abscess/sequestra, IV antibiotics followed by a course of oral antibiotics Septic Joint Infection of joint space (and synovium). Staph. aureus #1 organism. Hematogenous or extension of osteomyelitis common routes. Knee #1, hip #2 most common sites. Painful, warm swollen joint. Requires aspiration/surgical drainage IV antibiotics. Tetanus Neuroparalytic disorder caused from exotoxin from Clostridium tetani Vaccine prophylaxis: T etanus and diphtheria toxoid (Td); T etanus immunoglobulin (TIG) Previously vaccinated (5yrs), clean wound: no treatment Previously vaccinated (5yrs), clean or dirty wound: 0.5mg Td Unknown vaccination status or “dirty” wound: Td and TIG Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier IMAGING STUDY COMMENT X-ray (plain film) Standard study, multiple views needed, shows bones well, but soft tissues poorly. The joint above and below a fracture should always receive plain films. CT Best study for bony anatomy. Soft tissue seen, but not as well as MRI. Often used for comminuted fractures and preoperative planning. MRI Best study for soft tissues including intervertebral discs, ligaments, tendons. Also highly sensitive for osteonecrosis; T1 images weighted for fat (good for normal anatomy), T2 images weighted for water (better for pathology). Also used for preoperative planning Bone scan Radioactive isotope injected into blood. Imaging of the whole body allows visualization of areas of increased uptake. Good for identifying tumor, fractures, infections, and heterotopic bone activity (HO). Arthrography Contrast injected into joint followed by plain films to evaluate capsular integrity (e.g. used for rotator cuff tears) Myelography Contrast injected into epidural space; evaluates disc herniation, cord tumors Discography Contrast injected into nucleus pulposus to evaluate disc degeneration. Not a common procedure. Ultrasound Good for evaluating rotator cuff pathology Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com Thompson: Netter's Concise Atlas of Orthopaedic Anatomy, 1st ed. Copyright © 2001 Saunders, An Imprint of Elsevier ABBREVIA TIONS USED IN THIS BOOK A Abd abduct AC acromioclavicular ACL anterior cruciate ligament ADM abductor digitiminimi AGRAM arthrogram AIIS anterior inferior iliac spine AIN anterior interosseus nerve ALL anterior longitudinal ligament AMBRI atraumatic, multidirectional, bilateral instability ANA antinuclear antibody Ant. anterior AP anteroposterior APB abductor pollicis brevis APC anterior-posterior compression APL abductor pollicis longus ASIS anterior superior iliac spine AVN avascular necrosis B BR brachioradialis C Ca++ ion calcium CBC complete blood cell count CL capitate-lunate joint CMC carpal-metacarpal CPK creatine phosphokinase CRP C-reactive protein C-spine cervical spine CT computed tomography CTL capitotriquetral ligament CTS carpal tunnel syndrome D DDD degenerative disk disease DIO dorsal interossei DIP distal interphalangeal DISI dorsal intercalated segment instability DJD degenerative joint disease DRC dorsal radiocarpal ligament DRUJ distal radioulnar joint DVT deep vein thrombosis E ECRB extensor carpi radialis brevis ECRL extensor carpi radialis longus ECU extensor carpi ulnaris EDC extensor digitorum communis EDL extensor digitorum longus EDM extensor digiti minimi EHL extensor hallucis longus EIP extensor indicis proprius EMG electromyogram EPB extensor pollicis brevis EPL extensor pollicis longus ER external rotation ESR erythrocyte sedimentation rate F FCR flexor carpi radialis FCU flexor carpi ulnaris FDB flexor digitorum brevis FDL flexor digitorum longus FDMB flexor digiti minimi brevis FDP flexor digitorum profundus FDS flexor digitorum superficialis FHB flexor hallucis brevis FHL flexor hallucis longus FPB flexor pollicis brevis FPL flexor pollicis longus Fx fracture G GAG glycosaminoglycans GI gastrointestinal GU genitourinary H HNP herniated nucleus pulposus Hx history I ID incision and drainage IF index finger IJ internal jugular IM intramedullary Inf. inferior IP interphalangeal IR internal rotation ITB iliotibial band IV intravenous L Lat. lateral LBP low back pain LC lateral compression LCL lateral collateral ligament LE low er extremity LFCN lateral femoral cutaneous nerve LH long head LT lunotriquetral M MC metacarpal MCL medial collateral ligament MCP metacarpophalangeal MDI multidirectional instability Med. medial MF middle finger MRI magnetic resonance imaging MT metatarsal MVA motor vehicle accident N N. nerve NCS nerve conduction study NSAID non-steroidal anti-inflammatory drug O OA osteoarthritis OP opponens pollicis muscle ORIF open reduction, internal fixation P PAD palmar adduct PCL posterior cruciate ligament PCP percutaneous pinning PE physical examination PFCN posterior femoral cutaneous nerve PFS patellofemoral syndrome PIN posterior interosseus nerve PIP proximal interphalangeal PL palmaris longus PLC posterolateral corner complex PLL posterior longitudinal ligament PLRI posterolateral rotary instability PMHx past medical history PMRI posterolateral rotary instability PO postoperatively Post. posterior PQ pronator quadratus PSIS posterosuperior iliac spine PT pronator teres PTH parathyroid hormone PVNS pigmented villonodular synovitis Q Q quadriceps R RA rheumatoid arthritis RAD radiation absorbed dose RC rotator cuff RCL radioscaphocapitate ligament RF rheumatoid factor, ring finger RICE rest, ice, compression, and elevation ROM range of motion RSD reflex sympathetic dystrophy RSL radioscapholunate ligament RTL radiolunotriquetral ligament S SC sternoclavicular SCM sternocleidomastoid SF small finger SFA superficial femoral artery SH short head SI sacroiliac SL scapholunate SLAC scapholunate advanced collapse SLAP superior labrum anterior/posterior STT scaphotrapezoid-trapezial Sup. superior Sx symptom T TA tibialis anterior TCL transverse carpal ligament Td tetanus and diphtheria toxoid TFCC triangular fibrocartilage complex TFL tensor fascia lata THA total hip arthroplasty TIG tetanus immunoglobulin TLSO thoracolumbosacral orthosis TP tibialis posterior TTP tenderness to palpation TUBS traumatic, unilateral instability, and Bankart lesion U UE upper extremity UMN upper motor neuron V VIO volar interosseus VISI volar intercalated segment instability VMO vastus medialis obliquus W WB w eight bearing WBC w hite blood cell count X XR x-ray Copyright © 2008 Elsevier Inc. All rights reserved. - www.mdconsult.com
2837	https://maimonidesem.squarespace.com/s/Pediatric-Assessment-Triangle.pdf	Pediatric Assessment Triangle Dieckmann R et al. Pedriatr Emerg Care 2010. PMID 20386420 ER CAST: (Courtesy of Dr. Michelle Reina & Dr. Rob Bryant) The PAT functions as a rapid, initial assessment to determine “sick” or “not sick,” and should be immediately followed by/not delay the ABCDEs. It can be utilized for serial assessment of patients to track response to therapy. Appearance: The “Tickles” (TICLS) Mnemonic Characteristic Normal features T one Move spontaneously, resists examination, sits or stands (age appropriate) I nteractiveness Appears alert/engaged with clinician or caregiver, interacts well with people/environment, reaches for objects C onsolability Stops crying with holding/comforting by caregiver, has differential response to caregiver vs. examiner L ook/gaze Makes eye contact with clinician, tracks visually S peech/cry Uses age-appropriate speech Work of breathing Characteristic Abnormal features Abnormal airway sounds Snoring, muffled/hoarse speech, stridor, grunting, wheezing Abnormal positioning Sniffing position, tripoding, prefers seated posture Retractions Supraclavicular, intercostal, or substernal, head bobbing (infants) Flaring Flaring of the nares on inspiration Circulation to skin Characteristic Abnormal features Pallor White/pale skin or mucous membranes Mottling Patchy skin discoloration due to variable vasoconstriction Cyanosis Bluish discoloration of skin/mucous membranes Relationship of the PAT components to physiological categories and management priorities Presentation Appearance Work of breathing Circulat’n to skin Management priorities Stable Normal Normal Normal Specific therapy based on possible etiologies Respiratory distress Normal Abnormal Normal Position of comfort, O2/suction, specific therapy (e.g. albuterol, diphenhydramine, epinephrine), labs/x-rays Respiratory failure Abnormal Abnormal Normal or Abnormal Position head/open airway, BVM, FB removal, advanced airway, labs/x-rays Shock (compensated) Normal Normal Abnormal O2, peripheral IV, fluid resuscitation, specific therapy based on etiology (antibiotics, surgery, antidysrhythmics), labs/x-rays Shock (decompensated/ hypotensive) Abnormal Normal or Abnormal Abnormal O2, vascular access, fluid resuscitation, specific therapy based on etiology (antibiotics, vasopressors, blood products, surgery, antidysrhythmics, cardioversion), labs/x-rays CNS/Metabolic dysfunction Abnormal Normal Normal O2, POC glucose, consider other etiologies, labs/x-rays Cardiopulmonary failure/arrest Abnormal Abnormal Abnormal Position head/open airway, BMV with 100% O2, CPR, specific therapy based on etiology (defibrillation, epinephrine, amiodarone), labs/x-rays
2838	https://math.stackexchange.com/questions/3073896/expand-xz-frac11az-into-a-causal-sequence	convergence divergence - Expand $X(z) = \frac{1}{1+az}$ into a causal sequence - Mathematics Stack Exchange Join Mathematics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 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Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Expand X(z)=1 1+a z X(z)=1 1+a z into a causal sequence Ask Question Asked 6 years, 8 months ago Modified6 years, 8 months ago Viewed 159 times This question shows research effort; it is useful and clear 0 Save this question. Show activity on this post. For a HW problem, I'm told to expand 1 1+a z 1 1+a z into a causal and noncausal sequence. I found the noncausal sequence by long division (the result is 1−a z+(a z)2−…1−a z+(a z)2−…) and found the region of convergence, but I'm not sure how to find the causal sequence. The full question is : sequences-and-series convergence-divergence z-transform Share Share a link to this question Copy linkCC BY-SA 4.0 Cite Follow Follow this question to receive notifications edited Jan 14, 2019 at 23:46 Arnaud D. 21.5k 6 6 gold badges 40 40 silver badges 59 59 bronze badges asked Jan 14, 2019 at 23:32 Kevin SilkenKevin Silken 3 1 1 bronze badge Add a comment\| 2 Answers 2 Sorted by: Reset to default This answer is useful 1 Save this answer. Show activity on this post. If 1 1+a z=∑b n z−n 1 1+a z=∑b n z−n, then plugging in w=1/z w=1/z, we get 1 1+a/w=∑b n w n 1 1+a/w=∑b n w n. So let us rewrite 1 1+a/w=1 w+a w=w a+w=(w/a)1 1−(−w/a)=(w/a)∑(−w/a)n 1 1+a/w=1 w+a w=w a+w=(w/a)1 1−(−w/a)=(w/a)∑(−w/a)n. Now just expand and plug back in for z z. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications edited Jan 15, 2019 at 1:54 answered Jan 15, 2019 at 0:20 AaronAaron 25.7k 2 2 gold badges 53 53 silver badges 76 76 bronze badges Add a comment\| This answer is useful 0 Save this answer. Show activity on this post. HINT You have 1 1+a z=1 a z[1 1+1/(a z)]1 1+a z=1 a z[1 1+1/(a z)] Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications answered Jan 15, 2019 at 0:16 gt6989bgt6989b 55k 3 3 gold badges 40 40 silver badges 75 75 bronze badges 2 So do partial fraction decomposition on this? And it would chain into a series of these terms?Kevin Silken –Kevin Silken 2019-01-15 00:19:15 +00:00 Commented Jan 15, 2019 at 0:19 @KevinSilken no, the bracket is a geometric series, like the one you did, except in 1/z 1/z...gt6989b –gt6989b 2019-01-15 00:20:54 +00:00 Commented Jan 15, 2019 at 0:20 Add a comment\| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions sequences-and-series convergence-divergence z-transform See similar questions with these tags. 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2839	https://www.quora.com/What-is-the-correct-way-of-using-gregarious	What is the correct way of using 'gregarious'? - Quora Something went wrong. Wait a moment and try again. Try again Skip to content Skip to search Sign In English (language) Vocabulary Meaning Grammar Descriptive Adjective Usage English Language and Gram... Diction and Word Usage English Grammar Vocabulary Usage 5 What is the correct way of using 'gregarious'? All related (41) Sort Recommended Assistant Bot · Sep 6 "Gregarious" describes someone (or occasionally an animal) who seeks and enjoys the company of others; sociable, outgoing, and fond of groups. It primarily functions as an adjective. Core senses and examples Sociable, enjoys company: "She’s gregarious and makes friends easily at networking events." Prefers group living (biology/ecology): "Many bird species are gregarious outside the breeding season." Figurative use for environments or behaviors that encourage sociability: "The cafe’s gregarious atmosphere drew freelancers together." Grammatical notes Collocations: commonly used with people, personal Continue Reading "Gregarious" describes someone (or occasionally an animal) who seeks and enjoys the company of others; sociable, outgoing, and fond of groups. It primarily functions as an adjective. Core senses and examples Sociable, enjoys company: "She’s gregarious and makes friends easily at networking events." Prefers group living (biology/ecology): "Many bird species are gregarious outside the breeding season." Figurative use for environments or behaviors that encourage sociability: "The cafe’s gregarious atmosphere drew freelancers together." Grammatical notes Collocations: commonly used with people, personalities, animals, behavior, nature, temperament, or atmosphere. Correct: "a gregarious host," "a gregarious child," "gregarious animals" Awkward: "gregarious idea" (ideas are not social) Do not use as a verb or noun. The noun form is gregariousness (or gregariousity is incorrect). Correct: "His gregariousness made him popular." Register: neutral to slightly formal; suits both conversational and written contexts. Contrast with similar words Sociable vs. gregarious: largely interchangeable for people, but "gregarious" implies a stronger, innate preference for groups. Outgoing: emphasizes extroversion and ease in interactions; "gregarious" emphasizes enjoyment of company and group settings. Common pitfalls Avoid using about solitary people or non-social contexts. Incorrect: "She’s gregarious when alone." (contradictory) Avoid treating it as meaning "friendly" in every context; someone can be friendly but not gregarious. Quick usage templates "He is gregarious." (simple predicative) "A gregarious neighbor" (attributive) "Gregarious by nature" (descriptive phrase) "The species is gregarious" (biological statement) Summary: Use "gregarious" to describe persons or animals that prefer and thrive in social groups; pair it with nouns like person, personality, animal, behavior, or atmosphere, and use the noun form gregariousness when needed. Upvote · Sponsored by RHUZZALS You simply can't remain incifferent to the charm of this clothes! Refresh your look and stay in fashion! Shop Now 99 21 Related questions More answers below What is the opposite of “gregarious”? Is 'gregarious' a positive word? How can "gregarious" be used in a sentence? What is the correct way to use the word "gangster"? What is the use of "Meself"? Souvik Knowledge Mentor & Local Guide at FREELANCE (2016–present) · Author has 764 answers and 2.6M answer views ·6y Gregarious -'Adjective' - Adjective; (Ref to a person) Describing one who enjoys being in crowds and socializing i.e.(of a person) fond of company; sociable. (Zoology) Of animals that travel in herds or packs or loosely organised communities Plants growing in open clusters or in pure associations. Etymology; 1660s -from Latin gregarius "pertaining to a flock; of the herd, of the common sort, common," Continue Reading Gregarious -'Adjective' - Adjective; (Ref to a person) Describing one who enjoys being in crowds and socializing i.e.(of a person) fond of company; sociable. (Zoology) Of animals that travel in herds or packs or loosely organised communities Plants growing in open clusters or in pure associations. Etymology; 1660s -from Latin gregarius "pertaining to a flock; of the herd, of the common sort, common," from grex "flock, herd," from PIE reduplicated form of root ger- "to gather." Of persons, "sociable," first recorded 1789. Other forms: Gregariously; gregariousness. Use in a sentence: He's impassioned and animated, perceptive and entertaining, gregarious and charismatic. Like societies of many other gregarious mammals, social groups of spotted hyenas are structured by linear dominance... Upvote · 9 1 Ramesh Chandra Jha Professor in Department of English at MLSM College Darbhanga (1985–present) · Author has 4.3K answers and 8.3M answer views ·6y Gregarious means ' ( of animals ) tending to form a group with others of the same species . E.g. Elephants are gregarious animals . It also means ' ( of persons ) instinctively or temperamentally seeking and enjoying the company of others. Its signification can be expressed in different words as follows : Friendly, sociable , convivial , companionable , affable , amiable etc. Upvote · 9 1 Ipsita Mallick B.A. B.Ed in English Literature&English Literature, Regional Institute of Education, Bhubaneswar (Graduated 2019) ·6y Gregarious is an adjective that means sociable or company-loving. People who are gregarious are very friendly and outgoing, and they love being around people. So you may use gregarious like you use adjectives ‘good’ or ‘bad’ or ‘friendly’ as in : Sam was a gregarious man. This may be better understood when you give it a context, as in here: Sam was a gregarious man, almost never leaving the club. No wonder his family business in his unsure hands was nosediving. Animals can be gregarious too. So can be plants growing closely together. Animals that you see moving in herds or packs or troops may be Continue Reading Gregarious is an adjective that means sociable or company-loving. People who are gregarious are very friendly and outgoing, and they love being around people. So you may use gregarious like you use adjectives ‘good’ or ‘bad’ or ‘friendly’ as in : Sam was a gregarious man. This may be better understood when you give it a context, as in here: Sam was a gregarious man, almost never leaving the club. No wonder his family business in his unsure hands was nosediving. Animals can be gregarious too. So can be plants growing closely together. Animals that you see moving in herds or packs or troops may be called gregarious. The fish are a gregarious species. That there is both a fact and an example for you! Your response is private Was this worth your time? This helps us sort answers on the page. Absolutely not Definitely yes Upvote · Sponsored by Wordeep Proofreading Grammar correction solution using artificial intelligence. Have a look at our real-time AI based proofreading and grammar correction service. You will be amazed! Learn More 99 95 Related questions More answers below Where does the word gregarious originate from? Do you know the meaning of the word of the day: gregarious? What is the correct way in using the word "because"? What is the correct way of using the word 'officiate'? How do I properly use the word 'especially'? Sadasiva S Author has 3.9K answers and 1.9M answer views ·5y gregarious - adjective. fond of the company of others; sociable. living in flocks or herds, as animals. Botany: growing in open clusters or colonies; not matted together. pertaining to a flock or crowd. friendly; affable; clubby; companiable ; cordial; convivial. Someone who is gregarious enjoys being with other people. She is such a gregarious and outgoing person. Gregarious animals or birds normally live in large groups. Snow gees are very gregarious birds. Upvote · Sidhika Rathore Former Graduate · Author has 89 answers and 32.2K answer views ·5y gregarious means a person who is sociable or company loving you can use this word as a positive or a negative adjective Upvote · 9 1 Sponsored by Digital News Wire What Everyone Should Know in the Digital Age. Understanding the Evolving Concept of Financial Literacy in a Technology-Driven Economy. Learn More Raja Sarkar word enthusiast · Author has 447 answers and 617.7K answer views ·6y You see, the word “gregarious” has a very specific meaning. Derived from the Latin word “gregarius”, it means to show a tendency of flocking or associating with others of one’s kind. Now check out the below sentence as it’s the right implementation of this word: These victims might be lonely or distanced from their family, making it easy for a gregarious criminal to charm them and trick them into handing them a personal check. Upvote · Avinash Chandra Former Retired Section Officer · Author has 194 answers and 107.7K answer views ·5y Dear brother I am afraid, your would be wife will be annoyed with your gregarious company. Gregarious disposition is helpful in increasing marketing and trade,to make wealth. Upvote · Sponsored by JetBrains Become More Productive in Jakarta EE. Try IntelliJ IDEA, a JetBrains IDE, and enjoy productive Java Enterprise development! Download 999 649 Ragini Knows English · Author has 264 answers and 1.1M answer views ·5y Thanks for the A2A. His gregarious nature made him a popular person in the neighbourhood. To become a salesman, you should be gregarious and active. My brother is a gregarious person and there is not a single thing which I don’t share with him. Upvote · 9 1 Related questions What is the opposite of “gregarious”? Is 'gregarious' a positive word? How can "gregarious" be used in a sentence? What is the correct way to use the word "gangster"? What is the use of "Meself"? Where does the word gregarious originate from? Do you know the meaning of the word of the day: gregarious? What is the correct way in using the word "because"? What is the correct way of using the word 'officiate'? How do I properly use the word 'especially'? What word best describes someone who doesn't care about anything? Is the word 'gregarious' an adjective? What is the use of Wikoryl? 'Anyway' or 'anyways', which is the right form? What is the correct way to use phrases? Related questions What is the opposite of “gregarious”? Is 'gregarious' a positive word? How can "gregarious" be used in a sentence? What is the correct way to use the word "gangster"? What is the use of "Meself"? Where does the word gregarious originate from? Advertisement About · Careers · Privacy · Terms · Contact · Languages · Your Ad Choices · Press · © Quora, Inc. 2025
2840	https://en.wikipedia.org/wiki/Square_pyramid	Jump to content Search Contents (Top) 1 Special cases 1.1 Right square pyramid 1.2 Equilateral square pyramid 2 Applications 3 See also 4 References 4.1 Notes 4.2 Works cited 5 External links Square pyramid العربية Български Català Cymraeg Español Esperanto Euskara فارسی Français 한국어 Bahasa Indonesia Italiano Македонски Nederlands 日本語 Norsk bokmål Português Romnă Русский Slovenščina தமிழ் ไทย Türkçe Українська 中文 Edit links Article Talk Read Edit View history Tools Actions Read Edit View history General What links here Related changes Upload file Permanent link Page information Cite this page Get shortened URL Download QR code Print/export Download as PDF Printable version In other projects Wikimedia Commons Wikidata item Appearance From Wikipedia, the free encyclopedia Pyramid with a square base \| Square pyramid \| \| Type \| Pyramid,JohnsonJ92 – J1 – J2 \| \| Faces \| 4 triangles1 square \| \| Edges \| 8 \| \| Vertices \| 5 \| \| Vertex configuration \| \| \| Symmetry group \| \| \| Volume \| \| \| Dihedral angle (degrees) \| Equilateral square pyramid: triangle-to-triangle: 109.47° square-to-triangle: 54.74° \| \| Dual polyhedron \| self-dual \| \| Properties \| convex,elementary (equilateral square pyramid) \| \| Net \| In geometry, a square pyramid is a pyramid with a square base and four triangles, having a total of five faces. If the apex of the pyramid is directly above the center of the square, it is a right square pyramid with four isosceles triangles; otherwise, it is an oblique square pyramid. When all of the pyramid's edges are equal in length, its triangles are all equilateral and it is called an equilateral square pyramid, an example of a Johnson solid. Square pyramids have appeared throughout the history of architecture, with examples being Egyptian pyramids and many other similar buildings. They also occur in chemistry in square pyramidal molecular structures. Square pyramids are often used in the construction of other polyhedra. Many mathematicians in ancient times discovered the formula for the volume of a square pyramid with different approaches. Special cases [edit] Right square pyramid [edit] A square pyramid has five vertices, eight edges, and five faces. One face, called the base of the pyramid, is a square; the four other faces are triangles. Four of the edges make up the square by connecting its four vertices. The other four edges are known as the lateral edges of the pyramid; they meet at the fifth vertex, called the apex.. Otherwise, the pyramid has two or more non-isosceles triangular faces and is called an oblique square pyramid. The slant height of a right square pyramid is defined as the height of one of its isosceles triangles. It can be obtained via the Pythagorean theorem: where is the length of the triangle's base, also one of the square's edges, and is the length of the triangle's legs, which are lateral edges of the pyramid. The height of a right square pyramid can be similarly obtained, with a substitution of the slant height formula giving: A polyhedron's surface area is the sum of the areas of its faces. The surface area of a right square pyramid can be expressed as , where and are the areas of one of its triangles and its base, respectively. The area of a triangle is half of the product of its base and side, with the area of a square being the length of the side squared. This gives the expression: In general, the volume of a pyramid is equal to one-third of the area of its base multiplied by its height. Expressed in a formula for a square pyramid, this is: Many mathematicians have discovered the formula for calculating the volume of a square pyramid in ancient times. In the Moscow Mathematical Papyrus, Egyptian mathematicians demonstrated knowledge of the formula for calculating the volume of a truncated square pyramid, suggesting that they were also acquainted with the volume of a square pyramid, but it is unknown how the formula was derived. Beyond the discovery of the volume of a square pyramid, the problem of finding the slope and height of a square pyramid can be found in the Rhind Mathematical Papyrus. The Babylonian mathematicians also considered the volume of a frustum, but gave an incorrect formula for it. One Chinese mathematician Liu Hui also discovered the volume by the method of dissecting a rectangular solid into pieces. Like other right pyramids with a regular polygon as a base, a right square pyramid has pyramidal symmetry, the symmetry of cyclic group : the pyramid is left invariant by rotations of one-, two-, and three-quarters of a full turn around its axis of symmetry, the line connecting the apex to the center of the base; and is also mirror symmetric relative to any perpendicular plane passing through a bisector of the base. It can be represented as the wheel graph , meaning its skeleton can be interpreted as a square in which its four vertices connects a vertex in the center called the universal vertex. It is self-dual, meaning its dual polyhedron is the square pyramid itself. Equilateral square pyramid [edit] If all triangular edges are of equal length, the four triangles are equilateral, and the pyramid's faces are all regular polygons, it is an equilateral square pyramid. The dihedral angles between adjacent triangular faces are , and that between the base and each triangular face being half of that, . A convex polyhedron in which all of the faces are regular polygons is called a Johnson solid. The equilateral square pyramid is among them, enumerated as the first Johnson solid . Because its edges are all equal in length (that is, ), its slant, height, surface area, and volume can be derived by substituting the formulas of a right square pyramid: An equilateral square pyramid is an elementary polyhedron. This means it cannot be separated by a plane to create two small convex polyhedrons with regular faces. Applications [edit] The Egyptian pyramids are examples of square pyramidal buildings in architecture. One of the Mesoamerican pyramids, a similar building to the Egyptian, has flat tops and stairs at the faces In architecture, the pyramids built in ancient Egypt are examples of buildings shaped like square pyramids. Pyramidologists have put forward various suggestions for the design of the Great Pyramid of Giza, including a theory based on the Kepler triangle and the golden ratio. However, modern scholars favor descriptions using integer ratios, as being more consistent with the knowledge of Egyptian mathematics and proportion. The Mesoamerican pyramids are also ancient pyramidal buildings similar to the Egyptian; they differ in having flat tops and stairs ascending their faces. Modern buildings whose designs imitate the Egyptian pyramids include the Louvre Pyramid and the casino hotel Luxor Las Vegas. In stereochemistry, an atom cluster can have a square pyramidal geometry. A square pyramidal molecule has a main-group element with one active lone pair, which can be described by a model that predicts the geometry of molecules known as VSEPR theory. Examples of molecules with this structure include chlorine pentafluoride, bromine pentafluoride, and iodine pentafluoride. The base of a square pyramid can be attached to a square face of another polyhedron to construct new polyhedra, an example of augmentation. For example, a tetrakis hexahedron can be constructed by attaching the base of an equilateral square pyramid onto each face of a cube. Attaching prisms or antiprisms to pyramids is known as elongation or gyroelongation, respectively. Some of the other Johnson solids can be constructed by either augmenting square pyramids or augmenting other shapes with square pyramids: elongated square pyramid , gyroelongated square pyramid , elongated square bipyramid , gyroelongated square bipyramid , augmented triangular prism , biaugmented triangular prism , triaugmented triangular prism , augmented pentagonal prism , biaugmented pentagonal prism , augmented hexagonal prism , parabiaugmented hexagonal prism , metabiaugmented hexagonal prism , triaugmented hexagonal prism , and augmented sphenocorona . See also [edit] Regular octahedron or square bipyramid, a polyhedron constructed by attaching two square pyramids base-to-base; Square pyramidal number, a natural number that counts the number of stacked spheres in a square pyramid. References [edit] Notes [edit] ^ a b c d Johnson (1966). ^ Clissold (2020), p. 180. ^ O'Keeffe & Hyde (2020), p. 141; Smith (2000), p. 98. ^ Freitag (2014), p. 598. ^ Larcombe (1929), p. 177; Perry & Perry (1981), pp. 145–146. ^ Larcombe (1929), p. 177. ^ Freitag (2014), p. 798. ^ Alexander & Koeberlin (2014), p. 403. ^ Larcombe (1929), p. 178. ^ Cromwell (1997), pp. 20–22. ^ Eves (1997), p. 2. ^ Wagner (1979). ^ Pisanski & Servatius (2013), p. 21. ^ Wohlleben (2019), p. 485–486. ^ Hocevar (1903), p. 44. ^ Uehara (2020), p. 62. ^ Simonson (2011), p. 123; Berman (1971), see table IV, line 21. ^ Hartshorne (2000), p. 464; Johnson (1966). ^ Kinsey, Moore & Prassidis (2011), p. 371. ^ Herz-Fischler (2000) surveys many alternative theories for this pyramid's shape. See Chapter 11, "Kepler triangle theory", pp. 80–91, for material specific to the Kepler triangle, and p. 166 for the conclusion that the Kepler triangle theory can be eliminated by the principle that "A theory must correspond to a level of mathematics consistent with what was known to the ancient Egyptians." See note 3, p. 229, for the history of Kepler's work with this triangle. See Rossi (2004), pp. 67–68, quoting that "there is no direct evidence in any ancient Egyptian written mathematical source of any arithmetic calculation or geometrical construction which could be classified as the Golden Section ... convergence to , and itself as a number, do not fit with the extant Middle Kingdom mathematical sources"; see also extensive discussion of multiple alternative theories for the shape of the pyramid and other Egyptian architecture, pp. 7–56. See also Rossi & Tout (2002) and Markowsky (1992). ^ Feder (2010), p. 34; Takacs & Cline (2015), p. 16. ^ Jarvis & Naested (2012), p. 172; Simonson (2011), p. 122. ^ Petrucci, Harwood & Herring (2002), p. 414. ^ Emeléus (1969), p. 13. ^ Demey & Smessaert (2017). ^ Slobodan, Obradović & Ðukanović (2015). ^ Rajwade (2001), pp. 84–89. See Table 12.3, where denotes the -sided prism and denotes the -sided antiprism. Works cited [edit] Alexander, Daniel C.; Koeberlin, Geralyn M. (2014). Elementary Geometry for College Students (6th ed.). Cengage Learning. ISBN 978-1-285-19569-8. Berman, Martin (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute. 291 (5): 329–352. doi:10.1016/0016-0032(71)90071-8. MR 0290245. Clissold, Caroline (2020). Maths 5–11: A Guide for Teachers. Taylor & Francis. ISBN 978-0-429-26907-3. Cromwell, Peter R. (1997). Polyhedra. Cambridge University Press. ISBN 978-0-521-55432-9. Demey, Lorenz; Smessaert, Hans (2017). "Logical and Geometrical Distance in Polyhedral Aristotelian Diagrams in Knowledge Representation". Symmetry. 9 (10): 204. Bibcode:2017Symm....9..204D. doi:10.3390/sym9100204. Emeléus, H. J. (1969). The Chemistry of Fluorine and Its Compounds. Academic Press. ISBN 978-1-4832-7304-4. Eves, Howard (1997). Foundations and Fundamental Concepts of Mathematics (3rd ed.). Dover Publications. ISBN 978-0-486-69609-6. Feder, Kenneth L. (2010). Encyclopedia of Dubious Archaeology: From Atlantis to the Walam Olum: From Atlantis to the Walam Olum. ABC-CLIO. ISBN 978-0-313-37919-2. Freitag, Mark A. (2014). Mathematics for Elementary School Teachers: A Process Approach. Brooks/Cole. ISBN 978-0-618-61008-2. Hartshorne, Robin (2000). Geometry: Euclid and Beyond. Undergraduate Texts in Mathematics. Springer-Verlag. ISBN 9780387986500. Herz-Fischler, Roger (2000). The Shape of the Great Pyramid. Wilfrid Laurier University Press. ISBN 0-88920-324-5. Hocevar, Franx (1903). Solid Geometry. A. & C. Black. Jarvis, Daniel; Naested, Irene (2012). Exploring the Math and Art Connection: Teaching and Learning Between the Lines. Brush Education. ISBN 978-1-55059-398-3. Johnson, Norman W. (1966). "Convex polyhedra with regular faces". Canadian Journal of Mathematics. 18: 169–200. doi:10.4153/cjm-1966-021-8. MR 0185507. S2CID 122006114. Zbl 0132.14603. Kinsey, L. Christine; Moore, Teresa E.; Prassidis, Efstratios (2011). Geometry and Symmetry. John Wiley & Sons. ISBN 978-0-470-49949-8. Larcombe, H. J. (1929). Cambridge Intermediate Mathematics: Geometry Part II. Cambridge University Press. Markowsky, George (1992). "Misconceptions about the Golden Ratio" (PDF). The College Mathematics Journal. 23 (1). Mathematical Association of America: 2–19. doi:10.2307/2686193. JSTOR 2686193. Retrieved 29 June 2012. O'Keeffe, Michael; Hyde, Bruce G. (2020). Crystal Structures: Patterns and Symmetry. Dover Publications. ISBN 978-0-486-83654-6. Perry, O. W.; Perry, J. (1981). Mathematics. Springer. doi:10.1007/978-1-349-05230-1. ISBN 978-1-349-05230-1. Petrucci, Ralph H.; Harwood, William S.; Herring, F. Geoffrey (2002). General Chemistry: Principles and Modern Applications. Vol. 1. Prentice Hall. ISBN 978-0-13-014329-7. Pisanski, Tomaž; Servatius, Brigitte (2013). Configuration from a Graphical Viewpoint. Springer. doi:10.1007/978-0-8176-8364-1. ISBN 978-0-8176-8363-4. Rajwade, A. R. (2001). Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem. Texts and Readings in Mathematics. Hindustan Book Agency. doi:10.1007/978-93-86279-06-4. ISBN 978-93-86279-06-4. Rossi, Corinna (2004). Architecture and Mathematics in Ancient Egypt. Cambridge University Press. pp. 67–68. Rossi, Corinna; Tout, Christopher A. (2002). "Were the Fibonacci series and the Golden Section known in ancient Egypt?". Historia Mathematica. 29 (2): 101–113. doi:10.1006/hmat.2001.2334. hdl:11311/997099. Simonson, Shai (2011). Rediscovering Mathematics: You Do the Math. Mathematical Association of America. ISBN 978-0-88385-912-4. Slobodan, Mišić; Obradović, Marija; Ðukanović, Gordana (2015). "Composite Concave Cupolae as Geometric and Architectural Forms" (PDF). Journal for Geometry and Graphics. 19 (1): 79–91. Smith, James T. (2000). Methods of Geometry. John Wiley & Sons. ISBN 0-471-25183-6. Takacs, Sarolta Anna; Cline, Eric H. (2015). The Ancient World. Routledge. p. 16. ISBN 978-1-317-45839-5. Uehara, Ryuhei (2020). Introduction to Computational Origami: The World of New Computational Geometry. Springer. doi:10.1007/978-981-15-4470-5. ISBN 978-981-15-4470-5. S2CID 220150682. Wagner, Donald Blackmore (1979). "An early Chinese derivation of the volume of a pyramid: Liu Hui, third century A.D.". Historia Mathematica. 6 (2): 164–188. doi:10.1016/0315-0860(79)90076-4. Wohlleben, Eva (2019). "Duality in Non-Polyhedral Bodies Part I: Polyliner". In Cocchiarella, Luigi (ed.). ICGG 2018 – Proceedings of the 18th International Conference on Geometry and Graphics: 40th Anniversary – Milan, Italy, August 3–7, 2018. International Conference on Geometry and Graphics. Springer. doi:10.1007/978-3-319-95588-9. ISBN 978-3-319-95588-9. External links [edit] Wikimedia Commons has media related to Square pyramids. Weisstein, Eric W., "Square pyramid" ("Johnson solid") at MathWorld. Square Pyramid – Interactive Polyhedron Model Virtual Reality Polyhedra georgehart.com: The Encyclopedia of Polyhedra (VRML model Archived 7 October 2023 at the Wayback Machine) \| v t e Johnson solids \| \| Pyramids, cupolae and rotundae \| square pyramid pentagonal pyramid triangular cupola square cupola pentagonal cupola pentagonal rotunda \| \| Modified pyramids \| elongated triangular pyramid elongated square pyramid elongated pentagonal pyramid gyroelongated square pyramid gyroelongated pentagonal pyramid triangular bipyramid pentagonal bipyramid elongated triangular bipyramid elongated square bipyramid elongated pentagonal bipyramid gyroelongated square bipyramid \| \| Modified cupolae and rotundae \| elongated triangular cupola elongated square cupola elongated pentagonal cupola elongated pentagonal rotunda gyroelongated triangular cupola gyroelongated square cupola gyroelongated pentagonal cupola gyroelongated pentagonal rotunda gyrobifastigium triangular orthobicupola square orthobicupola square gyrobicupola pentagonal orthobicupola pentagonal gyrobicupola pentagonal orthocupolarotunda pentagonal gyrocupolarotunda pentagonal orthobirotunda elongated triangular orthobicupola elongated triangular gyrobicupola elongated square gyrobicupola elongated pentagonal orthobicupola elongated pentagonal gyrobicupola elongated pentagonal orthocupolarotunda elongated pentagonal gyrocupolarotunda elongated pentagonal orthobirotunda elongated pentagonal gyrobirotunda gyroelongated triangular bicupola gyroelongated square bicupola gyroelongated pentagonal bicupola gyroelongated pentagonal cupolarotunda gyroelongated pentagonal birotunda \| \| Augmented prisms \| augmented triangular prism biaugmented triangular prism triaugmented triangular prism augmented pentagonal prism biaugmented pentagonal prism augmented hexagonal prism parabiaugmented hexagonal prism metabiaugmented hexagonal prism triaugmented hexagonal prism \| \| Modified Platonic solids \| augmented dodecahedron parabiaugmented dodecahedron metabiaugmented dodecahedron triaugmented dodecahedron metabidiminished icosahedron tridiminished icosahedron augmented tridiminished icosahedron \| \| Modified Archimedean solids \| augmented truncated tetrahedron augmented truncated cube biaugmented truncated cube augmented truncated dodecahedron parabiaugmented truncated dodecahedron metabiaugmented truncated dodecahedron triaugmented truncated dodecahedron gyrate rhombicosidodecahedron parabigyrate rhombicosidodecahedron metabigyrate rhombicosidodecahedron trigyrate rhombicosidodecahedron diminished rhombicosidodecahedron paragyrate diminished rhombicosidodecahedron metagyrate diminished rhombicosidodecahedron bigyrate diminished rhombicosidodecahedron parabidiminished rhombicosidodecahedron metabidiminished rhombicosidodecahedron gyrate bidiminished rhombicosidodecahedron tridiminished rhombicosidodecahedron \| \| Other elementary solids \| snub disphenoid snub square antiprism sphenocorona augmented sphenocorona sphenomegacorona hebesphenomegacorona disphenocingulum bilunabirotunda triangular hebesphenorotunda \| \| (See also List of Johnson solids, a sortable table) \| Retrieved from " Categories: Elementary polyhedron Johnson solids Prismatoid polyhedra Pyramids (geometry) Self-dual polyhedra Hidden categories: Articles with short description Short description is different from Wikidata Good articles Use dmy dates from January 2024 Pages using multiple image with auto scaled images Commons category link from Wikidata Webarchive template wayback links Square pyramid Add topic
2841	https://www.vedantu.com/question-answer/joules-are-in-500-calories-class-11-physics-cbse-60139c43785e5d45c1c0f0cc	Talk to our experts 1800-120-456-456 How many joules are in $500$ calories? Repeaters Course for NEET 2022 - 23 © 2025.Vedantu.com. All rights reserved
2842	https://www.quora.com/How-can-I-prove-the-area-of-a-sphere-as-4pi-r2	How to prove the area of a sphere as 4pi r2 - Quora Something went wrong. Wait a moment and try again. Try again Skip to content Skip to search Sign In Mathematics Surface Area of Sphere Mathematics Homework Ques... Proofs (mathematics) Spatial Geometry Spheres Area (mathematics) Surface Area and Volume Maths Geometry 5 How can I prove the area of a sphere as 4pi r2? 18 Answers Sort Recommended Ron Davis I earn my living with mathematics. · Author has 6.7K answers and 18.8M answer views ·4y What is a good proof for showing why the surface area of a sphere is 4pir^2? Originally Answered: What is a good proof for showing why the surface area of a sphere is 4pir^2? · An interesting feature of this question is that the questioner doesn’t specify a criterion of “good”ness for a proof. There is already Lance Everett’s answer, in the form of a very thoroughly explained double integral, so it is “good” mathematically (indeed, better than I could have done it). As an interesting challenge, however, I here derive the surface area of a sphere without recourse to any integral, by geometry only. The geometrical derivation is not as good in a mathematical sense, because it lacks mathematical rigor; it depends on geometrical intuition. But, to some readers’ taste, its Continue Reading An interesting feature of this question is that the questioner doesn’t specify a criterion of “good”ness for a proof. There is already Lance Everett’s answer, in the form of a very thoroughly explained double integral, so it is “good” mathematically (indeed, better than I could have done it). As an interesting challenge, however, I here derive the surface area of a sphere without recourse to any integral, by geometry only. The geometrical derivation is not as good in a mathematical sense, because it lacks mathematical rigor; it depends on geometrical intuition. But, to some readers’ taste, its intuitiveness may be another form of “good”ness. Of course, the principle of integration is buried in my answer, because I work on a figure defined by straight lines, and note what happens when the number of lines increases indefinitely so that the figure becomes indefinitely close to a smooth sphere. However, the answer is intuitively clear from geometry alone, without knowledge of integrals, if one can visualize the odd shape I use, and if one accepts that phrase “indefinitely close” without a mathematically rigorous definition. To start with, we choose any two antipodal points on a sphere, and define them to be the poles for a notional set of lines on the sphere that denote what we call “latitude” and “longitude” on Earth’s surface. Figure 1: Sphere with Latitude and Longitude Lines Image Credit: Earth planet globe grid of meridians and parallels, or latitude.. The part of the surface of a sphere between any two lines of longitude is called a “gore”. We choose some lines of longitude to divide the sphere into gores. Figure 2 shows the sphere as seen from above one pole, and selected longitudes are shown as dashed lines. Figure 2: Sphere Divided into Gores, and Derived Flattened Gores Image Credit: Finding the interior angles of an irregular polygon inscribed on a circle Eventually, we will choose an indefinitely large number of gores, but the following steps are easier to visualize with the small number of random gores shown. The gores thus defined have bulgey surfaces, which cannot be flattened without stretching or compression, because their surfaces are parts of the surface of the original sphere. To distinguish these gores from related gores to be defined presently, we call them “curved gores”. Because there is no geometrical way to derive the areas of these bulgey surfaces, we derive from these gores “flattened gores”, so called because each such gore has a surface that can be deformed into a plane surface purely by bending, without any stretching or compression of any part of the surface. These dome tents Figure 3: Tents Shaped Like the Upper Parts of Spheres with Flattened Gores Image Credit: Outbound 5-Person 3-Season Instant Pop-Up Dome Tent with Carry Bag and Rainfly - Red and Greenhouse SUNBUBBLE Large 11,5', Clear show the general idea. The supporting rods have curved paths similar to lines of longitude near the poles. Each wall is made from flat fabric; it bends to meet the curved supporting rods, but one could gauge the area of each wall by cutting it out and laying it on flat ground, without deforming the cloth other than flexing it, without area-changing stretching or shrinking. (The word “gore”, as used in sewing, refers specifically to just such a piece of fabric.) To form such cooperative gores on a sphere, we suppose that we slice peels from the sphere, so that any cross section of the sphere looks like the inscribed buff-colored polygon shown in Figure 2, which is determined by the selected lines of longitude. Figure 2 can be considered to be a view of a cross section of the altered sphere, along any plane perpendicular to the axial diameter from pole to pole (although the dashed lines then do not denote any features visible on the cross section). If the cross section is through the equatorial plane, then the diameter represented in Figure 2 is the same as the diameter represented in Figure 1. As the cross section is taken closer to either pole, the diameter represented in Figure 2 becomes smaller, until it degenerates to a point at either pole. Other than this change of scale, the cross section has the same shape for any position along the polar axis. The selected longitudinal lines are the only parts of the original surface that remain. The surface of the thus-peeled sphere is somewhat like the surface of a spherical potato after it has been peeled with a straight-bladed knife. However, the operation considered here would leave the full thickness of peel along each selected line of longitude. (More precisely, the surfaces between selected longitudes coincide with the set of straight lines that run from each selected longitude line to the adjacent, selected longitude line, between pairs of points on the two longitude lines that are at the same latitude.) The next step is that we increase the number of gores. The insertion of new gores does not need to be regular, but the maximum width of any gore must decrease to an indefinitely narrow width as gores are added. The following are the results of this operation: The amount of the sphere that is sliced off decreases, getting indefinitely close to zero. The area of the peeled shape becomes indefinitely close to the area of the sphere, which is what we want, as the maximum width of any gore becomes indefinitely small. Thus, the main remaining necessity is to find the area of each flattened gore. The length of each flattened gore becomes indefinitely close to the length of each longitude line, i.e. π R,π R, in which R R is the radius of the sphere. Since two lines of longitude subtend a constant angle at the polar axis, the width of a gore, as a function of position along the gore, becomes proportional to distance from the polar axis. Therefore, if the width of gore number k k at the equator is w k,w k, the width at any other latitude becomes indefinitely close to w k×w k× the cosine of the latitude. Figure 4: The Width of a Gore Along its Length is Proportional to Distance from the Polar Axis, Therefore to Cosine of Latitude Image Credit: Your Humble Author The area, if expressed as an integral of the width along the length, thus becomes an integral of a cosine, which is well known, but we don’t use integrals in this answer. Instead, we can express the same area by projecting a semicircular band with width w k w k orthogonally onto a plane surface Figure 5: Orthogonal Projection Onto a Plane Yields the Same Cosine Dependence as in Figure 4 Image Credit: Your Humble Author This projection yields a rectangle with the same area as the idealized gore, because the cosine factor is supplied by the increasing foreshortening of the projection in relation to the projected, semicircular band with increasing distance from the equator. Thus, we find that, as the width of the widest gore becomes indefinitely small, the area of gore number k k becomes indefinitely close to 2 R w k.2 R w k. That means that the area of the peeled sphere is ∑k(2 R w k)=2 R∑k(w k).∑k(2 R w k)=2 R∑k(w k). As the width of the widest gore becomes indefinitely small, the shape of the peeled sphere becomes indefinitely close to the shape of the complete sphere, and the ∑k(w k)∑k(w k) becomes indefinitely close to the circumference of the equator of the sphere, 2 π R.2 π R. Therefore, as the peeled sphere becomes indefinitely close to the shape of the original sphere, the area becomes indefinitely close to 2 R×2 π R=4 π R 2.2 R×2 π R=4 π R 2. Q.E.D. Fun Fact Here is an example of flattened gores: a map of Earth drawn in 1507. Figure 6: Flattened Gores Used for a Map of a Globe (Ours) Image Credit: By Martin Waldseemüller - Badische Landesbibliothek, Public Domain, File:Waldseemüller-Globus.jpg - Wikimedia Commons This is how globes of a planet are made. The map is printed on flat paper or other substrate, the outline of the map is cut out, and the resulting map is glued onto a sphere. The gores are narrow enough that the substrate of the printing can deform to follow the sphere. Upvote · 9 3 Promoted by Unbiased Unbiased UK's leading platform connecting you with financial experts ·Sep 17 What are the best ways to invest money? Investing wisely means making your money work for you without taking on unnecessary risk. While there’s no ‘one best way’ for everyone, there are a few tried-and-tested principles that can help almost anyone build wealth more effectively. Start with a clear goal: Are you investing for retirement, buying a home, supporting your family, or passing on assets? Your goals will shape how much risk you can (and should) take. Diversify: Putting money into a mix of assets, such as shares, bonds, property, and cash, spreads your investment risk. This way, if one asset underperforms, your whole portf Continue Reading Investing wisely means making your money work for you without taking on unnecessary risk. While there’s no ‘one best way’ for everyone, there are a few tried-and-tested principles that can help almost anyone build wealth more effectively. Start with a clear goal: Are you investing for retirement, buying a home, supporting your family, or passing on assets? Your goals will shape how much risk you can (and should) take. Diversify: Putting money into a mix of assets, such as shares, bonds, property, and cash, spreads your investment risk. This way, if one asset underperforms, your whole portfolio isn’t affected. Use tax-efficient wrappers: Pensions and ISAs let your money grow free from tax. Every £80 you put in a pension becomes £100 thanks to basic-rate tax relief, and higher-rate taxpayers can claim back even more. (Unbiased) Balance growth and safety: Younger investors may consider taking more risk for higher potential returns, while those closer to retirement may prefer a steadier approach. Regularly reviewing your portfolio keeps your strategy in line with your life stage. Get professional advice: DIY investing is possible, but studies show that professional financial advice can leave people better off by tens of thousands of pounds over their lifetime. An adviser can help you avoid costly mistakes, minimise tax, and plan for the long term. (Unbiased) At Unbiased, we connect you with an FCA-regulated financial adviser who can tailor a plan to your needs. Your first consultation is free, so that you can explore your options with confidence. 👉 Ready to invest wisely? Get your free adviser match today. Find your expert adviser. Upvote · 99 15 Related questions More answers below How can I prove that the area of a sphere is 4pir2? Is 4pi R^2 the formula for determining the volume of a sphere? Is there a proof that the surface area of a sphere of radius R R is exactly 4 π R 2 4 π R 2? Why use a 4pi in a sphere? What is a good proof for showing why the surface area of a sphere is 4pir^2? Bill Crean Solved the Holey Cube problem without calculus. · Author has 5.1K answers and 6.4M answer views ·Updated 7y How can I prove that the area of a sphere is 4pir2? Originally Answered: How can I prove that the area of a sphere is 4pir2? · I think the best way is to do it the way the ancient mathematicians must have done it, however, it does involve a calculus-like procedure, as follows: Wrap a rectangular sheet around the sphere so that it exactly goes around the circumference. This makes the horizontal dimension of the rectangle 2pi.R.( R represents the radius of the sphere). The vertical dimension of the sheet you need to make as Continue Reading I think the best way is to do it the way the ancient mathematicians must have done it, however, it does involve a calculus-like procedure, as follows: Wrap a rectangular sheet around the sphere so that it exactly goes around the circumference. This makes the horizontal dimension of the rectangle 2pi.R.( R represents the radius of the sphere). The vertical dimension of the sheet you need to make as high as the sphere stands, i.e. 2R. We now project the surface of the sphere onto the sheet, similar to, but not the same as a Mercator projection from geography (strictly speaking the projection is called a horizontal cylindrical projection). Orient the sphere like the Earth with North at the top. Consider each arc of the sphere, which is to be projected onto the encircling sheet, as a parallel of latitude (also from geography). The radius of the parallel of latitude in terms of the radius of the sphere is (denoting the angle of latitude as t degrees) r = Rcos(t) .. (1) where r is radius of parallel of latitude. When the small length of arc is projected onto the sheet and swept around 360 degrees, an area of 2pi.Rcos(t) (=2pi.r) times the length of the arc (say, dR) is projected onto the flat sheet.. When part of a circular arc is projected onto a flat sheet in this way, the length on the sheet is dRcos(t). This length is then swept along the length of the sheet, which is 2.pi.R. But remember from above equation (1), that, R = r/(cos(t)) and so the area of projection on the sheet is 2.pi.(r/cos(t)) . dRcos(t) that is, the cos(t)s cancel, leaving 2.pi.r.dR All the small lengths of arc, dR, are summed from the south pole to north pole. In short, the area of the sphere is equal to the area of the rectangular sheet, which is set at 2.pi.R times 2R, which equals of course 4.pi.R^2. While I am at this point, I might as well do the volume of a sphere in the way the ancie... Upvote · 9 3 Parimal Kumar Ghosh International Society for Krishna Consciousness(I · Author has 2.3K answers and 3M answer views ·5y Why the surface area of sphere is 4pir^2? Originally Answered: Why the surface area of sphere is 4pir^2? · If a SOLID SPHERE is Diametrically cut into 4 Equal Pieces, the Internal Surface Areas of Each Piece (Two Sides Having Symmetrically Equal Areas equal to a Circle i.e. πr²) will represent the Total Surface Areas of a Solid Sphere. Please refer to the figure…hereunder: Its difficult to represent The Curved Surface Area on a flat surface like the Area of a Circle, but we can take the Advantage of a Hemisphere. Now lets bisect the Sphere (into Two Hemispheres) diametrically into Two Pieces. Since the curved Surface Area, opposite to the Central Cross Section of the Plain Area of the Circular Hemisp Continue Reading If a SOLID SPHERE is Diametrically cut into 4 Equal Pieces, the Internal Surface Areas of Each Piece (Two Sides Having Symmetrically Equal Areas equal to a Circle i.e. πr²) will represent the Total Surface Areas of a Solid Sphere. Please refer to the figure…hereunder: Its difficult to represent The Curved Surface Area on a flat surface like the Area of a Circle, but we can take the Advantage of a Hemisphere. Now lets bisect the Sphere (into Two Hemispheres) diametrically into Two Pieces. Since the curved Surface Area, opposite to the Central Cross Section of the Plain Area of the Circular Hemisphere, is equal to Twice areas of the Plain Circle of the Hemisphere, when this Hemisphere is further divided symmetrically into Two Equal Quarter Parts, the plain surface area of each 4th Quarter is also equal to the curved surface area of the Original Sphere. When the Hemisphere is fuether divided the Central Line being the Diameter, if equally divided we can find the out the Centre and Radius of the Sphere. Now the Areas of These Two Sides are Exactly Equal to the Plain Surface Area of a Circle having the Same Radius and Diameter of the SOLID SPHERE. IF we now proceed inversely, we will find that the … The Curved Surface Area of a Quarter Part represents the Same Area of a Circle, the Hemisphere twice the quarter parts, i.e. 2.πr², and the ENTIRE CURVED SURFACE AREA OF A SOLID SPHERE WILL REPRESENT FOUR TIMES THE SURFACE AREA OF A CIRCLE HAVING THE SAME DIAMETER. Therefore, surface area of a sphere is equal to … 4 x π x r² or 4πr². Note : Some of The Map Projections are represented in this manner, a photo attached. Thanks. Upvote · 9 4 9 1 Muthusamy Piramanayagam ( முத்துசாமி பிரமநாயகம்) M.Sc. Retired as Head of Department of Physics. Government of Tamilnadu, lndia. · Author has 11K answers and 7.8M answer views ·5y Why the surface area of sphere is 4pir^2? Originally Answered: Why the surface area of sphere is 4pir^2? · Consider a hemisphere in the figure. A small circular ring like surface is shown. The surface area of this ring is 2π x δt where δt is the thickness of the strip. The surface of the hemisphere is fully covered by many numbers of such strips whose areas sum is the surface area of the hemisphere. A little thought will show us that the sum is 2π r ( sum of all small δr s ) 2π r r = 2π r^2. For whole sphere it is 4π r ^2 Continue Reading Consider a hemisphere in the figure. A small circular ring like surface is shown. The surface area of this ring is 2π x δt where δt is the thickness of the strip. The surface of the hemisphere is fully covered by many numbers of such strips whose areas sum is the surface area of the hemisphere. A little thought will show us that the sum is 2π r ( sum of all small δr s ) 2π r r = 2π r^2. For whole sphere it is 4π r ^2 Upvote · 9 3 9 1 Sponsored by Grammarly Stuck on the blinking cursor? Move your great ideas to polished drafts without the guesswork. Try Grammarly today! Download 99 34 Related questions More answers below The volume of a sphere is equal to its surface area. What is the diameter of the sphere? How do I derive the formula for the surface area of the sphere? What is the formula for the surface area of a semi sphere? A sphere just fits inside a cube. What is the surface area of the sphere as a percentage of the surface area of the cube (rounded to the nearest whole percentage)? How do you calculate the surface area of a sphere? Assistant Bot · 1y To prove that the surface area A A of a sphere is given by the formula A=4 π r 2 A=4 π r 2, we can use calculus, specifically the method of integration. Here’s a step-by-step outline of the proof: Understanding the Sphere A sphere of radius r r can be defined in three-dimensional Cartesian coordinates as the set of points satisfying the equation: x 2+y 2+z 2=r 2 x 2+y 2+z 2=r 2 Using Spherical Coordinates We can express points on the surface of the sphere using spherical coordinates: x=r sin θ cos ϕ x=r sin⁡θ cos⁡ϕ y=r sin θ sin ϕ y=r sin⁡θ sin⁡ϕ z=r cos θ z=r cos⁡θ where: θ θ is the polar angle (from the positi Continue Reading To prove that the surface area A A of a sphere is given by the formula A=4 π r 2 A=4 π r 2, we can use calculus, specifically the method of integration. Here’s a step-by-step outline of the proof: Understanding the Sphere A sphere of radius r r can be defined in three-dimensional Cartesian coordinates as the set of points satisfying the equation: x 2+y 2+z 2=r 2 x 2+y 2+z 2=r 2 Using Spherical Coordinates We can express points on the surface of the sphere using spherical coordinates: x=r sin θ cos ϕ x=r sin⁡θ cos⁡ϕ y=r sin θ sin ϕ y=r sin⁡θ sin⁡ϕ z=r cos θ z=r cos⁡θ where: θ θ is the polar angle (from the positive z-axis) ranging from 0 0 to π π, ϕ ϕ is the azimuthal angle (in the x-y plane) ranging from 0 0 to 2 π 2 π. Calculating the Surface Area Element The surface area element d A d A on the sphere can be derived from the spherical coordinates. The differential area element on the surface is given by: d A=r 2 sin θ d θ d ϕ d A=r 2 sin⁡θ d θ d ϕ Integrating Over the Sphere To find the total surface area A A, we integrate d A d A over the entire sphere: A=∫2 π 0∫π 0 r 2 sin θ d θ d ϕ A=∫0 2 π∫0 π r 2 sin⁡θ d θ d ϕ Performing the Integration First, integrate with respect to θ θ: ∫π 0 sin θ d θ=[−cos θ]π 0=−cos(π)−(−cos(0))=1+1=2∫0 π sin⁡θ d θ=[−cos⁡θ]0 π=−cos⁡(π)−(−cos⁡(0))=1+1=2 Next, integrate with respect to ϕ ϕ: ∫2 π 0 d ϕ=2 π∫0 2 π d ϕ=2 π Combining Results Now, substituting back into the surface area integral: A=r 2⋅2⋅2 π=4 π r 2 A=r 2⋅2⋅2 π=4 π r 2 Conclusion Thus, we have shown that the surface area A A of a sphere with radius r r is: A=4 π r 2 A=4 π r 2 This proof combines geometric understanding with calculus, providing a clear derivation of the surface area formula for a sphere. Upvote · Jacob Palasek B.S. in Information Technology&Mathematics, Grand Valley State University (Graduated 2014) ·1y How can you derive the equation for the surface area of a sphere: A=4 π r 2 A=4 π r 2? Originally Answered: How can you derive the equation for the surface area of a sphere: A = 4 \pi r^2 ? · My first thought was that if you know the formula for the volume of the sphere then you could take two spheres where both radii are very close to each other and subtract the volume of one from the other. And keep doing this until those to radii are so close there is almost zero difference between them. Then I realized I was doing calculus by doing the above. So, when the difference between the two radii is zero then you have the surface area. And that is also the derivative of the formula for a sphere. Because the surface area is the rate of change of volume. Continue Reading My first thought was that if you know the formula for the volume of the sphere then you could take two spheres where both radii are very close to each other and subtract the volume of one from the other. And keep doing this until those to radii are so close there is almost zero difference between them. Then I realized I was doing calculus by doing the above. So, when the difference between the two radii is zero then you have the surface area. And that is also the derivative of the formula for a sphere. Because the surface area is the rate of change of volume. Upvote · 9 2 Parimal Kumar Ghosh International Society for Krishna Consciousness(I · Author has 2.3K answers and 3M answer views ·Updated 5y What are some interesting methods of proving that the surface area of a sphere is 4(pi) r^2? Originally Answered: What are some interesting methods of proving that the surface area of a sphere is 4(pi) r^2? · Please take a sphere and cut it in such a manner as we do for taking out the skin of an orange and place the whole skin on a flat surface. You would observe that the area covered by the skin is exactly the area 4 times the area of a circle whose radius is the same. It proves that surface area of a sphere is 4 times of a circle having same radius. Therefore, the formula is 4.πr^2. The Second Method… Take a Solid Sphere and bisect diametrically, we get Two equal halves and further bisect these Two perpendicularly from the outer surface through the Cente of the Sphere. We get four Quarter Parts of the Continue Reading Please take a sphere and cut it in such a manner as we do for taking out the skin of an orange and place the whole skin on a flat surface. You would observe that the area covered by the skin is exactly the area 4 times the area of a circle whose radius is the same. It proves that surface area of a sphere is 4 times of a circle having same radius. Therefore, the formula is 4.πr^2. The Second Method… Take a Solid Sphere and bisect diametrically, we get Two equal halves and further bisect these Two perpendicularly from the outer surface through the Cente of the Sphere. We get four Quarter Parts of the Sphere and all of them are identical in size, shape and areas. Lets take One Piece, and find out the internal surface area. It has Two distinct and identical half circles and combined together total surface Area is equal to the area of a circle having the same radius and diameter. If this Area is equal to 1/4th of the outer curved surface area then there are Four Such Units much conform with the Areas of the Four Slices i.e equal to the surface area of a circle. And in fact the the mathematical calculation establishes that the outer curved surface area of a Sphere is equal to the area of a circle having the same radius. Therefore the surface area of a sphere is 4 Times the Area of a circle or 4πr². Thanks. Upvote · 9 3 Sponsored by VAIZ.com Stop Fighting Your PM Tool — Switch to VAIZ VAIZ — Better Alternative to Your PM Tool. Boost Productivity with Built-in Doc Editor & Task Manager. Sign Up 9 6 Richard Kleinschmidt Math Teacher (1970–present) · Author has 113 answers and 110.5K answer views ·2y Is there any way to prove that the surface area of a sphere is greater than or equal to four times pi times the radius squared? Originally Answered: Is there any way to prove that the surface area of a sphere is greater than or equal to four times pi times the radius squared? · The one classic proof for this is called the Hatbox Theorem. You put a cylinder around the sphere that is exactly the same diameter as the sphere and exactly the same height as the sphere. The height of the cylinder is 2r and the radius of the cylinder is r so the lateral surface of the cylinder is the circumference (2pi) times the height (2pi). So the lateral area of the cylinder is 4pir^2. The tricky part is proving that the surface of the sphere is exactly the same as the lateral surface of the cylinder. You slice horizontally and show that each slice of the cylinder has the same area as eac Continue Reading The one classic proof for this is called the Hatbox Theorem. You put a cylinder around the sphere that is exactly the same diameter as the sphere and exactly the same height as the sphere. The height of the cylinder is 2r and the radius of the cylinder is r so the lateral surface of the cylinder is the circumference (2pi) times the height (2pi). So the lateral area of the cylinder is 4pir^2. The tricky part is proving that the surface of the sphere is exactly the same as the lateral surface of the cylinder. You slice horizontally and show that each slice of the cylinder has the same area as each slice of the sphere. The slices of the sphere will not be as long as the slices of the cylinder but will be thicker. Of course it requires that the slices be infinitely thin. If this seems a bit loose, it's all been formalized with calculus but Arcimedes figured out this proof way back. Upvote · 9 1 9 1 Gopal Menon B Sc (Hons) in Mathematics, Indira Gandhi National Open University (IGNOU) (Graduated 2010) · Author has 10.2K answers and 15.2M answer views ·4y The surface area of a sphere is 4πr^2. How is it derived? Originally Answered: The surface area of a sphere is 4πr^2. How is it derived? · The surface area of a sphere is 4πr^2. How is it derived? Let r r be the radius of the sphere. Consider a radius vector at an ∠θ∠θ to the horizontal. If this radius vector goes around the vertical axis maintaining the angle θ θ with the horizontal, it would trace a circle on the surface of the sphere of radius r cos θ.r cos⁡θ. The circumference of the circle would then be 2 π r cos θ.2 π r cos⁡θ. Let the radius vector move up slightly through an angle d θ.d θ. The arc formed due to this movement would be of length r d θ.r d θ. If this arc moves around the sphere, it would give us a ring of surface area Continue Reading The surface area of a sphere is 4πr^2. How is it derived? Let r r be the radius of the sphere. Consider a radius vector at an ∠θ∠θ to the horizontal. If this radius vector goes around the vertical axis maintaining the angle θ θ with the horizontal, it would trace a circle on the surface of the sphere of radius r cos θ.r cos⁡θ. The circumference of the circle would then be 2 π r cos θ.2 π r cos⁡θ. Let the radius vector move up slightly through an angle d θ.d θ. The arc formed due to this movement would be of length r d θ.r d θ. If this arc moves around the sphere, it would give us a ring of surface area 2 π r cos θ×r d θ=2 π r 2 cos θ d θ.2 π r cos⁡θ×r d θ=2 π r 2 cos⁡θ d θ. The surface area of the sphere can be obtained by integrating this. ⇒S=2 π/2∫0 2 π r 2 cos θ d θ=4 π r 2 π/2∫0 cos θ d θ.⇒S=2∫0 π/2 2 π r 2 cos⁡θ d θ=4 π r 2∫0 π/2 cos⁡θ d θ. =4 π r 2[sin θ]π/2 0=4 π r 2[sin⁡θ]0 π/2 ⇒S=4 π r 2[1−0]=4 π r 2.⇒S=4 π r 2[1−0]=4 π r 2. Upvote · 9 2 9 1 Promoted by JH Simon JH Simon Author of 'How To Kill A Narcissist' ·Updated Fri Narcissist abuse has lead me into a very dark depression. How do I overcome this? Pendulate between the darkness and the light. Depression is a healing tonic which restores the Self to a point of equilibrium. Remember that while in the narcissistic relationship you were identified with a grandiose construct, i.e. the false Self of the narcissist. Your old identity was demolished, and you were reprogrammed according to the narcissist’s tastes. This false identity is now crumbling, and your ego is undergoing a process of grief. That is what the depression is. Your ego drew a sense of identity from the narcissist, and it wants that identity back. It does not care what kind of id Continue Reading Pendulate between the darkness and the light. Depression is a healing tonic which restores the Self to a point of equilibrium. Remember that while in the narcissistic relationship you were identified with a grandiose construct, i.e. the false Self of the narcissist. Your old identity was demolished, and you were reprogrammed according to the narcissist’s tastes. This false identity is now crumbling, and your ego is undergoing a process of grief. That is what the depression is. Your ego drew a sense of identity from the narcissist, and it wants that identity back. It does not care what kind of identity you have; only that you should have one. It does not realise that you can rebuild yourself in a more actualised, empowered way. Before you can do that, however, you must grieve. Ideally you want to direct all of your awareness into the depression, to expand your consciousness and accept the depression in all of its intensity. However, that might be too overwhelming initially. Instead, take time each day to sit in an upright position and simply direct your consciousness toward the feeling of depression for as long as you can tolerate. Note its intensity. Where in the body does it manifest? In the chest? In your stomach? Let your face droop, let your body soften, let yourself be as sad and depressed as needed. Go with the flow. Do not think about it or analyse it, simply observe it and allow it to happen. This is how you allow the grieving process to complete itself. Just when you think it will never end, it will begin to transform. But that could be days, weeks or months away. For now, simply take time out each day to do this practice. When you become overwhelmed, and surely you will at the beginning, change up and do something that brings you relaxation and joy. Take a bath, spend time with a good friend, watch your favourite TV show, go for a walk, do exercise. When you are sufficiently filled, go back into the dark and sit there i.e. be conscious with it. You can be sure that when the work is done, the sun will shine again, and the darkness will recede back into the depths of your being. Then the spiritual growth can begin. Best of luck. If you have just started your narcissistic abuse recovery journey, check out How To Kill A Narcissist. Or if you wish to immunise yourself against narcissists and move on for good, take a look at How To Bury A Narcissist. Upvote · 999 507 99 18 99 12 Lance Everett Studied Nanoengineering&Mathematics at University of California, San Diego · Author has 1.1K answers and 1.5M answer views ·Updated 4y What is a good proof for showing why the surface area of a sphere is 4pir^2? Originally Answered: What is a good proof for showing why the surface area of a sphere is 4pir^2? · A sphere is a kind of two-dimensional submanifold of Euclidean space R 3 R 3. In other words, the sphere is essentially a map from a two-dimensional submanifold of R 2 R 2 into R 3 R 3 called an embedding. An embedding of a sphere of radius r r and center (0,0,0)(0,0,0) is given by Φ:[0,π)×[0,2 π)→R 3 Φ:[0,π)×[0,2 π)→R 3 (u,v)↦Φ(u,v)(u,v)↦Φ(u,v) =r(sin(u)cos(v),sin(u)sin(v),cos(u))=r(sin⁡(u)cos⁡(v),sin⁡(u)sin⁡(v),cos⁡(u)) u u and v v are the coordinates of this embedding, since they assign each point in the sphere a unique coordinate pair. Technically this embedding is missing a single point at the south pole of the sphere, but that doesn't matter if we are calcu Continue Reading A sphere is a kind of two-dimensional submanifold of Euclidean space R 3 R 3. In other words, the sphere is essentially a map from a two-dimensional submanifold of R 2 R 2 into R 3 R 3 called an embedding. An embedding of a sphere of radius r r and center (0,0,0)(0,0,0) is given by Φ:[0,π)×[0,2 π)→R 3 Φ:[0,π)×[0,2 π)→R 3 (u,v)↦Φ(u,v)(u,v)↦Φ(u,v) =r(sin(u)cos(v),sin(u)sin(v),cos(u))=r(sin⁡(u)cos⁡(v),sin⁡(u)sin⁡(v),cos⁡(u)) u u and v v are the coordinates of this embedding, since they assign each point in the sphere a unique coordinate pair. Technically this embedding is missing a single point at the south pole of the sphere, but that doesn't matter if we are calculating the surface area. The infinitesimal coordinate vectors at (u,v)(u,v) are given by the partial derivatives of this embedding with respect to the parameters u u and v v, multiplied by infinitesimal changes d u d u and d v d v. These vectors are the infinitesimal changes in the position of the point on the sphere as the coordinates are varied infinitesimally: D u Φ(u,v)d u D u Φ(u,v)d u =r(cos(u)cos(v),cos(u)sin(v),−sin(u))d u=r(cos⁡(u)cos⁡(v),cos⁡(u)sin⁡(v),−sin⁡(u))d u D v Φ(u,v)d v D v Φ(u,v)d v =r(−sin(u)sin(v),sin(u)cos(v),0)d v=r(−sin⁡(u)sin⁡(v),sin⁡(u)cos⁡(v),0)d v Note that these two vectors are orthogonal---their dot product is 0 0. They also have lengths r d u r d u and r sin(u)d v r sin⁡(u)d v, respectively. Thus the two vectors make two sides of an infinitesimal parallelogram (in this case, a rectangle), whose area is thus r 2 sin(u)d u d v r 2 sin⁡(u)d u d v Integration of these areas of the tiny parallelograms over u u and v v in their respective ranges gives the result: ∫2 π 0∫π 0 r 2 sin(u)d u d v∫0 2 π∫0 π r 2 sin⁡(u)d u d v =2 π r 2∫π 0 sin(u)d u=2 π r 2∫0 π sin⁡(u)d u =2 π r 2(−cos(π)+cos(0))=4 π r 2=2 π r 2(−cos⁡(π)+cos⁡(0))=4 π r 2 FYI, that is how surface area is defined---you integrate over the areas of infinitesimal parallelograms, just like arc length is integration over the lengths of infinitesimal line segments. Alternatively, if you know that the volume enclosed by a sphere of radius r r is 4 π r 3/3 4 π r 3/3, note that the surface area is actually the derivative of the volume with respect to the radius. This is not a coincidence, and is a consequence of a more general phenomenon. Lance Everett's answer to Why is 'pi' crucial to understanding the ways in which curves and lines interact? Upvote · 9 2 Gary Russell Former Professor at University of Iowa (1996–2025) · Author has 6K answers and 3.1M answer views ·1y How can you derive the equation for the surface area of a sphere: A=4 π r 2 A=4 π r 2? Originally Answered: How can you derive the equation for the surface area of a sphere: A = 4 \pi r^2 ? · One way is to realize that the surface area is just the derivative of the volume V = (4/3)(pi)(r^3) Then, A = dV/dr = 4(pi)(r^2) This result says that the amount of change to volume by slighting increasing the radius is equal to the surface area. A similar relationship exists between the area of a circle and the circumference. If A is the area of a circle and C is the circumference, then C = dA/dr where r is the radius of the circle. Upvote · 9 1 Douglas Porter B Sc in Mathematics, The Open University (Graduated 2011) · Author has 9.6K answers and 7.8M answer views ·6y What are some interesting methods of proving that the surface area of a sphere is 4(pi) r^2? Originally Answered: What are some interesting methods of proving that the surface area of a sphere is 4(pi) r^2? · Consider a light source moving down the axis of the sphere, projecting latitude/longitude lines on the surface of the sphere onto a cylinder wrapped around the sphere aligned with the vertical axis and just touching at the equator. The light source projects exactly horizontally. Now when the light source is aligned with the equator, any arbitrarily small square drawn at the equator will project onto the cylinder as a congruent square. But when the light source is north or south of the equator, such a square will appear distorted - compressed in the north-south direction but stretched in the eas Continue Reading Consider a light source moving down the axis of the sphere, projecting latitude/longitude lines on the surface of the sphere onto a cylinder wrapped around the sphere aligned with the vertical axis and just touching at the equator. The light source projects exactly horizontally. Now when the light source is aligned with the equator, any arbitrarily small square drawn at the equator will project onto the cylinder as a congruent square. But when the light source is north or south of the equator, such a square will appear distorted - compressed in the north-south direction but stretched in the east-west direction so that it appears as a small rectangle projected on the cylinder. However, you should be able to prove to yourself with the aid of a diagram or two (which I don’t have the tools or the time for here) that, between the compression and the stretching, the rectangle has the exact same area as the square. It therefore follows that this wrapping cylinder, exactly as tall as the sphere, has the same surface area as it — and you should be able to work out the area of the cylinder quite painlessly. Upvote · Girija Warrier Studied at Sufficiently Educated · Author has 5.9K answers and 13.9M answer views ·7y Area is the product of 2 length measures. If we divide a sphere into 2 hemispheres. And then find the area of each..… Area of 1 hemisphere = base length x height If we unfold the hemisphere, base length = 2 pi r & height = r => area of 1 hemisphere = 2 pi r r = 2 pi r² Area of 2nd hemisphere = 2 pi r² So total area of Sphere = 2 pi r² + 2 pi r² = 4 pi r² Upvote · 9 2 9 1 Related questions How can I prove that the area of a sphere is 4pir2? Is 4pi R^2 the formula for determining the volume of a sphere? Is there a proof that the surface area of a sphere of radius R R is exactly 4 π R 2 4 π R 2? Why use a 4pi in a sphere? What is a good proof for showing why the surface area of a sphere is 4pir^2? The volume of a sphere is equal to its surface area. What is the diameter of the sphere? How do I derive the formula for the surface area of the sphere? What is the formula for the surface area of a semi sphere? A sphere just fits inside a cube. What is the surface area of the sphere as a percentage of the surface area of the cube (rounded to the nearest whole percentage)? How do you calculate the surface area of a sphere? Why is the area of a sphere equal to the area Of a cylindrical surface with the same diameter and height as the sphere? What is a mathematical way to prove that the area of a circle is π r 2 π r 2? How do you prove that a sphere has the largest possible surface area of any object? What is the proof of the formula for the surface area of a sphere? How do you find a sphere's area and perimeter? Related questions How can I prove that the area of a sphere is 4pir2? Is 4pi R^2 the formula for determining the volume of a sphere? Is there a proof that the surface area of a sphere of radius R R is exactly 4 π R 2 4 π R 2? Why use a 4pi in a sphere? What is a good proof for showing why the surface area of a sphere is 4pir^2? The volume of a sphere is equal to its surface area. What is the diameter of the sphere? Advertisement About · Careers · Privacy · Terms · Contact · Languages · Your Ad Choices · Press · © Quora, Inc. 2025 Privacy Preference Center When you visit any website, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences or your device and is mostly used to make the site work as you expect it to. 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2843	https://www.splashlearn.com/math-vocabulary/roster-notation	Log inSign up Parents Parents Explore by Grade Preschool (Age 2-5)KindergartenGrade 1Grade 2Grade 3Grade 4Grade 5 Explore by Subject Math ProgramEnglish Program More Programs Homeschool ProgramSummer ProgramMonthly Mash-up Helpful Links Parenting BlogSuccess StoriesSupportGifting Educators Educators Teach with Us For Teachers For Schools and Districts Data Protection Addendum Impact Success Stories Resources Lesson Plans Classroom Tools Teacher Blog Help & Support More Programs SpringBoard Summer Learning Our Library Our Library All All By Grade PreschoolKindergartenGrade 1Grade 2Grade 3Grade 4Grade 5 By Subject MathEnglish By Topic CountingAdditionSubtractionMultiplicationPhonicsAlphabetVowels Games Games By Grade Preschool GamesKindergarten GamesGrade 1 GamesGrade 2 GamesGrade 3 GamesGrade 4 GamesGrade 5 Games By Subject Math GamesReading GamesArt and Creativity GamesGeneral Knowledge GamesLogic & Thinking GamesMultiplayer GamesMotor Skills Games By Topic Counting GamesAddition GamesSubtraction GamesMultiplication GamesPhonics GamesSight Words GamesAlphabet Games Worksheets Worksheets By Grade Preschool WorksheetsKindergarten WorksheetsGrade 1 WorksheetsGrade 2 WorksheetsGrade 3 WorksheetsGrade 4 WorksheetsGrade 5 Worksheets By Subject Math WorksheetsReading Worksheets By Topic Addition WorksheetsMultiplication WorksheetsFraction WorksheetsPhonics WorksheetsAlphabet WorksheetsLetter Tracing WorksheetsCursive Writing Worksheets Lesson Plans Lesson Plans By Grade Kindergarten Lesson PlansGrade 1 Lesson PlansGrade 2 Lesson PlansGrade 3 Lesson PlansGrade 4 Lesson PlansGrade 5 Lesson Plans By Subject Math Lesson PlansReading Lesson Plans By Topic Addition Lesson PlansMultiplication Lesson PlansFraction Lesson PlansGeometry Lesson PlansPhonics Lesson PlansGrammar Lesson PlansVocabulary Lesson Plans Teaching Tools Teaching Tools By Topic Math FactsMultiplication ToolTelling Time ToolFractions ToolNumber Line ToolCoordinate Graph ToolVirtual Manipulatives Articles Articles By Topic Prime NumberPlace ValueNumber LineLong DivisionFractionsFactorsShapes Log inSign up Skip to content Roster Notation (Roster Form of Set): Meaning, Examples Home » Math Vocabulary » Roster Notation (Roster Form of Set): Meaning, Examples What Is Roster Notation or Roster Form of a Set? Roster Form to Set Builder Form Roster Notation vs. Set Builder Notation Solved Examples on Roster Notation Practice Problems on Roster Notation Frequently Asked Questions about Roster Notation What Is Roster Notation or Roster Form of a Set? Roster notation or the roster form of a set is represented using curly brackets with elements separated by comma. Here, the order of elements does not matter. This is a simple and straightforward way to represent a set by listing the elements within curly brackets. Roster Notation Example: The set of natural numbers less than 10 can be written using roster form as In mathematics, a set is defined as a collection of distinct, well-defined objects. Each item in the set is known as an element of the set. We always write elements of a set within curly brackets. A set can be presented into two ways: Roster Notation Set Builder Form In this article, we will learn about the Roster Notation. Recommended Games Use Words to Identify Decimal Notations Game Play More Games Venn Diagram for Roster Form Roster form is the most simple method for representing the elements of a set. In the roster form, the elements are listed in a row. Drawing a Venn diagram with the help of roster notation is easy since we can easily read the elements and understand the number of elements present. Similarly, writing a set in the roster form using a given Venn diagram is equally not at all difficult. Example 1: Let A be the set of months starting with F. Thus, A contains a single element. In roster form, the set A can be represented as Example 2: Let B be the set of even numbers between 0 and 10. Set B can be represented in roster form as Example 3: Suppose we have a Venn diagram of set C. We can define the set using the Roster notation as Limitations of Roster Notation In the roster notation, we list the elements within curly brackets and separate the elements by comma. Now, imagine listing a large number of elements. It will be time consuming. It will also be difficult to get the idea of what the set actually represents. Example: Suppose we want to represent the first 1000 natural numbers in a set A. It is not convenient to write 1000 numbers in a single row. We can overcome this limitation by using an ellipsis or three periods (…), which indicates that the same pattern is continued. Write the first few elements followed by three dots and finally the last element. Separate these elements with a comma. Example 2: If a set has an infinite number of elements, like the set of whole numbers, it can be represented in roster form like: Roster Form to Set Builder Form We know that a set is a collection of well-defined objects. To represent a set in the set builder form, we first identify the unique property satisfied by all the elements in the given set and use a mathematical statement or a condition with a variable to define the set. Roster form to the set builder form examples: \| Roster Form \| Set Builder Form \| --- \| \| \| \| \| \| \| Roster Notation vs. Set Builder Notation \| Roster Form \| Set Builder Form \| --- \| \| It is represented by listing elements within curly brackets. Elements are separated by comma. \| It uses the unique properties or conditions satisfied by all the elements of the set to define the set. \| \| We actually write the elements of the set. \| We do not write the actual elements of the set, but a logical condition/statement/ formula that leads to the elements of the set. \| \| Convenient to use with the sets having fewer number of elements \| Convenient to use with the sets having a large or an infinite number of elements \| \| Easy to read and understand for a layman \| Can be tricky since a lot of math operators and symbols are used. Requires knowledge of math concepts to identify the elements of the set based on a given condition \| \| Example: Set of positive multiples of 6 \| Example: Set of positive multiples of 6 \| Facts about Roster Notation The roster form is also known as the enumeration notation as the enumeration is the process of listing things one after another. The Roster method is also known as the tabular method. Solved Examples on Roster Notation 1. Write the set of odd numbers less than 10 in a set notation form. What is the cardinality of the set? Solution: Set of odd numbers Set of odd numbers less than Cardinality of the set 2. Express the sets P and Q in the roster form. Solution: In the set P, there are 4 elements. Let’s list them using the roster notation. In the set P, there are 4 elements. Let’s list them using the roster notation. 3. Draw the Venn diagram for the given sets. Solution: Venn diagrams: 4. Express the set in the roster notation. Solution: When When When When 5. Write the given set in the roster notation to the set builder notation. Solution: Here, we have a set of cubes of natural numbers. Practice Problems on Roster Notation Roster Notation (Roster Form of Set): Meaning, Examples Attend this quiz & Test your knowledge. 1 Identify the roster form for the set of vowels in the word MATHEMATICS. CorrectIncorrect Correct answer is: Vowels in the word MATHEMATICS are A, E, and I. Note that repetition is not allowed in sets. 2 The roster form for the first three positive multiples of 9 is CorrectIncorrect Correct answer is: 3 The set can be written in the roster notation as CorrectIncorrect Correct answer is: Possible values of n are When , we have When , we have When , we have When , we have 4 The correct roster form for the sets in the following Venn diagram is: CorrectIncorrect Correct answer is: Frequently Asked Questions about Roster Notation What is a singleton set? How do you represent a null set in roster form? What is meant by the cardinality of the set? What is interval notation? What is the meaning of set notation ? What are the rules for the set roster notation?
2844	https://www.oldest.org/nature/mountain-ranges-in-north-america/	9 Oldest Mountain Ranges in North America The formation of mountain ranges in North America is a significant part of the history of the continent. Mountain ranges have been formed by several processes, including tectonic activity, erosion and deposition of rocks, and volcanism. These mountain ranges helped to shape the cultural landscape of North America by making it possible for early settlers to travel through them on their way westward. They also provided water for agriculture and made it easier for people to live off the land in more mountainous regions than flat ones like those found in most other parts of the world. In this article, we will explore the oldest mountain ranges in North America and some interesting facts about them. 9. Olympic Mountains Age: 35 million years State: Washington Peak: Mount Olympus Elevation: 2, 427 meters photo source: National Park Service The mountains now span the Olympic Peninsula south of the Juan de Fuca Strait and west of Puget Sound in northwest Washington, U.S. The Olympic Mountains are made up of a sizable chunk of oceanic crust that was elevated, folded, and subducted into the mainland during the course of the previous 35 million years along the edge of the Cascadia subduction zone. The sedimentary and metamorphic rocks that make up the majority of the range are heavily folded and fragmented, and they are surrounded on three sides by basalt from the Crescent Formation and overlying sedimentary strata that form a horseshoe-shaped barrier to the east. Young glacial and non-glacial sediments have locally buried these rocks. Did You Know? The Olympic summit with the most glaciers is Mount Olympus. It mostly absorbs the moisture carried by breezes from the Pacific Ocean. 8. Sierra Nevada Age: 40 million years State: California Peak: Mount Whitney Elevation: 4, 421 meters photo source: Live Science The high peaks and summits of the Sierra Nevada are clearly toward the east, making it an asymmetrical range. The Sierra Nevada has long been acknowledged as a piece of the Earth’s crust that is faulted and inclined. The large mass that formed the Sierra Nevada was raised and inclined westward along a major fault line that surrounds the block on the east. It is the longest uninterrupted and cohesive mountain chain. A number of rivers that empty into nearby bodies of water drain the Sierra Nevada, which is a portion of the American Cordillera. The Pacific Ocean is served by the Western Slope Watershed, which drains the Central Valley. Did You Know? The “Range of Light” is the most well-known moniker for the Sierra Nevada. This is based on the remarkably light-colored granite that glacial movements exposed on the summits. 7. Cascade Range Age: 45 million years State: Washington Peak: Mount Rainier Elevation: 4,392 meters photo source: Sunset Magazine Some of the most beautiful and intricately formed mountains in the United States may be found in Washington State’s northwest, in the North Cascade Range. With an average height of 7,000 feet and rough topography, the northern part of the range is home to several alpine glaciers. The range is heavily forested since it gets 80 inches of rain on average each year. Although the Cascade Mountains are officially known as such north of the Canada-U.S. The Cascades begin at the boundary and extend to the northernmost point of the range at Lytton Mountain. The northern section of the range, north of Mount Rainier, is referred to as the North Cascades in the United States. Did You Know? The Pacific Ring of Fire encompasses the Cascade Range as well. 6. Alaska Range Age: 60 million years State: Alaska Peak: Denali Elevation: 6,201 meters photo source: Britannica Kids Rock formed by volcanic eruptions makes up practically the entire Oregon part of the Cascade Range. The Alaska Range is a 600-mile-long arc of mountains that extends from the border between Alaska and Canada all the way to the Alaska Peninsula. The Denali National Park and Preserve, a large area with enormous glaciers and towering peaks, is where the range reaches its highest point. The inland tundra is divided from the Pacific coastal region by mountains that act as a climate barrier. Visitors are drawn to the summits by the massive glaciers and breathtaking Arctic environment, which provide several climbing obstacles. Did You Know? Alaska Range is the tallest mountain range in the globe that is not located in Asia and Andes 5. Sierra Madre Age: 60 millions years State: Mexico Peak: Cerro Potosí Elevation: 3,311 meters photo source: lacgeo.com The Sierra Madre.) is a mountain range that is a part of the American Cordillera, the mountainous network that runs across West Antarctica, Central America, North America, and South America. A large portion of northern and central Mexico is covered by the Mexican Plateau, which is surrounded by three major mountain ranges, including the Sierra Madre Oriental. In a northwest-southeast orientation, the Sierra Madre Occidental runs 1,250 km along Mexico’s Pacific coast, over its western and northern regions, and along the Gulf of California. Most of its topography is volcanic. Did You Know? Biodiversity is abundant in the Sierra Madre. It also includes several deserts and Taxa species. 4. Rocky Mountains Age: 80 million years State: Colorado, Wyoming, New Mexico, Montana, British Columbia, Idaho Peak: Mount Elbert Elevation: 4, 401 meters photo source: Wikipedia The massive upland system that dominates western North America is made up of a mountain range commonly referred to as the Rockies. It forms the cordilleran spine of the continent. A collection of mountain ranges known as the Rockies normally stretches from northern Alberta and British Columbia south to New Mexico. The current Rocky Mountain terrain was shaped by water in all of its forms. One-fourth of the country’s water supply is provided by the Rocky Mountain rivers and lakes through runoff and snowfall from the peaks. Did You Know? The museum collection at Rocky Mountain contains artifacts and specimens that document the history of the park, including antique photographs, household objects that were formerly part of park-era residences, and oil and watercolor paintings of the park’s beauty. 3. White Mountains Age: 100-124 million years State: New Hampshire Peak: Mount Washington Elevation: 1, 917 meters photo source: Boston Magazine The White Mountains are a mountain range that takes up roughly one-fourth of the state of New Hampshire and a tiny piece of western Maine. Most of the mountains are inside the White Mountain National Forest, with only the highest summits rising over the timberline. The area is a well-liked summer vacation destination due to its abundance of campgrounds and more than 1,000 miles of natural trails. There are extra winter sports facilities and excellent ski slopes in the mountains. The majority of the mountains are located inside the White Mountain National Forest, with only the highest summits rising beyond the timberline. Did You Know? Franklin Leavitt’s mountain map was one of the first two to be created for tourists. 2. Great Smoky Mountains Age: 200-300 million years State: North Carolina, Tennessee Peak: Clingmans Dome Elevation: 2, 025 meters photo source: National Geographic The Great Smoky Mountains in the southeast of the United States rise along the Tennessee-North Carolina border. The name of the range is occasionally abbreviated to the Smokies, or Smoky Mountains. The Great Smoky Mountains National Park, which safeguards the majority of the range, is the reason it is better known as the Great Smoky Mountains. The Smokies are renowned for their tremendous biodiversity. There are roughly 50 different species of fish, 240 different bird species, and more than 1,500 species of blooming plants. Did You Know? The land that is now the Great Smoky Mountains National Park was privately held until Tennessee and North Carolina contributed to the construction of Newfound Gap Road. The states also agreed that there would be no tolls or admission fees to be collected. 1. Appalachian Mountains Age: 480 million years State: Alabama, Georgia, Kentucky, Maryland, Mississippi, New York, North Carolina, Ohio, Pennsylvania, South Carolina, Tennessee, and Virginia, and all of West Virginia Peak: Mount Mitchell Elevation: 2, 037 meters photo source: Encyclopedia Britannica Eastern to northeastern North America is home to the Appalachian Mountains, sometimes known as the Appalachians. During the Ordovician Period, 480 million years ago, the Appalachians initially began to develop, making it the oldest mountain range in North America. Different geographic areas connected to the mountain range are referred to together as Appalachia. In its broadest sense, it refers to the entire mountain range, including the hills that surround it and the divided plateau area. They are renowned for the biological variety and magnificent beauty of their surroundings. The Appalachians have been significant throughout American history. The mountains, which had historically served as a natural barrier to the westward migration of European colonial settlers, served as a battleground throughout the French and Indian War, the American Revolution, and the American Civil War. Did You Know? The Appalachian Mountains attract an estimated 3 million hikers each year. Spread the love Somapika Dutta Somapika is a passionate writer for Oldest.org, specializing in world records, sports history, and the evolution of cinema. Her work dives into fascinating topics, from record-breaking athletes and historic sports moments to the origins of legendary movie franchises and the oldest surviving films. She has also explored the history of iconic car companies, ancient wars, and North America’s oldest landmarks. A dreamer at heart, Somapika aspires to travel the world, visiting historic stadiums, famous movie sets, and record-breaking landmarks. When she’s not writing, she enjoys painting, watching classic films, and analyzing sports documentaries. A true cinephile, she has a deep love for old Hollywood and international cinema. She also finds joy in nature, often unwinding with long walks or scenic hikes. Her blend of thorough research and engaging storytelling brings history and records to life, making her work both insightful and captivating for readers. Related Post 9 Oldest Redwood Trees in United States Posted by Pratik Patil 0 Redwood trees, in particular giant sequoias, are some of the oldest and largest trees in the world. Many specimens have… 8 Oldest Oak Trees Ever Found Posted by Damini R 0 Oaks are beautiful, robust trees that grow in many parts of the world where the climate is relatively mild. They… 9 Oldest Living Organisms in the World Posted by Damini R 0 Since the oldest living people only reach about 120 years of age, it can be hard to believe that other… 8 Youngest Planets Ever Observed Posted by Somapika Dutta 0 The Earth formed approximately 4.6 billion years ago, together with the Solar System. This may sound like an eternity to… 10 Oldest Rainforests in the World Posted by Damini R 0 Rainforests are some of the most beautiful natural places in the world and home to thousands of unique species of… Leave a comment Cancel reply
2845	https://openlearninglibrary.mit.edu/courses/course-v1:MITx+8.02.3x+1T2019/courseware/802-3x-Intro/seq-resources/	Textbook Chapters \| Textbook \| 8.02.3x Courseware \| MIT Open Learning Library Loading [a11y]/mathjax-sre.js Skip to main content All Courses About Contact Sign in Create Account Give All Courses About Contact Sign in Create Account Give Course , current location CourseCourse InformationTextbookTextbook Chapters Previous Next 1. other Textbook Chapters Textbook Chapters {"xmodule-type": "HTMLModule"} TextbookThe textbook for this course is "Electricity and Magnetism: MIT 8.02 Course Notes" by Peter Dourmashkin et al. It can be viewed from the textbook tab that appears at the top of all pages on this site. If you prefer to view the textbook using your own PDF reader or on a tablet device, the individual chapters are available for download at the links below. You can also download all of the chapters as a single zip file. Chapter 1. FieldsChapter 2. Coulomb's LawChapter 3. Gauss' LawChapter 4. Electric PotentialChapter 5. CapacitorsChapter 6. Current and ResistanceChapter 7. Direct Current CircuitsChapter 8. Magnetic FieldsChapter 9. Sources of Magnetic FieldsChapter 10. Faraday's LawChapter 11. Inductance and Energy in Magnetic FieldsChapter 12. Alternating Current CircuitsChapter 13. Maxwell's Equations and Electromagnetic WavesChapter 14. Interference and DiffractionMath ReviewIf you need a quick review of various math topics for this course, here are some helpful notes. (These notes are included in the zip file above.)General Math ReviewVector AnalysisCoordinate SystemsWork and Line IntegralsPotential Energy and the Conservation of Mechanical EnergySimple Harmonic Motion and Mechanical Energy Textbook Chapters Textbook The textbook for this course is "Electricity and Magnetism: MIT 8.02 Course Notes" by Peter Dourmashkin et al. It can be viewed from the textbook tab that appears at the top of all pages on this site. If you prefer to view the textbook using your own PDF reader or on a tablet device, the individual chapters are available for download at the links below. You can also download all of the chapters as a single zip file. Chapter 1. Fields Chapter 2. Coulomb's Law Chapter 3. Gauss' Law Chapter 4. Electric Potential Chapter 5. Capacitors Chapter 6. Current and Resistance Chapter 7. Direct Current Circuits Chapter 8. Magnetic Fields Chapter 9. Sources of Magnetic Fields Chapter 10. Faraday's Law Chapter 11. Inductance and Energy in Magnetic Fields Chapter 12. Alternating Current Circuits Chapter 13. Maxwell's Equations and Electromagnetic Waves Chapter 14. Interference and Diffraction Math Review If you need a quick review of various math topics for this course, here are some helpful notes. (These notes are included in the zip file above.) General Math Review Vector Analysis Coordinate Systems Work and Line Integrals Potential Energy and the Conservation of Mechanical Energy Simple Harmonic Motion and Mechanical Energy Previous Next © All Rights Reserved Enter equation Use the arrow keys to navigate the tips or use the tab key to return to the calculator For detailed information, see Entering Mathematical and Scientific Expressions in the edX Guide for Students. Tips: Use parentheses () to make expressions clear. You can use parentheses inside other parentheses. Do not use spaces in expressions. For constants, indicate multiplication explicitly (example: 5c). For affixes, type the number and affix without a space (example: 5c). For functions, type the name of the function, then the expression in parentheses. \| To Use \| Type \| Examples \| --- \| Numbers \| Integers Fractions Decimals \| 2520 2/3 3.14, .98 \| \| Operators \| + - / (add, subtract, multiply, divide) ^ (raise to a power) \|\| (parallel resistors) \| x+(2y)/x-1 x^(n+1) v_IN+v_OUT 1\|\|2 \| \| Constants \| c, e, g, i, j, k, pi, q, T \| 20c 418T \| \| Affixes \| Percent sign (%) and metric affixes (d, c, m, u, n, p, k, M, G, T) \| 20% 20c 418T \| \| Basic functions \| abs, exp, fact or factorial, ln, log2, log10, sqrt \| abs(x+y) sqrt(x^2-y) \| \| Trigonometric functions \| sin, cos, tan, sec, csc, cot arcsin, sinh, arcsinh, etc. \| sin(4x+y) arccsch(4x+y) \| \| \| Scientific notation \| 10^ and the exponent \| 10^-9 \| \| e notation \| 1e and the exponent \| 1e-9 \| Result Open Learning Library About Accessibility All Courses Why Support MIT Open Learning? Help Connect Contact Twitter Facebook Privacy Policy Terms of Service © Massachusetts Institute of Technology, 2025
2846	https://math.stackexchange.com/questions/1873493/how-to-find-a-derivative-of-fx-int-0x2ext-2dt	calculus - How to find a derivative of $f(x)=\int_0^{x^2}e^{xt^{-2}}dt$ - Mathematics Stack Exchange Join Mathematics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 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Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more How to find a derivative of f(x)=∫x 2 0 e x t−2 d t f(x)=∫0 x 2 e x t−2 d t Ask Question Asked 9 years, 2 months ago Modified9 years, 2 months ago Viewed 209 times This question shows research effort; it is useful and clear 0 Save this question. Show activity on this post. Let f(x)=∫x 2 0 e x t−2 d t f(x)=∫0 x 2 e x t−2 d t I want to find f′(x)f′(x) I tried to use taylor expansion: e x t−2=∑n=0∞x n t−2 n n!e x t−2=∑n=0∞x n t−2 n n! Indefinite integral gives, ∫e x t−2 d t=∫∑n=0∞x n t−2 n n!d t=∑n=0∞∫x n t−2 n n!d t=∑n=0∞x n t−2 n+1(−2 n+1)n!∫e x t−2 d t=∫∑n=0∞x n t−2 n n!d t=∑n=0∞∫x n t−2 n n!d t=∑n=0∞x n t−2 n+1(−2 n+1)n! Hence, f(x)=∑n=0∞x−3 n+2(−2 n+1)n!f(x)=∑n=0∞x−3 n+2(−2 n+1)n! Thus, f′(x)=∑n=0∞(−3 n+2)x−3 n+1(−2 n+1)n!f′(x)=∑n=0∞(−3 n+2)x−3 n+1(−2 n+1)n! And I am stuck here. Can you give me some tips to proceed from here? Indeed, I am not even sure what I've done so far is correct. Also, is there a better way to solve this problem (without using taylor expansion)? I'd really appreciate your help. Thank you. calculus real-analysis integration analysis derivatives Share Share a link to this question Copy linkCC BY-SA 3.0 Cite Follow Follow this question to receive notifications asked Jul 28, 2016 at 3:09 hotshotgghotshotgg 1 1 Note that for x>0 x>0, the integral blows up at the origin; furthermore, integrating negative integer powers of t t from 0 0 to x x is also going to blow up. Though this is more an issue of convergence than finding series.πr8 –πr8 2016-07-28 03:41:06 +00:00 Commented Jul 28, 2016 at 3:41 Add a comment\| 3 Answers 3 Sorted by: Reset to default This answer is useful 3 Save this answer. Show activity on this post. Hint: You just need the fundamental theorem of calculus (the so-called "second version"). It helps to rewrite the integral with the u u-substitution u=t/x−−√u=t/x. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered Jul 28, 2016 at 3:14 Ben GrossmannBen Grossmann 235k 12 12 gold badges 184 184 silver badges 358 358 bronze badges 2 Your substitution implies x>0,x>0, but the given integral equals ∞∞ for all such x.x.zhw. –zhw. 2016-07-28 16:35:54 +00:00 Commented Jul 28, 2016 at 16:35 Well, in the x<0 x<0 case, we can take u=t/\|x\|−−√u=t/\|x\|Ben Grossmann –Ben Grossmann 2016-07-29 01:09:59 +00:00 Commented Jul 29, 2016 at 1:09 Add a comment\| This answer is useful 1 Save this answer. Show activity on this post. As Omnomnomnom answered, what you need is to apply the fundamental theorem of calculus. Applied to I=∫a(x)0 f(x,t)d t I=∫0 a(x)f(x,t)d t it leads to d I d x=a′(x)f(x,a(x))+∫a(x)0 f′x(x,t)d t d I d x=a′(x)f(x,a(x))+∫0 a(x)f x′(x,t)d t Applied to your case a(x)=x 2 a(x)=x 2, f(x,t)=e x t−2 f(x,t)=e x t−2 this gives d I d x=2 x e−1 x 3−∫x 2 0 e−x t 2 t 2 d t=2 x e−1 x 3−π−−√x 3/2(1−erf(1 x 3/2))2 x 2 d I d x=2 x e−1 x 3−∫0 x 2 e−x t 2 t 2 d t=2 x e−1 x 3−π x 3/2(1−erf(1 x 3/2))2 x 2 d I d x=2 x e−1 x 3−π−−√erfc(1 x 3/2)2 x−−√d I d x=2 x e−1 x 3−π erfc(1 x 3/2)2 x Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered Jul 28, 2016 at 4:03 Claude LeiboviciClaude Leibovici 294k 55 55 gold badges 130 130 silver badges 316 316 bronze badges Add a comment\| This answer is useful 0 Save this answer. Show activity on this post. using Leibniz integral rule, f′(x)=∫x 2 0 e x t−2 t−2 d t+e x(x 2)−2 2 x f′(x)=∫0 x 2 e x t−2 t−2 d t+e x(x 2)−2 2 x The anti-derivative exists but it goes to infinite at 0. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered Jul 28, 2016 at 4:37 user115350user115350 1,311 8 8 silver badges 9 9 bronze badges Add a comment\| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions calculus real-analysis integration analysis derivatives See similar questions with these tags. Featured on Meta Introducing a new proactive anti-spam measure Spevacus has joined us as a Community Manager stackoverflow.ai - rebuilt for attribution Community Asks Sprint Announcement - September 2025 Report this ad Related 1Find lim n→∞n 2 2 n lim n→∞n 2 2 n 14Prove that ∫1 0 ln(1−x)ln 2 x x−1 d x=π 4 180∫0 1 ln⁡(1−x)ln 2⁡x x−1 d x=π 4 180 19Calculate ∫1 0 ln(1−x 2)x d x∫0 1 ln⁡(1−x 2)x d x without series expansion. 3Is this equality lim x→∞∫x 0 t 2 2(e t−1)d t=lim n→∞∑n k=1 1 k 3 lim x→∞∫0 x t 2 2(e t−1)d t=lim n→∞∑k=1 n 1 k 3 true? 2for f∈C 2(R)f∈C 2(R), finding the derivative of d d t∫∞0 f(x+t)⋅x d x d d t∫0∞f(x+t)⋅x d x 8How do I integrate ∫∞0 log(t+1)t 2+a 2 d t∫0∞log⁡(t+1)t 2+a 2 d t? 0Find the function f(x)=∑∞n=1 3(−1)n+1(3 x)n−1 f(x)=∑n=1∞3(−1)n+1(3 x)n−1 1How to tackle this integral ∫1 0 x p 1−x ln 1 x d x(p>−1)∫0 1 x p 1−x ln⁡1 x d x(p>−1) Hot Network Questions Where is the first repetition in the cumulative hierarchy up to elementary equivalence? 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2847	https://journals.lww.com/stdjournal/fulltext/2017/09000/low_prevalence_of_urethral_lymphogranuloma.6.aspx	Sexually Transmitted Diseases Log in or Register Subscribe to journal Subscribe Get new issue alerts Get alerts;;) Subscribe to eTOC;;) ### Secondary Logo Enter your Email address: Privacy Policy ### Journal Logo Articles Advanced Search Toggle navigation SubscribeRegisterLogin Browsing History Articles & Issues Current Issue Previous Issues Published Ahead-of-Print Collections Real World GC Surveillance For Authors Submit a Manuscript Information for Authors Language Editing Services Author Permissions Journal Info About the Journal About ASTDA Editorial Board Advertising Open Access Subscription Services Reprints Rights and Permissions Articles Advanced Search September 2017 - Volume 44 - Issue 9 Previous Article Next Article Outline Abstract METHODS Study Design and Setting Statistical Analysis and Data Collection RESULTS DISCUSSION REFERENCES Images Slideshow Gallery Export PowerPoint file Download PDF EPUB Cite Copy Export to RIS Export to EndNote Share Email Facebook X LinkedIn Favorites Permissions More Cite Permissions Image Gallery Article as EPUB Export All Images to PowerPoint FileAdd to My Favorites Email to Colleague Colleague's E-mail is Invalid Your Name: Colleague's Email: Separate multiple e-mails with a (;). 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Export to End Note Procite Reference Manager [x] Save my selection Note Low Prevalence of Urethral Lymphogranuloma Venereum Infections Among Men Who Have Sex With Men: A Prospective Observational Study, Sexually Transmitted Infection Clinic in Amsterdam, the Netherlands de Vrieze, Nynke H.N.; Versteeg, Bart†; Bruisten, Sylvia M.†‡; van Rooijen, Martijn S.†; van der Helm, Jannie J.†; de Vries, Henry J.C.†‡§ Author Information From the Department of Dermatology/Allergology, University Medical Center Utrecht, Utrecht; †Public Health Service of Amsterdam, STI Outpatient Clinic, Public Health Laboratory, Department of Infectious Diseases; ‡Amsterdam Infection and Immunity Institute, Academic Medical Centre, University of Amsterdam; and §Department of Dermatology, Academic Medical Centre, University of Amsterdam, Amsterdam, the Netherlands Conflict of Interest and Sources of Funding: None declared. Correspondence: H.J.C. de Vries, MD, PhD, Public Health Service of Amsterdam, STI Outpatient Clinic, Public Health Laboratory, Department of Infectious Diseases, P.O. Box 2200, 1000 CE Amsterdam, the Netherlands. E-mail: h.j.devries@amc.nl; N.H.N. de Vrieze, MD, Department of Dermatology/Allergology (G02.124), University Medical Center Utrecht, Heidelberglaan 100, 3584 CX Utrecht, The Netherlands. E-mail: n.h.n.devrieze@umcutrecht.nl. Received for publication March 15, 2017, and accepted April 24, 2017. This is an open-access article distributed under the terms of the Creative Commons Attribution-Non Commercial-No Derivatives License 4.0 (CCBY-NC-ND), where it is permissible to download and share the work provided it is properly cited. The work cannot be changed in any way or used commercially without permission from the journal. Sexually Transmitted Diseases 44(9):p 547-550, September 2017. \| DOI: 10.1097/OLQ.0000000000000657 Open Abstract In Brief In contrast to anorectal lymphogranuloma venereum (LGV), few urogenital LGV cases are reported in men who have sex with men. Lymphogranuloma venereum was diagnosed in 0.06% (7/12,174) urine samples, and 0.9% (109/12,174) anorectal samples. Genital-anal transmission seems unlikely the only mode of transmission. Other modes like oral-anal transmission should be considered. Among men who have sex with men, urethral lymphogranuloma venereum was diagnosed 15 times less often than anorectal LGV. Genital-anal contact seems not the only mode of transmission. Other modes like oral-anal transmission should be considered. Lymphogranuloma venereum (LGV) is an invasive ulcerative sexually transmitted infection (STI) caused by the Chlamydia trachomatis (Ct) LGV biovar (ompA genovars L1, L2 and L3).1 Since 2003 an LGV epidemic among men who have sex with men (MSM) is ongoing in Europe, North America, and Australia.2 LGV is associated with high-risk behavior, reflected in high rates of STI co-infections like human immunodeficiency virus (HIV) (in up to 82.2% of the cases) and hepatitis C.3–6 It is still unknown whether the increased frequency of STI coinfections found in HIV-infected MSM is due to increased susceptibility associated with immunodeficiency and/or immune restoration, or a network factor associated with risk behavior.4 Recent reports suggest an increase of LGV diagnoses in Europe in recent years.7–9 Mainly anorectal infections have been diagnosed, whereas the frequency of urogenital and pharyngeal LGV diagnoses seems rare.10–13 Assuming that most anorectal infections are caused by receptive genital-anal contact, the discrepancy between the frequency of anorectal and urethral infections remains unexplained. In a previous study, alternative transmission modes for the transmission of anorectal infections were suggested, such as fisting and/or sharing of sex toys.2 Yet, in a more recent systematic study, we found no evidence in support of this hypothesis.4 Tissue tropism of L2b (the most frequently found strain among MSM) with a predilection to infect anorectal mucosa as opposed to urogenital mucosa was thought to be another explanation for the discrepancy in the frequency of urethral as opposed to anorectal infections, yet we could not prove this.14 Currently, most guidelines do not recommend routine testing for urethral LGV,15,16 except for the European International Union against STI LGV guideline.17 Earlier, we reported 2.1% LGV positivity rate in MSM with a concurrent anorectal LGV infection and a 6.8% urethral LGV positivity in their sexual partners.18 This suggested that urethral LGV infections are a possible link in the ongoing transmission of LGV in MSM. To indicate its contribution to the current LGV epidemic, we aimed to determine the positivity rate of urethral LGV among MSM. METHODS Study Design and Setting In the STI outpatient clinic of Amsterdam, the Netherlands, MSM are offered free screening for Ct (urethral and pharyngeal), Neisseria gonorrhoeae (Ng) (urethral and pharyngeal), syphilis, hepatitis B, and using an opt-out strategy for HIV.19,20 If the client reports receptive anal sex in the previous 6 months, additionally, anorectal Ct and Ng infections are tested. All clinical findings, diagnoses, and subsequent treatment are recorded in an electronic patient database along with patient characteristics and information on sexual behavior. Prospectively, urine samples were collected from all MSM visiting the STI outpatient clinic between March 2014 and July 2015 and were screened for Ct by amplification with a very sensitive molecular screening assay (Aptima Combo test, Hologic, USA). Positive samples were genotyped using an in-house pmpH quantitative polymerase chain reaction to differentiate between LGV and non-LGV type infections.21,22 If the pmpH test was nontypable (mainly due to insufficient DNA), or tested positive for a non-LGV infection, the result was considered negative for an LGV infection. The same strategy was used for Ct-positive anorectal samples. MSM without an anorectal test were excluded. Statistical Analysis and Data Collection Statistical analyses were performed using SPSS V.19 (SPSS, Chicago, Ill). We analyzed whether determinants of MSM with an anorectal LGV infection deffered from MSM with a urethral LGV infection, using Fisher test. Tests were 2-sided and considered significant at P less than 0.05. Sexual preference and number of sexual partners referred to the period 6 months before consultation. HIV status was based on self-reported history of HIV. Urethral symptoms were defined as discharge, dysuria, and/or pruritus. Anorectal symptoms were defined as discharge and/or a burning sensation. A concurrent STI diagnosis was defined as Ct (non-LGV) or LGV at another anatomical location (eye/pharyngeal/urogenital/anorectal), Ng and/or infectious syphilis diagnosed at the same consultation. Men with a concurrent anorectal LGV and urogenital LGV infection were categorized in the urethral LGV infection group. RESULTS During the inclusion period, 12,564 screening tests were performed for urethral Ct infections (Fig. 1). In 383 visits, no anorectal test was performed, and in 7 Ct-positive tests, no pmpH quantitative polymerase chain reaction was executed; these were all excluded from the analyses. In 12,174 tests, 404 (3.3%) tested positive for urethral Ct, of which 319 (78.9%) were negative for an LGV type infection, 78 (19.3%) were nontypable and 7 (1.7%) tested LGV-positive. In total, 1010 (8.0%) samples tested positive for anorectal Ct, of which 674 (66.7%) were negative for an LGV type infection, 227 (22.5%) were nontypable and 109 (10.8%) tested LGV-positive. Overall, we found a urethral LGV positivity rate of 0.06% (7/12,174; 95% confidence interval [CI], 0.02–0.12) and an anorectal positivity rate of 0.9% (109/12,174; 95% CI, 0.74–1.08). Of those 7 with urethral LGV, 4 had urethral symptoms, 1 had a concurrent anorectal LGV infection, 3 were HIV co-infected, and 1 was notified for LGV. Of the 108 with anorectal LGV, 39 (36.1%) had anorectal symptoms, 91 (84.3%) were HIV coinfected, and 9 (8.3%) were notified for LGV (Table 1). Compared with MSM with urethral LGV, those with anorectal LGV were significantly more often HIV coinfected (P = 0,02). Figure 1: Study flowchart of 12,564 visits during which C. trachomatis (Ct) tests were performed in men who have sex with men at the STI outpatient clinic in Amsterdam, March 2014 to July 2015. One patient with anorectal LGV had a urethral LGV co-infection; therefore, he was included in the urethral LGV group. TABLE 1: Baseline Characteristics of 115 MSM With an LGV Infection Visiting the STI Outpatient Clinic in Amsterdam March 2014 to July 2015, by Anatomical Site DISCUSSION The observed positivity rate of urethral LGV infections (0.06%) is 15 times lower compared with the positivity rate of anorectal LGV infections of 0.9% found among MSM at the STI clinic. Therefore, it seems likely that other modes of transmission are needed to explain the current LGV epidemic in MSM. As thought earlier, transmission via toys is not supported based on epidemiological data.2 Nor does tissue tropism seem a likely explanation for the discrepancy found in the rate of anorectal and urethral LGV infections.14 A strength of this study is the high number and unbiased population of MSM that was prospectively tested for both urethral and anorectal LGV because all clients were included irrespective of symptoms, notification, or prioritization based on sexual risk assessment. The number of pmpH nontypable samples from the urethral and anorectal location was high: respectively, 19.3% (78/404) and 22.5% (227/1010). We considered nontypable samples LGV negative because it is known that these samples have a low bacterial load and LGV types are successful growers.23 Nevertheless, a chance remains that we have missed LGV infections. However, in an earlier study using a genotyping Reverse Hybridization Assay, we showed that at least one third of the pmpH nontypable samples were non-LGV types.23 Moreover, because there was no significant difference in the ratio of nontypable samples from urethral or anorectal locations, the discrepancy between anatomical locations remains unexplained. In a 2014 convenience sample study from Madrid, Spain, 13,585 samples, including 2420 urethral samples, were tested in 8407 clients of whom 3282 MSM. In total, 10 (2.6%) of 338 urethral Ct-positive samples were LGV positive.11 This is slightly higher than the 1.7% found here. In a 2013 German convenience sample study, 1883 MSM rectal and pharyngeal specimens were tested for LGV and 522 urethral samples were obtained; 8 were Ct-positive of which none were LGV-positive.12 In a 2009 prospective multicentre study from the United Kingdom, 4825 urethral and 6778 rectal samples from consecutive MSM attending for sexual health screening were screened for LGV.13 The LGV positivity in rectal samples was 0.90% (95% CI, 0.69%–1.16%) and in urethral samples 0.04% (95% CI, 0.01%–0.16%), very comparable to our results. None of the 3 studies tested such a number of MSM both for anorectal and urethral LGV infections as done here. Moreover, we compared characteristics of MSM with LGV at different anatomical locations. As found earlier,9 men with urethral LGV were significantly less often HIV coinfected as opposed to men with anorectal LGV, respectively, 42.9% and 84.3% (P = 0.02), indicating that the latter is a population with higher risk taking behavior. Unfortunately, the number of urethral LGV cases is too small to look into risk behavior in closer detail. The low prevalence of cases found here does not justify routine screening for urethral LGV. Yet, clinicians should be aware of possible treatment failures as we described earlier in a case series of MSM with advanced inguinal LGV with bubo formation that were likely caused by missed and/or undertreated urethral LGV infections.24 Apart from direct consequences for individual patients, missed urethral LGV infections likely also contribute to ongoing transmission. As we described earlier,9 a considerable part of LGV diagnosis found in this study were asymptomatic: approximately 36% of the anorectal infections, and 4 of the 7 urethral LGV infections. Because LGV requires prolonged treatment and follow-up compared with non-LGV chlamydia infections, this finding stresses the clinical importance to exclude LGV in high-risk groups, irrespective of complaints. With the skewed anorectal/urethral LGV ratio of 15:1, it seems unlikely that urethral LGV infections are responsible for all anorectal LGV transmissions. Recently, we suggested that oral infections may have a role in LGV transmission via ano-oral sex.25 Schachter et al26 demonstrated in early work that neonates with an initial chlamydia conjunctivitis or pneumonia, subsequently shedded Ct from the vagina and rectum. They suggested that the vagina and conjunctivae are exposed to chlamydia at birth and that pneumonia and gastrointestinal infection occur later via oropharyngeal transmission. This paradigm could possibly also account as an explanation for the unanswered findings in the current LGV epidemic in MSM. Apart from genital-anal transmission, oropharyngeal infection may occur via ano-oral sex (also known as rimming).25 Subsequently, LGV organisms pass through the gastrointestinal tract to cause anorectal LGV proctitis. Whether this paradigm proves right remains to be seen, and its contributing factor to the LGV epidemic in MSM needs to be addressed in future research. REFERENCES de Vrieze NH, de Vries HJ. Lymphogranuloma venereum among men who have sex with men. An epidemiological and clinical review. Expert Rev Anti Infect Ther 2014; 12:697–704. Cited Here \| Google Scholar Nieuwenhuis RF, Ossewaarde JM, Götz HM, et al. Resurgence of lymphogranuloma venereum in Western Europe: An outbreak of Chlamydia trachomatis serovar l2 proctitis in The Netherlands among men who have sex with men. Clin Infect Dis 2004; 39:996–1003. Cited Here \| Google Scholar Rönn MM, Ward H. The association between lymphogranuloma venereum and HIV among men who have sex with men: Systematic review and meta-analysis. BMC Infect Dis 2011; 11:70. Cited Here \| Google Scholar Van der Bij AK, Spaargaren J, Morré SA, et al. 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Cited Here \| Google Scholar Schachter J, Grossman M, Holt J, et al. Infection with Chlamydia trachomatis: Involvement of multiple anatomic sites in neonates. J Infect Dis 1979; 139:232–234. Cited Here \| Google Scholar View full references list Copyright © 2017 The Author(s). Published by Wolters Kluwer Health, Inc. on behalf of the American Sexually Transmitted Diseases Association. View full article text Source Low Prevalence of Urethral Lymphogranuloma Venereum Infections Among Men Who Have Sex With Men: A Prospective Observational Study, Sexually Transmitted Infection Clinic in Amsterdam, the... Sexually Transmitted Diseases44(9):547-550, September 2017. Full-Size Email Favorites Export View in Gallery Email to Colleague Colleague's E-mail is Invalid Your Name: Colleague's Email: Separate multiple e-mails with a (;). Message: Your message has been successfully sent to your colleague. Some error has occurred while processing your request. Please try after some time. 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2848	https://app.myeducator.com/reader/web/1811a/chapter01/hp4pw/	Skip to book tools 1.4 Forming an Equation of a Line In this section, we are going to learn three different forms of linear functions: point-slope form, slope-intercept form, and general form. Point-Slope Form In the previous section, we learned that the slope of a line is calculated as follows: Now, suppose you found the slope of a line. Remember from the last section that any two distinct points on a line can be used to compute the slope using the formula above. Let be a distinct point and be an arbitrary point on the line. The slope is calculated as follows: We can rewrite this as shown below: This equation is called the point-slope form or the point-slope equation. If you are given two distinct points, you can first find the slope of the line and then use one of the distinct points to find the equation of the line by plugging it into the point-slope equation. Of course, you can also use this form if the slope and a point on the line are given. Consider a line that has a slope of 2 and passes through the point (3, 4). We can find the equation of the line by plugging the known information into point-slope form: The graph of this equation is shown below. Figure 1.3: Line and Points Passed Through As we can see, the line passes through the points (-3, 8), (-2, -6), (-1, -4), (0, -2), (1, 0), (2, 2), (3, 4), (4, 6), (5, 8), and more. We can use any of these points to find the equation of the line, and all of them will produce the same equation. Also, notice that the line crosses the y-axis at the point (0, -2) and crosses the x-axis at the point (1, 0). The point where a line crosses the y-axis is called the The point where a line crosses the y-axis., and the point where a line crosses the x-axis is called the The point where a line crosses the x-axis.. We can find the x-intercept of a line by plugging x = 0 into the equation of the line: We can find the y-intercept by plugging y = 0 into the equation of a line: As stated earlier, the x-intercept of this line is (0, -2), and the y-intercept is (1, 0). Figure 1.4: Intercepts Examples Here are some examples to practice using the point-slope equation. Example 1 Suppose you have two points, (3, 2) and (5, 0). What is the equation of the line that goes through these points? First, we need to find the slope of the line, which is calculated as follows: Therefore, using the point-slope form with the point (3, 2), the equation of the line is as follows: Note that it is also correct to use the other point, (5, 0), as well: Example 2 Imagine you are a production manager at a manufacturing company, and you are trying to create an equation to calculate the total cost of production. Suppose the variable cost is $5 per unit produced. Last month, the company produced 300 units and incurred the total production cost of $1,900. Assume that the relationship between the variable cost and the production cost is linear. Write an equation that can be used to calculate the total cost of production. First, let’s define the variables given in the problem. The $5 per unit cost is the rate of change per unit produced, so this is the slope of the line. Last month, the production was , and the total cost was The resulting equation is as follows: Example 3 Suppose there is a line that goes through the two points (-1, -5) and (4, 0). You are looking for two other points (, -5) and (5, ) that are also points on the same line. Find the values of and . To find the values of and , we first need to find the equation that goes through the two points (-1, -5) and (4, 0). Given these two points, we can find the slope of the line: Now, plug one of the points and the slope we calculated into the point-slope formula: Then we can plug the points (, -5) and (5, ) separately into the above equation to find the values of and The value of is found as follows: The value of is found as follows: Slope-Intercept Form Consider the nonvertical line whose slope is and y-intercept is (0, ). We can form the equation of this line in slope-intercept form: This type of equation has many applications in different fields. For example, let’s assume that you are calculating the total cost of production. The cost is the composition of , the variable cost per unit produced, and , the fixed cost of production. If the company produces units, then it incurs the total cost of for production. If you can find the equation of a line in any form, you can convert it to slope-intercept form. In the first example from the previous section, we found the equation of a line that passes through points (3, 2) and (5, 0), which is We can rewrite the above equation by keeping on the left-hand side of the equation and the rest on the right-hand side: This equation tells us that the slope of the line is -1, and the y-intercept is (0, 5). While it is easy to identify the slope and y-intercept of a line, how can we find the x-intercept? Since we know that the value of is 0 at the x-intercept, assuming that the slope is not 0, Therefore, in general form, the x-intercept is ( , 0) where is the slope and is the y-coordinate of the y-intercept. Figure 1.5: Intercepts with Slope-Intercept Form What if the slope of the line is 0? If the slope is 0, then the equation of the line is that goes through the y-intercept (0, ). If the line is vertical, then the equation of the line is that goes through the x-intercept (, 0). Figure 1.6: Horizontal and Vertical Lines Examples Let’s run a few examples to become more familiar with slope-intercept form. Example 1 Suppose that a line with the slope of -4 crosses the y-intercept (0, 10). Find the equation of this line. Since the slope is -4, and the y-coordinate of the y-intercept is 10, the equation of the line using the slope-intercept formula is Example 2 Suppose a line passes through the two points (-4, 5) and (-1, -1). Find the equation of this line in slope-intercept form, the x-intercept, and the y-intercept. In order to find the equation given two points on the line, we first need to find the slope : Using point-slope form, we can express the equation as Further, the x-coordinate of the x-intercept is Therefore, we know that the x-intercept is (6, 0) and the y-intercept is (0, 3). Example 3 Suppose that a company is producing cell phones. The material cost of each cell phone is $30, and the company incurs a cost of $200,000 each year to keep up the production facilities no matter how many cell phones are produced. If the company is planning to produce number of cell phones, and the total annual costs are , find the cost function of the company’s production in slope-intercept form. Based on the scenario, we know that the rate of change per unit of cell phone production is $30. Even if the company does not produce any phones, it has to pay $200,000 a year (at = 0 production). Therefore, the equation to find the total production cost is General Form The point-slope form and the slope-intercept form are special cases of a more general linear equation. If A, B, and C are real numbers, the graph of the equation is a straight line, provided both A and B are not zero. An equation of this form is called a linear equation in two variables. The following equations are all linear equations in the general form: The slope-intercept equation and the point-slope equation are also called linear equations, and we can rewrite them in the general form. For example, suppose we have the following point-slope equation: We can rewrite this as a general linear equation: This is equivalent to the following slope-intercept equation: Examples Here are a couple of examples to sum up all the forms of a linear equation. Example 1 Suppose there is a line that passes through the two points (1, 2) and (-3, 6). Given this information, find the x-intercept, the y-intercept, and the general form of the linear equation. Since two points are given, we will first find the slope of the line. Using the point-slope formula, we can write the linear equation as follows: We can either set to find the x-intercept and to find the y-intercept, or rewrite the equation in slope-intercept form. The slope-intercept equation is found as follows: So the x-coordinate of the x-intercept, , is: The x-intercept is (3, 0), and the y-intercept is (0, 3). Finally, we can rewrite the equation in the general form: Example 2 Suppose the slope of a line that passes through the point (-4, 3) is . Find the x-intercept, the y-intercept, and the general form of the linear equation. Since a point and the slope are given, we can find the point-slope form of the linear equation: When , . When , . So the x-intercept is (, 0), and y-intercept is (0, 11). Now we can reorganize the above equation into the general form. Want to try our built-in assessments? Use the Request Full Access button to gain access to this assessment. Previous Next U U U U U U U Insert Insert Assessment Insert Material
2849	https://www.technologynetworks.com/cell-science/articles/prokaryotes-vs-eukaryotes-what-are-the-key-differences-336095	We've updated our Privacy Policy to make it clearer how we use your personal data. We use cookies to provide you with a better experience. You can read our Cookie Policy here. Skip to content or Skip to footer Cell Science Prokaryotes vs Eukaryotes: What Are the Key Differences? In this article, we explore prokaryotes and eukaryotes and outline the key differences separating them. Article Last Updated: January 29, 2025 Written by Nicole Gleichmann Nicole Gleichmann Freelance Writer Nicole is a freelance writer specializing in biology, health and technology. She received her undergraduate degree in Organismal Biology from Scripps College in California. Here, her passion for holistic health and wellness began, leading to a previous position as a nutrition and wellness coach and expanding to freelance health writing. Learn about our editorial policies Credit: Technology Networks. Facebook Twitter LinkedIn Reddit Share Listen with Speechify 0:00 Register or log In for free to listen to this article Read time: 9 minutes Every living organism falls into one of two groups: eukaryotes or prokaryotes, with cellular structure determining which group an organism belongs to. Prokaryotes are unicellular and lack a nucleus and membrane-bound organelles. They are generally smaller and simpler and include bacteria and archaea. Eukaryotes on the other hand are often multicellular and have a nucleus and membrane-bound organelles, which help to organize and compartmentalize cellular functions. They include animals, plants, fungi, algae and protozoans. In this article, we will explain in further detail what prokaryotes and eukaryotes are and outline the differences between the two. Contents Comparing prokaryotes and eukaryotes Key similarities between prokaryotes and eukaryotes What are the key differences between prokaryotes and eukaryotes? - Transcription and translation in prokaryotes vs eukaryotes Prokaryote definition - Prokaryotic cell features - Examples of prokaryotes - Do prokaryotes have a nucleus? - Do prokaryotes have mitochondria? Eukaryote definition - Eukaryotic cell features - Examples of eukaryotes Subscribe to Cell Science updates for FREE and get: Daily Breaking Science News Tailored newsletters Exclusive eBooks, infographics and online events Subscribe Now Comparing prokaryotes and eukaryotes Prokaryotes were the first form of life, scientists believe that eukaryotesevolved from prokaryotes around 2.7 billion years ago.1 Current doctrines on the origins of eukaryotes state that two prokaryotes formed a symbiotic relationship and merged in a process known as endosymbiosis.2 Endosymbiotic events are believed to have led to the development of membrane-bound organelles such as mitochondria. Thanks to mitochondria, eukaryotic ancestors had enough energy to develop into the more complex eukaryotic cells known today. However, research from the University of Jena, published in the journal mBio, has highlighted prokaryotic bacteria that can “eat” other cells.3 This contradicts previous beliefs that only eukaryotes could perform endocytosis (a cellular process in which substances are brought into the cell and form intracellular vesicles). These more recent findings suggest the need to re-evaluate theories regarding the origin of eukaryotes. The primary distinction between these two types of organisms is that eukaryotic cells have a membrane-bound nucleus, and prokaryotic cells do not. The nucleus is where eukaryotes store their genetic information. In prokaryotes, DNA is bundled together in the nucleoid region, but it is not stored within a membrane-bound nucleus. The nucleus is only one of many membrane-bound organelles in eukaryotes. Prokaryotes, on the other hand, have no membrane-bound organelles. Another important difference is the DNA structure and location.4 Eukaryote DNA consists of multiple molecules of double-stranded linear DNA found in the nucleus, while that of prokaryotes is double-stranded, often circular, and located within the cytoplasm. However, it is worth noting that linear plasmids and chromosomes have been found in certain prokaryotes.5 Key similarities between prokaryotes and eukaryotes Figure 1: A comparison showing the shared and unique features of prokaryotes and eukaryotes. Credit: Technology Networks. All cells, whether prokaryotic or eukaryotic, share these four features, as shown in Figure 1: DNA Plasma membrane Cytoplasm Ribosomes What are the key differences between prokaryotes and eukaryotes? Prokaryotes and eukaryotes vary in several important ways – these differences include structural variation – whether a nucleus is present or absent and whether the cell has membrane-bound organelles. The differences are summarized in Table 1, below. Table 1: Differences between prokaryotes and eukaryotes. \| \| \| \| --- \| \| Prokaryote \| Eukaryote \| \| Nucleus \| Absent \| Present \| \| Membrane-bound organelles \| Absent \| Present \| \| Cell structure \| Unicellular \| Mostly multicellular; some unicellular \| \| Cell size \| Typically smaller (0.1–5 μm), however, a much larger (centimeter-long) bacterium has recently been discovered in a mangrove swamp. \| Larger (10–100 μm) \| \| Complexity \| Simpler \| More complex \| \| DNA Form \| Often circular, however, linear plasmids and chromosomes have been found in certain prokaryotes. \| Linear \| \| Examples \| Bacteria, archaea \| Animals, plants, fungi, protists \| Create a Highly Efficient DNA Template for mRNA Therapeutic Applications This infographic demonstrates an easy, 3-step in vitro transcription workflow for mRNA therapeutic applications. View Infographic Advertisement Transcription and translation in prokaryotes vs eukaryotes In prokaryotic cells, transcription and translation are coupled, meaning translation begins during mRNA synthesis.6 In eukaryotic cells, transcription and translation are not coupled. Transcription occurs in the nucleus, producing mRNA. The mRNA then exits the nucleus, and translation occurs in the cell’s cytoplasm. Prokaryote definition Prokaryotes can be split into two domains, bacteria and archaea, and are unicellular organisms that lack membrane-bound structures. Prokaryotic cells tend to be small, simple cells, measuring around 0.1–5 μm in diameter.7 Figure 2: The key structures present in a prokaryote cell. Credit: Technology Networks. Enhancing Biotherapeutic Characterization With MALDI Mass Spectrometry This webinar introduces a refined matrix-assisted laser desorption/ionization time-of-flight (MALDI-TOF) MS workflow that streamlines analysis, enables high-throughput screening and integrates with existing development pipelines. View Webinar Advertisement While prokaryotic cells do not have membrane-bound structures, they do have distinct cellular regions. In prokaryotic cells, DNA bundles together in a region called the nucleoid (Figure 2). In prokaryotes, molecules of protein, DNA and metabolites are all found together, floating in the cytoplasm. Primitive organelles, found in bacteria, do act as micro-compartments to bring some sense of organization to the arrangement.8 Prokaryotic cell features Here is a breakdown of what you might find in a prokaryotic bacterial cell (Figure 2). Nucleoid: A central region of the cell that contains its DNA. Ribosome: Ribosomes are responsible for protein synthesis. Cell wall:The cell wall provides structure and protection from the outside environment. Most bacteria that include this have a rigid cell wall made from carbohydrates and proteins called peptidoglycans. Cell membrane: Every prokaryote has a cell membrane, also known as the plasma membrane, that separates the cell from the outside environment. Capsule: Some bacteria have a layer of carbohydrates that surrounds the cell wall called the capsule. The capsule helps the bacterium attach to surfaces and protects it from harmful substances or conditions. Pili: Pili, also referred to as fimbriae, are rod-shaped structures involved in multiple roles, including attachment and DNA transfer. Flagella: Flagella are thin, tail-like structures that assist in movement. Examples of prokaryotes Bacteria and archaea are the two types of prokaryotes. Do prokaryotes have a nucleus? Prokaryotes do not have a nucleus. Instead, prokaryote DNA can be found, bundled but free-floating, in a central region called the nucleoid. Prokaryote DNA is usually found as a single chromosome of circular DNA. These organisms also lack other membrane-bound structures such as the endoplasmic reticulum. Do prokaryotes have mitochondria? No, prokaryotes do not have mitochondria. Mitochondria are only found in eukaryotic cells. This is also true of other membrane-bound structures like the nucleus and the Golgi apparatus (more on these later). Eukaryote definition Eukaryotes are organisms whose cells have a nucleus and other organelles enclosed by a plasma membrane (Figure 3). Organelles are internal structures responsible for a variety of functions, such as energy production and protein synthesis. Figure 3: The key structures present in a eukaryote cell. Credit: Technology Networks. Innovative Approaches To Building Angiogenesis and Vascularized Systems This webinar will share approaches for creating complex, vascularized models that better reflect human biology, enabling more predictive and meaningful results for disease research and therapeutic testing. View Webinar Advertisement Eukaryotic cells are large (around 10–100 μm) and complex. While most eukaryotes are multicellular organisms, there are some single-cell eukaryotes.9 Eukaryotic cell features Within a eukaryotic cell, each membrane-bound structure carries out specific cellular functions. Here is an overview of many of the primary components of eukaryotic cells. Nucleus: The nucleus stores the genetic information in chromatin form. Nucleolus: Found inside of the nucleus, the nucleolus is the part of eukaryotic cells where ribosomal RNA is produced. Plasma membrane: The plasma membrane is a phospholipid bilayer that surrounds the entire cell and encompasses the organelles within. Cytoskeleton: The system of protein fibers and other molecules that gives shape to the cell, aiding in the correct positioning of organelles. Cell wall: The cell wall is only found in certain eukaryotes, such as plant cells, and is a rigid covering that provides structural support and protection to the cell. Ribosomes: Ribosomes are responsible for protein synthesis. Mitochondria:Mitochondria, also known as the powerhouses of the cell, are responsible for energy production. Cytoplasmic space: The cytoplasmic space is the region of the cell between the nuclear envelope and plasma membrane. Cytoplasm: The cytoplasm is defined as the total inner-cellular volume, with the exception of the nucleus, and includes the cytosol and all organelles. Cytosol: The cytosol, which consists of a gel-like substance, accounts for all the material in the cytoplasm, excluding the contents of the various membrane-bound organelles. Endoplasmic reticulum: The endoplasmic reticulum is an organelle dedicated to protein maturation and transportation. Vesicles and vacuoles: Vesicles and vacuoles are membrane-bound sacs involved in transportation and storage. Other common organelles found in many, but not all, eukaryotes include the Golgi apparatus, chloroplasts and lysosomes. Examples of eukaryotes Animals, plants, fungi, algae and protozoans are all eukaryotes. 1. Cooper GM. The Cell: A Molecular Approach. 2nd ed. Sunderland, MA: Sinauer Associates; 2000. Accessed January 29, 2025. 2. Archibald JM. Endosymbiosis and eukaryotic cell evolution. Curr Biol. 2015;25(19):R911-R921. doi: 10.1016/j.cub.2015.07.055 3. Wurzbacher Carmen E., Hammer Jonathan, Haufschild Tom, Wiegand Sandra, Kallscheuer Nicolai, Jogler Christian. “Candidatus Uabimicrobium helgolandensis”—a planctomycetal bacterium with phagocytosis-like prey cell engulfment, surface-dependent motility, and cell division. mBio. 2024;15(10):e02044-24. doi: 10.1128/mbio.02044-24 4. Karlin S, Mrázek J. Compositional differences within and between eukaryotic genomes. PNAS. 1997;94(19):10227-10232. doi: 10.1073/pnas.94.19.10227 5. Hinnebusch J, Tilly K. Linear plasmids and chromosomes in bacteria. Mol Microbiol. 1993;10(5):917-922. doi: 10.1111/j.1365-2958.1993.tb00963.x 6. Webster MW, Weixlbaumer A. The intricate relationship between transcription and translation. PNAS. 2021;118(21):e2106284118. doi: 10.1073/pnas.2106284118 7. Secaira-Morocho H, Chede A, Gonzalez-de-Salceda L, Garcia-Pichel F, Zhu Q. An evolutionary optimum amid moderate heritability in prokaryotic cell size. Cell Rep. 2024;43(6):114268. doi: 10.1016/j.celrep.2024.114268 8. Cole LA. Biology of Life. Academic Press; 2016:93-99. Accessed January 29, 2025. 9. Simon M, Plattner H. International Review of Cell and Molecular Biology. Academic Press; 2014:141-198. Accessed January 29, 2025. What characteristic best distinguishes prokaryotic cells from eukaryotic cells? Prokaryotes are unicellular and lack a nucleus and membrane-bound organelles. They are smaller and simpler and include bacteria and archaea. Eukaryotes are often multicellular and have a nucleus and membrane-bound organelles, which help to organize and compartmentalize cellular functions. They include animals, plants, fungi, algae and protozoans. How can you determine if a cell is prokaryotic or eukaryotic based on its structure? A prokaryotic cell lacks a nucleus and membrane-bound organelles, which are present in eukaryotic cells. In addition, prokaryotic cells tend to be simpler and smaller (0.1–5 μm) compared to eukaryotic cells that are more complex and larger (10–100 μm). What role does the nucleus play in differentiating eukaryotic cells from prokaryotic cells? Eukaryotic cells have a nucleus that separates the genetic material from the cytoplasm, while prokaryotic cells lack a nucleus and the genetic material is found in the cytoplasm. What organelles are exclusive to eukaryotic cells, and why are they absent in prokaryotic cells? The following organelles can be found in eukaryotic cells but not found in prokaryotic cells: nucleus, mitochondria, endoplasmic reticulum, lysosomes, Golgi apparatus and chloroplasts. Prokaryotic cells are simpler and smaller than eukaryotic cells and lack membrane-bound organelles. Scientists believe that eukaryotes evolved from prokaryotes around 2.7 billion years ago. Do prokaryotic cells and eukaryotic cells share any essential cellular processes? Prokaryotic and eukaryotic cells both share the processes of transcription and translation. However, in prokaryotic cells, transcription and translation are coupled, meaning translation begins during mRNA synthesis. In eukaryotic cells, transcription and translation are not coupled. Transcription occurs in the nucleus, producing mRNA. The mRNA then exits the nucleus, and translation occurs in the cell’s cytoplasm. How do the genetic materials of prokaryotic and eukaryotic cells differ in organization and location? The genetic material for eukaryotic cells is located within the nucleus, while the genetic material for prokaryotic cells is found in the cytoplasm. Eukaryote DNA consists of multiple molecules of double-stranded linear DNA, while that of prokaryotes is double-stranded and often circular. However, linear plasmids and chromosomes have been found in certain prokaryotes. What are the advantages of compartmentalization in eukaryotic cells over prokaryotic cells? In eukaryotic cells, specific cellular functions are compartmentalized into membrane-bound organelles. This compartmentalization improves the efficiency of many cellular functions and prevents potentially harmful molecules from freely roaming within the cell. What features in both prokaryotic and eukaryotic cells highlight their evolutionary connection? It is widely believed that mitochondria, the classical membrane-bound organelles of eukaryotic cells, evolved from prokaryotic cells by endosymbiosis. Mitochondria found within eukaryotic cells share several features with prokaryotic cells supporting theories of an evolutionary connection. These mitochondrial features include the presence of short and circular DNA, ribosomes that resemble prokaryotic ribosomes but differ from eukaryotic ribosomes and molecules that make up mitochondrial membranes resembling those in prokaryotic membranes. How do the reproduction mechanisms of prokaryotic and eukaryotic cells differ? Eukaryotic cells undergo mitosis and then cytokinesis. This involves the disintegration of the nuclear membrane followed by the sorting and separation of chromosomes to ensure that each daughter cell receives two sets of chromosomes. Following this, the cytoplasm divides, in a process known as cytokinesis, forming two genetically identical daughter cells. In contrast, prokaryotes undergo a simpler and faster process of binary fission. This involves DNA replication, chromosomal segregation and ultimately cell separation into two daughter cells genetically identical to the parent cell. Meet the Author Nicole Gleichmann Freelance Writer Nicole is a freelance writer specializing in biology, health and technology. She received her undergraduate degree in Organismal Biology from Scripps College in California. Here, her passion for holistic health and wellness began, leading to a previous position as a nutrition and wellness coach and expanding to freelance health writing. 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Δ Factors of 60 Updated: September 14, 2023 \| Category:Advanced Math, Factor We are writing a series of posts about factors to help you comprehend this mathematical topic and all of its complexities. In keeping with that, the focus of this blog will be on computing the factor of 60. We will explore two primary methods to determine them: Prime Factorization Factor Tree Method What Are the Factors of 60? The factors of 60 are whole numbers that yield a product of 60 when multiplied together. To elaborate, these numbers can be divided into 60 without leaving any remainder. \| Factors of 60 \| \| 1, 2, 3, 4, 5, 6, 10,12, 15, 20, 30 and 60 \| Recognizing these factors is indispensable as it breaks down numbers into their core components, simplifying numerous mathematical tasks. How to Determine the Factors of 60? Pinpointing the factors of 60 is a breeze with the following chief techniques: Using Prime Factorization This method breaks down a number into its prime components—numbers that are divisible only by 1 and themselves. Recommended Reading: Factors of 50 Through prime factorization, you can represent a number as a unique combination of prime numbers. Step 1: Start with the Smallest Prime Number Divide 60 by 2: 60 ÷ 2 = 30 Divide 30 by 2: 30 ÷ 2 = 15 Since 15 is not divisible by 2, divide 15 by 3: 15 ÷ 3 = 5 Step 2: Write Down the Prime Factorization The prime factorization of 60 is given by: 60 = 2 × 2 × 3 × 5 Step 3: List Down the Factors Factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 & 60. Using a Factor Tree A Factor Tree visually represents the decomposition of a number into its prime factors. Starting with the given number, keep dividing it until you reach prime numbers, constructing branches akin to a tree. Step 1: Begin with 60 and select 2 as a factor, as 60 is even: 60 / 2 30 Step 2: 30 is also even, so we divide it by 2: 60 / 2 30 / 2 15 Step 3: Now, split 15 into its prime factors: 60 / 2 30 / 2 15 / 3 5 From this tree, the prime factorization of 60 becomes evident: 2 × 2 × 3 × 5. Factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 & 60. Summary This guide has equipped you with the knowledge to adeptly discern the factors of 60 using the twin techniques of prime factorization and factor trees. Whether you’re an educator, student, or just someone with a penchant for mathematics, these strategies will surely augment your mathematical understanding and dexterity. Moonpreneur understands the needs and demands this rapidly changing technological world is bringing with it for our kids. Our expert-designedAdvanced Math course for grades 3rd, 4th, 5th, and 6th will help your child develop math skills with hands-on lessons, excite them to learn, and help them build real-life applications. 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Subscribe Notify of Label {}[+] Name Email [x] Δ Label {}[+] Name Email [x] Δ 0 Comments Inline Feedbacks View all comments RELATED ARTICALS Search Search Explore by Category Explore by Category MOST POPULAR Zero Factorial in Mathematics – Understanding 0! = 1 Read More » Rhombus vs Diamond: Key Differences Explained Simply Read More » Reciprocal in Mathematics \| Definition, Rules, and Solved Examples Read More » What is the Gauss Elimination Method? \| Definition, Steps & Examples Read More » GIVE A GIFT OF $10 MINECRAFT GIFT TO YOUR CHILD JOIN A FREE TRIAL CLASS Robotics Existing knowledge in programming/robotics No/Little Knowledge Fair Knowledge United States+1 United Kingdom+44 India (भारत)+91 Afghanistan (‫افغانستان‬‎)+93 Albania (Shqipëri)+355 Algeria (‫الجزائر‬‎)+213 American Samoa+1684 Andorra+376 Angola+244 Anguilla+1264 Antigua and Barbuda+1268 Argentina+54 Armenia (Հայաստան)+374 Aruba+297 Australia+61 Austria (Österreich)+43 Azerbaijan (Azərbaycan)+994 Bahamas+1242 Bahrain (‫البحرين‬‎)+973 Bangladesh (বাংলাদেশ)+880 Barbados+1246 Belarus (Беларусь)+375 Belgium (België)+32 Belize+501 Benin (Bénin)+229 Bermuda+1441 Bhutan (འབྲུག)+975 Bolivia+591 Bosnia and Herzegovina (Босна и Херцеговина)+387 Botswana+267 Brazil (Brasil)+55 British Indian Ocean Territory+246 British Virgin Islands+1284 Brunei+673 Bulgaria (България)+359 Burkina Faso+226 Burundi (Uburundi)+257 Cambodia (កម្ពុជា)+855 Cameroon (Cameroun)+237 Canada+1 Cape Verde (Kabu Verdi)+238 Caribbean Netherlands+599 Cayman Islands+1345 Central African Republic (République centrafricaine)+236 Chad (Tchad)+235 Chile+56 China (中国)+86 Christmas Island+61 Cocos (Keeling) Islands+61 Colombia+57 Comoros (‫جزر القمر‬‎)+269 Congo (DRC) (Jamhuri ya Kidemokrasia ya Kongo)+243 Congo (Republic) (Congo-Brazzaville)+242 Cook Islands+682 Costa Rica+506 Côte d’Ivoire+225 Croatia (Hrvatska)+385 Cuba+53 Curaçao+599 Cyprus (Κύπρος)+357 Czech Republic (Česká republika)+420 Denmark (Danmark)+45 Djibouti+253 Dominica+1767 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(‫الكويت‬‎)+965 Kyrgyzstan (Кыргызстан)+996 Laos (ລາວ)+856 Latvia (Latvija)+371 Lebanon (‫لبنان‬‎)+961 Lesotho+266 Liberia+231 Libya (‫ليبيا‬‎)+218 Liechtenstein+423 Lithuania (Lietuva)+370 Luxembourg+352 Macau (澳門)+853 Macedonia (FYROM) (Македонија)+389 Madagascar (Madagasikara)+261 Malawi+265 Malaysia+60 Maldives+960 Mali+223 Malta+356 Marshall Islands+692 Martinique+596 Mauritania (‫موريتانيا‬‎)+222 Mauritius (Moris)+230 Mayotte+262 Mexico (México)+52 Micronesia+691 Moldova (Republica Moldova)+373 Monaco+377 Mongolia (Монгол)+976 Montenegro (Crna Gora)+382 Montserrat+1664 Morocco (‫المغرب‬‎)+212 Mozambique (Moçambique)+258 Myanmar (Burma) (မြန်မာ)+95 Namibia (Namibië)+264 Nauru+674 Nepal (नेपाल)+977 Netherlands (Nederland)+31 New Caledonia (Nouvelle-Calédonie)+687 New Zealand+64 Nicaragua+505 Niger (Nijar)+227 Nigeria+234 Niue+683 Norfolk Island+672 North Korea (조선 민주주의 인민 공화국)+850 Northern Mariana Islands+1670 Norway (Norge)+47 Oman (‫عُمان‬‎)+968 Pakistan (‫پاکستان‬‎)+92 Palau+680 Palestine (‫فلسطين‬‎)+970 Panama (Panamá)+507 Papua New Guinea+675 Paraguay+595 Peru (Perú)+51 Philippines+63 Poland (Polska)+48 Portugal+351 Puerto Rico+1 Qatar (‫قطر‬‎)+974 Réunion (La Réunion)+262 Romania (România)+40 Russia (Россия)+7 Rwanda+250 Saint Barthélemy+590 Saint Helena+290 Saint Kitts and Nevis+1869 Saint Lucia+1758 Saint Martin (Saint-Martin (partie française))+590 Saint Pierre and Miquelon (Saint-Pierre-et-Miquelon)+508 Saint Vincent and the Grenadines+1784 Samoa+685 San Marino+378 São Tomé and Príncipe (São Tomé e Príncipe)+239 Saudi Arabia (‫المملكة العربية السعودية‬‎)+966 Senegal (Sénégal)+221 Serbia (Србија)+381 Seychelles+248 Sierra Leone+232 Singapore+65 Sint Maarten+1721 Slovakia (Slovensko)+421 Slovenia (Slovenija)+386 Solomon Islands+677 Somalia (Soomaaliya)+252 South Africa+27 South Korea (대한민국)+82 South Sudan (‫جنوب السودان‬‎)+211 Spain (España)+34 Sri Lanka (ශ්‍රී ලංකාව)+94 Sudan (‫السودان‬‎)+249 Suriname+597 Svalbard and Jan Mayen+47 Swaziland+268 Sweden 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2851	https://math.stackexchange.com/questions/1856042/what-are-some-standard-methods-for-solving-functional-equations	Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams What are some standard methods for solving functional equations? Ask Question Asked Modified 8 years ago Viewed 1k times 9 $\begingroup$ I have searched the internet for methods on solving functional equations, unfortunately, most of them consist mainly of substituting values for the variables or on guessing solutions. I think those methods sound like either an impossible task due to the infinite number of possible substitutions or a leap of faith due also to the infinite number of guesses there can be. How could one for example solve: $$f(1+x)=1+f(x)^2$$ I have tried many different ideas on this, from trying guesses to applying integrals, to eliminating the $1$ by calculating $f(2+x)$ and subtracting equations, then trying to simplify the answer through sums or products, but nothing seems to work. I think this is a rather interesting subject because $f(x)=f(x-1)^2$ has a solution that is very easy to find but somehow adding a $1$ makes it a much harder problem. Are there ways to solve this equation or even a polynomial functional equation? functions functional-equations Share asked Jul 11, 2016 at 15:17 GuPeGuPe 7,53055 gold badges3737 silver badges6868 bronze badges $\endgroup$ 6 1 $\begingroup$ For this example you can take f to be anything on [0,1), then the rule determines what f is on [1,2), etc... $\endgroup$ mathematician – mathematician 2016-07-11 15:32:37 +00:00 Commented Jul 11, 2016 at 15:32 $\begingroup$ But can it be solved? Can $f$ be expressed explicitely? $\endgroup$ GuPe – GuPe 2016-07-11 16:48:37 +00:00 Commented Jul 11, 2016 at 16:48 2 $\begingroup$ There are so many solutions that you have to be more specific about what you're looking for. For example, the equation has no polynomial solutions (Let $f(x)= a_nx^n+...+a_0$, you get a $2n$ degree polynomial equals an $n$ degree one, contradiction). You could plug in f(x)= a power series, and derive some relationship between the coefficients. $\endgroup$ mathematician – mathematician 2016-07-11 17:10:18 +00:00 Commented Jul 11, 2016 at 17:10 $\begingroup$ Perhaps the difference between your easily-solvable and unsolvable examples can be traced back to the differences between a homogeneous (easy case) and inhomogeneous (hard case) difference equation? I think your two examples are not exactly difference equations, but maybe you can find some insight from that point of view? More generally, I recently stumbled onto this website. $\endgroup$ jjstankowicz – jjstankowicz 2016-07-18 05:22:01 +00:00 Commented Jul 18, 2016 at 5:22 1 $\begingroup$ With @mathematician comments, the problem as stated is solved. If you add some smoothness condition, such as ask the function to be continuos or differentiable, I guess maybe by trying to set the "initial condition" on [0,1) of the same smootheness, and some "boundary condition" on the 0 and the 1 to match continuity/differentiability in the integers, that should be steps toward solution of that problem ,but the stated one in question, pointed how to solve it. $\endgroup$ Santropedro – Santropedro 2017-09-21 23:58:16 +00:00 Commented Sep 21, 2017 at 23:58 \| Show 1 more comment 1 Answer 1 Reset to default 4 $\begingroup$ As for iterating polynomials, there are two general forms of polynomials that we currently know how to iterate. They are polynomials of the form $$P(x)=a(x-b)^d+b$$ and $$T(x)=\cos(k\arccos(x))$$ and yes, I know the second one doesn't look like a polynomial, but for integer values of $k$, it is a polynomial where the inverse cosine is defined, and can be extended to other values (accurately, for our purposes) using the hyperbolic cosine function. For more info, see Chebyshev Polynomials. The iteration formulas for each of these is given by $$P^n(x)=a^{\frac{b^n-1}{b-1}}(x-b)^{d^n}+b$$ and $$T^n(x)=\cos(k^n\arccos(x))$$ In case you were wondering how this applies to your problem of finding $f$ given that $$f(1+n)=(Q\circ f)(n)$$ where $Q$ is a polynomial, here's how: if you assign a value for $f(0)$, say $y_0$, then you can say that $$f(n)=(Q^n\circ f)(0)=Q^n(y_0)$$ which allows you to find $f:\mathbb Z\to\mathbb Z$. Share answered Sep 21, 2017 at 23:22 Franklin Pezzuti DyerFranklin Pezzuti Dyer 41.1k99 gold badges8282 silver badges176176 bronze badges $\endgroup$ Add a comment \| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions functions functional-equations See similar questions with these tags. Featured on Meta Introducing a new proactive anti-spam measure Spevacus has joined us as a Community Manager stackoverflow.ai - rebuilt for attribution Community Asks Sprint Announcement - September 2025 Related 1 Solving Functional Equations How to solve this infinite set of equations? 4 Are there algorithms for solving simple functional equations? 9 Functional Equations with Taylor Series 2 Find all functions $ f: \mathbb{R} \to \mathbb{R}$ satisfying $f(x + y) = x + f(y)$ 1 What are some real-world applications of functional equations? 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2852	https://www.kenhub.com/en/library/anatomy/epidermis	Connection lost. Please refresh the page. #1 platform for learning anatomy Login Register Success stories Anatomy Basics Upper limb Lower limb Spine and back Thorax Abdomen Pelvis and perineum Head and neck Neuroanatomy Cross sections Radiological anatomy+ Histology Types of tissues Body systems+ Physiology Introduction Muscular system Nervous system Anatomy Basics Upper limb Lower limb Spine and back Thorax Abdomen Pelvis and perineum Head and neck Neuroanatomy Cross sections Radiological anatomy+ Histology Types of tissues+ Physiology Nervous system Get help How to study What's new? Kenhub in... Deutsch Português Español Français What's new? Get help How to study English English Deutsch Português Español Français Login Register #1 platform for learning anatomy Courses Anatomy Basics Upper limb Lower limb Spine and back Thorax Abdomen Pelvis and perineum Head and neck Neuroanatomy Cross sections Radiological anatomy Histology Types of tissues Body systems Physiology Introduction Muscular system Nervous system Articles Anatomy Basics Upper limb Lower limb Spine and back Thorax Abdomen Pelvis and perineum Head and neck Neuroanatomy Cross sections Radiological anatomy Histology Types of tissues Physiology Nervous system The #1 platform to learn anatomy 6,327,300 users worldwide Exam success since 2011 Serving healthcare students globally Reviewed by medical experts 2,907 articles, quizzes and videos ArticlesAnatomyBasicsIntroduction to the other systemsEpidermis Table of contents Ready to learn? Pick your favorite study tool Videos Quizzes Both Register now and grab your free ultimate anatomy study guide! ArticlesAnatomyBasicsIntroduction to the other systemsEpidermis Epidermis Author: Roberto Grujičić, MD • Reviewer: Dimitrios Mytilinaios, MD, PhD Last reviewed: October 30, 2023 Reading time: 2 minutes Recommended video: Integumentary system [19:28] Structure and layers of the skin. Epidermis 1/4 Synonyms: none The epidermis is the most superficial layer of the skin. The other two layers beneath the epidermis are the dermis and hypodermis. The epidermis is also comprised of several layers including the stratum basale, stratum spisosum, stratum granulosum, stratum lucidum, and stratum corneum. The number of layers and thickness of the epidermal layer depends on the location in the body. For example, the epidermis that covers the heel region is much thicker than the epidermis that covers the eyelid. The main cells of the epidermis are the keratinocytes. These cells originate in the basal layer and produce the main protein of the epidermis called the keratin. Other cells located in the epidermis are: Melanocytes (produce skin pigment) Langerhans’ cells (immune, antigen-presenting cells) Merkel’s cell (mechanoreceptors for light touch) The main function of the epidermis is to protect the deeper tissues from water, microorganisms, mechanical and chemical trauma, and damage from UV light. In addition, the epidermis continuously makes new skin that replaces the old skin cells and produces melanin that provides skin color. \| \| \| --- \| \| Terminology \| English: EpidermisLatin: Epidermis \| \| Definition \| Outermost layer of the skin \| \| Layers \| Stratum basale, stratum spinosum, stratum granulosum, stratum lucidum, stratum corneum \| \| Functions \| Protection, skin regeneration, skin color \| Learn everything about the components of the integumentary system with the following study unit: Learn faster Integumentary system Explore study unit Sources All content published on Kenhub is reviewed by medical and anatomy experts. The information we provide is grounded on academic literature and peer-reviewed research. Kenhub does not provide medical advice. You can learn more about our content creation and review standards by reading our content quality guidelines. Yousef H, Alhajj M, Sharma S. Anatomy, Skin (Integument), Epidermis. [Updated 2021 Nov 19]. In: StatPearls [Internet]. Treasure Island (FL): StatPearls Publishing; 2022. Moore, K. L., Dalley, A. F., & Agur, A. (2017). Clinically oriented anatomy (8th ed.). Lippincott Williams and Wilkins. Epidermis: want to learn more about it? Our engaging videos, interactive quizzes, in-depth articles and HD atlas are here to get you top results faster. What do you prefer to learn with? Videos Quizzes Both “I would honestly say that Kenhub cut my study time in half.” – Read more. 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2853	https://artofproblemsolving.com/wiki/index.php/Root-Mean_Square-Arithmetic_Mean-Geometric_Mean-Harmonic_mean_Inequality?srsltid=AfmBOoqfRQ1S8hkXWv2O2tMoKVHHogu8cAQgqlpIaVCVVrOIWilrKWjT	Art of Problem Solving Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality - AoPS Wiki Art of Problem Solving AoPS Online Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online ‚ Books for Grades 5-12Online Courses Beast Academy Engaging math books and online learning for students ages 6-13. Visit Beast Academy ‚ Books for Ages 6-13Beast Academy Online AoPS Academy Small live classes for advanced math and language arts learners in grades 2-12. Visit AoPS Academy ‚ Find a Physical CampusVisit the Virtual Campus Sign In Register online school Class ScheduleRecommendationsOlympiad CoursesFree Sessions books tore AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates community ForumsContestsSearchHelp resources math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten contests on aopsPractice Math ContestsUSABO newsAoPS BlogWebinars view all 0 Sign In Register AoPS Wiki ResourcesAops Wiki Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality Page ArticleDiscussionView sourceHistory Toolbox Recent changesRandom pageHelpWhat links hereSpecial pages Search Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality The Root-Mean Power-Arithmetic Mean-Geometric Mean-Harmonic Mean Inequality (RMP-AM-GM-HM) or Exponential Mean-Arithmetic Mean-Geometric Mean-Harmonic Mean Inequality (EM-AM-GM-HM) or Quadratic Mean-Arithmetic Mean-Geometric Mean-Harmonic Mean Inequality (QM-AM-GM-HM), is an inequality of the root-mean power, arithmetic mean, geometric mean, and harmonic mean of a set of positivereal numbers that says: , where , and is the . The geometric mean is the theoretical existence if the root mean power equals 0, which we couldn't calculate using radicals because the 0th root of any number is undefined when the number's absolute value is greater than or equal to 1. This creates the indeterminate form of . Then, we can say that the limit as x goes to 0 is the geometric mean of the numbers. The quadratic mean's root mean power is 2 and the arithmetic mean's root mean power is 1, as and the harmonic mean's root mean power is -1 as . Similarly, there is a root mean cube (or cubic mean), whose root mean power equals 3. When the root mean power approaches , the mean approaches the highest number. When the root mean power reaches , the mean approaches the lowest number. with equality if and only if . This inequality can be expanded to the power mean inequality, and is also known as the Mean Inequality Chain. As a consequence, we can have the following inequality: If are positive reals, then with equality if and only if ; which follows directly by cross multiplication from the AM-HM inequality. This is extremely useful in problem-solving. The Root Mean Power of 2 is also known as the quadratic mean, and the inequality is therefore sometimes known as the QM-AM-GM-HM Inequality. Proof The inequality is a direct consequence of the Cauchy-Schwarz Inequality; Alternatively, the RMS-AM can be proved using Jensen's inequality: Suppose we let (We know that is convex because and therefore ). We have: Factoring out the yields: Taking the square root to both sides (remember that both are positive): The inequality is called the AM-GM inequality, and proofs can be found here. The inequality is a direct consequence of AM-GM; , so , so . Therefore, the original inequality is true. Geometric Proofs The inequality is clearly shown in this diagram for Desmos SlidersDesmos Equation NOTE: The Desmos equation will not show the line when the numbers are negative. (Note how the RMS is "sandwiched" between the minimum and the maximum) Retrieved from " Categories: Algebra Inequalities Art of Problem Solving is an ACS WASC Accredited School aops programs AoPS Online Beast Academy AoPS Academy About About AoPS Our Team Our History Jobs AoPS Blog Site Info Terms Privacy Contact Us follow us Subscribe for news and updates © 2025 AoPS Incorporated © 2025 Art of Problem Solving About Us•Contact Us•Terms•Privacy Copyright © 2025 Art of Problem Solving Something appears to not have loaded correctly. Click to refresh.
2854	http://web.utk.edu/~rstrange/wfs556/html-content/04-buffering.html	Recirculation Aquaculture: Buffering 4. Water Quality: Buffering It Is All about the Carbon Cycle The interplay between the carbon-based life on earth with the physical biosphere results in the profoundly important carbon cycle, which is responsible for everything from the earth's food supply to free oxygen in the atmosphere to global warming. One aspect of the carbon cycle that is of paramount importance to fish is the relationship between inorganic carbon and water. The vast majority of carbon on earth (six times more than the carbon in all the organic matter, living and dead, including fossil fuels) resides in the earth's water in the form of carbonates (carbon/oxygen compounds). These form the basis of the carbonate buffering system, which is the key to successful fish life. As explained in an earlier chapter, (3. Water Quality: Gases) oxygen (and nitrogen for that matter) can dissolve in water and the amount of these gases in water is a matter of partial pressure and solubility. Carbon dioxide goes into simple solution, too (in fact, it is far more soluble than oxygen or nitrogen), but it also reacts chemically with the water molecule to form bicarbonate and carbonate. Once carbon dioxide turns into a carbonate it is no longer part of the gas pressure that governs the solubility and more carbon dioxide can dissolve. The form that inorganic carbon in water takes is a function of pH. As pH changes, the predominant species (to use the word in chemical sense) changes. It is a yin/yang relationship because when the ratio between the species of inorganic carbon changes the pH must change, as well. This is what makes the bicarbonate (the middle form) a buffer. What is buffering? A buffer is a chemical compound in solution that acts to keep water within a certain pH range. Acidity is the presence of "naked protons", that is, positively charged hydrogen atoms or hydroniums (H+). They are very reactive because they "steal" electrons from other compounds, thus damaging those molecules. An acid burn is the cumulative effect of these reactions. At the other extreme, base is the presence of a strong electron donor, the hydroxyl ion (OH -). A strong base is as reactive as a strong acid. Since biological processes are very sensitive to pH, it needs to be controlled in a recirculation system. Fortunately, nature provides a means through the carbonate buffering system. Carbonate buffering occurs to varying degrees in the waters of the world. In nature, it typically derives from limestone in the watershed. Carbonate buffering is measured in terms of alkalinity. This word may be misleading because, in this case, it doesn't mean the absence of acidity, but the presence of buffering. Buffering capacity is measured by titrating a water sample with a known concentration of acid in the presence of a pH indicator. As long as buffering capacity remains, the pH doesn't change. However, as soon as the buffering capacity is exhausted, the next drop of acid will cause a rapid drop in pH. Alkalinity is expressed as mg/L of CaCO 3 (these units assume that limestone is the source of the buffering, though this is not always the case). In natural waters, alkalinity ranges from as low as 10 to several hundred. Hardness is not the same as alkalinity, but is often similar in a given sample. In essence, carbonate buffering is the presence of the bicarbonate ion (HCO 3-) which has the ability to react with either hydroniums or hydroxyls, neutralizing them. pH Management As you will learn in the next chapter, it is important to keep the pH of a recirculation system between 7and 8. The carbonate buffering system coupled with control of CO 2 exchange allows the aquaculturist to do this relatively easily. Limestone is not a good source of alkalinity for recirculation aquaculture because the dissolution of limestone by carbonic acid, as shown in the game above, is slow. A much faster way to increase the bicarbonate content of water is to add sodium bicarbonate (NaHCO 3), also known as baking soda! Carbon dioxide, bicarbonate, and carbonate are all just different forms of inorganic carbon in water. Which species predominates is determined by pH. CO 2 predominates at low pH, HCO 3- at near neutral pH, and CO 3-- at high pH. In fact, alkalinity, pH, and CO 2 each may be calculated from the other two. In recirculation aquaculture, the tendency is for alkalinity to decline over time as it performs its buffering action. If buffering capacity falls too low, the CO 2 produced by the fish and bacteria will cause a decrease in pH. If pH falls below 7, it can cause problems. The solution is for the aquaculturist to periodically to replace lost buffering capacity by adding sodium bicarbonate, thus keeping alkalinity up and pH in the desired range. Scoopfuls of sodium bicarbonate can safely be added directly to the fish tank to correct a pH/alkalinity imbalance. The other part of pH management involves CO 2 stripping. When air is used as an oxygen source the aeration needed to increase the oxygen will also remove CO 2, so CO 2 stripping is essentially part of aeration. When liquid oxygen is employed, however, dedicated CO 2 stripping is often required. This is typically done prior to oxygenation so that any O 2 added from the air spares costly LOX. A typical CO 2 stripper is a cylinder filled with inert substrate and a flow of air running counter to the trickle of water over the substrate. The substrate breaks the water flow into a large surface area and the opposite flow of water and air optimizes the gas exchange. Measuring Alkalinity and pH Arguably, alkalinity is the single most important water quality measurement that can be made in regards to fish, yet a lot of confusion surrounds it. First, this word may be misleading because, in this case, it doesn't mean the absence of acidity, but the presence of buffering. Quite simply, alkalinity is a measure of buffering capacity. To measure it, a sample is titrated with dilute acid until the buffering is exhausted and the pH falls (indicated by a color change of a pH indicator). The units of alkalinity are typically mg/L as CaCO 3, though some kits yield results in antiquated units like grains per gallon. What mg/L as CaCO 3 means is that a given sample has the buffering capacity equivalent to the carbonate buffering that would come from the dissolution of that quantity of limestone. Water hardness is similar to alkalinity but not exactly the same. Hardness is a measure of the ions Mg and Ca (and some relatively rare metallic ones like Al, Fe, Mn) in the water and is directly relevant to lathering of soap. In nature, most buffering comes from Ca compounds, so hardness and buffering are usually highly correlated, but they don't have to be. For example, in a later chapter the use of baking soda, sodium bicarbonate (NaHCO 3), to achieve buffering without increasing hardness (no Mg or Ca) will be discussed. Another point of confusion is so-called "carbonate hardness", sometimes abbreviated KH (from the German?). It is the portion of total hardness that is equivalent to total alkalinity. If hardness is greater than total alkalinity the amount of hardness in excess of total alkalinity is the "noncarbonate hardness". If the total hardness is less than total alkalinity, as it would be if sodium bicarbonate was used to boost buffering, all the hardness is "carbonate hardness" and there is no "non-carbonate hardness". The important point here is that total alkalinity can exceed "carbonate hardness" (though carbonate hardness cannot exceed total alkalinity). Best to ignore hardness altogether and measure total alkalinity. Finally, total alkalinity can be further broken down into carbonate and bicarbonate alkalinity. Carbonate alkalinity only exists at pH's over 8 and is not generally a consideration in fish ponds. Consequently, the phenolphthalein endpoint in alkalinity titrations can be ignored. Bicarbonate alkalinity buffers our fish's water. Alkalinity is slowly lost over time as it reacts with acids or bases, but does not change according to season or time of day. A tank's pH may be measured colorimetrically or with a meter. As discussed above, pH and alkalinity are linked, so measuring one will give information about the other. Also, pH must be known to evaluate the meaning of the total ammonia value. Assignment 4
2855	https://petbioufma.wordpress.com/wp-content/uploads/2017/06/ana-carolina-loreti2016.pdf	Plant responses to ﬂooding stress Elena Loreti1, Hans van Veen2 and Pierdomenico Perata2 Most plant species cannot survive prolonged submergence or soil waterlogging. Crops are particularly intolerant to the lack of oxygen arising from submergence. Rice can instead germinate and grow even if submerged. The molecular basis for rice tolerance was recently unveiled and will contribute to the development of better rice varieties, well adapted to ﬂooding. The oxygen sensing mechanism was also recently discovered. This system likely operates in all plant species and relies on the oxygen-dependent destabilization of the group VII ethylene response factors (ERFVIIs), a cluster of ethylene responsive transcription factors. An homeostatic mechanism that controls gene expression in plants subjected to hypoxia prevents excessive activation of the anaerobic metabolism that could be detrimental to surviving the stress. Addresses 1 Institute of Agricultural Biology and Biotechnology, CNR, Pisa, Italy 2 PlantLab, Institute of Life Sciences, Scuola Superiore Sant’Anna, 56124 Pisa, Italy Corresponding author: Perata, Pierdomenico (p.perata@sssup.it) Current Opinion in Plant Biology 2016, 33:64–71 This review comes from a themed issue on Cell signalling and gene regulation Edited by Kimberley Snowden and Dirk Inze ´ 1369-5266/# 2016 Elsevier Ltd. All rights reserved. Introduction Although plants produce oxygen through photosynthesis, the lack of an efﬁcient system to transport oxygen to non-photosynthetic organs implies that these organs can be deprived of oxygen if their anatomy limits oxygen diffu-sion from outside [1,2]. Additionally, complete submer-gence of the plant by ﬂooding events may also lead to low-oxygen availability in the aboveground organs, especially when water turbidity limits photosynthesis . When oxygen becomes limiting for respiration plants experience hypoxia, whilst the complete absence of oxygen (anoxia) is even more detrimental to plant survival. Both hypoxia and anoxia trigger extensive reprogramming of gene expres-sion, with induction of the fermentative metabolism, allowing the plant to use glycolysis for ATP production . Climate changes will lead to extremes in water avail-ability that will cause severe drought in some areas, while ﬂooding due to extreme rainfall events will affect other geographical areas . Unless new crop varieties able to withstand abiotic stresses are developed, productivity will be gravely affected. Until a decade ago little was known about the genes that confer tolerance to submergence, and it is only during recent years that light has been shed on the molecular mechanisms behind oxygen sensing and signalling in plants . In this review we will highlight the most recent ﬁndings in the ﬁeld of plant anaerobiosis, from ecophysiology of plants growing in wetlands to the trans-lation of discoveries made in Arabidopsis to crops. Flooding in the wild Flooding is a natural occurrence in many ecosystems and therefore many wild species are superbly adapted to watery conditions. Here improved gas exchange with the environ-ment is essential to avoid hypoxia within the plant. To this end, plants can induce and/or constitutively develop aeren-chyma, longitudinal connected gas spaces, which provide a rapid means of aerial gas exchange over long distances within the plant . This is usually combined with a change in root architecture to minimize the distance (and therefore diffusive resistance) between the aerial surface and the ﬂooded root tips , for instance via adventitious roots, which can create a collection of air conducting snorkels originating from the hypocotyl or stem into the anaerobic substrate. Often aerenchyma are combined with a barrier that prevents oxygen leakage into the surrounding anaero-bic soil, which drastically improves ﬂooding tolerance . An extensive aerenchyma system is extremely effective under waterlogged conditions where the shoot remains in aerial contact and can thus funnel air down to the root. During complete submergence, however, the shoot does not make aerial contact oxygen, their effectiveness in funnelling air towards the roots is greatly compromised. In such cases, some wetland plant species, in an attempt to regain aerial shoot contact, display rapid vertical elonga-tion of leaves, internodes or petioles to snorkel for air. This escape strategy is observed in some rice varieties (see below), as well as in several other plant species . In an alternative strategy the plant aims to enter a state of inactivity (quiescence), to be revived once the ﬂood recedes [9,10]. This is also a difﬁcult tactic as energy and carbon utilisation should be kept to a minimum to make reserves last a long time, whilst they should simultaneously be sufﬁcient to maintain cellular integrity (Figure 1a). The submerged plant: low oxygen and high ethylene Because of its gaseous nature ethylene hardly leaves the plant under ﬂooded conditions and thus rapidly Available online at www.sciencedirect.com ScienceDirect Current Opinion in Plant Biology 2016, 33:64–71 www.sciencedirect.com accumulates inside the plant. It is therefore a highly reliable and rapid cue for plants to detect their predica-ment . Another signal is the oxygen availability. The internal level of these gases is a balance between con-sumption, production and diffusive resistance. Therefore active, heterotrophic or compact tissue, such as meristems and roots, will rapidly experience low oxygen upon ﬂood-ing. In photosynthetic tissue the consumption and pro-duction of oxygen is dependent on light conditions, and thus also the oxygen availability. Ethylene is the primary signal for most adaptations to ﬂooding. Ethylene modulates a hormonal cascade of ABA, GA and ultimately auxin to induce adventitious rooting in tomato, Solanum dulcamarum, and rice, and [12–14]. How-ever, root emergence also requires ethylene induce ROS formation in the epidermal cells, leading to their cell death to allow root penetration . Similarly, lysigenous aeren-chyma formation, which is formed by apoptosis of speciﬁc cells in the cortex, involves an ethylene dependent drop in antioxidant activity. The subsequent increase in ROS leads to the required cell death [16–18]. Interestingly, the important suberin based oxygen barrier is not affected by ethylene, but likely causal genes involved in its forma-tion have been identiﬁed . The escape strategy to reach the water surface is also ethylene driven. However, downstream signalling is considered divergent in the plant kingdom, as it was found to act via group VII ERFs in rice (see below), but via genes typical of low light induced elongation in Rumex palustris [20–22]. Remarkably, ethyl-ene pre-treatment induced anoxia tolerance of Rumex palustris was associated with enhanced hypoxia related gene expression. A behaviour that was absent in Rumex acetosa, a species that experience fewer ﬂooding events and employs a quiescence instead of an escape strategy . This highlights the importance of a link between ethylene and hypoxic signalling pathways. The high levels of ethylene associated with ﬂooding inhibit root elongation, but through the formation of aerenchyma the excessive ethylene is easily removed. However, species that are ineffective in producing aeren-chyma therefore experience strong root growth reduction under ﬂooded conditions . The strong dose depen-dency of ethylene signalling might play an important role in its contrasting developmental roles during ﬂooding (Figure 2). To avoid detrimental effects associated with high levels of ethylene, some of the species that continu-ously occupy aquatic or ﬂood-prone environments have Plant responses to flooding stress Loreti, van Veen and Perata 65 Figure 1 (a) flooding ethylene entrapment adventitious root formation root elongation Iysigenous aerenchyma ↑ ROS localized cell death in root cortex photosynthesis escape escape quiescence quiescence short term long term reserve mobilisation glycolytic flux shoot elongation carbon starvation adv. roots/ aerenchyma O2 levels ↑ NADPH-oxidase ↓ metallothionein ↓ ABA ↑GA, IAA ↑ epidermal ROS (b) Current Opinion in Plant Biology Ethylene is a pivotal regulator growth survival strategies (a) and root development (b) during submergence and waterlogging. During complete submergence, ethylene induced growth strategies are paramount to survival (a), but both have different short term and long term effects on plant performance, especially since under long term submergence escaping plants will have regained aerial contact. Naturally, photosynthesis is severely reduced by flooding, but through an escape strategy some photosynthesis can be recovered through the re-establishment of aerial contact. This subsequently reduces the need for reserve mobilisation and limits oxygen shortage via aerenchyma. Initially, escaping plants will have low internal O2 levels, due to their high metabolic activity which is fuelled by a strong glycolytic flux and reserve mobilisation. These high demands, generally mean escaping plants suffer strongly from carbon starvation. Though all these effects are ameliorated once aerial contact is made. Because quiescent plants have low activity, their requirements on reserves and energy are limited. Subsequently, internal O2 levels would be at a higher steady state and carbon shortage would be considerable lower. However, quiescent plants still rely on reserve mobilisation to sustain cellular functions, both during short term and long term flooding. Root development is also essential to survive flooded conditions. The aqueous environment prevents ethylene to readily leave the plant tissues through gas diffusion. This ethylene entrapment starts a cascade that leads a change in root architecture (b), through for instance the formation of adventitious roots. This includes a hormonal cascade and ROS induced epidermal cell death. Simultaneously, high levels of ethylene in submerged roots, inhibit root elongation. However, ethylene also leads to a drop in the antioxidant metallothionein and an increase NADPH oxidase, which together leads to an accumulation of ROS. ROS acts as a signal for programmed cell death of specific cortex cells, eventually leading to the formation of lysigenous aerenchyma. As a result, the improved gas diffusion can remove high ethylene levels and thus releases the inhibition on root elongation. www.sciencedirect.com Current Opinion in Plant Biology 2016, 33:64–71 lost or reduced their capacity to either produce, sense or respond to ethylene [24,25]. Darkness, a typical component of ﬂooding in murky water, is responsible for a large portion of the transcriptomic changes observed during complete submergence in the dark in Arabidopsis . This indicates that acclimation to ﬂooding, at least in dark conditions, predominately occurs via sugar and energy signalling, as also was shown in rice . Where the contribution of hypoxia in regulating gene expression in dark submergence acclimation can be minor, hypoxia regulated gene expression is correlated, either positively or negatively, to ﬂooding tolerance in natural variation of Arabidopsis, Rumex and Rorippa, which makes it an important area of study [20,26,28]. Flooding in the ﬁeld: rice Rice is remarkably well adapted to submergence (Figure 2) and can even germinate in the complete absence of oxygen . This anaerobic germination (AG) includes a lengthening of the coleoptile, that, anal-ogous to the escape strategy, aims to make aerial contact but considerable variation exists among rice genotypes in coleoptile extension during anoxia . Differently from other cereal seeds that fail to induce the a-amylase enzymes required for starch degradation under anoxia, rice caryopses produce this enzyme, which allows starch degradation coupled to the fermentative metabolism and subsequent germination [30–32]. The rapid depletion of soluble carbohydrates occurring during the ﬁrst hours of germination under anoxia, together with a possible low-oxygen dependent change in calcium levels, leads to a signalling cascade that ﬁnally leads to a-amylase 66 Cell signalling and gene regulation Figure 2 (a) O2 O2 Gibberellin α-Amylase α-Amylase Starch Germination Sugar starvation T6P Trehalose Sucrose Glucose Anaerobic metabolism Germination Air Submergence Ethylene Gibberellin Elongation Growth SK1, SK2 SUB1A Starch CIPK15 OsTPP7 Glucose Aerobic Respiration (b) (c) Current Opinion in Plant Biology Rice germination and growth under aerial (a) and submerged (b) conditions is regulated at different levels, depending on the genotype as well as the growth-stage. Rice germination under anoxia is very peculiar, with rapid coleoptile elongation (a, b). Only once the water surface is reached and the coleoptile can act as a snorkel do the root and primary leaf develop. Germination under anoxia is extremely challenging because ATP can only be produced through the activity of glycolysis coupled with ethanolic fermentation, which yields only a fraction of the ATP produced by mitochondrial respiration, ready access to starch reserves is thus essential. Under anoxia or hypoxia starch degradation through the gibberellin-induced a-amylase pathway cannot occur because oxygen is required for gibberellin synthesis (a) and also because rice fails to respond to gibberellins under low-oxygen conditions. In anaerobically germinating rice varieties the low-oxygen conditions (b) require starch degradation through the action of a-amylases, some of which are induced by sugar starvation, rather than gibberellin, and in a feed-back manner is repressed by increased availability of sugars. This feedback loop between sugar starvation and a-amylase acts via a pathway requiring CIPK15. Viable anaerobic germination requires OsTPP7 to reduce the perception of sugars so that sugar induced inhibition a-amylase is prevented, resulting in a strong flux sugars released via starch degradation. This allows rice to feed the anaerobic metabolism with sugars and obtain enough ATP to support germination. In adult rice plants (c) different strategies are observed that allow the rice plant to survive submergence. Submergence results in ethylene accumulation, that induces SUB1A in genotypes possessing this gene. SUB1A represses growth of submerged plants, thus allowing the plant to preserve carbon reserves, which in turn will allow re-growth of the plant when the water recedes. Instead in deepwater rice varieties ethylene induces the SK genes, which induce fast stem elongation. This results in an ‘escape’ strategy that allows the plant to keep its leaves above the water surface, thus allowing oxygen to be transported to the submerged parts of the plant through the aerenchyma. Rice varieties that do not possess either SUB1A or SK genes display an intermediate phenotype, with slow stem elongation that depletes the plant from carbon resources without allowing to gain aerial contact. Current Opinion in Plant Biology 2016, 33:64–71 www.sciencedirect.com production. This process begins with the activation of a Calcineurin B-like (CBL), which targets the protein ki-nase CIPK15, which in turn triggers the SnRK1A pathway that induces the MYBS1 transcription factor which acti-vates the starvation-inducible a-amylase gene RAmy3D . There is considerable variation amongst rice varieties in their ability to successfully germinate and establish when submerged in the ﬁeld, though most activate RAmy3D during this anaerobic germination. AG of rice allows direct sowing instead of transplanting, which is of great importance as it makes rice cultivation more eco-nomically sustainable . A QTL analysis identiﬁed OsTPP7 as the locus responsible for efﬁcient AG. OsTPP7 encodes a trehalose-6-P-phosphate (T6P) phosphatase, which is non-functional in rice varieties that are unable to establish under submerged conditions . The pres-ence of the OsTPP7 in rice accessions was correlated with increased sink strength of elongating coleoptiles, resulting in prolonged tolerance to complete submergence. High sucrose results in high T6P levels and consequently in repression of SnrK1 and downregulation of a-amylases. During anaerobic germination OsTPP7 misleads the seed-ling about its sugar status by converting T6P into treha-lose. Subsequently the rice seedling can maintain a relative high sugar availability but low T6P levels, which, if high, would repress a-amylases (Figure 2b). The subse-quent intense ﬂux of glucose from starch degradation is essential for fuelling glycolysis and lengthening of the coleoptile. Rice germination under anoxia is therefore the consequence of clever sugar management, that allows adept access to starch reserves . To this aim the ﬁne tuning of sugar sensing by keeping lower T6P levels for a given sucrose concentration by the low-oxygen inducible OsTPP7 appears to be essential . In some areas of Asia submergence occurs very rapidly and lasts for months, here rice varieties named ‘deep-water rice’ are grown. The adult plant continues to snorkel for air and keeps up with the increasing water level. This trait relies on two group VII ERF genes: SNORKEL1 and SNORKEL2 (SK1, SK2) . Only pres-ent in deep water rice varieties, they activate a gibberel-lin-dependent internode elongation, up to 25 cm per day, sufﬁcient to maintain an aerial contact with some of the leaves which allow air transfer to the submerged parts of the plant via aerenchyma (Figure 2c). Clearly the success of rice in ﬂooded habitats is due to its ability to rapidly regain aerial contact . Interest-ingly, only a few rice varieties can survive complete submergence for an extended period of time, a phe-nomena that regularly occurs in so-called ﬂash-ﬂoods. These varieties survive thanks to the group VII ERF gene SUB1A , whose product positively regulates the fermentation capacity, but represses plant growth by restricting gibberellin-signalling [37,38]. Therefore, rice varieties that survive complete submergence activate, through SUB1A, a quiescence strategy that allows them to reduce carbohydrate use to the mini-mum required for keeping the plant alive, while it waits for water to recede, to continue aerial growth (Figure 2c). Flooding in the lab: Arabidopsis and the N-end rule pathway for oxygen sensing Arabidopsis is not highly tolerant to submergence , nevertheless it made the discovery of oxygen sensing and signalling mechanisms possible . Besides the classical anaerobic genes, several HYPOXIA-RESPONSIVE UN-KNOWN PROTEIN (HUP) genes were identiﬁed , representing possibly interesting elements in the anaero-bic response pathway. Furthermore, an atlas of hypoxic-dependent gene expression in speciﬁc cell types was produced and revealed a set of approximately 50 genes that were activated regardless of their cellular identity . This provides an enormous amount of information that could be exploited to elucidate the signalling path-way behind the response of plants to low oxygen. The role of group VII ERFs in rice prompted research on this gene-family in Arabidopsis, in which the group VII ERFs are ﬁve , with the initial identiﬁcation of two HYP-OXIA-RESPONSIVE ERFs (HRE1 and HRE2) which contribute to hypoxia tolerance and signalling . RAP2.12, another group VII ERF, is not induced by hypoxia, but nevertheless activates ADH . RAP2.12 is regulated by oxygen at the protein level, with oxygen provoking its degradation [47,48]. Only under low oxygen are RAP2.12 and the other two constitutively expressed group VII ERFs, RAP2.2 and RAP2.3, stable and redun-dantly activate the core anaerobic response [49,50]. This oxygen sensing mechanism relies on the oxygen-depen-dent oxidation of the group VII ERF N-terminal cysteine (Cys), mediated by the PLANT CYSTEINE OXIDASE (PCO) enzymes . The oxidised Cys targets RAP2.12 to the proteasome through an N-end-rule pathway of ubiquitin mediated proteolysis (Figure 3). Interestingly, also nitric oxide (NO) is able to induce group VII ERF degradation, indicating that this pathway might also be involved in other processes including seed germination, stomatal closure, and hypocotyl elongation . Re-markably, oxygen sensing through group VII ERFs was shown to coordinate photomorphogenesis during seedling development . It is presently unknown whether and how group VII ERF cysteine oxidation requires both PCOs and NO. The RAP2.12 dependent activation of the downstream genes is essential to survive submergence, but also needs to be ﬁnely tuned. The HYPOXIA-RESPONSE AT-TENUATOR1 (HRA1) is a trihelix transcription factor that represses the action of RAP2.12. HRA1 gene expres-sion is itself activated by RAP2.12 stabilization under hypoxia, indicating the existence of an homeostatic mechanism for regulating the anaerobic response, such Plant responses to flooding stress Loreti, van Veen and Perata 67 www.sciencedirect.com Current Opinion in Plant Biology 2016, 33:64–71 that it does not harmfully exceed the needs of the plant . Interestingly, also hydrogen peroxide production under anoxia occurs during the early phases of the stress . Recently, a protein interconnecting the oxygen-sensing machinery with ROS production was identiﬁed. HYPOXIA-RESPONSIVE UNIVERSAL STRESS PROTEIN 1 (HRU1) is induced by the oxygen-respon-sive N-end-rule pathway and affects ROS production, possibly through an interaction with a membrane-local-ized NADPH-oxidase (RBOHD) and its regulator ROP2 . Overall these recent ﬁndings suggest that hypoxia-dependent signalling is tightly controlled via various signals and proteins in a highly connected network. It is tempting to speculate that excessive activation of the fermentative pathway by RAP2.12 may deplete sugars to a level that induces severe starvation, hampering long term survival and recovery from hypoxia. A highly coor-dinated network, including HRA1 and HRU1, could prevent such a detrimental scenario. Translating lab research into better crops The identiﬁcation of SUB1A as the determinant for submergence tolerance in rice allowed the breeding of ﬂood-tolerant rice varieties, often called ‘scuba rice’ [4,57,58]. These varieties showed the same yield and quality traits as their non-Sub1 counterparts when grown under non-ﬂooded conditions, but displayed yield advan-tages of 1 to more than 3 t ha1 after complete submer-gence for various durations . This is a great example of rapid translation of a scientiﬁc discovery into agricultural improvements in less than ten years since the discovery of SUB1A in 2006 . Experimental evidence showing that SUB1A also contributes to drought tolerance in rice suggests that this trait will contribute to the development of rice varieties better adapted to climate changes . Incorporating ﬂooding tolerance into crops other than rice will be very challenging, given the lack of accessions with ﬂooding tolerance traits. However, the discovery of the oxygen sensing mechanism in Arabidopsis could show 68 Cell signalling and gene regulation Figure 3 Anaerobic Gene Expression O2 O2 Sucrose Starch Glucose Hypoxia NAD Glycolysis NAD NAD NADH ATP ATP Pyruvate Ethanol Acetaldehyde ADP ADP Aerobic respiration NADH H2O2 RAP2.12 SuSy RAP2.12 RAP2.12 PROTEASOME PCO NO HRA1 PDC ADH HRU1 RBOHD ROS Gene Expression Current Opinion in Plant Biology Anaerobic signalling (left) and metabolism (right) in Arabidopsis. Under conditions of hypoxia or anoxia respiration in the mitochondria is severely impaired. NADH regeneration to NAD, required to allow glycolysis to proceed, thus occurs through the activation of PDC and ADH. Starch metabolism and sucrose metabolism through sucrose synthase (SuSy) provide the carbon units required for glycolysis. SuSy, PDC, ADH are examples of enzymes encoded by anaerobic genes, whose activation is triggered by hypoxia. Oxygen sensing occurs through ERF-VII genes such as RAP2.12 and RAP2.2 (the latter not shown in figure) that are unstable under aerobic conditions, because PCO enzymes oxidise the N-terminal Cys residue, resulting in degradation of RAP2.12 by the proteasome. Nitric oxide (NO) also induces degradation of ERF-VII proteins. RAP2.12 induces the expression of anaerobic genes, among which is also HRU1, which controls hydrogen peroxide production by RBOHD. The interaction of RAP2.12 with HRA1 dampens the action of RAP2.12. Current Opinion in Plant Biology 2016, 33:64–71 www.sciencedirect.com great promise for crop improvements. However, both strong and weak hypoxic signalling, that is very large versus moderate induction of group VII ERF targets, has been connected to ﬂooding tolerance [20,26,28,47,48,61]. Nevertheless, barley with reduced expression of the N-end-rule pathway E3 ligase PROTEOLYSIS6 (PRT6) shows increased tolerance to waterlogging . Incorporating traits from the superbly adapted wetland species will invariably be challenging, but could provide big leaps in ﬂooding tolerance. Aerenchyma formation is a developmentally complex trait and so far we have been unable to import this trait into a species that did not possess it already. Moreover, maize develops aerenchyma upon waterlogging, but despite this ability it still suffers strongly from soil ﬂooding. Other changes in root devel-opment, such as enhanced adventitious rooting, might be more promising and pliable to our crops, as these traits are often already present to some extent. Tolerance to submergence includes the delicate balance between the induction of the fermentative mechanism, that represents a requirement for basal tolerance, and other mechanisms preventing carbon starvation and oxi-dative stress. Only after we have a complete picture of the many tolerance traits will the development of crop varie-ties tolerant to waterlogging or submergence be feasible. Acknowledgement This work was supported by Scuola Superiore Sant’Anna, Pisa, Italy. References and recommended reading Papers of particular interest, published within the period of review, have been highlighted as: of special interest of outstanding interest 1. Perata P, Alpi A: Plant responses to anaerobiosis. Plant Sci 1993, 93:1-17. 2. van Dongen JT, Licausi F: Oxygen sensing and signaling. Annu Rev Plant Biol 2015, 66:345-367. 3. Voesenek LACJ, Colmer TD, Pierik R, Millenaar FF, Peeters AJM: How plants cope with complete submergence. New Phytol 2006, 170:213-226. 4. Bailey-Serres J, Lee SC, Brinton E: Waterprooﬁng crops: effective ﬂooding survival strategies. Plant Physiol 2012, 160:1698-1700. 5. Takahashi H, Yamauchi T, Colmer TD, Nakazono M: Aerenchyma formation in plants. In Low Oxygen Stress in Plants. 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Transl Genom Crop Breed 2013, 2:9-30. 59. Ismail AM, Singh US, Singh S, Dar HD, Mackill DJ: The contribution of submergence-tolerant (Sub1) rice varieties to food security in ﬂood-prone rainfed lowland areas in Asia. Field Crops Res 2013, 152:83-93. 60. Fukao T, Yeung E, Bailey-Serres J: The submergence tolerance regulator SUB1A mediates crosstalk between submergence and drought tolerance in rice. Plant Cell 2011, 23:412-427. 61. Riber W, Mu ¨ ller JT, Visser EJ, Sasidharan R, Voesenek LACJ, Mustroph A: The greening after extended darkness1 is an N-end rule pathway mutant with high tolerance to submergence and starvation. Plant Physiol 2015, 167:1616-1629. 62. Mendiondo GM, Gibbs DJ, Szurman-Zubrzycka M, Korn A, Marquez J, Szarejko I, Maluszynski M, King J, Axcell B, Smart K, Corbineau F, Holdsworth MJ: Enhanced waterlogging tolerance in barley by manipulation of expression of the N-end rule pathway E3 ligase PROTEOLYSIS6. Plant Biotechnol J 2016, 14:40-50. Transgenic RNAi barley plants with reduced expression of PROTEOLY-SIS6 were produced. As expected from studies carried out in Arabidop-sis, the transformed barley plants showed increased expression of hypoxia-associated genes. In response to waterlogging, transgenic plants showed enhanced yield. Plant responses to flooding stress Loreti, van Veen and Perata 71 www.sciencedirect.com Current Opinion in Plant Biology 2016, 33:64–71
2856	https://www.cancer.gov/publications/dictionaries/cancer-terms/def/adoptive-cell-therapy	Definition of adoptive cell therapy - NCI Dictionary of Cancer Terms - NCI Skip to main content An official website of the United States government Español Menu Search Search About Cancer Cancer Types Research Grants & Training News & Events About NCI Home Publications NCI Dictionaries NCI Dictionary of Cancer Terms Publications Patient Education Publications PDQ® Fact Sheets NCI Dictionaries Dictionary of Cancer Terms Drug Dictionary Dictionary of Genetics Terms Blogs and Newsletters Health Communications Publications Reports adoptive cell therapy Listen to pronunciation (uh-DOP-tiv sel THAYR-uh-pee) A type of immunotherapy in which T cells (a type of immune cell) are given to a patient to help the body fight diseases, such as cancer. In cancer therapy, T cells are usually taken from the patient's own blood or tumor tissue, grown in large numbers in the laboratory, and then given back to the patient to help the immune system fight the cancer. Sometimes, the T cells are changed in the laboratory to make them better able to target the patient's cancer cells and kill them. Types of adoptive cell therapy include chimeric antigen receptor T-cell (CAR T-cell) therapy and tumor-infiltrating lymphocyte (TIL) therapy. Adoptive cell therapy that uses T cells from a donor is being studied in the treatment of some types of cancer and some infections. Also called adoptive cell transfer, cellular adoptive immunotherapy, and T-cell transfer therapy. More Information T-cell Transfer Therapy click to view video titled T-Cell Transfer Therapy Immunotherapy uses the body’s immune system to fight cancer. This animation explains one type of immunotherapy called T-cell transfer therapy that is used to treat cancer. Search NCI's Dictionary of Cancer Terms Starts with Contains Browse: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z # About About This Website en Español Reuse & Copyright Social Media Resources Contact Us Publications Dictionary of Cancer Terms Find a Clinical Trial Policies Accessibility FOIA Privacy & Security Disclaimers Vulnerability Disclosure Sign up for email updates Enter your email address Enter a valid email address Sign up National Cancer Institute at the National Institutes of Health Contact Us Live Chat 1-800-4-CANCER NCIinfo@nih.gov Site Feedback Follow us U.S. Department of Health and Human ServicesNational Institutes of HealthNational Cancer InstituteUSA.gov
2857	https://mascentigrados.com/eficiencia-y-rendimiento-en-equipos-termicos-importancia-del-cop/	EFICIENCIA Y RENDIMIENTO EN EQUIPOS TÉRMICOS: IMPORTANCIA DEL COP Cuando hablamos de un sistema energético, una de las principales características que se deben tener en cuenta a la hora de adquirirlo o utilizarlo, es la eficiencia que el sistema posee en su principio de funcionamiento. El término de eficiencia en equipos de uso de energía está relacionado con la cantidad de trabajo útil que el sistema brinda, en contraste con la energía que el mismo consume. Existe una ley física, que siempre se cumple para cualquier tipo de sistema, conocida como la primera ley de la termodinámica o ley de la conservación de la energía, que profesa explícitamente eso, que en todo sistema la energía global siempre se conserva. En términos prácticos se acostumbra a definir esta ley de manera en que la energía no se puede crear ni destruir, pero sí puede ser transformada. De acuerdo con esto, cualquier sistema que produzca un trabajo o una función, sea mecánico, químico o térmico, debe tener también un consumo de alguna fuente de energía y lo que el equipo hace es transformar la energía de esta fuente, en otro tipo de energía útil para cumplir una función. Teniendo en cuenta la primera ley de la termodinámica la eficiencia de un equipo se define como esa cantidad de energía útil que el equipo brinda para cumplir con su función, en relación con la cantidad de energía que el equipo se consume haciendo esa transformación. En un equipo mecánico, por ejemplo, como una motobomba, la eficiencia se define como la relación o división algebraica de la energía mecánica del movimiento de la bomba entre la energía eléctrica consumida por el motor que la impulsa. Normalmente en este tipo de sistemas la eficiencia toma un valor entre 0 y 1, y se expresa en porcentaje; un sistema ideal, es decir, sin pérdidas de energía, debería tener una eficiencia de 1 que equivale al 100%, pero debido a pérdidas mecánicas y térmicas en los sistemas este valor del 100% de eficiencia es un límite teórico. En equipos de uso final de energía que tienen como principio de funcionamiento el ciclo termodinámico de compresión de vapor, como pueden ser neveras, aires acondicionados, bombas de calor, deshumidificadores, entre otros, no es convencional hablar en términos de eficiencia. El ciclo de refrigeración por compresión de vapor es un tipo de sistema térmico cíclico, donde la energía útil ya no es energía mecánica sino energía térmica (en forma de calor), en el cual circula un fluido refrigerante que es capaz de transferir el calor desde una fuente de baja temperatura hasta una fuente de alta temperatura, por medio de un c onsumo de energía eléctrica que mueve un compresor. Como los procesos que ocurren dentro del sistema obtienen energía térmica del medio ambiente, la energía útil (energía de frío o calor), es mayor a la energía eléctrica que el sistema consume, por lo cual ya no se habla de eficiencia sino de COP, o coeficiente de rendimiento, por sus siglas en inglés. El COP es una medida de alta importancia en sistemas de calefacción o de refrigeración, ya que es una muestra de la capacidad que tiene el sistema de aprovechar la energía ilimitada disponible en el medio ambiente para cumplir con un propósito de utilidad que puede ser el calentar o enfriar un espacio o un material. El COP de un sistema que funcione bajo el ciclo termodinámico de refrigeración por compresión siempre será un valor superior a 1, y entre más alto sea este valor, mejor será el sistema, debido a que puede entregar más energía útil, con un consumo menor de energía eléctrica, lo cual es mucho más ecológico, y a su vez genera ahorros significativos en el costo energético. El COP de un sistema varía de acuerdo a varios factores de incidencia, desde factores de diseño del propio sistema como geometrías, tipo de compresor o tipo de gas refrigerante, hasta condiciones medioambientales como la temperatura ambiente o la humedad relativa. Por ejemplo, para una bomba de calor, como su propósito es calentar un espacio o material, el COP será mayor entre mayor sea la temperatura ambiente y la humedad relativa. Con un refrigerador pasa lo contrario. Si por ejemplo una bomba de calor para calentar una piscina tiene un COP de 5, quiere decir que por cada 5 unidades de calor que el equipo le entregue al agua, se va a consumir solo 1 unidad de energía eléctrica, y el resto de la energía será tomada del ambiente de manera ilimitada. A la hora de cotizar su sistema de climatización, siempre debe tener en cuenta el COP nominal de los sistemas, ya que esto le ahorrará costos energéticos que pueden llegar a ser muy significativos en corto, mediano y largo plazo. Debido a los altos valores de COP que tienen las bombas de calor, son consideradas la mejor opción para el calentamiento de fluidos, o espacios en general. En contraste con sistemas convencionales como quemadores a gas o resistencias eléctricas, una bomba de calor puede ahorrarle incluso hasta más de 5 veces los costos calentamiento, y brindándole el mismo resultado de calidad y confianza. IM. Santiago Valencia Cañola. Departamento de Ingeniería, Más Centígrados S.A.S Referencias F. Incropera, D. Dewitt, T. Bergman y A. Lavine, Fundamentals of heat and mass transfer, John Wiley & Sons, Inc, 2007. R. Monasterio, P. Hernandez, and J. Sainz, La bomba de calor. Fundamentos, técnicas y aplicaciones, 1st ed. Madrid, España: McGraw Hill, 1993. Y. A. Cengel and M. A. Boles, Termodinâmica, 7th ed. 2012. Todos los derechos reservados © 2024 Más Centigrados ® Política de Privacidad y Tratamiento de Datos Personales.
2858	https://www.extendoffice.com/excel/functions/excel-decimal-function.html	How to use Excel DECIMAL function Skip to main content ExtendOffice Solutions Office Tab Kutools for Excel Kutools for Outlook Kutools for Word Kutools for PowerPoint Download Office Tab Kutools for Excel Kutools for Outlook Kutools for Word Setup Made Simple User License Agreement Pricing Office Tab Kutools for Excel Kutools for Outlook Kutools for Word 4 Software Bundle 60-Day Refund Office Tips (5300) Howtos Excel (3000+) Howtos Outlook (1200+) Howtos Word (300+) Excel Functions (498) Excel Formulas (350) Excel Charts Outlook Tutorials Support Changelog Office Tab Kutools for Excel Kutools for Outlook Kutools for Word Feature Tutorials Office Tab Kutools for Excel Kutools for Outlook Kutools for Word Get Help? Lost License? Bug Report About Us Search Kutools for Office — One Suite. Five Tools. Get More Done. ExtendOffice Solutions Kutools for Office One Suite. 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Report a Bug Bug Report Contact Us 24/7 email Search Excel DECIMAL function Author Sun•Last modified 2024-10-11 Description The DECIMAL function converts a text representation of a number in a base into its equivalent decimal number. Take a instance, 11 is a text representation of 3 in base 2, using the formula =DECIMAL(11,2) convert 11 into decimal number 3. Syntax and arguments Formula syntax =DECIMAL(text,radix)) Arguments Text: Required. The text representation of the number that you want to convert to decimal number. The length of the text must be less than 256. Radix: Required. The base of the supplied number, must be an integer ≥ 2 and ≤ 36. Return Value The DECIMAL function returns a decimal number. Notes: The DECIMAL function is unavailable in the earlier versions of Excel 2013. The Text argument is no case sensitive. About Errors NUM! error value: Occurs if either one appears: The Radix argument is < 2 or > 36 The Text argument and Radix argument are not a pair. VALUE! error value: Occurs if either one appears: The Text argument is longer than 255 characters The Radix argument is nonnumeric value. Usage and Examples Before using the DECIMAL function, you should read below table firstly, which list usually used text representation of numbers and the bases. BaseAlpha-Numeric Characters Binary (2)0,1 Octal (8)0-7 Decimal (10)0-10 Hexidecimal (16)0-9 then A-F Before using the DECIMAL function, you should read below table firstly, which list usually used text representation of numbers and the bases. FormulaDescriptionResult =DECIMAL(1111,2)Convert the binary (base 2) value 111 to the equivalent decimal (base 10) value 15 =DECIMAL("C2",16)Convert the hexadecimal (base 16) value C2 to its equivalent decimal (base 10) value 194 =DECIMAL(12,8)Convert the octal (base 8) value 12 to the equivalent decimal (base 10) value 10 Sample File Relative Formulas Relative Articles Convert decimal to times This article introduce two different methods to convert decimal number to time format. Take instance, convert decimal hour 31.23 to time format 31:13:48. The Best Office Productivity Tools Kutools for Excel - Helps You To Stand Out From Crowd 🤖Kutools AI Aide: Revolutionize data analysis based on: Intelligent Execution\|Generate Code\|Create Custom Formulas\|Analyze Data and Generate Charts\|Invoke Kutools Functions… Popular Features: Find, Highlight or Identify Duplicates\|Delete Blank Rows\|Combine Columns or Cells without Losing Data\|Round without Formula... 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2859	https://brilliant.org/wiki/applications-of-completing-the-square/	Reset password New user? Sign up Existing user? Log in Applications of Completing the Square Already have an account? Log in here. Contents Completing the Square - Applications Completing the square is converting a quadratic equation from (ax^2 + bx + c =0 ) to (a(x-p)^2 + q = 0), where (a, b, c, p, q) are constants. This is a very helpful technique with several uses. Finding the Vertex The vertex of a parabola can be found by completing the square in its equation. Consider a parabola whose equation is (y=ax^2+bx+c.) Completing the square gives [\begin{align} y&=ax^2+bx+c \ y&=a\left(x^2+\frac{b}{a}x\right)+c\ y&=a\left(x^2+2\cdot\frac{b}{2a}x+\frac{b^2}{4a^2}\right)-\frac{b^2}{4a}+c\ y&=a\left(x+\frac{b}{2a}\right)^2-\frac{b^2-4ac}{4a}. \end{align}] According to the properties of a square, the value of the square minimizes to zero when the variable being squared equals zero, and the value of the square increases as the absolute value of the variable being squared increases. Thus, our whole expression (ax^2+bx+c,) or (a\left(x+\frac{b}{2a}\right)^2-\frac{b^2-4ac}{4a}) will have an extreme value when (x+\frac{b}{2a}=0.) Plugging (x=-\frac{b}{2a}) into the equation gives (y=-\frac{D}{4a},) and now we know the coordinates of the vertex: (\left(-\frac{b}{2a},-\frac{D}{4a}\right),) where (D) is the quadratic's discriminant. Finding the Range of Quadratic Functions Finding the range of (f(x) = ax^2 + bx + c) may require quite a bit of effort. However, once completing the square is done, the range can be found in just a matter of seconds. Upon completing the square, we have [\begin{align} y&=ax^2+bx+c \ y&=a\left(x^2+\frac{b}{a}x\right)+c\ y&=a\left(x^2+2\cdot\frac{b}{2a}x+\frac{b^2}{4a^2}\right)-\frac{b^2}{4a}+c\ y&=a\left(x+\frac{b}{2a}\right)^2-\frac{b^2-4ac}{4a}. \end{align}] We've discussed in the prior section that this has an extreme value when (x=-\frac{b}{2a}.) Whether this is a maximum or minimum depends on the sign of (a.) We know that (\left(x+\frac{b}{2a}\right)^2\geq0) (since it's a square). Hence if (a>0,) the extreme value will be a minimum, and the range will be (\left[-\frac{D}{4a},\infty\right).) When (a<0,) the function will have a maximum, and its range will be (\left(-\infty,-\frac{D}{4a}\right].) Find the range of (f(x) = a(x-p)^2 + q), where (a) is a positive real number. The variable term in (f) is (a(x-p)^2). Since it is a perfect square, its range is all non-negative real numbers, that is ([0, \infty)). After adding (q), which is a constant, the range of (f) becomes ([q, \infty)). (_\square) Note: We can also find the value of (x) at which the minimum occurs. This is when (x - p = 0) , that is, (x = p). Find the range of (f(x) = a(x-p)^2 + q), where (a) is a negative real number. Now the range of (a(x-p)^2) is ((-\infty, 0]), since (a) is negative. After adding (q), the range of (f) becomes ((-\infty, q]), and we are done. (_\square) Note: Again, at the extreme point, the (x)-coordinate is (p). Find the range of (f(x) = x^2 + 2x + 4). After completing the square, we get (f(x) = (x+1)^2 + 3), implying that its range is ([3, \infty)). (_\square) Factorization Many times, it becomes difficult to factorize a quadratic expression, especially when the roots are irrational or complex. This is where completing the square helps. Factorization of any quadratic can be done by completing the square followed by using the identity (a^2-b^2=(a+b)(a-b).) So given a quadratic equation (ax^2+bx+c=0,) if we succeed in bringing it to the form ((x+m)^2-n^2=0,) then we can factorize it with the difference of two squares identity to obtain both of its roots easily. The procedure is as follows: [\begin{align} ax^2+bx+c&=0 \ x^2+\dfrac{b}{a}x+\dfrac{c}{a}&=0 \ x^2+\dfrac{b}{a}x+\dfrac{b^2}{4a^2}+\dfrac{c}{a}&=\dfrac{b^2}{4a^2}\ \Bigl( x+\dfrac{b}{2a}\Bigr) ^2 +\dfrac{c}{a} - \dfrac{b^2}{4a^2} &=0 \ \Bigl( x+\dfrac{b}{2a}\Bigr) ^2 -\dfrac{b^2-4ac}{4a^2} &=0. \end{align} ] Observe that now the equation has come into the form ((x+m)^2-n^2 =0.) Then we apply the difference of two squares identity, and we're done! Factorize (x^2 - x - 1.) First, we complete the square: (\displaystyle \left(x- \frac{1}{2}\right)^2 - \frac{5}{4} .) Now using the identity (a^2 - b^2 = (a+b)(a-b),) we get [\begin{align} \left(x- \frac{1}{2}\right)^2 - \frac{5}{4} &= \left(x- \frac{1}{2}\right)^2 - \left(\frac{\sqrt{5}}{2}\right)^2 \ &= \left(\left(x - \frac{1}{2}\right) + \frac{\sqrt{5}}{2}\right)\left(\left(x - \frac{1}{2}\right) - \frac{\sqrt{5}}{2}\right)\ &= \left(x - \frac{1 - \sqrt{5}}{2}\right)\left(x - \frac{1 + \sqrt{5}}{2}\right). \ _\square \end{align}] Factorize (x^2 + 2x + 5.) Using the same method as in the above example, we get [\begin{align} x^2 + 2x + 5 &= (x+1)^2 + 4\ &= (x+1)^2 + 2^2\ &= (x+1)^2 - (2i)^2\ &=(x + 1 + 2i)(x + 1 - 2i). \ _\square \end{align}] Reset password New user? Sign up Existing user? Log in
2860	https://web.williams.edu/Mathematics/sjmiller/public_html/BrownClasses/153/SarahM_Mersenne.pdf	MERSENNE PRIMES SARAH MEIKLEJOHN AND STEVEN J. MILLER ABSTRACT. A Mersenne prime is a prime that can be written as 2p −1 for some prime p. The ﬁrst few Mersenne primes are 3, 7 and 31 (corresponding respectively to p = 2, 3 and 5). We give some standard conditions on p which ensure that 2p −1 is prime, and discuss an application to even perfect numbers. The proof requires us to study the ﬁeld Z/qZ[ √ 3], where q ̸= 3 is a prime. 1. INTRODUCTION If n ≥2 and an −1 is prime, we call an −1 a Mersenne prime. For which integers a can an −1 be prime? We take n ≥2 as if n = 1 then a is just one more than a prime. We know, using the geometric series, that an −1 = (a −1)(an−1 + an−2 + · · · + a + 1). (1) So, a−1 \| an−1 and therefore an−1 will be composite unless a−1 = 1, or equivalently unless a = 2. Thus it sufﬁces to investigate numbers of the form 2n −1. Further, we need only examine the case of n prime. For assume n is composite, say n = mk. Then 2n = 2mk = (2m)k, and 2n −1 = (2m)k −1 = (2m −1)((2m)k−1 + (2m)k−2 + · · · + (2m)2 + 2m + 1). (2) So if n = mk, 2n −1 always has a factor 2m −1, and therefore is prime only when 2m −1 = 1. This immediately reduces to 2m = 2, or simply m = 1. Thus, if n is composite, 2n −1 is composite. Now we know we are only interested in numbers of the form 2p −1; if this number is prime then we call it a Mersenne prime. As it turns out, not every number of the form 2p −1 is prime. For example, 211 −1 = 2047, which is 23 · 89. 2. STATEMENT OF THE LUCAS-LEHMER TEST How do we determine which p yield Mp = 2p −1 prime? An answer is the Lucas-Lehmer test, which states that Mp is prime if and only if Mp \| sp−2, where we recur-sively deﬁne si = ( 4 if i = 0, s2 i−1 −2 if i ̸= 0. (3) We prove one direction of this statement, namely that if Mp \| sp−2, then Mp is prime. We start by deﬁning u = 2 − √ 3 and v = 2 + √ 3. Some immediate properties are • u + v = 4 = s0; • uv = 1, implying that (uv)x = uxvx = 1 (so uv to any power equals one). Date: December 4, 2005. 1 2 SARAH MEIKLEJOHN AND STEVEN J. MILLER We will show that sn = ut(n) + vt(n), where we have deﬁned t(s) = 2s. We shall see later that this is a useful way to write sn. Two properties of t(s) that we need are • t(0) = 1; • t(s + 1) = 2s+1 = 2t(s). We prove by induction that sn = ut(n) + vt(n). Base Case: Clearly the base case is true, as we have already seen that s0 = u1+v1 = 4. Inductive Case: Assuming sn = ut(n)+vt(n), we must show sn+1 = ut(n+1)+vt(n+1) = s2 n −2. To do this, we look at sn+1 = s2 n −2 = (ut(n) + vt(n))2 −2 = u2t(n) + v2t(n) + 2ut(n)vt(n) −2. (4) But we already know that ut(n)vt(n) = (uv)t(n) = 1 and 2t(n) = t(n + 1), so we have sn+1 = ut(n+1) + vt(n+1) + 2 −2 = ut(n+1) + vt(n+1), (5) which shows that sn+1 = ut(n+1) + vt(n+1). 3. PROOF OF THE LUCAS-LEHMER TEST We prove one direction of the Lucas-Lehmer test. Speciﬁcally, we prove by contra-diction that if Mp\|sp−2 then Mp is prime. 3.1. Preliminaries. We assume that sp−2 is divisible by Mp, but that Mp is not prime. By direct calculation we may assume that p > 5. There is therefore an integer R > 1 such that sp−2 = ut(p−2) + vt(p−2) = RMp. (6) If we multiply both sides by ut(p−2), we obtain ut(p−2) · (ut(p−2) + vt(p−2)) = ut(p−1) + 1 = RMp · ut(p−2). (7) Subtracting one from each side gives ut(p−1) = RMp · ut(p−2) −1. (8) We square both sides. As (ut(p−1))2 = ut(p), we obtain that ut(p) = (RMp · ut(p−2) −1)2. (9) Note ut(p) is not necessarily an integer. Let us choose some prime factor q > 1 of Mp such that q ≤ p Mp, or equivalently so that q2 ≤Mp. Does such a q exist? There is no problem with assuming q > 1, but what about q ≤ p Mp? If Mp = bc then either b or c is at most p Mp, for if both were larger then the product would exceed Mp. Note we are not claiming that q < p Mp, just that q ≤ p Mp. We use below the fact that q ̸= 3; we need q ̸= 3 so that 3 will have a multiplicative inverse in Z/qZ. We are assuming p > 5 (as the other cases can be handled by direct MERSENNE PRIMES 3 computation). Thus we may write p as 4n + a, where n is an integer and a ∈{1, 3}. Thus Mp = 2p −1 = 24n+a −1 = 24n · 2a −1 = (24)n · 2a −1 ≡ 2a −1 mod 3, (10) since 24 = 16 ≡1 mod 3. If a = 1 then 24n+a −1 ≡1 mod 3, while if a = 3 then 24n+a −1 ≡1 mod 3. Thus 3 does not divide Mp, and we may assume that q ̸= 3 below. The proof is completed by analyzing the order of ut(p) in the ﬁeld Z/qZ[ √ 3], where q is a prime dividing Mp. There are two different cases, depending on whether or not 3 is a square modulo q. Note that if 3 is a square modulo q, then this ﬁeld is actually just Z/qZ. 3.2. 3 is not a square modulo q. We ﬁnish the proof in the case that 3 is not a square modulo q. This means that t2 −3 does not have a root in Z/qZ, or equivalently that t2 −3 is irreducible in Z/qZ. Proof. Consider the ring Z/qZ[ √ 3] = © a + b √ 3 : a, b ∈Z/qZ ª ; note there are q2 elements, and q2 −1 non-zero elements. As q ̸= 3 is prime, Z/qZ is a ﬁeld. Further, Z/qZ[ √ 3] is a ﬁeld as √ 3 is invertible in Z/qZ[ √ 3]; the inverse is b √ 3, where b ∈ Z/qZ is such that 3b ≡1 mod q. More generally, let p(t) = t2 −3 ∈Z/qZ[t] be the irreducible monic polynomial for √ 3 over Z/qZ. Given any a+b √ 3 ∈Z/qZ[ √ 3] with a and b not both zero, consider the linear polynomial g(t) = a + bt. Then p(t) and g(t) are relatively prime (since p(t) is monic and irreducible). Thus there are polynomials such that h1(t)g(t) + h2(t)p(t) = 1; letting t = √ 3 yields h1( √ 3)g( √ 3) = 1, so we have found an inverse to g( √ 3) = a + b √ 3, proving Z/qZ[ √ 3] is a ﬁeld. We may study the subset of elements with multiplicative inverses, ¡ Z/qZ[ √ 3] ¢∗. The order of this multiplicative group is q2 −1; thus by Lagrange’s theorem every element x ∈Z/qZ[ √ 3] satisﬁes xq2−1 = 1; note that here by equals 1 we mean with respect to the multiplication operation of Z/qZ[ √ 3] (which includes multiplication modulo q and √ 3 · √ 3 = 3). From (7), we see that ut(p−1) ≡RMp · ut(p−2) −1 mod q. (11) As q\|Mp, Mp ≡0 mod q. Therefore ut(p−1) ≡−1 mod q. (12) Similarly, looking at (9), we see that ut(p) ≡(RMp · ut(p−2) −1)2 mod q, (13) which implies that ut(p) ≡(0 −1)2 ≡1 mod q. (14) 4 SARAH MEIKLEJOHN AND STEVEN J. MILLER The order of an element g in our multiplicative group ¡ Z/qZ[ √ 3] ¢∗is the smallest positive k such that gk = 1; we often denote this by ord(g). By Lagrange’s theorem, k\|q2 −1. Further, by (14) we know that ord(u) \| t(p). We now show that ord(u) is exactly t(p). From (14) we see that ord(u)\|t(p). As t(s) = 2s, if ord(u) ̸= t(p) then ord(u)\|t(p −1). But if ord(u) divided t(p −1) then ut(p−1) ≡1 mod q, (15) which contradicts (12). Thus ord(u) = t(p) = 2p. However, since the order of any element is at most the order of the group, we have ord(u) = 2p ≤q2 −1 < Mp = 2p −1, (16) where the second inequality follows from q2 ≤Mp. We thus obtain the contradiction 2p < 2p −1, (17) which proves that Mp is prime. □ 3.3. 3 is a square modulo q. We ﬁnish the proof in the case that 3 is a square modulo q. This means that t2 −3 has a root in Z/qZ, or equivalently that t2 −3 factors into two linear terms in Z/qZ. For example, if q = 13 then t2 −3 ≡(t −4)(t −9) mod q. Proof. We now assume that 3 is a square modulo q; for deﬁniteness, let b2 = 3. In §3.1 we showed that ut(p−1) = RMp · ut(p−2) −1 (18) and ut(p) = (RMp · ut(p−2) −1)2. (19) Note ut(n) is not necessarily an integer. We may regard these equations modulo q. Doing so, we replace √ 3 with b. Reducing these equations modulo q yield ut(p−1) ≡−1 mod q (20) and ut(p) ≡1 mod q. (21) Arguing as in §3.2, ord(u) = 2p; the only difference is that now there are q−1 non-zero elements in our ﬁeld Z/qZ, and not q2 −1. We therefore have ord(u) = 2p ≤q −1 ≤Mp = 2p −1, (22) and this contradiction completes the proof. □ 4. MERSENNE PRIMES AND PERFECT NUMBERS Another interesting fact about Mersenne primes is their correspondence with perfect numbers. Perfect numbers are integers whose proper divisors (all divisors except the number itself) sum to the number. For example, 6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7 + 14. There is a one-to-one correspondence between even perfect numbers and Mersenne primes. While it can be shown that every even perfect number is of the form (2p −1) · 2p−1, where 2p −1 is a Mersenne prime, we content ourselves with showing that any number of the form (2p −1) · 2p−1 is perfect when 2p −1 is a Mersenne prime. MERSENNE PRIMES 5 Let q = 2p −1 be a Mersenne prime; we show that q · 2p−1 is perfect. We know that the proper divisors break up into two disjoint sets: {1, 2, 4, . . . , 2p−1} ∪{q, 2q, 4q, . . . , 2p−2q}. (23) So, using the geometric formula 1 + x + x2 + · · · + xn−1 = xn −1 x −1 , (24) we see that the ﬁrst set sums to 1 + 2 + 4 + · · · + 2p−1 = 2p −1 2 −1 = 2p −1 = q, (25) and the second set sums to q+2q+4q+· · ·+2p−2q = q(1+2+4+· · ·+2p−2) = q µ2p−1 −1 2 −1 ¶ = q(2p−1−1). (26) Thus the sum of the proper divisors is q + q(2p−1 −1) = q + 2p−1q −q = 2p−1q, (27) proving that (2p −1) · 2p−1 is perfect. E-mail address: sjmiller@math.brown.edu DEPARTMENT OF MATHEMATICS, BROWN UNIVERSITY, PROVIDENCE, RI 02912
2861	https://www.uofmhealth.org/conditions-treatments/rheumatology/pulmonary-hypertension-scleroderma	Jump to content Updated visitor policies Other Michigan Medicine Sites About Michigan Medicine UofMHealth.org Medical School Nursing Find a Clinical Trial Michigan Medicine Federated Search Page Form block Quick Links Patient Portal Login For Health Providers Maps & Directions Contact Us Pulmonary Hypertension in Scleroderma Pulmonary hypertension means high blood pressure in the lungs. This is an extremely important issue in scleroderma occurring in up to 40% of patients. Patients with limited scleroderma have the risk of developing progressive blood vessel narrowing in the lungs frequently in the absence of lung scarring and inflammation. This complication is called pulmonary arterial hypertension (PAH). PAH is recognizedby the World Health Organization (WHO) as a distinct medical syndrome that shares common tissue features of non-inflammatory blood vessel narrowing. PAH occurs in scleroderma but also in lupus, as an isolated disease called idiopathic PAH, and as a complication of liver failure, HIV infection and use of diet pills. PAH can occur in diffuse scleroderma as well but a more common scenario is for progressive lung scarring to lead to loss of microvasculature in the lung again leading to elevated lung blood pressure. This syndrome is recognized by the WHO as pulmonary hypertension (PH) secondary to intrinsic lung disease. Symptoms The most common symptom of PAH and PH is shortness of breath with physical activity. Fainting or near fainting with physical activity may occur as well as other symptoms such as fluid retention and chest discomfort. The diagnosis of PAH and PH are often overlooked by the community physician. It is sufficiently common and of such high impact that all patients with scleroderma should be screened for its presence on a regular basis. Pathophysiology Blood returning to the lungs is pumped through the lungs by the right ventricle of the heart. Blood pressure in the lungs is ordinarily rather low, for example 20/10, in contrast to body blood pressure which is usually around 110/70. Resistance to blood flow through the lungs puts a strain on the right ventricle. At early stages, the right ventricle is able to compensate but as the resistance increases and the pressures go higher, the right ventricle cannot keep up. At early stages, shortness of breath occurs with moderate physical activity but as it worsens, it takes less and less physical activity to cause shortness of breath. The pressure in the lungs can be elevated for several different reasons in scleroderma. In scleroderma, the left ventricle can become stiff or weakened, leading to back pressure into the lungs (red arrow). This is called heart failure. In patients with lung scarring (see Lung Involvement), the capillaries can be damaged (lavender arrow) which leads to PH. The distinctive problem in scleroderma is narrowing of the small lung arteries (blue arrow). This is the problem that leads to PAH. How is PAH/PH Diagnosed? The pulmonary function tests can be helpful. Patients in whom the diffusing capacity is reduced either as an isolated finding or out of proportion to changes in their forced vital capacity are more likely to have pulmonary hypertension. Blood pressure in the lung can be estimated and the size and function of the right ventricle assessed by echocardiography with Doppler. This non-invasive test uses sound waves to image the chambers of the heart. However, Doppler echocardiograms are notoriously inaccurate when the pulmonary hypertension is mild or when there is simultaneous presence of lung scarring. A small protein released by stretched heart muscle (BNP-brain natriuretic peptide) can be measured by a simple blood test. Elevated levels are a strong clue for suspecting pulmonary hypertension. PULMONARY HYPERTENSION CANNOT BE DIAGNOSED WITHOUT RIGHT HEART CATHETERIZATION This is an invasive test performed by specialists in cardiology wherein a slender tube is introduced into the circulation, advanced carefully to the right side of the heart and into the lungs. Careful measurements of blood flows and pressures define the diagnosis of pulmonary hypertension. What Causes Pulmonary Hypertension? As with many features of scleroderma, the basic issue is progressive scarring of the inner lining of the small artery. The changes in the lung blood vessels look remarkably similar to those in the fingers, kidneys and gastrointestinal tract. Many in the research community think of scleroderma as a blood vessel narrowing disease and view the immune system activation and tissue scarring as secondary events. We do not yet understand the specific triggers of blood vessel injury or just how the blood vessel damage progresses. We do understand that injury to the lining of the blood vessels leads to specific chemical imbalances that participate in pulmonary hypertension. Injury to the endothelial cells which line healthy blood vessels leads to overproduction of endothelin – a key cause of blood vessel scarring and spasm – and to reduced production of nitric oxide and prostacyclins – two key body chemicals which keep blood vessels relaxed and open. Treatment The treatment options for PAH complicating scleroderma represent the most active area of progress in the history of the disease. The only drugs approved by the Food & Drug Administration for use in scleroderma are agents recently approved for pulmonary hypertension. These include Flolan® (an intravenous prostacyclin), Treprostinil® (a prostacyclin that is given either subcutaneously or intravenously), Ventavis® (an inhaled prostacyclin), Tracleer ® (a pill taken by mouth) and Revatio ® (a pill containing the active ingredient in Viagra). Other similar drugs are at late stages of preapproval clinical trials and there is great interest in studying the benefits of various combination therapies. All of these agents are uniquely and specifically suited to the blood vessel issues of PAH. Tracleer® blocks the actions of endothelin. Revatio® accentuates the actions of nitric oxide. Flolan®, Treprostinil® and Ventavis® are artificial prostacyclins designed to replace what the damaged blood vessel no longer can produce. These agents are very complex and choosing the correct initial treatment requires considerable expertise. The University of Michigan Scleroderma Program works closely with the Pulmonary Hypertension Program in identifying pulmonary hypertension and in choosing the correct path in treatment. These drugs are extraordinarily expensive, ranging from $15,000-$150,000 per year. Pharmaceutical benefit issues influence choice of therapy. All of these drugs were developed under new government programs that foster drug development for rare or “orphan” diseases. In part, because the “market” is small, the cost per prescription is high. All of these treatments reduce shortness of breath, improve exercise capacity, and slow the rate of clinical worsening. How is Treatment Assessed? The most important elements in following response to treatment are the level of shortness of breath and the exercise capacity. A simple test called the “six minute walk” is done in the pulmonary function laboratory. This measures how far one can walk and what happens to blood oxygen levels during exertion. It is commonly used in assessing and adjusting treatments for pulmonary hypertension. Other tests include serial assessments of right ventricle health by echocardiogram and by BNP blood test. Repeat catheterization studies are done when clinically indicated. For more information, contact: Pulmonary Hypertension Association This national organization is a model for collaboration between patients, physicians and other care givers and the pharmaceutical industry. This webpage features rich information for patients and health professionals. Click here to access a copy of PHA's recently-published patient brochure.
2862	https://brilliant.org/wiki/solving-logarithmic-inequalities/	Solving Logarithmic Inequalities \| Brilliant Math & Science Wiki HomeCourses Sign upLog in The best way to learn math and computer science. Log in with GoogleLog in with FacebookLog in with email Join using GoogleJoin using email Reset password New user? Sign up Existing user? Log in Solving Logarithmic Inequalities Sign up with FacebookorSign up manually Already have an account? Log in here. Rasha Yagami contributed This wiki is incomplete. This is a placeholder wiki page. Replace this text with information about the topic of this page. For further help in starting a wiki page, check out Wiki Guidelines and Wiki Formatting or come chat with us. Section Heading Add explanation that you think will be helpful to other members. Example Question 1 This is the answer to the question, with a detailed solution. If math is needed, it can be done inline: x 2=144 x^2 = 144 x 2=144, or it can be in a centered display: x 2 x+3=4 y \frac{x^2}{x+3} = 4y x+3 x 2=4 y And our final answer is 10. □ _\square □ Cite as: Solving Logarithmic Inequalities. Brilliant.org. Retrieved 22:43, September 28, 2025, from Join Brilliant The best way to learn math and computer science.Sign up Sign up to read all wikis and quizzes in math, science, and engineering topics. Log in with GoogleLog in with FacebookLog in with email Join using GoogleJoin using email Reset password New user? Sign up Existing user? Log in
2863	http://www.bgcryst.com/symp12/PDF_BCC_2013/21_BCC_2013_Kossev_27.pdf	543 To whom all correspondence should be sent: E-mail: k_kossev@yahoo.com © 2013 Bulgarian Academy of Sciences, Union of Chemists in Bulgaria Bulgarian Chemical Communications, Volume 45, Number 4 (pp. 543–548) 2013 Synthesis and crystal structure of magnesium chlorate dihydrate and magnesium chlorate hexahydrate K. Kossev, L. Tsvetanova, L. Dimowa, R. Nikolova, B. Shivachev 1 Institute of Mineralogy and Crystallography “Acad. Iv. Kostov”, Bulgarian Academy of Sciences, Acad. G. Bonchev str., building 107, 1113 Sofia, Bulgaria Received February, 2013; Revised May, 2013 Two magnesium chlorate hydrates, Mg(ClO3)2×6(H2O) (1) and Mg(ClO3)2×2(H2O) (2), have been synthesized by slow evaporation from water and ethanol, respectively. The structures were determined by single-crystal X-ray diffrac tion at 150 K due to the dehydration-rehydration at room temperature leading to multiple phase transitions. Both com pounds crystallized in the monoclinic space group P21/c (SG 14) with respective unit cell parameters of a = 6.3899(3), b = 6.5139(3), c = 13.8963(6)Å, β = 100.319(5)°, V = 569.05(5) Å3, Z = 2, R= 0.0210 and a = 6.3707(5), b = 5.4092(3), c = 9.8208(6) Å, β = 97.338(6)°, V = 335.66(4) Å3, Z = 2, R = 0.0201. The structure solution shows an octahedral coor dination of the Mg2+ for both compounds 1 and 2. In the case of Mg(ClO3)2×6(H2O) the coordination is achieved by the water molecules, while for Mg(ClO3)2×2(H2O) the coordination involves two water molecules and is complemented by four oxygen atoms from the chlorate moiety. Key words: magnesium chlorate, hydrates, single crystal. Introduction The coordination chemistry of magnesium is well studied because of its role and participation in a multitude of reactions in the living organisms. Magnesium is an essential component of many enzymes . It binds and thus activates ATP and participates in the process of energy transfer and construction of nucleic acids . The preferred co ordination number of magnesium is six . In the majority of known crystal phases involving the participation of magnesium it favors the octahedral coordination. With the discovery of perchlorates ClO4 – on Mars by NASA Phoenix Lander the interest in study ing and modeling the oxidized forms of chlorine in creased. Between the chloride (oxidation state –1) and perchlorate (oxidation state +7) there are three other ions – hypochlorite ClO – (oxidation state +1), chlorite ClO2 – (oxidation state +3) and chlorate ClO3 – (oxidation state +5). Chlorates are of peculiar inter est due to their stability , though, their structural and crystallographic characteristics are similar to those of perchlorates. Alkali or alkaline earth metal chlorates are intensively studied, mostly sodium and magnesium chlorates, which is reasoned by the distribution of those elements in nature and the low eutectic temperatures of the aqueous solutions of Mg(ClO3)2. Magnesium chlorates have different industrial applications: in paper production; in agro chemistry as herbicide and defoliant; in pyrotech nics; and as antiseptic agent . The first communication about magnesium chlo rate hexahydrate was made by Wachter , who ob tained it in 1841 from the reaction of barium chlorate and magnesium sulfate. Later, Meusser deter mined the temperature at which (Mg(ClO3)2).6H2O melts in its crystallization water to be 35 °С. In addition to the hexahydrate, magnesium chlorate forms two other crystal hydrates with two (com pound 2) and four water molecules. The phase di agram of the system magnesium chlorate – water was reported by Linke in 1965 . The tetrahydrate form (Mg(ClO3)2).4H2O is stable in the temperature range 35–65 °С, while above that temperature the stable form is (Mg(ClO3)2).2H2O. While the chemical and physicochemical prop erties of the anhydrous magnesium chlorate, as well as its hydrate forms have been well studied, the crystal structure(s) of none the salts were de termined. In this study we report the crystal struc tures of two of the three magnesium hydrates 544 namely Mg(ClO3)2×6(H2O), (compound 1) and Mg(ClO3)2×2(H2O) (compound 2). Materials and methods Synthesis Magnesium chlorates were obtained via the re action of barium chlorate monohydrate and mag nesium sulfate heptahydrate in equimolar ratio, followed by recrystallization in ethanol solution. Synthesis of compound 1 Barium chlorate monohydrate (0.322 g, 1.0 mmol) was dissolved in 20 ml distilled water. Magnesium sulfate heptahydrate (0.246 g, 1.0 mmol) was dis solved in 20 ml distilled water. The water solution of magnesium sulfate heptahydrate was slowly add ed to the barium one under constant stirring. After three hours the mixture is centrifuged for 30 min at 5000 rpm. The barium sulfate pellet was discard ed while the supernatant is transferred to a rotary evaporator. The obtained magnesium chlorate was recrystallized in 5 ml ethanol. Colorless single crystals of magnesium chlorate hexahydrate, Mg(ClO3)2.6H2O (compound 1), were grown by slow evaporation from an aqueous solu tion at room temperature. Synthesis of compound 2 The synthesis of compound 2 followed the same steps as described for compound 1. Colorless sin gle crystals of magnesium chlorate dehydrate, Mg(ClO3)2.2H2O (compound 2), were grown by slow evaporation from absolute ethanol at room temperature. Single crystal X-ray diffraction study Crystals of compounds 1 and 2 suitable for sin gle crystal XRD analysis were placed on a glass fiber and mounted on an Agilent, SuperNovaDual four-circle diffractometer equipped with Atlas CCD detector and using mirror-monochromatized MoKα (λ = 0.7107 Å) radiation from a micro-focus source. The crystals were flash frozen at 150 K in an N2 gas stream (Cobra, Oxford cryosystems) and diffraction data were collected at this temperature by ω-scan technique. The determination of cell parameters, data integration, scaling and absorption correction were carried out using the CrysAlisPro program package . The structures were solved by direct methods using ShelxS and refined by full-ma trix least-square procedures on F2 with ShelxL-97 . The hydrogen atoms were located from dif ference Fourier map and refined as riding on their parent atoms, with Uiso(H) = 1.2Ueq(O). Results and discussion The crystal structures of the magnesium chlorate hydrates (di-, tetra- and hexa- hydrates) have not been reported although the synthesis of these three magnesium chlorates has been published . The performed check (ICDD-PDF and ICSD) revealed that similar magnesium chlorates (where the water is replaced by another small highly polar molecule e.g. urea) have been characterized. The structures of some magnesium oxychlorides have also been reported (Mg(ClO4)2.6H2O) and (Mg(ClO2)2.6H2O) ). The problem with the crys tal structure determination of magnesium chlorates is associated with their relative instability at ambient temperature. Actually, the performed room temper ature data collection resulted in good diffraction of the crystals for 10–15 minutes after what diffraction disappeared almost instantly. The attempted X-ray powder data collection was also unsuccessful. Thus we performed single crystal data collection by flash freezing the crystals in N2 at 150 K. An ORTEP view with 50% probability of the molecular structures of compounds 1 and 2 and the atom numbering scheme is shown in Figure 1. The experimental conditions are summarized in Table 1. Fig. 1. View of the molecular structures of compounds 1 and 2 with atomic num bering scheme. Displacement ellip soids for the non-H atoms are drawn at the 50% proba bility level. The H atoms are presented with spheres with arbitrary radii K. Kossev et al.: Synthesis and crystal structure of magnesium chlorate dihydrate and magnesium chlorate hexahydrate 545 Selected bond distances and bond angles are listed in Table 2. Hydrogen bonding geometry is presented in Table 3. The data for publication were prepared with WinGX , ORTEP , and Mercury program packages. As expected, the crystal structure of the hexahy drate consists of discrete [Mg(H2O)6]2+ octahedra and chlorate anions (Fig. 2). The [Mg(H2O)6]2+ oc tahedra are connected via hydrogen bonds to chlo rate anions, where every H atom of the six water Table 1. Crystal data and most important refinement indicators for compounds 1 and 2 1 2 Empirical formula Cl2H12MgO12 Cl2H4MgO8 Molecular weight 299.31 227.24 Crystal size (mm) 0.32 × 0.30 × 0.28 0.23 × 0.21 × 0.18 Crystal habit, color prism, colorless prism, colorless Crystal system Monoclinic Monoclinic Space group P21/c P21/c T(K) 150 150 Radiation wavelength (Е) 0.71073 (Mo Ka) 0.71073 (Mo Ka) a (Е) 6.3899(3) 6.3707(5) b (Е) 6.5139(3) 5.4092(3) c (Е) 13.8963(6) 9.8208(6) α (o) 90 90 β (o) 100.319(5) 97.338(6) γ (o) 90 90 V (Е3) 569.05(5) 335.66(5) Z 2 2 d (mg. m–3) 1.747 2.248 μ (mm–1) 0.67 1.06 diffractometer Agilent SupernovaDual Agilent SupernovaDual Detector, resolution mm–1 Atlas CCD, 10.3974 pixels Atlas CCD, 10.3974 pixels radiation source, wavelength (Å) Mova(Mo) X-ray source, λ = 0.7107 Mova(Mo) X-ray source, λ = 0.7107 Absorption correction multi-scan, CrysAlisPro multi-scan, CrysAlisPro Refinement, Least-squares matrix F2, Full F2, Full Reflections collected/I>2σ (I) 3955/1288 1388/788 parameters 94 61 R1 (F2 > 2σ (F2)) 0.021 0.02 wR2 (all data) 0.057 0.055 GOF 1.08 0.83 Extinction correction none 0.049(5) Δρmax/ Δρmin (e Å–3) 0.32/–0.45 0.23/–0.33 Table 2. Selected geometrical parameters for compounds 1 and 2 (Å, °) Bond distance compound 1 compound 2 Cl1 — O2 1.4923(8) Cl1 — O1 1.5019 (9) Cl1 — O3 1.4808(9) Cl1 — O3 1.4793(11) Cl1 — O6 1.4843(8) Cl1 — O4 1.4850 (9) Mg2 — O1 2.0481(8) Mg1 — O1 2.1039(9) Mg2 — O4 2.0455(8) Mg2 — O4 2.0733(10) Mg2 — O5 2.0703(8) Mg2 — O5 2.0429(10) Bond angle O1— Mg2 — O5 88.07(3)/91.93(3) i O5— Mg1 — O1 89.48(4)/90.52(4)ii O4— Mg2 — O1 90.62(4)/89.38(4) i O5— Mg1 — O4 88.96(4)/91.04(4)ii O4— Mg2 — O5 90.45(4)/89.55(4) i O4— Mg1 — O1 88.05(4)/91.95(4)ii O3 —Cl1 — O2 107.25(5) O1— Cl — O3 105.98(6) O3 —Cl1 — O6 106.62(5) O1— Cl — O4 106.48(6) O6 —Cl1 — O2 107.19(5) O3— Cl — O4 107.41(6) Symmetry operations: (i) −x, −y, −z+1; (ii) x, y, z−1. K. Kossev et al.: Synthesis and crystal structure of magnesium chlorate dihydrate and magnesium chlorate hexahydrate 546 molecules is in contact with a chlorate anions, with a O…O distances in the range of 1.95–2.15 Å (Table 3), thus arranging 10 chlorate anions around the octahedral unit (Fig. 2). The result is the ap pearance of a complex three-dimensional hydro gen-bonding network comprising layers of chlo rates anions and [Mg(H2O)6]2+ octahedra (Fig. 3). In compound 2 the Mg coordination is also oc tahedral. However, in compound 2 a chlorate oxy gen participates in the Mg coordination sphere. The magnesium atom (ion) and four chlorate ions lie in one plane, while the water molecules are in axial Table 3. Hydrogen bond for compounds 1 and 2 (Å, °) D—H···A D—H d(H···A) d(D···A) <(DHA) Compound 1 O1—H1A···O6i 0.801 2.084 2.879(5) 172.0 O1—H1B···O3i 0.804 1.957 2.763(5) 160.7 O4—H4A···O6ii 0.804 1.951 2.739(4) 166.6 O4—H4B ···O2iii 0.809 2.038 2.845(5) 175.3 O5—H5A ···O3i 0.775 2.120 2.886(5) 170.3 O5—H5B···O2iv 0.822 2.087 2.901(5) 177.5 Compound 2 O5—H1···O3v 0.751 2.094 2.843(5) 168.8 O5—H2···O3vi 0.745 2.267 2.942(5) 151.2 Symmetry codes : (i) –x, –1/2+y, 1/2–z; (ii) –1+x, 1/2–y, 1/2+z; (iii) 1–x, -y, 1–z; (iv) –1+x, 1/2–y, 1/2+z; (v) –1+x, y, z (vi) 1–x, 1–y, 2–z. Fig. 2. Hydrogen bonding motif of Mg(ClO3)2×6(H2O) Fig. 3. Three-dimensional hydrogen-bond networks com prising layers of chlorates anions and [Mg(H2O)6]2+ octahedra positions and Mg–O bonds are nearly perpendicular to this plane (89.44(5)°). The chlorate molecule acts as a bridge between two magnesium atoms (Mg– O–ClO–O–Mg) and thus produces layers that are stacked along a. The layers are stabilized by internal hydrogen bonds involving water molecules, O5 and chlorate O3. The three-dimensional stabilization of the structure is achieved by a bicyclic hydrogen O5-H…O3 between the adjacent layers (Fig. 4). Conclusions The crystal structures of two elusive magnesium compounds, Mg(ClO3)2×6(H2O) and Mg(ClO3)2× 2(H2O) were determined. They will help in the un derstanding of the rapid hydration processes and multiple phase transitions associated with magne sium hydrates and solvates. K. Kossev et al.: Synthesis and crystal structure of magnesium chlorate dihydrate and magnesium chlorate hexahydrate 547 Supplementary Materials ICSD 425637 and 425637 contain the supplemen tary crystallographic data for this paper. Further de tails of the crystal structure investigation(s) may be obtained from Fachinformationszentrum Karlsruhe, 76344 Eggenstein-Leopoldshafen, Germany (fax: (+49)7247-808-666; e-mail: crysdata(at)fiz-karl sruhe.de, deposited_data.html) on quoting the appropriate ICSD number. Fig. 4. Three-dimensional stabilization of the structure is achieved by a bicyclic hydrogen O5-H…O3 between the layers Acknowledgment: This work was supported by the National Science Fund of Bulgaria, contract DRNF 02/1. References D. Rogolino, M. Carcelli, M. Sechi, N. Neamati 1. , Coord. Chem. Rev., 256, 3063 (2012). J. P. Glusker, A. K. Katz, Ch. W. Bock, 2. The Rigaku Journal, 16(2), 8 (1999). C. Hsiao, M. Tannenbaum, H. VanDeusen, E. Hersh 3. kovitz, G. Perng, A. Tannenbaum, L. D. Williams, Complexes of Nucleic Acids with Group I and II Cations. Nucleic Acid Metal Ion Interactions, N. Hud (ed.), The Royal Society of Chemistry, London, 2008, p. 1–35. M. H. Hecht et 4. al., Science, 325, 64 (2009). J. Hanley et al. 5. Geophys. Res. Lett., 39, L08201, 5 (2012). W. Robbins, A. S. Crafts, R. N. Raynor, Weed 6. Control, Mc Graw-Hill, Inc., New York and London, 1942; Pulp and Paper Canada, 97(10), 11 (1996). A. Wachter, 7. J. Prakt. Chem., (1) 30, 325 (1841). A. Meusser, 8. Ber., 35, 1415 (1902). W. F. Linke, Solubilities: Inorganic and Metal-Or 9. gan ic Compounds, American Chemical Society, 4th ed., 1965, p. 1914. Agilent. CrysAlisPro (version 1.171.35.15). Agilent 10. Technologies Ltd, Yarnton England, (2010). G. M. Sheldrick, 11. Acta Cryst. A, 64, 112 (2008). T. Todorov, R. Petrova, K. Kossev, J. Macicek, O. 12. An gelova, Acta Cryst. C, 54, 927 (1998). C. D. West, 13. Zeitschrift fuer Kristallographie, Kris tallgeometrie, Kristallphysik, Kristallchemie, 91, 480 (1935). Marsh, R.E., 14. Acta Cryst., 46, 1755 (1990). L. J. Farrugia, 15. J. Appl. Cryst., 32, 837 (1999). L. J. Farrugia, 16. J. Appl. Cryst., 30, 565 (1997). I. J. Bruno, J. C. Cole, P. R. Edgington, M. Kessler, 17. C. F. Macrae, P. McCabe, J. Pearson, R. Taylor, Acta Cryst. B, 58, 389 (2002). K. Kossev et al.: Synthesis and crystal structure of magnesium chlorate dihydrate and magnesium chlorate hexahydrate 548 Синтез и кристална структура на магнезиев хлорат дихидрат и магнезиев хлорат хексахидрат К. Косев, Л. Цветанова, Л. T. Димова, Р. Николова, Б. Л. Шивачев Институт по Минералогия и кристалография, БАН, ул. „Акад. Георги Бончев“, бл. 107, София 1113, България Постъпила февруари, 2013 г.; приета май, 2013 г. (Резюме) Получени са монокристални образци на магнезиев хлорат Mg(ClO3)2×6(H2O) (1) и Mg(ClO3)2×2(H2O) (2) при условията на бавно изпарение, съответно от вода (1) и етанол (2). Поради ниската устойчивост на криста лите на стайна температура монокристалният експеримент е осъществен на 150 K. Рентгеноструктурният ана лиз разкрива, че двете съединения кристализират в моноклинната P21/c пространствена група (No 14) с пара метри на елементарната клетка a = 6.3899(3), b = 6.5139(3), c = 13.8963(6) Å, β = 100.319(5)°, V = 569.05(5) Å3 и a = 6.3707(5), b = 5.4092(3), c = 9.8208(6) Å, β = 97.338(6)°, V = 335.66(4) Å3. Рафинирането на кристалната структура показа, че Mg2+ е октаедрично координиран и при двете съединения. При Mg(ClO3)2×6H2O коорди нацията е само от водни молекули, докато при Mg(ClO3)2×2(H2O) координационният октаедър включва две молекули вода, а останалите четири позиции се заемат от кислородни атоми на хлоратни йони. K. Kossev et al.: Synthesis and crystal structure of magnesium chlorate dihydrate and magnesium chlorate hexahydrate
2864	https://math.stackexchange.com/questions/3538965/function-equation-fxfy-f-frac-xy-1xy	function equation $f(x)+f(y)=f(\frac {x+y} {1+xy})$ - Mathematics Stack Exchange Join Mathematics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Loading… Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about Stack Overflow the company, and our products current community Mathematics helpchat Mathematics Meta your communities Sign up or log in to customize your list. more stack exchange communities company blog Log in Sign up Home Questions Unanswered AI Assist Labs Tags Chat Users Teams Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Try Teams for freeExplore Teams 3. Teams 4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams Hang on, you can't upvote just yet. You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more function equation f(x)+f(y)=f(x+y 1+x y)f(x)+f(y)=f(x+y 1+x y) [closed] Ask Question Asked 5 years, 7 months ago Modified5 years, 7 months ago Viewed 162 times This question shows research effort; it is useful and clear 0 Save this question. Show activity on this post. Closed. This question needs details or clarity. It is not currently accepting answers. Want to improve this question? As written, this question is lacking some of the information it needs to be answered. If the author adds details in comments, consider editing them into the question. Once there's sufficient detail to answer, vote to reopen the question. Closed 5 years ago. Improve this question Is there exist f:(−1,1)→R f:(−1,1)→R such that f(x)+f(y)=f(x+y 1+x y)f(x)+f(y)=f(x+y 1+x y)? how to find f f? functional-equations Share Share a link to this question Copy linkCC BY-SA 4.0 Cite Follow Follow this question to receive notifications asked Feb 8, 2020 at 12:17 추민서추민서 560 2 2 silver badges 7 7 bronze badges 0 Add a comment\| 2 Answers 2 Sorted by: Reset to default This answer is useful 4 Save this answer. Show activity on this post. The functional equation for hyperbolic tangent is g(x):=tanh(x),g(x+y)=g(x)+g(y)1+g(x)g(y).g(x):=tanh⁡(x),g(x+y)=g(x)+g(y)1+g(x)g(y). The Wikipedia article Hyperbolic functions is one source for this. The functional equation for its inverse function is f(x):=tanh−1(x),f(x)+f(y)=f(x+y 1+x y).f(x):=tanh−1⁡(x),f(x)+f(y)=f(x+y 1+x y). Note that, just like with Cauchy's functional equation, without any other assumptions such as continuity, and so on, there could be many strange solutions. Consult the Wikipedia article for details. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications edited Feb 8, 2020 at 13:16 answered Feb 8, 2020 at 12:51 SomosSomos 37.6k 3 3 gold badges 35 35 silver badges 85 85 bronze badges Add a comment\| This answer is useful 3 Save this answer. Show activity on this post. I will assume f′(0)f′(0) exist. for \|x\|<1\|x\|<1, we can find relatively small H H such that \|h\|<H\|h\|<H implies 1−x 2−x h≠0 1−x 2−x h≠0 . I will define y = h 1−x 2−x h h 1−x 2−x h. Then y−x 2 y=h+x y h=h(1+x y)⇒y−x 2 y 1+x y=x+y 1+x y−x=h.y−x 2 y=h+x y h=h(1+x y)⇒y−x 2 y 1+x y=x+y 1+x y−x=h. Now, f(x+h)−f(x)=f(y)=f(h 1−x 2−x h)f(x+h)−f(x)=f(y)=f(h 1−x 2−x h). Moreover, f(0)=0 f(0)=0 is easily checked. (x=y=0 x=y=0 on original equation) Thus f′(x)=lim h→0 f(x+h)−f(x)h=lim h→0 f(h 1−x 2−x h)−f(0)h=lim h→0 f(h 1−x 2−x h)−f(0)h 1−x 2−x h 1 1−x 2−x h=f′(0)1−x 2 f′(x)=lim h→0 f(x+h)−f(x)h=lim h→0 f(h 1−x 2−x h)−f(0)h=lim h→0 f(h 1−x 2−x h)−f(0)h 1−x 2−x h 1 1−x 2−x h=f′(0)1−x 2 Of course, h h varies in the range \|h\|<H\|h\|<H. now, for \|x\|<1\|x\|<1 , f(x)=f(x)−f(0)=∫x 0 f′(t)d t=f′(0)∫x 0 1 1−t 2 d t=C 2(ln\|1+x\|−ln\|1−x\|),f(x)=f(x)−f(0)=∫0 x f′(t)d t=f′(0)∫0 x 1 1−t 2 d t=C 2(ln⁡\|1+x\|−ln⁡\|1−x\|), Where C∈R C∈R. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications edited Feb 8, 2020 at 14:14 answered Feb 8, 2020 at 12:42 needmoremathneedmoremath 1,127 5 5 silver badges 12 12 bronze badges Add a comment\| Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions functional-equations See similar questions with these tags. Featured on Meta Introducing a new proactive anti-spam measure Spevacus has joined us as a Community Manager stackoverflow.ai - rebuilt for attribution Community Asks Sprint Announcement - September 2025 Report this ad Linked 2Determine all the continuous functions f:(−1,1)→R f:(−1,1)→R such that ∀x,y∈(−1,1)∀x,y∈(−1,1), f(x)+f(y)=f(x+y 1+x y)f(x)+f(y)=f(x+y 1+x y) Related 10Existence of a function 5Functional equation involving sine function: sin x+f(x)=2–√f(x−π 4)sin⁡x+f(x)=2 f(x−π 4) 9Find the function f f such that f(x)=f(2 x)x+1 f(x)=f(2 x)x+1. 1Solution to the functional equation (x(t))p(1−t 2)=x(2 t 1−t 2)(x(t))p(1−t 2)=x(2 t 1−t 2) for x(t).x(t). 9Prove there are no other solutions of functional equation f(x+y)=f(x)+f(y)1−f(x)f(y)f(x+y)=f(x)+f(y)1−f(x)f(y) 1Rational Functional Equation: 3 f(1 x)+2 f(x)x=x 2 3 f(1 x)+2 f(x)x=x 2 1Functional equation in one variable: f(x)=1 4 f(2 x)+1 4 f(x 3)+1 2 f(x 5)f(x)=1 4 f(2 x)+1 4 f(x 3)+1 2 f(x 5) 2Solving the functional equation f(x)=2−x 2 2 f(x 2 2−x 2)f(x)=2−x 2 2 f(x 2 2−x 2) 2Is f(x)=x a f(x)=x a with a=−1±5√2 a=−1±5 2 the only solution to the equation f(x)f(f(x))=x f(x)f(f(x))=x? 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2865	https://math.stackexchange.com/questions/718279/explanation-for-the-wilson-score-interval	statistics - Explanation for the Wilson Score Interval? - Mathematics Stack Exchange Join Mathematics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Loading… Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about Stack Overflow the company, and our products current community Mathematics helpchat Mathematics Meta your communities Sign up or log in to customize your list. more stack exchange communities company blog Log in Sign up Home Questions Unanswered AI Assist Labs Tags Chat Users Teams Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Try Teams for freeExplore Teams 3. Teams 4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams Hang on, you can't upvote just yet. You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Explanation for the Wilson Score Interval? Ask Question Asked 11 years, 6 months ago Modified4 years, 1 month ago Viewed 31k times This question shows research effort; it is useful and clear 7 Save this question. Show activity on this post. I'm looking at this blog to try to understand the Wilson Score interval. I understand it somewhat, but I'm confused by the part under the title "Excerpt". In particular, I don't understand what he's calling the "Interval equality principal" and how he arrived at the below graph: Could someone elaborate on it, or really just explain how/why the Wilson Score Interval is arrived at from the basic Wald Interval (normal approximation)? Basically, what I'm trying to understand is why the Wilson Score Interval is more accurate than the Wald test / normal approximation interval? statistics Share Share a link to this question Copy linkCC BY-SA 3.0 Cite Follow Follow this question to receive notifications asked Mar 19, 2014 at 13:29 u3lu3l 201 1 1 gold badge 3 3 silver badges 11 11 bronze badges 1 2 Another notable article that mentions it evanmiller.org/how-not-to-sort-by-average-rating.html with SQL implementation but no derivation.Ciro Santilli OurBigBook.com –Ciro Santilli OurBigBook.com 2015-10-07 09:04:40 +00:00 Commented Oct 7, 2015 at 9:04 Add a comment\| 3 Answers 3 Sorted by: Reset to default This answer is useful 6 Save this answer. Show activity on this post. The explanation of "interval equality principle" was impossible for me to readily understand. However, it is not needed to know why the Wilson score interval works. The Wilson interval is derived from the Wilson Score Test, which belongs to a class of tests called Rao Score Tests. It relies on the asymptotic normality of your estimator, just as the Wald interval does, but it is more robust to deviations from normality. Case in point: Wald intervals are always symmetric (which may lead to binomial probabilties less than 0 or greater than 1), while Wilson score intervals are assymetric. Wilson intervals get their assymetry from the underlying likelihood function for the binomial, which is used to compute the "expected standard error" and "score" (i.e., first derivative of the likelihood function) under the null hypotheisis. Since these values will change as you very your null hypothesis, the interval where the normalized score (score/expected standard error) exceeds your pre-specified Z-cutoff for significance will not be symmetric, in general. In basic terms, the Wilson interval uses the data more efficiently, as it does not simply aggregate them into a a single mean and standard error, but uses the data to develop a likelihood function that is then used to develop an interval. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications edited Jul 29, 2019 at 0:47 Bill Clark 133 6 6 bronze badges answered Mar 20, 2014 at 13:34 user76844 user76844 Add a comment\| This answer is useful 2 Save this answer. Show activity on this post. The difference between the Wald and Wilson interval is that each is the inverse of the other. This is how the Wilson interval is derived! As a result we have the following type of equality, which I referred to as the interval equality principle to try to get this idea across. Wald(tail, α, Wilson(¬tail, α, p)) = p, and, correspondingly, Wilson(tail, α, Wald(¬tail, α, P)) = P, where tail ε {0=lower, 1=upper}, α represents the error level (e.g. 1 in 100 = 0.01), and p is an observed probability ε [0, 1]. The Wald interval is a legitimate approximation to the Binomial interval about an expected population probability P, but (naturally) a wholly inaccurate approximation to its inverse about p (the Clopper-Pearson interval). In fitting contexts it is legitimate to employ a Wald interval about P because we model an ideal P and compute the fit from there. But when we plot observed p, we need to employ the Wilson interval. I would encourage people to read the paper, not just the excerpt! Sean Wallis Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered Jun 1, 2014 at 9:11 Sean WallisSean Wallis 21 1 1 bronze badge Add a comment\| This answer is useful 0 Save this answer. Show activity on this post. I think the plot in question originally comes from Wallis (2021) so I recommend you have a look at that book for further explanation on the particulars of that graphical representation. In any case, the main reason why the Wilson score interval is superior to the classical Wald interval is that is is derived by solving a quadratic inequality for the proportion parameter that leads to an interval that respects the true support of the parameter. Contrarily, the Wald interval can go outside the true support, and it also has worse coverage properties (see Brown, Cai and DasGupta (2001) for further discussion). It might help here to show you the derivation of the interval in algebraic terms. Suppose we have n n binary data values giving the sample proportion p n p n (which we will treat as a random variable) and let θ θ be the true proportion parameter. The classical Wald interval uses the asymptotic pivotal distribution: n−−√⋅p n−θ θ(1−θ)−−−−−−−√∼Approx N(0,1).n⋅p n−θ θ(1−θ)∼Approx N(0,1). For the Wilson score interval we first square the pivotal quantity to get: n⋅(p n−θ)2 θ(1−θ)∼Approx ChiSq(1).n⋅(p n−θ)2 θ(1−θ)∼Approx ChiSq(1). Let χ 2 1,α χ 1,α 2 denote the critical point of the chi-squared distribution with one degree-of-freedom (with upper tail area α α). For any confidence level 1−α 1−α we then have the probability interval: 1−α≈P(n(p n−θ)2⩽χ 2 1,α θ(1−θ))=P(n(p 2 n−2 p n θ+θ 2)⩽χ 2 1,α(θ−θ 2))=P((n+χ 2 1,α)θ 2−(2 n p n+χ 2 1,α)θ+n p 2 n⩽0)=P(θ 2−2⋅n p n+1 2 χ 2 1,α n+χ 2 1,α⋅θ+n p 2 n n+χ 2 1,α⩽0)=P((θ−n p n+1 2 χ 2 1,α n+χ 2 1,α)2⩽χ 2 1,α(n p n(1−p n)+1 4 χ 2 1,α)(n+χ 2 1,α)2)=P(θ∈[n p n+1 2 χ 2 1,α n+χ 2 1,α±χ 1,α n+χ 2 1,α⋅n p n(1−p n)+1 4 χ 2 1,α−−−−−−−−−−−−−−−−√]),1−α≈P(n(p n−θ)2⩽χ 1,α 2 θ(1−θ))=P(n(p n 2−2 p n θ+θ 2)⩽χ 1,α 2(θ−θ 2))=P((n+χ 1,α 2)θ 2−(2 n p n+χ 1,α 2)θ+n p n 2⩽0)=P(θ 2−2⋅n p n+1 2 χ 1,α 2 n+χ 1,α 2⋅θ+n p n 2 n+χ 1,α 2⩽0)=P((θ−n p n+1 2 χ 1,α 2 n+χ 1,α 2)2⩽χ 1,α 2(n p n(1−p n)+1 4 χ 1,α 2)(n+χ 1,α 2)2)=P(θ∈[n p n+1 2 χ 1,α 2 n+χ 1,α 2±χ 1,α n+χ 1,α 2⋅n p n(1−p n)+1 4 χ 1,α 2]), and substitution of the observed sample proportion (for simplicity I will use the same notation for this value) then leads to the Wilson score interval: CI θ(1−α)=[n p n+1 2 χ 2 1,α n+χ 2 1,α±χ 1,α n+χ 2 1,α⋅n p n(1−p n)+1 4 χ 2 1,α−−−−−−−−−−−−−−−−√].CI θ(1−α)=[n p n+1 2 χ 1,α 2 n+χ 1,α 2±χ 1,α n+χ 1,α 2⋅n p n(1−p n)+1 4 χ 1,α 2]. As you can see, solving the quadratic inequality in the probability interval leads to an interval that respects the true space of possible values of the proportion parameter (i.e., it is between zero and one). This is a major advantage of this method but it also has better coverage properties in general. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications answered Aug 8, 2021 at 16:14 BenBen 4,534 15 15 silver badges 32 32 bronze badges Add a comment\| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions statistics See similar questions with these tags. Featured on Meta Introducing a new proactive anti-spam measure Spevacus has joined us as a Community Manager stackoverflow.ai - rebuilt for attribution Community Asks Sprint Announcement - September 2025 Report this ad Linked 1Confusion over Wald and Wilson coverage probabilities Related 4Working out the "best" score based on quantity and ratio 0Solution verification for hypothesis testing and confidence interval problem 2The difference between a Z-Score and a Z-statistic? Why do we divide by n−−√n for the latter? 0Question about the Z 2How can I evaluate average with negative values, given I will apply this weight for both negative and positive values 1How is the threshold determined in the Monobit Test, and why is it larger for a 1% significance level than for 5%? 0How to cheaply guess if a probability is less or greater than half Hot Network Questions Xubuntu 24.04 - Libreoffice ConTeXt: Unnecessary space in \setupheadertext Is there a way to defend from Spot kick? 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2866	https://tasks.illustrativemathematics.org/blueprints/A2/4/4	Illustrative Mathematics Typesetting math: 100% Engage your students with effective distance learning resources. ACCESS RESOURCES>> Course High School - A2 Unit Polynomials and Rational Functions Section Rational Functions Section: A2.4.4 Rational Functions • Build a rational function that describes a relationship between two quantities (F-BF.A.1⋆). • Graph rational functions (A-SSE.A.1a⋆, F-IF.C.7d⋆). • Interpret the graph of a rational function in terms of a context (F-IF.B.4⋆). In this section students study simple rational functions. The emphasis is on rational functions that arise naturally out of a real-world context, and on interpreting features of their graphs in terms of that context. Students experiment with graphs using technology to learn the relationship between features of the graph and the structure of the expression defining the function. Continue Reading Tasks 1 The Canoe Trip, Variation 1 View Details WHAT: Students are guided through the construction of a simple rational function that models the time it takes someone to paddle upstream a certain distance as a function of the speed of the current. They interpret the vertical intercept and the vertical asymptote of the graph in terms of the context. WHY: The purpose of this task is to introduce the idea of a vertical asymptote for a rational function and provide a context where the behavior of the function near the asymptote makes sense in terms of the context. Although this task doesn’t incorporate the entire modeling cycle, students engage in aspects of (MP.4) because they create a function to represent a given situation, and then interpret features of its graph in a context. 2 Average Cost View Details WHAT: Students are guided through the construction of a simple rational function that models the average cost per DVD of producing a number of DVDs as a function of the number of DVDs produced. They explain why the values of the function level off and visualize this behavior as a horizontal asymptote of the graph. They describe the domain of the function. WHY: The purpose of this task is to give students a context where they understand numerically why a horizontal asymptote occurs and interpret the long term behavior represented by the asymptote in terms of a context. The task also provides an opportunity for students to express regularity in repeated reasoning as they construct the function by generalizing from calculations in a table (MP.8). 3 Graphing Rational Functions View Details WHAT: Students use sliders in the online graphing calculator Desmos to make connections between expressions for rational functions and their asymptotes. This task starts with an exploration of the graphs of two functions whose expressions look very similar but whose graphs behave completely differently. At first glance this is surprising but can be explained by a closer look at the functions’ expressions. WHY: The purpose of this task is to build on the work in the previous two activities to increase students’ understanding of the connection between expressions for rational functions and the asymptotes of their graphs. In this task there is now context and students focus purely on the structure of the expressions (MP.7). External Resources 1 Carpe Donut Go to resource Description WHAT: Students read a description of a donut shop with a pricing scheme that makes it cheaper to buy donuts in bulk. They model the cost as a linear function of the number of donuts purchased and then model the average cost per donut as a rational function of the number of donuts purchased. An extension of the activity includes using piecewise linear and piecewise rational functions to model a more complicated scheme in which the 13th donut is free. WHY: This activity brings together several ideas from this and previous units, including building (F-BF.A.1⋆) and interpreting the graphs of and interpreting rational, linear and piecewise defined functions (F-IF.C.7ab⋆, F-IF.C.7d⋆(+)), and relating the domain of a function to its graph (F-IF.B.5⋆). Students are presented with a context and must construct and interpret a mathematical model (MP.4). By performing several computations and then generalizing with an equation, students are expressing regularity in repeated reasoning (MP.8). Note that a paid subscription is required to access this resource. Standards Typeset May 4, 2016 at 18:58:52. Licensed by Illustrative Mathematics under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
2867	https://www.unicode.org/notes/tn39/	UTN #39: Bidi Brackets for Dummies Technical Notes Unicode® Technical Note #39 Bidi Brackets for Dummies Version2 Authors Ken Whistler Date 2020-08-06 (originally 2014-04-01) This Version Previous Version Latest Version Summary This document is the Bidi Brackets for Dummies course. Status This document is a Unicode Technical Note. Sole responsibility for its contents rests with the author(s). Publication does not imply any endorsement by the Unicode Consortium. For information on Unicode Technical Notes, including criteria for acceptance, see Contents Introduction Chapter 1. Brackets: What are they? Chapter 2. Brackets: What are pairs? Chapter 3. Brackets: The syntax of pairing Chapter 4. Brackets: Trial Runs Chapter 5. Unicode Bracket Club Chapter 6. SNIPE Hunting Chapter 7. SNARK Hunting Review Module References Modifications Introduction This is the Bidi Brackets for Dummies course. When you finish this course, you will be able to identify bracket pairs according to the Unicode Bidirectional Algorithm (UBA for short). You will also be able to add to your resume that you have completed the BFD course. There are no real prerequisites for this course. You don't even have to understand the concept of bidirectional text, or why the UBA even exists. We'll just focus on what you need to know to match up bracket pairs for rule N0 in the UBA. (And no, that is not en oh "NO" — it is en zero "N0".) Chapter 1. Brackets: What are they? So what is a "bracket", anyway? — Um, a "["? Yes, correct. But actually, in the UBA, many other things are also identified as "brackets". So a parenthesis "(" is considered a bracket, and a curly brace "{" is also considered a bracket. The "[" bracket is called a square bracket, so we can tell it apart from all the other "brackets". Basically, any of the punctuation marks that come in pairs, and which are used to enclose stuff in text, to set it off from other stuff outside the marks, are considered brackets. There are many others, besides the ones people typically recognize on their keyboard, including such oddities as U+29FC ⧼ LEFT-POINTING CURVED ANGLE BRACKET and U+2E26 ⸦ LEFT SIDEWAYS U BRACKET. There are even brackets in Unicode that aren't even called brackets or parentheses or braces — characters like U+29DA ⧚ LEFT DOUBLE WIGGLY FENCE and U+27C5 ⟅ LEFT S-SHAPED BAG DELIMITER. Don't ask me why — they are all just a bunch of weird squigglies used in pairs as punctuation. — I don't see any weird squigglies there. That's just a bunch of boxes! Get yourself some better fonts. — So is the Arabic U+FD3E ﴾ ORNATE LEFT PARENTHESIS a bracket, too? No. — Why not? Because I said so. — How about quotation marks? They come in pairs and enclose stuff inside, separating it from all the stuff outside. Some of them, like that French thingie ‹ even look like brackets. No, they aren't "brackets", either. — Why not? Because I said so. — How about U+300C 「 LEFT CORNER BRACKET? Is that a "bracket"? Yes, that is a bracket. — But I thought that was a Japanese quotation mark. How come it is a bracket, but the other quotation marks aren't? Because I said so. Oh, and U+230A ⌊ LEFT FLOOR is a "bracket", too. — But that doesn't make any sense. What does "FLOOR" have to do with brackets? It's probably something to do with maths, but it's a "bracket", anyway. Besides, if you were a decent carpenter, you would know what a floor bracket was. — How about the left angle bracket "<"? Is that at least a "bracket"? No. — Why not? Because it's a LESS-THAN SIGN. And enough with all these silly "Why?" and "Why not?" questions. Homework Your homework is to memorize the list of code points and character names in BidiBrackets.txt [Brackets]. That list defines all the brackets for UBA. And it doesn't matter what else you might think is a "bracket". That's the list and that's just the way it is. Chapter 2. Brackets: What are pairs? If you were paying attention in the Chapter 1 lesson, you may have noticed that all the "brackets" had "LEFT" in their names. It turns out, not too surprisingly, that there are also corresponding "brackets" with "RIGHT" in their names. The left and right versions of brackets form pairs with each other. So "[" is the left version of "]", and vice versa. They form a pair. And usually they are graphical mirrors of each other. The left of the pair points right, and the right of the pair points left. — Wait! That's too confusing. O.k., think of it this way. The left of the pair occurs to the left side of the text it encloses, and the right of the pair occurs to the right side of the text it encloses. Is that clearer? — Yes, much! Only, sometimes in bidirectional text, it is just the opposite. — What?! Well, don't worry about that now. Remember, you don't need to understand bidirectional text to understand the identification of bracket pairs. So forget I even mentioned that. The important thing is to know that the "LEFT" ones pair with the "RIGHT" ones, and it doesn't really matter, in the end, which way they point or where the text they enclose actually is, o.k.? — Now wait a minute. I did my homework, and there was something in that list called TIBETAN MARK GUG RTAGS GYON. That doesn't have "LEFT" in its name. What's up with that? Well, "GYON" means left in Tibetan. Same difference. — You can't fool me though — I didn't see "LEFT" in OGHAM FEATHER MARK, either. And you can't convince me that "FEATHER" means left. That one forms a pair with the OGHAM REVERSED FEATHER MARK. And you know what they say about foolish consistency, right? Anyway, the point is that all of the "brackets" come in pairs. And even if they don't all actually have "LEFT" or "RIGHT" in their names, you can always tell which one is paired with which other one. If this isn't always obvious, then the way to figure it out is to go back to BidiBrackets.txt. The second number in each line of that file is the character that forms the pair. So you look at a line like: 005B; 005D; o # LEFT SQUARE BRACKET And that tells you that U+005D is the pair for U+005B LEFT SQUARE BRACKET. The next line in the file is: 005D; 005B; c # RIGHT SQUARE BRACKET And that tells you that U+005B is the pair for U+005D RIGHT SQUARE BRACKET. — But isn't that redundant? Why use two lines to tell me the same thing? Yes. And because if I tell you something two times, it is true. — Where did the "U+" come from? I don't see that in BidiBrackets.txt. Go back to the Unicode for Dummies course. So, for those of you who are still following along, the next thing to know about is that the "LEFT" brackets in the pairs are also known as "opening" and the "RIGHT" brackets in the pairs are also known as "closing". The mysterious "o" and "c" that you see on each line of BidiBrackets.txt stand for "o"pening and for "c"losing. The "o" or "c" values are how you tell which one of a pair is which, even if the characters don't actually have "LEFT" or "RIGHT" in their names. It is really, really important, it turns out, to know which is the opening one and which is the closing one, because figuring out which particular bracket can pair up with which particular other bracket in a long line of text depends on being able to find all the opening ones and all the closing ones. But there is another complication here. It turns out that some brackets are canonical equivalents of other brackets, and those count as pairs, too. — Come again? Well, according to the Unicode Normalization Algorithm, a few of the angle brackets are canonically equivalent to other angle brackets, and because canonical equivalents cannot really be distinguished in most text processing, it isn't a good idea to separate them when it comes to identifying bracket pairs. — Wait, I thought angle brackets weren't actually "brackets"! Whatever. I'm not talking about "angle brackets" — I'm talking about U+2329 〈 LEFT-POINTING ANGLE BRACKET and U+3008 〈 LEFT ANGLE BRACKET, which are the same as each other, but which aren't the "<" angle bracket you are thinking about. — Ugh. I hated the Unicode Normalization for Dummies course — it was so confusing. Well, you are right about that, at least! For now, just memorize that U+3008 pairs with U+3009 and U+2329 pairs with U+232A, but that U+3008 also pairs with U+232A and U+3009 also pairs with U+2329. It is just like having to know that "pair" is also spelled "pear" — only that the pairs we are talking about here aren't pears, but angle brackets, even though they aren't "angle brackets". Homework Your homework for this chapter is to go back to BidiBrackets.txt [Brackets] and memorize all the "o" and "c" values in the list, so you know which character in each pair is the opening or the closing one. — Will that be on the midterm? Yes. Chapter 3. Brackets: The syntax of pairing In Chapter 2 we learned all about which brackets pair with which brackets. But it turns out that so far that information is all abstract. We can think of that as the "semantics" of bracket pairs, because it is the truth of their meaning, independent of any use of them in text. An "opening bracket" is opening, just because of what it is and what it means. It doesn't matter where it occurs in text. But determining which bracket pairs with which other bracket is more complicated than just having memorized all the information in BidiBrackets.txt and knowing which ones are opening and which ones are closing. So what am I talking about? Let's start by looking at a few examples. In the following short text: Voltage equals current multiplied by resistance (Ohm's Law) The opening "(" is paired with the closing ")". And we then interpret the text "Ohm's Law" as being "inside the parentheses" — or, more generally as "inside the brackets", since we are talking generically about bracket pairs here. Our concern is not the Chicago Manual of Style [CMS], or exactly which brackets should be used where, but rather just figuring out which opening bracket goes with which closing bracket. It doesn't matter whether they are the right (or, um, "correct") ones or not. So for our purposes, it would be equally valid to consider: Voltage equals current multiplied by resistance [Ohm's Law] And see that the opening "[" is paired with the closing "]". It is crucial, however, to note the exact order in which these pairs occur. If the text is changed to: Voltage equals current multiplied by resistance ]Ohm's Law[ We would no longer consider that particular "]" to be paired with that particular "[". They are in the wrong order here. In principle, the closing "]" might pair with some opening "[" off the to the left of the line, but not visible here. And the "[" might pair with some other "]" from somewhere continuing on ahead in the text. But these two do not pair with each other, because they appear here in the wrong order to be considered a matching pair. — Wait! In this case, can't I just redefine "]" as opening and "[" as closing? I kinda like the way "]Ohm's Law[" looks! No. Well, maybe you could in an alternative universe, I suppose, but not with the UBA we have. Remember, I told you to memorize the opening and closing values in BidiBrackets.txt — not to question them. At any rate, the point here is that for any particular bracket in any particular text, it totally matters which order you encounter them in. "(^.^)" contains a bracket pair — well, it might also contain an emoticon, but that is the topic for another course. ")^.^(", on the other hand, does not contain a bracket pair (but might contain an Anpanman). Also, if I have two lefts, as in "(^.^(", those two brackets (well, parentheses, but whatever!) also do not pair with each other. They might pair up with some closing parentheses much later on in your text, but not with each other in just the text as we have it there. Remember, the general principle here is that two lefts do not make a right and two rights do not make a left. :-) :-) Homework Your homework for this chapter is to meditate on the epistemological implications of missing text and on The Order of Things. Chapter 4. Brackets: Trial Runs Now that we have come to grips with the fact that the order of brackets matters, we need to move on to dealing with the implications of more than two brackets occurring together. When people use multiple brackets altogether in a line, whether they are individually matching pairs or not, how do we figure out which particular bracket pairs up with which particular other bracket? In the case of just two brackets, this is pretty simple to determine. But once the brackets start multiplying (with or without multiplication signs), the situation gets exponentially more complicated. I'll start the discussion first using just sets of three brackets, which I like to call "trials" — because, after all, they are sets of three, kinda like triads. But also because we have to try various combinations to figure them out, and because it is a trial to explain all this to you. And before I go further, there is something else you need to know. All of this work to match up brackets only counts if you are inside a bidirectional "isolating run sequence". Those are kinda like long distance training runs, only different. If you don't know what they are, don't worry — you can always buy my two-part course, Unicode Bidirectional Runs for Dummies and the sequel, Unicode Bidirectional Isolating Run Sequences for Dummies VI.III: The X10 Also Rises. For now, we'll just consider that somebody has already figured out what they are, and all the examples here are inside a single one of them, whatever they are. So let's look at some examples. We'll stay simple to start with and use brackets all of the same kind — just square brackets. Bud [Gus [Hank [Xerxes O.k., there are three opening brackets in that line, but no closing brackets. So there are no bracket pairs, which require an opening bracket and a closing bracket in the correct order. How about: Bud ]Gus [Hank [Xerxes Still three brackets, but now one is closing and two are opening. They are still in the wrong order to make any pairs, however. But look at: Bud [Gus ]Hank [Xerxes Now the first two brackets are opening and closing in the correct order, so we can figure out that "[Gus ]" contains a bracket pair, which, in turn encloses the text "Gus ", including the space. The last opening bracket has no closing bracket to match with. Similarly: Bud ]Gus [Hank ]Xerxes Now the first closing bracket has nothing to match, but "[Hank ]" contains a bracket pair, enclosing the text "Hank ". Where things start to get interesting is in the following: Bud [Gus [Hank ]Xerxes In this case "[Hank ]" still contains a bracket pair. The first opening bracket on the line doesn't match the closing bracket, because there is an opening bracket after it which is closer to the closer. So the closer opener wins... wait, now I'm confusing myself! In any case, the outside brackets in "[Gus [Hank ]" don't match for a bracket pair — only the inside ones do. Bud [Gus ]Hank ]Xerxes That example is similar to the last one, only this time the bracket pair is in "[Gus ]", and the second closing bracket doesn't match any opening bracket. Now let's throw a curveball in here — in particular, a curly brace. This changes things, and if you just swing away without keeping your eye on the ball, you are going to miss how things work: Bud [Gus {Hank ]Xerxes That example has the same pattern of opening-opening-closing, but now the second opening bracket (the left curly brace) no longer can match the closing bracket, because it is a different kind of bracket. So in this case, the outer pair matches, and we find a bracket pair in "[Gus {Hank ]", enclosing the text "Gus {Hank ", including the left curly brace, as well as the spaces. As we'll see in the next chapter, the effect on that curly brace is pretty drastic, because the bracket pair (of square brackets) has taken square aim at and effectively snuffed the curly brace they enclose. That curly brace can't match any other curly brace which might have come later, because it has been enclosed already [in a {-coffin, as it were]. Be that as it may, remember that this behavior isn't inherent to square brackets and curly braces. It isn't as if square brackets are inherently "stronger" than curly braces and always win when they come into contention for matching. If the situation were reversed, the curly braces could turn the tables and ice the bracket, instead: Bud {Gus [Hank }Xerxes In this case there is a "bracket" pair in "{Gus [Hank }", enclosing the text "Gus [Hank ", including the left square bracket, as well as the spaces. So here the curly braces bury the single opening square bracket — along with Gus and Hank. So as far as finding bracket pairs go, no single type of "bracket" has priority over any other. They are all equals. The important thing is which one comes first and then in what order all the rest occur on the line. Homework Your homework is to try to dig up more triads with square brackets and curly braces. If you find any "little problems" I have forgotten about, it doesn't matter. Chapter 5. Unicode Bracket Club O.k., in the last chapter we move on to the most difficult part of matching bracket pairs for UBA: what to do for long sequences of multiple brackets of different types. If you haven't washed out of the course by now, you might have what it takes to join Unicode Bracket Club. So what is the first rule of Unicode Bracket Club? — Um..., don't talk about brackets? No. The first rule of Unicode Bracket Club is: Matching bracket pairs for UBA is not expression evaluation. — Huh? Well, yes. Most people's notion of how multiple parenthesis matching works comes from their high school algebra class. Remember those equations: ((a + b) (b + 9a)) + ((2a – 1) (2b + 17)) = –1 You had to go through and evaluate all the expressions and combine the inner expressions by applying the operators to form other expressions, and so on. And what happened if you were missing a balancing parenthesis? — Uh, the algebra teacher hit us with a ruler. Right, and the equations didn't work, because they had what is called an expression error. If you use parentheses (or brackets for matrices, and so forth), they have to balance, or the expression has a syntax error, and it cannot be evaluated. But in Unicode Bracket Club, if a "balancing parenthesis" is missing, meh! It just changes what "bracket" matches what other "bracket" in the line. And a missing match isn't an error — it's just "interesting". So if you're all ready with your BNF for brackets and have an LL-grammar ready and want to get busy with an LALR parsing strategy for bidirectional brackets, you're probably in the wrong class. Maybe Compilers for Dummies is what you wanted. Instead, in Unicode Bracket Club, we need to be able to figure out what to do with: (((a[(])b We need to identify the matching bracket pairs and the brackets that don't match, but always come out smiling. Because there are no errors, and everyone is a winner in Unicode Bracket Club! So how do we find matching bracket pairs? You already have some clues from Chapter 4 and the earlier chapters: An opener for a match has to come before its closer. An opener for a match has to be the same kind of bracket as its closer. If an unmatched opener or closer ends up inside a different matched pair, then it is erased from contention and stays unmatched. So let's extend the examples for bracket trials and move into full-blown bracket matches. How can we describe the moves which will lead to a good match? First of all, always start from the left and move systematically to the right. This is known as the logical order for a match. (Members of Unicode Bracket Club living in the Middle East tend to think of this as always starting from the right and moving systematically to the left, but they are still using the same logical order — it just looks different when you watch their matches. Think of it as watching from the other side of the ring.) Then take the following steps: If you encounter a closer before you have any opener to match it, just discard it. That is known as a feint. If you encounter an opener, remember it. Those count for scoring. Also remember what kind of opener it was and exactly where you encountered it. Move on. If the next bracket you encounter is a closer, check to see if it matches one of the openers you remembered. You start from the most recently encountered opener and think back to the first opener that is of the same type. If you find one, congratulations, you have identified a bracket pair. Mark it down for keeping. This is known as a combination. And by the way, if you identified a bracket pair, but while you were remembering back to it, you passed over any brackets of different types, whether opening or closing, you can now forget about them. Those are all known as misses. However, if the next bracket you encounter is another opener, just remember it, too, and where you found it. Those are known as keepers. If the next character you encounter is not a bracket at all, just skip on by. That is known as fancy footwork. Now check if you have moved all the way systematically in logical order to the end. If so, you are done, and it is time for scoring. But if you aren't at the end, then go back to step 3 and move on again. Remember to keep bobbing and weaving as you continue the match. Once the match is done, it is time for scoring. Bring all the combinations you found to the scoring table and lay them out for the judges to examine. The judges will then reorder all the combinations by the positions they occurred in. The number of combinations you found, reordered in neat ascending order by position, constitute your score. Congratulations! O.k., let's watch the progress of a particular match closely, to see all the moves one-by-one. The contender is: (((a[(])b 123456789 Note that the positions are labeled, starting at 1, beneath each character. We are going to work our way through, systematically, in logical order, from 1 until we get all the way to the end at 9. Position 1: Found an opener "(". Remember it: "(" at 1. Position 2: Found an opener "(". Remember it: "(" at 2. Position 3: Found an opener "(". Remember it: "(" at 3. Position 4: Not a bracket. Skip on by. Position 5: Found an opener "[". Remember it: "[" at 5. Position 6: Found an opener "(". Remember it: "(" at 6. Position 7: Found a closer (finally!) "]". Think back to the first matching opener before. That would be "[" at 5. Save the combination: "[" at 5 and "]" at 7. And since we passed over an unmatched opener "(" at 6, just forget that one now. It is a miss. Position 8: Found a closer ")". Think back to the first matching opener before. Since we already forgot about "(" at 6, the correct match will be "(" at 3. Save the combination: "(" at 3 and ")" at 8. Position 9: Not a bracket. Skip on by. But hey, we are at the end, so we're done with this match. Scoring: Lay out the combinations you found: "[" at 5 and "]" at 7 "(" at 3 and ")" at 8 The judges rearrange those to: "(" at 3 and ")" at 8 "[" at 5 and "]" at 7 And count up your matches: your score is 2! All the other brackets you found during the match don't count for matching bracket pairs. They can be ignored from now on, because you have your neatly arranged and scored list of actual matched bracket pairs. So lets go back to the start. The contender was: (((a[(])b And we have discovered that the first bracket pair in it is "(a[(])", which encloses the text "a[(]". But that enclosed text itself contains the second bracket pair: "[(]", which encloses the text "(", itself an unmatched "bracket". And the two parentheses at the very start are also unmatched "brackets". So matched bracket pairs can be inside other matched bracket pairs. We just have to be very careful in finding them, because they aren't always immediately obvious. In particular, for a different contender: [(]) The "[(]" contains a matching bracket pair, but "(])" does not. Homework Go back in the ring with a new contender: a[(])b[)] 123456789 Chapter 6. SNIPE Hunting How did your homework go? Are you getting the hang of it? I hope you scored another 2 with that contender I sent you home with. — Well, I guess I understand sorta. But I've suffered all the way through these explanations and I still don't know what a matching bracket pair is when I see one. You've given me instructions for how to join Unicode Bracket Club and then find combinations that count as matched bracket pairs as I step through all the moves to deal with a contender. Why can't you just define what a goldarn matching bracket pair is, instead of sending me through all this Bracket Club folderol to try to find them?? Fair enough. I suppose we could try it that way. First add a few definitions: BD14. An opening paired bracket is a character whose Bidi_Paired_Bracket_Type property value is Open. BD15. A closing paired bracket is a character whose Bidi_Paired_Bracket_Type property value is Close. BD16a. A bracket pair is a pair of an opening paired bracket and a closing paired bracket character such that the Bidi_Paired_Bracket property value of the former character or its canonical equivalent equals the latter character or its canonical equivalent. Note that you already memorized the list of those for your homework for Chapter 2. But let's add a new definition: BD16b. A SNIPE is a bracket pair that has been selected from among possible bracket pairs in an isolating run sequence. (SNIPE is short for "Systematically Normalized and Identified Paired Enclosure".) Now we've already agreed to ignore the fact that we don't know exactly how to find an "isolating run sequence". And this new term "SNIPE" is just another way of saying "a bracket pair which has been resolved by some selection process to be a pair that matches". In other words, it means a "matching bracket pair" in a syntactic context. — But that just gets me back to where I was before. I know what "matching bracket pair" means — I just don't know how to find the darn things! Now you're just saying that SNIPEs are "selected from among possible bracket pairs". How do I find them? Relax, relax. The answer to that is coming. We'll just define a rule to find them: Rx. For each isolating run sequence, bracket characters are determined to be SNIPEs (or not) as follows: Starting at the beginning of the run sequence, when a closing bracket character is encountered, find the nearest preceding opening character that forms a bracket pair, but is not already determined to be a SNIPE, and not ignored for bracket pair selection. If one exists, mark it as a SNIPE, and mark any enclosed opening brackets of any kind as not part of a bracket pair and ignored for further bracket pair selection. Otherwise, if no pair can be determined to be a SNIPE, mark the closing bracket as not part of a pair and ignored for further pair selection. — Gah! That's it?! How is that any clearer than the rules for Unicode Bracket Club matches? Can you show me how that works? Sure. Let's go back to our original contender: (((a[(])b 123456789 Let's use Rule Rx and select all the resolved bracket pairs. Scan forward to the first closing bracket character. That is "]" at 7. Scan backwards to the nearest opening bracket character that forms a bracket pair. That is "[" at 5. Is "[" at 5 already part of a resolved bracket pair? (Oops! Reminder to self: remember to first scan through and set all brackets to "not-in-resolved-bracket-pair" before starting to apply Rx.) Let's go with the answer: No. Is "[" at 5 not ignored for bracket pair selection? (Oops! Reminder to self: remember to first scan through and set all brackets to "not-ignored-for-bracket-selection" before starting to apply Rx.) Let's go with the answer: No. O.k., we've determined that "[" at 5 meets the criterion, so we now have our first resolved bracket pair. Set "[" at 5 and "]" at 7 to "in-resolved-bracket-pair". Find any enclosed opening bracket in the resolved bracket pair. That means scanning between 5 and 7. We find "(" at 6. Mark that as "not-in-resolved-bracket-pair" and "ignored-for-bracket-selection". Go back to where we left off at step #1 and scan forward to the next closing bracket character. That is ")" at 8. Scan backwards to the nearest opening bracket character that forms a bracket pair. That is "(" at 6. Is "(" at 6 part of a resolved bracket pair? No. Is "(" at 6 ignored for bracket pair selection? Yes. O.k., then we need to keep scanning back. Scan backwards to the next nearest opening bracket character that forms a bracket pair. That is "(" at 3. Is "(" at 3 part of a resolved bracket pair? No. Is "(" at 3 ignored for bracket pair selection? No. O.k., we've determined that "(" at 3 meets the criterion, so we now have our second resolved bracket pair. Set "(" at 3 and ")" at 8 to "in-resolved-bracket-pair". Find any enclosed opening bracket in the resolved bracket pair. That means scanning between 3 and 8. We find "[" at 5 and (" at 6. Mark them as "not-in-resolved-bracket-pair" and as "ignored-for-bracket-selection". (Oops! Reminder to self: Update rule Rx, because we really didn't want to change "[" at 5 to "not-in-resolved-bracket-pair" and "ignored-for-bracket-selection". For now, we'll just pretend the rule is already patched up.) Go back to where we left off at step #7 and scan forward to the next closing bracket character. O.k., there aren't any. We are done. Now we can go on to scoring. What are all the brackets identified as "in-resolved-bracket-pair"? Those would be: "(" at 3, "[" at 5, "]" at 7, and ")" at 8 But what are the actual resolved pairs? Oops! We forgot to keep track. Reminder to self: Add requirement in rule to keep exact list of each resolved bracket pair as it is identified, for later reference. O.k., let's go back and keep track as we resolve them, and we get: "[" at 5 and "]" at 7 "(" at 3 and ")" at 8 That's two resolved bracket pairs, so our final score is 2! We're done, right? Oops! We forgot to reorder the list in ascending order by position, rather than in order of selection by rule Rx. Reminder to self: Add requirement in rule to do post-selection reordering by position of the first character in each resolved bracket pair in the list. Well, there were a few things that needed patching up here and there, but the process of selection is ever so much clearer expressed this way, right? We don't have to "remember" a stack of openers. All we have to "remember" is the process status of all the brackets, where we left off to restart the forward scan, the spans we have to check (and recheck) each time we identify a new resolved bracket pair, and the list of resolved bracket pairs as we select them. Homework Find all the SNIPEs hiding in the following swamp: (S[N{I(the deep, dark swamp}P]E)!) Make sure you get yourself a sturdy pair of waders, because it is likely you will end up walking back and forth through the swamp many times. Chapter 7. SNARK Hunting In this chapter, I will be introducing yet another way to look for bracket pairs. This method is called SNARK hunting. — Wait! I can't follow the first two methods, and I did get lost in that deep, dark swamp. How am I going to learn a third method? Well, this method is loads easier. Instead of hunting for SNIPEs, we'll be hunting for SNARKs. And this is a super-special hunting method — it is recursive SNARK hunting. It's called that because you get to curse and re-curse each time you fail to find a SNARK! — O.k. That sounds fun at least. But what is a SNARK? Well, in general, you can use all the same definitions we already used for SNIPE hunting, but add one new definition: BD16c. A SNARK is a bracket pair that has been selected from among possible bracket pairs in an isolating run sequence when you are self-isolating and running on a beach. (SNARK is short for "Systematically Normalized and Algorithmically Reconstituted KPairedBracketsForMock🐢Soup".) — That's a really stupid acronym — especially the "K" part. Whatever. It works. If I had gone with "Bracket Pairs", it would have come out SNARBP! Wouldn't you rather hunt SNARKs than SNARBPs?! — Well, I guess. So how does SNARK hunting work? It is all explained in snark.c, a self-documenting bit of C code. — What is C code? Is that just short for character code? That explanation is outside the scope of this course. Go read C for Dummies. Homework Find the bracket pairs in this new contender: ({([{([){()([])]}}[{(X)})[)])})}}{([])[]}) For extra credit do the exercise thrice, once with each method described in Chapters 5, 6, and 7, and compare and contrast your results. Review Module for Bidi Brackets for Dummies "[" is a bracket. "(" is a "bracket", too. "[" is opening, and pairs with "]", which is closing. "{..}" contains a bracket pair. "}..{" does not. "(..[)..]" → (..[), but neither "[" nor "]" is part of a pair. "[(]x[)]" → [(] and [)], but neither "(" nor ")" is part of a pair. References [Brackets]BidiBrackets.txt [CMS]The Chicago Manual of Style: The Essential Guide for Writers, Editors, and Publishers (14th Edition) University of Chicago Press (Trd); ISBN: 0226103897 Also see their FAQ at [FAQ]Unicode Frequently Asked Questions answers to common questions on technical issues. [Glossary]Unicode Glossary explanations of terminology used in this and other documents. [Reports]Unicode Technical Reports information on the status and development process for technical reports, and for a list of technical reports. [Versions]Versions of the Unicode Standard For details on the precise contents of each version of the Unicode Standard, and how to cite them. Modifications The following summarizes modifications from the previous version of this document. 2 Split Chapter 5 into two more manageable chunks, for easier learning. Restored the chapter on snark hunting which had been removed by the Unicode Censors. Minor editorial corrections. 1 First version © 2014–2020 Ken Whistler. This publication is protected by copyright, and permission must be obtained from the author and Unicode, Inc. prior to any reproduction, modification, or other use not permitted by the Terms of Use. Use of this publication is governed by the Unicode Terms of Use. The authors, contributors, and publishers have taken care in the preparation of this publication, but make no express or implied representation or warranty of any kind and assume no responsibility or liability for errors or omissions or for consequential or incidental damages that may arise therefrom. This publication is provided “AS-IS” without charge as a convenience to users. Unicode and the Unicode Logo are registered trademarks of Unicode, Inc., in the United States and other countries.
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2869	https://www.scienceabc.com/pure-sciences/what-is-midpoint-theorem.html	Midpoint Theorem: Definition, Explanation, Proof And Formula Physics Astrophysics Theoretical Physics Sports Super Heroes Earth Science Chemistry Biology Botany Zoology Medicine Neuroscience Engineering Technology Artificial Intelligence Computing Mathematics Social Science Psychology History Sociology Geography Philosophy Economics Linguistics Art Videos About Us Physics Astrophysics Theoretical Physics Sports Super Heroes Earth Science Chemistry Biology Botany Zoology Medicine Neuroscience Engineering Technology Artificial Intelligence Computing Mathematics Social Science Psychology History Sociology Geography Philosophy Economics Linguistics Art Videos About Us Home»Pure Sciences What Is The Midpoint Theorem? Written by Sarthak Singh GaurLast Updated On: 2 Jun 2024 Published On: 22 Sep 2019 Table of Contents (click to expand) Midpoint Theorem Practical Understanding Conclusion The midpoint theorem states that the line segment connecting the midpoints of two sides of a triangle is parallel to the third side and is congruent to one half of the third side. Whereas, its converse states that the line drawn through the midpoint of one side of a triangle and parallel to another side bisects the third side. Imagine sitting in a baseball stadium and watching your favorite team play!! It was a good game, right? Baseball has one of the most uniquely shaped fields in sports, much different than the most common rectangular or circular playing fields, and it is referred to as a Baseball Diamond. Now, someone with even a bit of mathematical knowledge might wonder how that field was made? Curiosity regarding the construction of the diamond-shaped, field according to the specified dimensions and the theorems involved in its construction, must make one feel inquisitive. If you take a closer look towards the field or see a birds’ eye view of the field, you will notice that the diamond looks like the sector of a circle with a triangle in it (joining the endpoints of the grass line with the help of a straight line).With that image in mind, let’s explore the mathematics behind the construction of the legendary baseball diamond. Baseball Diamond (Photo Credit: Frank Romeo/Shutterstock) Recommended Video for you: Play Why is a Circle 360 Degrees, Why Not a Simpler Number, like 100? Midpoint Theorem The midpoint theorem states that “For a given triangle ∆ABC, let D and E be the midpoints of AC and AB, respectively. Then the segment DE is parallel to BC and its length is one half the length of segment BC.” Or, in simple words, it can be stated as The line segment connecting the midpoints of two sides of a triangle is parallel to the third side and is congruent to one half of the third side. Midpoint Theorem Illustration Figure Midpoint Theorem Proof Any theorem must have a mathematical proof for it to be valid and the midpoint theorem also has one. To Prove- DE = (1/2) BC and DE\|\|BC In the above figure, extend the line segment DE to a point F in such a way that DE = EF and also joins F to point C. In triangle ADE and ECF, we have – DE = EF (by construction), ∠ AED = ∠ CEF (since they are vertically opposite angles)and EC = AE (since E is the midpoint of AC). According to the above results, we can say that the triangles AED and CEF are congruent. Therefore, we can say that ∠ ADE = ∠CFE (alternate interior angles), and similarly, ∠DAE = ∠FCE (alternate interior angles) and AD=CF. Therefore, we can say that CF\|\|AB, so CF\|\|BD. Since opposite sides of the quadrilateral BDFC are parallel and equal, BDFC is a parallelogram, hence BC\|\|DF, i.e., BC\|\|DE and DE = (1/2) BC. Hence, the midpoint theorem is proved. This is the general textbook explanation that students tend to understand, but never question in terms of its application to real-world problems. Now, before this gets boring, we’ll shift back into baseball to make this concept more interesting and easy to understand. Practical Understanding To understand any theorem, it’s essential to understand its practical importance and application. So, we’re coming back to the baseball field for a practical understanding of the theorem.Below are the dimensions of a baseball field (listing only the important/relevant dimensions to prove the practical application of the midpoint theorem). Home to first base – 27.43 m Third base to home – 27.43 m Home to left-field foul pole – 99.06 m Home to right field foul pole – 99.06 m Major League Baseball Field Dimensions The known distance between the two foul poles is 140.09 meters; we can now use the midpoint theorem to calculate this and find out whether the theorem is practically valid or not. Baseball field geometrical representation Considering the triangle formed by the two foul poles and home plate, we have two sides of the triangle, both having their length equal to 99.06 m, and the third side, i.e., the distance between the foul poles, which is 140.09 m. The midpoints of the equal sides (from home plate to the left and right foul poles) are at a distance of 49.53 m from home and poles. Now, if you join the two midpoints with the help of a line segment, the length of the line segment is unknown, but can be easily determined using basics or trigonometry and triangle congruency. Here we have A (home plate), B (right foul pole) and C (left foul pole). O is the perpendicular dropped from A to line segment DE. We will consider AB = AC since in a baseball field, the distance of the two foul poles from home plate is the same. Now, we know that in the triangle AOD, we can calculate DO by – Cosine = Base/Hypotenuse, ∠ ADO = 45 (since a baseball field is symmetrical) Therefore, cos 45= Base/49.53, which gives us the length of the base DO, i.e., 35.02 m. Also, since the triangles ADO and AEO are congruent by RHS congruency, line segment MN = 70.04 m, which is one half of LR, hence proving the midpoint theorem! Conclusion If you pay attention, you will see that we are surrounded by real-world examples that can help us learn subjects in a much more unique and fun way. However, it is up to us to find them! Mathematical theorems have their applications in various fields, but who would have thought that even their favorite sport would have applications from a subject which is a nightmare to most! References (click to expand) 1. Mid-point theorem. 2. Department of Mathematics and Statistics Homepage : Florida .... 3. Midpoint Theorem - timganmath.edu.sg 4. Baseball - www.dsr.wa.gov.au Sarthak Singh Gaur Sarthak Singh Gaur is a third-year civil engineering undergraduate at Ramaiah Institute of Technology, Bengaluru. He has done his internship at IIT, Kanpur in the field of GNSS and Navigation Messages and is interested in mathematics and its applications in various fields. He also is a huge football fan and has represented his school in various national level football tournaments. Hidden Cameras Found in 312 Hotel Rooms Last Month Victims lost $847,000 total. All were watched for days before being robbed. Harold and Barbara barely escaped. 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2870	https://blog.csdn.net/u013279723/article/details/106265948	判断点在多边形内的算法（Winding Number详解）-CSDN博客博客下载学习社区 GitCode InsCodeAI 会议搜索 AI 搜索登录登录后您可以：复制代码和一键运行与博主大V深度互动解锁海量精选资源获取前沿技术资讯立即登录会员·新人礼包消息历史创作中心创作判断点在多边形内的算法（Winding Number详解）最新推荐文章于 2025-05-18 15:59:13 发布原创于 2020-06-30 20:13:08 发布·置顶·1.4w 阅读 · 22 · 77· CC 4.0 BY-SA版权版权声明：本文为博主原创文章，遵循CC 4.0 BY-SA版权协议，转载请附上原文出处链接和本声明。文章标签： #winding number 探讨了计算几何中判定点是否位于多边形内的两种主要方法：交叉数（Crossing Number）和环绕数（Winding Number）。介绍了两种方法的工作原理、优缺点及适用场景，特别强调了Winding Number在自相交多边形判定中的优势。在计算几何中，判定点是否在多边形内，是个非常有趣的问题。通常有两种方法： 1.Crossing Number（交叉数）它计算从点P开始的射线穿过多边形边界的次数。当“交叉数”是偶数时，点在外面;当它是奇数时，点在里面。这种方法有时被称为“奇-偶”检验。 2.Winding Number（环绕数）它计算多边形绕着点P旋转的次数。只有当“圈数”wn = 0时，点才在外面; 否则，点在里面。如果一个多边形是不自交的（称为“简单多边形”），那么这两种方法对任意点都给出相同的结果。但对于非简单多边形，这两种方法在某些情况下会给出不同的答案。如下图所示，当一个多边形与自身重叠时，对于重叠区域内的点，如果使用交叉数判断，它在外面；而使用环绕数判断则在里面。顶点按次序编号:0 1 2 3 4 5 6 7 8 9 在上图中，绿色区域中的点，wn= 2，表示在多边形中重叠了2次。相比于Crossing number，winding number给出了更内蕴性的答案。尽管如此，早些时候，crossing number方法应用的更广泛，因为最初计算几何专家们错误地认为crossing number比winding number计算起来更加高效。但事实并非如此，两者的时间复杂度完全一样。Franklin在2000年给出一个计算winding number的非常快的实现。因此，为了几何正确性和效率的原因，在确定一个多边形中的一个点时，wn 算法应该总是首选的。 The Crossing Number 该方法计算从点P开始的射线穿过多边形边界的次数（不管穿过的方向）。如果这个数是偶数，那么点在外面;否则，当交叉数为奇数时，点在多边形内。其正确性很容易理解，因为每次射线穿过多边形边缘时，它的内外奇偶性都会发生变化(因为边界总是分隔内外)。最终，任何射线都在边界多边形之外结束。所以，如果点在多边形内，那么对边界的穿过次序一定是：out>...>in>out,因此交叉数一定是奇数；同样地，如果点在多边形外，那么对边界的穿过次序一定是in> out...>in>out,因此交叉数必是偶数。 < 最低0.47元/天解锁文章 200万优质内容无限畅学确定要放弃本次机会？福利倒计时 : : 立减 ¥ 普通VIP年卡可用立即使用 SimpleFelix 关注关注 22点赞踩 77 收藏觉得还不错? 一键收藏 2评论分享复制链接分享到 QQ 分享到新浪微博扫一扫举报举报专栏目录判断点在多边形内的算法 gghhb12的博客 04-01 2838 在计算几何中，判定点是否在多边形内，是个非常有趣的问题。 2 条评论您还未登录，请先登录后发表或查看评论开源项目 Winding Number 常见问题解决方案 9-20 问题描述:用户可能想要将 Winding Number 项目集成到自己的应用程序中,但不知道如何操作。解决步骤: 确保你的应用程序支持C++。将 Winding Number 项目的源文件和头文件添加到你的项目中。如果你的项目使用CMake,将 Winding Number 的CMakeLists.txt文件集成到你的主CMake文件中。确保链接了Intel TBB库和任何其他必要的库。一个判断点是否在参数方程连续可微的封闭曲线界定的区域之内的好概念... 9-16 本文介绍了一种基于卷绕数 ( winding number ) 的算法来判断平面内的点是否位于特定闭合曲线内部的方法。该方法不仅适用于椭圆,还能应对更复杂的C∞曲线及多边形,包括五角星形状的判定。先把标题放在这里,内容我慢慢准备,一点点添加1. 一个简单的例子从最简单的情况出发,连续可微的封闭曲线为边界的,最简单例子是”圆”... 判断一个坐标点是否在不规则多边形内部的算法 weixin_34082789的博客 01-22 3250 在GIS（地理信息管理系统）中，判断一个坐标是否在多边形内部是个经常要遇到的问题。乍听起来还挺复杂。根据W. Randolph Franklin 提出的PNPoly 算法，只需区区几行代码就解决了这个问题。假设多边形的坐标存放在一个数组里，首先我们需要取得该数组在横坐标和纵坐标的最大值和最小值，根据这四个点算出一个四边型，首先判断目标坐标点是否在这个四边型之内，如果在这个四边型之外，那可以... 山东大学计算机图形学期末复习11——CG13上最新发布 qq_66878808的博客 05-18 890 在三维场景中，多个物体可能重叠观察者只能看到最前面的面所以需要消除被遮挡的“隐藏面”，避免不必要的渲染。点在多边形内判断方法matlab实现 ( 点在凸包内、任意形状内判定 ) _matlab... 9-27 1 射线交叉点法 ( Crossing Number ) 2 环绕数法 ( Winding Number ) 3 角度法 ( 转角法 ) 3.1 角度相加法 3.2 改进角度法 ( matlab自带inpolygon函数方法 ) 4 叉积法 ( 只适用于凸多边形 ) 5 网格法 6 二分法 ( O ( logn ) 算法 ) 7 多个区域之间惯例声明:本人没有相关的工程应用经验,只是纯粹对相关算法感兴趣才写此博客。所... 中除了某个数以外的_卷绕数和高斯-博内定理 9-21 卷绕数 ( turning number, winding number )。平面上的闭曲线关于某个点的卷绕数,是一个整数,它表示了曲线绕过该点的总次数。卷绕数与曲线的定向有关,如果曲线依顺时针方向绕过某个点,则卷绕数是负数。这条曲线关于点 p的卷绕数是2 ( cr:wikipedia ) 经典算法 \| 计算几何 \| 判断点是否在多边形内部两个算法 u012737193的博客 01-08 4592 判断点 P是否在多边形中时计算几何中一个非常重要基本的算法。方法一是：用带符号的三角形面积之和与多边形面积进行比较，这种算法由于使用浮点运算所以会带来一定的误差，首先取目标点和多边形任意一条边构成三角形，三角形的符号这样确定，假设以多边形每个顶点逆时针顺序为正方向，按照这个方向每条边都是一个向量，当目标点在多边形某条边向量左边时，目标点和这条边构成的三角形的面积为正，当目标点在多边形某条 Winding Number 开源项目教程 gitblog_00221的博客 08-31 593 Winding Number 开源项目教程项目介绍 Winding Number 是一个在 GitHub 上托管的开源项目，地址为 Number。该项目主要实现并提供了计算二维平面上点相对于简单闭合曲线的绕数（Winding Number）的功能。绕数是图形学和计算机几何中一个重要的概念，常用于判断点是否位于曲线内或确... VB实现点与多边形位置关系的程序设计 9-8 本项目的目标是实现算法以确定点是否位于多边形内部,主要涉及射线法和 Winding Number 算法。提供的文件包括可执行文件、窗体文件、项目文件、工作区文件和版本控制文件。算法实现需要定义数据结构、处理向量运算和条件判断语句,并集成到用户界面中,以允许用户输入点坐标和多边形顶点,显示判断结果。点在多边形内算法——判断一个点是否在一个复杂多边形的内部热门推荐 hjh2005的专栏 07-04 11万+ 新页面（new page）介绍了将样条曲线添加到此技术的内容。也可以访问多边形内最短路径页（shortest-path-through-polygonpage）！图 1 图1显示了一个具有14条边的凹多边形。我们要判断红色点是否在多边形内。解决方案是将测试点的Y坐标与多边形的每一个点进行比较，我们会得到一个测试点所在的行与多边形边的交点的列表。在这个例子中有8条边与测试点所在的行点在多边形内算法的实现恒 09-24 8853 点在多边形内算法的实现cheungmine2007-9-22 本文是采用射线法判断点是否在多边形内的C语言程序。多年前，我自己实现了这样一个算法。但是随着时间的推移，我决定重写这个代码。参考周培德的《计算几何》一书，结合我的实践和经验，我相信，在这个算法的实现上，这是你迄今为止遇到的最优的代码。这是个C语言的小算法的实现程序，本来不想放到这里。可是，当我自己要实现这样点在多边形内的算法 QQ18959549054的博客 02-21 354 算法：过点向左作射线，射线与多边形相交的点的个数为奇数，则点在多边形内，否则点不在多边形内。（如下图）代码： public static boolean pointInPoly ( Point p,Point[] pointArr ){ //判断点是否在多边形内 double py=p.getY ( ); double px=p.getX ( ); ... 判断点是否在多边形内部 12-21 1910 判断点是否在多边形内部转载自：如何判断一个点是否在多边形内部？（1）面积和判别法：判断目标点与多边形的每条边组成的三角形面积和是否等于该多边形，相等则在多边形内部。（2）夹角和判别法：判断目标点与所有边的夹角和是否为360度，为360度则在多边形内部。（3）引射线法：从目标点出发引一条射... 易语言判断点在多边形内外 07-22 在计算机科学中，判断点在多边形内外通常有几种常见的方法，如射线法、 winding number（环绕数）法以及基于向量叉乘的算法。这里我们将重点讨论后两种方法，因为它们在易语言中比较常见且实用。 1. 环绕数法... 点在多边形内的判断算法 11-02 c++实现的。判断点在多边型内的判断算法验证点是否在多边形内的算法 12-22 一个验证点是否在多边形内的算法，浮点运算尽可能转换为整数运算。易语言源码易语言判断点在多边形内外源码.rar 03-30 压缩包内的“易语言判断点在多边形内外源码”应该包含了以上或类似算法的实现。易语言语法简洁，其程序结构清晰，适合初学者阅读和学习。通过分析这段源码，你可以了解到如何在易语言中处理几何计算，以及如何构建和... 【计算几何】判断一个点是否在多边形内部 zsjzliziyang的博客 09-26 1万+ 注：本文属转载原文链接判断一个点是否在多边形内部射线法思路、判断一个点是否在多边形内部射线法实现比如说，我就随便涂了一个多边形和一个点，现在我要给出一种通用的方法来判断这个点是不是在多边形内部（别告诉我用肉眼观察……）。首先想到的一个解法是从这个点做一条射线，计算它跟多边形边界的交点个数，如果交点个数为奇数，那么点在多边形内部，否则点在多边形外。这个结论很简单，那它是怎么来的？下面就简单讲解一下。首先，对于平面内任意闭合曲线，我们都可以直观地认为，曲线把平点在多边形内经典算法 ( 转 ) y391770118的专栏 10-15 4393 再经典不过的算法了： // 功能：判断点是否在多边形内 // 方法：求解通过该点的水平线与多边形各边的交点 // 结论：单边交点为奇数，成立! //参数： // POINT p 指定的某个点 // LPPOINT ptPolygon 多边形的各个顶点坐标（首末点可以不一致） // int nCount 多边形定点的个数判断点在多边形内部加油 (•̀ᴗ•́)و ̑̑ 03-24 255 int pnpoly ( int nvert, float vertx, float verty, float testx, float testy ) //testx,testy 是要判断的点 { int i, j, c = 0; for ( i = 0, j = nvert-1; i < nvert; j = i++) //对于多边形相邻的两点 { if ( ( ( vert... 点在多边形内算法夜空中明亮的星的专栏 08-09 1554 /// /// 射线相交算法 1 /// /// /// /// public static bool IsInPolygon ( List poly, RgPoint aPoint ) { bool flag = false; int 点在多边形内的判断算法及实现这是编译后生成的可执行文件，用户可以通过这个程序来运行点在多边形内判断的算法。 5. Resource.h, PointInPolygon.rc, PointInPolygon.sln：这些文件涉及到资源文件的管理，以及解决方案文件，它们是项目... 关于我们招贤纳士商务合作寻求报道 400-660-0108 kefu@csdn.net 在线客服工作时间 8:30-22:00 公安备案号11010502030143 京ICP备19004658号京网文〔2020〕1039-165号经营性网站备案信息北京互联网违法和不良信息举报中心家长监护网络110报警服务中国互联网举报中心 Chrome商店下载账号管理规范版权与免责声明版权申诉出版物许可证营业执照 ©1999-2025北京创新乐知网络技术有限公司 SimpleFelix 博客等级码龄12年 54 原创108 点赞 570 收藏 63 粉丝关注私信热门文章判断点在多边形内的算法（Winding Number详解） 14040 凹包求解之滚球法 10205 平面点集的凸包算法 9770 多边形快速凸包算法(Melkman‘s Algorithm) 9316 CAD—dxf & dwg格式解析库 7372 分类专栏 Delphi学习5篇读书5篇其他1篇计算机图形学11篇数据结构1篇算法学习8篇 Poco库学习笔记 C++基础16篇计算几何算法与实现10篇展开全部收起上一篇： Delphi I/O Errors（Delphi读写错误）下一篇：寻找强连通分量的Tarjan算法最新评论判断点在多边形内的算法（Winding Number详解） qq_41923024:最后一点和第一点的边没有进行判断，部分情况会检测出错判断点在多边形内的算法（Winding Number详解） weixin_42564068:没搞懂，wn为0才是外面，但是wn会得出大于零的数，那么这个算法难道不是错误的吗？多边形快速凸包算法(Melkman‘s Algorithm) SimpleFelix:本算法适用于没有自相交的封闭的简单多边形多边形快速凸包算法(Melkman‘s Algorithm) SimpleFelix:前言的第一句话，已经说明了多边形快速凸包算法(Melkman‘s Algorithm) 王思笨:请问这个算法需要对点集进行排序吗？如果不排序会不会朝着错误的方向生长大家在看 Python智能电子相册系统 764 AcWing 1172：祖孙询问 ← 倍增法求LCA（BFS预处理） ssm基于Springboot+的球鞋销售商城网站 387 【计算机毕业设计】235ssm基于关联分析羽毛球比赛会员积分系统vue 464 最新文章《启示录》读书笔记（3）分治算法设计：切割篱笆问题《聪明的投资者》读书笔记 2021年 7篇 2020年 42篇 2017年 8篇上一篇： Delphi I/O Errors（Delphi读写错误）下一篇：寻找强连通分量的Tarjan算法分类专栏 Delphi学习5篇读书5篇其他1篇计算机图形学11篇数据结构1篇算法学习8篇 Poco库学习笔记 C++基础16篇计算几何算法与实现10篇展开全部收起登录后您可以享受以下权益：免费复制代码和博主大V互动下载海量资源发动态/写文章/加入社区 ×立即登录评论 2 被折叠的条评论为什么被折叠?到【灌水乐园】发言查看更多评论添加红包祝福语请填写红包祝福语或标题红包数量个红包个数最小为10个红包总金额元红包金额最低5元余额支付当前余额 3.43 元前往充值 > 需支付：10.00 元取消确定成就一亿技术人! 领取后你会自动成为博主和红包主的粉丝规则 hope_wisdom 发出的红包实付元使用余额支付点击重新获取扫码支付钱包余额 0 抵扣说明： 1.余额是钱包充值的虚拟货币，按照1:1的比例进行支付金额的抵扣。 2.余额无法直接购买下载，可以购买VIP、付费专栏及课程。余额充值确定取消举报选择你想要举报的内容（必选）内容涉黄政治相关内容抄袭涉嫌广告内容侵权侮辱谩骂样式问题其他原文链接（必填）请选择具体原因（必选）包含不实信息涉及个人隐私请选择具体原因（必选）侮辱谩骂诽谤请选择具体原因（必选）搬家样式博文样式补充说明（选填）取消确定点击体验 DeepSeekR1满血版下载APP 程序员都在用的中文IT技术交流社区公众号专业的中文 IT 技术社区，与千万技术人共成长视频号关注【CSDN】视频号，行业资讯、技术分享精彩不断，直播好礼送不停！客服返回顶部
2871	https://macaulay2.com/doc/Macaulay2/share/doc/Macaulay2/EdgeIdeals/html/_complete__Multi__Partite.html	completeMultiPartite -- returns a complete multipartite graph Macaulay2 » Documentation Packages » EdgeIdeals :: completeMultiPartite next \| previous \| forward \| backward \| up \| index \| toc completeMultiPartite -- returns a complete multipartite graph Usage: macaulay2 K = completeMultiPartite(R,n,m) macaulay2 K = completeMultiPartite(R,L) Inputs: R, a ring, n, an integer, number of partitions m, an integer, size of each partition L, a list, of integers giving the size of each partition, or a list of partitions that are lists of variables Outputs: K, a graph, the complete multipartite graph on the given partitions Description A complete multipartite graph is a graph with a partition of the vertices such that every pair of vertices, not both from the same partition, is an edge of the graph. The partitions can be specified by their number and size, by a list of sizes, or by an explicit partition of the variables. Not all variables of the ring need to be used. macaulay2 i1 : R = QQ[a,b,c,x,y,z]; ```macaulay2 i2 : completeMultiPartite(R,2,3) o2 = Graph{"edges" => {{a, x}, {a, y}, {a, z}, {b, x}, {b, y}, {b, z}, {c, x}, {c, y}, {c, z}}} "ring" => R "vertices" => {a, b, c, x, y, z} o2 : Graph macaulay2 i3 : completeMultiPartite(R,{2,4}) o3 = Graph{"edges" => {{a, c}, {a, x}, {a, y}, {a, z}, {b, c}, {b, x}, {b, y}, {b, z}}} "ring" => R "vertices" => {a, b, c, x, y, z} o3 : Graph macaulay2 i4 : completeMultiPartite(R,{{a,b,c,x},{y,z}}) o4 = Graph{"edges" => {{a, y}, {a, z}, {b, y}, {b, z}, {c, y}, {c, z}, {x, y}, {x, z}}} "ring" => R "vertices" => {a, b, c, x, y, z} o4 : Graph ``` When n is the number of variables and M = 1, we recover the complete graph. macaulay2 i5 : R = QQ[a,b,c,d,e]; ```macaulay2 i6 : t1 = completeMultiPartite(R,5,1) o6 = Graph{"edges" => {{a, b}, {a, c}, {a, d}, {a, e}, {b, c}, {b, d}, {b, e}, {c, d}, {c, e}, {d, e}}} "ring" => R "vertices" => {a, b, c, d, e} o6 : Graph macaulay2 i7 : t2 = completeGraph R o7 = Graph{"edges" => {{a, b}, {a, c}, {a, d}, {a, e}, {b, c}, {b, d}, {b, e}, {c, d}, {c, e}, {d, e}}} "ring" => R "vertices" => {a, b, c, d, e} o7 : Graph macaulay2 i8 : t1 == t2 o8 = true ``` See also completeGraph -- returns a complete graph Constructor Overview -- a summary of the many ways of making graphs and hypergraphs Ways to use completeMultiPartite: completeMultiPartite(Ring,List) completeMultiPartite(Ring,ZZ,ZZ) For the programmer The object completeMultiPartite is a method function. The source of this document is in EdgeIdeals.m2:2168:0.
2872	https://stackoverflow.com/questions/57410117/how-to-find-the-foot-of-a-perpendicular-given-one-point-and-a-line-with-two-poin	Skip to main content Stack Overflow About For Teams Stack Overflow for Teams Where developers & technologists share private knowledge with coworkers Advertising Reach devs & technologists worldwide about your product, service or employer brand Knowledge Solutions Data licensing offering for businesses to build and improve AI tools and models Labs The future of collective knowledge sharing About the company Visit the blog How to find the foot of a perpendicular given one point and a line with two points Ask Question Asked Modified 6 years ago Viewed 862 times This question shows research effort; it is useful and clear 0 Save this question. Show activity on this post. All points are 3-d vector, suppose one point is a, the line with two points (b and c), now draw a perpendicular line from a to the line bc, the foot is, e.g. d, how to find the coordinate of point d with numpy? Thanks supposing: ``` a {x1, y1, z1} b {x2, y2, z2} c {x3, y3, z3} ``` I guess cross product will help, however, still have no idea, coule someone shed a light on this, Thanks! ``` 3 points def getfoot(a, b, c): foot = np.cross(a-b, b-c).... # no idea return root ``` the d is either inside or outside of bc. python numpy Share CC BY-SA 4.0 Improve this question Follow this question to receive notifications edited Aug 8, 2019 at 10:47 James Hao asked Aug 8, 2019 at 10:07 James HaoJames Hao 89322 gold badges1414 silver badges4545 bronze badges 4 2 Can't you just use the formula to calculate the line equation from two points and then use the formula to calculate the "foot"? I think this is a maths question. – Ardweaden Commented Aug 8, 2019 at 10:36 1 That looks a lot like homework. – Nikolas Rieble Commented Aug 8, 2019 at 11:37 3 I'm voting to close this question as off-topic because it isn't a programming question, it is a mathematics question – talonmies Commented Aug 8, 2019 at 11:37 The math has been done, all you need is here: en.wikipedia.org/wiki/Vector_projection .Other than that you should only need to know numpy.vdot(a, b) , which is vector dot product. – Peter Commented Aug 8, 2019 at 11:44 Add a comment \| 1 Answer 1 Reset to default This answer is useful 0 Save this answer. Show activity on this post. get the answer from here just project a to the line ab, not sure this is the simplest one. Post my code here. ``` def get_foot(p, a, b): ap = p - a ab = b - a result = a + np.dot(ap, ab)/np.dot(ab, ab) ab return result ``` Share CC BY-SA 4.0 Improve this answer Follow this answer to receive notifications edited Aug 8, 2019 at 13:16 answered Aug 8, 2019 at 11:34 James HaoJames Hao 89322 gold badges1414 silver badges4545 bronze badges 1 1 Answers will be downvoted if they do not add value beyond a link. – Nikolas Rieble Commented Aug 8, 2019 at 11:37 Add a comment \| Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions python numpy See similar questions with these tags. The Overflow Blog Research roadmap update, August 2025 Robots in the skies (and they use Transformer models) Featured on Meta Upcoming initiatives on Stack Overflow and across the Stack Exchange network... 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2873	https://www.scaler.com/topics/sum-of-first-n-even-natural-numbers/	Find the Sum of First n Even Natural Numbers Data Structures Cheatsheet Find the Sum of First n Even Natural Numbers By Abhinav Kaushik 3 mins read Last updated: 26 Mar 2024 253 views Learn via video course FREE View all courses DSA Problem Solving for Interviews using Java by Jitender Punia 1000 4.9 Start Learning Start Learning View all courses DSA Problem Solving for Interviews using Java by Jitender Punia 1000 4.9 Start Learning Topics Covered Problem Statement Given a number n. Find the sum of the first n even natural numbers. Example Input : 4 Output : 20 Explaination The Sum of the first 4 natural numbers is 20 because 2+4+6+8 = 20 Constraints 1 <= N <= 10000 Approach 1: Naive Approach (Using a for Loop) This is the Brute-Force method to find the sum of the first n even numbers. In this approach, we will be using a loop. Algorithm Create a variable named sum to keep the track of the sum of the numbers till now in the loop Run a loop from i=1 to i=n do sum = sum+i2 Finally return the sum variable C++ Implementation Java Implementation Python Implementation Complexity Analysis Time Complexity Analysis In this approach, we are running a loop from i=1 to i=n, which will take O(N) time. Inside the loop, we are doing sum = sum+i2, which will take O(1) time. So the overall time complexity of this solution will be O(N)+O(1) = O(N) Space Complexity Analysis In this approach, we are using only 1 variable i.e. sum, which will take O(1) space. So the overall space complexity of this solution will be O(1) Approach 2: Calculating the Nth Term This is the best approach for calculating the sum of first n even numbers without using a loop and without using extra space. In this approach, we will be using a direct formula to calculate the sum of n first even numbers. Algorithm This algorithm has a simple mathematical formula, we can calculate the sum of the first n numbers by using this formula sum = (n (n+1))/2 We also know that sum of the first n even numbers is two times the sum of the first n numbers, so the formula for sum of first n even numbers become sum = n (n+1) So we have to just assign sum = n(n+1), and return this variable C++ Implementation Java Implementation Python Implementation Complexity Analysis Time Complexity Analysis In this approach, we are doing sum = n(n+1), which will take O(1) time. So the overall time complexity of this solution will be O(1) Space Complexity Analysis In this approach, we are using only 1 variable i.e. sum, which will take O(1) space. So the overall space complexity of this solution will be O(1) Explore Scaler Topics Data Structures Tutorial and enhance your DSA skills with Reading Tracks and Challenges. Conclusion In this quick tutorial, we have discussed 2 different approaches for calculating the sum of first n-even numbers in C++, Java, and Python. In Approach 1, we used a for loop in which the time complexity is O(N) and space complexity is O(1). In Approach 2, we used the formula for calculating the sum of first n even numbers, and both time and space complexity for this approach became O(1). Related Blogs Sum of first N natural numbers Sum of n natural numbers in C++ Sum of n natural numbers in Python Frequently Asked Questions (FAQs) What Is the Formula for Calculating the Sum of the First n Even Numbers? The formula for the sum of the first n even numbers is sum = n (n + 1). This formula is derived as follows: The sum of an arithmetic progression is given by sum = (n/2) (2a + (n-1) d), where a is the first term and d is the common difference. For the first n even numbers, the first term a=2 (since the first even number is 2) and the common difference d=2 (as even numbers are 2 units apart). Substituting these values into the formula, we get: sum = (n/2) (2 2 + (n-1) 2) sum = (n/2) (4 + 2n - 2) sum = (n/2) (2 + 2n) sum = n (n + 1) Example: For n = 4 (the first 4 even numbers are 2, 4, 6, 8): Using the Formula: n (n + 1) = 4 (4 + 1) = 20 Manually: 2 + 4 + 6 + 8 = 20, thus verifying our formula is correct. What Will be the Formula for Calculating the Sum of First n Odd Numbers? A: The formula for the sum of the first n odd numbers will be sum = nn This formula can be derived as follows: As we know the sum of n numbers which are in the Arithmetic progression is given by sum = (n/2) (2a+(n-1) d) So in the sum of the first n odd numbers, the first term is a=1 and the common difference is d=2. So by putting these values in the original formula, we get sum = (n/2) (2 1 + 2 n - 2) sum = (n/2) (2 n) sum = n n Example: Let n = 4, then sum = (4/2) (2 + 2 4 -2) sum = (4/2) (2 4) sum = 4 4 sum = 16 ( which is equal to nn i.e. 4 4 ) Good job on finishing the article! Its time to level up with a Challenge CHALLENGE Master the concepts you discovered from this article in 3 Questions! Boost your memory and retention Take on challenges and level up your skills Showcase your achievements and earn exciting rewards Start Challenge! Over 10k+ learners have completed this challenge and improved their skills
2874	https://link.springer.com/article/10.1007/s00454-011-9353-9	Optimally Decomposing Coverings with Translates of a Convex Polygon 382 Accesses 14 Citations Explore all metrics Abstract We show that any k-fold covering using translates of an arbitrary convex polygon can be decomposed into Ω(k) covers. Such a decomposition can be computed using an efficient (polynomial-time) algorithm. Article PDF Download to read the full article text Similar content being viewed by others Covering Polygons with Rectangles Covering Convex Polygons by Two Congruent Disks Curiosities and counterexamples in smooth convex optimization Explore related subjects Avoid common mistakes on your manuscript. References Aloupis, G., Cardinal, J., Collette, S., Langerman, S., Orden, D., Ramos, P.: Decomposition of multiple coverings into more parts. In: SODA’09: Proceedings of the Nineteenth Annual ACM–SIAM Symposium on Discrete Algorithms, pp. 302–310. Society for Industrial and Applied Mathematics, Philadelphia (2009) Google Scholar Buchsbaum, A.L., Efrat, A., Jain, S., Venkatasubramanian, S., Yi, K.: Restricted strip covering and the sensor cover problem. In: SODA’07: Proceedings of the Eighteenth Annual ACM–SIAM Symposium on Discrete Algorithms, pp. 1056–1063. Society for Industrial and Applied Mathematics, Philadelphia (2007) Google Scholar Mani, P., Pach, J.: Decomposition problems for multiple coverings with unit balls. Manuscript (1986) Pach, J.: Covering the plane with convex polygons. Discrete Comput. Geom. 1, 73–81 (1986) Article MATH MathSciNet Google Scholar Pach, J., Tóth, G.: Decomposition of multiple coverings into many parts. Comput. Geom. 42(2), 127–133 (2009) Article MATH MathSciNet Google Scholar Pach, J., Tardos, G., Tóth, G.: Indecomposable coverings. In: Akiyama, J., Chen, W.Y.C., Kano, M., Li, X., Yu, Q. (eds.) CJCDGCGT. Lecture Notes in Computer Science, vol. 4381, pp. 135–148. Springer, Berlin (2005) Google Scholar Pálvölgyi, D.: Indecomposable coverings with concave polygons. Discrete Comput. Geom. 44(3), 577–588 (2010). doi:10.1007/s00454-009-9194-y Article MATH MathSciNet Google Scholar Pálvölgyi, D., Tóth, G.: Convex polygons are cover-decomposable. Discrete Comput. Geom. 43(3), 483–496 (2010) Article MATH MathSciNet Google Scholar Pandit, S., Pemmaraju, S.V., Varadarajan, K.R.: Approximation algorithms for domatic partitions of unit disk graphs. In: Dinur, I., Jansen, K., Naor, J., Rolim, J.D.P. (eds.) APPROX-RANDOM. Lecture Notes in Computer Science, vol. 5687, pp. 312–325. Springer, Berlin (2009) Google Scholar Pemmaraju, S.V., Pirwani, I.A.: Energy conservation via domatic partitions. In: MobiHoc’06: Proceedings of the 7th ACM International Symposium on Mobile Ad Hoc Networking and Computing, pp. 143–154. ACM, New York (2006) Chapter Google Scholar Tardos, G., Tóth, G.: Multiple coverings of the plane with triangles. Discrete Comput. Geom. 38(2), 443–450 (2007) Article MATH MathSciNet Google Scholar Download references Author information Authors and Affiliations Department of Computer Science, The University of Iowa, Iowa City, IA, 52242-1419, USA Matt Gibson & Kasturi Varadarajan Search author on:PubMed Google Scholar Search author on:PubMed Google Scholar Corresponding author Correspondence to Matt Gibson. Additional information A preliminary version of the results in this article appears in Gibson and Varadarajan, Decomposing Coverings and the Planar Sensor Cover Problem, Proceedings of the 50th IEEE Symposium on Foundations of Computer Science (FOCS), 2009. Rights and permissions Reprints and permissions About this article Cite this article Gibson, M., Varadarajan, K. Optimally Decomposing Coverings with Translates of a Convex Polygon. Discrete Comput Geom 46, 313–333 (2011). Download citation Received: 23 March 2010 Revised: 09 March 2011 Accepted: 06 April 2011 Published: 29 June 2011 Issue Date: September 2011 DOI: Share this article Anyone you share the following link with will be able to read this content: Sorry, a shareable link is not currently available for this article. Provided by the Springer Nature SharedIt content-sharing initiative Keywords Avoid common mistakes on your manuscript. Advertisement Search Navigation Discover content Publish with us Products and services Our brands 54.198.238.231 Not affiliated © 2025 Springer Nature
2875	https://artofproblemsolving.com/wiki/index.php/Diophantine_equation?srsltid=AfmBOopVE5b8kDm4z3wgrKsk0z8DkrGJ9C9k3cIrxHmR5V5PlpDvAXBU	Art of Problem Solving Diophantine equation - AoPS Wiki Art of Problem Solving AoPS Online Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online ‚ Books for Grades 5-12Online Courses Beast Academy Engaging math books and online learning for students ages 6-13. Visit Beast Academy ‚ Books for Ages 6-13Beast Academy Online AoPS Academy Small live classes for advanced math and language arts learners in grades 2-12. Visit AoPS Academy ‚ Find a Physical CampusVisit the Virtual Campus Sign In Register online school Class ScheduleRecommendationsOlympiad CoursesFree Sessions books tore AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates community ForumsContestsSearchHelp resources math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten contests on aopsPractice Math ContestsUSABO newsAoPS BlogWebinars view all 0 Sign In Register AoPS Wiki ResourcesAops Wiki Diophantine equation Page ArticleDiscussionView sourceHistory Toolbox Recent changesRandom pageHelpWhat links hereSpecial pages Search Diophantine equation A Diophantine equation is an equation relating integer (or sometimes natural number or whole number) quanitites. Finding the solution or solutions to a Diophantine equation is closely tied to modular arithmetic and number theory. Often, when a Diophantine equation has infinitely many solutions, parametric form is used to express the relation between the variables of the equation. Diophantine equations are named for the ancient Greek/Alexandrian mathematician Diophantus. Contents [hide] 1 Linear Combination 2 Pythagorean Triples 2.1 Method of Pythagoras 2.2 Method of Plato 2.3 Babylonian Method 3 Sum of Fourth Powers 4 Pell Equations 5 Methods of Solving 5.1 Coordinate Plane 5.2 Modular Arithmetic 5.3 Induction 5.4 General Solutions 6 Fermat's Last Theorem 7 Problems 7.1 Introductory 7.2 Intermediate 7.3 Olympiad 8 References 9 See also Linear Combination A Diophantine equation in the form is known as a linear combination. If two relatively prime integers and are written in this form with , the equation will have an infinite number of solutions. More generally, there will always be an infinite number of solutions when . If , then there are no solutions to the equation. To see why, consider the equation . is a divisor of the LHS (also notice that must always be an integer). However, will never be a multiple of , hence, no solutions exist. Now consider the case where . Thus, . If and are relatively prime, then all solutions are obviously in the form for all integers . If they are not, we simply divide them by their greatest common divisor. See also: Bézout's identity. Pythagorean Triples Main article: Pythagorean triple A Pythagorean triple is a set of three integers that satisfy the Pythagorean Theorem, . There are three main methods of finding Pythagorean triples: Method of Pythagoras If is an odd number, then is a Pythagorean triple. Method of Plato If , is a Pythagorean triple. Babylonian Method For any (), we have is a Pythagorean triple. Sum of Fourth Powers An equation of form has no integer solutions, as follows: We assume that the equation does have integer solutions, and consider the solution which minimizes . Let this solution be . If then their GCD must satsify . The solution would then be a solution less than , which contradicts our assumption. Thus, this equation has no integer solutions. If , we then proceed with casework, in . Note that every square, and therefore every fourth power, is either or . The proof of this is fairly simple, and you can show it yourself. Case 1: This would imply , a contradiction. Case 2: This would imply , a contradiction since we assumed . Case 3: , and We also know that squares are either or . Thus, all fourth powers are either or . By similar approach, we show that: , so . This is a contradiction, as implies is odd, and implies is even. QED [Oops, this doesn't work. 21 (or ) are equal to and not even...] Pell Equations Main article: Pell equation A Pell equation is a type of Diophantine equation in the form for natural number. The solutions to the Pell equation when is not a perfect square are connected to the continued fraction expansion of . If is the period of the continued fraction and is the th convergent, all solutions to the Pell equation are in the form for positive integer . Methods of Solving Coordinate Plane Note that any linear combination can be transformed into the linear equation , which is just the slope-intercept equation for a line. The solutions to the diophantine equation correspond to lattice points that lie on the line. For example, consider the equation or . One solution is (0,1). If you graph the line, it's easy to see that the line intersects a lattice point as x and y increase or decrease by the same multiple of and , respectively (wording?). Hence, the solutions to the equation may be written parametrically (if we think of as a "starting point"). Modular Arithmetic Sometimes, modular arithmetic can be used to prove that no solutions to a given Diophantine equation exist. Specifically, if we show that the equation in question is never true mod , for some integer , then we have shown that the equation is false. However, this technique cannot be used to show that solutions to a Diophantine equation do exist. Induction Sometimes, when a few solutions have been found, induction can be used to find a family of solutions. Techniques such as infinite Descent can also show that no solutions to a particular equation exist, or that no solutions outside of a particular family exist. General Solutions It is natural to ask whether there is a general solution for Diophantine equations, i.e., an algorithm that will find the solutions for any given Diophantine equations. This is known as Hilbert's tenth problem. The answer, however, is no. Fermat's Last Theorem Main article: Fermat's Last Theorem is known as Fermat's Last Theorem for the condition . In the 1600s, Fermat, as he was working through a book on Diophantine Equations, wrote a comment in the margins to the effect of "I have a truly marvelous proof of this proposition which this margin is too narrow to contain." Fermat actually made many conjectures and proposed plenty of "theorems," but wasn't one to write down the proofs or much other than scribbled comments. After he died, all his conjectures were re-proven (either false or true) except for Fermat's Last Theorem. After over 350 years of failing to be proven, the theorem was finally proven by Andrew Wiles after he spent over 7 years working on the 200-page proof, and another year fixing an error in the original proof. Problems Introductory Two farmers agree that pigs are worth dollars and that goats are worth dollars. When one farmer owes the other money, he pays the debt in pigs or goats, with "change" received in the form of goats or pigs as necessary. (For example, a dollar debt could be paid with two pigs, with one goat received in change.) What is the amount of the smallest positive debt that can be resolved in this way? (Source) Intermediate Let be a polynomial with integer coefficients that satisfies and Given that has two distinct integer solutions and find the product . (Source) Olympiad Determine the maximum value of , where and are integers satisfying and . (Source) Solve in integers the equation . References Proof of Fermat's Last Theorem See also Number Theory Pell equation Retrieved from " Category: Number theory Art of Problem Solving is an ACS WASC Accredited School aops programs AoPS Online Beast Academy AoPS Academy About About AoPS Our Team Our History Jobs AoPS Blog Site Info Terms Privacy Contact Us follow us Subscribe for news and updates © 2025 AoPS Incorporated © 2025 Art of Problem Solving About Us•Contact Us•Terms•Privacy Copyright © 2025 Art of Problem Solving Something appears to not have loaded correctly. 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2876	https://www.cdc.gov/sti/about/about-genital-hpv-infection.html	A .gov website belongs to an official government organization in the United States. A lock ( ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites. Related Topics: About Genital HPV Infection Key points Overview What is human papillomavirus (HPV)? HPV is the most common STI. There are many different types of HPV. While most do not cause any health problems, some types can cause genital warts and cancers. Vaccines can stop these health problems from happening. HPV is a different virus than HIV and HSV (herpes). Signs and symptoms How do I know if I have HPV? Genital HPV often has no symptoms, but it can cause serious health problems, even without symptoms. Most people with HPV never develop symptoms or health problems from it. In most cases (9 out of 10), HPV goes away on its own within two years without health problems. When HPV does not go away, it can cause health problems like genital warts and cancer. The types of HPV that cause warts do not cause cancer. Talk to your healthcare provider if you notice any of the following in your genital area, mouth, or throat: What are the symptoms of genital warts? Genital warts usually appear as a small bump or group of bumps in the genital area. They can be small or large, raised or flat, or shaped like a cauliflower. The warts may go away, stay the same, or grow in size or number. A healthcare provider can usually diagnose genital warts by looking at them. Genital warts can come back, even after treatment. Risk factors Am I at risk for HPV? You can get HPV by having vaginal, anal, or oral sex with someone who has the virus. A person with HPV can pass the infection to someone even when they have no signs or symptoms. If you are sexually active, you can get HPV, even if you have had sex with only one person. You also can develop symptoms years after having sex with someone who has the infection. This makes it hard to know when you first got it. How it spreads How is HPV spread? HPV is most commonly spread during vaginal or anal sex. It also spreads through close skin-to-skin touching during sex. A person with HPV can pass the infection to someone even when they have no signs or symptoms. Prevention How can I avoid HPV and the health problems it can cause? You can do several things to lower your chances of getting HPV and avoiding the health problems it can cause. Get vaccinated. The HPV vaccine is safe and effective. It can protect against diseases (including cancers) caused by HPV when given in the recommended age groups. Get screened for cervical cancer. Routine screening for women aged 21 to 65 years old can prevent cervical cancer. If you are sexually active: If you or your partner have genital warts, stop having sex until you no longer have warts. We do not know how long a person is able to spread HPV after warts go away. I'm pregnant. Will having HPV affect my pregnancy? Pregnant women with HPV can get genital warts or develop abnormal cell changes on the cervix. Routine cervical cancer screening can help find abnormal cell changes. You should get routine cervical cancer screening even when you are pregnant. Testing and diagnosis How will my healthcare provider know if I have HPV? There is no test to find out a person's "HPV status." Also, there is no approved HPV test to find HPV in the mouth or throat. There are HPV tests that can screen for cervical cancer. Healthcare providers only use these tests for screening women aged 30 years and older. HPV tests are not recommended to screen men, adolescents, or women under the age of 30 years. Most people with HPV do not know they have the infection. They never develop symptoms or health problems from it. Some people find out they have HPV when they get genital warts. Women may find out they have HPV when they get an abnormal Pap test result (during cervical cancer screening). Others may only find out once they've developed more serious problems from HPV, such as cancers. Treatment and recovery Is there treatment for HPV or health problems that develop from HPV? There is no treatment for the virus itself. However, there are treatments for the health problems that HPV can cause: Related conditions Does HPV cause cancer? HPV can cause cervical and other cancers, including cancer of the vulva, vagina, penis, or anus. It can also cause cancer in the back of the throat (called oropharyngeal cancer). This can include the base of the tongue and tonsils. Cancer often takes years, even decades, to develop after a person gets HPV. Genital warts and cancers result from different types of HPV. There is no way to know who will develop cancer or other health problems from HPV. People with weak immune systems (including those with HIV) may be less able to fight off HPV. They may also be more likely to develop health problems from HPV. Resources STI Fact Sheets STIs during Pregnancy Fact Sheet Primary Prevention Methods (Condoms) CDC-INFO CDC National Prevention Information Network (NPIN) American Sexual Health Association (ASHA) On This Page STI STIs are common. Many are curable - all are preventable. For Everyone Health Care Providers Public Health Languages Language Assistance
2877	https://www.youtube.com/watch?v=ScdzmbHtxcI	Art of Problem Solving: Counting Objects in a Circle Art of Problem Solving 103000 subscribers 156 likes Description 17433 views Posted: 28 Dec 2011 Art of Problem Solving's Richard Rusczyk explains how to count the number of ways to arrange an objects in a circle. This video is part of our AoPS Counting & Probability curriculum. Take your math skills to the next level with our Counting & Probability materials: 📚 AoPS Introduction to Counting & Probability Textbook: 🖥️ AoPS Introduction to Counting & Probability Course: 🔔 Subscribe to our channel for more engaging math videos and updates Transcript:
2878	https://math.stackexchange.com/questions/1509809/counting-permutations-with-conditions	combinatorics - counting permutations with conditions - Mathematics Stack Exchange Join Mathematics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 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Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more counting permutations with conditions Ask Question Asked 9 years, 11 months ago Modified9 years, 11 months ago Viewed 431 times This question shows research effort; it is useful and clear 2 Save this question. Show activity on this post. Given the set S=1,2,3,4 S=1,2,3,4, how many ways can I make w=(x i,x 2,...,x N)w=(x i,x 2,...,x N) such that x i∈S x i∈S with a rule that m 1's appear before k 2's in a word of length N? For example if m=2, k=3 then I want to know how many words of length N have the form (..,1,..1,..,2,..,2,2,..)(..,1,..1,..,2,..,2,2,..). I have already tried brute forcing the solution using a computer, but with 4 N 4 N possibilities to check this quickly becomes untenable. Apologies if this has been asked before, but I couldn't find anything helpful searching. Clarification: The full problem I am considering is looking for sequences which at some point m−k>r m−k>r and in the full final sequence m+k>d m+k>d, but I think and answer to the initial question would give me some more insight to figure this out. combinatorics permutations combinations Share Share a link to this question Copy linkCC BY-SA 3.0 Cite Follow Follow this question to receive notifications edited Nov 2, 2015 at 18:33 user1984528user1984528 asked Nov 2, 2015 at 18:14 user1984528user1984528 143 4 4 bronze badges 5 1 From the example you give it appears that you want all the 1′s 1′s to appear before the 2′s 2′s start. So (1,2,2,1,2)(1,2,2,1,2) would not be a good example. Is this correct? Also: do you want exactly m m 1′s 1′s or would it be ok if there are more than m m?lulu –lulu 2015-11-02 18:19:00 +00:00 Commented Nov 2, 2015 at 18:19 @lulu yes, I want the 1's to appear before the 2's start so (1,2,2,1,2) is not a good example. Hmm, I'm not 100% sure if more than m m would work with what I am trying to do. I am trying to count the number of combinations such that at some point in the sequence m−k>r 1 m−k>r 1 and that in the full sequence m+k>d m+k>d.user1984528 –user1984528 2015-11-02 18:31:20 +00:00 Commented Nov 2, 2015 at 18:31 Also, after the k t h k t h 2 2 appears, can more 1′s 1′s and/or 2′s 2′s follow? That is, is (1,1,2,2,2,1,2)(1,1,2,2,2,1,2) a good example?lulu –lulu 2015-11-02 18:35:27 +00:00 Commented Nov 2, 2015 at 18:35 more 1's and 2's can follow as long as there are equal numbers of them or more 2's than 1's. but if you can solve the case for exactly m 1's and k 2's I can code a loop over the possible combinations of m and k that meet my condition user1984528 –user1984528 2015-11-02 18:43:07 +00:00 Commented Nov 2, 2015 at 18:43 Ok, let's say exactly m m and k k. Makes life a lot simpler. I'll post something below.lulu –lulu 2015-11-02 18:44:30 +00:00 Commented Nov 2, 2015 at 18:44 Add a comment\| 1 Answer 1 Sorted by: Reset to default This answer is useful 1 Save this answer. Show activity on this post. As per the discussion in the comments: we are assuming that, amongst the n n symbols in our list, we get exactly m m 1′1′ and exactly k k 2′s 2′s and that all the 1 1's precede the first 2 2. In that case, all we need to do is to choose m+k m+k slots ((n m+k)(n m+k) ways to do that). Then there is a unique way to populate those slots (the 1′s 1′s come first, then the 2′s 2′s). then we need to populate the remaining n−m−k n−m−k slots each with 3 3 or 4 4. (there are 2 n−m−k 2 n−m−k ways to do that). putting it all together we see that there are (n m+k)2 n−m−k(n m+k)2 n−m−k ways to make lists of your type. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications edited Nov 2, 2015 at 18:55 answered Nov 2, 2015 at 18:48 lulululu 77.4k 6 6 gold badges 89 89 silver badges 140 140 bronze badges Add a comment\| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions combinatorics permutations combinations See similar questions with these tags. Featured on Meta Introducing a new proactive anti-spam measure Spevacus has joined us as a Community Manager stackoverflow.ai - rebuilt for attribution Community Asks Sprint Announcement - September 2025 Report this ad Related 5Counting the number of sequences with these properties. 0In how many ways the letters of the word RAINBOW be arranged such that A is always before I and I is always before O. 0Number of distinguishable permutations with repetition and restrictions 1Combinatorial Analysis question 2From sets A and B, find enough distinct pairs with unique appearances of the elements 8Re-arrangements of (1,1,2,2,3,3,4,4)(1,1,2,2,3,3,4,4) with (1,2,3,4)(1,2,3,4) as a subsequence 3number of "single-peak" permutations Hot Network Questions Alternatives to Test-Driven Grading in an LLM world Clinical-tone story about Earth making people violent Can I go in the edit mode and by pressing A select all, then press U for Smart UV Project for that table, After PBR texturing is done? 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2879	https://www.astro.org/ASTRO/media/ASTRO/AffiliatePages/arro/PDFs/ARROCase_GlioMulti.pdf	Vestibular Schwannoma Kelli B. Pointer MD, PhD University of Chicago Chicago, IL Faculty: Steven J. Chmura MD, PhD March 24, 2020 Case • 67 year old F presented with a ringing sensation in her ears for the past few years as well as progressive left sided hearing loss. • For the past month she noted progressive vertigo causing her to be unable to drive. • She also endorsed a posterior headache that felt like an earache. March 24, 2020 Common presentations • Symptoms are due to cranial nerve involvement and tumor progression: • Acoustic Nerve (VIII): 95% hearing loss (only 2/3 realize it) • Vestibular nerve (VIII): 61% unsteadiness • Facial nerve (VII): 6% facial paresis and taste disturbances • Trigeminal nerve (V): 17% facial numbness / pain. • Posterior fossa: rare compression on cerebellum or brainstem, results in ataxia / hydrocephalus March 24, 2020 Differential Diagnoses • Vestibular schwannoma • Meningioma (4-10%) • Facial nerve schwannoma • Glioma • Cholesteatoma • Epidermoid inclusion cyst • Glomus jugulare • Lymphoma • Hemangioblastoma (VHL) • Brain metastases • Ependymoma • Arachnoid cyst • Lipoma Non-oncologic: hemangioma, aneurysm, basilar artery ectasia March 24, 2020 Workup • Physical exam: – Rinne test (tuning fork on mastoid bone, air conduction > bone conduction is normal, with sensorineural both are depreciated) • In conductive hearing loss, bone conduction > air conduction – Weber test (assessed sensorineural hearing loss; vibratory sound louder on “good” side); – Cranial nerve test (facial weakness, facial numbness, corneal reflex) – Vestibular testing: May see decreased or absent caloric response on affected side. – Romberg, Dix Hall-Pike, and balance are typically normal • Audiometry: Best initial screening test, since only 5% will have normal initial test. – Look for asymmetrical high frequency hearing loss. – Speech discrimination loss is often out of proportion to measured hearing loss March 24, 2020 March 24, 2020 MRI demonstrated a 14 x 7 mm left cerebellopontine angle cistern mass most likely representing an acoustic neuroma Imaging Typical imaging findings • well circumscribed T1-gad enhancing lesions arising near porus acusticus. T2 isointense – “Ice cream on a cone” or “Dumbbell” in IAC • CPA angle tumors: 80% are vestibular schwannomas. Of remaining 20%, majority are meningiomas, cholesteatoma, etc. • MRI sensitivity: 98% (miss some due to small size) • MRI specificity – approaches 100% March 24, 2020 Vestibular Schwannoma Meningioma Cholesteatoma CT scan Usually iso intense and contrast enhancing Greater contrast than VS Hypodense with irregular, lobulated margins. No contrast enhancement MRI T1 Isointense (compared to pons) Iso- or minimally hyper-Hypo, ~CSF-like MRI T2 “filling defect” – heterogeneously hyperintense Usually hypointense hyperintense MRI post contrast Enhancing: may be homogeneous (50%), hetero (30%), or cystic (5-15%) Strongly enhancing Other Obtuse angle with petrous ridge vs acute for VS Keratinizing squamous epithelium, erosive (best seen on CT) Audiometry • Non-serviceable≥50dB audiogram and <50% speech discrimination • Gardner-Robertson scale used March 24, 2020 Gardner-Robertson scale Grade Description Pure-tone Average (decibels) Speech-discrimination Score (percent) 1 Good to excellent 0-30 70-100 2 Serviceable 31-50 50-69 3 Non-serviceable 51-90 5-49 4 Poor 91-100 (maximum) 1-4 5 None Not testable 0 Epidemiology • Overall incidence: 1-2/100,000 (increased over past the 30-40 years with improved non-invasive diagnostic studies) • 3000 cases in US per year • Size has decreased as incidence has increased • Autopsy results have shown that subclinical acoustic neuromas are present in up to 1% of people • Account for 8% of intracranial tumors and 80-90% of CPA tumors • Age at presentation: 30 to 50 • Almost always unilateral, with exception of NF-2 March 24, 2020 Risk Factors • Acoustic trauma - OR of 2.2 if 10 years exposure to extremely loud noise. OR 13.1 if 20 or more years of exposure but subject to RECALL BIAS (studies looking at occupational exposure negative) • Parathyroid adenoma- OR of 3.4 for acoustic neuroma (cause/association is unknown) • Childhood radiation exposure- 20x higher compared to normal population • NF-2 (bilateral)- Accounts for 10% of patients with acoustic neuroma • Cell phone use (controversial)- ipsilateral use >1640 hr – OR 2.55 (1.5-4.4) March 24, 2020 Neurofibromatosis (NF1 and NF2) NF1 NF2 Inheritance Autosomal dominant Autosomal dominant Incidence 1:3000 1:40,000 Chromosome 17q11.2 22q12.2 Gene product Neurofibromin Merin Presentation Café-au-lait macules, axillary/inguinal freckling, cutaneous neurofibromas, subcutaneous neurofibromas Hearing loss or vestibular dysfunction at young age, cataracts, juvenile posterior subcapsular lenticular opacity cutaneous schwannomas Intracranial tumors Optic path gliomas, other astrocytomas/gliomas Vestibular schwannomas, meningiomas Cognitive IQ mildly decreased Normal Other tumors CML, pheochromocytoma None March 24, 2020 Diagnosis of NF2: 1.Bilateral eighth nerve masses with imaging OR 2. A first degree relative with NF2 with either a unilateral eight nerve mass or 2 of the following: glioma, meningioma, schwannoma, neurofibroma, or juvenile posterior subcapsular lenticular opacity Pathology • Arise from the Schwann cell: perineural elements of the affected nerve • Occur with equal frequency on the superior and inferior branches of the vestibular nerve (rarely affect cochlear portion CN VIII) • Obersteiner-Redlich zone: Arise at junction of central myelin produced by glial cells and peripheral myelin from Schwann cells – This zone is where the CNS meets the PNS so change from oligodendrocyte myelin to Schwann cell myelin • Antoni A and B areas: Microscopically, zones of alternately dense (A) and sparse (B) cellularity are characteristic of AN’s. • Stain positive for S100 March 24, 2020 Koos grading system • Grade I: small intracanalicular tumor • Grade II: small tumor with protrusion into the cerebellopontine angle • Grade III: Tumor occupying the cerebellopontine cistern with no brainstem displacement • Grade IV: Large tumor with brainstem and nerve displacement March 24, 2020 Anatomy • Common to have CN deficits in CN V, VII and VIII March 24, 2020 House-Brackmann Score • Scores the degree of facial nerve palsy • Measurement determined by measuring the upwards (superior) movement of the mid-portion of the top of the eyebrow, and the outward (lateral) movement of the angle of the mount • 1 point per 0.25 cm movement, up to a max. of 1 cm. Scores added together to give a number out of 8 March 24, 2020 Grade Description Measurement Function % Est. Function % I Normal 8/8 100 100 II Slight 7/8 76-99 80 III Moderate 5/8-6/8 51-75 60 IV Moderately Severe 3/8-4/8 26-50 40 V Severe 1/8-2/8 1-25 20 VI Total 0/8 0 0 Treatment Options • Observation • Surgical resection • Radiotherapy (EBRT, FSRT, SRS) March 24, 2020 Observation • MRI recommended every 6 to 12 months in patients without baseline hearing loss and stable or slow growth rates, especially elderly patients – Beware 15-30% are lost to follow-up. • Growth rate of 2.9 mm/year per UCSF lit review (others say 1.2-1.9mm/yr) • 30%-50% of tumors show no growth or regression on serial imaging studies • Growth rate is highest for those that grow in first year. Progressive decrease in rate of growth if growth starts later. • No predictive relationship between growth rate and tumor size at presentation March 24, 2020 Surgical Resection • Typically performed by ENTs and neurosurgeons. Learning curve is steep 20 to 60 cases. • Often recommended for tumors > 3-4 cm or for salvage after RT. • CN VII, VIII damage is significant • French series also shows good CN VII function (preservation not correlated with surgical approach used), but 1/3 pts had disabling vestibular symptoms at 1 year. March 24, 2020 Surgical Resection Approaches (based on size, location, and consideration of hearing preservation): • Retromastoid suboccipital (retrosigmoid): An incision is made behind the ear and mastoid bone and some inner ear structures are removed. Often best hearing preservation – Advantages: decreased risk of facial nerve damage, ability to attempt hearing preservation, any size tumors, good visualization of CPA – Disadvantage: If tumor extends distally into IAC, complete resection may not be possible; long term postop headaches; cerebellar retraction may increase rates of ataxia • Middle cranial fossa (transtemporal): Incision anterior to the ear with removal of the underlying bone to expose the area of interest; used primarily for small tumors confined to IAC (allows for complete exposure of IAC) – Advantage: Hearing preservation attempted (preservation of inner ear structures), only approach for IAC fundus – Disadvantage: complete tumor removal may not be feasible due to poor visualization of CPA; risk of facial nerve palsies due to increased manipulation of nerve within the auditory canal (puts facial nerve between surgeon and tumor). Need to be less than 1.5 -2 cm. • Translabyrinthine: This approach goes directly through the inner ear and invariably sacrifices hearing, but preserves CN VII – Recommended for large vestibular schwannomas (> 3 cm) in young pts without serviceable hearing – Can be used for smaller tumors if hearing preservation is not important. – Associated with better post operate gait stability because there is minimal retraction of the cerebellum, lowest incidence of postoperative headaches • Retrolabyrinthine: – Allows excision from both the CPA and the IAM, regardless of the histological nature of the tumor and size March 24, 2020 SRS Data • First used by Leksell at Karolinska institute in Sweden to treat vestibular schwannoma in 1969. – Tumor control 81% at 3.7y med f/u. – Transient CN V and VII in 18% and 14% of pts, respectively. • Initially used for older patients, recurrence after surgery, bilateral tumors, and medically inoperable pts. • Early Gamma Knife data (Pitt, Mayo) had significant CN toxicity (33-80% preservation). Prescribed to higher dose than given now. • Initial Linac results (Florida, Cleveland Clinic) showed better toxicity profile. Likely because prescribing a lower dose (~12 Gy). Modern SRS series: • lower doses 12-13Gy, MRI-based planning, improved conformality in plans (multiple isocenters, improved planning systems) • PFS: 92-100% • CN V preservation: 92-100% • CN VII preservation: 94-100% • Hearing preservation: 60-68% March 24, 2020 SRS Data • Flickinger Pittsburgh (Gamma Knife)- 6 year follow-up after 12-13 Gy –Tumor control= 98.6% –Facial nerve function preservation=100% –Normal trigeminal function=95.6% –Unchanged hearing level- 70.3% –Useful hearing preservation=78.6% March 24, 2020 SRS vs. FSRT • Jefferson (Andrews) (Gamma Knife) = – 12 Gy SRS vs. 50 Gy in 2 Gy fractions – Tumor control: 98% SRS vs 97% SRT – Hearing preservation was significantly higher in FSRT=81 % vs SRS=33% – Criticized for short follow-up time and worse patients in SRS group • Heidelberg (Combs) (Linac) = both safe/effective – FSRT= 57.6 Gy/1.8 Gy fractions vs. median SRS of 13 Gy – SRS ≤ 13 Gy for smaller lesions (< 3cm), FSRT for larger lesions. – Local control: SRS=90% vs. FSRT=95% (NS) – Hearing at 5 years SRS=60% vs. FSRT 78% (p=0.02) • BC (Chung) (Linac)= FSRT gives comparable tumor control, good rates of hearing preservation (all SRS patients were already had non-serviceable hearing) – 12 Gy SRS vs. 45 Gy/1.8 Gy fractions – Local control 100% in both groups March 24, 2020 Meijer Netherlands IJROBP 2003 Tumor Control CN V Preservation CN VII Preservation Hearing Preservation SRS 100% 92% 93% 75% FSRT 94% 98% 97% 61% Surgery vs. SRS March 24, 2020 Regis JNS France 2002 Facial motor disturbance CN V disturbance Preserved Hearing Overall functional disturbance Hospital Stay (Days) Mean days missed from work Surgery 37% 29% 37.5% 39% 23 130 Gamma Knife 0% 4% 70% 9% 3 7 Observation vs. FSRS • Shirato Japan IJROBP– Observation vs. FSRT as Initial Management for Vestibular Schwannoma – No permanent facial or trigeminal neuropathy observed in the SRT group – SRT: Transient VII in 5%, Transient V in 12% – Obs: 4% permanent facial palsy (after salvage surgery) and 4% w/transient V palsy (after salvage surgery) – Hearing: No SS difference in G & R class preservation rates for patients with measurable hearing • Hearing preservation: 60.9% (3yr) and 31% (5yr) observation vs 53% SRT (at 3yr and 5yr) • Analysis of Hearing preservation excludes/censors patients in observation group at time of salvage. • In reality 4/6 sx salvaged pts and 1/4 RT salvaged pts became DEAF but were NOT included in actuarial curve! • In FSRT group, 1 pt (3%) became deaf March 24, 2020 Hearing Preservation after SRS March 24, 2020 Johnson et al. Pittsburgh. J Neurosurg. Predicting hearing outcomes before primary radiosurgery for vestibular schwannomas. -Retrospective study of 307 patients with serviceable hearing at time of SRS Dose and CN Toxicity March 24, 2020 • CN V and VII: tumor size and prescription dose correlate with toxicity. – Friedman UF JNS 2006 (p.24)CN VII tox: 1cc increase in tumor = 17% increase, 2.5Gy increase in dose = 8.1x increase – Boegle UF JNS 2007 Dosimetric variables: Conformity and dose gradient: no effect on outcomes – Generally no SRS if >3cm tumors • CN VIII: Fukuoka Japan Prog Neurol Surg 2009 17% transient dizziness/gait imbalance post SRS, 2% persistent dizziness post SRS • Hearing: Range of hearing preservation 32-71%. Hearing can decline long-term (>10yrs) after SRS. Dose matters. – Prasad UVa JNS 2000 – no decline in first 2 years, then progressive decline – Chopra Pitt IJROBP 2007 – 3y 75% G-R I/II, 10y 44% – Combs Heidelberg RadOnc 2013 –@10yr, 72% if ≤13Gy (if >13Gy, 36%) – Yang UCSF J NSGY 2010 – Review: N=4234, Preservation of hearing about 50% overall, but can be increased to 60% if <13 Gy Treatment • Sim: Scan vertex to C2, 1mm thickness • With IV contrast, upper alpha cradle, and stereotactic frame • Contour: Contour GTV=PTV (in this case due to frame) – PTV expansion based on immobilization • Framed cases may use 0 mm expansion, while frameless SRS cases may use 2-3 mm expansion March 24, 2020 Treatment • Linac based framed SRS to L-sided vestibular schwannoma 12.5 Gy to 80% isodose line using 6 MV photons, FFF March 24, 2020 Representative Isodose lines March 24, 2020 OAR Constraints March 24, 2020 • Brainstem: 12.5 Gy (<5% neuropathy or necrosis) • Chiasm: 8 Gy (<10% optic neuropathy) • Cochlea: 14 Gy (<25% sensory-neural hearing loss), Ideally keep cochlea/modiolus <5.3 Gy (possibly <4.2 Gy) • Spinal cord: 13 Gy (1% myelopathy) • Brain: V12 <5-10 cc (<20% symptomatic necrosis) • Conformality index: (Rx isodose volume)/(tumor volume) should be <2 • Homogeneity index: (maximum dose)/(peripheral dose) should be <2 • Gamma-knife: Rx to 50% IDL • Linac: Rx to 80% IDL Follow-up • ~ 30% vestibular schwannomas will show a transient increase in volume after SRS (mean time to max tumor is roughly 13 months) • Hearing may continue to decline long term after SRS (even >10 years) • 36% of tumors shrink, 58% remain unchanged after SRS • Imaging can be performed at 3-6 months followed by yearly or if changes in symptoms March 24, 2020 Summary • SRS is a good treatment option for vestibular schwannoma with excellent control rates • SRS has decreased rates of CN toxicities and increased rates of hearing preservation when compared to surgery • Surgery may be a better option if there is brain stem compression, due to minimal change in size of tumor after SRS or transient increase in tumor size • Observation with imaging every 6-12 months may be appropriate for some patients due to many vestibular schwannomas remaining stable or regressing without treatment March 24, 2020 References 1.Babu R, Sharma R, Bagley JH, Hatef J, Friedman AH, Adamson C. Vestibular schwannomas in the modern era: epidemiology, treatment trends, and disparities in management. J Neurosurg 2013;119: 121-30. 2.Backlund LM, Grander D, Brandt L, Hall P, Ekbom A. Parathyroid adenoma and primary CNS tumors. Int J Cancer 2005;113:866-9. 3.Baschnagel AM, Chen PY, Bojrab D, et al. Hearing preservation in patients with vestibular schwannoma treated with Gamma Knife surgery. J Neurosurg 2013;118:571-8. 4.Carlson ML, Link MJ, Wanna GB, Driscoll CL. Management of sporadic vestibular schwannoma. Otolaryngol Clin North Am 2015;48:407-22. 5.Chopra R, Kondziolka D, Niranjan A, Lunsford LD, Flickinger JC. Long-term follow-up of acoustic schwannoma radiosurgery with marginal tumor doses of 12 to 13 Gy. Int J Radiat Oncol Biol Phys 200 7;68:845-51. 6.Chopra R, Kondziolka D, Niranjan A, Lunsford LD, Flickinger JC. Long-term follow-up of acoustic schwannoma radiosurgery with marginal tumor doses of 12 to 13 Gy. Int J Radiat Oncol Biol Phys 200 7;68:845-51. 7.Chung HT, Ma R, Toyota B, Clark B, Robar J, McKenzie M. Audiologic and treatment outcomes after linear accelerator-based stereotactic irradiation for acoustic neuroma. Int J Radiat Oncol Biol Phy s 2004;59:1116-21. 8.Combs SE, Welzel T, Schulz-Ertner D, Huber PE, Debus J. Differences in clinical results after LINAC-based single-dose radiosurgery versus fractionated stereotactic radiotherapy for patients with vesti bular schwannomas. Int J Radiat Oncol Biol Phys 2010;76:193-200. 9.Edwards CG, Schwartzbaum JA, Nise G, et al. Occupational noise exposure and risk of acoustic neuroma. Am J Epidemiol 2007;166:1252-8. 10.Flickinger JC, Kondziolka D, Niranjan A, Maitz A, Voynov G, Lunsford LD. Acoustic neuroma radiosurgery with marginal tumor doses of 12 to 13 Gy. Int J Radiat Oncol Biol Phys 2004;60:225-30. 11.Gardner G, Robertson JH. Hearing preservation in unilateral acoustic neuroma surgery. Ann Otol Rhinol Laryngol 1988;97:55-66. 12.Group IS. Acoustic neuroma risk in relation to mobile telephone use: results of the INTERPHONE international case-control study. Cancer Epidemiol 2011;35:453-64. 13.House JW, Brackmann DE. Facial nerve grading system. Otolaryngol Head Neck Surg 1985;93:146-7. 14.Kondziolka D, Lunsford LD, McLaughlin MR, Flickinger JC. Long-term outcomes after radiosurgery for acoustic neuromas. N Engl J Med 1998;339:1426-33. 15.Koos WT, Day JD, Matula C, Levy DI. Neurotopographic considerations in the microsurgical treatment of small acoustic neurinomas. J Neurosurg 1998;88:506-12. 16.Lanman TH, Brackmann DE, Hitselberger WE, Subin B. Report of 190 consecutive cases of large acoustic tumors (vestibular schwannoma) removed via the translabyrinthine approach. J Neurosurg 999;90:617-23. 17.Lanman TH, Brackmann DE, Hitselberger WE, Subin B. Report of 190 consecutive cases of large acoustic tumors (vestibular schwannoma) removed via the translabyrinthine approach. J Neurosurg 1999;90:617-23. 18.Leksell L. A note on the treatment of acoustic tumours. Acta Chir Scand 1971;137:763-5. 19.Prasad D, Steiner M, Steiner L. Gamma surgery for vestibular schwannoma. J Neurosurg 2000;92:745-59. 20. Regis J, Pellet W, Delsanti C, et al. Functional outcome after gamma knife surgery or microsurgery for vestibular schwannomas. J Neurosurg 2002;97:1091-100. 21.Schneider AB, Ron E, Lubin J, et al. Acoustic neuromas following childhood radiation treatment for benign conditions of the head and neck. Neuro Oncol 2008;10:73-8. 22.Shirato H, Sakamoto T, Sawamura Y, et al. Comparison between observation policy and fractionated stereotactic radiotherapy (SRT) as an initial management for vestibular schwannoma. Int J Radi at Oncol Biol Phys 1999;44:545-50. 23.Shore-Freedman E, Abrahams C, Recant W, Schneider AB. Neurilemomas and salivary gland tumors of the head and neck following childhood irradiation. Cancer 1983;51:2159-63. 24.Sobel RA. Vestibular (acoustic) schwannomas: histologic features in neurofibromatosis 2 and in unilateral cases. J Neuropathol Exp Neurol 1993;52:106-13. 25.Sughrue ME, Yang I, Aranda D, et al. The natural history of untreated sporadic vestibular schwannomas: a comprehensive review of hearing outcomes. J Neurosurg 2010;112:163-7. 26.West N, Sass H, Caye-Thomasen P. Sporadic and NF2-associated vestibular schwannoma surgery and simultaneous cochlear implantation: a comparative systematic review. Eur Arch Otorhinolaryn gol 2020;277:333-42. 27.Yang I, Huh NG, Smith ZA, Han SJ, Parsa AT. Distinguishing glioma recurrence from treatment effect after radiochemotherapy and immunotherapy. Neurosurg Clin N Am 2010;21:181-6. Please provide feedback regarding this case or other ARROcases to arrocase@gmail.com March 24, 2020
2880	https://brainly.com/question/44989372?source=next+question	[FREE] What is the antonym of "haggard"? A) Exhausted B) Healthy C) Haggard D) Tired - brainly.com 3 Search Learning Mode Cancel Log in / Join for free Browser ExtensionTest PrepBrainly App Brainly TutorFor StudentsFor TeachersFor ParentsHonor CodeTextbook Solutions Log in Join for free Tutoring Session +56,8k Smart guidance, rooted in what you’re studying Get Guidance Test Prep +45,2k Ace exams faster, with practice that adapts to you Practice Worksheets +5,8k Guided help for every grade, topic or textbook Complete See more / English Textbook & Expert-Verified Textbook & Expert-Verified What is the antonym of "haggard"? A) Exhausted B) Healthy C) Haggard D) Tired 1 See answer Explain with Learning Companion NEW Asked by erikacastro3542 • 12/21/2023 0:00 / -- Read More Community by Students Brainly by Experts ChatGPT by OpenAI Gemini Google AI Community Answer This answer helped 37154570 people 37M 0.0 0 Upload your school material for a more relevant answer The antonym of "haggard" is Healthy, thus the correct option is B. Explanation "Haggard" refers to a worn-out, exhausted, or fatigued appearance often associated with tiredness, illness, or extreme stress. In contrast, "healthy" stands as the antonym, representing a state of well-being, vitality, and robust physical and mental condition. The term "healthy" conveys a sense of vigor, soundness, and overall good health, opposite to the drained and weary connotations of "haggard." When someone is healthy, they exhibit energy, freshness, and a lack of weariness or fatigue, providing a clear contrast to the tired and depleted state described by the term "haggard." The antonym "healthy" stands in opposition to the concept of "haggard," presenting a state of being that is vibrant, robust, and free from the signs of fatigue, exhaustion, or distress. A healthy individual displays vitality, a glow of well-being, and an absence of the worn-out appearance associated with being haggard. The contrast between "haggard" and "healthy" exemplifies the dichotomy between an exhausted, drained state and a state of optimal physical and mental well-being, highlighting the stark difference between the two conditions (option B). Answered by damonsalvator7878 •38.8K answers•37.2M people helped Thanks 0 0.0 (0 votes) Textbook &Expert-Verified⬈(opens in a new tab) This answer helped 37154570 people 37M 0.0 0 Simplified Signs A Manual Sign-Communication System for Special Populations, Volume 2. - John D. Bonvillian; Nicole Kissane Lee; Tracy T. Dooley; Filip T. Loncke Surface and Subtext: Literature, Research, Writing - Claire Carly-Miles; Dorothy Todd; Frances Thielman; James Francis, Jr.; Kathy Christie Anders; Kimberly Clough; Matt McKinney; Nicole Hagstrom-Schmidt; R. Paul Cooper; Sarah LeMire; and Travis Rozier Mindful Technical Writing: An Introduction to the Fundamentals - Dawn Atkinson, Stacey Corbitt Upload your school material for a more relevant answer The antonym of "haggard" is "healthy," so option B is the correct choice. 'Haggard' refers to looking worn out or fatigued, while 'healthy' conveys robustness and energy. This shows a clear contrast between the two states. Explanation The antonym of "haggard" is "healthy," making option B the correct choice. The term "haggard" describes someone who looks worn out, exhausted, or fatigued, often due to stress, illness, or lack of sleep. When someone is haggard, they may appear pale, tired, and lack the vitality associated with good health. On the other hand, the word "healthy" represents a state of well-being, depicting someone who feels energetic, vigorous, and free from signs of fatigue or stress. The contrast between the two terms illustrates how one refers to a depleted, drained state, whereas the other conveys a sense of robust physical and mental condition. To summarize, while "haggard" suggests tiredness and exhaustion, "healthy" embodies the opposite qualities of vitality and well-being, thus solidifying B as the correct answer in this multiple-choice question. Examples & Evidence For example, a person who has just returned from a long trip and has not slept well may appear haggard with dark circles under their eyes, whereas someone who has had a restful weekend might appear healthy, with a clear complexion and bright eyes. The definitions of the words 'haggard' and 'healthy' can be verified in credible English dictionaries, confirming that they are indeed antonyms. Thanks 0 0.0 (0 votes) Advertisement erikacastro3542 has a question! Can you help? Add your answer See Expert-Verified Answer ### Free English solutions and answers Community Answer 9 The criteria retailer must meet to receive a reduced penalty and/or protect the license/permit if an illegal alcohol sale takes place at the establishment is often referred to Community Answer 4.1 240 Based on the passage you just read, what conclusion can you draw about the cultural values of the iroquois? “The right-handed twin accused his brother of murdering their mother, and their quarrels continued until it was time to bury their mother. With the help of their grandmother, they made her a grave. From her head grew the three sister plants: corn, beans, and squash. From her heart grew tobacco, which people still use to give thanks in ceremony. She is called ‘our mother’ and the people dance and sing to her to make the plants grow.” Community Answer 2 Making them clean the floors would be a(n) because it would be outside their usual duties, Community Answer 5 a letter to the Ambassador highlighting hardship faced by Nigerian students and soliciting for assistance Community Answer 19 What does Stitch say in the beginning of "Lilo and Stitch" in Alien to make the robot throw up Community Answer 2 Then, from heaven, the voice of the god called to Gilgamesh: "Hurry, attack, attack Humbaba while the time is right, before he enters the depths of the forest, before he can hide there and wrap himself in his seven auras with their paralyzing glare. He is wearing just one now. Attack him! Now!" –Gilgamesh: A New English Translation, Stephen Mitchell Identify a feature of epic poetry in the passage. In three to four sentences, explain its impact on the epic's plot. Community Answer 4.5 487 Write two to three sentences explaining how Gilgamesh demonstrates the characteristics of an epic hero. Use evidence from the text to support your answer Community Answer What do you call a dead polar bear? if you know this one don't say anything New questions in English The root "sens" means "feel, hear, see, or perceive." Which of these words means "able to be felt"? A. insensible B. insensitive C. sensational D. sensible You will demonstrate your ability to make inferences and support them with textual evidence by creating a realistic text message conversation based on the story 'Thank You, Ma'am' by Langston Hughes. Read the passage from "Daughter of Invention" by Julia Alvarez. "What did you do to provoke them? It takes two to tangle, you know." In it, the mother misuses the English idiom "It takes two to tango." What does her language show about her as a character? A. She has picked up expressions and uses them like a native English speaker. B. She is at ease in her new home. C. She is embarrassed to speak. D. She has trouble communicating clearly in English. Which of these steps is crucial when doing a close reading of nonfiction? A. scanning the text for information B. writing notes in the margins C. summarizing the text D. creating a glossary Brandon is writing a research paper on this topic: changes in the game of football in the twentieth century. He needs to use a secondary source. Which source should he choose? A. a photograph of a football player taken in 1948 B. a player's account of what it was like to play in the National Football League C. a biography about quarterback Joe Namath D. a piece of video footage of the 1979 Super Bowl Previous questionNext question Learn Practice Test Open in Learning Companion Company Copyright Policy Privacy Policy Cookie Preferences Insights: The Brainly Blog Advertise with us Careers Homework Questions & Answers Help Terms of Use Help Center Safety Center Responsible Disclosure Agreement Connect with us (opens in a new tab)(opens in a new tab)(opens in a new tab)(opens in a new tab)(opens in a new tab) Brainly.com Dismiss Materials from your teacher, like lecture notes or study guides, help Brainly adjust this answer to fit your needs. Dismiss
2881	http://www.hieronymus.us.com/latinweb/Grammatica/The_Latin_Perfect.htm	The Latin Perfect The Latin Perfect General Remarks: a) In Latin the perfect tense is the “narrative tense”; that is, it appears mostly in texts which describe past events and, in general, does so more frequently than the imperfect.The perfect expresses punctual (completed) actions, whereas the imperfect describes states of long duration, repeated or attempted actions. b)When translating into English, there is often no clear distinction between the perfect and imperfect— laudavi can often be translated either with “I praised” or “I was praising.” Since in English the preterite (simple past or perfect) is the “narrative tense,” the usual translation is “I praised” rather than “I was praising.” In the ACTIVE VOICE, the perfect uses only the following personal endings: -ī(I …) -istī(thou …) -it(he/she/it …) -imus(we …) -istis(ye …) -ērunt(they …) There are differing paradigms of the perfect stem. Hence it is impossible to predict which verbs follow which model. As a result, for those that do not follow the “regular” so-called “v-perfect,” the first person singular must be learnt as part of the basic verb itself. 1) The “regular” v-perfect applies primarily to verbs of the ā- and ī-conjugations, but also to some of the other conjugations: laudāre audīre dēlēre laudāvī audīvī dēlēvī laudāvistī audīvistī dēlēvistī laudāvit audīvit dēlēvit laudāvimus audīvimus dēlēvimus laudāvistis audīvistis dēlēvistis laudāvērunt audīvērunt dēlēvērunt 2) The u-perfect 3) The s-perfect 4) The lengthend-grade perfect monēre (to warn)scrībere (to write)venīre (to come) monuī scrīpsī vênī monuistī …scrīpsistī …vênistī … 5) The stem-perfect 6) The ablaut-perfect 7) The reduplication perfect dēfendere (to defend)facere (to write)currere (to run) dēfendī fēcī cucurrī dēfendistī …fēcistī …cucurristī … 8) The perfect of esse (“to be”)9) The perfect of ferre (“to carry”) fuī (“I was/have been”)tulī fuistī …tulistī … The Perfect PASSIVE In Latin, the passive is constructed out of two components: out of the so-called perfect passive participle (PPP) and a present-tense form of the helping verb esse. The formation has some similarity to the English: I was praised sum laudātus Although in Latin the participle usually comes first:laudātus sum. For the passive, the active forms of the perfect are mostly irrelevant; instead, when learning the “irregular” verbs, the PPP has to be memorized in addition to the present and perfect active. Thus the so-called “four principle parts” have to be committed to memory as, for example, habeō - habēre – habuī – habitum - “to have.” The PPP is given in the neuter singular nominative. The perfect passive participle (PPP) is simply the fourth principal part of a transitive verb. The literal translation is “having been + verb + -ed” (or its equivalent). Thus it corresponds to English forms such as “(having been) praised,” “(having been) held.” “(having been) meant,” “(having been) sought,” “(having been) paid” it is declined as a regular “2-1-2” adjective of the a-/o-declension (like magnus, -a, -um). Hence it has singular and plural forms as well as forms for masculine, feminine and neuter gender along with nominative, genitive, dative cases, and so forth. Most Latin PPPs end in –tus, –ta, –tum, a few in –sus, –sa, –sum. Examples:laudātus (“praised,” from laudāre), habitus (“had/held,” from habēre), monitus (“warned” from monēre), vīsus (“seen,” from vidēre),jussus (“commanded,” from jubēre). ➠MasculineFeminine laudātus sum laudāta sum“I was/have been praised” laudātus es laudāta es“thou wast/hast been praised” laudātus est laudāta est“he/she/it was/has been praised” laudāti sumus laudātæ sumus“we were/have been praised” laudāti estis laudātæ estis“ye were/have been praised” laudāti sunt laudātæ sunt“they were/have been praised” esse has no perfect passive participle; The PPP offerre is lātus, lāta, lātum. The principle parts to be learned, thus, are ferre – ferō – tulī – lātum - “to carry.” For a good overview of all participles (present, future, etc.), go to ->>>>>>⇈⇑⇈<<<<<<- Deus vult!— Brian Regan (Inscriptio electronica:Theedrich@harbornet.com) Dies immutationis recentissimæ: die Martis, 2021 Junii 8
2882	https://www.cuemath.com/algebra/cubes-and-cube-roots/	Cube Root The cube root of a number is that number which is multiplied 3 times to get the original number. Whenever a number (x) is multiplied three times, then the resultant number is known as the cube of that number. The cube for the number (x) is represented as x3 and is read as 'x-cubed' (or) 'x to the power of 3' (or) 'x raised to 3. For example, let us take the number 5. We know that 5 × 5 × 5 = 125. Hence, 125 is called the cube of 5. While on the other hand, finding the cube root of a number involves the reverse process of the cube of a number and is denoted by ∛. Considering the same example, 5 is called the cube root of the number 125 and is written as ∛125 = 5. On this page, we will learn more about the cubes and cube roots of a number. \| \| \| --- \| \| 1. \| Cube Root Definition \| \| 2. \| How to Find Cube Root of a Number? \| \| 3. \| Cube Root Formula \| \| 4. \| What is Cube of a Number? \| \| 5. \| List of Cube Root of Numbers \| \| 6. \| FAQs on Cube Root \| Cube Root Definition When we think about the words cube and root, the first picture that might come to our mind is a literal cube and the roots of a tree. Isn't it? Well, the idea is similar. Root means the primary source or origin. So, we just need to think 'cube of which number should be taken to get the given number'. In mathematics, the definition of cube root says that Cube root is the number that needs to be multiplied three times to get the original number. Now, let us look at the cube root formula, where y is the cube root of x. ∛x = y. The radical sign ∛ is used as a cube root symbol for any number with a small 3 written on the top left of the sign. Another way to denote cube root is to write 1/3 as the exponent of a number. Cube root is an inverse operation of the cube of a number. Cube Root Symbol The cube root symbol which is used to represent the cube root of a number is ∛. This means, if we want to write the cube root of 27, we use the cube root symbol in the following way. ∛27 = 3 Perfect Cubes A perfect cube is an integer that can be expressed as the product of three equal integers. For example, 125 is a perfect cube because 53 = 5 × 5 × 5 = 125. However, 121 is not a perfect cube because there is no number, which when multiplied three times gives the product 121. In other words, a perfect cube is a number whose cube root is an integer. It is also known as the 'cube number'. The following table shows the perfect cubes of the first 10 natural numbers. \| Number/Cube root \| Perfect cube \| --- \| \| 1 \| 1 \| \| 2 \| 8 \| \| 3 \| 27 \| \| 4 \| 64 \| \| 5 \| 125 \| \| 6 \| 216 \| \| 7 \| 343 \| \| 8 \| 512 \| \| 9 \| 729 \| \| 10 \| 1000 \| How to Find Cube Root of a Number? The cube root of a number can be determined by using the prime factorization method. In order to find the cube root of a number: Example: Let us see how to find the cube root of 15625. Cube Root Formula The cube root formula helps in the calculation of the cube root of any given number that is expressed in a radical form using the symbol ∛. It can be calculated by first finding out the prime factorization of the given number and then later applying the cube root formula. Suppose, x is any number such that, x = y × y × y. The formula to calculate the cube root is given as: where, y is the cube root of any number x. This also means that the number x would be a perfect cube if y has an integer value. Applications of Cube Root Formula Given below are a few major applications of the cube root formula: What is the Cube of a Number? When we multiply a number three times by itself, the resultant number (product) is known as the cube of the original number. We call it a cube because it is used to represent the volume of a cube. In other words, a number raised to exponent 3 is called the cube of that number. For example, the cube of 3 is 27. That means 3 × 3 × 3 = 27, and it can be written as 33. Similarly, the cube of 4 is 64, and the cube of 5 is 125, and so on. To find the cube of a number, first, multiply that number by itself, then multiply the product obtained with the original number again. Let us find the cube of 7 through the same process. We know that the cube of a number N is N × N × N. So, the cube of 7 is 7 × 7 × 7. Now, in order to find the cube of 7, we will first find the value of 7 × 7. This value is 49. Now, we will find 49 × 7. This is equal to 343. Hence, we can say that cube of the number 7 is 343. Cube of a Fraction Similar to the cube of a number, the cube of a fraction can be found by multiplying it three times. For example, the cube of the fraction (2/5) can be written as 2/5 × 2/5 × 2/5. Simplifying it further, we get the value of the cube as (2 × 2 × 2) / (5 × 5 × 5). This is equal to (23/ 53) = 8/125. Cube of Negative Numbers The process to find the cube of a negative number is the same as that of a whole number and fraction. Here, always remember that the cube of a negative number is always negative, while the cube of a positive number is always positive. For example, let us try finding the cube of -7. We know that the cube of -7 is (-7) × (-7) × (-7). Now, in order to find the cube of (-7), we will first find the value of (-7) × (-7). This value is 49. Now, we will find 49 × (-7). This is equal to -343. Hence, we can say that cube of the number -7 is -343. The cube root formula for negative numbers is: ☛ Related Topics Check these interesting articles related to the cube root. List of Cube Root of Numbers \| \| \| --- \| \| Cube Root of 1 \| Cube Root of 2 \| \| Cube Root of 3 \| Cube Root of 4 \| \| Cube Root of 5 \| Cube Root of 6 \| \| Cube Root of 8 \| Cube Root of 9 \| \| Cube Root of 10 \| Cube Root of 12 \| \| Cube Root of 16 \| Cube Root of 18 \| \| Cube Root of 24 \| Cube Root of 25 \| \| Cube Root of 27 \| Cube Root of 32 \| \| Cube Root of 54 \| Cube Root of 64 \| \| Cube Root of 81 \| Cube Root of 100 \| \| Cube Root of 125 \| Cube Root of 216 \| \| Cube Root of 256 \| Cube Root of 343 \| \| Cube Root of 512 \| Cube Root of 729 \| \| Cube Root of 1000 \| Cube Root of 1331 \| \| Cube Root of 1728 \| Cube Root of 4096 \| Cube Root of 1 Cube Root of 2 Cube Root of 3 Cube Root of 4 Cube Root of 5 Cube Root of 6 Cube Root of 8 Cube Root of 9 Cube Root of 10 Cube Root of 12 Cube Root of 16 Cube Root of 18 Cube Root of 24 Cube Root of 25 Cube Root of 27 Cube Root of 32 Cube Root of 54 Cube Root of 64 Cube Root of 81 Cube Root of 100 Cube Root of 125 Cube Root of 216 Cube Root of 256 Cube Root of 343 Cube Root of 512 Cube Root of 729 Cube Root of 1000 Cube Root of 1331 Cube Root of 1728 Cube Root of 4096 Cube Root Examples Example 1: Is 729 a perfect cube? Solution: Yes, 729 is a perfect cube because when we multiply 9 × 9 × 9 = 93 = 729. This means the cube root of 729 is 9. Answer: Yes. Example 2: Emily's father's age is 27 years. Find the age of Emily if her age is the cube root of her father's age. Solution: Given, age of Emily's father = 27 years. Therefore, Emily's age= cube root of 27 = ∛27 years = 3 years. Answer: 3 years. Example 3: Check whether 512 is a perfect cube or not. Solution: To find: Whether 512 is the perfect cube or not. Using the cube root formula, Cube root of 512 = ∛512=∛(2×2×2×2×2×2×2×2×2)=∛(8×8×8) = 8, which is an integer. Therefore, 512 is a perfect cube. Therefore, 512 is a perfect cube. Answer: Yes, it is. go to slidego to slidego to slide Book a Free Trial Class Practice Questions on Cube Root go to slidego to slide FAQs on Cube Root What is Cube Root of a Number? The cube root is the reverse of the cube of a number and is denoted by ∛. For example, ∛216, that is, the cube root of 216 = 6 because when 6 is multiplied thrice with itself, it gives 216. In other words, since 63 = 216, we have ∛216 = 6. What is the Difference Between Cube and Cube Root? What is Cube Root Used For? Cube root is used to solve cubic equations. They are also used to find the side length of a cube if its volume is given. How to Find the Cube Root of a Number? Cube root can be simplified using the prime factorization method. First, do the prime factorization of the given number, then take out the common factors in groups of 3. Multiply these common factors taking only one from each group to get the answer. Is it Possible to Simplify the Negative Cube Root? Yes, the simplification of negative cube roots is the same as positive cube roots. The only difference is the presence of a negative sign with the cube root of a negative number. What is Not a Perfect Cube? A number is not a perfect cube if we cannot make 3 equal groups of factors of the number after doing the prime factorization. For example, 144 is not a perfect cube because there is no number that, when multiplied 3 times with itself, gives 144 as the product. In other words, if the cube root of a number is not an integer, then it is not a perfect cube. What is the Cube of an Odd Natural Number? The cube of an odd natural number is always an odd number. For example, 53 = 125, 73 = 343, 93 = 729, etc. Can the Cube Root of Any Odd Number be Even? No, the cube root of an odd number is always odd. For example, the cube root of 27 = (27)1/3 = 3. Here, both 3 and 27 are odd numbers. How to Calculate Cube Root of Any Number? The easiest and basic method to find the cube root of any number is the prime factorization method. What is the Cube Root Formula in Algebra? In math, the cube root formula is used to represent any number in the form of its cube root, such as for any number x, its cube root will be 3√x = x1/3. For example, the cube root of 125 is 5 1251/3 = (53)1/3 = 5. What is the Cube Root Formula for Negative Numbers? Yes, it is possible to find the cube root of negative numbers. For example, -64 = (-4) × (-4) × (-4). We can write -64 as the product of 3 negative 4's. Thus, ∛-64 = -4 because the product of three negative values gives us a negative result. How to Use Cube Root Formula? In order to use the cube root formula, we use the following steps: How to Write Cube Root Formula in Words? The cube root formula can be written in words in the following way. The cube root of any number is the number raised to the power of 1/3. What is the Cube Root Symbol? The cube root symbol which is used to represent the cube root of a number is ∛. This means, if we want to write the cube root of 125, we use the cube root symbol in the following way. ∛125 = 5
2883	https://stackoverflow.com/questions/48082681/algorithm-for-determine-the-arc-mid-point	Skip to main content Stack Overflow About For Teams Stack Overflow for Teams Where developers & technologists share private knowledge with coworkers Advertising Reach devs & technologists worldwide about your product, service or employer brand Knowledge Solutions Data licensing offering for businesses to build and improve AI tools and models Labs The future of collective knowledge sharing About the company Visit the blog Algorithm for determine the Arc Mid Point Ask Question Asked Modified 7 years, 7 months ago Viewed 2k times This question shows research effort; it is useful and clear 2 Save this question. Show activity on this post. I am currently looking to implement an algorithm that will be able to compute the arc midpoint. From here on out, I will be referring to the diagram below. What is known are the start and end nodes (A and B respectively), the center (point C) and point P which is the intersection point of the line AB and CM (I am able to find this point without knowing point M because line AB is perpendicular to line CM and thus, the slope is -1/m). I also know the arc angle and the radius of the arc. I am looking to find point M. I have been looking at different sources. Some suggest converting coordinates to polar, computing the mid point from the polar coordinates then reverting back to Cartesian. This involves sin and cos (and arctan) which I am a little reluctant to do since trig functions take computing time. I have been looking to directly computing point M by treating the arc as a circle and having Line CP as a line that intersects the circle at Point M. I would then get two values and the value closest to point P would be the correct intersection point. However, this method, the algebra becomes long and complex. Then I would need to create special cases for when P = C and for when the line AB is horizontal and vertical. This method is ok but I am wondering if there are any better methods out there that can compute this point that are simpler? Also, as a side note, I will be creating this algorithm in C++. c++ algorithm geometry computational-geometry Share CC BY-SA 3.0 Improve this question Follow this question to receive notifications asked Jan 3, 2018 at 17:55 philmphilm 91511 gold badge99 silver badges3333 bronze badges 5 2 Do you know about math.stackexchange.com ? – Drew Dormann Commented Jan 3, 2018 at 17:57 Yes, I am aware of the place – philm Commented Jan 3, 2018 at 17:57 You know vector CP and want to renormalize it to give it the same length as CA? Btw P is just the midpoint of AB. – Marc Glisse Commented Jan 3, 2018 at 18:10 1 This is a math question, not a programming question. – François Andrieux Commented Jan 3, 2018 at 18:12 4 Seems like the vector CM is just the vector CP with it's length equal to the arc radius. CM = CP / \|CP\| [arch radius]? – François Andrieux Commented Jan 3, 2018 at 18:15 Add a comment \| 2 Answers 2 Reset to default This answer is useful 3 Save this answer. Show activity on this post. A circumference in polar form is expressed by ``` x = Cx + R cos(alpha) y = Cy + R sin(alpha) ``` Where alpha is the angle from center C to point x,y. The goal now is how to get alpha without trigonometry. The arc-midpoint M, the point S in the middle of the segment AB, and your already-calculated point P, all of them have the same alpha, they are on the same line from C. Let's get vector vx,vy as C to S. Also calculate its length: ``` vx = Sx - Cx = (Ax + Bx)/2 - Cx vy = Sy - Cy = (Ay + By)/2 - Cy leV = sqrt(vx vx + vy vy) ``` I prefer S to P because we can avoid some issues like infinite CP slope or sign to apply to slope (towards M or its inverse). By defintions of sin and cos we know that: ``` sin(alpha) = vy / leV cos(alpha) = vx / leV ``` and finally we get ``` Mx = Cx + R vx / leV My = Cy + R vy / leV ``` Note: To calculate Ryou need another sqrt function, which is not quick, but it's faster than sin or cos. For better accuracy use the average of Ra= dist(AC) and Rb= dist(BC) Share CC BY-SA 3.0 Improve this answer Follow this answer to receive notifications edited Jan 3, 2018 at 20:04 answered Jan 3, 2018 at 19:55 Ripi2Ripi2 7,21611 gold badge1919 silver badges3535 bronze badges 6 So the R here, is that the radius? If so, that is a known parameter – philm Commented Jan 3, 2018 at 21:05 @philm Yes, R= radius. I added the note for the case you have only points (A, B, Center) – Ripi2 Commented Jan 3, 2018 at 21:07 Cool thanks, So the point S in your description, is sounds like that this is point P in my diagram. If so, then this is another known parameter. – philm Commented Jan 3, 2018 at 21:10 S != P S is on AB segment and on CM segment. P is on CM, but may be not on AB. My solution works well with S, I don't know if it will also do with P. Read the line "I prefer..." – Ripi2 Commented Jan 3, 2018 at 21:19 Point P has to be on line segment AB and CM because point P is the intersection point of line segment AB and CM which are perpendicular to each other. Either way, I will attempt both methods (when S = P and S != P) and see if it matters. But I do like this algorithm – philm Commented Jan 3, 2018 at 21:42 \| Show 1 more comment This answer is useful 0 Save this answer. Show activity on this post. I would then get two values This is algebraically unavoidable. and the value closest to point P would be the correct intersection point. Only if the arc covers less than 180°. Then I would need to create special cases for when P = C This is indeed the most tricky case. If A, B, C lie on a line, you don't know which arc is the arc, and won't be able to answer the question. Unless you have some additional information to start with, e.g. know that the arc goes from A to B in a counter-clockwise direction. In this case, you know the orientation of the triangle ABM and can use that to decide which solition to pick, instead of using the distance. and for when the line AB is horizontal and vertical Express a line as ax + by + c = 0 and you can treat all slopes the same. THese are homogeneous coordinates of the line, you can compute them e.g. using the cross product (a, b, c) = (Ax, Ay, 1) × (Bx, By, 1). But more detailed questions on how best to compute these lines or intersect it with the circle should probably go to the Math Stack Exchange. if there are any better methods out there that can compute this point that are simpler? Projective geometry and homogeneous coordinates can avoid a lot of nasty corner cases, like circles of infinite radius (also known as lines) or the intersection of parallel lines. But the problem of deciding between two solutions remains, so it probably doesn't make things as simple as you'd like them to be. Share CC BY-SA 3.0 Improve this answer Follow this answer to receive notifications answered Jan 3, 2018 at 18:20 MvGMvG 61.6k2525 gold badges157157 silver badges293293 bronze badges Add a comment \| Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions c++ algorithm geometry computational-geometry See similar questions with these tags. 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2884	https://wiki.math.ntnu.no/linearmethods/innerproductspaces	Institutt for matematiske fag, NTNU [ \newcommand{R}{\mathbb{R}} \newcommand{C}{\mathbb{C}} \newcommand{K}{\mathbb{K}} \newcommand{N}{\mathbb{N}} \newcommand{defarrow}{\quad \stackrel{\text{def}}{\Longleftrightarrow} \quad} ] Inner-product spaces Let (X) be a vector space over (\K \in {\R,\C}). Inner-product spaces An inner product (\langle \cdot, \cdot \rangle) on (X) is a map (X \times X \to \K), ((x,y) \mapsto \langle x,y \rangle), that is conjugate symmetric [ \langle x,y \rangle = \overline{\langle y,x \rangle}, ] linear in its first argument, [ \begin{align} &\langle \lambda x,y \rangle = \lambda \langle x,y \rangle,\ &\langle x + y, z \rangle = \langle x , z \rangle + \langle y, z \rangle, \end{align} ] and non-degenerate (positive definite), [ \langle x, x \rangle > 0 \quad \text{ for }\quad x \neq 0, ] with (x,y,z \in X) and (\lambda \in \K) arbitrary. The pair ((X,\langle \cdot, \cdot \rangle)) is called an inner-product space. Ex. The canonical inner product is the dot product in (\R^n): [ \langle x, y \rangle := x \cdot y = \sum_{j=1}^n x_j y_j. ] For matrices in (M_{n \times n}(\R)) one can define a dot product by setting [ \langle A, B \rangle := \mathrm{tr}(B^t A), ] where (\mathrm{tr}(C) = \sum_{j=1}^n c_{jj}) is the trace of a matrix (C), and (B^t) is the transponse of (B). Then [ B^t A = \sum_{j=1}^n b^t_{ij} a_{jk} = \sum_{j=1}^n b_{ji} a_{jk}, ] and [\mathrm{tr}(B^t A) = \sum_{k=1}^n \sum_{j=1}^n b_{jk} a_{jk} = \sum_{1 \leq j,k \leq n} a_{jk} b_{jk}] coincides with the dot product on (\R^{nn} \cong M_{n \times n}(\R)). Properties of the inner product An inner product satisfies [ \begin{align} &\text{(i)} \qquad &\langle x, y+ z \rangle &= \langle x, y \rangle + \langle x, z \rangle,\ &\text{(ii)} \qquad &\langle x, \lambda y \rangle &= \bar\lambda \langle x, y \rangle,\ &\text{(iii)} \qquad &\langle x,0 \rangle &= \langle 0, x \rangle = 0,\ &\text{(iv)} \qquad & \text{ If } \langle x, z \rangle &= 0 \text{ for all } z \in X \quad \text{ then } \quad x = 0. \end{align} ] N.b. By linearity, the last property implies that if (\langle x, z \rangle = \langle y, z \rangle) for all ( z \in X), then (x = y). Proof Proof (i) [ \langle x, y + z \rangle = \overline{\langle y +z, x \rangle} = \overline{\langle y, x \rangle + \langle z, x \rangle} = \overline{\langle y, x \rangle} + \overline{\langle z, x \rangle} = \langle x,y \rangle + \langle x,z \rangle. ] (ii) [ \langle x, \lambda y \rangle = \overline{\langle \lambda y, x \rangle} = \overline{\lambda \langle y, x \rangle} = \bar \lambda \langle x,y \rangle. ] (iii) [ \langle {\bf 0}, x \rangle = \langle 0 x, x \rangle = 0 \langle x, x \rangle = 0, ] and [ \langle x, 0 \rangle = \overline{ \langle 0, x \rangle} = 0. ] (iv) [ \langle x, z \rangle = 0 \text{ for all } z \in X \quad \Longrightarrow\quad \langle x, x \rangle = 0 \quad \Longrightarrow\quad x = 0. ] Inner-product spaces as normed spaces An inner-product space ((X,\langle \cdot, \cdot \rangle)) carries a natural norm given by (\\|x\\| := \langle x, x \rangle^{1/2} ). To prove this, we need: > The Cauchy–Schwarz inequality For all (x,y \in (X, \langle \cdot, \cdot \rangle)), [ \| \langle x, y \rangle \| \leq \\| x\\| \\|y\\|, ] with equality if and only if (x) and (y) are linearly dependent. Proof Proof Linearly dependent case: Without loss of generality, assume that (x = \lambda y) (if (y = \lambda x) we can always relabel the vectors). Then [ \begin{align} \| \langle x, y \rangle \| &= \| \langle \lambda y, y \rangle \| = \|\lambda\| \langle y, y \rangle \ &=\|\lambda\| \\|y\\|^2 = \\| \lambda y\\| \\|y\\| = \\|x\\| \\|y\\|. \end{align} ] Linearly independent case: If (x - \lambda y \neq 0) and ( y - \lambda x \neq 0) for all (\lambda \in \K), then also (x,y \neq 0), and [ \begin{align} 0 &< \langle x + \lambda y, x + \lambda y \rangle\ &= \langle x, x + \lambda y \rangle + \lambda \langle y, x + \lambda y\rangle \ &= \langle x, x\rangle + \langle x, \lambda y \rangle + \lambda \langle y, x \rangle + \lambda \langle y, \lambda y \rangle \ &= \\| x \\|^2 + \bar\lambda \langle x, y \rangle + \lambda \overline{\langle x, y \rangle} + \lambda \bar\lambda \\| y \\|^2\ &= \\| x \\|^2 + 2 \Re \big( \bar\lambda \langle x, y \rangle \big) + \|\lambda\|^2 \\| y \\|^2. \end{align} ] If (\langle x,y \rangle = 0) the Cauchy–Schwarz inequality is trivial, so assume that (\langle x, y \rangle \neq 0). Let (\lambda := tu) with (u := \frac{\langle x,y \rangle}{\|\langle x, y \rangle\|}), so that [ \bar\lambda \langle x, y \rangle = t \frac{\overline{\langle x,y \rangle}\langle x, y \rangle}{\|\langle x, y \rangle\|} = t \|\langle x,y \rangle\| \qquad\text{ and }\qquad \|\lambda\|^2 = t^2. ] Hence, [ 0 < \\|x\\|^2 + 2 t \|\langle x,y\rangle\| + t^2\\|y\\|^2 = \Big(\\|y\\| t + \frac{\|\langle x, y \rangle\|}{\\|y\\|}\Big)^2 + \\|x\\|^2 - \Big( \frac{\|\langle x, y \rangle\|}{\\|y\\|}\Big)^2. ] By choosing (t = - \|\langle x,y\rangle\|/\\|y\\|^2), we obtain that [ \frac{\|\langle x,y \rangle\|^2}{\\|y\\|^2} < \\|x\\|^2, ] which proves the assertion. > Inner-product spaces are normed If ((X,\langle \cdot, \cdot \rangle)) is an inner-product space, then ( \\|x\\| = \langle x, x \rangle^{1/2}) defines a norm on (X). Proof Proof Positive homogeneity: [ \\|\lambda x\\| = \langle \lambda x, \lambda x \rangle^{1/2} = \big( \lambda \bar\lambda \langle x, x \rangle \big)^{1/2} = \big( \|\lambda\|^2 \\|x\\|^2 \big)^{1/2} = \|\lambda \| \\|x\\|. ] Triangle inequality: By the Cauchy–Schwarz inequality, [ \begin{align} \\| x + y \\|^2 &= \\|x\\|^2 + 2 \Re \langle x,y\rangle + \\|y\\|^2\ &\leq \\|x\\|^2 + 2 \|\langle x,y\rangle\| + \\|y\\|^2\ &\leq \\|x\\|^2 + 2 \\|x\\| \\|y\\| + \\|y\\|^2\ &= \big( \\|x\\| + \\|y\\| \big)^2. \end{align} ] Non-degeneracy: [ \\|x\\| = 0 \quad\Longleftrightarrow \quad \\|x\\|^2 = 0 \quad\Longleftrightarrow \quad \langle x, x \rangle = 0 \quad\Longleftrightarrow \quad x = 0, ] according to the positive definiteness of the inner product. Parallelogram law and polarization identity Let ((X, \\|\cdot\\|)) be a normed space. Then the parallelogram law [ \\| x + y \\|^2 + \\| x- y\\|^2 = 2 \\|x\\|^2 + 2 \\|y\\|^2 ] holds exactly if (\\|\cdot\\| = \langle \cdot, \cdot \rangle^{1/2}) can be defined using an inner product on (X). If so, [ \langle x, y \rangle = \frac{1}{4} \big( \\| x + y \\|^2 - \\|x - y\\|^2 \big), ] if (X) is real, and [ \langle x , y \rangle = \frac{1}{4} \sum_{k=0}^3 i^k \\| x + i^k y\\|^2, ] if (X) is complex. Proof Proof We only show that the parallelogram law and polarization identity hold in an inner product space; the other direction (starting with a norm and the parallelogram identity to define an inner product) is left as an exercise. Parallelogram law: If (X) is an inner-product space, then [ \\|x \pm y \\|^2 = \\|x\\|^2 \pm 2 \Re \langle x, y \rangle + \\|y\\|^2; ] the parallelogram law follows from adding these two equations to each other. Polarization identity: When (X) is a real inner-product space, it follows directly that [ \\| x + y\\|^2 - \\|x - y \\|^2 = \big( \\|x\\|^2 + 2 \langle x, y \rangle + \\|y\\|^2 \big) - \big(\\|x\\|^2 - 2 \langle x, y \rangle + \\|y\\|^2 \big) = 4 \langle x, y \rangle. ] If (X) is complex, the corresponding calculcation yields that [ \begin{align} \sum_{k=0}^3 i^k \\| x + i^k y \\|^2 &= \sum_{k=0}^3 i^k \big( \\|x\\|^2 + 2 \Re \langle x, i^k y \rangle + \\|i^k y\\|^2\big)\ &= \big( \\|x\\|^2 + 2 \Re \langle x, y \rangle + \\|y\\|^2\big) - \big( \\|x\\|^2 - 2 \Re \langle x, y \rangle + \\|y\\|^2\big)\ &+ i\big( \\|x\\|^2 - 2 \Re i \langle x, y \rangle + \\|y\\|^2\big) - i\big( \\|x\\|^2 + 2 \Re i \langle x, y \rangle + \\|y\\|^2\big). \end{align} ] Since (\Re iz = - \Im z) for any (z \in \C), we obtain [ \sum_{k=0}^3 i^k \\| x + i^k y \\|^2 = 4 \Re \langle x,y \rangle + 4 \Im \langle x,y \rangle = 4 \langle x, y \rangle. ] Ex. Pythagoras' theorem: If (\langle x,y \rangle = 0 ) in an inner-product space, then [ \\|x + y \\|^2 = \\|x\\|^2 + \\|y\\|^2, ] which, in (\R^2), we recognize as [ a^2 + b^2 = c^2, ] with (a,b,c) the sides of a right-angled triangle. If we define ( \langle x, y \rangle := \frac{1}{4} \left( \\| x+y\\|^2 - \\| x - y\\|^2 \right) ) in (\R^2) using the polarization identity , we see that [ \begin{align} \langle x, y \rangle &= \frac{1}{4} \left( ( x_1 + y_1 )^2 + ( x_2 + y_2)^2 \right) - \frac{1}{4} \left( ( x_1 - y_1 )^2 + ( x_2 - y_2)^2 \right) \ & =\frac{1}{4} \left( x_1^2 + 2x_1y_1 + y_1^2 + x_2^2 + 2x_2 y_2 + y_2^2 \right) - \frac{1}{4} \left( x_1^2 - 2x_1y_1 + y_1^2 + x_2^2 - 2x_2 y_2 + y_2^2 \right)\ &=x_1 y_1 + x_2 y_2 \end{align}] is the standard dot product. Hilbert spaces A complete inner-product space is called a Hilbert space. Similarly, inner-product spaces are sometimes called pre-Hilbert spaces. Ex. The Banach spaces (\R^n), (l_2(\R)) and (L_2(I,\R)), as well as their complex counterparts (\C^n), (l_2(\C)) and (L_2(I,\C)), all have norms that come from inner products: [ \langle x, y \rangle_{\C^n} = \sum_{j=1}^n x_j \bar{y_j} \quad \text{ in }\quad \C^n, ] [ \langle x, y \rangle_{l_2} = \sum_{j=1}^\infty x_j \bar{y_j} \quad \text{ in }\quad l_2, ] and [ \langle x, y \rangle_{L_2} = \int_I x(s) \overline{y(s)}\,ds \quad \text{ in }\quad L_2. ] (If the spaces are real, there are no complex conjugates.) Thus, they are all Hilbert spaces. In particular, this proves the (l_2)- and (L_2)-norms defined earlier in this course are indeed norms. The space of real-valued bounded continuous functions on a finite open interval, (BC((a,b),\R)), can be equipped with the (L_2)-inner product. This is a pre-Hilbert space, the completion of which is (L_2((a,b),\R)). Convex sets and the closest point property Let (X) be a linear space. A subset (M \subset X) is called convex if [x, y \in M \quad \Longrightarrow \quad tx + (1-t)y \in M \quad \text{ for all } \quad t \in (0,1),] i.e., if all points in (M) can be joined by line segments in (M). Ex. Any hyperbox ( { x \in \R^n \colon a_j \leq x_j \leq b_j} ) is convex. Intuitively, any region with a 'hole', like ( \R^n \setminus B_1 ), is not convex. Linear subspaces are convex: [ x, y \in M \quad \Longrightarrow\quad \mu x + \lambda y \in M \quad\text{ for all scalars } \mu, \lambda, ] clearly implies that ( t x + (1-t) y \in M) for all (t \in (0,1)). > Closest point property (Minimal distance theorem) Let (H) be a Hilbert space, and (M \subset H) a non-empty, closed and convex subset of (H). For any (x_0 \in H) there is a unique element (y_0 \in M) such that [ \\| x_0 - y_0 \\| = \inf_{y \in M} \\| x_0 - y \\|. ] N.b. The number (\inf_{y \in M} \\| x_0 - y \\|) is the distance from (x_0) to (M), denoted (\mathrm{dist}(x_0,M)). Proof Proof A minimizing sequence: Since (M \neq \emptyset), the number (d := \inf_{y \in M} \\| x_0 - y \\|) is finite and non-negative, and by the definition of infinimum, there exists a minimizing sequence ({y_j}_{j \in \N} \subset M) such that [ \lim_{j \to \infty} \\| x_0 - y_j\\| = d. ] ({y_j}_{j \in \N}) is Cauchy: By the parallelogram law applied to (x_0 - y_n), (x_0 - y_m), we have [ \\| 2 x_0 - (y_m + y_n)\\|^2 + \\| y_m - y_n\\|^2 = 2 \\| x_0 - y_m\\|^2 + 2 \\| x_0 - y_n\\|^2 \to 4 d^2, \qquad m,n \to \infty. ] In view of that (M) is convex and (d) minimal, we also have that [ \\| 2 x_0 - (y_m + y_n)\\|^2 = 4 \Big\\| x_0 - \frac{y_m + y_n}{2}\Big\\|^2 \geq 4d^2. ] Consequently, [ \\| y_m - y_n\\|^2 \to 0 \quad \text{ as }\quad m,n \to \infty. ] Since (M \subset H) is closed and (H) is complete, there exists [ y_0 = \lim_{j \to \infty} y_j \in M \quad\text{ with }\quad \\| x_0 - y_0 \\| = \lim_{j \to \infty}\\|x_0 - y_j\\| = d. ] Uniqueness: Suppose that (z_0 \in M) satisfies (\\| x_0 - z_0 \\| = d). Then (\frac{y_0 + z_0}{2} \in M) and the parallelogram law (applied to (x_0 - y_0), (x_0 - z_0)) yields that [ \\| y_0 - z_0 \\|^2 = 2\\|x_0 - y_0\\|^2 + 2\\|x_0 - z_0\\|^2 -4\Big\\| x_0 - \frac{y_0 + z_0}{2}\Big\\|^2 \leq 2d^2 + 2d^2 - 4d^2 = 0, ] so that (z_0 = y_0). Ex. In the Hilbert space (\R^2): The closed unit disk ({ x_1^2 + x_2^2 \leq 1}) contains a unique element that minimizes the distance to the point ((2,0)) (namely ((1,0))). The subgraph ({ x_2 \leq x_1^2 }) is closed but not convex; it has more than one point minimizing the distance to the point ((0,1)). The open unit ball ({ x_1^2 + x_2^2 < 1}) is convex but not closed; it has no element minimizing the distance to a point outside itself. Let [ M_n := \mathrm{span} { e^{ikx}}_{k=-n}^n] be the closed linear span of trigonometric functions (1, e^{ix}, e^{-ix} \ldots, e^{inx}, e^{-inx} \in L_2((-\pi,\pi),\C)). For any (n \in \N) and any (f \in L_2((-\pi,\pi),\C)) there is a unique linear combination of such functions that minimizes the (L_2)-distance to (f): [ \int_{-\pi}^\pi \big\| f(x) - \sum_{k=-n}^n c_k e^{ikx} \big\|^2\,dx = \min_{g \in M_n} \int_{-\pi}^\pi \big\| f(x) - g(x) \big\|^2\,dx. ] The coefficients (c_k) are known as (complex) Fourier coefficients of the function (f). © Mats Ehrnström. This material is free for private use. Public sharing, online publishing and printing to sell or distribute are prohibited.
2885	https://www.fao.org/4/ae906e/ae906e17.htm	FAO/WHO Ad Hoc Committee of Experts on Energy and Protein: Requirements and Recommended Intakes, 22 March - 2 April 1971, Rome THE ADOPTION OF JOULES AS UNITS OF ENERGY prepared by FAO Since early days the calorie or kilocalorie has been used as a unit of energy. In some circles, however, it is realised that this cannot be continued indefinitely and that in due course the joule is to be substituted for the calorie as the unit of energy in all nutritional work. Calories should then fall into disuse. In the International System of Units (Système International d'Unités) called S.I., there are 6 basic units as adopted in 1954: the meter (M) for length, the kilogramme (Kg.) for mass, the second (s) for time, the ampere (A) for electric current, the kelvin (K) for thermodynamic temperature and the candela (cd) for luminous intensity (1). All the other units are derived from these 6 basic units such as the unit for force as the newton (N) (Kgm/s 2), the unit of energy in any form is the joule (J) (Nm) and the unit of power as the watt (W) (J/s). The joule was adopted as the unit for electrical work, heat, mechanical work and energy in 1948 at the 9th General Conference on Weights and Measures, avoiding the calorie as far as possible. This unit was also formally approved in 1960 in the International System of Units (SI). Also the International Standards Organization (ISO) has recommended its adoption as the preferred unit for energy, as well as the USA National Bureau of Standards, the British Standards Institution and the Royal Society. Much consideration has already been given on the advisability of substituting joules for calories. The calorie cannot be derived directly from the basic SI units without using an experimentally determined factor, whereas the joule corresponds with the energy measurements in all branches of science on which the nutritional sciences depend. It is well defined in terms of the basic unit of mass, length and time. Efforts to retain the calorie as the unit of energy will tend to isolate the nutritional sciences from the advances in the fields of physics and chemistry. The calorie will disappear in future courses of the basic sciences. The calorie as used in nutritional science has also been defined in different ways which lends to confusion: The "15° calorie" is defined as the amount of heat required to increase the temperature of water from 14.5 to 15.5 C with the specific heat of the water at 15° C and the constant pressure being defined as unity. The electrical energy required to increase the temperature of water this 1° has been experimentally determined but with a degree of uncertainty. Values for the "15° calorie", therefore, range from 4.1852 J/cal (15° C) to 4.1858 J/cal (15° C), with a mean value recorded by ISO as 4.1855 J. The nutritional calorie is therefore not the 15° calorie. The International Steam Table Calorie (I T calorie) was defined in 1929 at the International Steam Table Conference in terms of electric units namely 1/860 x international watt hour or joule. Because of differences in terms of absolute joule the USA and European International joule differ one I T equals 4.1868 J in European publications and 4.1867 J in American journals. The "Thermochemical calorie" was defined by Rossini simply as 4.1833 international joules in order to avoid the difficulties associated with uncertainties about the heat capacity of water (it has been redefined as 4.1840 J exactly). The thermochemical calorie actually approximates a "17° calorie" in contrast to the "15° calorie". Since calorimetrical measurements are based on electrical calibration Rossini argued that it would seem logical to calibrate in energy units rather than in thermal units. Another type of calorie, the "Bunsen calorie" also is uncertain about the mean value of the heat capacity of water over the temperature range considered. While the nutritional calorie has not been defined, basically it is the thermochemical calorie. The standards used in calorimetric work in nutrition is ultimately the heat of combustion of an internationally graded standard benzoic acid. This is primarily expressed as joules per gramme mole and secondarily as thermochemical calories per mole derived by dividing by 4.182, a factor which has been approved by the Committee on Nomenclature of the IUNS. Although not generally realized nutritionists actually have been measuring energy in joules and thus applying various factors to convert to the various calories. Calorimeters are also calibrated by measuring the amount of electrical energy that is necessary to duplicate the thermal effect (heat of combustion of an amount of standard benzoic acid). The input in electrical energy is measured in the units, volts × amperes × seconds (or watts × seconds). One watt second = 1 Newton meter = 1 J. The specific energy of food whether these are heat of combustion or metabolizable fuels or other physiologically defined entities should be expressed for convenience in terms of kilojoules per kilogrammes (kJ/kg) or megajoules per kilogramme (MJ × kg), a transition from calories and kilocalories to Joules and kiloJoules. Among these lines more accurate information and improved concepts are substituted for the less accurate and less adequate (2). The Joint FAO/WHO Expert Committee on Nutrition (3) considered the implications of substituting joules for calories in 1966. In view of its importance it was decided to refer this matter to the Committee on Nomenclature of the International Union of Nutritional Sciences (IUNS) and to other appropriate organizations. The conversion from calories to joules was considered by the IUNS Committee at its pre-congress meetings, in Belgrade on 20 to 23 August 1969 and again in Prague on 26 August 1969 (4). It was formally recommended that the change from the calorie to joule should be made, however, such change should be gradual. The conversion factor to be used is debatable; for those engaged in research on energy change involving calorimetry an exact figure of 4.1840 J = 1 calorie is required, whereas for most nutritionists and dietitians a less accurate ratio of 4.19 or even 4.2 kilojoules per kilocalorie will probably suffice. Also a Working Group of the British National Committee for Nutritional Sciences of the British Royal Society (5) agreed that the calorie should fall into disuse and the joule be adopted as the unity for energy in all nutritional work. To convert calories into joules the conversion factor 4.1840 J = 1 calorie is recommended, which also identifies the calorie used by nutritionists with the thermochemical calorie. It should be expressed as MJ/kg, as a matter of convenience. The pronounciation of the unit joul shall be according to the Concise Oxford Dictionary "Jool". The change over will take a few years to be completed. It has already been started in British schools and should be completed in the early 1970's. International agreement, however, on this matter is important. The Committee of Dietary Allowance of the US Food and Nutrition Board of the National Research Council also proposed to express energy allowances as both kilocalorie and kiloJoules in their next revision. Conversion Procedures No doubt the conversion of tables of energy content of foods and of energy allowances and requirements from calories to joules (J) will take time. The Royal Society Conference of Editors proposed that initially journals and other publications should give the nonmetric units plus the metric equivalent in parenthesis, e.g. "55 kcall (230 kJ)" and to reverse the procedure somewhat later as "230 kJ (55 kcal)". Eventually the values will be given only in joules. Initially conversion will be made merely by multiplying the present caloric values by the conversion factor. In the meantime the metabolizable energy values of food can be re-investigated and can be improved. The factors used today to convert food proteins, fats and carboydrates into utilizable calories are based on work done mainly decades ago and different values are also being used in different countries. Conversion Factor The Committee on Nomenclature of the International Union of Nutritional Sciences concluded that most nutritionists use the thermochemical calorie which equals 4.1840 J exactly. An exact number cannot really be rounded off especially by those engaged in research on energy changes involving calorimetry. For most nutritional and dietetic requirements a less accurate conversion figure between the Calorie and the kiloJoule will probably be adequate, such as the figure of 4.19 or even 4.2 kiloJoules per kilocalorie. It was, therefore, recommended that as a first step the conversion factor 4.19 between the calorie and the Joule should be included in the literature whatever unit is mentioned - the change over should be gradual. Physiological Energy Values In calculating the metabolizable energy value of food the future 4 kcal per gramme for carbohydrate and protein and 9 kcal per gramme of fat are used in rough estimates. If such values are converted from kilocalories to kiloJoules rounded off figures of 17 kJ per gramme for carbohydrate and protein and 38 kJ per gramme for fats are suggested. Also converting calorie requirements and allowances into joules, figures may be rounded off as has already been done in the latest revision of the Recommended Intakes of Nutrients for the United Kingdom (6) in which energy allowances are given in both kilocalories and megajoules, for example men from 18 to 35 years, sedentary, weighing 65 kg., 2700 kcal or 11.3 MJ (megajoules) and women 18 to 55 years for most occupations, weighing 55 kg., 2200 kcal or 9.2 MJ. No change is envisaged for terms such as calorimeter, calorimetry and calorific, but "energy value" can be substituted for "calorie value". References (1) Ames, S.R. The Joule - Unit of Energy. J. Amer. Diet. Assoc., 57, 415 (1970). (2) Harper, A.E. Remarks on the Joule. J. Amer. Diet. Assoc., 57, 416 (1970). (3) FAO/WHO Expert Committee on Nutrition. Seventh Report, Rome 12 to 20 December 1966. page 8. Food and Agriculture Organization of the United Nations, Rome, 1967. (4) International Union Nutritional Sciences, Committee on Nomenclature. (1969). The Status of the Joule and the Calorie as Units of Energy. Personal Communication. (5) Working Party, British National Committee on Nutritional Sciences, Royal Society (1969). The Adoption of the Joule as the Unit of Energy and of the SI Units in Nutritional Sciences. Personal Communication. (6) Recommended Intakes of Nutrients for the United Kingdom. (1969). Report of the Panel on Recommended Allowances of Nutrients. London, H.M.S.O., 1969.
2886	https://www.turito.com/learn/math/proportionality-theorems-grade-9	Proportionality Theorem: Detailed Explanation \| Turito Need Help? Get in touch with us 1-646-564-2231 Tutoring 1-on-1 Tutoring Math English Science Coding Robotics College Admission Prep College Admission Prep SAT ACT AP More AI Academy Free resources Blogs Home SAT AP ACT PSAT College guide Score guide Physics Chemistry Biology 1-on-1 Tutoring Coding Robotics Learn Home Maths English Physics Chemistry Biology Ask a doubt +14708451137 ##### Reach us at: 1-646-564-2231 Book a free demo Sign in Are you sure you want to logout? Yes No Please select your grade Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Grade 6 Grade 7 Grade 8 Grade 9 Grade 10 Grade 11 Grade 12 MathsEnglishPhysicsChemistryBiologyScienceEarth and space Maths English Physics Chemistry more.. Biology Science Earth and space MathProportionality Theorem Proportionality Theorem Grade 9 Sep 13, 2022 Key Concepts Proportionality theorem for triangles Use of proportionality theorem for triangles Converse of Proportionality theorem for triangles Use of the converse of proportionality theorem of triangles Different Postulates of Similarity of Triangles Angle-Angle (AA) Similarity Postulate If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. Side Side Side (SSS) Similarity Postulate If the corresponding side lengths of two triangles are proportional, then the triangles are similar. Side Angle Side (SAS) Similarity Postulate If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides, including these angles, are proportional, then the triangles are similar. Triangle Proportionality Theorem If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally. Proof: In the triangle ABC, in which DE is drawn parallel to BC. Point D lies in BA, and E is in BC. Prove ∠BAC = ∠BDE (Corresponding angles) ∠BCA = ∠BED (Corresponding angles) ∠B is common Then by AA similarity postulate Triangle BAC is similar to triangle BDE. (Corresponding sides of similar triangles are proportional) (Subtracting 1 from both sides) Hence proved. In ΔABC, D and E are points on the sides AB and AC, respectively, such that DE \|\| BC If and AC = 15 cm. find AE. (Proportionality theorem for triangles) 15 – AE = 5AE 6AE = 15 AE = 15/6 = 5/2 = 2.5 AE = 2.5 cm Use the proportionality theorem of triangles to find the value of x. DE \|\| BC, D is a point on side AB of triangle ABC, and E is a point on side AC of triangle ABC. 5(9 – x) = 4x 45 – 5x = 4x 45 = 4x + 5x 45 = 9x x = = 5 Converse of Triangle Proportionality Theorem If a line divides two sides of a triangle proportionally, then it is parallel to the third side. Proof A triangle ABC and line DE intersects AB at D and AC at E such that, To Prove: DE is parallel to BC Assume that DE is not parallel to BC. Draw a line DE’ parallel to BC. Then by Basic Proportionality Theorem, DE’ \|\| BC, But Adding 1 on both sides But EC = E’C is possible only if E and E’ coincide. i.e., DE and DE’ are the same line. DE \|\| BC, Hence proved. If D and E are the points on the sides AB and AC of a triangle ABC, respectively, such that AD = 6 cm, BD = 9 cm, AE = 8 cm, and EC = 12 cm. Show that DE is parallel to BC. By the converse proportionality theorem of triangles, DE will be parallel to BC if Therefore, DE is parallel to BC. Theorem If three parallel lines intersect two transversals, then they divide the transversals proportionally. Three lines k 1, k 2, k 3 such that k 1 \|\| k 2 \|\| k 3. Two transversals, t 1 and t 2, intersect at k 1, k 2, k 3 at points A, B, C, D, E, F. To prove CONSTRUCTION: Draw an auxiliary line through the points A and D, and it intersects k 2 at M. Proof Consider the triangle ADF, then (By proportionality theorem of triangles) ———(1) Consider the triangle ADC, then (By proportionality theorem of triangles) ———(2) or (Reciprocal property of proportions) ———–(3) From (1) and (3) Hence proved. Theorem If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides. To prove Given ∠YXW = ∠WXZ CONSTRUCTION Draw a line AZ parallel to the bisector XW. Extend segment XY to meet AZ. Proof Given ∠YXW = ∠WXZ ∠WXZ = ∠XZA (Alternate interior angles are equal since XW \|\| AZ) ∠WXZ = ∠XAZ (Corresponding angles are equal since XW \|\| AZ) Therefore, base angles are equal. Then AX=XZ In triangle YAZ, XW \|\| AZ, then by proportionality theorem of triangles, But AX= XZ Hence proved. Find the value of x. If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides. x= 10 Exercise Use proportionality theorem for finding the missing length. Solve for x using proportionality theorem for triangles. Find the missing length. ABCD is a trapezium with AB parallel to DC, P and Q are points on non-parallel sides AD and BC, respectively, such that PQ parallel to AB Show that. n the given figure shows quadrilateral ABCD in which AB is parallel to DC, OA = 3x – 19, OB = x – 3, OC = x – 5 and OD = 3 find the value of x. If the bisector of an angle of a triangle bisects the opposite side, prove that the triangle is isosceles. In the given figure PS/SQ = PT/TR and PST = PRQ. Prove that PQR is an isosceles triangle. In the given figure, XY is parallel to MN if a. LX = 4cm, XM = 6cm and LN = 12.5 cm. find LY b. LX : XM = 3:5 and LY = 3.6 cm. Find LN Concept Map What we have learnt Proportionality Theorem for Triangles If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally. Converse of Proportionality Theorem for Triangles If a line divides two sides of a triangle proportionally, then it is parallel to the third side. Theorem If three parallel lines intersect two transversals, then they divide the transversals proportionally. Theorem based on angle bisection If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides. Comments: Submit Related topics Addition and Multiplication Using Counters & Bar-Diagrams Introduction: We can find the solution to the word problem by solving it. Here, in this topic, we can use 3 methods to find the solution. 1. Add using counters 2. Use factors to get the product 3. Write equations to find the unknown. Addition Equation: 8+8+8 =? Multiplication equation: 3×8=? Example 1: Andrew has […] Read More >> Dilation: Definitions, Characteristics, and Similarities Understanding Dilation A dilation is a transformation that produces an image that is of the same shape and different sizes. Dilation that creates a larger image is called enlargement. Describing Dilation Dilation of Scale Factor 2 The following figure undergoes a dilation with a scale factor of 2 giving an image A’ (2, 4), B’ […] Read More >> How to Write and Interpret Numerical Expressions? Write numerical expressions What is the Meaning of Numerical Expression? A numerical expression is a combination of numbers and integers using basic operations such as addition, subtraction, multiplication, or division. The word PEMDAS stands for: P → Parentheses E → Exponents M → Multiplication D → Division A → Addition S → Subtraction Some examples […] Read More >> System of Linear Inequalities and Equations Introduction: Systems of Linear Inequalities: A system of linear inequalities is a set of two or more linear inequalities in the same variables. The following example illustrates this, y < x + 2…………..Inequality 1 y ≥ 2x − 1…………Inequality 2 Solution of a System of Linear Inequalities: A solution of a system of linear inequalities […] Read More >> Other topics #### How to Find the Area of Rectangle? Mar 3, 2022 #### How to Solve Right Triangles? 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2887	https://ocw.mit.edu/courses/18-01sc-single-variable-calculus-fall-2010/resources/clip-1-slope-of-tangent-to-circle-direct/	Clip 1: Slope of Tangent to Circle: Direct \| Single Variable Calculus \| Mathematics \| MIT OpenCourseWare Browse Course Material Syllabus 1. Differentiation Part A: Definition and Basic Rules Part B: Implicit Differentiation and Inverse Functions Exam 1 2. Applications of Differentiation Part A: Approximation and Curve Sketching Part B: Optimization, Related Rates and Newton's Method Part C: Mean Value Theorem, Antiderivatives and Differential Equa Exam 2 3. The Definite Integral and its Applications Part A: Definition of the Definite Integral and First Fundamental Part B: Second Fundamental Theorem, Areas, Volumes Part C: Average Value, Probability and Numerical Integration Exam 3 4. Techniques of Integration Part A: Trigonometric Powers, Trigonometric Substitution and Com Part B: Partial Fractions, Integration by Parts, Arc Length, and Part C: Parametric Equations and Polar Coordinates Exam 4 5. Exploring the Infinite Part A: L'Hospital's Rule and Improper Integrals Part B: Taylor Series Final Exam Course Info Instructor Prof. David Jerison Departments Mathematics As Taught In Fall 2010 Level Undergraduate Topics Mathematics Calculus Differential Equations Learning Resource Types grading Exams with Solutions notes Lecture Notes theaters Lecture Videos assignment_turned_in Problem Sets with Solutions laptop_windows Simulations theaters Problem-solving Videos Download Course menu search Give Now About OCW Help & Faqs Contact Us searchGIVE NOWabout ocwhelp & faqscontact us 18.01SC \| Fall 2010 \| Undergraduate Single Variable Calculus Menu More Info Syllabus 1. Differentiation Part A: Definition and Basic Rules Part B: Implicit Differentiation and Inverse Functions Exam 1 2. Applications of Differentiation Part A: Approximation and Curve Sketching Part B: Optimization, Related Rates and Newton's Method Part C: Mean Value Theorem, Antiderivatives and Differential Equa Exam 2 3. The Definite Integral and its Applications Part A: Definition of the Definite Integral and First Fundamental Part B: Second Fundamental Theorem, Areas, Volumes Part C: Average Value, Probability and Numerical Integration Exam 3 4. Techniques of Integration Part A: Trigonometric Powers, Trigonometric Substitution and Com Part B: Partial Fractions, Integration by Parts, Arc Length, and Part C: Parametric Equations and Polar Coordinates Exam 4 5. Exploring the Infinite Part A: L'Hospital's Rule and Improper Integrals Part B: Taylor Series Final Exam Session 14: Examples of Implicit Differentiation Clip 1: Slope of Tangent to Circle: Direct » Accompanying Notes (PDF) From Lecture 5 of 18.01 Single Variable Calculus, Fall 2006 Video Player is loading. Play Video Play Mute Current Time 0:00 / Duration 49:01 Loaded: 0.00% 08:57 Stream Type LIVE Seek to live, currently behind live LIVE Remaining Time-40:04 1x Playback Rate Chapters Chapters Descriptions descriptions off, selected Captions captions settings, opens captions settings dialog captions off, selected English Captions Audio Track Picture-in-Picture download button Download video Fullscreen playback speed Playback speed 0.25 0.5 0.75 1, selected 1.25 1.5 This is a modal window. Beginning of dialog window. Escape will cancel and close the window. 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Transcript Download video Download transcript 0:00 The following content was created under a Creative 0:02 Commons License. 0:03 Your support will help MIT OpenCourseWare 0:05 continue to offer high quality educational resources for free. 0:09 To make a donation or to view additional materials 0:12 from hundreds of MIT courses, visit MIT OpenCourseWare 0:16 at ocw.mit.edu. 0:19 Professor: So, we're ready to begin the fifth lecture. 0:24 I'm glad to be back. 0:25 Thank you for entertaining my colleague, Haynes Miller. 0:33 So, today we're going to continue 0:35 where he started, namely what he talked about was the chain 0:40 rule, which is probably the most powerful technique 0:43 for extending the kinds of functions 0:45 that you can differentiate. 0:47 And we're going to use the chain rule in some rather clever 0:50 algebraic ways today. 0:54 So the topic for today is what's known 0:57 as implicit differentiation. 1:10 So implicit differentiation is a technique 1:15 that allows you to differentiate a lot of functions you didn't 1:17 even know how to find before. 1:20 And it's a technique - let's wait for a few people 1:24 to sit down here. 1:28 Physics, huh? 1:29 Okay, more Physics. 1:36 Let's take a break. 1:37 You can get those after class. 1:40 All right, so we're talking about implicit differentiation, 1:46 and I'm going to illustrate it by several examples. 1:53 So this is one of the most important and basic formulas 1:57 that we've already covered part way. 1:59 Namely, the derivative of x to a power is ax^(a-1). 2:06 Now, what we've got so far is the exponents, 0, plus or minus 2:15 1, plus or minus 2, etc. 2:19 You did the positive integer powers in the first lecture, 2:24 and then yesterday Professor Miller 2:30 told you about the negative powers. 2:32 So what we're going to do right now, 2:35 today, is we're going to consider 2:39 the exponents which are rational numbers, ratios of integers. 2:44 So a is m/n. 2:46 m and n are integers. 2:53 All right, so that's our goal for right now, 2:55 and we're going to use this method 2:56 of implicit differentiation. 2:58 In particular, it's important to realize that this 3:01 covers the case m = 1. 3:03 And those are the nth roots. 3:04 So when we take the one over n power, 3:07 we're going to cover that right now, 3:09 along with many other examples. 3:13 So this is our first example. 3:16 So how do we get started? 3:17 Well we just write down a formula for the function. 3:20 The function is y = x^(m/n). 3:24 That's what we're trying to deal with. 3:26 And now there's really only two steps. 3:30 The first step is to take this equation to the nth power, 3:38 so write it y^n = x^m. 3:42 Alright, so that's just the same equation re-written. 3:46 And now, what we're going to do is 3:50 we're going to differentiate. 3:52 So we're going to apply d/dx to the equation. 4:01 Now why is it that we can apply it to the second equation, not 4:05 the first equation? 4:06 So maybe I should call these equation 1 and equation 2. 4:10 So, the point is, we can apply it to equation 2. 4:13 Now, the reason is that we don't know how to differentiate 4:17 x^(m/n). 4:18 That's something we just don't know yet. 4:21 But we do know how to differentiate integer powers. 4:24 Those are the things that we took care of before. 4:29 So now we're in shape to be able to do the differentiation. 4:32 So I'm going to write it out explicitly 4:34 over here, without carrying it out just yet. 4:37 That's d/dx of y^n = d/dx of x^m. 4:46 And now you see this expression here 4:51 requires us to do something we couldn't do before yesterday. 4:55 Namely, this y is a function of x. 4:58 So we have to apply the chain rule here. 5:01 So this is the same as - this is by the chain rule now - 5:06 d/dy of y^n times dy/dx. 5:13 And then, on the right hand side, we can just carry it out. 5:15 We know the formula. 5:17 It's mx^(m-1). 5:21 Right, now this is our scheme. 5:24 And you'll see in a minute why we win with this. 5:29 So, first of all, there are two factors here. 5:32 One of them is unknown. 5:33 In fact, it's what we're looking for. 5:35 But the other one is going to be a known quantity, 5:38 because we know how to differentiate y 5:40 to the n with respect to y. 5:42 That's the same formula, although the letter 5:44 has been changed. 5:46 And so this is the same as - I'll write it underneath here - 5:53 n y^(n-1) dy/dx = m x^(m-1). 6:07 Okay, now comes, if you like, the non-calculus part 6:14 of the problem. 6:15 Remember the non-calculus part of the problem 6:17 is always the messier part of the problem. 6:20 So we want to figure out this formula. 6:22 This formula, the answer over here, 6:25 which maybe I'll put in a box now, 6:29 has this expressed much more simply, only in terms of x. 6:33 And what we have to do now is just solve for dy/dx 6:36 using algebra, and then solve all the way in terms of x. 6:39 So, first of all, we solve for dy/dx. 6:44 So I do that by dividing the factor on the left-hand side. 6:47 So I get here mx^(m-1) divided by ny^(n-1). 6:56 And now I'm going to plug in-- so I'll write this as m/n. 7:02 This is x^(m-1). 7:04 Now over here I'm going to put in for y, x^(m/n) times n-1. 7:15 So now we're almost done, but unfortunately we 7:18 have this mess of exponents that we have to work out. 7:22 I'm going to write it one more time. 7:25 So I already recognize the factor a out front. 7:28 That's not going to be a problem for me, 7:30 and that's what I'm aiming for here. 7:31 But now I have to encode all of these powers, 7:34 so let's just write it. 7:36 It's m-1, and then it's minus the quantity (n-1) m/n. 7:46 All right, so that's the law of exponents applied to this ratio 7:50 here. 7:50 And then we'll do the arithmetic over here on the next board. 7:58 So we have here m - 1 - (n-1) m/n = m - 1. 8:08 And if I multiply n by this, I get -m. 8:12 And if the second factor is minus minus, that's a plus. 8:15 And that's +m/n. 8:18 Altogether the two m's cancel. 8:21 I have here -1 + m/n. 8:23 And lo and behold that's the same thing as a - 1, 8:27 just what we wanted. 8:29 All right, so this equals a x^(n-1). 8:31 Okay, again just a bunch of arithmetic. 8:39 From this point forward, from this substitution 8:42 on, it's just the arithmetic of exponents. 8:51 All right, so we've done our first example here. 8:58 I want to give you a couple more examples, 9:00 so let's just continue. 9:04 The next example I'll keep relatively simple. 9:08 So we have example two, which is going to be the function x^2 + 9:15 y^2 = 1. 9:18 Well, that's not really a function. 9:21 It's a way of defining y as a function of x implicitly. 9:29 There's the idea that I could solve for y if I wanted to. 9:34 And indeed let's do that. 9:36 So if you solve for y here, what happens is you get y^2 = 1 - 9:42 x^2, and y is equal to plus or minus the square root of 1 - 9:47 x^2. 9:52 So this, if you like, is the implicit definition. 9:58 And here is the explicit function y, 10:00 which is a function of x. 10:04 And now just for my own convenience, 10:06 I'm just going to take the positive branch. 10:09 This is the function. 10:13 It's just really a circle in disguise. 10:15 And I'm just going to take the top part of the circle, 10:19 so we'll take that top hump here. 10:24 All right, so that means I'm erasing this minus sign. 10:27 I'm just taking the positive branch, just 10:35 for my convenience. 10:36 I could do it just as well with the negative branch. 10:40 Alright, so now I've taken the solution, 10:46 and I can differentiate with this. 10:49 So rather than using the dy/dx notation over here, 10:53 I'm going to switch notations over here, 10:55 because it's less writing. 10:56 I'm going to write y' and change notations. 10:59 Okay, so I want to take the derivative of this. 11:04 Well this is a somewhat complicated function here. 11:11 It's the square root of 1 - x^2, and the right way always 11:15 to look at functions like this is to rewrite them using 11:21 the fractional power notation. 11:26 That's the first step in computing 11:28 a derivative of a square root. 11:32 And then the second step here is what? 11:38 Does somebody want to tell me? 11:40 Chain rule, right. 11:43 That's it. 11:44 So we have two things. 11:45 We start with one, and then we do something else to it. 11:47 So whenever we do two things to something, 11:50 we need to apply the chain rule. 11:52 So 1 - x^2, square root. 11:55 All right, so how do we do that? 11:57 Well, the first factor I claim is 11:58 the derivative of this thing. 12:01 So this is 1/2 blah to the -1/2. 12:06 So I'm doing this kind of by the advanced method 12:09 now, because we've already graduated. 12:11 You already did the chain rule last time. 12:14 So what does this mean? 12:15 This is an abbreviation for the derivative with respect 12:20 to blah of blah ^ 1/2, whatever it is. 12:27 All right, so that's the first factor that we're going to use. 12:30 Rather than actually write out a variable for it 12:34 and pass through as I did previously 12:36 with this y and x variable here, I'm 12:39 just going to skip that step and let 12:41 you imagine it as being a placeholder for that variable 12:45 here. 12:45 So this variable is now parenthesis. 12:48 And then I have to multiply that by the rate of change of what's 12:52 inside with respect to x. 12:55 And that is going to be -2x. 12:58 The derivative of 1 - x^2 is -2x. 13:02 And now again, we couldn't have done this example two 13:09 before example one, because we needed 13:11 to know that the power rule worked not just for a integer 13:17 but also for a = 1/2. 13:19 We're using the case a = 1/2 right here. 13:22 It's 1/2 times, and this -1/2 here is a-1. - 13:29 So this is the case a = 1/2. 13:33 a-1 happens to be -1/2. 13:39 Okay, so I'm putting all those things together. 13:41 And you know within a week you have 13:44 to be doing this very automatically. 13:45 So we're going to do it at this speed now. 13:47 You want to do it even faster, ultimately. 13:49 Yes? 13:50 Student: [INAUDIBLE] 13:53 Professor: The question is could I have done it implicitly 13:56 without the square roots. 13:58 And the answer is yes. 13:59 That's what I'm about to do. 14:02 So this is an illustration of what's 14:04 called the explicit solution. 14:07 So this guy is what's called explicit. 14:13 And I want to contrast it with the method 14:17 that we're going to now use today. 14:18 So it involves a lot of complications. 14:20 It involves the chain rule. 14:21 And as we'll see it can get messier and messier. 14:23 And then there's the implicit method, 14:27 which I claim is easier. 14:29 So let's see what happens if you do it implicitly 14:36 The implicit method involves, instead of writing 14:41 the function in this relatively complicated way, 14:43 with the square root, it involves leaving it alone. 14:47 Don't do anything to it. 14:50 In this previous case, we were left with something which was 14:52 complicated, say x^(1/3) or x^(1/2) or something 14:56 complicated. 14:57 We had to simplify it. 14:59 We had an equation one, which was more complicated. 15:01 We simplified it then differentiated it. 15:03 And so that was a simpler case. 15:05 Well here, the simplest thing us to differentiate 15:09 is the one we started with, because squares are practically 15:13 the easiest thing after first powers, or maybe zeroth powers 15:16 to differentiate. 15:18 So we're leaving it alone. 15:19 This is the simplest possible form for it, 15:21 and now we're going to differentiate. 15:23 So what happens? 15:24 So again what's the method? 15:26 Let me remind you. 15:27 You're applying d/dx to the equation. 15:30 So you have to differentiate the left side of the equation, 15:33 and differentiate the right side of the equation. 15:35 So it's this, and what you get is 2x + 2yy' is equal to what? 15:51 0. 15:52 The derivative of 1 0. 15:56 So this is the chain rule again. 15:58 I did it a different way. 16:00 I'm trying to get you used to many different notations 16:02 at once. 16:04 Well really just two. 16:05 Just the prime notation and the dy/dx notation. 16:10 And this is what I get. 16:14 So now all I have to do is solve for y'. 16:19 So that y', if I put the 2x on the other side, is -2x, 16:24 and then divide by 2y, which is -x/y. 16:30 So let's compare our solutions, and I'll apologize, 16:34 I'm going to have to erase something to do that. 16:39 So let's compare our two solutions. 16:44 I'm going to put this underneath and simplify. 16:46 So what was our solution over here? 16:48 It was 1/2(1-x^2)^(-1/2) (-2x). 16:56 That was what we got over here. 17:02 And that is the same thing, if I cancel the 2's, and I change it 17:06 back to looking like a square root, 17:08 that's the same thing as -x divided by square root of 1 - 17:11 x^2. 17:13 So this is the formula for the derivative 17:18 when I do it the explicit way. 17:21 And I'll just compare them, these two expressions here. 17:29 And notice they are the same. 17:32 They're the same, because y is equal to square root of 1 - 17:37 x^2. 17:40 Yeah? 17:40 Question? 17:41 Student: [INAUDIBLE] 17:46 Professor: The question is why did the implicit method 17:48 not give the bottom half of the circle? 17:50 Very good question. 17:53 The answer to that is that it did. 17:57 I just didn't mention it. 17:59 Wait, I'll explain. 18:00 So suppose I stuck in a minus sign here. 18:05 I would have gotten this with the difference, so 18:08 with an extra minus sign. 18:10 But then when I compared it to what was over there, 18:12 I would have had to have another different minus sign over here. 18:15 So actually both places would get an extra minus sign. 18:19 And they would still coincide. 18:20 So actually the implicit method is a little better. 18:22 It doesn't even notice the difference 18:23 between the branches. 18:24 It does the job on both the top and bottom half. 18:28 Another way of saying that is that you're 18:31 calculating the slopes here. 18:33 So let's look at this picture. 18:35 Here's a slope. 18:36 Let's just take a look at a positive value 18:39 of x and just check the sign to see what's happening. 18:42 If you take a positive value of x over here, x is positive. 18:46 This denominator is positive. 18:48 The slope is negative. 18:49 You can see that it's tilting down. 18:52 So it's okay. 18:53 Now on the bottom side, it's going to be tilting up. 18:59 And similarly what's happening up here 19:01 is that both x and y are positive, and this x and this y 19:05 are positive. 19:06 And the slope is negative. 19:07 On the other hand, on the bottom side, x is still positive, 19:10 but y is negative. 19:11 And it's tilting up because the denominator is negative. 19:15 The numerator is positive, and this minus sign 19:17 has a positive slope. 19:19 So it matches perfectly in every category. 19:23 This complicated, however, and it's easier 19:26 just to keep track of one branch at a time, 19:30 even in advanced math. 19:32 Okay, so we only do it one branch at a time. 19:37 Other questions? 19:43 Okay, so now I want to give you a slightly more 19:47 complicated example here. 19:49 And indeed some of the-- so here's 19:52 a little more complicated example. 19:54 It's not going to be the most complicated example, 19:56 but you know it'll be a little tricky. 20:17 So this example, I'm going to give you a fourth order 20:22 equation. 20:23 So y^4 + xy^2 - 2 = 0. 20:31 Now it just so happens that there's 20:35 a trick to solving this equation, 20:38 so actually you can do both the explicit method 20:41 and the non-explicit method. 20:46 So the explicit method would say okay well, 20:50 I want to solve for this. 20:51 So I'm going to use the quadratic formula, but on y^2. 20:55 This is quadratic in y^2, because there's a fourth power 20:59 and a second power, and the first and third powers are 21:02 missing. 21:03 So this is y^2 is equal to -x plus or minus the square root 21:09 of x^2 - 4(-2) divided by 2. 21:19 And so this x is the b. 21:22 This -2 is the c, and a = 1 in the quadratic formula. 21:29 And so the formula for y is plus or minus the square root of -x 21:37 plus or minus the square root x^2 + 8 divided by 2. 21:45 So now you can see this problem of branches, 21:47 this happens actually in a lot of cases, 21:50 coming up in an elaborate way. 21:53 You have two choices for the sign here. 21:54 You have two choices for the sign here. 21:56 Conceivably as many as four roots for this equation, 21:59 because it's a fourth degree equation. 22:02 It's quite a mess. 22:02 You should have to check each branch separately. 22:06 And this really is that level of complexity, 22:09 and in general it's very difficult 22:11 to figure out the formulas for quartic equations. 22:17 But fortunately we're never going to use them. 22:21 That is, we're never going to need those formulas. 22:24 So the implicit method is far easier. 22:31 The implicit method just says okay I'll 22:35 leave the equation in its simplest form. 22:38 And now differentiate. 22:40 So when I differentiate, I get 4y^3 y' plus - 22:47 now here I have to apply the product rule. 22:50 So I differentiate the x and the y^2 separately. 22:56 First I differentiate with respect to x, so I get y^2. 22:59 Then I differentiate with respect to the other factor, 23:03 the y^2 factor. 23:04 And I get x(2 y y'). 23:08 And then the 0 gives me 0. 23:10 So minus 0 equals 0. 23:16 So there's the implicit differentiation step. 23:21 And now I just want to solve for y'. 23:26 So I'm going to factor out 4y^3 + 2xy. 23:32 That's the factor on y'. 23:35 And I'm going to put the y^2 on the other side. 23:39-y^2 over here. 23:43 And so the formula for y' is -y^2 divided by 4y^3 + 2xy. 23:55 So that's the formula for the solution. 24:01 For the slope. 24:06 You have a question? 24:07 Student: [INAUDIBLE] 24:16 Professor: So the question is for the y would 24:18 we have to put in what solved for in the explicit equation. 24:22 And the answer is absolutely yes. 24:24 That's exactly the point. 24:25 So this is not a complete solution to a problem. 24:30 We started with an implicit equation. 24:32 We differentiated. 24:33 And we got in the end, also an implicit equation. 24:36 It doesn't tell us what y is as a function of x. 24:39 You have to go back to this formula 24:43 to get the formula for x. 24:45 So for example, let me give you an example here. 24:49 So this hides a degree of complexity of the problem. 24:54 But it's a degree of complexity that we must live with. 24:58 So for example, at x = 1, you can see that y = 1 solves. 25:10 That happens to be-- solves y^4 + xy^2 - 2 = 0. 25:16 That's why I picked the 2 actually, 25:18 so it would be 1 + 1 - 2 = 0. 25:21 I just wanted to have a convenient solution there 25:23 to pull out of my hat at this point. 25:25 So I did that. 25:26 And so we now know that when x = 1, y = 1. 25:30 So at (1, 1) along the curve, the slope is equal to what? 25:41 Well, I have to plug in here, -1^2 / (41^3 + 211). 25:52 That's just plugging in that formula over there, 25:54 which turns out to be -1/6. 25:59 So I can get it. 26:00 On the other hand, at say x = 2, we're 26:13 stuck using this formula star here to find y. 26:32 Now, so let me just make two points 26:37 about this, which are just philosophical points for you 26:40 right now. 26:42 The first is, when I promised you 26:45 at the beginning of this class that we 26:47 were going to be able to differentiate 26:48 any function you know, I meant it very literally. 26:53 What I meant is if you know the function, 26:56 we'll be able give a formula for the derivative. 26:58 If you don't know how to find a function, 27:00 you'll have a lot of trouble finding the derivative. 27:02 So we didn't make any promises that if you 27:05 can't find the function you will be 27:06 able to find the derivative by some magic. 27:09 That will never happen. 27:10 And however complex the function is, 27:12 a root of a fourth degree polynomial 27:16 can be pretty complicated function of the coefficients, 27:20 we're stuck with this degree of complexity in the problem. 27:23 But the big advantage of his method, notice, 27:27 is that although we've had to find star, 27:29 we had to find this formula star, 27:31 and there are many other ways of doing these things numerically, 27:34 by the way, which we'll learn later, 27:36 so there's a good method for doing it numerically. 27:39 Although we had to find star, we never had to differentiate it. 27:43 We had a fast way of getting the slope. 27:46 So we had to know what x and y were. 27:48 But y' we got by an algebraic formula, 27:50 in terms of the values here. 27:54 So this is very fast, forgetting the slope, 27:57 once you know the point. yes? 28:02 Student: What's in the parentheses? 28:03 Professor: Sorry, this is-- Well let's see if I can manage this. 28:06 Is this the parentheses you're talking about? 28:16 Ah, "say". 28:17 That says "say". 28:17 Well, so maybe I should put commas around it. 28:19 But it was S A Y, comma comma, okay? 28:24 Well here was at x = 1. 28:28 I'm just throwing out a value. 28:33 Any other value. 28:34 Actually there is one value, my favorite value. 28:36 Well this is easy to evaluate right? x = 0, 28:39 I can do it there. 28:42 That's maybe the only one. 28:45 The others are a nuisance. 28:55 All right, other questions? 29:03 Now we have to do something more here. 29:06 So I claimed to you that we could differentiate 29:10 all the functions we know. 29:11 But really we can learn a tremendous 29:13 about functions which are really hard to get at. 29:17 So this implicit differentiation method 29:20 has one very, very important application 29:30 to finding inverse functions, or finding derivatives 29:38 of inverse functions. 29:40 So let's talk about that next. 29:51 So first, maybe we'll just illustrate by an example. 29:55 If you have the function y is equal to square root x, 29:59 for x positive, then of course this idea 30:04 is that we should simplify this equation 30:07 and we should square it so we get this somewhat simpler 30:10 equation here. 30:11 And then we have a notation for this. 30:14 If we call f(x) equal to square root of x, and g(y) = x, 30:21 this is the reversal of this. 30:25 Then the formula for g(y) is that it should be y^2. 30:33 And in general, if we start with any old y = f(x), 30:48 and we just write down, this is the defining relationship 30:52 for a function g, the property that we're saying is that 30:57 g(f(x)) has got to bring us back to x. 31:01 And we write that in a couple of different ways. 31:04 We call g the inverse of f. 31:08 And also we call f the inverse of g, 31:13 although I'm going to be silent about which variable 31:15 I want to use, because people mix them up a little bit, 31:19 as we'll be doing when we draw some pictures of this. 31:31 So let's see. 31:32 Let's draw pictures of both f and f inverse 31:42 on the same graph. 31:50 So first of all, I'm going to draw the graph of f(x) 32:02= square root of x. 32:06 That's some shape like this. 32:11 And now, in order to understand what g(y) is, 32:16 so let's do the analysis in general, 32:20 but then we'll draw it in this particular case. 32:23 If you have g(y) = x, that's really 32:31 just the same equation right? 32:34 This is the equation g(y) = x, that's y^2 = x. 32:37 This is y = square root of x, those are the same equations, 32:40 it's the same curve. 32:43 But suppose now that we wanted to write down what g(x) is. 32:49 In other words, we wanted to switch the variables, 32:51 so draw them as I said on the same graph with the same x, 32:55 and the same y axes. 32:59 Then that would be, in effect, trading the roles of x and y. 33:04 We have to rename every point on the graph which 33:07 is the ordered pair (x, y), and trade it for the opposite one. 33:12 And when you exchange x and y, so 33:15 to do this, exchange x and y, and when 33:23 you do that, graphically what that looks 33:27 like is the following: suppose you have a place here, 33:30 and this is the x and this is the y, 33:33 then you want to trade them. 33:35 So you want the y here right? 33:39 And the x up there. 33:41 It's sort of the opposite place over there. 33:44 And that is the place which is directly opposite this point 33:51 across the diagonal line x = y. 33:55 So you reflect across this or you flip across that. 33:58 You get this other shape that looks like that. 34:01 Maybe I'll draw it with a colored piece of chalk here. 34:10 So this guy here is y = f^(-1)(x). 34:24 And indeed, if you look at these graphs, 34:26 this one is the square root. 34:27 This one happens to be y = x^2. 34:34 If you take this one, and you turn it, 34:36 you reverse the roles of the x axis and the y axis, 34:39 and tilt it on its side. 34:43 So that's the picture of what an inverse function is, and now I 34:51 want to show you that the method of implicit differentiation 34:56 allows us to compute the derivatives 34:59 of inverse functions. 35:03 So let me just say it in general, 35:05 and then I'll carry it out in particular. 35:07 So implicit differentiation allows 35:16 us to find the derivative of any inverse function, 35:32 provided we know the derivative of the function. 35:53 So let's do that for what is an example, which 35:58 is truly complicated and a little subtle here. 36:02 It has a very pretty answer. 36:04 So we'll carry out an example here, 36:09 which is the function y is equal to the inverse tangent. 36:19 So again, for the inverse tangent 36:25 all of the things that we're going to do 36:30 are going to be based on simplifying 36:32 this equation by taking the tangent of both sides. 36:36 So, us let me remind you by the way, 36:38 the inverse tangent is what's also known as arctangent. 36:41 That's just another notation for the same thing. 36:45 And what we're going to use to describe 36:49 this function is the equation tan y = x. 36:55 That's what happens when you take 36:56 the tangent of this function. 36:59 This is how we're going to figure out 37:01 what the function looks like. 37:19 So first of all, I want to draw it, 37:23 and then we'll do the computation. 37:26 So let's make the diagram first. 37:32 So I want to do something which is 37:33 analogous to what I did over here with the square root 37:35 function. 37:38 So first of all, I remind you that the tangent function 37:43 is defined between two values here, which are pi/2 and -pi/2. 37:52 And it starts out at minus infinity 37:55 and curves up like this. 37:58 So that's the function tan x. 38:08 And so the one that we have to sketch 38:11 is this one which we get by reflecting this 38:14 across the axis. 38:21 Well not the axis, the diagonal. 38:25 This slope by the way, should be less - a little lower here so 38:33 that we can have it going down and up. 38:37 So let me show you what it looks like. 38:42 On the front, it's going to look a lot like this one. 38:44 So this one had curved down, and so the reflection 38:50 across the diagonal curved up. 38:52 Here this is curving up, so the reflection 38:54 is going to curve down. 38:56 It's going to look like this. 38:58 Maybe I should, sorry, let's use a different color, 39:02 because it's reversed from before. 39:04 I'll just call it green. 39:10 Now, the original curve in the first quadrant 39:15 eventually had an asymptote which was straight up. 39:17 So this one is going to have an asymptote which is horizontal. 39:24 And that level is what? 39:27 What's the highest? 39:29 It is just pi/2. 39:33 Now similarly, the other way, we're 39:35 going to do this: and this bottom level 39:40 is going to be -pi/2. 39:42 So there's the picture of this function. 39:47 It's defined for all x. 39:50 So this green guy is y = arctan x. 39:57 And it's defined all the way from minus infinity 39:59 to infinity. 40:05 And to use a notation that we had from limit notation 40:11 as x goes to infinity, let's say, x is equal to pi/2. 40:21 That's an example of one value that's of interest in addition 40:24 to the finite values. 40:28 Okay, so now the first ingredient 40:31 that we're going to need, is we're 40:34 going to need the derivative of the tangent function. 40:37 So I'm going to recall for you, and maybe you 40:40 haven't worked this out yet, but I hope that many of you have, 40:43 that if you take the derivative with respect to y of tan y. 40:48 So this you do by the quotient rule. 40:55 So this is of the form u/v, right? 40:59 You use the quotient rule. 41:00 So I'm going to get this. 41:06 But what you get in the end is some marvelous simplification 41:09 that comes out to cos^2 y. 41:12 1 over cosine squared. 41:14 You can recognize the cosine squared from the fact that you 41:17 should get v^2 in the denominator, 41:19 and somehow the numerators all cancel and simplifies to 1. 41:26 This is also known as secant squared y. 41:32 So that something that if you haven't done yet, 41:38 you're going to have to do this as an exercise. 41:48 So we need that ingredient, and now we're 41:50 just going to differentiate our equation. 41:59 And what do we get? 42:00 We get, again, (d/dy tan y) times dy/dx is equal to 1. 42:15 Or, if you like, 1 / cos^2 y times, in the other notation, 42:22 y', is equal to 1. 42:30 So I've just used the formulas that I just wrote down there. 42:35 Now all I have to do is solve for y'. 42:37 It's cos^2 y. 42:44 Unfortunately, this is not the form 42:47 that we ever want to leave these things in. 42:49 This is the same problem we had with that ugly square root 42:52 expression, or with any of the others. 42:54 We want to rewrite in terms of x. 42:58 Our original question was what is d/dx of arctan x. 43:05 Now so far we have the following answer to that question: 43:08 it's cos^2 (arctan x). 43:15 Now this is a correct answer, but way too complicated. 43:31 Now that doesn't mean that if you 43:33 took a random collection of functions, 43:35 you wouldn't end up with something this complicated. 43:37 But these particular functions, these beautiful circular 43:41 functions involved with trigonometry all 43:42 have very nice formulas associated with them. 43:45 And this simplifies tremendously. 43:48 So one of the skills that you need 43:50 to develop when you're dealing with trig functions 43:54 is to simplify this. 43:56 And so let's see now that expressions like this all 44:03 simplify. 44:07 So here we go. 44:10 There's only one formula, one ingredient 44:12 that we need to use to do this, and then we're 44:14 going to draw a diagram. 44:15 So the ingredient again, is the original defining relationship 44:18 that tan y = x. 44:22 So tan y = x can be encoded in a right triangle 44:27 in the following way: here's the right triangle and tan 44:34 y means that y should be represented as an angle. 44:38 And then, its tangent is the ratio 44:40 of this vertical to this horizontal side. 44:43 So I'm just going to pick two values that work, 44:46 namely x and 1. 44:48 Those are the simplest ones. 44:51 So I've encoded this equation in this picture. 44:57 And now all I have to do is figure out what the cosine of y 45:01 is in this right triangle here. 45:03 In order to do that, I need to figure out what the hypotenuse 45:06 is, but that's just square root of 1 + x^2. 45:13 And now I can read off what the cosine of y is. 45:18 So the cosine of y is 1 divided by the hypotenuse. 45:23 So it's 1 over square root, whoops, yeah, 1 + x^2. 45:32 And so cosine squared is just 1 / 1 + x^2. 45:39 And so our answer over here, the preferred answer which is way 45:43 simpler than what I wrote up there, 45:45 is that d/dx of tan inverse x is equal to 1 over 1 + x^2. 46:04 Maybe I'll stop here for one more question. 46:06 I have one more calculation which I can do even 46:10 in less than a minute. 46:11 So we have a whole minute for questions. 46:16 Yeah? 46:20 Student: [INAUDIBLE] 46:26 Professor: What happens to the inverse tangent? 46:34 The inverse tangent-- Okay, this inverse tangent 46:41 is the same as this y here. 46:44 Those are the same thing. 46:46 So what I did was I skipped this step here entirely. 46:50 I never wrote that down. 46:52 But the inverse tangent was that y. 46:54 The issue was what's a good formula 46:56 for cos y in terms of x? 47:01 So I am evaluating that, but I'm doing it using the letter y. 47:04 So in other words, what happened to the inverse 47:06 tangent is that I called it y, which 47:10 is what it's been all along. 47:15 Okay, so now I'm going to do the case 47:17 of the sine, the inverse sine. 47:20 And I'll show you how easy this is 47:22 if I don't fuss with-- because this one has an easy trig 47:27 identity associated with it. 47:29 So if y = sin^(-1) x, and sin y = x, 47:37 and now watch how simple it is when I do the differentiation. 47:40 I just differentiate. 47:42 I get (cos y) y' = 1. 47:50 And then, y', so that implies that = 1 / cos y, 48:00 and now to rewrite that in terms of x, 48:03 I have to just recognize that this is the same as this, 48:10 which is the same as 1 / square root of 1 - x^2. 48:14 So all told, the derivative with respect to x of the arcsine 48:19 function is 1 / square root of 1 - x^2. 48:30 So these implicit differentiations 48:32 are very convenient. 48:34 However, I warn you that you do have 48:38 to be careful about the range of applicability of these things. 48:42 You have to draw a picture like this one 48:44 to make sure you know where this makes sense. 48:47 In other words, you have to pick a branch for the sine function 48:50 to work that out, and there's something 48:52 like that on your problem set. 48:53 And it's also discussed in your text. 48:56 So we'll stop here. 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2888	https://www.hpmuseum.org/cgi-bin/archv019.cgi?read=171248	The Museum of HP Calculators HP Forum Archive 19 [ Return to Index \| Top of Index ] \| \| \| \| --- \| \| \| \| When was 1 a prime number? Message #1 Posted by Palmer O. Hanson, Jr. on 29 Aug 2010, 9:21 p.m. In my first response to Bill's Bowling Challenge Variation challenge I used 1 as a prime number. Don Shepherd noted that 1 isn't a prime number, but Katie came to my rescue by noting that 1 had been considered to be a prime number in the past. Even so, I wasn't sure what I might have relied on from the olden days to accept 1 as a prime number. Then I found that Table 24.9 on page 870 of my copy of AMS-55 published in 1964 didn't include 1 in the list of prime numbers. But yesterday I was looking up something else in my copy of Mathematical Tables from the Handbook of Chemistry and Physics (1959). I looked at the table of factors and primes starting on page 242. A note at the top of the table states "If n is prime the mantissa of its logarithm is given." The mantissa is given as 0000000 for 1. \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #2 Posted by Egan Ford on 29 Aug 2010, 9:42 p.m., in response to message #1 by Palmer O. Hanson, Jr. Quote: --- When was 1 a prime number? --- In hindsight it never was. However, according to the 18th century, 1956, and 1941 (death of the last believer). \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #3 Posted by htom trites jr on 30 Aug 2010, 1:12 a.m., in response to message #2 by Egan Ford Always was, never was, depending on how you defined "prime number". These days it's mostly not, but the question shows up sometimes when non-professional people are asked about prime numbers. We had this just last week at our health club, the daily question was "What is the sum of the first five prime numbers?" and the "correct" answer was ( 1 2 3 5 7 11 ) either 18 or 28, depending on whether you started with 1 or 2. (The definition "a natural number divisible, without remainder, only by itself and one" leads to the conflict; there's an implication that that the word "and" means "itself <> one" but not everyone sees or agrees with that.) Edited: 30 Aug 2010, 1:18 a.m. \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #4 Posted by Don Shepherd on 30 Aug 2010, 7:39 a.m., in response to message #3 by htom trites jr The definition of prime number that the school textbooks use today is "a number with exactly 2 factors, 1 and itself", and a composite number is "a number with more than 2 factors". Since 1 has only 1 factor, it can't be prime, nor can it be composite. \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #5 Posted by Gerson W. Barbosa on 30 Aug 2010, 8:48 a.m., in response to message #4 by Don Shepherd Quote: --- The definition of prime number that the school textbooks use today is "a number with exactly 2 factors, 1 and itself", --- Shouldn't this be "a number with exactly 2 distinct factors, 1 and itself"? Otherwise one could argue that 1 is prime because 1 = 1 1. I was taught composite numbers are those that can be expressed as a finite product of prime numbers. This would exclude 1 from the set of prime numbers because 6, for instance, could be expressed by the infinite product 2 3 1 1 ... \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #6 Posted by Don Shepherd on 30 Aug 2010, 12:27 p.m., in response to message #5 by Gerson W. Barbosa Gerson, I see what you are saying, but I don't know that you need the word "distinct." I think the definition works because 1 only has 1 factor, which is 1 (you can't count 1 twice), and a prime number needs to have exactly 2 factors, and 1 doesn't. We teach it that way. We teach that both 0 and 1 are neither prime nor composite, they are special cases. I'm sure kids couldn't care less. \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #7 Posted by Egan Ford on 30 Aug 2010, 1:42 p.m., in response to message #6 by Don Shepherd IMHO, "exactly 2" implies a quantity with no mention of uniqueness. Distinct implies uniqueness. I know I can find a person on the street that will insist that 1 factors into exactly 2 numbers. 1 x 1. Because, most children spend a lot of time on a x b = c and remember that as adults. From Wikipedia: A prime number (or a prime) is a natural number that has exactly two distinct natural number divisors: 1 and itself. \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #8 Posted by Don Shepherd on 30 Aug 2010, 3:18 p.m., in response to message #7 by Egan Ford Hey Egan. Well, I have no doubt that you could find a person on the street who would agree to almost anything; that doesn't mean they are right. I'm sure there are some today who would say man never landed on the moon. And, please everyone, let's not steer this thread into an examination of did man, in fact, ever land on the moon! The Wikipedia definition is fine. Middle school textbooks typically don't say "distinct", but that's OK, most kids are able to correctly distinguish prime and composite numbers, at least long enough to demonstrate that knowledge on a test. And, believe me, kids have no interest in debating whether or not 1 is prime. I'd bet that if you took a survey of people on the street and asked what is the difference between a prime and a composite number, maybe 5% would know. Things like that are just not relavent in people's lives. Edited: 30 Aug 2010, 3:19 p.m. \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #9 Posted by David Hayden on 31 Aug 2010, 4:14 p.m., in response to message #6 by Don Shepherd Quote: --- Gerson, I see what you are saying, but I don't know that you need the word "distinct." I think the definition works because 1 only has 1 factor, which is 1 (you can't count 1 twice), and a prime number needs to have exactly 2 factors, and 1 doesn't. --- Don, So, does that mean that 12 has only 2 prime factors? ... I think you'll confuse the kids greatly if you don't say when duplicates matter. Regarding the primality of 1, I think the key thing is that defining it as non-prime makes some other mathematical concepts work out better. I wish I could remember what those concepts were though.... :) This is an example of a key concept that I think should be taught explicitly in school - that sometimes in math we define things a particular way to make the model work better or easier. \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #10 Posted by Don Shepherd on 31 Aug 2010, 6:08 p.m., in response to message #9 by David Hayden Well, 12 has 6 factors, which makes it a composite number (>2 factors). That's as far as middle school kids get into this subject. The non-primality of 1 (and 0, for that matter) is just something we don't get into with these kids. And most of them understand which numbers are prime and which are not. And they LOVE making factor trees, which is nice. Then we move from prime/composite into greatest common factor and least common multiple, which of course are used for simplifying and adding/subtracting fractions with different denominators. But by the time they graduate high school, they've probably forgotten it all. Which, honestly, doesn't disturb me. They've exercised their minds and learned how to think logically, and that's really the important part. \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #11 Posted by Palmer O. Hanson, Jr. on 31 Aug 2010, 9:51 p.m., in response to message #10 by Don Shepherd Don: You wrote: Quote: --- But by the time they graduate high school, they've probably forgotten it all. --- That pretty much agrees with my experience. Somewhere in my early education I was exposed to prime numbers, Mersenne numbers, perfect numbers, et al, but in forty years of work with military electronics systems I can't recall a single instance in which I had any use for any of them. Is work with these numbers primarily limited to mathematicians? Are there engineering disciplines depend on them in some way? Palmer \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #12 Posted by Don Shepherd on 31 Aug 2010, 10:10 p.m., in response to message #11 by Palmer O. Hanson, Jr. Quote: --- Is work with these numbers primarily limited to mathematicians? Are there engineering disciplines depend on them in some way? --- Palmer, I can't speak for the engineering disciplines regarding their possible use of prime numbers, but I do know that the entire security system behind the Internet is based upon the impossibility of factoring certain very large numbers into their two prime factors. I believe this is the RSA public key encryption system. I'm certainly no expert in that, but I do tell my students that if they can figure out a way to factor huge integers, they just might put the Internet out of business or, more likely, earn a few million dollars by agreeing to keep their discovery secret! RSA at one time offered lots of money to anyone who could factor certain large numbers; I believe they discontinued that. I printed out one of those numbers and showed it to my students and told them how much they could win if they could factor it. One student actually spent the weekend trying to divide the number by 2, and when he reached the last digit he realized that there would be a remainder. On Monday morning he brought in his work, and I remember the really bad feeling he had when I pointed out that he could have saved himself all that work if he had only realized that the number ended in 3 and therefore could not be divisible by 2! \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #13 Posted by David Hayden on 1 Sept 2010, 7:02 a.m., in response to message #10 by Don Shepherd Don, do you teach Euclid's algorithm to find the greatest common factor? I think it would be a great way to show kids the joy of a good algorithm and a nice way to introduce them to the concept of programming. I'm not suggesting that they would write a program, but you could explain the algorithm and then say "this is the sort of step-by-step procedure that can be done by a computer." \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #14 Posted by Don Shepherd on 1 Sept 2010, 7:50 a.m., in response to message #13 by David Hayden For the sixth graders, we generally teach it this way. For the seventh and eighth graders (if it is included in their curriculum), I might teach them Euclid's algorithm like this if I think they are capable of understanding it. With some classes, the simplest method is the best, but a good teacher challenges inquisitive students with methods they are capable of understanding and using. I try to be a "good teacher," of course. It is much easier to tailor your instruction to the individual student if you have 13 students a day, as opposed to 150 students a day. Fortunately, I have 13. \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #15 Posted by Mike on 1 Sept 2010, 9:18 a.m., in response to message #5 by Gerson W. Barbosa I disagree with the belief that 1 isn't a prime number the same as I disagree with Pluto not being a planet. However, "exactly 2 factors" means 2 and no more or less. 1 = 1 1 is two factors but not "exactly" 2 factors because 1 = 1 1 1 also. However, I think that definition is also wrong. Definition changes are done for one reason and that reason is to try and exclude 1, for no particular purpose. 1 being a prime violates nothing in mathematics and fits the old definition of a prime number. \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #16 Posted by Don Shepherd on 1 Sept 2010, 10:30 a.m., in response to message #15 by Mike Mike, you are free to believe what you like, of course. But the reality is that 1 is not a prime number, and that's what kids are taught. \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #17 Posted by Gerson W. Barbosa on 1 Sept 2010, 12:40 p.m., in response to message #15 by Mike Ultimately, 1 is defined as non-prime to keep a pattern intact, according to Dr. Math: \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #18 Posted by Katie Wasserman on 1 Sept 2010, 12:51 p.m., in response to message #17 by Gerson W. Barbosa I've been staying out of this since my initial comment on Palmer's use of 1 as a prime number. But if you want to debate something more relevant to HP calculators, why does the 20b and 30b define 0^0 as 1? I think that this is wrong but there are plenty of examples of this, try the Google calculator. Edited: 1 Sept 2010, 12:55 p.m. \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #19 Posted by Gerson W. Barbosa on 1 Sept 2010, 4:40 p.m., in response to message #18 by Katie Wasserman For 0^0 the 33s returns INVALID yx, the 48GX returns 1.The HP 50g returns ?, that is, a mathematical indetermination, however for 0.^0. it returns 1. Perhaps there's a practical reason for this behavior, but I can only guess. \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #20 Posted by Martin Pinckney on 1 Sept 2010, 5:45 p.m., in response to message #18 by Katie Wasserman Maybe this sheds some light (from Wikipedia): Quote: --- Treatment on computers The IEEE 754-2008 floating point standard is used in the design of most floating point libraries. It recommends a number of different functions for computing a power: pow treats 0^0 as 1. This is the oldest defined version, it checks if the power is an exact integer and uses the value defined by pown in that case otherwise the value is as for powr except for some extra exceptional cases. pown treats 0^0 as 1. The power must be an exact integer. The value is defined for negative bases, e.g. pown(-3,2) is 9. powr treats 0^0 as NaN (Not-a-Number - undefined). The value is also NaN for cases like powr(-3,2) where the base is less than zero. Programming languages Most programming language with a power function are implemented using the IEEE pow function and therefore evaluate 0^0 as 1. The later C and C++ standards, and the Java standard mandate this behavior. The .NET Framework method System.Math.Pow also treats 0^0 as 1. --- \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #21 Posted by Katie Wasserman on 1 Sept 2010, 6:20 p.m., in response to message #20 by Martin Pinckney There's a good, short discussion of the issue here. The vast majority of HP calculators (and I think all of them from the Pioneer series backward) return an error for 0^0 as do most other calculators. I wonder if there's much thought given to this or whether it's just a matter of convenience as to what the calculator returns. Just curious, has TI been consistent over the years as to how they handle this? (I don't have enough TI calculators to check.) \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #22 Posted by Gerson W. Barbosa on 1 Sept 2010, 7:53 p.m., in response to message #21 by Katie Wasserman Quote: --- Just curious, has TI been consistent over the years as to how they handle this? --- TI-82: ERR:DOMAIN TI-83+: ERR:DOMAIN TI-85: ERROR 04 DOMAIN TI-86: ERROR 04 DOMAIN TI-89: 1. TI-92+: 1. \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #23 Posted by Don Shepherd on 1 Sept 2010, 8:22 p.m., in response to message #22 by Gerson W. Barbosa Some more TI's: NSpire CAS undef 34ii error 34 Multiview error 30XIIS error 30XS error 36X error 15 error Sharp EL531W gives error (the only calculator I've seen with a built-in base 5). Edited: 1 Sept 2010, 8:27 p.m. \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #24 Posted by Katie Wasserman on 1 Sept 2010, 8:37 p.m., in response to message #23 by Don Shepherd It seems like TI and HP have been following a similar path, I think it pretty strange (and a bit disturbing) that both companies chose to abandon consistency on this with some of their recent high-end calculators. Pretty soon they'll start returning answers when you divide by zero like some of the earliest calculators did. \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #25 Posted by Gerson W. Barbosa on 1 Sept 2010, 8:45 p.m., in response to message #24 by Katie Wasserman Quote: --- Pretty soon they'll start returning answers when you divide by zero like some of the earliest calculators did. --- Actually on the HP 48/49/50 this is achieved by setting flag -22. Edited: 1 Sept 2010, 8:46 p.m. \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #26 Posted by Gerson W. Barbosa on 1 Sept 2010, 8:39 p.m., in response to message #23 by Don Shepherd Quote: --- the only calculator I've seen with a built-in base 5 --- An interesting feature for former abacus users? \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #27 Posted by Palmer O. Hanson, Jr. on 1 Sept 2010, 9:50 p.m., in response to message #21 by Katie Wasserman Katie: You wrote: Quote: --- Just curious, has TI been consistent over the years as to how they handle this? (I don't have enough TI calculators to check.) --- Here are some results from older TI's: TI Business Analyst I 1 TI-30 1 SR-40 1 TI-55 1 TI-59 1 TI-68 Error TI-80 ERR:DOMAIN TI-81 ERROR 04 MATH TI-83+ ERR: DOMAIN TI-85 ERROR 04 DOMAIN TI-86 ERROR 04 DOMAIN TI-89 Titanium undef voyage200 undef where my result for the TI-89 Titanium is different from the result reported by Gerson. I don't know how to explain the difference unless there is a difference between the TI-89 and the TI-89 Titanium. I get the "undef" result whether I am in AUTO, EXACT, or APPROXIMATE. Using my results there is a consistency of sorts. Before the TI-68 the result is 1, and from the TI-68 forward the result is an error. Here is an interesting one from the HP-19BII: ERROR: 0^0 where it has separate error messages for 0^NEG, 0/0, 0^0, and /0. Palmer Edited: 1 Sept 2010, 9:55 p.m. \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #28 Posted by Chuck on 1 Sept 2010, 11:20 p.m., in response to message #18 by Katie Wasserman 0^0 = 1 is somewhat logical if you look at the graph of y=x^x, and then take the limit as x->0 from the right. Obviously the limit is 1. Weird #$@%$#!! happens from the left, but I'm comfortable with the right-hand limit value 95% of the time. :) \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #29 Posted by Mike on 31 Aug 2010, 7:18 p.m., in response to message #4 by Don Shepherd Quote: --- "a number with exactly 2 factors, 1 and itself", --- Factor 1: 1 Factor 2: itself Factors imply multiplication 1 itself = 1 IF one is not a prime, then the definition was changed along the way. The definition I was taught (in olden times) was "a prime is a number that is divisible only by itself and one". And I'm sticking with that, unless someone asks me to bet on that sum of first five primes question. Edited: 31 Aug 2010, 7:19 p.m. \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #30 Posted by Don Shepherd on 31 Aug 2010, 8:18 p.m., in response to message #29 by Mike One is not prime today. It and zero are neither prime nor composite. That's what we teach the kids. \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #31 Posted by Mike on 1 Sept 2010, 9:40 p.m., in response to message #30 by Don Shepherd Yeah! But what we teach kids, isn't the the criteria for deciding whether or not 1 should be prime. It's not a prime today, for only one reason... to be consistent with a new definition. Mathematics has been around for eons. Up until a hundred years ago, and even into the 20th century, many professional mathemiticians considered 1 a prime. Up until the 50s and even in the 60s when I was in school, '1' was a prime number. What has changed is a definition; not mathematics. Hell, "what we teach kids" is not only often wrong, but irrlevant. It wouldn't surprise me that some history revisionists "teach" that the U.S. lost the Vietnam war and offer as proof, the helicopters landing on the embassy in Saigon. Nevermind that the U.S. had alread been out of Vietnam for 2+ years and the war ended in a peace agreement. Edited: 1 Sept 2010, 9:42 p.m. \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #32 Posted by Don Shepherd on 1 Sept 2010, 10:09 p.m., in response to message #31 by Mike Quote: --- But what we teach kids, isn't the the criteria for deciding whether or not 1 should be prime. --- I agree. In 2010, 1 is not regarded as a prime number. That is the mathematically-accepted truth. We teach kids the mathematically-accepted truth. Quote: --- "what we teach kids" is not only often wrong, but irrlevant. --- You've got it partially right. We don't teach kids "wrong" things. We teach the curriculum, and the curriculum is defined by the state educational organizations. Now they are not perfect, but at least where math is concerned I can't think of a single fact that we teach our kids that is "wrong." Some of what we teach is, in my opinion, irrelevant. Box-and-whiskers plots and stem and leaf plots are examples. Kids will never see one of these once they leave middle school. But, truth be told, most kids will never see a quadratic equation once they leave high school either. But we teach them, because they make the kids think, and that's a value that never goes out of style. \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #33 Posted by Egan Ford on 30 Aug 2010, 10:39 a.m., in response to message #3 by htom trites jr Quote: --- Always was, never was, depending on how you defined "prime number". --- I was taught that a prime was a positive integer that was only divisible by 1 and itself. 1 does fit that definition. And there may have been a period of time in my youth that I also believed it. The missing term distinct I read from a book later on. I believe that confusion is still being spread because that term is being dropped from casual (and perhaps formal) conversation. 1 is neither prime or composite--it is truly the loneliest number. 1 does have the unique distinction of being the unit, i.e. unit = 1/unit (or unit == 1/unit for TW, et al :-). Edited: 30 Aug 2010, 10:40 a.m. \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #34 Posted by John B. Smitherman on 31 Aug 2010, 12:00 a.m., in response to message #1 by Palmer O. Hanson, Jr. Palmer, in my opinion defining 1 as non-prime is similar to defining 0! = 1. It seems to be an arbitrary way to get the math to work. Regards, John \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #35 Posted by John Mosand on 31 Aug 2010, 8:27 a.m., in response to message #34 by John B. Smitherman One reason why 1 is not a prime is that it would be in conflict with the Fundamental Theorem of Arithmetic. \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #36 Posted by Ken Shaw on 31 Aug 2010, 10:13 a.m., in response to message #35 by John Mosand This is elegantly expressed in the video found here: \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #37 Posted by Mike on 1 Sept 2010, 9:25 a.m., in response to message #35 by John Mosand I must be in a disagreeable mood today. How does 1 violate the Fundamental Theory of Arithmetic? If what you say is true, then making 1 a prime number would violate that theorem. But if 1 was prime, it fits perfectly with the definition of Fundamental Theory of Arithmetic. Fundamental Theory of Arithmetic "In number theory and algebraic number theory, the Fundamental Theorem of Arithmetic (or Unique-Prime-Factorization Theorem) states that any integer greater than 1 can be written as a unique product of prime numbers." In fact, in my view if that is the definition, you it is flawed without 1 being a prime. For instance, according to that theory, what is the factors of 2? 2 must have uniquie product of prime numbers. 2 = 2 1 According to the theory, those factors must be prime. In order for that theory to hold, 1 must be a prime. If 1 is not a prime number, then there is no product of factors at all for 2. Product implies multiplication of 2 numbers. The problem I see, is that all of these definitions are designed to exclude 1 from the list of primes, for no particularly good reason. The definitions are flawed. 1 being a prime voilates no rules of mathematics that I know of, unless it's related to a flawed definition. So here is a test 1) What is the oldest known list showing 1 as a prime? (no definitions please; just lists of numbers) 2) What is the oldest known list showing 2 as the first prime? (no definitions please; just lists of numbers) 3) Which is older? I know "todays" answer as to whether or not 1 is prime. I know when I was going to school, many, many moons ago 1 was prime. I also know that when I was going to school, Pluto was a planet. From where I sit now; nothing has changed but definitions. Edited: 1 Sept 2010, 9:43 a.m. \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #38 Posted by Palmer O. Hanson, Jr. on 1 Sept 2010, 1:46 p.m., in response to message #37 by Mike Mike: You asked Quote: --- 1) What is the oldest known list showing 1 as a prime? (no definitions please; just lists of numbers) 2) What is the oldest known list showing 2 as the first prime? (no definitions please; just lists of numbers) 3) Which is older? --- If you look at the first item in this thread you will see that my AMS 55 of 1964 does not show 1 as prime. My Mathematical Tables from Handbook of Chemistry and Physics does show 1 as prime. It is fourth printing of the eleventh edition with the first printing in December 1959. The book was originally copyrighted in 1931. In its definitions of "prime' my Webster's Collegiate Dictionary (Fifth Edition 1946) states Quote: --- 5. math a. divisible by no number except itself or unity; as, 7 is a prime number. b. Having no common divisor but 1; -- used with to; as, 12 is prime to 25. --- My 1969 issue of Encyclopedia Britannica says Quote: --- PRIME NUMBER, a positive integer (whole number) greater than 1 that cannot be expressed as the product of two positive integers neither of which is one. --- Those are the oldest references that I have here in North Carolina. I have some older references in Florida including the "Mathematical Tables from ..." that I purchased in 1946 when I was a freshman in college. I also have the algebra and geometry texts that my father used in about 1918 where my memory says that the copyrights were in the very early 1900's. I will be able to look at those in about two weeks. I told my wife that those old books would come in handy sometime! Palmer \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| "Is 1 a prime number" and "0! = 1"? Message #39 Posted by Karl Schneider on 2 Sept 2010, 12:50 a.m., in response to message #34 by John B. Smitherman Quote: --- Palmer, in my opinion defining 1 as non-prime is similar to defining 0! = 1. It seems to be an arbitrary way to get the math to work. --- There's been much parsing in this thread about the reasons for or against the primality of 1, but the fundamental point seems clear to me: The purpose of integer factorization is to decompose an integer into a product of factors (smaller integers). A prime factor is one that itself cannot be decomposed. Any number -- integer or not -- can be expressed as the product of itself and unity (1), which renders 1 a "sub-factor" that cannot be used for decomposition. Thus, 1 would be trivial as a prime number. It follows that two (2) cannot possibly be broken down, due to an absence of any smaller factors. However, to divide a positive even number by 2 will render an integer quotient that is smaller than the original number, so 2 is a valid factor that is prime. --- As for the factorial of 0 (0! = 1): Yes, it seems arbitrary, but it fits the inductive mathematical definition and is useful in practice: 1! = 1 (by definition) k! = k(k-1)! {k is integer} --> (k-1)! = (k!)/k Let k = 1, --> (1-1)! = (1!)/1 --> 0! = 1/1 = 1 --> (-1)! = (0!)/0 = 1/0 (undefined) -- KS Edited: 2 Sept 2010, 1:50 a.m. \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #40 Posted by Don Shepherd on 31 Aug 2010, 9:06 p.m., in response to message #1 by Palmer O. Hanson, Jr. I can verify that 1 is not prime from the highest authority. TI-89 Titanium isprime(1) = false TI-NSpire isprime(1) = false Surely this should convince everyone. : ) big smiley Don \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #41 Posted by Gerson W. Barbosa on 31 Aug 2010, 9:21 p.m., in response to message #40 by Don Shepherd HP 50g: 1 ISPRIME? -> 0 Now, I am convinced! :-) Edited: 31 Aug 2010, 9:23 p.m. \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #42 Posted by Mike on 1 Sept 2010, 9:46 p.m., in response to message #40 by Don Shepherd And I wager the algorithm is different for '1' than any other number. I bet the algorithm is something like: isprime (n) { if (n==1) { return "false" } is_modernconvention_prime(n); } It's simply one extra step to the actual algorithm, to eliminate '1' by convention and nothing more. I also bet you think Pluto isn't a planet? Edited: 1 Sept 2010, 9:52 p.m. \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #43 Posted by Don Shepherd on 1 Sept 2010, 10:54 p.m., in response to message #42 by Mike Mike, slavery is also no longer accepted in the United States. Things change. If you want to consider 1 prime or Pluto a planet, hey, it's a free country. But I've got to teach prime numbers to kids, and I want to teach the truth, and the truth is 1 is not a prime number. Maybe it once was, fine, but today it's not. If you really feel strongly about this, perhaps you should write the American Mathematical Association and plead your case. I wish you luck. Don \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #44 Posted by Martin Pinckney on 1 Sept 2010, 11:29 p.m., in response to message #43 by Don Shepherd I think I understand both viewpoints. There appear to be valid reasons why today 1 is defined as not prime, whether we understand them or not; that is mathematical convention today, so that is what is taught in the schools. However, "convention" is not the same as "truth". As David Hayden said: Quote: --- ...sometimes in math we define things a particular way to make the model work better or easier. --- By "we", I mean "me". To me, it seems intuitive that 1 is prime. However, intuition often fails to maintain rigor in mathematics. \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #45 Posted by htom trites jr on 1 Sept 2010, 11:46 p.m., in response to message #44 by Martin Pinckney It seems that the natural numbers are divided these days into four classes: 0; 1; primes; and composites. Glad I'm not a mathematician, and I still call Pluto a planet. Wrong, wrong. Anyway, MathWorld has a discussion of this, with dates and sources. It's one of those arbitrary "math things". Primacy of One, and Primes The Fundamental Theorem thing doesn't do it for me. If N is a prime > 1, then it has two factors, itself and one, and is the product of those two primes. But one is not a prime, so the theorem seems to fail. I suspect this is one of those "don't do math in English" problems, which I'm frequently reduced to in attempts to explain quantum and relativity physics. Edited: 1 Sept 2010, 11:50 p.m. \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #46 Posted by Gerson W. Barbosa on 2 Sept 2010, 1:20 p.m., in response to message #45 by htom trites jr It's just occurred to me that the Eratosthenes Sieve would not work if 1 were defined as prime. In that case all multiples of 1 (that is, the whole set of integer numbers but 1) would be striken from the list and 1 would be the only prime number. Perhaps 1 should be called a primary number to distinguish it from the primes, but I'm not sure this is another Mathematics reserved word. Including 1 in the set of prime numbers would spoil the mnemonic for the first seven primes in your reference. Likewise the exclusion of Pluto from the planetary family has ruined "My Very Educated Mother Just Served Us Nine Pizzas" :-) \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #47 Posted by Dieter on 2 Sept 2010, 1:46 p.m., in response to message #46 by Gerson W. Barbosa Quote: --- Likewise the exclusion of Pluto from the planetary family has ruined "My Very Educated Mother Just Served Us Nine Pizzas" :-) --- Ah, very interesting - in German it's "Mein Vater Erklärt Mir Jeden Sonntag Unsere Neun Planeten", so the loss of Pluto is even worse since both the number nine and the word "planet" are explicitely mentioned. ;-) And what about other languages? How do you memorize the planets in French, Spanish or in other parts of the the world? Dieter \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #48 Posted by Gerson W. Barbosa on 2 Sept 2010, 5:25 p.m., in response to message #47 by Dieter In Portuguese it is, or used to be, "Meu velho terno marrom joguei sábado último no porão" which means "Last saturday I threw my old brown suit in the basement". P.S.: I don't know if this is widespread as I cannot find any reference on the web. I rembember my father told it to me once (not every Sunday :-) Another one, which takes into account the exclusion of Pluto, is "Meu Velho Tio Me Jurou Ser Um Netuniano", that is, "My old uncle has sworn me to be from Neptune". But who needs this to memorize only eight planets anyway? Edited: 2 Sept 2010, 5:48 p.m. \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #49 Posted by Etienne Victoria on 3 Sept 2010, 6:17 a.m., in response to message #47 by Dieter Hi all, The one I was taught in french: "Mais Viendras-Tu Manger Jeudi Sur Une Nappe Propre ?" and when Pluton was demoted, this was changed to: "Mais Viendras-Tu Manger Jeudi Sur Une Nappe ?" Which respectively mean: "Will You Come To eat Thursday On a Clean Tablecloth?" "Will You Come To eat Thursday On a Tablecloth?" When Pluton was dropped, the tablecloth went dirty... Edited: 3 Sept 2010, 6:20 a.m. \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Message #50 Posted by Gerson W. Barbosa on 3 Sept 2010, 9:52 a.m., in response to message #49 by Etienne Victoria Salut Etienne! It appears these are not so difficult to create. Par éxample: "M'amie Voulait Toujours Me Jeter Sur Une Noce". However, does this one make any sense? :-) Gerson. \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Solar system, Granma, weddings... Message #51 Posted by Etienne Victoria on 3 Sept 2010, 4:35 p.m., in response to message #50 by Gerson W. Barbosa Dear Gerson, Well...it would make sense if my Grandmother could "throw me on a wedding" :-) Even though she passed away a couple of years ago, she lived long enough to meet and appreciate my wife-to-be. So...no question "she would have thrown me on a wedding" :-) All the best Etienne \| \| \| \| \| \| \| \| \| --- --- \| \| \| \| \| \| Re: When was 1 a prime number? Solar system, Granma, weddings... Message #52 Posted by Gerson W. Barbosa on 3 Sept 2010, 6:32 p.m., in response to message #51 by Etienne Victoria The following isn't any better: "Mers Vertes, Tremblez Maintenant ! Je Suis Un Navigant !" Also, there is rhyme but no rhythm. I quit, this is not so easy as I imagined ... The best so far is the German one, which is related to the (planets) subject. Regards, Gerson. \| \| \| \| [ Return to Index \| Top of Index ] Go back to the main exhibit hall
2889	https://www.youtube.com/watch?v=dtP2__hS1QE	pH and pKa - Analyzing Titration Curves - AP Chem Unit 8, Topic 7 Jeremy Krug (krugslist) 44900 subscribers 413 likes Description 19751 views Posted: 2 Feb 2024 Learn AP Chemistry with Mr. Krug! Get the AP Chemistry Ultimate Review Packet: Guided notes for these AP Chem videos are now included in the Ultimate Review Packet! Find them at the start of each unit. If you need help reviewing for your other AP Courses, such as Biology, Physics 1, Environmental Science, Calculus AB, Pre-Calculus, Psychology, Macroeconomics, World History, US History, and more, check out #apchemistry #apchem In this video, Mr. Krug shows students how to analyze a titration curve. Using a titration curve, it is possible to describe what kind of titration is being carried out, the approximate pH of the equivalence point, and even the pKa and Ka of the weak acid involved in the titration. Mr. Krug will cover all of these points in this video. Finally, he will show how to choose an appropriate indicator for an acid-base titration. 00:00 Introduction 00:13 Titration Curve Analysis 03:36 Acid-Base Indicators 05:46 Conclusion 11 comments Transcript: Introduction hi there I'm Jeremy Krug and in this video we're going to be moving right along to AP Chemistry Unit 8 section 7 which is about the relationship between pH and PKA especially in acidbase titrations so let's start by just Titration Curve Analysis looking at the titration curve for a strong base added to a weak acid so once again we have the volume of Base that's been added on the x-axis and the change in pH is represented on the the Y AIS and as you can see we start out with that pH Rising fairly slowly and then all of a sudden we get to the point where it just shoots up and then it starts to continue up not quite as fast but it goes up uh fairly fairly quickly after that as well so I want you to notice that just like we said before the inflection point denotes the equivalence point now even if they didn't tell you you should be able to tell that this is a strong base weak acid titration and that's because the equivalence point that that inflection point right there has a pH greater than seven so that greater than seven implies that the base is going to predominate that's why we say strong base weak acid in that case now we also learned another point about this graph already we know that at halfway to the equivalence point the concentration of the weak acid is equivalent or equal to the concentration of the conjugate base and because of that we'll talk about that in our next video but because of that at that particular point the pH of the solution is equal to the pka or the negative log of the KA of that acid and so at that point ha the the acid concentration equals the con base concentration now let's look at what predominates on either side of that halfway point because of that you know when you have any point before the halfway point what predominates is the acid you have more ha more of the weak acid you only have a little bit of ha in fact we can take that to its logical beginning there at the very beginning of the titration we haven't added any base at all have we so the only thing you have is ha now let's take that to the other side if you go past the halfway point and before the equivalence point in this this region right here what you have is more conjugate base than the acid so that's what's going to predominate once you get to the equivalence point you just have have the weak base there now what's going to predominate after the equivalence point well let's think about it notice where the pH is it's very high and think about what we're adding to this as well so because of that I think it's safe to say that the substance that predominates here is the hydroxide and so that's what's going to predominate from basically past the equivalence point lots of hydroxides still being added there that's when you have what's called overshooting your titration now when you are carrying out Acid-Base Indicators an acidbase titration always make sure to use the most appropriate acidbase indicator and in your chemical closet or your Chemical stock room there's a good chance that you have several acidbase indicators available to you there are dozens if not more out there on the market every acidbase indicator has a a specific pH at which it transitions from one color to another uh for example litmus which you've heard of before it transitions around pH 7.0 that means that at pH is below 7.0 it's red at pH is greater than 7.0 it's blue its PKA is 7.0 for methyl Orange it's about 3.4 for phenol thaline it's around 9.4 so that transition that transition pH that's the pka so let's imagine that we're trying to titrate a strong acid with a strong base which of the three indicators would you choose well remember a strong acid and a strong base are going to have an equivalence point right in the middle right at seven right so that means that we're looking at a indicator or trying to find an indicator that has a pka of around seven so litmus is your best choice on that one now what about a weak acid and a strong base which indicator would you use then well if you have a strong base and a weak acid well then that the base is going to predominate and your equivalence point is going to be on the basic side probably around nine somewhere in that area so you'll want a an acidbase indicator that transitions around fine and so the best choice here is phenol phine that's the one that you'd want to use so use the pka of these indicators to determine which indicator to use in a certain titration not always going to be the Conclusion same hope you learned something about pH and PKA in this video if you did go ahead and slam that like button I'm Jeremy kug and our next video we're going to move on to uh Unit 8 sections 8 n and 10 and wrap up this unit by talking about buffer Solutions hope to see you then
2890	https://www.law.cornell.edu/wex/egregious	Please help us improve our site! No thank you Cornell Law School Search Cornell egregious Egregious, from the Latin egregius, meaning “illustrious” or literally “standing out from the flock” is a term used to describe a conduct that is flagrant, or outrageous in comparison to a normal standard of conduct. In general, the term egregious is used to describe a conduct by a person intentionally committing an act or omission that involves knowing the violation of law. In a legal context, egregious conduct refers to an action that is obviously wrong, beyond a reasonable degree. The term can be used to describe a conduct of a party to a legal action, its attorney(s) or any other legal professional, or the court itself. Egregious behavior is considered in order to bring to an end the person’s actions, or to introduce a request for increased damages. [Last reviewed in April of 2022 by the Wex Definitions Team] ACADEMIC TOPICS legal education and writing accidents and injuries criminal law criminal procedure wex definitions civil procedure criminal law and procedure legal education and practice legal theory
2891	https://www.youtube.com/watch?v=ea3WvtrszWk	Classification of Fungi \| Biological Classification \| Biology \| Khan Academy Khan Academy India - English 546000 subscribers 784 likes Description 50366 views Posted: 1 Dec 2022 In this video, we talk about the different phyla of fungi and how the classification system of fungi is ever-changing and ever-improving. Timestamps: 00:00 - Introduction 01:18 - Chytridiomycota 02:52 - Zygomycota 07:21 - Ascomycota 10:09 - Basidiomycota 12:11 - Other phyla Practice this concept: Master the concept of Fungi through practice exercises and videos: Check out more videos and exercises on Biological Classification: To get you fully ready for your exam and help you fall in love with Biology, find the complete bank of exercises and videos for Class 11 Biology here: Khan Academy is a free learning platform for Class 1-12 students with videos, exercises, and tests for maths, science, and more subjects. 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Donate here: Created by Sulagna Das 18 comments Transcript: Introduction what if I tell you that fungi are so widely distributed on our planet that you can find them pretty much everywhere yes really in fact you might not even have to step out of your house in order to see one the greenish black patches on that two-week old bread fungi that bottle on the kitchen shelf which is marked activated East fungi the mushrooms growing on those damp logs in your garden you know it fungi and you've probably noticed by now that they're wildly diverse too each of these fungi has its own set of unique features and using these features scientists were able to classify them over the years but the thing is that in all this time fungi classification has changed dramatically many scientists nowadays refer to newer more advanced systems of classification so in this video that's what we're gonna do as well we're gonna talk about some of the major phyla in kingdom fungi and their distinct characteristics and while we are doing so we'll try to stick to the more recent versions of fungi classification rather than the older ones so let's get into it the first phylum we have over here is Chytridiomycota Sky tridiomycota now I'll be honest I didn't get that pronunciation right the first time and neither did I know that this phylum existed up until I actually started looking into it and it turns out that this phylum houses all of the simplest most primitive fungi to ever exist on our planet most of these skytrids that's what you call the members of this phylum most of this guy these kitrids are unicellular organisms that are often found in water bodies or wet soils and like every other fungi they also have chitinus cell walls and Exhibits saprotrophic or parasitic nutrition however what truly sets skytrids apart from the rest of the fungi are there a motile zeuspores the spores that the kitrids produce they are called zoos Sports and each of them come equipped with a flagellum like this and using this flagellum the Zeus pore can actively move around or swim around all on their own and mind you no other fungal spore can do this they are the only more tile spores to exist and kitrids are the ones that can produce these pores a very famous example or a very famous kitrid is this organism right here which is called aloe my sis Zygomycota next up we have the zygomyces members of the phylum zygomycota and we've already come across a zygomycin before remember the stale moldy bread at the beginning of this video well the fungus that grows on stale bread it's called the common bread mold and that is scientifically called rhizopus tolonifer which is this fungus right over here so if you take a piece of that moldy bread and place it under a microscope this is what you're gonna see now the most fascinating thing about zygomycetes is how they reproduce sexually normally they reproduce asexually with the help of something called sporangiospores which are these pores that are formed inside a structure called a sporangium you can actually see a sporangium right over here so this bulb-like structure here this is the sporangium sporangium and inside of this is where the sporangiospores are formed and sporangiospores are the way to go normally when things are fine but when the environmental conditions become unfavorable the fungus or these fungi they resort to sexual reproduction that is when sexual reproduction takes place so let's take a look at this process so during sexual reproduction what really happens is that two compatible haploid hyphae will start extending towards each other till their tips touch kind of like this and the minute the tip touches what happens is that it kind of walls off a section or a portion of this tip so let's say it starts to wall off somewhere here by building by forming something called a septum which you can think of it like a partition so this is a septum I'll just write it down over here so it starts to wall off a certain section by forming this septum and when this once this partition is done or this Walling off is done what will happen is that everything in this portion right over here will fuse together even the nuclei now over here I have you can see that there are only two two nuclei but usually what happens like this is just an image that I've drawn but uh in real life like in nature when this fusing happens there are many haploid nuclei which are present in this area depending on where this wall of thing happens where the partition begins to appear So based on where the partition appears many haploid nuclei are present in between or they are trapped in between this uh section of the extension or the tips basically so now what happens these nuclei they will also fuse along with the tips so after all of the fusion what really happens what we get at the end of that is this zygospore so this uh haploid nuclei will fuse with this haploid nuclei and give us this purple one and again the same thing happens with these two nuclei and we have two basically uh diploid nuclei because the haploid nuclei diffused right so in the end we finally have a zygospore which contains multiple diploid nuclei now what happens after this this zygospore will eventually undergo meiosis and because of meiosis What will happen this diploid nuclei will become haploid nuclei because of the meiosis that it happens and each of these haploid nuclei each of them separately will become a haploid Spore and that is how uh through sexual reproduction these haploid sexual spores are formed now this type of sexual reproduction where the tips of the hyphy fuse together to give the zygospore which eventually gives us a haploid spores this entire process is called conjugation and because of this because of this conjugation process zygomyces are often referred to as conjugated fungi moving on the third Ascomycota phylum on our list is ascomycota these fungi are characterized by the presence of something called acai or an Asus and ascus is a sac-like structure which contains the haploid sexual spores called ascospores so these bead-like structures that you can see inside this sac-like thingy so these are the ascospores inside the ascus and because of the sac-like structure these fungi are also called Sac fungi during sexual reproduction ascospores are formed inside thousands of acai which in turn are found inside of a Spore bearing structure called an ASCO ascocarp now this so basically what happens these ASCO spores they are formed inside this ascus or inside the acai and these acai is in turn found inside something called the Asco carp which is a Spore bearing structure so inside this you will find the acai and inside the acai you will find the ASCO spores now the process of sexual reproduction goes something like this plasmogamy or Fusion of cytoplasm takes place between two compatible haploid cells and the fused cell enters a diaryotic stage where the nuclei the haploid nuclei they remain free eventually these nuclei will also fuse that means kariogami will take take place and we will end up with a diploid zygote inside the ascus this zygote will further undergo meiosis and that will give us the haploid ascospores these ASCO spores are then released into the environment and they end up germinating at suitable places as for my seats also undergo asexual reproduction by producing asexual spores called kunidia ascomyces are incredibly helpful to us as well especially commercially and a very important example of that is East remember how we talked about uh dry activated yeast being something that we can find in our kitchen so we use East to East in a variety of different things like baking Brewing fermenting wine and a bunch of different things so East is an ascomycet which is extremely important to us other examples of ascomyces include truffles uh then there are morals now these are treated as Delicacies uh in different parts of the world then there's also aspergillus which is something you might have heard of uh which is also uh an ascomycet so these are some examples of ascomyces the next major phylum is that Basidiomycota of the mushrooms AKA phylum basidio mycota bacidio my seeds are easily recognized by their club-shaped Spore bearing structures called basedia now there's only one structure here so this will be a basidium that's the singular form and because of this club shaped basidium or basidia these uh fungi are also called Club fungi now where will you find this basidium or where will you find all the biseria you will find them on the gills of the mushroom so these lines that you can see these are the gills of the mushroom which are found on the underside of the cap and this whole protruding thing with the cap and everything that is the fruiting body of the mushroom called the basidio carp so this entire thing let's pick a different color we can't really see it in here so this entire thing the protruding part of the mushroom which is you know on the top of the soil that we can see this is the fruiting body which is called The Busy do carp so that means the basidio carp Bears the basidia on the gills now inside each basidium beside your spores which are these sexual spores by the way they are produced and these basidio spores they are produced in it's very similar to how the ascospores are produced now once these haploid basidio spores they are formed they will be released into the environment from the basidios carp and then they will germinate into new fungi now remember all mushrooms toadstools uh then shelf fungi then smarts and rusts all these types of fungi they belong to phylum basidio mycota so they're all basidiomyces now if you're wondering why I didn't mention an asexual reproduction here that's because usually all basidio my seeds reproduce sexually so that's why we are not discussing about the asexual reproduction here Other phyla now these are just some of the major phyla which has been included in kingdom fungi now that that doesn't mean that these are the only phyla by the way there are other different phyla as well for example glomeromycota is one such phylum which is there in kingdom fungi but one that we didn't really discuss here uh this includes fungi which live in very close association with uh plant roots there's another group that you might have heard of called deuteromycetes aka the imperfect fungi you know the fungi that do not show a sexual phase in their reproductive Cycles at all so those are the deuteronomyces but the thing is that deuteronomycota is not considered to be a true phylum why because their members are more close to closely related to organisms from other phyla than each other let me explain this to you with an example aspergillus was earlier thought to be a Deuter my seat but after a lot of research and a lot of studies and molecular analysis they finally found out that aspergillus is more closely related to ascomyces rather than deuteronomyces so that's a completely different phylum right ascomycota in fact we we literally wrote down aspergillus in the lesson today under ascomycota so that's how aspergillus was moved from deuteromycota to ascomycota and that is why ASCO um that is why deuteronomycota or deuteronomyces they are not considered to be a true uh a true member or a true phylum for that matter so you see fungi classification is an ever-changing Ever improving ordeal it's not set in stone who knows that maybe a few decades from now will have a completely new system of classification for all of these fungi it is that dynamic
2892	https://artofproblemsolving.com/wiki/index.php/1996_AHSME_Problems/Problem_24?srsltid=AfmBOopT0d3kNZPALzqLJO_WZFxTNSH6MkwSyb57ITnBzIqwLoYqHW3B	Art of Problem Solving 1996 AHSME Problems/Problem 24 - AoPS Wiki Art of Problem Solving AoPS Online Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online ‚ Books for Grades 5-12Online Courses Beast Academy Engaging math books and online learning for students ages 6-13. Visit Beast Academy ‚ Books for Ages 6-13Beast Academy Online AoPS Academy Small live classes for advanced math and language arts learners in grades 2-12. Visit AoPS Academy ‚ Find a Physical CampusVisit the Virtual Campus Sign In Register online school Class ScheduleRecommendationsOlympiad CoursesFree Sessions books tore AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates community ForumsContestsSearchHelp resources math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten contests on aopsPractice Math ContestsUSABO newsAoPS BlogWebinars view all 0 Sign In Register AoPS Wiki ResourcesAops Wiki 1996 AHSME Problems/Problem 24 Page ArticleDiscussionView sourceHistory Toolbox Recent changesRandom pageHelpWhat links hereSpecial pages Search 1996 AHSME Problems/Problem 24 Contents 1 Problem 2 Solution 3 Solution 2 (Alcumus) 4 See also Problem The sequence consists of ’s separated by blocks of ’s with ’s in the block. The sum of the first terms of this sequence is Solution The sum of the first numbers is The sum of the next numbers is The sum of the next numbers is In general, we can write "the sum of the next numbers is ", where the word "next" follows the pattern established above. Thus, we first want to find what triangular numbers is between. By plugging in various values of into , we find: Thus, we want to add up all those sums from "next number" to the "next numbers", which will give us all the numbers up to and including the number. Then, we can manually tack on the remaining s to hit . We want to find: Thus, the sum of the first terms is . We have to add more s to get to the term, which gives us , or option . Note: If you notice that the above sums form , the fact that appears at the end should come as no surprise. Solution 2 (Alcumus) The th appearance of 1 is at position . Then there are 1's and 2's among the first numbers, so the sum of these terms is . When , , and when , . The sum of the first 1225 terms is . The numbers in positions 1226 through 1234 are all 2's, so their sum is . Therefore, the sum of the first 1234 terms is . See also 1996 AHSME (Problems • Answer Key • Resources) Preceded by Problem 23Followed by Problem 25 1•2•3•4•5•6•7•8•9•10•11•12•13•14•15•16•17•18•19•20•21•22•23•24•25•26•27•28•29•30 All AHSME Problems and Solutions These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Retrieved from " Art of Problem Solving is an ACS WASC Accredited School aops programs AoPS Online Beast Academy AoPS Academy About About AoPS Our Team Our History Jobs AoPS Blog Site Info Terms Privacy Contact Us follow us Subscribe for news and updates © 2025 AoPS Incorporated © 2025 Art of Problem Solving About Us•Contact Us•Terms•Privacy Copyright © 2025 Art of Problem Solving Something appears to not have loaded correctly. Click to refresh.
2893	https://arxiv.org/html/2007.07780v12/	SPECIAL RELATIVITY Evgeny Saldin Final draft Applications to astronomy and the accelerator physics Evgeny Saldin Final draft Current draft was completed by a new Chapter 17. If you have any comments, please send an email to evgueni.saldin@desy.de. Many books have been written on the classical subject of special relativity. However, after years of experience in both relativistic engineering and research, I have come to believe that there is room for a new perspective. This book is not quite like the others, as it aims to shed light on often-overlooked aspects of the subject. To clarify, this is not a textbook on relativity theory. Instead, it focuses on the nature of special relativistic kinematics, its connection to space and time, and the operational interpretation of coordinate transformations. Every theory contains a number of quantities that can be measured by experiment and expressions that cannot possibly be observed. Whenever we have a theory containing an arbitrary convention, we should examine what parts of the theory depend on the choice of that convention and what parts do not. Unfortunately, this distinction is often overlooked, leading some authors to mistakenly classify certain data as observable when, in reality, they rely on arbitrary choices rather than physical experiments. This oversight results in inconsistencies and paradoxes that should be avoided. The practical approach adopted in the book should appeal to astronomers, space engineers, accelerator engineers, and more broadly, relativistic engineers. This approach, unusual in the relativistic literature, may be clarified by quoting one of the problems discussed in the text: the new light beam kinematics for rotating frames of references. Since we live on such a rotating (Earth-based) frame of reference, the difference in relativistic kinematics between rotating and non-rotating frames of reference is of great practical as well as theoretical significance. A correct solution of this problem requires the use of relativistic principles even at low velocities since the first-order terms in , where is the orbital velocity, play a fundamental role in the non-inertial relativistic kinematics of light propagation. All the results presented here are derived from the first principles, with every step involving physical reasoning explicitly detailed. To maintain a self-consistent style, auxiliary derivations are provided in the appendices. To encourage readers to develop their own perspectives, each chapter includes a suggested bibliography with relevant remarks. The references list only papers I personally consulted, and while many valuable works remain unmentioned, I apologize for any omission. I am deeply grateful to my longtime friends Gianluca Geloni and Vitaly Kocharyan for years of discussions on much of the material covered in this book. I also extend my sincere thanks to DESY (Deutsches Electronen-Synchrotron) for the opportunity to work in this fascinating field. Contents 1 Introduction 2 A Critical Survey of Present Approaches to Special Relativity 2.1 What is Special Relativity? 2.2 Different Approaches to Special Relativity 2.2.1 The Usual Einstein’s Approach 2.2.2 The Usual Covariant Approach 2.2.3 The Space-Time Geometric Approach 2.3 The Myth About the Incorrectness of Galilean Transformations 2.4 The Non-Relativistic Limit of Lorentz Transformations 2.5 The Myth about the Constancy of the Speed of Light 2.6 Convention-Dependent Aspects of the Theory 2.7 Relativistic Time Dilation and Length Contraction 2.8 Relativistic Particle Dynamics 2.9 Commonly Used Method of Coupling Fields and Particles 3 Space-Time and Its Coordinatization 3.1 Choosing a Coordinate System in an Inertial Frame 3.2 The Inertial Frame Where a Light Source is at Rest 3.3 Motion of a Light Source Relative to an Inertial Frame 3.4 Discussion 3.5 A Clock Re-Synchronization Procedure 3.6 A Moving Light Source: Peculiarities of Collinear Geometry 4 Aberration of Light in an Inertial Frame of Reference 4.1 The ”Plane Wave” Emitter 4.2 A Moving Emitter: Galilean Transformations-Based Explanation 4.3 A Moving Emitter: Lorentz Transformations-Based Explanation 4.4 Reflection from a Mirror Moving Parallel to Its Surface 4.5 Solving the Emitter-Mirror Problem in (3+1) Space-Time 4.6 A Moving Mirror: Lorentz Transformations-Based Explanation 4.7 Discussion 4.8 Large Aperture Mirror 4.9 Analysis of Transmission through a Hole in an Opaque Screen 4.10 Spatiotemporal Transformation of the Transmitted Light Beam 4.11 Applicability of Ray Optics 4.12 Moving Large Aperture Emitter 4.13 A Point Source in an Inertial Frame of Reference 4.14 Experimental Test: Reflection from a Moving Grating 4.15 Differences between the Light and Sound Aberration 5 Aberration of Light: Non-inertial Frame of Reference 5.1 Absolute Time Coordinatization in Accelerated Systems 5.2 Asymmetry Between Inertial and Accelerated Frames 5.3 Electrodynamic Explanation of Light Aberration 5.4 A Simple Explanation of Light Aberration in Accelerated Frames 5.5 Clock Resynchronization in Accelerated Systems 5.6 Composition of Motions in Non-Inertial Frames 5.7 Discussion 5.8 A Single Moving Emitter in an Accelerated Frame 5.9 A Point Source in an Accelerated Frame 5.10 Aberration of a Tilted Incoming Plane Wave 5.11 Non-Reciprocity in the Aberration of Light Theory 5.12 Two Metrics 5.13 Ether Theory and the Aberration of Light Effect 5.13.1 Lorentz’s Theory of Corresponding States 5.13.2 Analogy with the Sound Theory 6 Stellar Aberration 6.1 The Corpuscular Model of Light and Stellar Aberration 6.2 The Wave Theory of Stellar Aberration 6.3 Stellar aberration in the Context of Special Relativity 6.4 Defining a Physical Coordinate System in Space 6.5 Limits of Applicability 7 The Concept of Ordinary Space in Special Relativity 7.1 Inertial frame view of Observations of a Non-Inertial Observer 7.2 Relativity of Simultaneity 7.3 The Non-Existence of Instantaneous Three-Dimensional Space 7.4 Acceleration of a Rigid Body in Special Relativity 7.5 Discussion 8 Kinematics of Wigner Rotation 8.1 Composition of Lorentz Boosts 8.2 Wigner Rotation 8.3 Measuring Wigner rotation 8.4 Discussion 9 Aberration of Relativistic Particles 9.1 Explanation of Electron Aberration Based on Electrodynamics 9.2 Magnetic Field Measurements in an Accelerated Electron Source 9.3 Aberration of Particles from a Moving Source 9.4 Inertial Frame View of Observations by a Non-Inertial Observer 10 Earth-Based Setups to Detect the Aberration Phenomena 10.1 The Potential of Earth-Based Electron Microscopes 10.2 A Thought Experiment with a Point-Like Source 10.3 Practical Setup Using an Incoherent Optical Source 11 The Aberration of Light from a Laser Source 11.1 Theory of a Laser with an Optical Resonator 11.2 Absent of Light Aberration from an Earth-Based Laser 11.3 A Moving Laser Source in an Inertial Frame of Reference 12 Special Relativity and Reciprocal Symmetry 12.1 Pseudo-Gravitation Fields and the Langevin Metric 12.2 The Equivalence Principle and Gravitational Physics 12.3 Time Dilation 12.4 The Light-Clock and Observations by a Non-Inertial Observer 12.5 Metrics in Accelerated Frames 13 Relativistic Measurements 13.1 The Concept of Time in Special Relativity 13.2 Interference Phenomena 14 A Preferred Inertial Frame and Second Order Experiments 14.1 Second-Order Optical Experiments and Special Relativity 14.2 Features of Second Order Optical Experiments 14.3 Wigner Rotation and the Trouton-Noble Experiment 14.4 Time Dilation in Moving Atomic Clock 15 The Principle of Relativity and Modern Cosmology 16 Relativistic Particle Dynamics 16.1 Manifestly Covariant Formulation 16.2 Conventional Particle Tracking 16.3 Incorrect Expansion of the Relation for Arbitrary Motion 16.4 Standard Integration of the 4D Covariant Equation of Motion 16.5 Convention-Invariant Particle Tracking 16.6 Einstein’s Velocities Addition Vs. Covariant Addition 17 Mathematical Analysis of Relativistic Velocity Composition 17.1 Parametrization of Lorentz Transformations 17.2 Observer-Independence and Lorentz Covariance 17.3 Einstein Velocity Addition and Conventional Particle Tracking 17.4 Velocity Addition and the Wigner Rotation 17.5 Lorentz-Covariant Particle Tracking 18 Relativistic Spin Dynamics 18.1 Magnetic Dipole at Rest in an Electromagnetic Field 18.2 Derivation of the Covariant (BMT) Equation of Spin Motion 18.3 Changing Spin Variables 18.4 An Alternative Approach to BMT Theory 18.5 Spin Tracking 19 Relativity and Electrodynamics 19.1 The Dipole Approximation 19.2 An Illustrative Example 19.3 Electron Motion Accelerated by a Kicker in a Bending Magnet 19.4 Redshift of the Synchrotron Radiation Critical Frequency 20 Synchrotron Radiation 20.1 Introductory Remarks 20.2 Paraxial Approximation of the Radiation Field 20.3 Undulator Radiation 20.3.1 Conventional Theory 20.3.2 Why Did the Error in Insertion Device Theory Remain Undetected so Long? 20.3.3 Influence of the Kick According to Conventional Theory 20.3.4 Influence of the Kick According to Correct Coupling of Fields and Particles 20.3.5 Experimental Test of SR Theory in 3rd Generation Light Source 20.4 Synchrotron Radiation from Bending Magnets 20.4.1 Conventional Theory 20.4.2 Why Did the Error in Synchrotron Radiation Remain Undetected so Long? 20.4.3 Influence of the Kick According to Conventional Theory 20.4.4 Influence of the Kick According to Correct Coupling of Fields and Particles 20.5 Problem-Solving: Multiple Trajectory Kicks 20.6 Helical Trajectories and Synchrotron Radiation 20.6.1 Existing Theory 20.6.2 Methodology of Solving Problems Involving Boosts 20.6.3 On the Advanced ”Paradox” Related to the Coupling Fields and Particles 21 Relativity and X-Ray Free Electron Lasers 21.1 Introductory Remarks 21.2 Modulation Wavefront Orientation 21.3 XFEL Radiation Configuration 21.4 Modulation Wavefront Tilt and Maxwell’s Theory 21.5 Discussion 21.6 Experimental Test: Electrodynamics - Dynamics Coupling 21.7 Wavefront Tilt and Degradation of Electron Beam Modulation 22 Appendix 22.1 A1. Radiation from Moving Charges 22.2 A2. Undulator Radiation in the Far Zone 22.3 A3. Approximating the Electron Path 22.4 A4. Self-Fields of a Modulated Electron Beam 1 Introduction This book begins with a critical examination of current approaches to special relativity. Traditionally, the theory is framed around Einstein’s two postulates: the principle of relativity and the constancy of the speed of light. However, in a more general geometric approach, the principle of relativity is not a fundamental axiom but rather a consequence of space-time geometry. We emphasize that the core of special relativity lies in the following fundamental postulate: all physical processes occur in four-dimensional space-time, whose geometry is pseudo-Euclidean. This perspective shapes the presentation of the subject in this book, where the four-dimensional geometric formulation takes precedence over conventional approaches. The space-time geometric approach accommodates all possible choices of coordinates for reference frames, making the second Einstein postulate—the constancy of the coordinate speed of light—unnecessary in this broader framework. The coordinate speed of light remains isotropic and constant only in Lorentz coordinates, where Einstein’s synchronization of distant clocks and Cartesian space coordinates are applied. Thus, the basic elements of the space-time geometric formulation of the special relativity and the usual Einstein’s formulation, are quite different. It is important to emphasize that, in practical applications, there are two useful choices for clock synchronization conventions: (a) Einstein’s convention, which leads to the Lorentz transformations between reference frames. (b) The absolute time convention, which leads to the Galilean transformations between reference frames. Absolute time (or simultaneity) can be introduced in special relativity without altering either the logical structure or the convention-independent predictions of the theory. While this choice may seem unconventional within relativity, it is often the most practical when connecting theoretical results to laboratory reality. It is widely believed that only philosophers of physics discuss the issue of distant clock synchronization. Indeed, a typical physics laboratory does not contain a predefined space-time grid. It is important to recognize that a rule-clock structure exists only in our minds, and the use of hypothetical clocks in special relativity is a necessary prerequisite for applying dynamics and electrodynamics in a coordinate-based framework. This situation often leads physicists to assume that the theory of relativity can be applied to physical processes without a detailed understanding of the clock synchronization procedure. However, many problems in special relativity can only be adequately addressed using a non-standard absolute time synchronization approach. A distinctive feature of this book is its exploration of absolute time coordinatization—specifically, the use of Galilean transformations—within the framework of special relativity. Chapter 3 presents an ”operational interpretation” of Lorentz and absolute time coordinatizations, making it arguably the most important chapter of this book. Today, asserting the validity of Galilean transformations is considered a ”shocking heresy,” conflicting with the prevailing relativistic intuition and the widely accepted interpretation of special relativity among physicists. The distinction between absolute time synchronization and Einstein’s time synchronization, from an operational perspective, will be a significant revelation for any expert in special relativity. To our knowledge, neither the operational interpretation of absolute time synchronization nor this key distinction has been explored elsewhere in the literature. We begin by examining the phenomenon of the aberration of light. Light, as a special case of electromagnetic waves, is governed by electrodynamics. It is well known that electrodynamics fully complies with the principles of relativity and, therefore, must accurately describe the properties of light—an inherently relativistic entity. In Chapters 4 and 5, we undertake a critical reexamination of the existing theory of the aberration of light. Even at the first-order approximation in , the phenomenon remains complex, and the literature contains numerous incorrect results. To analyze optical phenomena involving the relative motion of two (or more) light sources, it becomes essential to employ electrodynamics within an absolute time framework. When the sources are independent, introducing Lorentz coordinates poses no issue, as each source can be assigned an individual coordinate system with its own set of clocks. However, the situation changes when a secondary source interacts with the light emitted by a primary source. A key challenge from the perspective of special relativity is that, in this case, a single inertial frame cannot accommodate a common Lorentz coordinatization for both sources. This fundamental difficulty necessitates a deeper examination of how electrodynamics should be applied in such scenarios. It is widely assumed that when a mirror moves tangentially to its surface, the law of reflection remains the same as for a stationary mirror. However, this common textbook assertion is incorrect. The first fundamental error in textbook explanations is conceptual. Observers typically describe the light beam from a stationary emitter using the Minkowski metric. Textbooks assume that the same metric applies to a moving mirror, incorrectly implying that both the emitter and mirror share a common Lorentz coordinatization within a single inertial frame. This assumption is flawed. The second error arises from the use of an idealized plane wave and an infinite plane mirror in the context of tangential motion. In this scenario, ”mirror motion” is not a physically observable effect. If the mirror were truly infinite, the problem would be time-independent. Only the motion of a finite mirror has physical significance. Experimental methods for measuring aberration increments rely on light signals and thus do not directly measure phase velocity (i.e., frequency and wavevector) but rather the group velocity. What most textbooks overlook is that energy transport is well-defined only when the apertures of both the source and the mirror are explicitly specified. Consider a point source and an observer, along with their respective measuring devices, all at rest in an inertial laboratory frame. We show that when a finite-aperture mirror moves tangentially at a constant velocity and interacts with radiation in the far field—where the point source effectively produces a plane wave at the mirror—the energy transport of the reflected light experiences a measurable deviation. This effect has important practical implications. According to Babinet’s principle, our theory of light aberration predicts that an analogous deviation occurs for light transmitted through a hole in a moving opaque screen, or equivalently, through the open end of a moving telescope barrel. Questions related to transmission through a transversely moving pupil detection system (e.g., the moving end of a telescope barrel) often lead to serious misunderstandings. These arise from an inadequate grasp of several complex aspects of statistical optics. Modern explanations suggest that the phenomenon of aberration of light can be interpreted using a ray model. A particularly interesting case is the transmission of light through a moving telescope tube. The rays of light from a star impinge on the telescope tube, and according to the prevailing literature, they do not interact with its sides. One might naively assume—following textbook reasoning—that the domain of ray optics applies to all spatially incoherent radiation. However, this is a misconception. A completely incoherent spatial source (such as an incandescent lamp or a star) consists of elementary, statistically independent point sources with different offsets. The characteristic dimension of each elementary statistically independent source is approximately , where is the wavelength of visible radiation. The radiation field generated by such a source can be considered a linear superposition of fields from individually incoherent sources. Each elementary source effectively produces a plane wave in front of the pupil detection system. Moreover, any measuring instrument inevitably influences the detected radiation due to the unavoidable diffraction of a plane wave by an aperture. It is important to note that a linear superposition of radiation fields from elementary point sources preserves fundamental single-point source characteristics, including their independence from source motion. Chapter 6 explores astronomical applications. Stellar aberration is often considered one of the simplest phenomena in astronomical observations. However, despite its apparent simplicity, it remains one of the most intricate effects of special relativity. A common belief is that stellar aberration depends on the relative velocity between the source (star) and the observer. However, observations clearly indicate that aberration is independent of the relative motion between stars and telescopes on Earth. The lack of symmetry, between the cases when either the source or telescope is moving is shown clearly on the basis of the separation of binary stars. The relative motion of these stars with respect to each other (and hence, with respect to the Earth) is never followed by any aberration, although the motion of these stars is, sometimes, much faster than that of the Earth around the Sun. It should be stressed that it is the telescope and not the star that must change its velocity (relative to the fixed stars) to cause aberration. The contradiction is so evident that some physicists argue stellar aberration conflicts with the special theory of relativity. Currently, no explanation accounts for why observational data on stellar aberration align with a moving Earth, yet the symmetric case—where the star has relative transverse motion—fails to produce compatible predictions. We show that the absence of widely separated binary stars does not necessitate a fundamental revision of physical principles. Instead, it highlights the need to interpret stellar aberration within a space-time geometric framework. The phenomenon of the aberration of light is often interpreted, within the corpuscular model of light, as analogous to the oblique fall of raindrops observed by a moving observer. This classical kinematic approach has been employed in astronomy for nearly three centuries to compute stellar aberration. This book presents a relativistic theory of the aberration of light in rotating frames. To explore the effects of relativistic modifications to stellar aberration, we begin with the classical aberration increment, given by , where is the observer’s velocity and is the velocity of the particles. It is assumed that . According to the conventional approach, by neglecting terms of order in comparison to 1, the stellar aberration increment simplifies to the elementary formula . This result is particularly notable, as the study of stellar aberration traditionally relies on classical kinematics, specifically the Galilean vectorial law of velocity addition. We conclude that the standard analysis of stellar aberration fails to account for the fundamental distinction between the velocity of light and the velocity of raindrops. A satisfactory treatment of relativistic modifications should be based on two key relativistic parameters. For an observer on Earth, the velocity relative to the solar reference frame is approximately 30 km/s, corresponding to Earth’s motion around the Sun. In the theory of stellar aberration, we typically consider the small expansion parameter , neglecting terms of the order of . However, in addition to this parameter, we must also account for the relativistic quantity . For light, , and according to the special theory of relativity, stellar aberration cannot be treated purely within the framework of classical mechanics. Light is always an ”ultra-relativistic” phenomenon, regardless of how small the ratio may be. Since physical processes occur within the metric structure of space-time, as dictated by the special theory of relativity, the geodesic interval must be used to accurately describe the phenomenon of stellar aberration. The challenges associated with Earth-based measurements are resolved by recognizing a fundamental asymmetry between Earth-based and Sun-based observers—specifically, the acceleration of the traveling Earth-based observer relative to the fixed stars. We demonstrate that explaining the aberration of light in a rotating frame of reference does not require modifying special relativity or invoking general relativity. Instead, a rigorous application of special relativity is sufficient. All phenomena in non-inertial reference frames should be analyzed within the space-time geometric framework using the metric tensor. Employing the Langevin metric in a rotating frame of reference allows us to account for all Earth-based experiments. A correct treatment of this problem in a rotating frame necessitates the use of the metric tensor, even for first-order effects in . This is because the cross term in the Langevin metric—which represents the first-order deviation of the metric tensor from its Minkowski form—plays a crucial role in the non-inertial kinematics of light beam propagation. The historical progression of studies on optical effects in rotating frames is quite unusual. The Sagnac effect was discovered in 1913 and later described by Langevin in 1921. It is intriguing that the Langevin metric has never been previously applied to stellar aberration theory. We demonstrate that the aberration of light is a complex phenomenon that must be categorized into various types based on their origin. These categories depend on the cause of the aberration—whether it arises from a change in the observer’s velocity (relative to the fixed stars) or the velocity of the light source. Furthermore, aberration can be further subdivided based on the physical influence of the optical instrument on the measurement. A theory of stellar aberration within the Earth-based frame of reference must account for two key observations: (1) the annual apparent motion of fixed stars relative to their positions, and (2) the absence of apparent aberration in rotating binary systems (they exhibit aberrations not different from other stars). We present a theory that addresses both of these phenomena while also providing new insights. All Earth-based experiments can be explained by considering the effect of the measuring instrument (i.e., the inevitable physical influence of the telescope on the measurement) and the acceleration of the Earth-bound observer relative to the fixed stars. In Chapter 10, we analyze the potential for using Earth-based sources to confirm predictions of the relativistic aberration of light theory. We introduce a simple scaling model for stellar aberration, emphasizing that the motion of stars relative to Earth is not accompanied by aberration. Notably, an aberration shift occurs even when a star moves at the same velocity as Earth. We derive a condition for optical similarity between the aberration of light from a distant star moving with Earth’s velocity and that from an Earth-based incoherent source. The proposed method for measuring the aberration angle leverages Earth-based sources and offers a significant advantage: Earth’s rotation should induce a corresponding shift in the observed image. This aberration shift depends solely on , the component of Earth’s orbital velocity perpendicular to its rotation axis. The apparent position of the source image is thus always a little displaced in the direction of the Earth’s motion around the Sun at that moment, and hence describes a small elliptical figure during the annular revolution of the Earth around its axis. In principle, observations could be recorded within a single day. Special relativity is generally considered a reciprocal theory. However, the aberration of light in an accelerated frame reveals a fundamental asymmetry between accelerated and inertial observers. Notably, without referencing anything external to the Earth-based accelerated frame, one could determine Earth’s velocity relative to the Sun-based frame by measuring the aberration of an Earth-based incoherent light source. While no such experiment has been conducted, astronomical observations confirm what the outcome would be. Many people who learn special relativity in the usual way find this argument unsettling. The reasoning behind the claim that aberration shift must be symmetrical typically proceeds as follows: A fundamental principle of special relativity states that the metric contains all the necessary information about the physics of a given situation as described in the chosen coordinates. The direction of an observer’s acceleration is encoded in the cross term of the Langevin metric. It is always possible to choose a coordinate system in which the metric of an accelerated frame becomes diagonal. This follows from the pseudo-Euclidean geometry of space-time. Maxwell’s equations hold in all inertial frames, ensuring that light propagates at the same velocity in each of them. This eliminates any privileged frame, implying that the concept of absolute motion has no physical meaning. In the argument above, steps (2) and (3) are correct, but step (1), and consequently step (4), are incorrect. Step (3) asserts that all inertial frames are equivalent with respect to physical laws, but not necessarily with respect to physical facts. At first glance, the diagonalization of the Langevin metric might seem to establish symmetry between inertial frames, but where does the asymmetry arise? The electrodynamics equations alone do not provide a complete description of the physical situation. To solve them, one must also specify the initial conditions. The time after diagonalization is readily obtained by introducing the time offset factor. This time shift introduces a rotation of the plane of simultaneity, which in turn leads to a rotation of the plane wavefront in the accelerated frame. Consequently, after metric diagonalization, the information about the observer’s acceleration is no longer contained in the metric but is instead encoded in the initial conditions—specifically, in the orientation of the radiation wavefront. One can conclude that not everything is relative in relativity, as the theory also encompasses certain absolute features. Many people assume that, because time and distance in the Lorentz coordinatization have direct physical significance, there must be an underlying physical (dynamical) reason for wavefront rotation. Some believe that the dynamics governed by physical fields are merely concealed within the language of pseudo-Euclidean geometry. However, the paradoxical asymmetry between inertial and accelerated reference frames is best explained through a dynamical perspective. The resolution of this asymmetry paradox identifies the inertial (pseudo-gravitational) force within the accelerated system as the cause of the asymmetry. By applying the principle of equivalence, one can address non-inertial kinematic problems using dynamical methods. In this context, wavefront rotation, which arises when transforming from an inertial frame to an accelerated frame (relative to the fixed stars), can be understood as the effect of a pseudo-gravitational potential gradient during the acceleration process. Several key points arise from the above results. One interesting question is why we are not discussing the aberration of light emitted by a laser. When considering light aberration, it is essential to distinguish between incoherent and laser sources. Naively, one might expect Earth’s motion around the Sun to introduce an aberration significant enough to affect precise laser-based observations. However, it is important to emphasize that laser light aberration cannot be measured. Inside a laser resonator, the electromagnetic wave travels back and forth, reflecting between mirrors. This repeated reflection effectively cancels out any asymmetry over successive round trips, preventing the manifestation of aberration. A crucial distinction exists between laser and incoherent sources. The radiation from an incoherent source depends on initial conditions, influenced by a mix of spatial and temporal factors—an effect akin to pseudo-gravity experienced by an accelerated observer. In contrast, in an optical resonator, the (3-space) boundary conditions dictate the direction of energy propagation. Many books claim that no experiment can determine which observer remains ”at rest” during acceleration because, according to the principle of relativity, only relative motion has physical significance. From this, they conclude that any observed asymmetry would contradict the principle of relativity. However, this argument is incorrect. There is no conflict between the fundamental structure of special relativity and the aberration of light phenomena. The principle of special relativity, as stated in Einstein’s writings and every textbook on the subject, asserts that the laws of physics are the same in all inertial frames. The principle of special relativity states that velocity is irrelevant to physical laws, but not to everything. It is widely assumed that covariant equations must yield covariant solutions. This notion, commonly found in textbooks, is incorrect. For example, from a mathematical perspective, the Lorentz covariance of electrodynamics equations does not necessarily ensure covariant solutions. Here, we emphasize a crucial point: the principle of special relativity, which states that all physical equations must remain invariant under Lorentz transformations (i.e., treating the pseudo-Euclidean geometry of space-time as an axiom of the theory), does not inherently impose reciprocal symmetry in nature. Textbook authors mistakenly assume that all inertial frames are equivalent. Relativistic (or reciprocal) symmetry is commonly associated with symmetry under Lorentz boosts. However, it is well known that Lorentz boosts alone do not form a group—the composition of two boosts in different directions results in a combination of a pure boost and a spatial (Wigner) rotation. From a mathematical perspective, symmetry between inertial frames requires that the relevant transformations form a group. Since Lorentz boosts alone do not satisfy this criterion, true symmetry between inertial frames is absent. Inertial frames are defined by motion at a constant velocity relative to the fixed stars. A fundamental distinction exists between an accelerated inertial frame (relative to the fixed stars) and one without an acceleration history. In the case of the aberration of light, this difference is closely linked to the Wigner rotation effect. It is important to note that all well-known tests of special relativity rely on second-order optical (interference) measurements, with the Michelson-Morley experiment being the most prominent example. In such experiments, light from a source is split into two beams, which later recombine to produce an interference pattern. It is not possible to detect the orbital velocity using second-order (interference) experiments. Phase, as a 4D invariant, is merely a number—it remains unchanged regardless of the chosen inertial frame or coordinate system. This invariance stems from the fundamental geometry of space-time. Consequently, in interference experiments, neither the Doppler effect nor the aberration of light exist as independent, well-defined physical phenomena; only phase has physical significance. This reasoning explains why special relativity predicts a null fringe shift in Earth-based frames. A careful analysis of all second-order optical experiments within an inertial frame confirms that observed phenomena remain unaffected by uniform motion relative to the fixed stars. While the standard formulation of relativity incorporates the principle of velocity irrelevance, this principle strictly applies in the limiting case of optical second-order measurements. Beyond interference-based experiments, several other second-order tests exist that do not involve light interference. This raises the question of whether it is possible to determine the motion of an initial inertial frame from within an accelerated frame using time dilation effects. We demonstrate that the slowing of a physical clock accelerated relative to the fixed stars is independent of the reference frame in which this effect is measured. In other words, time dilation is an absolute effect, not a relative one—a fundamental property of pseudo-Euclidean space-time. Now, let us return to the topic of metric diagonalization. In textbooks, time dilation is typically derived from the diagonal (Minkowski) metric. After diagonalization, these textbooks assert that coordinates in inertial frames should be related through the Lorentz transformation. At first glance, the problem appears completely symmetrical, suggesting that time dilation is a relative effect. However, something is evidently amiss. In Chapter 12, we explore the connection between time dilation and the Langevin metric. Perhaps the only way to demonstrate that time dilation is absolute is through a mathematical perspective. Mathematically, the argument for the absoluteness of time dilation follows this reasoning: The metric tensor must be a continuous quantity. The Langevin metric is obtained by smoothly transitioning between the metric tensors of an accelerated frame and an inertial frame. To achieve this smooth transition, it is sufficient to relate the coordinates and time of the accelerated observer to those of the inertial observer using a Galilean boost. Under these conditions, the inertial frame’s metric naturally evolves into the Langevin metric of the accelerated frame. However, after diagonalization, the metric tensor in the accelerated frame must abruptly shift from the Langevin metric to the Minkowski (diagonal) form. This discontinuity leads to a critical conclusion: when applying the Lorentz transformation, if the transformation fails to preserve the continuity of the metric tensor, then symmetry between inertial frames does not hold. Thus, any change in velocity or acceleration—relative to the fixed stars—has an absolute significance. The formalism of relativistic physics is fundamentally tied to the concept of an absolute space-time structure. This structure is inherently linked to an initial inertial frame, which, in turn, serves as an absolute rest frame. Consequently, uniform motion is not relative, and a particle’s absolute velocity can be determined within this initial frame. The key distinction between the initial inertial frame and Newton’s concept of absolute space lies in the lessons of special relativity, which introduced a unified space-time model. In this framework, Newton’s separate notions of absolute space and absolute time are merged into a single, cohesive space-time continuum. The proper time interval can be associated with various physical phenomena, including particle lifetimes, atomic transition periods, and nuclear half-life. Until now, all tests of time dilation using atomic clocks in an Earth-based frame have involved circular path geometries. For instance, atomic clocks flown on commercial jets were compared with a reference clock upon returning to the laboratory, ensuring that both were at rest relative to the Earth. According to special relativity, any time dilation experiment investigating the effects of orbital velocity should use a linear (one-way) path geometry. This is typically observed in experiments involving the decay of rapidly moving unstable particles. However, current experimental techniques lack the sensitivity required to detect the influence of orbital velocity on the average distance traveled by a decaying particle. In contrast, we describe (see Chapter 10) a simple, first-order optical experiment accessible to most undergraduate physics departments. This experiment illustrates key principles in the foundations of special relativity. Specifically, it demonstrates that the Earth’s speed relative to a Sun-based reference frame could be determined using measurements of aberration from an Earth-based incoherent light source. In Chapter 16, we describe the particle dynamics method in the Lorentz lab frame, using lab time as the evolution parameter. The study of relativistic particle motion in a constant magnetic field, as traditionally approached in accelerator engineering, remains closely tied to classical (Newtonian) kinematics. Specifically, it relies on the Galilean vectorial law for velocity addition. This approach, which is non-covariant, implicitly assumes an absolute time coordinatization—though this remains hidden within the formalism. The absolute time synchronization convention is self-evident, which is why it is rarely discussed in accelerator physics. Standard textbooks suggest that when only a single frame is considered, a detailed understanding of relativity is unnecessary. In this conventional approach, the only modification to classical mechanics is the introduction of relativistic mass, thereby maintaining a distinction between time and space. From the lab frame perspective, conventional particle tracking treats a trajectory as the result of successive Galilean boosts that follow the motion of an accelerated particle. The preference for a non-covariant approach within the framework of dynamics stems from its simplicity and practicality. By avoiding the intermixing of spatial coordinates and time, the (3+1)-dimensional non-covariant tracking method remains intuitive, transparent, and well-suited to laboratory conditions. It can be demonstrated that this approach poses no fundamental difficulties in mechanics or electrodynamics. It is both effective and consistent, as the choice of transformation does not alter the underlying physical reality. What matters is that once a particular convention is adopted, it must be applied consistently across both dynamics and electrodynamics. A common mistake in accelerator physics arises from an incorrect algorithm for solving electromagnetic field equations. When using Maxwell’s equations, only the covariant formulation of the dynamics equations (i.e., in Lorentz coordinates) ensures the correct coupling between Maxwell’s equations and particle trajectories in the lab frame. Interestingly, employing the conventional coupling of Maxwell’s equations with a corrected Newtonian equation to calculate radiation from a moving source does not always lead to errors. In cases where the source moves rectilinearly along with the emitted light beam, both covariant and non-covariant approaches yield identical trajectories, making Maxwell’s equations consistent with conventional particle tracking. However, this method was mistakenly extended to non-collinear geometries, where it breaks down. To better understand this issue, we first examine the reasoning presented in textbooks. It is generally assumed that, in Lorentz coordinates, the magnetic field can only alter an electron’s direction of motion, not its speed. However, there is a counterargument to this widely accepted derivation of velocity composition. Consider the composition of two velocities that do not lie along a straight line. Conventional particle tracking, following Newtonian mechanics, treats velocity as an ordinary vector and combines them geometrically using the parallelogram method. However, this approach fails to account for relativistic effects and leads to inconsistencies in non-collinear motion. Solving problems related to covariant particle tracking presents two key challenges. First, conventional particle tracking within a single inertial frame is typically based on classical Newtonian mechanics, which does not incorporate Lorentz transformations. Second, there is considerable confusion regarding the combination of non-parallel velocities in special relativity. The standard textbook treatment of relativistic velocity transformation implicitly assumes that the axes of a moving observer remain parallel to the axes of the lab frame. In other words, it presumes that both observers share a common three-dimensional space—a misconception. Textbooks often overlook the fact that in Lorentz coordinatization, two observers following different trajectories experience different three-spaces. Special relativity lacks an absolute notion of simultaneity, leading to a fundamental mixing of spatial and temporal coordinates. For one observer, spatial measurements inherently include a contribution from the time coordinate as perceived by another observer. Consequently, there is no well-defined concept of an instantaneous three-space. At first glance, textbooks on special relativity typically derive the relativistic velocity addition formula analytically from the combination of Lorentz boosts. Many authors argue that the principle of relativity requires the velocity addition law to obey a group composition structure. However, the non-associativity of Einstein’s velocity addition for non-collinear velocities is often overlooked, despite being demonstrable through straightforward algebra. Consequently, constructing a fully covariant trajectory using only Einstein’s velocity addition is impossible—additional structure is needed. We emphasize that the addition of non-collinear velocities in Lorentz-coordinatized spacetime is governed by the Wigner rotation. A key consequence of the non-commutativity of non-collinear Lorentz boosts is the lack of a shared, global ”ordinary” space. It is crucial to recognize that Einstein’s velocity addition does not directly follow from the full structure of Lorentz transformations. Textbook derivations of transverse (perpendicular) velocity addition often invoke the relation , yet fail to account for the Wigner rotation, which introduces a coupling between transverse position and time. Only by solving the equations of motion within a fully covariant framework—one that includes the Wigner rotation in the treatment of non-collinear velocities—can the coupling between Maxwell’s equations and particle trajectories in the laboratory frame be properly described. Our analysis highlights the distinction between the concepts of path and trajectory. A path is a purely geometric notion, representing the spatial course of a particle without reference to time. In contrast, a trajectory provides more information, as it specifies not only the particle’s spatial position but also the corresponding time instant. The path has a precise, objective meaning. For example, in a magnetic field, the curvature radius of the path—and consequently, the three-momentum—is convention-invariant and thus objectively defined. However, the trajectory is convention-dependent, reflecting the inherent conventionality of velocity, and therefore lacks an exact objective meaning. While dynamical theory involves the concept of a particle’s trajectory, direct verification of it is unnecessary. Instead, the trajectory serves as a theoretical tool in the analysis of electrodynamics problems. In Chapter 16, we derived specific quantitative results for relativistic velocity addition using only elementary principles of special relativity. However, readers seeking a more mathematically rigorous treatment may prefer Chapter 17, where we present a comprehensive account of the mathematical structure underlying the composition of Lorentz boosts. In Chapter 19, we present a critical reexamination of radiation theory. To evaluate radiation fields generated by external sources, we must determine the particle velocity and position as functions of the lab-frame time . As discussed earlier, Maxwell’s equations should be solved in the lab frame, incorporating the current and charge density produced by particles moving along their covariant trajectories. For arbitrary values of , covariant calculations of radiation processes become highly complex. However, in certain cases, significant simplifications are possible. One such case is a non-relativistic radiation setup. The non-relativistic assumption enables the dipole approximation, which is of great practical importance. When considering only the dipole component of radiation, we disregard all details of the electron trajectory. Consequently, dipole radiation is insensitive to differences between covariant and non-covariant particle trajectories. However, this is merely the first and most practically relevant term in the expansion. To compute higher-order corrections beyond the dipole approximation, detailed knowledge of the electron trajectory is required. In such cases, relying on the non-covariant approach is insufficient, and the covariant trajectory must be used instead. It is generally assumed that Maxwell’s equations and the corrected form of Newton’s second law can explain all experiments conducted within a single inertial (laboratory) frame. According to standard textbooks, if only one frame is involved, knowledge of relativity is deemed unnecessary. Many physicists, having learned special relativity from textbooks, might say, “For those who want just enough understanding to solve problems, special relativity merely modifies Newton’s laws by introducing a correction factor to mass.” However, this is a misconception. The velocity of light transforms like that of a particle, and one cannot simply apply classical kinematics to mechanics while using relativistic kinematics for electrodynamics. We highlight errors in the standard approach to coupling fields and particles through a relatively simple example, ensuring that the core physical principles remain clear and unobscured by unnecessary mathematical complexity. We hope this example will draw greater attention to the crucial role of Wigner rotation in the transformation of non-collinear velocities. Accelerator physicists studying the application of relativity theory to synchrotron radiation often face a fundamental issue: the distinction between covariant and non-covariant particle trajectories has never been fully understood. Consequently, the contribution of relativistic kinematic effects to synchrotron radiation has been overlooked. Traditionally, accelerator physics has relied on Newtonian kinematics, which is incompatible with Maxwell’s equations. This raises an important question: if storage rings are designed without accounting for relativistic kinematics, how do they function? In reality, electron dynamics in storage rings are significantly influenced by radiation emission. We address this question in detail in Chapter 20, focusing on spontaneous synchrotron radiation from bending magnets and undulators. Just as the non-relativistic asymptote simplifies calculations, the ultrarelativistic asymptote does the same for covariant analysis, as it naturally leads to the paraxial approximation. The motion of electrons in a bending magnet exhibits cylindrical symmetry, which gives rise to several remarkable effects. We demonstrate that covariant particle tracking is unnecessary for deriving bending magnet radiation. However, there is one case where the conventional approach fails: the covariant framework predicts a nonzero redshift of the critical frequency due to perturbations of electron motion along the magnetic field. A direct laboratory test of synchrotron radiation theory could highlight the incompatibility between standard relativistic electrodynamics—based on Maxwell’s equations—and non-covariant particle tracking. Despite decades of measurements, synchrotron radiation theory remains experimentally unverified in certain aspects. We explore the potential of synchrotron radiation sources to test the predictions of a revised synchrotron radiation theory, proposing experiments that have never been conducted at existing facilities. The theory of relativity demonstrates that the relativity of simultaneity—a fundamental effect of relativistic kinematics—is closely linked to extended relativistic objects. Until the 21st century, no macroscopic objects were known to travel at relativistic speeds, and it was widely believed that only microscopic particles in experiments could reach velocities close to the speed of light. However, the 2010s saw rapid advancements in laser light sources in the X-ray wavelength range. An X-ray free electron laser (XFEL) is an example where improvements in accelerator technology make it possible to develop ultrarelativistic macroscopic objects with an internal structure (modulated electron bunches), and the relativistic kinematics plays an essential role in their description. In Chapter 21, we critically reexamine existing XFEL theory, particularly for readers with limited knowledge of accelerator physics. We focus on the production of coherent undulator radiation by a modulated ultrarelativistic electron beam. Fortunately, understanding this process does not require prior knowledge of undulator radiation theory from Chapter 20, as it can be explained in a straightforward manner. Relativistic kinematics plays a fundamental role in XFEL physics, particularly through the rotation of the modulation wavefront, which, in the ultrarelativistic approximation, is closely linked to the relativity of simultaneity. When particle trajectories are analyzed in a Lorentz reference frame—an inertial frame where time coordinates are assigned using the Einstein synchronization procedure—relativistic kinematic effects, such as the relativity of simultaneity, must be considered. In the ultrarelativistic limit, the orientation of the modulation wavefront, i.e. the orientation of the plane of simultaneity, is always perpendicular to the electron beam velocity when the evolution of the beam’s evolution is described using Lorentz coordinates. It is important to recognize that Maxwell’s equations are valid only within Lorentz reference frames. Consequently, Einstein’s concept of time ordering must be applied consistently in both dynamics and electrodynamics. According to Maxwell’s equations, the wavefront of a laser beam is always orthogonal to its direction of propagation. Notably, in the framework of special relativity, a modulated electron beam in the ultrarelativistic (asymptotic) limit exhibits the same kinematic behavior as a laser beam when described in Lorentz coordinates. Experimental evidence confirms the accuracy of this prediction. The theory of relativity, formulated within a space-time framework based on pseudo-Euclidean geometry, has been developed for over a century. Recently, it has experienced a rapid expansion in practical applications, particularly in XFEL physics. 2 A Critical Survey of Present Approaches to Special Relativity 2.1 What is Special Relativity? Special relativity is founded on the principle that the laws of physics are invariant under Lorentz transformations. This principle is inherently restrictive: it does not dictate the specific form of the dynamics involved, but rather constrains any physical theory to be consistent with Lorentz symmetry. Understanding the postulates of special relativity is conceptually similar to understanding the principle of energy conservation. Initially, we encounter energy conservation as a general rule. Later, we seek deeper, microscopic theories—such as classical mechanics or quantum field theory—that not only comply with this principle but also explain the underlying mechanisms. These deeper theories must, of course, recover energy conservation as a consequence of their more fundamental laws. The utility of the energy conservation principle lies in its ability to guide analysis, even when the full fundamental theory is not yet known. A similar methodological perspective applies to the postulates of special relativity. For example, suppose we do not fully understand the microscopic mechanism behind muon decay. However, if we know the decay law in the muon’s rest frame, we can treat this as a phenomenological law. By applying the principles of special relativity, we can generalize this decay law to any Lorentz frame and make testable predictions. In particular, if we transform the decay law to the laboratory frame using a Lorentz boost, we find that the population of muons is reduced to half its original value after they travel a distance of , where is the Lorentz factor and is the muon’s proper lifetime. This implies that, from the lab’s perspective, the muon’s lifetime appears to be extended to . Such predictions are direct consequences of the relativistic transformation of time intervals. Despite its successes, special relativity is not a complete theory. It is a restrictive framework—more a set of symmetry constraints than a constructive (microscopic) explanation. Constructive theories, such as electrodynamics or quantum field theory, provide a more detailed understanding of physical phenomena. Special relativity, in this context, serves primarily as a kinematical interpretation of how physical processes appear from different inertial frames. Importantly, all relativistic effects can, in principle, be derived directly from underlying physical laws—without invoking relativity as an independent framework—provided those laws are Lorentz covariant. For instance, muons in motion exhibit relativistic behavior because the quantum field equations governing their interactions and decay are Lorentz invariant. In the ”microscopic” view of relativistic phenomena, the Lorentz covariance of the fundamental laws of physics remains, much like energy conservation, an empirical and unexplained fact—but ultimately, all explanations must terminate at some foundational level. 2.2 Different Approaches to Special Relativity In literature, three main approaches to special relativity are discussed: Einstein’s approach, the standard covariant approach, and the space-time geometric approach. Einstein’s formulation is based on two fundamental postulates: the principle of relativity and the constancy of the speed of light. The standard covariant formulation describes special relativity in terms of pseudo-Euclidean space-time geometry and the invariance of the space-time interval . However, this interpretation is often limited to cases where the metric is strictly diagonal. Assuming a diagonal metric implicitly imposes Lorentz coordinates, ensuring that different inertial frames are related by Lorentz transformations. The space-time geometric approach, on the other hand, places primary emphasis on the geometry of space-time. In this framework, space-time is assumed to have a pseudo-Euclidean structure, and only four-tensor quantities are considered physically meaningful. Unlike Einstein’s formulation, where the principle of relativity is postulated, in this approach, it naturally follows from the geometric properties of space-time. Since the space-time geometric approach accommodates all possible coordinate choices within a given reference frame, Einstein’s second postulate—asserting the constancy of the coordinate speed of light—does not generally hold. The coordinate speed of light remains isotropic and constant only in Lorentz coordinates, where Einstein’s synchronization of distant clocks and Cartesian space coordinates are used. 2.2.1 The Usual Einstein’s Approach Traditionally, the special theory of relativity is founded on two key postulates: Principle of relativity. The laws of nature are the same (or take the same form) in all inertial frames Constancy of the speed of light. Light propagates with constant velocity independently of the direction of propagation, and of the velocity of its source. However, the constancy of the speed of light in all inertial frames is not a fundamental statement of relativity. The core principle of special relativity is the Lorentz covariance of all fundamental physical laws. Contrary to common textbook presentations, the second postulate is not an independent physical assumption but rather a convention that cannot be experimentally tested. By assuming the constancy of the speed of light in all inertial frames, we implicitly adopt Lorentz coordinates and assume that different inertial frames are related by Lorentz transformations. In this restrictive interpretation of relativity, only Lorentz transformations are used to map event coordinates between inertial observers. 2.2.2 The Usual Covariant Approach In the standard covariant approach, special relativity is formulated as a theory of space-time with pseudo-Euclidean geometry. Physical quantities are represented by tensors in four-dimensional space-time, making them covariant, and the laws of physics are expressed as four-tensor equations to ensure manifest covariance. To develop space-time geometry, a metric or measure of space-time intervals must be introduced. The choice of measure determines the nature of the geometry. In this framework, an event is mathematically represented by a point in space-time, called a world-point, while the evolution of a particle is depicted as a curve in space-time, known as a world-line. If represents the infinitesimal displacement along a particle’s world-line, then \| \| \| --- \| \| \| (1) \| where we have selected a special type of coordinate system (a Lorentz coordinate system), defined by the requirement that Eq. (1) holds. To simplify our writing we will use, instead of variables , variables . Then, by adopting the tensor notation, Eq. (1) becomes , where Einstein summation is understood. Here are the Cartesian components of the metric tensor and by definition, in any Lorentz system, they are given by , which is the metric canonical, diagonal form. As a consequence of the space-time geometry, Lorentz coordinate systems are connected by Lorentz transformations. Physical quantities are represented by space-time geometric (tensor) entities. When a basis is introduced, a tensor as a geometric object consists of both its components and the basis itself. In the conventional covariant approach, one typically considers only the components of tensors in a Lorentz coordinate system—i.e., under the assumption that the basis four-vectors are orthogonal. Consequently, physical laws are expressed exclusively through four-tensor equations written in component form. However, in this approach, the definition of a tensor relies on the transformation properties of its components. For instance, in standard covariant formalism, the electromagnetic ”tensor” is not actually a full tensor; rather, it consists only of component values implicitly taken in an orthogonal basis. These components are coordinate-dependent quantities and do not fully describe the physical entity, as they lack explicit reference to the space-time basis itself. This limitation is not an issue when restricting transformations to those between orthogonal bases, such as Lorentz transformations, which are assumed to correctly map event coordinates. However, according to the conventional covariant approach, transformations from a standard orthogonal basis to a non-orthogonal one—such as Galilean transformations—are considered ”incorrect.” 2.2.3 The Space-Time Geometric Approach We emphasize the significant freedom in choosing a coordinate system for Minkowski space-time. The space-time continuum, defined by the interval in Eq. (1), can be described using arbitrary coordinates, not just Lorentz coordinates. When transitioning to an arbitrary coordinate system, the geometry of four-dimensional space-time remains unchanged. In special relativity, there are no restrictions on the choice of coordinate system: the spatial coordinates can be any parameters defining the position of particles, while the time coordinate can correspond to any arbitrary clock. The components of the metric tensor in the coordinate system can be determined by performing the transformation from the Lorentz coordinates to the arbitrary variables , which are fixed as . One then obtains \| \| \| --- \| \| \| (2) \| In textbooks and monographs, special relativity is typically presented in terms of an interval expressed in the Minkowski form of Eq. (1), whereas Eq. (2) is often associated with general relativity. However, in the space-time geometric approach, special relativity is fundamentally a theory of four-dimensional space-time with pseudo-Euclidean geometry. In this formulation, the space-time continuum can be described equivalently in any coordinate system, provided that remains invariant. In contrast, the conventional presentation of special relativity also relies on the invariance of but interprets it in a more restricted sense, assuming a strictly diagonal metric. Textbook presentations of the special theory of relativity typically follow Einstein’s approach or, more generally, the standard covariant approach, which, as discussed above, deals only with the components of 4-tensors in specific (orthogonal) Lorentz bases. However, these presentations often overlook the fact that transitioning to arbitrary coordinates does not alter the underlying geometry of space-time. As a result, a common misconception among experts is that transforming from an orthogonal Lorentz basis to a nonorthogonal one is incorrect, while a Lorentz transformation—between two orthogonal Lorentz bases—is valid. This belief is unfounded. Physics can be described in any coordinate system; changes in coordinates merely affect how the components of 4-tensors are expressed, without altering the tensors themselves. Although Einstein synchronization (i.e., the choice of Lorentz coordinates) is preferred by physicists for its simplicity and symmetry, it is no more ”physical” than any other coordinate system. In this book, we will explore an especially unconventional choice: the absolute time coordinate system. 2.3 The Myth About the Incorrectness of Galilean Transformations The role of Galilean transformations within the framework of relativity requires careful reconsideration. Many physicists continue to view Galilean transformations—specifically, transformations from an orthogonal Lorentz basis to a non-orthogonal one—as outdated and incorrect when applied to spatial coordinates and time. A common argument against their validity is that they fail to preserve the form-invariance of Maxwell’s equations under a change of inertial frame. This reasoning is frequently found in well-known textbooks and monographs. 111It is widely believed that Galilean transformations, being pre-relativistic, are incompatible with the special theory of relativity. For instance, Bohm states: “… the Galilean law of addition of velocities implies that the speed of light should vary with the speed of the observing equipment. Since this predicted variation is contrary to fact, the Galilean transformations evidently cannot be correct.” Similar assertions appear in more recent pedagogical literature. Drake and Purvis , for example, write: “One of the great insights of relativity theory was the realization that Galilean transformations are incorrect. The proper way to translate space-time measurements between inertial frames is through Lorentz transformations.” However, the assertion that Galilean transformations are inherently incorrect is a misconception. Special relativity is fundamentally a theory of four-dimensional space-time with a pseudo-Euclidean geometry. From this perspective, the principle of relativity naturally emerges as a geometric property of space-time, which can be described using arbitrary coordinate systems. 222The mathematical claim that transitioning to arbitrary coordinates leaves the geometry of space-time unchanged is considered erroneous in standard textbooks. As L. Landau and E. Lifshitz note : ”This formula is called the Galilean transformation. It is easy to verify that this transformation, as was to be expected, does not satisfy the requirements of the theory of relativity; it does not leave the interval between events invariant.” This misconception is attributed to a failure to distinguish between the convention-dependent and convention-invariant aspects of the theory. In pseudo-Euclidean geometry, the interval between events remains invariant under arbitrary coordinate transformations. Thus, contrary to the standard presentation in most textbooks, Galilean transformations are in fact compatible with the principle of relativity, even though they modify the explicit form of Maxwell’s equations. 2.4 The Non-Relativistic Limit of Lorentz Transformations It is commonly assumed that the Lorentz transformation reduces to the Galilean transformation in the non-relativistic limit. However, we argue that this widely accepted textbook claim is incorrect and misleading. Kinematics inherently involves a comparative study between two coordinate systems, requiring the assignment of time coordinates to both. The method of clock synchronization determines these time coordinates, and different synchronization conventions lead to different time assignments. In Minkowski space-time, choosing a synchronization convention is equivalent to selecting a specific coordinate system within an inertial frame of reference. Practical considerations may favor one coordinate system over another in different contexts. In relativistic engineering, we typically choose between the absolute time coordinate and the Lorentz time coordinate. Importantly, the space-time continuum can be equally well described in both frameworks. This implies that, for arbitrary particle speeds, Galilean coordinate transformations effectively characterize the change in reference frame from a laboratory inertial observer to a comoving inertial observer, even within the framework of relativity. Now, let us examine the non-relativistic limit. For small enough that terms of order can be neglected, the Lorentz transformation simplifies to , . This infinitesimal Lorentz transformation differs from the infinitesimal Galilean transformation , . The difference lies in the additional term in the Lorentz transformation for time, which is a first-order correction. We wish to emphasize the following point: An infinitesimal Lorentz transformation differs from the Galilean transformation only by incorporating the relativity of simultaneity. This is the sole relativistic effect that appears in the first order of . Other higher-order effects, such as Lorentz-Fitzgerald contraction and time dilation, arise from higher-order terms and can be derived mathematically by iterating the infinitesimal transformation. 333There is a common misconception that the Lorentz transformation reduces to the Galilean transformation in the non-relativistic limit. For example, French states, ”The reduction of to the Galilean relation requires as well as ”. Similar statements are found in recent textbooks, such as Rafelski , who asserts: ”A wealth of daily experience shows that the Galilean coordinate transformation (GT) is correct in the nonrelativistic limit in which the speed of light is so large that it plays no physical role. Any coordinate transformation replacing the GT must also agree with this experience, and thus must contain the GT in the nonrelativistic limit.” This conclusion is mathematically flawed. As Baierlein points out: ”If the Lorentz transformation for infinitesimal were to reduce to the Galilean transformation, then the iterative process could never generate a finite Lorentz transformation that is radically different from the Galilean transformation.” The primary distinction between Lorentz coordinatization and absolute time coordinatization lies in the transformation laws that relate coordinates and time between relatively moving systems. It is incorrect to assert, as some textbooks do, that the expression reduces to the Galilean relation in the non-relativistic limit. Such a claim would imply that infinitesimal Lorentz transformations become identical to infinitesimal Galilean transformations in this limit, which is not the case. The essence of Lorentz (or Galilean) transformations is captured in their infinitesimal form: relativistic kinematic effects cannot be derived simply by iterating infinitesimal Galilean transformations through a mathematical procedure. 2.5 The Myth about the Constancy of the Speed of Light It is widely believed that experiments demonstrate the independence of the speed of light in a vacuum from both the source and the observer 444Many textbook authors still treat conventional quantities as if they are empirically measurable. For instance, Cristodoulides states: ”The fact that the Galilean transformation does not preserve Maxwell’s equations has already been mentioned […] On the other hand, experiments show that the speed of light in a vacuum is independent of the source or observer.”. This claim is commonly presented in textbooks but is, in fact, incorrect. The constancy of the speed of light is not an empirical fact but rather a consequence of the chosen synchronization convention, making it impossible to test experimentally 555Since we can empirically access only the round-trip average speed of light, any claims about the magnitude and isotropy of the one-way speed of light depend on the chosen time coordinate system. Such values vary with the synchronization scheme used. As Anderson, Vetharaniam, and Stedman state: ”No experiment, then, is a ’one-way’ experiment. An empirical test of any property of the one-way speed of light is not possible. Such quantities as the one-way speed of light are irreducibly conventional in nature, and recognizing this aspect is to recognize a profound feature of nature.”. To measure the one-way speed of light, one must first synchronize an infinite set of clocks distributed across space to enable time measurements. However, an inherent circularity arises when clocks are synchronized under the assumption that the one-way speed of light is . If the synchronization process is based on this assumption, then any measurement using these clocks will necessarily yield , as the measurement framework has been defined accordingly. Thus, the one-way speed of light is a matter of convention rather than a physical quantity with empirical meaning. In contrast, the two-way (or round-trip) speed of light has physical significance because it is directly measurable without reliance on clock synchronization. Round-trip experiments involve observing simultaneity (or lack thereof) at a single spatial location, making them independent of any synchronization convention. All well-established methods for measuring the speed of light are, in fact, round-trip measurements. A prime example is the Michelson-Morley experiment, which employs an interferometer to compare light beams in a two-way manner. Apparent paradoxes emerge when Galilean transformations are applied within electrodynamics. Many sources claim that Galilean velocity transformations are inconsistent with the electron-theoretic explanations of optical phenomena such as refraction and reflection. This notion is widespread in standard literature. Pauli, for instance, states : ”It is essential that the spherical waves emitted by the dipoles in the body interfere with the incident wave. If we consider the body at rest while the light source moves relative to it, the waves emitted by the dipoles will have a velocity different from that of the incident wave, making interference impossible.” However, such claims stem from a misunderstanding. It is often asserted that when a light source moves relative to a stationary medium (like glass), the resulting wave emitted by the oscillating electrons in the glass cannot interfere properly with the incident wave due to differing velocities. This is incorrect. Regardless of the light’s velocity, the frequency of the incident wave determines the oscillation frequency of the electrons in the medium. These electrons then re-emit light of the same frequency. Therefore, both the incident and re-emitted waves maintain the same frequency and can interfere. The velocity difference merely results in a spatially varying phase difference, not a breakdown in interference. 2.6 Convention-Dependent Aspects of the Theory Consider the motion of a charged particle in a given magnetic field. According to the theory of relativity, the trajectory of the particle in the lab frame depends on the chosen synchronization convention for clocks in that frame. Whenever a theory involves an arbitrary convention, it is essential to examine which aspects of the theory depend on that choice and which do not. The former are convention-dependent, while the latter are convention-invariant. Clearly, physically meaningful measurement results must be convention-invariant. Now, consider the motion of two charged particles in a given magnetic field, which controls their trajectories. Suppose there are two apertures located at points and . From the solution of the equations of motion, we conclude that the first particle passes through aperture while the second passes through aperture simultaneously. The occurrence of these two events—particles passing through and —has an exact objective meaning, making them convention-invariant. However, the simultaneity of these events is convention-dependent and lacks an exact objective meaning. It is important to emphasize that, consistent with the conventionality of simultaneity, the value of the particle’s speed is also a matter of convention and does not possess a definite objective meaning. To determine which aspects of dynamical theory depend on a chosen convention and which do not, we examine the distinction between the notions of path and trajectory. Consider the motion of a particle in three-dimensional space, described by the vector-valued function The particle follows a prescribed curve, or path, as it moves. This motion along the path is parameterized by , where represents a certain parameter—specifically, in our case of interest, the arc length. Now, if we take the origin of a Cartesian coordinate system and connect it to a point on the path, the resulting vector is the position vector . 666For a more detailed discussion on the distinction between path and trajectory, see . The key difference between trajectory and path is fundamental: the path has an exact, objective meaning—it is independent of any convention. In contrast, and consistent with the inherent conventionality of velocity, the trajectory is convention-dependent and lacks an exact objective interpretation. To illustrate this distinction, consider experiments in accelerator physics. Suppose we wish to measure a particle’s momentum. A uniform magnetic field can be used to construct a momentum analyzer for high-energy charged particles. Importantly, this method of determining momentum is convention-independent. The curvature radius of the path within the magnetic field—and consequently, the three-momentum—has an objective, convention-invariant meaning. While dynamical theory includes the concept of a particle’s trajectory, this quantity is not directly measured but instead serves as a tool in the analysis of electrodynamics problems. 2.7 Relativistic Time Dilation and Length Contraction Experts in the theory of relativity often mistakenly conflate the fundamental properties of Minkowski space-time with the specific forms that certain convention-dependent quantities take under standard Lorentz coordinatization. These quantities, commonly referred to as ”relativistic kinematic effects,” include time dilation, length contraction, and Einstein’s velocity addition. There is a widespread misconception that such effects have direct physical significance, rather than being artifacts of the chosen synchronization convention. Relativistic kinematic effects are inherently coordinate-dependent and lack absolute physical meaning. As noted by Leubner, Auflinger, and Krumm , many physicists incorrectly equate the standard formulations of these effects with relativity itself. Their study demonstrates that adopting a different synchronization convention can lead to significant changes in the appearance of these quantities. For instance, under Lorentz coordinatization, one observes time dilation. However, in a framework based on absolute time coordinatization, relativistic kinematic effects disappear—no time dilation is observed. Despite such variations, all coordinate-independent quantities, such as the particle path and momentum , remain unaffected by changes in clock synchronization. It is a misconception to believe that physics must rely solely on concepts that are directly measurable. Special relativity, when formulated in Lorentz coordinates, includes constructs such as time dilation and length contraction that are not directly observable. Nevertheless, these constructs are essential for making theoretical predictions. The strength of a theory lies in its predictive power, not in the observability of every individual element. This idea can be further illustrated. Both the velocity of an electron and the speed of light are convention-dependent quantities. In absolute-time coordinatization, the speed of light differs from the electrodynamic constant , as Maxwell’s equations are not invariant under Galilean transformations. Conversely, in Lorentz coordinatization, the speed of light equals , while the electron’s velocity is changed due to relativistic effects. Yet the dimensionless ratio of the electron’s velocity to the speed of light remains convention-independent. This ratio, unaffected by changes in clock synchronization or rhythm, forms the basis for physically meaningful predictions in electrodynamics—for example, in the analysis of synchrotron radiation emitted by bending magnets. 2.8 Relativistic Particle Dynamics The accelerated motion of a relativistic charged particle is governed by a covariant equation of motion under the influence of the four-force in the Lorentz lab frame. The particle’s trajectory, , as observed in this frame, results from a sequence of infinitesimal Lorentz transformations. Importantly, the lab frame time in the equation of motion cannot be treated as independent of the spatial coordinates. This dependence arises because Lorentz transformations inherently mix space and time coordinates, with relativistic kinematic effects manifesting as a consequence of the relativity of simultaneity. Let us consider the conventional approach to particle tracking. It is generally accepted that, to describe the dynamics of relativistic particles in an inertial laboratory reference frame, one only needs to account for the relativistic dependence of momentum on velocity. The treatment of relativistic particle dynamics is based on a modified form of Newton’s second law. In a given lab frame, there is an electric field and magnetic field . They push on a particle in accordance with \| \| \| --- \| \| \| (3) \| \| \| (4) \| where here the particle’s mass, charge, and velocity are denoted by , , and respectively. The Lorentz force law, together with measurements of the acceleration components of test particles, serves to define the components of the electric and magnetic fields. Once these field components are determined from the acceleration of test particles, they can be used to predict the accelerations of other particles. This solution to the dynamics problem in the lab frame does not rely on Lorentz transformations. Conventional particle tracking treats space-time in a non-relativistic manner, as a (3+1) manifold. In this approach, the only modification to classical mechanics is the introduction of relativistic mass, while time remains distinct from space. There is no mixing of spatial and temporal coordinates. 777Clarification of the true content of the non-covariant theory can be found in various advanced textbooks. To quote e.g. Ferrarese and Bini : ”… within a single inertial frame, the time is an absolute quantity in special relativity also. As a consequence, if no more than one frame is involved, one would not expect differences between classical and relativistic kinematics. But in the relativistic context, there are differences in the transformation laws of the various relative quantities (of kinematics or dynamics), when passing from one reference frame to another.” These authors emphasize the unique role of a ”single inertial frame.” This term suggests that a defining characteristic of non-covariant theory is the absence of relativistic kinematics in describing particle motion. A similar viewpoint is expressed by Friedman : ”Within any single inertial frame, things look precisely the same as in Newtonian kinematics: there is an enduring Euclidean three-space, a global time , and the law of motion.” Many fascinating phenomena arise when charges move under the influence of electromagnetic fields, often in highly complex situations. However, here we focus on a simpler case: the accelerated motion of particles in a constant magnetic field. In a non-covariant treatment, a magnetic field can change only the direction of a particle’s motion, not its speed (or mass). The study of relativistic particle motion in a constant magnetic field—common in accelerator physics—appears similar to its non-relativistic counterpart in Newtonian dynamics and kinematics. The trajectory of a particle, , derived from the corrected form of Newton’s second law, does not account for relativistic kinematic effects, as it relies on the Galilean vector addition of velocities and disregards the relativity of simultaneity. Now, let us address the important problem of velocity addition in relativity. It is evident that the velocity of light transforms just as a particle’s velocity does. We cannot use one set of kinematics for a particle’s velocity and another for the velocity of light. Consider a scenario where a light source in the lab frame is accelerated from rest to a velocity . Suppose that, in the Lorentz inertial frame where the light source is at rest, an observer sees light propagating with velocity . How will this appear to an observer in the lab frame? According to the relativistic velocity addition formula, the answer remains . Maxwell’s equations retain their form under Lorentz transformations. However, these transformations lead to non-Galilean velocity transformation rules. Consequently, special relativity dictates that if Maxwell’s equations are to hold in the lab frame, particle trajectories must incorporate relativistic kinematic effects. That is, Maxwell’s equations can be consistently applied in the lab frame only if particle trajectories are described as the result of successive infinitesimal Lorentz transformations. The absence of relativistic kinematic effects is a fundamental flaw in conventional (3+1) non-covariant particle tracking, making it clearly inconsistent with Maxwell’s electrodynamics. The relativity of simultaneity—i.e., the interdependence of spatial positions and time—is entirely absent from this framework. Consequently, the approach of coupling fields and particles within a ”single inertial frame” is not just flawed but fundamentally and profoundly incorrect. 2.9 Commonly Used Method of Coupling Fields and Particles The electrodynamics problem is generally considered solvable within a ”single inertial frame” framework, without requiring Lorentz transformations. Standard derivations assume that Maxwell’s equations, along with a corrected version of Newton’s second law, can fully account for all experiments conducted within a single inertial frame, such as the laboratory reference frame. Going to the electrodynamics problem, the differential form of Maxwell’s equations describing electromagnetic phenomena in the same inertial lab frame (in cgs units) is given by the following expressions: \| \| \| --- \| \| \| (5) \| \| \| (6) \| \| \| (7) \| \| \| (8) \| Here the charge density and current density are written as \| \| \| --- \| \| \| (9) \| \| \| (10) \| where is three-dimensional delta function, , and denote the mass, charge, position, and the velocity of the th particle, respectively. To evaluate radiation fields arising from external sources Eq. (8) we need the velocity and the position as a function of lab frame time . It is generally accepted by the physics community that the equation of motion, which describes how coordinates of the particle carrying the charge change with time , is described by corrected Newton’s second law Eq. (4). This coupling of Maxwell’s equations with the corrected Newtonian equation is widely recognized as a valuable approach in accelerator physics, particularly in analytical and numerical studies of radiation properties. This method of treating relativistic dynamics and electrodynamics often leads accelerator physicists to assume that designing particle accelerators is possible without a deep understanding of the theory of relativity. However, a common misconception in accelerator physics arises from the distinction between the trajectories and . To examine this issue from the perspective of electrodynamics of relativistically moving charges, consider the evaluation of fields generated by external sources. This requires knowledge of their velocity and position as functions of the lab-frame time . Now, suppose one aims to compute the properties of radiation. Given our previous discussion, an important question emerges: Should one solve Maxwell’s equations in the lab frame using the current and charge density associated with a particle moving along the non-covariant trajectory ? We argue that the answer is no. The conventional approach—solving Maxwell’s equations in the lab frame with currents and charge densities based on non-covariant trajectories—is widely regarded as relativistically correct. However, this method contradicts the fundamental principles of special relativity. Remarkably, this critical issue has largely gone unnoticed in the physics community. The only correct way to ensure consistency between Maxwell’s equations and particle trajectories in the lab frame is to solve the equations of motion in fully covariant form. 3 Space-Time and Its Coordinatization 3.1 Choosing a Coordinate System in an Inertial Frame Let us explore an ”operational interpretation” of Lorentz and absolute time coordinatizations. Every physical phenomenon occurs in both space and time, and a concrete means of representing these dimensions is a frame of reference. A single space-time continuum can be described using different coordinate-time grids, corresponding to various frames of reference. Even the simplest space-time coordinate systems require careful definition. Clocks play a crucial role in tracking a particle’s motion through a coordinate-time grid. The general approach to determining a particle’s motion is as follows: at any instant, a particle possesses a well-defined velocity, , as measured in a laboratory frame of reference. But how is this velocity determined? It is found once coordinates are assigned in the lab frame and then measured over appropriate time intervals along the particle’s trajectory. However, measuring a time interval between events occurring at different spatial points poses a fundamental challenge. To do so—and thereby measure a particle’s velocity within a single inertial lab frame—it is first necessary to synchronize distant clocks. The concept of synchronization is central to understanding special relativity. There are multiple methods to synchronize distant clocks.888According to the thesis of the conventionality of simultaneity [13, 14, 15, 16, 17], simultaneity between distant events is a matter of convention, as it can be legitimately defined in different ways within any given inertial reference frame. Moeller , for example, states: ”All methods for the regulation of clocks meet with the same fundamental difficulty. The concept of simultaneity between two events in different places has no exact objective meaning at all since we cannot specify any experimental method by which this simultaneity could be ascertained. The same is therefore true also for the concept of velocity.” The choice of a synchronization convention is ultimately a matter of selecting a coordinate system within an inertial frame of reference in Minkowski space-time. The space-time continuum can be described using arbitrary coordinates. However, altering these coordinates does not change the underlying geometry of four-dimensional space-time. In special relativity, we are therefore unrestricted in our choice of a coordinate system. By leveraging the geometric structure of Minkowski space-time, one can define a class of inertial frames and adopt a Lorentz frame with orthonormal basis vectors. Within this chosen Lorentz frame, Einstein’s synchronization procedure—which relies on the constancy of the speed of light in all inertial frames—is applied, along with Cartesian spatial coordinates. 3.2 The Inertial Frame Where a Light Source is at Rest Let us provide an operational interpretation of Lorentz coordinatization. The fundamental laws of electrodynamics are governed by Maxwell’s equations, which state that light propagates at a constant velocity in all directions. This isotropy arises naturally from the structure of Maxwell’s theory, which lacks any intrinsic directional dependence. It is often asserted that Maxwell’s equations, in their original form, are strictly valid within inertial frames. However, these equations can only be expressed in coordinate form once a space-time coordinate system has been properly defined. Assigning Lorentz coordinates to the laboratory frame when a light source is in motion presents a complex challenge. To approach this problem systematically, we begin with the simpler case: assigning space-time coordinates to an inertial frame in which the light source remains at rest. Our goal is to provide a practical and operational procedure for this assignment. The most natural synchronization method involves first bringing all ideal clocks to the same spatial location, where they can be synchronized. Once synchronized, the clocks are then slowly transported to their designated positions—a process known as slow clock transport. 999Eddington, in a text first published in 1923, was apparently the first to describe a synchronization procedure based on the slow transport of clocks . Further details can be found in the review (see also ). Maxwell’s equations in their usual form are valid in any inertial frame where the sources are at rest, and the time coordinate is assigned using the slow clock transport procedure. The same principles apply when charged particles move in a non-relativistic manner. In particular, oscillating charged particles emit radiation. In the non-relativistic case, where the velocities of the oscillating charges are much smaller than , the emitted radiation is predominantly dipole in nature and is well-described by Maxwell’s equations in their standard form. Let us now examine in more detail how the dipole radiation term arises. The retardation time in the integrands of the radiation field amplitude expression can be neglected when the charge’s trajectory changes insignificantly during this time. The conditions for this approximation can be determined as follows. Denote by the characteristic size of the system. The retardation time is then of the order . To ensure that the charge distribution remains nearly unchanged during this period, it is required that , where is the radiation wavelength. This condition implies that the system’s dimensions must be small relative to the radiation wavelength. Expressing this constraint differently, we obtain , where represents the characteristic velocity of the charges. Since dipole radiation theory accounts only for the dipole component of the emitted radiation, it disregards detailed information about the electron trajectory. Consequently, it is not surprising that the fields predicted by dipole radiation theory closely resemble those obtained from instantaneous approximations. The theory of relativity provides an alternative method for clock synchronization, based on the fundamental principle that the speed of light remains constant in all inertial frames. While this is often regarded as a postulate, as we have seen, it is actually a matter of convention. The synchronization process that follows adheres to the standard Einstein synchronization procedure. Consider a dipole radiation source. When the dipole source is at rest, the governing field equations are given by Maxwell’s equations. In dipole radiation theory, we introduce a small expansion parameter, , and neglect terms of order . This effectively means that the theory employs a zero-order nonrelativistic approximation. Einstein synchronization is defined using light signals emitted by the dipole source at rest, under the assumption that light propagates isotropically with velocity in all directions. Applying the Einstein synchronization procedure in the rest frame of the dipole source allows us to establish a Lorentz coordinate system. Slow transport synchronization is equivalent to Einstein’s synchronization in an inertial system where the dipole light source is at rest. 101010Many textbook authors still attribute a reality status to the one-way speed of light. To quote Hrasko : ”It is sometimes claimed that Einstein’s synchronization of distant clocks and is circular. … This argument, however, is fallacious. It is true that a one-way measurement of light velocity can be performed only if clocks at the endpoints are correctly synchronized. However, since they need not show the correct coordinate time, they can be synchronized without light signals by transporting them symmetrically from a common site. … As we see, the described thought experiment is capable of proving the constancy of the speed of light if it is true or disproving it if it is false. It therefore provides a solid logical foundation for Einstein’s synchronization prescription.” This reasoning, however, is incorrect. Slow clock transport synchronization is equivalent to Einstein’s synchronization only in an inertial system where the light source is at rest. In other words, suppose we have two sets of synchronized clocks spaced along the -axis. Suppose that one set of clocks is synchronized by using the slow clock transport procedure and the other by light signals. If we would ride together with any clock in either set, we could see that it has the same time as the adjacent clocks, with which its reading is compared. This is because in our case of interest when the light source is at rest, field equations are the usual Maxwell’s equations and Einstein synchronization is defined in terms of light signals emitted by a source at rest assuming that light propagates with the same velocity in all directions. Applying either synchronization method in the rest frame establishes a Lorentz coordinate system. In this system, the metric of the light source takes the Minkowski form. We now turn to the topic of electromagnetic waves. In free space, the electric field of an electromagnetic wave emitted by a stationary dipole source satisfies the wave equation: . This result is closely related to the Minkowski metric . 3.3 Motion of a Light Source Relative to an Inertial Frame We now consider the case in which a light source, initially at rest in the lab frame, is accelerated to a velocity along the -axis. In other words, we examine an active (physical) boost. When we say that the source undergoes acceleration while the inertial observer (equipped with measuring devices) does not, we mean acceleration relative to the fixed stars. Any acceleration relative to the fixed stars—that is, any active boost of velocity—has an absolute meaning. In the case of an active boost, we analyze the motion of the same physical system over time, as observed from a single reference frame. A key question is whether the lab’s clock synchronization method depends on the motion of the light source. The answer to this is a matter of convention. The simplest synchronization method involves maintaining the same set of uniformly synchronized clocks used when the light source was at rest—essentially preserving clock transport synchronization (or Einstein synchronization, which is defined using light signals from a stationary dipole source). This choice is the most practical from a laboratory perspective, as it preserves simultaneity and follows the absolute time (or absolute simultaneity) convention. Absolute simultaneity can be incorporated into special relativity without altering either the logical structure or the (convention-independent) predictions of the theory. We begin with the metric as the true measure of space-time intervals for an inertial observer with coordinates . Applying a Galilean boost, we transform the spatial coordinate as while keeping time unchanged. Substituting this into the Minkowski metric, yields \| \| \| --- \| \| \| (11) \| This metric describes the electrodynamics of the moving light source from the viewpoint of the inertial observer measurements. From the structure of the equation, it is evident that the measurement of the electromagnetic field configuration—expressed in terms of the coordinates and of a measuring device at rest in the inertial frame—yields the same result as that obtained at the spatial position at time . To our knowledge, the metric in Eq. 6 is presented here for the first time. This result has not been previously reported in the literature. The agreement between this theoretical framework and empirical observations lends sup- port to our revised formulation of special relativity incorporating absolute simultaneity (see Section 4.14 for further details). By contrast, existing literature often assumes—incorrectly—that an inertial observer and an accelerated light source share the same Minkowski metric, treating this assumption as self-evident. To understand the physical interpretation of Eq. 11, it is important to recognize that space-time points can be labeled in infinitely many ways using different coordinate systems. These systems are related through coordinate transformations. A new frame of reference can always be introduced by relabeling coordinates and describing physical phenomena using the new labels—a process known as a passive (coordinate) transformation. For instance, in a laboratory frame, one can define a comoving coordinate system to analyze radiation from a moving source in terms of the new (comoving) coordinates. In this system, fields are expressed as functions of the independent variables , which are related to the original coordinates via a passive Galilean transformation. Specifically, after the passive transformation, the Cartesian coordinates of the source transform as . This transformation completes with the invariance of simultaneity, . The transformation of time and spatial coordinates of any event has the form of a Galilean transformation. One must take into account that are coordinates of the rule clock structure at rest in the inertial frame. The equivalence of the active and passive perspectives arises because shifting a system in one direction is equivalent to shifting the coordinate system in the opposite direction by the same amount. However, there is no a priori guarantee that the physical world must behave this way. For example, Newtonian mechanics is not invariant under passive rotations, as demonstrated by Newton’s bucket experiment. The empirical principle states that within a single inertial frame, every passive Galilean boost is postulated to correspond to an active Galilean boost. According to this equivalence of passive and active boosts, Maxwell’s equations remain valid in the comoving frame. In this frame, the fields are expressed as functions of the independent variables , and . The electric field of an electromagnetic wave satisfies the equation . However, the variables can be expressed in terms of the independent variables using a Galilean transformation, so that fields can be written in terms of . From the Galilean transformation , after partial differentiation, one obtains , . Hence the wave equation transforms into \| \| \| --- \| \| \| (12) \| This last result is closely related to the metric equation Eq. 11. The solution of this equation is the sum of two arbitrary functions, one of argument and the other of argument . Here we obtained the solution for waves that move in the direction by supposing that the field does not depend on and . The first term represents a wave traveling forward in the positive direction, and the second term represents a wave traveling backward in the negative direction. We conclude that the speed of light emitted by a moving source, as measured in the laboratory frame , depends on the relative velocity between the source and the observer. In other words, in this scenario, the speed of light follows the Galilean law of velocity addition. Specifically, the coordinate velocity of light along the -axis is given by in the positive direction and in the negative direction. This apparent deviation from the electrodynamics constant arises because the clocks are synchronized according to an absolute time convention. Once such a convention is chosen, it must be consistently applied to both mechanics and electrodynamics. Under this absolute time coordinatization, the velocity of light transforms analogously to that of a particle, as expected. 3.4 Discussion So far, we have examined the Galilean transformation of the electrodynamics equations. Now, we highlight a crucial distinction between fundamental and phenomenological theories. For instance, Newton’s equation in a co-moving frame can be regarded as a phenomenological law. It does not provide a microscopic interpretation of a particle’s inertial mass; instead, rest mass is introduced in an ad hoc manner. This suggests that the particle may possess internal variables that remain unknown. Any phenomenological law valid in the Lorentz rest frame can be embedded in four-dimensional space-time only through Lorentz coordinatization. Now, let us turn our attention to electrodynamics. It is essential to emphasize that electrodynamic laws do not rely on any hidden internal variables or underlying mechanisms. Electromagnetic fields are fundamental, and electrodynamics fully complies with the principles of relativity. As a result, any synchronization scheme can be used to describe electromagnetic field dynamics. The flexibility in describing physical systems allows us to choose the most suitable representation for each problem. A new frame of reference can always be introduced by relabeling coordinates, enabling the discussion of physical phenomena through passive transformations. In contrast, an active transformation involves the actual motion of a physical system—where its characteristics change due to internal or external interactions. Here, we are concerned with the evolution of the same physical system over time, analyzed from the perspective of a single reference frame. Let us now explore the correspondence between passive and active transformations, drawing a comparison between special relativity and Newtonian mechanics. In Newton’s non-relativistic theory, the formalism is grounded in an absolute space structure. However, there is relativity concerning uniform motion, meaning we cannot determine a particle’s absolute velocity. Passive Galilean boosts are dynamical symmetries of Newtonian mechanics. The equivalence between active and passive Galilean boosts arises from the fact that moving a system in one direction is equivalent to shifting the coordinate system in the opposite direction by the same amount, as illustrated by Galileo’s famous ship experiment. This symmetry, known as the Galilean group, exists because the equations of motion in Newtonian mechanics do not explicitly depend on velocity. The internal dynamics of the system remain unchanged under a Galilean boost. However, Newtonian mechanics is not invariant under passive rotations. The form of the equations of motion changes due to the introduction of pseudo-forces. As a general discussion of the correspondence between passive and active transformations, we refer the reader to the paper . Newton’s bucket argument contrasts two situations: one where a vessel of water is at rest, and another where the vessel is gradually rotated to a state of uniform angular velocity. The argument against the equivalence of active and passive rotations is based on the observation that, while there is no relative motion between the water and the bucket in both cases, the shape of the water differs. There is no relativity of accelerations in Newtonian mechanics, and we can determine the absolute acceleration of a particle in an inertial frame. Absolute acceleration refers to acceleration relative to the fixed stars. Let us now examine the correspondence between active and passive transformations in special relativity. Within a single inertial frame, kinematical symmetry corresponds to dynamical symmetry, and the symmetry principle (i.e., the equivalence of the active and passive Galilean boosts) holds. It is important to note that a passive Galilean boost, when considered within a single inertial frame, is merely a different parametrization of the observations made by the inertial observer. Under the passive transformation (, ), the motion of fixed stars relative to the observer and their devices remains unchanged. Note that represent the coordinates of the clock structure at rest in the inertial frame. In Chapter 5, we will address the problem of light aberration in accelerated systems using the framework of special relativity. This issue can be adequately tackled only by adopting an approach that employs absolute time coordinatization. To express the laws of electrodynamics in an accelerated frame, we define the metric of the accelerated frame using the inverse Galilean boost. It is crucial to highlight that, when applying an inverse Galilean boost in velocity, we are accounting for the effects of interaction in terms of the acceleration of motion relative to the fixed stars. This implies that the passive Galilean transformation within a single inertial frame is fundamentally distinct from the (passive) inverse Galilean transformation, , . This transformation describes the acceleration of an observer (with their measuring instruments) relative to the fixed stars. It’s important to note that are the coordinates of the rule-clock structure at rest in the accelerated frame. The transition of the coordinates of the light source does not signify a mere change in representation but reflects a genuine change in the absolute motion of the light source (and rule-clock structure) with respect to the fixed stars. It is important to recognize that in the actual process of transforming to an accelerated frame (relative to the fixed stars), the observer will experience inertial forces, which are not accounted for in standard treatments. We argue that the reciprocity of the inertial frames, which is considered in textbooks as relativistically correct, is at odds with special relativity. The arguments concerning the relativity of motion in our case of interest cannot be applied, since the inertial and accelerated reference systems are not equitable. According to special relativity, there is a significant difference between an accelerated inertial frame (relative to the fixed stars) and an inertial frame that has not undergone any acceleration, with this distinction being intimately tied to inertial forces. While the duration of the acceleration period has a negligible effect on anisotropy in the accelerated frame, the overall impact of inertial forces fundamentally determines the problem. A useful way to conceptualize the asymmetry between inertial and accelerated frames is to view it as the result of pseudo-gravity experienced by the accelerated observer. A uniformly accelerating frame can be treated as an inertial frame supplemented by a uniform pseudo-gravitational field. This concept is at the heart of the equivalence principle, which allows us to apply it to solve non-inertial kinematics problems using dynamics methods. There is an alternative approach to solving the problem. The formalism of relativistic physics assumes an absolute space-time structure, where there is no relativity of uniform motion, allowing us to determine the system’s absolute velocity in the initial inertial frame. From a space-time geometric perspective, the solution involves recognizing that the metric tensor must be a continuous quantity. When an accelerated system transitions to constant velocity, we can smoothly adjust the metric tensor. The (non-diagonal) Langevin metric arises by matching the metric tensors of the accelerated frame and the initial inertial frame. This perspective may be unsettling for those who have learned relativity from standard textbooks. According to conventional teachings, the problem is entirely symmetric, and both frames are considered equivalent. The widespread belief is that the coordinates of these frames are related by Lorentz transformations, and the metric in both frames is diagonal. However, upon diagonalizing the metric, we find that in the accelerated frame, the metric tensor must abruptly shift from the Langevin form to the Minkowski form. This reveals that, contrary to the textbook understanding, Lorentz transformations do not preserve the continuity of the metric tensor ( see Chapter 5 for further details). At this point, a reasonable question arises: Why address this topic now? Why not wait until Chapter 5? The answer is that the subject is subtle and complex, and the best way to understand it is through gradual exposure. The first step is to develop an intuitive sense of what should occur in various situations. Later, with a more solid grasp of relativistic effects, we will revisit the topic and refine our understanding with greater precision. 3.5 A Clock Re-Synchronization Procedure Most existing studies assume that an observer at rest and a moving source share the same Minkowski metric, thereby neglecting the need for any clock re-synchronization procedure. However, this standard approach is flawed, as it fails to account for the principle of equivalence between active and passive boosts within a single inertial frame. Our analysis shows that after applying a Galilean boost under an absolute time coordinatization, the homogeneous wave equation for the field in the lab frame appears nearly—but not exactly—in its standard form (valid in the absence of acceleration). The key deviation lies in the presence of a mixed derivative term, , which complicates the equation’s solution. To resolve this, we note that a simplification is always possible by introducing a suitable change of the time variable. If we limit our analysis to first-order terms in , a time transformation of the form can be applied. This time shift effectively eliminates the mixed derivative term, reducing the equation to the standard wave equation form. Let us now consider the case of an arbitrary velocity along the -axis. We have, then, a general method for finding a solution of the electrodynamics problem in the case of absolute time coordinatization. The new independent variables can be expressed in terms of the old independent variables : \| \| \| --- \| \| \| (13) \| Since the change variables completed by the Galilean transformation is mathematically equivalent to the Lorentz transformation, it follows that transforming to new variables leads to the usual Maxwell’s equations. In particular, the wave equation Eq. (12) transforms into \| \| \| --- \| \| \| (14) \| In the new variables, the velocity of light emitted by a moving source is constant in all directions, and equal to the electrodynamics constant . The combined effect of the Galilean boost and variable changes results in the Lorentz transformation under absolute time coordinatization in the lab frame. However, in this context, the transformation should be regarded merely as a mathematical tool that simplifies solving the electrodynamics problem while maintaining absolute time synchronization. This raises an interesting question: Is it necessary to transform the solution back into the original variables? We argue that the variable changes introduced above have no intrinsic significance; their meaning is purely conventional. Notably, the time shift is directly related to the issue of clock synchronization. Moreover, the final rescaling of time and spatial coordinates remains physically unobservable. Crucially, the convention-independent results remain unchanged in the new variables. Consequently, there is no need to revert the solution to its original form. Consider two independent light sources labeled ”1” and ”2”. Suppose that in the lab frame the velocities of ”1” and ”2” are , and . The question now arises of how to assign a time coordinate to the lab reference frame. We have two choices: an absolute time coordinate or a Lorentz time coordinate. The most natural choice, in terms of connecting to laboratory reality, is absolute time synchronization. In this framework, simultaneity is absolute, meaning that a single set of synchronized clocks is sufficient for both sources in the lab frame. However, Maxwell’s equations are not form-invariant under Galilean transformations, meaning their mathematical form differs in the lab frame. Using absolute time synchronization leads to more complex field equations, which also vary for each source. To introduce Lorentz coordinates, we must adopt a different approach. The only viable method in this scenario is to assign an individual coordinate system—i.e., an independent set of synchronized clocks—to each source. The situation becomes more complex when a secondary source interacts with light emitted by a primary source. A fundamental challenge in special relativity is that both sources cannot, in general, be described using a single shared Lorentz frame within a common inertial reference system. This issue, and its implications for electrodynamics, is discussed in more detail in the next chapter. 3.6 A Moving Light Source: Peculiarities of Collinear Geometry So far, we have considered both covariant and non-covariant approaches to solving the problem of radiation emitted by a moving source in an inertial frame. Now, let us examine the acceleration of a dipole source in the laboratory inertial frame, where it accelerates to a velocity along the -axis. This raises the question of how to assign synchronization in the lab frame after the source has undergone acceleration. Before acceleration, we adopted a Lorentz coordinate system. However, if synchronization in the lab frame remains unchanged after the acceleration, the electrodynamics of moving charges becomes significantly more complex. In this case, the transformation of time and spatial coordinates does not retain the standard Lorentz form. Maxwell’s equations remain valid in the lab inertial frame only when Lorentz coordinates are properly assigned. To preserve Lorentz coordinatization after acceleration, it is necessary to redefine the space-time coordinate system by introducing new variables , as given in Eq. (13). In this new coordinate system, the velocity of light emitted by the moving source remains equal to the electrodynamic constant in all directions. The aberration of light and the Doppler effect serve as practical examples to illustrate the distinction between covariant and non-covariant approaches. The most effective way to calculate radiation is by applying the Lorentz transformation formulas between inertial reference frames. Specifically, consider a radiation source at rest in the frame , emitting dipole radiation with frequency . In the laboratory frame , where the source moves with velocity and the radiation propagates along the direction of motion, we observe the so-called radial Doppler effect. This effect is governed by the well-known formula: . The effect of the factor can be summarized in the following statement: on the moving object, time is flowing slower than expected (time dilation). It is worth noting that using the conventional coupling of Maxwell’s equations and the corrected Newton’s equation to calculate radiation from a moving source does not necessarily lead to errors. For rectilinear motion of both the source and the emitted light beam, non-covariant and covariant approaches yield identical trajectories, ensuring compatibility between Maxwell’s equations and conventional particle tracking. However, this method was incorrectly extended to non-collinear geometries. This raises a reasonable question: why does the same method produce incorrect results when the source moves perpendicular to the direction of its emitted radiation? The key distinction in collinear geometry—where the source moves along the same line as the radiated beam—is that the velocity is perpendicular to the plane of the radiation wavefront (i.e., the plane of simultaneity). Consequently, for collinear motion, the plane of simultaneity in absolute time coordinatization retains the same orientation in Lorentz coordinatization. In the transverse case, the motion is perpendicular to this direction, and the source undergoes acceleration along the radiation wavefront (i.e., the initial plane of phase simultaneity). Since the orientation of the radiation wavefront is no longer absolute, it depends on the chosen clock synchronization convention. To analyze this, let us first consider a Lorentz transformation. Specifically, we use a Lorentz boost to describe the uniform translational motion of the light source in the lab frame. It is evident that wavefront phases, which are simultaneous in but occur at different -locations, are not simultaneous in . When applying a Lorentz boost, a time transformation is introduced: . The effect of this transformation is equivalent to a rotation of the radiation wavefront in the lab frame by an angle of approximately in the first-order approximation. According to Maxwell’s electrodynamics, radiation is always emitted in a direction normal to the radiation wavefront. Consequently, the radiated light beam propagates at an angle , leading to the phenomenon known as the aberration of light. This remarkable effect will be the focus of our discussion in the next chapter. In the conventional (3+1) approach, simultaneity is considered absolute, and there is no mixing of spatial coordinates and time when sources change their velocities in an inertial frame. Thus, it appears that the conventional (3+1) approach cannot account for the geometric phenomenon of aberration. By applying Maxwell’s equations alongside an absolute time transformation, neither the wavefront plane nor the direction of energy transport undergoes deflection. Let us now return to the conventional coupling between Maxwell’s equations and particle trajectories within the ”single lab frame.” According to the theory of special relativity, the standard description of optical phenomena related to the motion of a light source remains valid when radiation propagates along the direction of velocity. For instance, in a collinear geometry, the conventional coupling of fields and particles can provide a useful framework for analyzing the Doppler effect. To understand the relativistic redshift of a light source in a moving system, we must analyze the mechanism of the source and observe its behavior while in motion. Since this can be quite challenging, we will consider a simplified source that captures the fundamental principles. In a vacuum, an electron emits radiation only when it undergoes acceleration. In the non-relativistic case, this radiation exhibits a dipole pattern. When a non-relativistic electron moves within a magnetic field, the emitted radiation is commonly known as cyclotron radiation. The frequency of the cyclotron radiation (the dipole radiation) is of course equal to the frequency of electron rotation in the magnetic field , i.e. . In the case of circular motion (with the velocity component parallel to the field ) the radius of the orbit is , where is the wavelength of the cyclotron radiation and is the electron velocity. When we clearly always have , which means that the dipole approximation is applicable. The relativistic motion of an electron is described by with . This equation can be written as where is the relativistic Larmor frequency. When the source is accelerated, the speed of electrons is increased, and therefore the mass is also increased and the electron is heavier. The emissivity presents a spectrum in which the frequency is given by . The frequency is multiplied by the relativistic Larmor frequency accounting at the same time through the denominator for the Doppler effect caused by the motion parallel to the magnetic field. We can thus conclude that the frequency of an electron’s oscillations decreases when the source moves with velocity . At first glance, the mechanism of the source described may not seem to involve relativistic time dilation. However, it is implicitly present in the assumption that the mass of a moving object corresponds to its relativistic mass, given by . According to the covariant approach, various relativistic kinematic effects relevant to the dipole radiation setup emerge in successive orders of approximation. The first-order kinematic term plays a crucial role only in the description of dipole radiation in the perpendicular geometry. In the case of collinear geometry, however, the motion of the dipole source affects only the second-order kinematic terms , in accordance with the theory of relativity. In the non-covariant approach, solving the dynamics problem in the lab frame does not involve Lorentz transformations. This means that, within the lab frame, particle motion appears exactly as predicted by classical mechanics, which assumes an absolute time framework. This absolute-time approach is well-suited for explaining experimental outcomes in collinear geometry. However, we argue that this method of solving Maxwell’s equations in the lab frame is not applicable in the transverse case. The use of electrodynamics in absolute-time coordinatization becomes essential when adopting a non-covariant (3+1)-dimensional approach (i.e., ”old kinematics”) to relativistic particle dynamics. Conventional particle tracking relies on the lab time as the independent variable, thereby excluding relativistic kinematic effects from the description. This approach is based on the assumption of absolute time synchronization in the lab frame. If one intends to use corrected Newtonian equations—where time serves as the independent variable—the electrodynamic equations must be formulated in a non-covariant manner (i.e., within the absolute-time coordinatization). Only in this framework does the coupling between electrodynamic equations and particle trajectories remain consistent within a single reference frame. 4 Aberration of Light in an Inertial Frame of Reference In this chapter, we explore several practical applications of the concepts introduced earlier, using the spacetime metric given in Eq. (11). Our first topic addresses the phenomenon of light aberration caused by a light source moving perpendicular to the direction of radiation. Light aberration in an inertial frame is generally understood as an apparent shift in the direction of light propagation due to the motion of the source. While often treated as a first-order effect in , a rigorous description is not trivial—even at this level of approximation. Many standard textbooks contain inaccurate or oversimplified accounts, underscoring the subtlety of the underlying physics. To keep the analysis manageable, we will limit our discussion to first-order effects in . Our focus will be on the physical influence of optical components on the observed aberration of light. We begin by examining the case of a finite-aperture mirror moving parallel to its surface, with an incident plane wave striking it normally. We will show that the reflected light exhibits aberration, observable as a deviation in the direction of energy transport. We then turn to a more practically significant scenario: the transmission of light through an aperture in a moving opaque screen, invoking Babinet’s principle. A particularly important application is the passage of light through the moving entrance of a telescope barrel—a configuration we will analyze in detail. 4.1 The ”Plane Wave” Emitter To build intuitive insight into the phenomenon of aberration, we begin by analyzing a simple, idealized case: a single ”plane-wave” emitter. We model this emitter as a two-dimensional array of identical, coherent elementary sources (dipoles), uniformly distributed over a fixed (–) plane. These sources are assumed to begin radiating simultaneously in the lab frame, where the plane is at rest. All dipoles oscillate with the same frequency , amplitude, and phase, and their motion is confined to the plane. By decreasing the spacing between adjacent sources to a value much smaller than the radiation wavelength, , this arrangement effectively approximates an ideal plane-wave emitter. The resulting wavefronts are planar, and the emitted radiation resembles an ideal plane wave. While the concept of an (infinitely extended) plane wave is a useful mathematical abstraction—owing to its simplicity and exact correspondence with solutions to Maxwell’s equations—it is not physically realizable, as such a wave would carry infinite energy. In any real scenario, the emitter has a finite aperture. It is also assumed that any detector used to measure the radiation direction is sensitive to energy flow (rather than to phase fronts, for instance) and has an aperture large enough compared to the beam size to capture the overall direction of energy propagation. What is commonly described as aberration, in this context, is in fact a shift in the direction of energy transport. 4.2 A Moving Emitter: Galilean Transformations-Based Explanation Let us consider the case of an emitter lying in the plane of an inertial laboratory frame, which is accelerated from rest to a velocity along the -axis. Suppose an observer at rest in this inertial frame measures the direction of energy transport. There are two possible approaches to analyzing the aberration of light emitted by a single moving source. The first is the covariant approach, commonly used in the literature, which explains the aberration effect by applying a Lorentz boost to determine how the direction of a light beam depends on the velocity of the source relative to the inertial frame. The second is a non-covariant approach, which employs an old kinematics description without explicit reference to Lorentz transformations. Both approaches, whether formulated using Einstein’s synchronization convention or an absolute time synchronization scheme, yield the same result in the case of a single moving emitter. Thus, the choice between them is a matter of pragmatism rather than fundamental necessity. In this chapter, we present both approaches, beginning with the non-covariant one. It is important to emphasize that the (3+1) non-covariant formulation in relativistic electrodynamics is entirely valid and provides a consistent description. The aberration of light can be treated within a single-inertial-frame framework, without invoking Lorentz transformations. In this context, time remains an absolute quantity in special relativity. But what does ”absolute time” mean? It signifies that simultaneity is preserved within the inertial frame, and there is no intermixing of spatial positions and time coordinates when sources change velocity within this frame. A distinctive feature of this non-covariant approach is the absence of relativistic kinematics in the description of light aberration. For a tangentially moving emitter (relative to its surface), the direction of energy transport of the emitted light undergoes a deviation. Within the ”single-frame” approach, this effect arises due to the Doppler effect, which induces angular frequency dispersion of the emitted light waves (see Fig. 1, left). The current approach to the moving emitter problem employs the Fourier transform method. In the analysis of linear systems, it is often useful to break down a complex input into simpler components, determine the system’s response to each elementary function, and then superimpose these responses to obtain the total outcome. Fourier analysis serves as a fundamental tool for this decomposition. 111111For a broader discussion on Fourier transform methods in spatial filtering theory or Abbe diffraction theory, we recommend referring to . Consider the inverse transform relationship which express the profile function in terms of its wavenumber spectrum. We may regard this expression as a decomposition of the function into a linear combination (in our case into an integral) of elementary functions, each with specific form . From this, it is clear that the number is simply a weighting factor that must be applied to the elementary function of wavenumber to synthesize the desired . An emitter with a finite aperture is a kind of active medium that breaks up the radiated beam into a number of diffracted beams of plane waves. Each of these beams corresponds to one of the Fourier components into which an active medium can be resolved. Let us assume that the current density of the elementary dipole sources varies according to the law . From this, we conclude that the active medium of the emitter is sinusoidally space-modulated. Let us examine the electrodynamics of a moving source. To explain the phenomenon of radiation in our case of interest, we apply an (active) Galilean boost to describe the uniform translational motion of the source in the inertial lab frame. According to equivalence of passive and active boosts, Maxwell’s equations remains valid in the comoving frame. The measurement of the electromagnetic field configuration - expressed in terms of coordinates and of a measuring device at rest - yields the same result that obtained at the spatial position at time . Applying a Galilean boost, we transform the spatial coordinate as , while keeping time unchanged. Since Maxwell’s equations are not invariant under Galilean transformations, their form is not preserved. Starting from the Galilean transformation and applying partial differentiation, we obtain the wave equation Eq.(12). Our goal is to demonstrate how the additional terms introduced in the field equations due to the Galilean transformation lead to the prediction of the Doppler effect. Consider as a possible solution a radiated plane wave . With a plane wave with the wavenumber vector and the frequency equation Eq.(12) becomes: . The wavevector is determined by the initial conditions before the acceleration and subsequent Galilean transformation. Specifically, we set , where corresponds to the wavenumber of the sinusoidally space-modulated dipole density, and , with denoting the frequency of the emitted radiation prior to acceleration. The phase of the wave at the world point is invariant at a change of the reference system (when the field is zero, everyone measures the field as zero). This invariance holds regardless of the specific coordinate transformation used to describe the change in the reference system. Therefore, the phase must be invariant of the Galilean transformation. Consequently, . Substituting the Galilean transformation formulae into the phase equality formula we obtain . This transformation completes with the invariance of wavenumber vector , . Substituting these into dispersion relation confirms that it remains satisfied, as expected. Let us first remind the reader that the velocity of waves is typically defined in terms of the phase difference between oscillations observed at two different points in a free plane wave. This concept is primarily used to compute interference fringes, which visually highlight phase differences. In a plane wave, the phase velocity is given by the ratio . In addition to phase velocity, another type of velocity—known as the group velocity or energy propagation velocity—can be defined by considering the propagation of a disturbance, such as a change in amplitude, superimposed on a wave train. A simple example of a wave group is formed by the superposition of two waves with frequencies and wave numbers and . This combination represents a carrier wave with frequency and a modulation with frequency . The resulting wave can be described as a succession of moving beats (or groups), where the carrier wave travels with velocity , and the group velocity is given by , which, in the limit, becomes . Many textbooks on electromagnetic theory discuss the phenomenon of light aberration in the context of plane waves. However, using a plane wave model to explain light aberration leads to an incorrect understanding. When an infinite sinusoidal wave propagates, it maintains a uniform average energy density throughout space. The question then arises: does this energy remain stationary, or does it propagate through space? It is impossible to determine this with certainty based on the plane wave model. All experimental methods used to measure the aberration of light rely on light signals, which measure not the phase velocity, but the signal velocity. In our case of interest, this signal velocity coincides with the group velocity. In this example, plane waves with different wavenumber vectors propagate from the moving emitter, each with its own frequency. The relation holds for each scattered wave, regardless of the sign or magnitude of the radiated angle. Specifically, in this case, represents the Doppler shift, given by , and is simply the transverse component of the radiated wavenumber vector, . These equations indicate that a light beam with a finite transverse size propagates along the -direction with a group velocity . In our discussion, we assume that the aberration angle is significantly larger than the divergence of the emitted beam. Mathematically, this is expressed as (i.e. ), where is the characteristic size of the emitter and the is the reduced wavelength. 4.3 A Moving Emitter: Lorentz Transformations-Based Explanation An alternative and insightful approach to understanding the aberration of light from a moving source is through Lorentz coordinatization. This method provides a relativistically consistent explanation of the phenomenon within the framework of Maxwell’s equations. This explanation utilizes a clock-re-synchronization procedure. The time coordinate in the lab frame, under Lorentz coordinatization, is obtained by introducing a time offset of the form given in Eq. (13). Although Maxwell’s equations are invariant under Lorentz transformations, the Lorentz boost introduces a transformation of the time coordinate of the form . This spatially varying time shift introduces a delay in the emission of radiation across the source plane, effectively tilting the phase front of the emitted wave. To first order in , this corresponds to a rotation of the phase front by an angle approximately equal to . This rotation manifests physically in the fact that features of the wave, such as the electric field maximum, reach different spatial positions at different times in the lab frame. For instance, the field maximum at position along the -axis is delayed relative to the field maximum at , with a time shift given by . In accordance with Maxwell’s electrodynamics, coherent radiation is always emitted perpendicular to the phase front. Since Maxwell’s equations contain no intrinsic directional bias, the tilt in the wavefront—induced by the Lorentz time transformation—results in a corresponding shift in the direction of energy propagation. Within this relativistic description, the moving emitter exhibits a phase chirp across its aperture, given by , which means that the individual elementary sources now oscillate with position-dependent phase. As a result, the wavefront undergoes a rotation, and the emitted radiation propagates at an angle , capturing the essence of the aberration effect (see Fig. 1, right). We now examine the group velocity of the emitted beam. For a plane wave described by , the dispersion relation in vacuum, derived from Maxwell’s equations, is . In special relativity, the four quantities form a four-vector and transform accordingly under Lorentz transformations. We will restrict our analysis to terms up to first order in . Using the initial conditions and the Lorentz transformation, we obtain the following expression: , , , where is the wavenumber of sinusoidally space-modulated current density, is the frequency of the emitter radiation before the acceleration. Substituting these into the dispersion relation confirms that it remains satisfied, as expected. As a consequence of the Doppler effect in Lorentz coordinatization, we observe an angular frequency dispersion in the light waves emitted from a moving source with a finite aperture. The Doppler shift, , of a radiated light wave (in the first order approximation) is given by , where is the transverse component of the radiated wavenumber vector. This equation implies that a light beam with a finite transverse extent propagating along the -direction has a group velocity given by . It is important to recognize the significance of the fact that both covariant (Lorentz-based) and non-covariant (absolute time-based) approaches yield the same group velocity. This agreement underscores the physical objectivity of the result: group velocity is a measurable, convention-independent quantity, and must therefore remain invariant regardless of the chosen coordinatization. The distinction between absolute time coordinatization and Lorentz coordinatization, as discussed in Chapter 3, is particularly illuminating. For non-relativistic velocities, the transition between these two frameworks can be understood as a simple redefinition of the time variable: . When combined with the Galilean transformation, this leads naturally to the Lorentz transformation, even within the context of an absolute-time formalism. However, this change of variables has no intrinsic physical significance—it represents merely a reparametrization of time, not a real physical effect. This coordinate transformation is closely tied to the issue of clock synchronization. Crucially, the physical content of electrodynamics remains unchanged under such transformations. As a result, once a solution is obtained in the transformed variables, there is no need to revert to the original (3+1) variables; the results remain valid and physically meaningful. It is commonly assumed that the theory of relativity can be applied to physical processes without a detailed understanding of the clock synchronization procedure. Most textbooks suggest that an operational interpretation of Lorentz coordinatization within a single inertial frame is unnecessary. Nevertheless, distinguishing between absolute-time and Lorentz coordinatization (from an operational standpoint) deepens conceptual understanding—even in the simple case of a single moving source. While some physicists have expressed concern about the lack of a ”dynamical” explanation for wavefront rotation in Lorentz coordinates, Chapter 3 offers a compelling perspective: the rotation arises naturally from the transformation of the time coordinate, which alters the simultaneity structure of spacetime events. Within the absolute-time framework, this provides an intuitive—though coordinate-dependent—explanation of the aberration phenomenon. 4.4 Reflection from a Mirror Moving Parallel to Its Surface Until now, the previous two sections have not introduced any fundamentally new results. As previously discussed, when the sources are independent, adopting Lorentz coordinates presents no difficulty—each source can be assigned its own coordinate system, along with an independent set of clocks. The situation becomes more complex, however, when a secondary source interacts with light emitted by a primary source within a single inertial frame. This challenge in special relativity can be effectively addressed using an electrodynamics-based approach framed within an absolute time perspective using metric Eq. (11). As noted earlier, the metric in Eq. (11) has not been previously reported in the literature. Existing studies generally assume—incorrectly—that both sources share the same Minkowski metric and are governed by Maxwell’s equations within a single inertial (laboratory) frame. This assumption is often treated as self-evident. In effect, the conventional approach relies on a hidden premise: that the principle of equivalence between active and passive boosts does not holds within a single inertial frame. Our formulation challenges this assumption. The agreement between our theoretical framework and experimental observations lends strong support to our revised formulation of special relativity, which incorporates the metric of Eq. (11) (see Section 4.14 for further discussion). With the results of Chapter 3 now established, we are prepared to examine the intriguing phenomenon of radiation from moving sources. We begin by analyzing the intensity of light reflected when a stationary source illuminates a moving mirror. It is commonly assumed that if the mirror moves tangentially to its surface, the law of reflection remains unaffected—just as it does for a stationary mirror (see Fig. 2). According to this view, a monochromatic plane wave incident normally on a small aperture of a moving mirror is reflected as an oblique beam. In other words, it is assumed that the energy transport velocity coincides with the phase velocity. This assumption, frequently found in textbooks (see, for example, and ), is, however, incorrect. To examine this claim, we analyze the reasoning found in standard texts. 121212 Many sources claim that no aberration occurs when light reflects off a mirror moving parallel to its surface. For example, Sommerfeld states: ”For a mirror moving tangentially to its surface, the law of reflection that holds for a stationary mirror is preserved.” Textbooks typically analyze reflection using two Lorentz reference frames. The fixed (lab) frame is at rest relative to the plane-wave emitter, while the moving frame travels at velocity , in which the mirror is at rest. Both frames use a Cartesian coordinate system with the plane tangent to the mirror’s surface. The -axis aligns with the mirror’s velocity . For simplicity, we consider light incident from the -direction in the lab frame. The incident wave is described by its four-dimensional wave vector, where the time-like component represents the angular frequency , and the space-like components define the propagation direction. In the lab frame , the wave vector has components , with the negative sign indicating propagation toward the mirror (see Fig. 3a). Our goal is to determine the wave vector of the reflected beam. The argument that light reflected from a tangentially moving mirror experiences no aberration proceeds as follows. It is most straightforward to analyze the reflection in the mirror’s rest frame (Fig. 3b), where the surface is stationary and the standard laws of optical reflection apply. We consider aberration effects only up to the first order in . In this frame, an observer at rest with respect to the mirror perceives the incoming wave vector as . The effect of reflection is to reverse the sign of the component of the wave vector, . We now obtain the reflected wave vector in the lab frame by applying the inverse Lorentz transformation: . This vector represents a light beam traveling away from the mirror, having the same frequency as the incoming beam. This confirms that the reflection follows the usual laws of geometrical optics and that the beam undergoes no aberration. There are two fundamental errors in solving problems related to the reflection of light from a mirror moving tangentially to its surface. The first error arises from applying the concept of a plane wave and an infinite plane mirror to the case of tangential motion. This assumption leads to an inherent contradiction: if the mirror were truly infinite, the problem would not be time-dependent. This hidden assumption—namely, the absence of time dependence—eliminates aberration, which is unsurprising. However, only the motion of a finite mirror is physically meaningful. From an electrodynamics perspective, only the velocity of a finite mirror has physical significance. Textbook treatments often overlook the crucial interaction between light and the moving edges of the mirror. A well-defined energy transport problem requires specifying both the source and mirror apertures—an essential detail frequently neglected in conventional explanations. We consider the case of a finite-aperture mirror moving tangentially to its surface. For simplicity, we assume that the mirror’s size is much smaller than that of the ”plane-wave” emitter, as illustrated in Fig. 2. Notably, we analyze a scenario where the aberration angle is significantly larger than the divergence of the reflected radiation. Mathematically, this is expressed as , where and are the sizes of the emitter and mirror, respectively. Classical optics textbooks state that energy transport in the reflected light beam remains unchanged. The second error in the textbook argument is conceptual. Consider an observer at rest in the inertial frame who measures the direction of energy transport. This observer describes the light beam emitted from a stationary source using the Minkowski metric. According to textbooks, the metric of the moving mirror is assumed to be the same Minkowski metric as that of the stationary emitter. The textbook authors incorrectly assume that a common Lorentz time coordinate axis can be assigned to both the emitter and the mirror. This is a misconception. A fundamental question arises: how should synchronization be established in the laboratory frame after the mirror has undergone acceleration? A distinctive feature of this problem in the context of relativistic kinematics is that the emitter (along with the observer and measuring devices) remains at rest in the lab’s inertial frame, while the mirror moves at a constant velocity relative to the lab frame, interacting with the radiated light beam. How can we address the issue of the emitter-mirror relative velocity? One approach is to assign a Lorentz time coordinate to describe the emitter’s radiation. However, for the moving mirror, this time coordinate would act as an absolute time reference, and its motion would be described by a Galilean transformation in this framework. Without altering synchronization in the lab frame, a common set of synchronized clocks for both the mirror and the emitter can only be established under absolute time coordinatization—that is, when simultaneity is absolute. Suppose we re-synchronize the clocks in the lab frame to define the Lorentz time coordinate for the boosted mirror. In this framework, the reflection of light is described using Maxwell’s equations. However, this time transformation merely results in a rotation of the radiation phase front of the incoming plane wave by an angle . Consequently, in this new coordinatization, Maxwell’s equations no longer directly apply to describing the emitter’s radiation. Notably, a key feature of this coordinatization is that the energy transport velocity differs from the phase velocity of the incoming light beam. This problem in special relativity can be effectively analyzed using an absolute time coordinatization approach. We demonstrate that when a mirror with a finite aperture moves tangentially while a plane wave of light falls normally onto it, the energy transport of the reflected light deviates. In this case, the aberration of light arises directly from the time dependence of the mirror’s position, necessitated by its finite aperture. Notably, in absolute time coordinatization, the wave equation explicitly depends on the velocity vector. Consequently, the solution naturally involves light beams of different frequencies, incorporating the Doppler effect. For a tangentially moving mirror with a finite aperture, angular frequency dispersion—an effect of first order in —cannot be neglected. We have already emphasized that Chapter 3 is arguably the most important part of this book. To our knowledge, the operational interpretation of absolute time coordinatization, as well as the distinction between absolute time synchronization and Einstein’s time synchronization from an operational perspective, has not been explicitly addressed in the literature. Applying the Galilean boost, we substitute while keeping time unchanged in the Minkowski metric, , leading to the metric given in Eq. 11. This metric characterizes the electrodynamics of the moving mirror as observed from an inertial reference frame, representing the measurements from the perspective of an inertial observer. There is an intuitively plausible reason why textbook authors conclude that light reflected from a tangentially moving mirror experiences no aberration. A comparison between conventional aberration theory for light and sound theory can provide insight. In Section 4.15, we will consider a sound emitter at rest in the atmospheric frame and examine a mirror moving tangentially along its surface with velocity . According to sound theory, the reflected beam’s energy transport does not deviate because the group velocity equals the phase velocity. When a plane wave falls the mirror normally, it produces an oblique sound beam. The (diagonal) wave equation for the moving mirror in the atmospheric frame is identical to that of an emitter at rest. Textbooks often draw important parallels between sound and light aberration in the first-order approximation. 131313We should make one additional remark regarding two common errors when solving problems related to reflection from a mirror moving tangentially to its surface. These errors are not independent. Textbooks typically assume that the metric of the moving mirror is diagonal, similar to the metric of a ’plane wave’ emitter at rest, and that the group velocity equals the phase velocity. As a result, regardless of the mirror’s size, there is no deviation in the energy transport of the reflected light beam. This, however, is a misconception. There is a fundamental difference between the propagation of light and sound. A proper treatment of light aberration must be based on the pseudo-Euclidean geometry of space-time. 4.5 Solving the Emitter-Mirror Problem in (3+1) Space-Time The aberration of light problem can be addressed within a ”single inertial frame” framework, without invoking Lorentz transformations. When an object is illuminated by a monochromatic, spatially coherent source, a particularly simple method exists for calculating the reflected intensity. This approach, rooted in Fourier transform techniques from spatial filtering theory—specifically, Abbe diffraction theory—offers conceptual elegance. In this context, Abbe’s method treats the mirror as a diffraction grating that decomposes an incident plane wave into multiple diffracted beams, each representing a Fourier component of the reflected power. Since a finite-aperture mirror is a non-periodic object, it produces an infinite continuum of diffracted beams. A basic example of a diffraction grating is illustrated in Fig. 4. Assuming the grating’s reflectance follows the function , the reflectance is sinusoidally modulated in space. It is important to note that the permanent reflectance distribution grating discussed here is only our mathematical model and we do not need to discuss how it can be created. The wave vectors shown in Fig.4 represent the propagation vector of the incident plane wave, which is assumed to be directed perpendicularly to the surface. The vectors and indicate the scattered light. The Bragg condition , describes the relationship of the incident and scattered waves. Since the scattering occurs from a sinusoidal grating rather than a set of discrete planes (grooves), the first-order maximum dominates. We assume that the grating vector lies parallel to the surface of the grating, while the incident wave is normal to the surface, as illustrated in Fig. 4. Under the Bragg condition for small angles, the scattering angle is approximately given by . Let us analyze the reflection of light from a mirror moving tangentially (i.e., parallel to its surface), relying solely on relativistic kinematics. To describe this scenario, we adopt the framework of absolute time coordinatization and apply a Galilean boost. In this context, we utilize the metric given by Eq. 11, which characterizes the electrodynamics of a moving light source as observed from an inertial frame. From the structure of Eq. 11, it becomes evident that the electromagnetic field configuration—expressed in terms of the coordinates and of a detector at rest in the inertial frame—coincides with the field configuration at position at time . Using a Fourier analysis, we show that the problem can be reduced to that of reflection from a tangentially moving diffraction grating. Consider a plane wave of the form where is the wavevector and the angular frequency. In the absolute time coordinatization, the dispersion relation takes the form: . Within a single inertial frame, the equivalence between active and passive Galilean boosts implies that translating the system in one direction is indistinguishable from shifting the coordinate system in the opposite direction by the same amount. This equivalence ensures that Maxwell’s equations retain their form in the comoving coordinate system. In the comoving frame, the wavevector of the diffracted wave is determined by the initial conditions. Specifically, we have: , , where denotes the spatial frequency of the sinusoidally space-modulated reflectance. Transforming back to the lab frame using a Galilean transformation, we find that the frequency shifts according to: , while the components of the wavevector remain unchanged. Substituting these expressions into the dispersion relation yields: , which is readily verified to be satisfied, as expected. This analysis leads to the conclusion that the frequency shift and the spatial frequency are related by: . This relation is recognized as a manifestation of the Doppler effect, arising from reflection off a moving (as opposed to stationary) surface. A crucial aspect of describing a ”single inertial frame” cannot be overstated. If the light source is at rest while the mirror is in motion, it follows that the electrodynamic equations must remain identical for all electromagnetic waves. In other words, the dispersion equation in absolute time coordinatization must be consistently applied to both incoming and scattered waves (Fig. 4). In our previous discussion on absolute time coordinatization, we established that an emitter at rest must still be governed by Maxwell’s electrodynamics. In this framework, the dispersion equation simplifies to . From the initial conditions, we derive and . However, an apparent contradiction arises—one that is resolved through a geometric analysis of light reflection. A key feature of this geometry is that even after applying a Galilean transformation along the -axis, the dispersion equation in absolute time coordinatization retains its diagonal form: for the incident wave. To clarify the fundamental physical principles, we examined a simple case. One of the consequences of the Doppler effect is the angular frequency dispersion of light waves reflected from a moving mirror with a finite aperture. If denotes a unit vector in the direction of the wave normal, and is the mirror velocity vector relative to the lab frame, we get the equation . The Doppler effect is responsible for angular frequency dispersion to the first order of even when (i.e when ). In fact, at . We can rewrite this equation in a different way. The differential of the scattered angle is given by . With the help of this relation and account for that we have . One of the key conclusions of the preceding discussion is a striking prediction in the theory of the aberration of light—specifically, the deviation of energy transport for light reflected from a mirror moving tangentially to its surface. When a plane wave (in absolute time coordinatization) falls such a mirror perpendicularly, the reflected beam exhibits a deviation in energy transport (see Fig. 5). This phenomenon can be understood as a straightforward consequence of the Doppler effect141414The Doppler effect serves merely as an intermediate mathematical description; when the full calculation is performed, only the geometric (aberration) effect remains.. A common misconception in the literature asserts that the direction of propagation in reflection from a tangentially moving mirror is not determined by the normal to the wavefront.151515A widespread misunderstanding stems from assuming that the orientation of radiation wavefronts has objective significance. As Norton notes: ”One might try to escape the problem by supposing that the direction of propagation is not always given by the normal to the wavefront. We might identify the direction of propagation with the direction of energy propagation, supposing the latter to transform differently from the wave normal under Galilean transformation. Whatever may be the merits of such proposals, they are unavailable to some trying to implement a principle of relativity. If the direction of propagation of a plane wave is normal to the wavefronts in one inertial frame, then that must be true in all inertial frames.” This misunderstanding arises from a failure to distinguish between convention-dependent and convention-invariant aspects of the theory. The direction of energy transport is an objective, convention-invariant quantity. In contrast, the phase front orientation—corresponding to an observer’s plane of simultaneity—lacks objective meaning, as no experimental method can precisely determine this orientation due to the finite speed of light. Since the phase front orientation has no physical reality within the angular range , one might ask: why must we account for its exact orientation in electrodynamics calculations? The answer lies in the fact that, when analyzing radiation beam evolution within a single inertial frame, the phase front orientation appears unchanged. While this invariance lacks objective physical meaning, it serves as a useful tool in solving electrodynamics problems. A useful analogy can be drawn with gauge transformations in Maxwell’s electrodynamics. 4.6 A Moving Mirror: Lorentz Transformations-Based Explanation Another clear way to explain the aberration of light from a moving mirror is by using a Lorentz boost, which accounts for the uniform translational motion of the mirror in the laboratory frame. Maxwell’s equations are invariant under Lorentz transformations; however, the Lorentz boost introduces a space-time coordinates transformation of the form Eq. (13). To minimize the mathematical complexity of the discussion, we will restrict our analysis to terms up to first order in . This entails only a time shift of the form . This time shift corresponds to an effective rotation of the phase front of the incoming radiation by an angle of . Within this framework, the mirror’s reflection is analyzed using Maxwell’s equations. For a plane wave of the form , the dispersion equation in Maxwell’s electrodynamics simplifies to . In the Lorentz-synchronized coordinates, the electric field of the incident wave transforms to: . Using a Fourier decomposition, the problem reduces to that of wave reflection from a grating moving tangentially with velocity . By invoking the equivalence between passive and active Lorentz transformations, we find that in the lab frame the diffracted wave components are given by: , , . Substituting these expressions into the dispersion relation confirms that it remains satisfied, as expected. As a result of the Doppler effect under Lorentz coordinatization, the reflection from a finite-sized moving mirror exhibits angular frequency dispersion. This implies that the reflected light beam has a group velocity along the -direction given by: , consistent with the prediction from Maxwell’s equations that the group velocity equals the phase velocity. It is also insightful to analyze the electrodynamics of an emitter at rest in the case of Lorentz coordinatization. A rotation of the -axis without changing the orientation of the -axis is described by a Galilean transformation , . Conversely, a rotation of the -axis without changing the orientation of the -axis is described by the transformation , . To first-order, this transformation mathematically is similar to the Galilean transformation , . This time transformation introduces a new crossed term in the wave equation which describe emitter radiation, predicting a group velocity . However, since there is an angle between the wave vector and the -axis, and considering the initial condition, the radiated light beam from the stationary source propagates along the -axis with a group velocity , as expected. This outcome is unsurprising—convention-independent physical results remain unchanged under the new variables. 4.7 Discussion Physicists investigating the textbook approach to the aberration of light often find themselves puzzled by an apparent inconsistency. When analyzing the reflection of light from an infinite mirror moving normally to its surface, the standard textbook reasoning—based on two Lorentz reference frames—correctly predicts the variation in light frequency upon reflection. Relativistic kinematics successfully describes this phenomenon. However, a natural question arises: why does the same method yield incorrect results when the mirror moves tangentially to its surface? The key to understanding this discrepancy lies in the nature of the motion. In the case of a normally moving mirror, the velocity is perpendicular to the plane of oscillating elementary sources (dipoles). As a result, the plane of simultaneity in absolute time coordinates retains the same orientation in Lorentz coordinates. For collinear motion, relativistic kinematics affects the system only at the second order. Now, let us consider frequency measurement within the framework of special relativity. Suppose we use a Fabry-Perot interferometer or a grating spectrometer. In such cases, the measured frequency corresponds to the wavelength of the standing wave. This principle applies universally to all frequency measurements, which inherently obey the relationship between light frequency and interference patterns. Importantly, while the deviation of energy transport direction is a geometric effect, interference patterns themselves are independent of the chosen metric. In other words, in space-time geometry, phase remains a four-dimensional invariant. For collinear motion, both the Minkowski metric and the metric given by Eq. 11 yield identical predictions (see Chapter 13 for further details). Another crucial aspect to consider is the size of the mirror. In the case of an infinite mirror moving normally to its surface, the motion of the mirror is a physically observable effect. Even for an infinite mirror, the problem remains time-dependent, and its velocity carries physical significance. This makes the infinite mirror a useful conceptual tool for analyzing the Doppler effect in such a scenario. 4.8 Large Aperture Mirror In this chapter, we focus on a specific range of problem parameters where the effects of emitter edges are neglected. Although this treatment of aberration in light theory is an approximation, it holds significant practical importance. We also consider the case where the transverse size of the emitter is much smaller than that of the moving mirror. It is important to note that this scenario does not occur in stellar aberration measurements. However, analyzing this case allows us to gain a clearer understanding of the underlying physical principles. Notably, it can be shown that in this situation, energy transport deviation is absent (see Fig. 6). To address the moving mirror problem, we employ the Fourier transform method as a direct approach. 4.9 Analysis of Transmission through a Hole in an Opaque Screen Previously, we demonstrated that when a finite-aperture mirror moves tangentially to its surface and a plane wave of light is incident normally on the mirror, an aberration (deviation in energy transport) occurs in the reflected light. In this section, we consider a more practically significant problem: a screen moving with velocity along its surface in the lab frame. It is commonly assumed that light passing through a hole in such a moving opaque screen does not experience any deviation in energy transport (see Fig. 7). However, a common mistake in relativistic optics arises from aberration effects associated with a tangentially moving screen containing a hole. We analyze this system using a Fourier transform method similar to the one employed earlier. The screen with a hole acts as a diffraction grating, decomposing the incident plane wave into multiple diffracted plane-wave components. Each of these components corresponds to a Fourier mode of the transmitted light beam. The gratings discussed thus far modulate the amplitude of an incident plane wave through a periodic reflection function. However, our analysis can be readily extended to gratings that modulate the amplitude of incident light using a periodic transmission function. Consider a grating whose transmittance varies according to the function , as illustrated in Fig. 8. In this case, the transmittance exhibits a sinusoidal spatial modulation. Notably, all previously derived equations remain valid for the forward-scattered beams. Our approach reveals a remarkable prediction of the theory of aberration of light concerning the deviation of energy transport for light passing through a hole in a moving screen. Specifically, when a screen with a hole moves tangentially while a plane wave of light falls normally on it, the transmitted light exhibits a deviation in energy transport (see Fig. 9). 4.10 Spatiotemporal Transformation of the Transmitted Light Beam Suppose the transmitted light pulse propagates in the plane. Our focus is on the space-time intensity distribution within this plane. Spatiotemporal coupling naturally emerges in the transmitted radiation behind the screen due to the angular-frequency dispersion introduced during the transmission process. In most cases, the emitted light beam can be accurately represented as the product of independent spatial and temporal factors. However, this assumption breaks down when the light must pass through an aperture in a moving opaque screen, as the transmission process disrupts the simple separation of space and time dependencies. We begin by expressing the field of an emitted pulse as . The initial amplitude distribution, , in front of the moving screen is the optical replica of the emitter’s aperture. The electric field of the transmitted pulse can be represented in the reciprocal domain as , which corresponds to the Fourier component of a beam with angular frequency dispersion, where . Taking the inverse Fourier transform, we obtain . This represents the field immediately behind the moving screen. The beam profile, , is the optical replica of the moving aperture. Now, consider an observer’s screen positioned at rest at a distance behind the moving aperture. For simplicity, we assume that the Fresnel number is large, , allowing us to neglect diffraction effects. Here, is the characteristic aperture size. From this, we conclude that the light spot on the observer’s screen moves with the same velocity as the moving screen. Additionally, we observe that a particular quantity remains unchanged: the combination is the same for both the spot and the aperture. What does this imply? A careful examination reveals that events occur simultaneously at two separate locations along the -axis. 161616Spatiotemporal coupling is typically discussed in the literature in the context of ultrashort laser pulse propagation through a grating monochromator. In this scenario, in addition to considering phase fronts, one must also account for planes of constant intensity—known as pulse fronts. In a grating monochromator, the different spectral components of the outgoing pulse propagate in different directions. The electric field of a pulse with angular dispersion can be expressed in the Fourier domain as . Its inverse Fourier transform to the space-time domain results in , representing a pulse with a pulse-front tilt. In our case of interest, we consider light diffracted by a tangentially moving set of gratings, where the Doppler effect induces frequency dispersion given by . Consequently, the spatiotemporal coupling due to light transmission through a hole in a moving screen differs significantly from the conventional pulse-front tilt distortion. 4.11 Applicability of Ray Optics Today, it is commonly suggested that the phenomenon of light aberration can be explained using a corpuscular model of light. According to this model, light particles (corpuscles) fall a moving screen normally, generating an oblique light beam as shown in Fig. 10. However, this argument remains incorrect and continues to persist. A satisfactory treatment of light aberration should be based on the electromagnetic wave theory.” Some experts argue that the applicability of the corpuscular model in the theory of light should be reinterpreted as the applicability of ray optics. 171717To quote Brillouin : ”Fig. 5 explains the situation assuming a simplified device consisting of a parallel plate moving with uniform velocity in the horizontal direction. Monochromatic light falls normally on the plate and generates an oblique ray.” This oblique effect is demonstrated in Fig. 10. Let us examine what happens, according to ray optics, in our case of interest. When light rays fall normally on a moving screen, they generate an oblique ray beam, as illustrated in Fig. 10. In this scenario, treating light rays as tiny particles leads to a well-known effect. Next, we discuss the region of applicability of ray optics. The use of ray optics in the theory of light aberration presents complexities. One might intuitively expect, based on textbook reasoning, that ray optics should apply to any spatially incoherent radiation. However, this assumption leads to incorrect results. Specifically, a completely incoherent source (such as an incandescent lamp or a star) consists of elementary point sources that are statistically independent and have varying offsets. The radiation field produced by a completely incoherent source can be understood as a linear superposition of the fields of individual elementary point sources. Each elementary source generates an effective plane wave in front of an aperture. Consequently, the transmission process inevitably involves the diffraction of a plane wave through the aperture. Notably, any linear superposition of radiation fields from elementary sources retains fundamental single-source characteristics, such as deviations in the energy transport direction. This reasoning explains why ray optics is not applicable to the aberration theory of light originating from (spatially) completely incoherent sources. To illustrate the applicability of ray optics in light aberration theory, we consider a specific class of spatially incoherent light beams. A method can be proposed to generate a ray beam from primary sources using an array of randomly phased lasers. Such a planar source produces rays that fall normally on the moving screen (within the laser Rayleigh range) and generate an oblique transmitted ray beam, as shown in Fig. 10. Intuitively, a tangentially moving screen with a hole acts as a switcher for the lasers. A luminous spot moving at velocity can, in principle, be created more simply—so to speak, ”manually.” By arranging laser-like sources along the -axis and activating them sequentially from left to right with a given time lag, a luminous spot can be made to move at any desired velocity, even exceeding . However, it is crucial to note that in this process, no information is transmitted along the -axis, as each source radiates independently. 4.12 Moving Large Aperture Emitter Let us now consider a ”plane-wave” emitter that starts at rest in the lab inertial frame and is then accelerated to a velocity along the -axis. An emitter with a finite aperture acts as an active medium, breaking the radiated beam into multiple diffracted plane-wave components. Each of these beams corresponds to a Fourier component of the emitted radiation. As discussed earlier in this chapter, the energy transport of coherent light from a tangentially moving emitter deviates from the expected path—a well-known phenomenon related to the aberration of light in an inertial frame of reference (Fig. 1). In our case of interest, the screen remains at rest in the lab frame, while the emitter moves tangentially to its surface at a constant velocity. For simplicity, we assume that the moving ”plane-wave” emitter is significantly larger than the hole in the screen. Now, consider an observer at rest relative to the screen, measuring the direction of energy transport (Fig. 11). If the screen is at rest while the light source is in motion, it is evident that the equations of electrodynamics must remain identical for all electromagnetic waves. In our previous discussion, we established that a stationary screen must be described by Maxwell’s electrodynamics. Consequently, the dispersion equation in the Lorentz coordinatization must be consistently applied to both incoming and scattered waves. In Maxwell’s electrodynamics, the dispersion equation is given by: . A key characteristic of the discussed geometry is that, even after applying a Galilean transformation along the -axis, the dispersion equation in the absolute time coordinatization retains the same diagonal form for a normally incident wave: . A closer examination of the physics reveals that, in the inertial laboratory frame where the screen is stationary, the problem corresponds to a steady-state transmission. Since the Doppler effect is absent, the transmitted beam travels vertically, having lost its horizontal group velocity component. As a result, the transmission behaves as depicted in Fig. 11. It is important to emphasize the following point: when light passes through a small aperture, the resulting beam undergoes diffraction, perturbing its fields. This diffraction effect removes any information about the emitter’s motion, making the transmitted light independent of its original source movement. 4.13 A Point Source in an Inertial Frame of Reference Above, we considered a single moving ”plane wave” emitter in an inertial frame of reference. When analyzing the aberration of light in such a frame, two types of sources are particularly useful to consider: (a) A ”plane wave”-like emitter (b) A point-like source (or, more generally, a spatially incoherent source) A defining characteristic of a ”plane wave” emitter is its ability to produce highly directional fields. Within the near zone, where , the wavefront remains nearly planar with minimal change. This allows for the analysis of light aberration from a single ”plane wave” emitter without significant detector influence, provided the detector is sufficiently large. In contrast, point-like sources emit radiation isotropically, forming spherical wavefronts. As a result, the measuring instrument invariably influences the detected radiation. The diffraction of a source field is typically categorized into Fresnel (near-zone) and Fraunhofer (far-zone) diffraction. In Fraunhofer diffraction, the wave phase is assumed to vary linearly across the detector aperture, which occurs, for example, when a plane wave impinges on the aperture at an angle to the optical axis. In contrast, Fresnel diffraction replaces this linear phase variation with a quadratic one. In the far zone, both types of sources effectively produce a plane wave in front of the pupil detection system. Let us now consider the case of a moving point source. Specifically, we examine a setup where the screen is at rest, while the point source moves tangentially. A key takeaway from the earlier discussion is that no aberration is observed for a large-aperture plane wave emitter moving tangentially in an inertial frame (see Fig. 11). Remarkably, the same conclusion holds for a point source. This stems from the linearity of Maxwell’s equations: the governing laws of electrodynamics are linear differential equations. As discussed previously, the most effective approach to analyzing a moving emitter is via the Fourier transform method. A point source can be decomposed into a superposition of spatial Fourier components—each corresponding to a radiated plane wave. The current density associated with these components can be modeled as . Each elementary dipole source radiates a plane wave of the form , where the wavevector is determined by the initial spatial modulation. Near normal incidence, we set , where is the transverse spatial frequency corresponding to the modulated dipole density. For small angles, the incidence angle is approximately given by . The radiation transmitted through the aperture undergoes diffraction with a divergence angle on the order of , where is the diameter of the hole. Within this diffraction angle, individual elementary emitters cannot be distinguished. Consequently, the problem of a tangentially moving point source reduces to that of a tangentially moving, large-aperture plane wave emitter (see Fig. 12). This leads to a remarkable conclusion: the transmitted beam carries no information about the source’s tangential motion. This principle lies at the heart of the binary star paradox discussed in Chapter 6. 4.14 Experimental Test: Reflection from a Moving Grating Textbook treatments suggest there is strong theoretical evidence for the absence of an aberration effect when light reflects from a mirror moving transversely (parallel to its surface). Although this result was established long ago by Pauli and Sommerfeld , a direct experimental verification remains absent. This raises a natural question: why has it not been possible to experimentally detect the aberration effect in this configuration? Consider a mirror moving tangentially to its surface, assuming the mirror size is smaller than the transverse size of the incoming light beam. Notably, we analyze a setup where the aberration angle is significantly larger than the diffraction divergence of the reflected radiation. Let us estimate the parameters for the Earth-based setup using tangentially moving mirror and an optical laser. For instance with a light wavelength of and a mirror size of 10 cm, the mirror speed must satisfy . Clearly, current experimental capabilities are insufficient to meet these demanding conditions. Fortunately, there exists a more practical alternative to this otherwise formidable experimental challenge. Previously, we analyzed the reflection of light from a mirror moving parallel to its surface, relying solely on relativistic kinematics. After applying a Galilean boost in the case of absolute time coordinatization, we employed the metric given by Eq. 6, which describes the electrodynamics of a moving light source as observed from an inertial frame. Using a Fourier approach, we showed that the problem could be reduced to the case of reflection from a tangentially moving grating. From the resulting dispersion relation, we derived a Doppler shift in the diffraction maxima due to the grating’s motion. The consistency between our theoretical framework and experimental ob- servations—such as the measured frequency shift by a tangentially moving grating —provides further support for the validity of our approach. In contrast, special relativity textbooks commonly conclude that no Doppler effect occurs for light reflected from a tangentially moving grating. This argument is typically made for a tangentially moving mirror but applies equally to gratings. The issue stems from the widely accepted assumption that the metric of a moving grating (or mirror) is the same as the Minkowski metric of the stationary emitter. We now present a simple explanation of our central result. The clearest understanding arises when we accept, as an experimental fact, the presence of a Doppler frequency shift in radiation diffracted from a moving grating. Within our relativistically consistent framework—based on the equivalence of active and passive boosts within a single inertial frame—the diffracted radiation must satisfy the dispersion relation: . The frequency shift is exactly compensated by the cross term , ensuring that the dispersion relation remains valid. In contrast, conventional treatments in special relativity textbooks describe diffraction from a moving grating using Maxwell’s equations, leading to the dispersion relation: . This equation is isotropic and independent of velocity. As a result, it contradicts experimental observations: the Doppler frequency shift cannot be accounted for or compensated. However, experimental evidence clearly demonstrates that the frequency of the reflected radiation depends on the tangential velocity of the grating. Doppler frequency shifts can be detected with high precision using advanced spectroscopic techniques, even at very low grating velocities. For instance, the authors of used a Mach-Zehnder-type interferometer to measure these shifts. This striking discrepancy suggests that the conclusions reached by Pauli and Sommerfeld are inconsistent with the principles of special relativity. 4.15 Differences between the Light and Sound Aberration Having completed our discussion on the aberration of light in an initial inertial frame, we now turn to optical phenomena in an accelerated frame. However, before proceeding, we will first explore optical-acoustic analogies. Sound is a periodic motion of air caused by vibrations of a source. The effect of sound aberration is best understood using a model of a single plane-wave emitter. Specifically, we consider a two-dimensional array of identical, coherent elementary sources uniformly distributed on a given plane , all emitting waves simultaneously. We analyze a scenario in which a finite-aperture mirror moves tangentially to its surface. For simplicity, we assume the mirror’s size is small compared to the plane-wave emitter. An observer, at rest in the atmosphere frame, describes the emitted sound beam using the diagonal wave equation: , where is the speed of sound. The emitter radiates a plane wavefront in the vertical -direction, and the moving mirror obeys the same wave equation as the stationary emitter. The amplitude of the beam emitted by the oscillating sources can be understood as a superposition of spherical wavelets, as dictated by Huygens’ principle. The wavefront at any given instant is the envelope of these wavelets. Since the diagonal wave equation exhibits no intrinsic anisotropy, the energy transport of the reflected sound beam remains unchanged. A plane wave incident normally on the finite-aperture mirror generates a reflected oblique beam. Notably, in the atmosphere frame, the phase and group velocities remain equal. Textbooks generally state that, to first order in , there is no fundamental difference between the aberration of light and sound. The geometry of reflection from a moving mirror is identical (Fig. 2). Now, consider a screen moving tangentially to its surface with velocity in the atmosphere frame. According to sound theory, energy transport for a sound beam transmitted through a hole in a moving screen remains undeviated. Consequently, it is commonly assumed that the energy transport for light transmitted through a hole in a moving opaque screen follows the same geometric behavior (Fig. 7). In the next chapter, we will continue our discussion on the aberration of sound. The conventional treatment of the aberration of light often leads physicists to draw misleading parallels between the aberration of sound and light in the first-order approximation. This is largely due to the fact that the wave equation governing a moving mirror (or screen) retains the same diagonal form as the wave equation of an emitter at rest. However, in sound theory, we operate within Newtonian space and time rather than the pseudo-Euclidean geometry of space-time. This distinction is crucial: the standard analysis of light aberration within a single inertial frame fails to account for the fundamental difference between the velocity of light and that of sound. A rigorous relativistic treatment must be grounded in a space-time geometric approach. The Minkowski metric remains valid in an inertial frame where the observer is at rest, with time coordinates assigned through the slow clock transport method or Einstein synchronization, which is defined via light signals emitted by a dipole source at rest. In this Lorentz coordinate system, the Minkowski metric accurately describes the electrodynamics of a stationary light source from the perspective of an inertial observer. Now, consider the case where a mirror in an inertial frame accelerates from rest to velocity along the -axis. Synchronization is maintained using the same set of synchronized clocks, preserving the Minkowski metric for both the inertial observer and the stationary light source. In this scenario, the mirror undergoes a Galilean boost, represented by the transformation , while keeping time unchanged in the observer’s Minkowski metric. This results in Eq. 11, which describes the electrodynamics of the moving mirror from the inertial observer’s perspective. In contrast, prior literature has often made the incorrect assumption that an inertial observer (i.e., an observer at rest in an inertial frame) and an accelerated mirror within the same frame share a common Minkowski metric—an assumption long treated as self-evident. Consequently, the widely accepted approach to special relativity in the first-order approximation effectively reduces to a classical theory akin to sound propagation. 5 Aberration of Light: Non-inertial Frame of Reference In this chapter, we revisit the problem of light transmission in non-inertial frames, focusing on the aberration of light. Using the Langevin metric in general relativity, we derive the aberration for a pulse of light traveling in an accelerated system. The aberration problem is fundamentally resolved by recognizing the inherent asymmetry between inertial and non-inertial observers. This asymmetry, akin to the well-known Sagnac effect, arises from the absolute nature of acceleration. As Langevin noted in 1921, ”any change of velocity, or any acceleration, has an absolute meaning” . Mathematically, however, there is no distinction between calculations in general relativity and special relativity when space-time curvature is absent. The usual arguments for the relativity of motion do not apply here, as inertial and non-inertial reference frames are not equivalent. The asymmetry paradox is typically resolved by recognizing acceleration as the defining factor. Although the duration of acceleration has a negligible effect on anisotropy within the accelerated frame, the acceleration itself fundamentally determines the problem. Interestingly, the lack of a dynamical explanation for this asymmetry in special relativity has puzzled some physicists. A useful perspective is to interpret the asymmetry as a consequence of the pseudo-gravitational effects experienced by an accelerated observer. From a pedagogical standpoint, treating the accelerated frame via the equivalence principle provides valuable insight, a topic we will explore further in Chapter 12. A natural question arises: how can different observers—such as those based on Earth and the Sun—determine which one experienced acceleration? The surprising answer is that this determination can be made solely through observations of the ”fixed stars.” In principle, acceleration is defined relative to these distant celestial objects, implying an implicit absolute reference frame. 5.1 Absolute Time Coordinatization in Accelerated Systems We investigate the phenomenon of light aberration in accelerated systems within the framework of special relativity. In particular, we demonstrate that explaining optical effects in rotating frames does not require modifying special relativity or invoking general relativity. A rigorous application of special relativity is sufficient. However, to express electrodynamics in a non-inertial frame, one must take the additional step of defining the metric of that frame. Crucially, the metric tensor must remain a continuous function, leading naturally to the concept of smoothly tailoring the metric. This issue in special relativity can be effectively addressed using an approach based on absolute time coordinatization. Toward the end of this section, we outline how special relativity may be mathematically extended to describe accelerating systems. In some situations, however, direct physical reasoning can yield rapid insights while remaining consistent with formal derivations. Consider an inertial frame , assumed to be at rest relative to the fixed stars. A second frame, , along with an observer and measuring instruments, is accelerated from rest in to a velocity along the -axis. In such accelerated systems, the notion of absolute simultaneity remains logically consistent with the smooth tailoring of the metric tensor. Synchronization in this context involves preserving the same uniformly synchronized clocks that were used when was initially at rest. It is well known that during acceleration relative to the fixed stars, Einstein’s synchronization procedure cannot be directly applied. As a result, by the time reaches uniform motion, the metric in its reference frame will generally acquire a non-diagonal form. We begin by treating the metric as the true measure of spacetime intervals for an accelerated observer in frame , with coordinates . To express this in the coordinates of an inertial observer in , moving at velocity relative to , we apply the inverse Galilean transformation , keeping the time coordinate unchanged: . Substituting into the Minkowski metric, , we obtain the transformed (Langevin) metric: \| \| \| --- \| \| \| (15) \| It is crucial to note that Langevin metric, in fact, reflects measurements performed by an inertial observer using the Minkowski metric, subsequently re-expressed via the transformation , . This metric describes the electrodynamics of the accelerated light source with the viewpoint of the accelerated observer measurements as viewing this of the inertial observer. Inspecting Eq.(15) we can find the components of the metric tensor in the coordinate system of . We obtain , , . Note that the metric in Eq. (15) is not diagonal, since, , and this implies that time is not orthogonal to space. This result represents the Langevin metric, derived by matching the metric tensors of the accelerated frame and the initial inertial frame. To interpret Eq. (15) physically, recall that a new frame of reference can always be introduced via a passive coordinate transformation—a mere relabeling of coordinates used to describe physical events. For example, an observer in the inertial frame can introduce a comoving coordinate system to analyze radiation from a moving source. In this system, fields are expressed as functions of , which relate to the original coordinates through a passive Galilean transformation. From the structure of the Langevin metric, it is evident that the electromagnetic field measured at position and time by a device at rest in the accelerated frame will match the field measured by an identical device at rest in the inertial frame at position and time . Substituting this coordinate relationship into the Minkowski metric of the inertial observer leads directly to Eq. (15). In describing physical phenomena in an accelerated frame, one must distinguish between coordinate quantities and physical quantities. The Langevin metric allows us to determine how these relate. For instance, the length of a physical rod and the proper time interval in the inertial frame coincide with the coordinate length and time in the accelerated frame: , . For a more detailed discussion on how to relate coordinate and physical quantities in non-inertial frames, refer to Chapter 12. Let us now extend the mathematical framework of the special theory of relativity to include accelerated motion. To keep the mathematical complexity minimal, we assume that the reference frame starts at and moves with a constant acceleration until . Our goal is to determine an expression for velocity from the perspective of an inertial (non-accelerated) observer. In relativistic mechanics, uniformly accelerated motion satisfies the equation which integrates to . This leads to the velocity expression . We emphasize that represents the velocity of the accelerating frame as observed from the inertial reference frame. Applying the inverse Galilean transformation, we have , . With this transformation, the metric in the uniformly accelerating observer’s frame takes the form: \| \| \| --- \| \| \| (16) \| Since the metric tensor of space-time must be a continuous quantity, the coordinates in the accelerated and inertial stages must be matched accordingly. At , the metric naturally coincides with that of the frame, giving . During acceleration, we adopted an absolute time coordinatization. To ensure continuity of the metric tensor as the reference frame transitions from non-inertial to inertial motion, we employ the metric given in Eq. (16). Thus, at , the metric of the non-inertial frame must smoothly transition into the Langevin metric. This can be achieved by relating the coordinates and time of the accelerated observer, , to those of the inertial observer, , via the Galilean boost: , , where . For coordinate times , the reference frame of the accelerated system follows the metric given by Eq.(15). Importantly, represents the velocity of the frame as observed by an inertial observer during the second (inertial) stage of motion. It is evident that at , the metric of the non-inertial frame continuously transitions into that of the (formerly accelerated) inertial frame. For a broader discussion on the mathematical extension of special relativity to accelerated motion, we refer the reader to . Textbooks describe inertial frames as moving at constant velocity relative to the fixed stars, with different inertial frames connected by Lorentz transformations. Consequently, one might expect the coordinates and time in frames and (during the inertial stage) to be related by a Lorentz transformation. However, during the inertial segment of the trajectory, the metric in both frames remains diagonal in Lorentz coordinates. Thus, at , the metric tensor of the accelerated frame, according to textbooks, must undergo an abrupt change from to . 5.2 Asymmetry Between Inertial and Accelerated Frames The velocity of light emitted by a source at rest in the coordinate system of the inertial frame is . The Minkowski metric, given by Eq. (1), implies a symmetry in the one-way speed of light in this frame. However, in the coordinate system of an accelerated frame , the speed of light emitted by a source at rest in differ from This is because is related to via a Galilean transformation. This is readily verified if one recalls that the velocity of light in the reference system is equal to . If is the infinitesimal displacement along the world line of a ray of light, then and we obtain . In the accelerated reference system, since and , this expression takes the form , as seen by setting in Eq. (15). Consequently, in the accelerated coordinate system , the velocity of light parallel to the x-axis, is in the positive direction, and in the negative direction, as expected. We have identified a fundamental asymmetry between inertial and accelerated frames: Maxwell’s equations do not hold from the perspective of an observer at rest with respect to an accelerated frame . The metric given by Eq. (15), which corresponds to the accelerated frame , predicts an asymmetry in the one-way speed of light along the direction of relative velocity. Accelerations (with respect to the fixed stars) influence the propagation of light. In an accelerated system , the velocity of light emitted by a source at rest must be modified by the acceleration—either added to or subtracted from—resulting in different speeds in opposite directions. In contrast, in an inertial frame , the velocity of light emitted by a source at rest remains . One crucial aspect of describing an accelerated reference frame cannot be overstated: the metric applies to physical laws, not to physical facts. In other words, it is always possible to choose a set of variables in which the metric of the accelerated source remains diagonal. We interpret the Langevin metric as indicating that the laws of electrodynamics, when expressed in an accelerated frame, take the form of anisotropic field equations. Here, ”physical facts” refer specifically to the aberration of light emitted by a single ”plane wave” source within the accelerated frame. To account for this effect, the electrodynamics equation must be integrated using the appropriate initial conditions for the radiation wavefront. After the boost we can see that acceleration does not affect the wavefront orientation. In fact, the variables can be expressed in terms of the variables by means of Galilean transformation , . This transformation ensures that, following acceleration, the emitted light beam’s wavefront remains perpendicular to the vertical direction , as illustrated in Fig. 13. We will explore this topic further in Section 5.5. It is useful to begin with an overview of some key results. Consider an observer in the accelerated frame performing an aberration measurement. To predict to the outcome, the observer must use the non-diagonal metric given in Eq. (15). Due to the inherent asymmetry between inertial and accelerated frames, an intriguing prediction arises in the theory of light aberration. Specifically, if an opaque screen with a hole is initially at rest relative to the fixed stars and then set into motion, the apparent angular position of a ”plane-wave” emitter observed through the aperture in the accelerated frame will shift by an angle . This effect is illustrated in Fig. 13. Notably, the source of this asymmetry is not the relative acceleration between the two frames but rather the difference in their individual acceleration histories with respect to the fixed stars. In Chapter 4, we analyzed the emitter-screen problem in the reference frame , which remains at rest (i.e., without any history of acceleration) relative to the fixed stars. We found that, in the case of a small aperture hole, no aberration arises from the emitter, regardless of its motion. It is important to emphasize that the aberration of the light beam transmitted through the small aperture hole also remains independent of the emitter’s motion in the accelerated frame (see Fig. 14). This is because the cross-term in the metric (Eq. 15), which induces aberration in this particular case, depends solely on the velocity change of the accelerated frame relative to the initial inertial frame. 5.3 Electrodynamic Explanation of Light Aberration It is well known that electrodynamics fully complies with the principles of relativity. For instance, we can derive the velocity of light in an accelerated frame without relying on the metric equation (15). In an inertial frame, electromagnetic fields are expressed as functions of the independent variables , and . According to special relativity, Maxwell’s equations remain valid in any Lorentz reference frame, and the electric field of an electromagnetic wave satisfies the wave equation: . By applying a Galilean transformation, the variables can be expressed in terms of the coordinates in an accelerated frame. Consequently, the electromagnetic field of a wave emitted by a source at rest in the accelerated frame can be rewritten in terms of these transformed variables. As a result, the wave equation transforms into \| \| \| --- \| \| \| (17) \| where coordinates and time are transformed according to a Galilean transformation. Acceleration affects the propagation of light. The coordinate velocity of light parallel to the -axis is given by in the positive direction, and in the negative direction. We now consider the transmission through the hole in the opaque screen in the accelerated frame , Fig. 14. We describe the aberration of light based on the observations made by an observer in the same accelerated frame as the screen. We have already discussed the explanation of the aberration based on electrodynamics. Our earlier discussion is really about as far as anyone would normally need to go with the subject, but we are going to do it again. One reason is that one should know how to deal with what happens to waves in the accelerated frame from the viewpoint of electrodynamics equations. The screen containing a hole is a kind of diffraction grating that breaks up the radiated beam into a number of diffracted beams of plane waves. Each of these beams corresponds to one of the Fourier components into which a transmittance can be resolved. Let us assume that the transmittance of the grating varies according to the law . Acceleration influences the field equations, transforming the wave equation into Eq. (17). Consider a transmitted plane wave of the form . With a plane wave with the wavenumber vector and the frequency equation Eq.(17) becomes: . The wavevector is determined by the initial conditions before the acceleration. Transforming to the accelerated frame using Galilean transformation , we find that the frequency shifts according to: , while the components of wavevector remain unchanged. Substituting these into dispersion relation confirms that it remains satisfied, as expected. In our example, plane waves with different wavenumber vectors propagate away from the screen at different frequencies. Each transmitted wave satisfies the equation , regardless of the sign or magnitude of the transmission angle. This implies that a transmitted light beam with a finite transverse size moves along the -direction with a group velocity of (see Fig. 14). Now, let us address an apparent paradox. We derived the electrodynamics equations in an accelerated frame using the Galilean transformation of Maxwell’s equations. From the Galilean transformation , after partial differentiation, one obtains , . Intuitively, one might expect everything to be at rest in the accelerated frame. However, special relativity dictates the presence of a time derivative, , since is nonzero. This situation differs fundamentally from electrodynamics in an inertial frame. We will explore this topic further in Chapter 9. 5.4 A Simple Explanation of Light Aberration in Accelerated Frames Previously, we analyzed the aberration of light in an accelerated reference frame. Calculating this effect in a non-inertial frame is inherently complex. To tackle this challenge, we employed a metric tensor to derive the electrodynamics equations within the accelerated frame, using the Fourier transform method. A key consequence of the cross term in the wave equation is the emergence of a group velocity, which explains the aberration. Now, we present a straightforward explanation of light aberration in an accelerated frame using electrodynamics. To determine the aberration increment, we apply Galilean transformation laws to the electric and magnetic fields. We have an expression for the energy flow vector of the electromagnetic field. This vector, , is called ”Poynting’s vector”. It tells us the rate at which the field energy moves around in space. To observe the effect of light aberration within an accelerated frame, it is essential to establish a coordinate system with a well-defined reference direction. We define the reference directions as those perpendicular to the direction of motion in both the inertial and accelerated frames. For simplicity, we assume that the motion of the aberrated light beam remains confined to a single plane, with its angular position described by a single angle. The analysis is much simplified if we treat separately the case of a radiated beam with its -vector parallel to the ”plane of incidence” (that is -plane) and the case of a radiated beam with the -vector perpendicular to the -plane. We will carry through the analysis for an incoming beam polarized perpendicular to the plane of incidence, but the principle is the same for both. So we take that , has only - component. For a magnetic field, we get , where is the direction of Poynting vector in the inertial frame. Above, we demonstrated that the metric tensor must be matched in both the accelerational and inertial segments of the trajectory. To achieve this, it is sufficient to relate the coordinates and time of the accelerated observer to those of the inertial observer using a Galilean boost. The electrodynamics equations in an accelerated frame can be obtained by applying a Galilean transformation (with velocity ) to Maxwell’s equations. The electromagnetic fields and observed in the accelerated frame differ from the fields and in the inertial frame prior to the active Galilean boost. The transformation of the fields under the Galilean boost is given by , . This expression accounts only for first-order effects in . Here, the velocity vector of the accelerated frame relative to the inertial frame is . If the electric field is perpendicular to the -plane, the magnetic field in the inertial frame has only an -component and we have immediately , . This results in a Poynting vector, . Thus, the radiated beam propagates at an angle with respect to the -axis, leading to the phenomenon of aberration of light in the accelerated frame. We find that the energy transport direction remains the same whether analyzed using electrodynamics equations in an accelerated frame or the Galilean transformation law for electromagnetic fields. In the first approach, the equations show that a transmitted light beam with a finite transverse size propagates along the -direction with a group velocity given by . In the second, light aberration is interpreted as a change in the direction of the Poynting vector. Both methods yield the same result. In this section, we demonstrate that the group velocity direction and the Poynting vector direction are fundamentally equivalent—two sides of the same coin. This is a good point to make a general remark about the physical reality of the energy flow, represented by the Poynting vector. In any practical aberration measurement, one must account for a finite radiation beam size. The direction of energy transport—and thus the Poynting vector—is well-defined and convention-invariant. However, a common misconception in electrodynamics arises when considering energy transport in plane waves. In such cases, determining the exact direction of energy flow is fundamentally impossible. 5.5 Clock Resynchronization in Accelerated Systems It should be noted, however, that there is an alternative satisfactory approach to explaining the effect of light aberration in the accelerated frame . This explanation utilizes a clock re-synchronization procedure. When the system begins to move with constant velocity, the standard procedure for Einstein’s clock synchronization can be applied. This synchronization method is based on the assumption that light signals emitted by a source at rest propagate with the same velocity in all directions. By using this synchronization procedure, we can select a Lorentz coordinate system for the screen. In this framework, the transmission through the aperture is described using Maxwell’s equations. The interval in the accelerated reference frame will then take the diagonal form given by Eq. (1) for the transmitted light beam. The time coordinate in the frame, under Einstein synchronization, is obtained by introducing a time offset of the form given in Eq. 13, where the velocity is replaced with : specifically . Applying this offset yields the first-order approximation . This time adjustment effectively rotates the plane of simultaneity by an angle . Consequently, after re-synchronization, the radiation wavefront undergoes rotation in the accelerated frame, as illustrated in Fig. 15. The new time coordinate in the accelerated frame is interpreted in accordance with Maxwell’s equations, ensuring their applicability to light propagation. As a result, the transmitted light beam (passing through the aperture) propagates at an angle , demonstrating the phenomenon of the aberration of light. Both approaches yield the same outcome. The choice between these clock synchronization conventions is ultimately pragmatic. While changing the (four-dimensional) coordinate system does not introduce new physical phenomena, it provides a more consistent and coherent description of existing ones. We restricted our analysis to the diagonalization procedure for nonrelativistic velocities. However, the method can be readily generalized for an arbitrary parameter . A full diagonalization can be achieved using the transformation: , . Here, the physical time determines the flow of time in a physical process in the accelerated frame , while denotes the physical distance between two points of three-dimensional space. 5.6 Composition of Motions in Non-Inertial Frames Until now, we have considered only the case of nonrelativistic motion. We now extend our analysis to arbitrary velocities. As discussed earlier, the metric tensor must be a continuous quantity. The Langevin metric is derived by matching the metric tensors of the accelerated and inertial frames. For a smooth transition of the metric tensor, it is sufficient to relate the coordinates and time of the accelerated observer to those of the inertial observer using the inverse Galilean boost. Under this transformation, the inertial frame metric transitions into the Langevin metric of the accelerated frame. Next, we consider the transformation of the direction of light propagation. The group velocity of light transforms similarly to that of a particle, following the Galileo velocity addition theorem. Let us assume that the accelerated frame moves with velocity , then the angle of aberration is such that . If the angle of aberration is small, being approximated by . It is always possible to choose such variables, in which the metric of an accelerated source will be diagonal. In the Lorentz coordinatization, light propagates with constant velocity independently of the direction of propagation, and of the velocity of its source. If we look at the transformation , , we see that velocity along the -axis is the same in the Lorentz coordinatization too: . Substituting into this equation confirms that . The group velocity of the light beam remains , but when we use the new velocity of light this becomes . This represents a correction to the classical aberration of light law within Lorentzian coordinates. Now, suppose an accelerated observer measures the aberration of light. Since time and distance in Lorentzian coordinates retain direct physical significance, the observer finds that the aberration angle satisfies the relation . 5.7 Discussion Let us now return to the topic of metric diagonalization and deepen our understanding of the relationship between the Lorentz coordinate systems and . We have already noted that the old coordinates are found by matching the accelerated frame and inertial frame metrics; what does mismatching of the coordinates and mean, in terms of measurements made by the accelerated observer? In both reference frames, the metric remains diagonal, and, according to standard textbooks, the coordinates should be related by the Lorentz transformation. At first glance, the problem appears entirely symmetrical, suggesting that both frames should be equivalent. However, as we demonstrated earlier, there is a fundamental distinction between an accelerated inertial frame and an inertial frame without a history of acceleration. Where does the asymmetry originate? The electrodynamics equation must be integrated with initial conditions. According to Maxwell’s electrodynamics, coherent radiation is always emitted in the direction normal to the radiation wavefront. In the inertial frame, the wavefront of the emitted light beam is perpendicular to the vertical -axis. However, when considering Lorentz coordinatization, the wavefront orientation is affected in the accelerated frame . According to relativistic kinematics, the extra phase chirp is introduced as a consequence of this, the plane wavefront rotates after the acceleration. Then, the radiated light beam is propagated at the angle with respect to the -axis, yielding the phenomenon of the aberration of light in the accelerated frame . This phenomenon is purely kinematic and involves no forces. Many may find this counterintuitive. We explore the resolution of this paradox in detail in Chapter 12, using a dynamical approach to explain the asymmetry between inertial and accelerated reference frames. Without proof, we state the key result: the apparent asymmetry arises from the presence of inertial (pseudo-gravitational) forces in the accelerated frame . The principle of equivalence allows non-inertial kinematic problems to be addressed using dynamical methods. Wavefront rotation, which occurs when transitioning from an inertial to an accelerated frame (relative to the fixed stars), can be attributed to the pseudo-gravitational potential gradient during acceleration. A clock’s rate depends on the local pseudo-gravitational potential, leading to an accumulated time difference between spatially separated clocks. Suppose (for simplicity) that there is a constant acceleration along the -axis during the time . The pseudo-gravitational acceleration is simply equal to the gradient of scalar potential: . A clock at a higher gravitational potential (aligned with the acceleration direction) runs faster. The pseudo-gravitational potential difference between two points along the -axis is given by . Once the fixed stars move with constant velocity, the potential gradient in becomes zero. The accumulated time difference between two spatially separated clocks is . This time shift effectively rotates the plane of simultaneity by an angle . 5.8 A Single Moving Emitter in an Accelerated Frame In Chapter 4, we examined the single-emitter problem within the initial inertial frame of reference, . Our analysis revealed a deviation in the energy transport of light emitted by an accelerated source, which corresponds to a well-known phenomenon: the aberration of light from a tangentially moving emitter in an inertial frame of reference (see Fig. 1). We demonstrated that this aberration can be analyzed using two distinct approaches: the covariant and the non-covariant formulations. We now extend our analysis to the aberration of light emitted by a ”plane wave” source moving within an accelerated reference frame. In this case, two different methods of coordinatization are possible: (a) Consider a system , initially at rest in the inertial frame , which undergoes acceleration along the -axis until it reaches a velocity . The simplest synchronization method, based on absolute time, involves maintaining the same set of uniformly synchronized clocks as when was at rest. However, acceleration relative to the fixed stars influences the propagation of light. As a result, by the time attains a constant velocity, the velocity of light emitted from a source at rest in the accelerated frame will exhibit anisotropy along the -axis (see Eq. (15)). (b) Once the system begins moving with a constant velocity, Einstein’s standard clock synchronization procedure can be applied. This synchronization method is defined using light signals emitted by a source at rest in the accelerated frame, under the assumption that light propagates isotropically with velocity . By employing this Einstein synchronization procedure, we describe the light emitted from the source at rest (in the accelerated frame) using the standard Maxwell equations. We now consider the case where the emitter is at rest in the initial inertial frame (i.e., at rest relative to the fixed stars), while the observer, who is at rest in the accelerated frame , performs the measurement of energy transport direction. It is important to emphasize that the aberration of light emitted by a single source also depends on the motion of the emitter within the accelerated frame . Due to the fundamental asymmetry between inertial and accelerated frames, the theory of light aberration predicts a remarkable effect. Specifically, if the emitter remains at rest relative to the fixed stars while the observer transitions from rest to motion with respect to the fixed stars, the apparent angular position of the ”plane-wave” emitter, as seen in the accelerated frame, would jump by an angle of . This aberration effect in the frame is illustrated in Fig. 16. Let us describe the radiation emitted by a source moving with velocity in an accelerated frame, as observed by an accelerated observer. The acceleration relative to the accelerated frame influences the propagation of light. Assuming an absolute time coordinatization, we apply a Galilean boost by substituting while keeping the time coordinate unchanged. Applying this transformation to the Langevin metric yields \| \| \| --- \| \| \| (18) \| This metric characterizes the electrodynamics of a stationary light source in an inertial frame from the perspective of measurements made by an accelerated observer. From the formula, it is evident that the measurement of the electromagnetic field configuration in the moving system—expressed in terms of the coordinate and time of a measuring device at rest in the accelerated frame—yields the same result as that of the configuration at the point and time . A similar situation arises in the inertial frame. In Chapter 3, we analyzed the electrodynamics of a moving emitter using the metric given in Eq. (11). Within a single accelerated frame, during the inertial phase, the symmetry principle—namely, the equivalence of active and passive Galilean boosts—is upheld. Let us now examine the fundamental asymmetry between inertial and accelerated frames in absolute time coordinatization. An inertial observer describes the electrodynamics of a stationary light source using the Minkowski metric. By substituting and into the Minkowski metric, we obtain the Langevin metric (Eq. 15) in the comoving frame. A natural question arises: what is the metric of the inertial observer from the perspective of the accelerated observer? To address this, we transform the coordinates , which correspond to the accelerated observer , using the inverse Galilean transformation. Specifically, we substitute while keeping time unchanged ( ) into the Langevin metric (Eq. 15). This transformation yields the Minkowski metric, which correctly describes the electrodynamics of a stationary light source in the inertial frame. Importantly, this transformation highlights how an inertial observer perceives the measurements of an accelerated observer. In particular, it illustrates the time dilation effect experienced by a physical clock at rest in the accelerated frame. Crucially, the slowdown of the accelerated clock is independent of the reference frame in which this effect is observed (see Chapter 12 for further details). Another approach to solving the problem involves the Langevin metric diagonalization procedure. In this framework, the new coordinate system in the accelerated frame is interpreted such that an observer perceives a diagonal metric: for an emitter at rest in the accelerated frame, and for a moving emitter. This reveals a symmetry (or reciprocity) in the metrics—and consequently in the electrodynamic equations—between the accelerated and inertial frames. This symmetry arises naturally from the pseudo-Euclidean geometry of space-time. The only difference between this metric and Eq. (11) is the sign of the velocity. After diagonalization, it becomes evident that any apparent asymmetry stems from the initial conditions. Notably, the plane radiation wavefront of a moving emitter (regardless of its velocity) undergoes a rotation by an angle of in the accelerated frame as a result of the metric diagonalization. This implies that the radiated light beam has a group velocity along -direction given by . To predict the outcome of aberration measurements, one can alternatively analyze the metric of the moving emitter directly within the absolute-time coordinate system, given by Eq. (18). According to electrodynamics, the group velocity emerges as a direct consequence of the cross term . Now, let us consider the most general case in which the system , initially at rest in the inertial frame , is accelerated to velocity along the -axis. Simultaneously, an emitter in the accelerated frame is accelerated from rest to velocity along the -axis. Suppose an observer in the accelerated frame performs an aberration measurement. To describe the radiation emitted by a source moving with velocity in the accelerated frame, we must account for the electrodynamics in terms of the accelerated observer. This can be done by employing the metric . A closer examination of the physics reveals that the aberration increment is related to the system parameters by the expression . The key point is that the cross term in the metric, which induces anisotropy in the accelerated frame, depends solely on the velocity of the accelerated frame relative to the initial inertial frame. 5.9 A Point Source in an Accelerated Frame Now, let us return to observations of an accelerated observer. A key characteristic of point-like sources is that radiation emitted at a given instant forms a spherical wave around the source, and any measuring instrument inevitably influences the observed radiation (see Fig. 17 - Fig. 18). The aberration of a point source is independent of the source’s speed and is attributed solely to the observer’s motion relative to the fixed stars. The cross term in metric Eq. (15) introduces anisotropy in the accelerated frame, altering the direction of transmitted radiation. As a result, a point source exhibits the same aberration pattern as fixed stars, meaning its apparent position shifts by an angular displacement identical to that of the fixed stars. In conventional aberration theory, an intriguing puzzle arises concerning stellar aberration. Some binary star systems undergo velocity changes on timescales ranging from days to years. At certain times, the components of such systems can have significantly different velocities relative to Earth. However, it is well established that both components always exhibit the same aberration angle. Rotating binary systems follow the same aberration pattern as all fixed stars, appearing to shift annually by the same universal aberration angle. This observation strongly supports our theoretical prediction for the aberration of light from a point source in a rotating frame. In the next chapter, we explore this experimental test in greater detail. 5.10 Aberration of a Tilted Incoming Plane Wave We now return to the scenario depicted in Fig. 17 – Fig. 18. A point source located in the far zone effectively generates a plane wave in front of the aperture. Above, we considered only the case where the point source is positioned on the optical axis (which is parallel to the -axis), producing a normally incident plane wave. The fundamental electrodynamic equations must hold for all electromagnetic waves. In other words, the dispersion equation in the accelerated frame should be consistently applied to both the incoming and scattered waves. In Maxwell’s electrodynamics, the dispersion relation simplifies to . A notable feature of this case is that, even after a Galilean transformation along the -axis, the dispersion relation in the accelerated frame retains the same diagonal form, , for a normally incident wave. Now, let us examine the behavior of an incoming wave with a phase gradient along . In other words, there is an angle between the wavevector and the -axis. For small angles , we can approximate . Our goal is to express the parameters of the incoming wave in terms of the source offset relative to the optical axis. The wave tilt angle can be written as , where is the distance from the source to the screen. Next, we analyze the aberration of light emitted by a stationary point source in an accelerated frame. The wavevector of the incoming plane wave is determined by the initial conditions. Assuming that the emitted wavefront is initially tilted by an angle , we obtain , . Since acceleration modifies the field equations, we substitute the plane wave solution into Eq. (17), yielding: . A screen with a hole acts as a diffraction grating, splitting the incident plane wave into multiple diffracted components. The wavenumber of the transmitted plane wave along the -direction is , where is the wavenumber associated with the sinusoidally modulated transmittance. The dispersion equation then simplifies to . This result indicates that the transmitted light beam propagates along with group velocity . Thus, the apparent angular position of the point source is related to the problem parameters by 181818A similar effect arises if the screen moves with velocity along the -axis in an inertial frame. In Chapter 4, we considered the special case of a normally incident plane wave, but this approach generalizes to tilted wavefronts as well. The deviation in energy transport for light passing through a hole in the moving screen leads to an aberration increment given by .. 5.11 Non-Reciprocity in the Aberration of Light Theory In this section, we continue our discussion on the aberration of light emitted by a single moving source. Now, let us consider a scenario with two identical emitters. Suppose the first emitter remains at rest in an inertial frame, while the second emitter is initially at rest but then accelerates to a velocity along the -axis. An observer, who is at rest in the inertial frame, measures the direction of energy transport. Let us assume that, at the outset, the velocity component of the emitted light along the -axis is zero. The question then arises: How does the light beam from the moving emitter appear? For the inertial observer, the angular displacement of the emitted radiation is given by . This effect, illustrated in Fig. 19, is a well-known example of the phenomenon of aberration of light. Let us analyze the scenario in which an accelerated observer redirects an accelerated emitter. We consider the specific case where, after redirection, the (group) velocity component of the light beam along the -axis becomes zero. Following this redirection, an inertial observer would measure an angular displacement of . Now, suppose the accelerated observer also conducts an aberration measurement. As shown in Fig. 20, the aberration increment is related to the system’s parameters by the expression . . To further explore the aberration of light in an accelerated frame, it is useful to consider a slightly modified source—a ”plane wave” emitter. We adopt the model described in Section 4.2 because it is both relatively simple and sufficiently general to serve as a prototype for understanding aberration phenomena. While no one has performed all the ”thought experiments” exactly as described here, the outcomes can be confidently predicted based on the laws of special relativity, which are themselves grounded in experimental evidence. Figure 21 illustrates the modified ”plane wave” emitter setup. Though it appears more complex at first glance, it is practically feasible to construct. The setup consists of a point source of light positioned in the front focal plane of a lens. The source’s dimensions are assumed to be on the order of , where is the wavelength of optical radiation. Let us revisit the redirection procedure for an accelerated source. A key question arises: where is the information about the source’s acceleration recorded when redirecting light from an accelerated emitter? To understand the redirection of a light source in an accelerated system, we must examine the source’s internal mechanics and observe what occurs during the redirection process. Figure 21 illustrates a schematic of a ”plane wave” emitter. In electrodynamics, redirection can be achieved by introducing an offset between the point source and the optical axis (which is aligned with the - axis), as shown in Figure 22. This offset encodes the information about the source’s acceleration. 5.12 Two Metrics In special relativity, an important theorem states that applying a Galilean boost yields the metric of an accelerated emitter in the initial inertial frame, given by Eq. 11. In this framework, a non-accelerated light source is described using the Minkowski (diagonal) metric. To transform the coordinates of an inertial observer, we use the equivalence of active and passive pictures. By substituting into the Minkowski metric, we obtain Eq. 11, which describes the electrodynamics of a moving emitter from the perspective of an inertial observer. On the other hand, the Langevin metric describes the electrodynamics of an accelerated emitter from the viewpoint of an accelerated observer measurements. A key principle is that the metric tensor must remain continuous. This problem in special relativity can be effectively addressed using an absolute time coordinatization, which relies on Galilean boosts. To transform the inertial observer’s coordinates to the accelerated frame, we apply the inverse Galilean transformation: substituting into the Minkowski metric yields the Langevin metric, Eq. (15). Notably, the Langevin metric Eq.15 of the accelerated emitter in the accelerated frame is the same as metric Eq.11 of the accelerated emitter in the inertial frame. The difference is only that the sign of velocity . These two metrics are two sides of the same coin. Notably, this theorem is absent from conventional formulations of special relativity. Here, for the first time, we present the metric given in Eq. 11. It is surprising that this result has not previously been reported, especially considering that the Langevin metric—derived by matching metric tensors across frames using Galilean boosts—is already widely used in applications involving optical phenomena in rotating frames. In the problem under discussion, a puzzle remains. We introduce a comoving coordinate system in the initial inertial frame and analyze radiation from the moving source in terms of the new (comoving) coordinate labels. After performing a passive transformation, the source coordinates transform as . According to the equivalence of active and passive transformations within a single inertial frame, Maxwell’s equations remain valid in the comoving coordinate system. Consequently, the electric field of an electromagnetic wave satisfies the equation . This implies that the Minkowski metric, , governs the electrodynamics of the accelerated light source in the comoving coordinate system. However, a conceptual tension emerges: Why does the Minkowski metric apply in the comoving description (within an inertial frame), while the Langevin metric governs the physics in the truly accelerated frame? This apparent inconsistency requires careful examination. The key lies in recognizing that inertial and accelerated observers do not share the same three-dimensional space in Minkowski spacetime. Although this might seem paradoxical if both observers were assumed to share a common spatial geometry, special relativity resolves the issue: observers following different worldlines inhabit different instantaneous 3-spaces due to the relativity of simultaneity. To analyze this, we begin by examining the relationship between passive and active transformations. A passive Galilean boost within a single inertial frame, given by , simply re-expresses the results of measurements made by the inertial observer using a new set of coordinates. In the case of active transformations within the same inertial frame, the motion of fixed stars relative to the observer and their instruments remains unchanged. This establishes the equivalence between active and passive Galilean boosts in inertial frames. In contrast, the Langevin metric describes measurements made by an accelerated observer. Under an inverse Galilean boost, the motion of fixed stars relative to the accelerated observer’s space-time grid is no longer preserved. This highlights the fundamental difference in how motion is perceived in non-inertial frames. When observers adopt absolute-time coordinatization, relativistic kinematics becomes remarkably simple. In this approach, all asymmetries between the initial inertial and accelerated frames are absorbed into their respective frame metrics. Alternatively, consider observers using Lorentz coordinatization. Here, the frame metrics become diagonal, and the complexities of special relativity manifest as counterintuitive kinematic effects. Notably, in this coordinatization, the inertial and accelerated observers experience different three-dimensional spatial geometries. On the other hand, the standard textbook treatment of Lorentz transformations implicitly assumes that the -axes of the accelerated observer remain parallel to the -axes of the inertial observer. In other words, it presumes that both observers share a common three-dimensional space. This assumption, often taken for granted, suggests that the simultaneous acceleration of a rigid reference frame has direct physical significance. However, this is a misconception. For example, conventional derivation of the aberration of light effect overlooks the conventionality of distant simultaneity. There exists an inherent ambiguity in the spatial position along the -axis (in the -direction) due to uncertainties in the timing of acceleration. This uncertainty can be quantified as , revealing that the associated events are space-like separated. Consequently, the orientation of the -axis lacks an exact objective meaning, due to the relativity of simultaneity. The idea that the direction of the -axis relative to the inertial -axis is physically well-defined is, at best, an approximation. This relationship can be visualized as a vertical axis with an angular uncertainty (or ”blurring”) given by: . This inclination angle has no precisely defined physical meaning, since—given the finite speed of light—there is no experimental method to determine such a spatial displacement. Let us illustrate these ideas with the example of a plane-wave emitter moving along the -axis with velocity in an initial inertial frame. Assume that clocks are synchronized according to the Einstein synchronization procedure, using light signals from a stationary dipole source. The electrodynamics of the moving source in this inertial frame is described by the metric in Eq. 11. As previously shown, the radiated beam exhibits aberration: due to the motion of the source, the direction of energy propagation is altered. However, when we introduce a comoving coordinate system via a passive Galilean boost, this aberration disappears. In this comoving system, the light beam propagates along the -axis. Crucially, because the boost is performed within a single inertial frame, the comoving coordinate axes remain parallel to the original inertial axes . That is, both systems inhabit the same three-dimensional space. Let us now examine the relationship between the coordinate system of the inertial frame and that of the comoving frame. To detect aberration in the inertial frame, one must first establish a coordinate system, which requires a physical, operational procedure. In this sense, the spatial grid of the inertial frame represents a physical reality. By contrast, the comoving coordinate grid introduced through the passive Galilean boost is a purely mathematical construct—there is no need to consider how it would be physically realized. It is important to emphasize that a passive Galilean boost within a single inertial frame merely provides an alternative parametrization of the same observations made by an inertial observer using real, physical reference axes. Now consider the perspective of an accelerated observer. As before, to detect any aberration effect within the accelerated frame, a coordinate system must be established. This requires the observer to define a practical, operational method for assigning spatial coordinates . The accelerated observer finds that the angular displacement of the beam is negative, given by . This result leads to a key insight: from the viewpoint of the inertial observer, the -axis of the accelerated observer is not parallel to the inertial -axis. In other words, the inertial and accelerated observers inhabit different three-spaces within Minkowski spacetime. This subtle but important distinction reflects the breakdown of simultaneity in relativistic physics and anticipates deeper discussions that will follow in Chapter 7. 5.13 Ether Theory and the Aberration of Light Effect The framework of classical physics is rooted in the concept of absolute space, often linked to the hypothetical ”ether”—a substance once thought to provide an absolute frame of rest. In this section, we will show that the initial inertial frame discussed earlier is not physically equivalent to absolute space or the ether. Instead, it represents an abstract idealization: the concept of absolute space-time. To deepen our analysis, we revisit the classical ether theory in its original form, which posits the existence of a stationary ether that determines the speed of light. Since the ether is presumed immobile, any frame moving relative to it would experience anisotropy in light propagation. A non-inertial frame, accelerating with respect to the ether, would thus detect an ”ether wind” when measuring the speed of light. We begin by examining the influence of the ether wind on light speed. According to the ether hypothesis, the velocity of light observed from an accelerated frame would be for a beam propagating in the same direction as the accelerated system and for a beam propagating in the opposite direction, where represents the ether velocity in the accelerated reference frame. The aberration effect arises as a consequence of the ether wind’s influence on the speed of light (Fig. 23). 5.13.1 Lorentz’s Theory of Corresponding States In 1895, Hendrik Lorentz derived a first-order version of the so-called ”theorem of corresponding states” to provide a general explanation for the null results of first-order optical interference experiments. This theorem establishes a relationship between pairs of electromagnetic field configurations: one in a system at rest in the ether and the other in uniform motion through it. Lorentz demonstrated that if a particular field configuration is allowed by the laws of electrodynamics, its corresponding configuration in the moving system must also be valid. The theorem works as follows. The mathematical description of the configuration in the moving system in terms of ”local time” (instead of ”real time” ) is the same as the description of its corresponding state in the system at rest in terms of real time. Lorentz showed that two field configurations related to one another in many of their observable properties, in particular, they give rise to identical interference patterns. Thus, it is not possible to detect the orbital velocity using interference (e.g. Michelson-Morley) experiments 191919For discussion of the relationship between the generally accepted way of looking at special relativity and Lorentz’s ether theory we suggest reading the paper .. We now examine the application of Lorentz’s pre-relativistic theory to the aberration of light in a non-inertial frame of reference. Previously, we analyzed the aberration of light in accelerated systems within the framework of special relativity, considering only first-order effects. Since light is inherently a relativistic phenomenon, its behavior in non-inertial frames requires careful treatment. In this chapter, we have already discussed the asymmetry between inertial and accelerated frames. A key observation was that the metric tensor must remain continuous. In special relativity, this issue is best addressed through an absolute time coordinatization approach. Maxwell’s equations hold in the initial inertial frame, where the optical system (e.g., a light source and lens, as shown in Fig. 21) is at rest, and time coordinates are assigned via slow clock transport. To ensure continuity, the metric of a non-inertial frame associated with an accelerating optical system (relative to the initial inertial frame) must transition smoothly into the Langevin metric. This requires relating the accelerated observer’s coordinates to those of the inertial observer using the inverse Galilean transformation: , . Under this transformation, the wave equation takes the form of Eq. 17, where the coordinates and time adhere to Galilean transformation rules. The Langevin metric, along with the initial conditions, then governs optical effects in the accelerated frame. In the preceding discussion, we analyzed the aberration of light in an accelerated frame using the Fourier transform method. The fundamental asymmetry arises from the electrodynamics equation (Eq. 17). One consequence of the cross-term in the wave equation is the emergence of group velocity, which contributes to the aberration. However, an alternative approach provides an equally satisfactory explanation. This involves clock resynchronization, wherein Einstein’s synchronization method is used to define a Lorentz coordinate system for the optical setup at rest in the accelerated frame. In this framework, Maxwell’s equations describe the field configuration, and the resynchronized time is given by in the first-order approximation. After resynchronization, the asymmetry shifts to the initial condition: the plane wavefront of the emitter undergoes a rotation by an angle in the accelerated frame. At first glance, Lorentz’s treatment of an optical setup in an inertial frame moving through the ether appears similar to our approach to such a setup in an accelerated frame (relative to the fixed stars). We now analyze this similarity. One key observation is that our initial inertial frame—one without a history of acceleration relative to the fixed stars—can be interpreted as what Lorentz considered a frame at rest in the ether. In Lorentz’s theory, and more broadly in pre-special relativity physics, the relationship between an observer at rest (using coordinates )) and an observer bound to a moving body (using ) was governed by the Galilean transformation: . In Lorentz’s framework, a Galilean boost is empirically equivalent to a smooth adjustment of the metric tensor. Under this transformation, the metric of the initial inertial frame transitions into the Langevin metric of the accelerated frame. Of course, Lorentz did not conceptualize his theory in this way, as Minkowski spacetime was not yet available to him. Due to this Galilean boost, the homogeneous wave equation for the field in the moving frame (Eq. 17) lacks a standard form. The primary complication arises from the presence of a cross-term, which makes solving the equation more challenging. To circumvent this difficulty, Lorentz observed that simplification is always possible. In solving the wave equation, he introduced the variable , which he called ”local time.” In Lorentz’s theory, this time shift was a mathematical device used to transform the electrodynamics equations into Maxwell’s form. At first sight, Lorentz’s theorem of corresponding states for first-order phenomena seems sufficient to provide an exact treatment of light aberration in an accelerated frame. One might naively expect that Lorentz’s theory would predict, in full generality, that observations of light aberration from an earth-based incoherent source should reveal Earth’s orbital motion (Fig. 22). However, this leads to an important question: Why did Lorentz conclude, based on his theorem of corresponding states, that no first-order effects of Earth’s orbital motion could be detected? 5.13.2 Analogy with the Sound Theory A comparison between Lorentz ether theory and the theory of sound may be insightful in this context. In Section 4.15, we analyzed the problem of a sound emitter in the atmosphere frame, specifically considering a screen moving along its surface with velocity . According to the theory of sound, energy transport remains unaffected when sound passes through a hole in the moving screen. When a plane wave falls the screen normally, it generates a transmitted oblique sound beam, as illustrated in Fig. 7. In this scenario, the group velocity matches the phase velocity. The diagonal wave equation, , is always valid in the atmosphere frame. However, this equation is not applicable from the perspective of an observer at rest in an accelerated frame. Accelerations relative to the atmosphere frame influence the propagation of sound, leading to a transformation of the wave equation into \| \| \| --- \| \| \| (19) \| where coordinates and time are transformed according to the Galilean transformation , . To describe sound propagation, the wave equation must be integrated with the initial conditions of the sound wavefront. Notably, acceleration does not influence the wavefront’s orientation, meaning the emitted sound beam remains perpendicular to the axis. Since the air remains stationary in the atmospheric frame, any reference frame moving relative to it experiences anisotropy in sound propagation. As a result, an observer in such a moving frame perceives an ”air wind” when measuring the speed of sound. According to the wave equation (Eq. 19), the sound velocity observed from an accelerated reference frame is for a beam propagating in the same direction as the accelerated system and for a beam propagating in the opposite direction. Here, represents the wind velocity in the accelerated frame. This effect, known as aberration, arises due to the influence of the perceived air wind on the speed of sound (Fig. 23). Let us now analyze the aberration of sound radiated by a single ”plane-wave” emitter in the atmospheric frame. The wavefront radiated by the plane-wave emitter at a given instant time is constructed as an envelope of all the spherical wavelets from the point sources on the emitter that have radiated until this instant. It is not difficult to see that when the emitter is moving at a constant velocity along the axis, the wavefront propagates at a speed at an angle from the vertical. This phenomenon arises due to the motion of the emitter, which causes constructive interference to occur when the projection of the phase velocity onto the -axis matches the velocity of the emitter. According to classical kinematics, this results in an additional phase chirp given by . Consequently, the plane wavefront undergoes rotation as the emitter accelerates. As a result, the radiated sound beam propagates at an angle , manifesting as the aberration of sound in the atmospheric frame—analogous to the aberration of light. Now, consider the perspective of an accelerated observer. The wave equation must be integrated with the initial condition set by the sound wavefront. A Galilean boost does not affect the orientation of the wavefront, meaning that the wavefront of the emitted sound beam remains tilted at an angle relative to the -axis. The wave vector of the radiated plane wave is determined by this initial condition, leading to . Furthermore, the non-diagonal wave equation (Eq. 19) implies that the radiated sound beam moves along the -axis with a group velocity given by . For further details, refer to Section 5.10. Thus, we derive the same addition theorem for sound beam velocities as for particle velocities in classical physics. According to Lorentz, the wave equation for a moving light (or sound) emitter in the ether (or atmospheric) frame is identical to that of an emitter at rest. Lorentz concluded that Earth’s orbital motion could not be determined using aberration measurements from Earth-based sources. Consequently, ether and sound theories were merely variations of the same classical framework. However, this analysis overlooks a fundamental distinction: the velocity of light differs inherently from the velocity of sound. When Lorentz formulated his pre-relativistic theory he assumed a space-time background with a well-defined geometrical structure: Newtonian space-time. A close look at the physics of this subject shows that there is a real difference between the inertial frame without an accelerational history relative to the fixed stars and the Lorentz classical ether at rest. The main difference between the initial inertial frame and the absolute space of Newton is that special relativity taught us to think in terms of a unified space-time model. In special relativity, the absolute space and the absolute time of Newton are fused into absolute space-time. 6 Stellar Aberration 6.1 The Corpuscular Model of Light and Stellar Aberration It is commonly understood that the aberration of light can be interpreted through the corpuscular model of light, drawing an analogy to how a moving observer perceives the oblique fall of raindrops. This classical kinematic approach to the calculation of stellar aberration has been employed in astronomy for nearly three centuries 202020The phenomenon of the annual apparent motion of celestial objects, known as stellar aberration, was first discovered by Bradley in 1727, who also explained it using the corpuscular model of light .. However, in the 20th century, it was discovered that Newton’s dynamical laws for light were incorrect, necessitating the introduction of electromagnetic wave theory to rectify these errors. As is well-known in textbooks, the physical basis of stellar aberration arises from the finite speed of light and the change in its direction when observed from a different reference frame. This can be explained as a consequence of the velocity addition formula applied to a light beam when the observer shifts between reference frames. For an observer on Earth, the velocity relative to the solar reference frame is approximately , corresponding to Earth’s motion around the Sun. To describe the effect of stellar aberration, it is sufficient to consider only the first-order approximation, . The conventional treatment of stellar aberration remains deeply connected to classical (Newtonian) kinematics, relying on the erroneous assumption that observers on Earth and the Sun share a common 3-space. 6.2 The Wave Theory of Stellar Aberration Most authors treat light propagating through the telescope barrel as a stream of photons rather than as a wave. 212121A common misconception is that the rays of light coming from a star simply fall onto the telescope tube without interacting with its sides. As French states: ”Regarding light as composed of a rain of photons, we can easily calculate the change in the apparent direction of a distant object such as a star.” Similar views are also found in recent textbooks. For instance, Rafelski writes: ”We consider a light ray originating from a distant star… The following discussion addresses the observation of well-focused light rays, not (spherical) plane wave light… This view of the experimental situation is accurate because the light emitted by a star consists of an incoherent flux of photons produced in independent atomic processes.” This conclusion, however, is incorrect. We emphasize that the design of a telescope’s optical system is based on classical diffraction theory, which stems directly from classical electromagnetic theory. The resolution of a telescope is fundamentally limited by the diffraction of light waves, and the telescope always affects the measured radiation due to the unavoidable diffraction of starlight by the telescope aperture. In a well-corrected optical system with a circular pupil, the size of the Airy disk (i.e., the size of the image of a point source) is inversely proportional to the pupil’s diameter. By questioning the validity of this standard reasoning, we argue that a proper treatment of stellar aberration should rely on coherent wave optics. It is easy to show that light from a distant star is approximately coherent over a circular area whose diameter, in practical terms, is far larger than the diameter of the telescope. Thus, the telescope samples only a tiny portion of the star’s coherent light, meaning the waveforms can be approximated as (flat) plane waves. 222222While stars are considered incoherent sources, the mutual intensity function produced by an incoherent source is fully described by the Van Cittert-Zernike theorem . Any star can be considered as very distant from the Sun, and telescopes typically operate in the far zone of such sources. For instance, consider Sirius, one of the closest stars to Earth. The coherent area of light from Sirius has a diameter of about 6 meters, a correlation observed by Brown and Twiss in 1956 . Let’s verify whether this assertion holds true. The spatial coherence of a light beam generally refers to the coherence between two points in the field illuminated by the light source. To understand spatial coherence more clearly, we can refer to Young’s two-pinhole experiment (see Fig. 24). In its simplest form, the degree of coherence between two points is described by the contrast of the interference fringes obtained when these points are treated as secondary sources. Consider a source illuminating two pinholes, and , as shown in Fig. 24. If the source is perfectly incoherent, no interference fringes can be observed by placing two pinholes in the plane of the source. However, it has been demonstrated that when the two pinholes are positioned sufficiently far from the incoherent source, interference fringes with good contrast can be observed. It is often stated that the spatial coherence of light beams increases with distance ”simply through propagation.” It would be valuable to find an elementary explanation that allows us to visualize the physical process behind this phenomenon. Consider a quasi-monochromatic wave incident on an aperture in an opaque screen, as depicted in Fig. 24. In general, this wave may exhibit partial coherence. As the wave propagates through space, its detailed structure evolves, and similarly, the spatial coherence structure also changes. In this context, the transverse coherence function is said to ”propagate.” Given the spatial coherence at the aperture, our goal is to determine the spatial coherence on the observation screen, which is located at a distance beyond the aperture. Stellar radiation is inherently stochastic, and for any starlight beam, there exists a characteristic linear dimension, , which defines the scale of spatially random fluctuations. Fig. 25 illustrates the spiky pattern that appears on the aperture in an opaque screen. When , where is the aperture size, the radiation beyond the aperture remains partially coherent, as shown in Fig. 25. In this case, corresponds to the typical linear dimension of the spikes. First, we aim to calculate the instantaneous intensity distribution observed across a parallel plane located a distance beyond the aperture. The observed intensity distribution can be determined by taking a two-dimensional Fourier transform of the field. The radiation field across the aperture can be expressed as a superposition of plane waves, all with the same wavenumber . The ratio represents the sine of the angle between the -axis and the propagation direction of the plane wave. In the paraxial approximation, we have . If the radiation beyond the aperture is partially coherent, a spiky angular spectrum is expected. The characteristics of these spikes can be easily described using Fourier-transform notation. The typical width of the angular spectrum envelope is expected to be on the order of , where is the coherence length. Additionally, the angular spectrum of a source with transverse size will exhibit spikes with a typical width of about , which follows from the reciprocal width relations of Fourier transform pairs (see Fig. 26). It is the linear dimension of the source that determines the coherent area of the observed wave, which is given by . Therefore, if the screen is placed sufficiently far from the incoherent source, such that , a coherent area of large linear dimension can be observed. A star is an incoherent source, characterized by spatial fluctuations on the scale of . Such a source radiates in all directions. Additionally, the angular spectrum of the starlight contains spikes with a typical width of , where represents the star’s diameter. In astronomical observations using a telescope, each spike observed on the Earth’s surface has a finite thickness, approximately equal to the spatial coherence length of the starlight. Therefore, the coherent area of the observed starlight on the Earth’s surface can be estimated to be of the order of , where is the distance from the star to the Earth. The star chosen by Bradley for this observation was Draconis, one of the nearest stars. For this case, the diameter of the coherent area on the Earth’s surface is roughly estimated to be around 100 meters. 6.3 Stellar aberration in the Context of Special Relativity It is widely accepted that the theory of relativity aligns with Bradley’s discovery of stellar aberration, interpreting it as a consequence of an observer’s motion relative to a light source. 232323 In his paper on relativity, Einstein derived the aberration formula by considering the velocity as the relative velocity of the star-Earth system. This interpretation has been adopted by numerous textbook authors. As Moeller states: ”This phenomenon, known as aberration, was observed by Bradley, who noticed that stars appear to undergo an annual collective motion in the sky. This apparent motion arises because the observed direction of a light ray from a star depends on the Earth’s velocity relative to the star.” The asymmetry between the cases where either the star or the Earth-based telescope is in motion becomes evident when considering the separation of binary stars. Spectroscopic binaries exhibit velocities exceeding that of Earth’s orbit around the Sun. These stars revolve around their common center of mass within just a few days—a period during which Earth’s motion remains practically constant. If the changing velocities of the binary components were comparable to Earth’s orbital velocity, their separation should be easily detectable. However, this is not observed (Fig. 27-Fig. 28) 242424In 1950, Ives was the first to highlight a significant challenge that binary star observations pose to the theory of relativity. He argued that the idea of aberration being purely a function of relative motion is contradicted by the existence of spectroscopic binaries with velocities comparable to Earth’s orbital velocity. Yet, these binaries exhibit no aberration effects different from other stars. For instance, the spectroscopic binary Mizar A has well-established orbital parameters. If aberration were solely due to relative velocity, its angular separation should be observable at 1’10”. However, empirical measurements show a value below 0.01”, which is clearly inconsistent with the perspective presented in standard textbooks [37, 38].. In Chapter 4, we critically reexamined the textbook claim that wavefronts and raindrops experience the same aberration. Using the theory of relativity, we demonstrated that when a transversely moving mirror reflects a normally incident plane wave of light, the energy transport deviates. This effect arises because, in the absolute time coordinatization, the wave equation depends on the velocity vector. As a result, the energy transport velocity differs from the phase velocity. According to Babinet’s principle, this remarkable prediction should also apply to light transmitted through a hole in a moving opaque screen or, consequently, through the moving open end of a telescope barrel. The binary star paradox is resolved by recognizing that when light passes through the end of a telescope barrel, diffraction perturbs its fields. As a result, the light beam no longer carries information about the star’s motion relative to the fixed stars. If the telescope were at rest relative to the fixed stars and the observed star began moving, its apparent position in the telescope would not suddenly shift by any angle. However, a different challenge arises when explaining Earth-based observations—specifically, the apparent shift in the positions of fixed stars as the Earth’s telescope changes its motion relative to them. It is crucial to emphasize that aberration is caused by changes in the telescope’s velocity, not the star’s. The resolution of this issue lies in the fundamental asymmetry between Earth-based and Sun-based observers—namely, the acceleration of the Earth-based observer relative to the fixed stars. Some experts, such as Selleri, have explicitly recognized this, noting that aberration can be explained in terms of variations in the Earth’s absolute velocity due to its orbital motion, while the star–Earth relative velocity is irrelevant. Thus, planetary acceleration plays a key role in the phenomenon . We derive stellar aberration using the Langevin metric. Although stellar aberration exhibits asymmetry, it does not contradict special relativity. This is because the heliocentric (Sun-based) reference frame is inertial, whereas the geocentric (Earth-based) reference frame is non-inertial. In the discussion above, we demonstrated that when a small hole is present in an opaque screen at rest relative to the fixed stars, and a point source moves tangentially to the screen, there is no aberration—no deviation in the transport of energy—for light passing through the hole. The transmission appears as illustrated in Fig. 12. The absence of any effect from the motion of the light source relative to the fixed stars in this setup suggests a fundamental issue for stellar aberration theory in the heliocentric frame of reference. Now, consider an observer at rest with respect to the telescope who measures the direction of energy transport. A careful analysis reveals that in the heliocentric frame, where the telescope is at rest, we are dealing with a steady-state transmission problem. How does the transmitted light beam behave in this case? The beam appears to travel along the telescope’s axis because it has lost its horizontal group velocity component. This situation corresponds to a telescope aligned perpendicularly to the phase front of incoming starlight—pointed directly at the star. If the star’s motion is parallel to the phase front (i.e., perpendicular to the telescope axis), then starlight entering the telescope’s aperture will propagate through its full length without deviation. A key conclusion from this discussion is that stellar aberration is absent in this configuration. Specifically, binary star components remain unresolved, meaning their velocities do not affect the aberration phenomenon. The analogy between the obliquity of raindrops and stellar aberration is incorrect. A complete description of all Earth-based experimental observations of stellar aberration is possible only through the theory of relativity and wave optics. Our theory predicts stellar aberration effects that are in full agreement with Bradley’s results (Fig. 29). Due to the asymmetry between inertial and rotating frames, the theory of light aberration makes a striking prediction: if a telescope is at rest relative to the Earth while the Earth rotates relative to the fixed stars, the observed direction of a star from Earth differs from that seen by a hypothetical observer at rest with respect to the Sun. Specifically, the apparent angle is smaller than the actual angle. The difference between the actual and apparent angles, , is given by the relation: , where is the Earth’s orbital velocity around the Sun. This effect arises from the anisotropy introduced by the cross-term in the metric equation (Eq. 15), which influences the direction of radiation (aberration) in the rotating frame. The transmission through the telescope aperture in this frame is illustrated in Fig. 17 - Fig. 18. In the geocentric frame of reference, stellar aberration is independent of the star’s speed and originates solely from the observer’s motion relative to the Sun. According to our interpretation, there are multiple types of aberration, with stellar aberration in the Earth-based frame being just one of them. We have demonstrated that all Earth-based experiments can be explained by two factors: the effect of the measuring instrument (i.e., the physical influence of the telescope on the measurement) and the acceleration of the Earth-based observer relative to the fixed stars. 6.4 Defining a Physical Coordinate System in Space The aberration of light is a geometric phenomenon. To detect its effects within an Earth-based frame, a well-defined coordinate system with a reference direction is necessary. A conventional approach to the aberration of light relies on an implicit assumption of a reference direction, though this assumption often goes unrecognized. Traditionally, the physical interpretation of aberration is based on measurements using rods that remain at rest in the observer’s frame. This local frame of reference is so intuitively accepted that it is rarely questioned or explicitly discussed in aberration theory. However, a more fundamental spatial description exists beyond the reliance on measuring rods. In an Earth-based frame, the natural reference axis is defined by the gravitational field vector. The plumb line, pointing toward the Earth’s center, establishes the nadir, forming the most fundamental local coordinate system. For instance, James Bradley used a vertically mounted telescope to study stellar aberration. He selected the star Draconis because it transited almost exactly at the zenith. The traditional plumb line provided a sufficiently accurate zenith reference for observing stellar aberration. 6.5 Limits of Applicability The applicability of our stellar aberration theory is broader than one might expect. Previously, for simplicity, we assumed that a star image is represented by a point spread function in the image plane of a telescope—in other words, that the input signal is effectively a plane wave. Now, suppose a telescope can resolve details in the star image. A complete understanding of the object-image relationship requires accounting for diffraction effects. Diffraction convolves the ideal image with the Fraunhofer diffraction pattern of the telescope pupil. A star is a spatially incoherent source 252525In statistical optics, stellar emission is described classically at the level of Maxwell’s equations, while the emitting medium—an ensemble of atoms—is treated using quantum mechanics. A star consists of elementary point sources, each statistically independent, with characteristic dimensions on the order of the radiation wavelength, . Within an elementary source volume , a vast number of atoms contribute to emission. Semi-classical theory models these atoms as coherent, interacting radiating dipoles. The induced macroscopic dipole moment in an elementary source produces classical electromagnetic radiation. In statistical optics, electromagnetic fields are treated classically until they interact with the atoms of a photosensitive material, where their interaction is quantized. This avoids the need to quantize the electromagnetic field itself—only the field-matter interaction is quantized. The photodetector converts the continuous cycle-averaged classical intensity into a sequence of discrete photocounts.. Because a star is composed of statistically independent elementary point sources at different spatial offsets, each elementary source effectively produces a plane wave at the telescope pupil. An offset in an elementary source tilts the far-zone field. The total radiation field from a completely incoherent source is thus a linear superposition of the fields from these individual point sources. Since the image of each elementary source is a point spread function, the telescope inevitably influences the measurement of any completely incoherent source. Importantly, any linear superposition of radiation fields from elementary point sources preserves single-source characteristics, such as independence from source motion. This reasoning confirms that our stellar aberration theory remains valid for imaging any completely incoherent source. 7 The Concept of Ordinary Space in Special Relativity 7.1 Inertial frame view of Observations of a Non-Inertial Observer In this chapter, we introduce a novel approach to the theory of light aberration in non-inertial reference frames, offering a fresh perspective on this complex subject. We begin by emphasizing a key principle: the laws of physics in an initial inertial frame must fully account for all physical phenomena, including those observed by non-inertial observers. This principle was already applied in Chapter 5, where we analyzed phenomena in non-inertial (e.g., rotating) reference frames using the Langevin metric. This metric captures the electrodynamics of an accelerated light source from the viewpoint of measurements made by an accelerated observer—but interpreted from the perspective of an inertial observer. Our findings there can be summarized as follows: Inertial Frame Setup We assume that the inertial frame has no history of acceleration and that spacetime, as perceived by an observer at rest, is described by the standard diagonal Minkowski metric: . Applying a Galilean transformation , we obtain the metric of an accelerated emitter in the inertial frame, as presented in Eq. (11). In this coordinate system , the devices of the inertial observer remain governed by the standard diagonal metric. Non-Inertial Frame Transformation To describe an accelerated observer with coordinates , we apply the inverse Galilean transformation: , with unchanged. Substituting this into the Minkowski metric yields the Langevin metric, as shown in Eq. (15). However, we argue that physical phenomena in accelerated frames can also be effectively analyzed within an inertial frame using the framework of standard Einsteinian special relativity and its well-established kinematic effects. For instance, consider the Sagnac effect. From the perspective of an inertial laboratory frame, its interpretation becomes straightforward. The phase difference between counter-propagating waves arises from the relativistic velocity addition formula. That is, as viewed from the lab frame, the Sagnac effect emerges purely as a kinematic consequence of special relativity. This analysis assumes the speed of light in the lab frame is , which—according to special relativity—is independent of the source’s motion when using Lorentz coordinates. In contrast, analyzing the Sagnac effect within a rotating frame is more intricate. In such cases, authors typically employ a spacetime approach using a metric tensor—namely, the Langevin metric—within a four-dimensional Minkowski framework, to compute the propagation time differences between counter-propagating waves. This leads us to a natural question: Can we analyze the aberration of light in the same way? Indeed, we propose applying the Langevin metric to describe light aberration in non-inertial frames as well. That said, this approach requires a clock resynchronization procedure in the inertial frame. To maintain Lorentz coordinates for a moving light source, one must diagonalize the metric given in Eq. (11), as detailed in Section 3.5. The Lorentz coordinates can then be expressed in terms of the original coordinates , using the transformation given in Eq. (13). 7.2 Relativity of Simultaneity Let us first apply the classical kinematic method to compute the aberration of light effect in a non-inertial frame, as observed from an inertial frame. Textbook authors derive the same velocity addition theorem for light beams as for particles in classical physics. This result is particularly striking, as the study of light aberration is deeply rooted in classical kinematics. When terms of order are neglected, the Galilean vectorial velocity addition law is used. However, applying classical kinematics to aberration calculations leads to a significant error. According to our approach, a remarkable prediction emerges regarding light aberration in an accelerated frame (see Fig. 19). To correctly predict the aberration measurement, the accelerated observer must use the Langevin metric. The cross-term in this metric induces anisotropy in the accelerated frame, which alters the radiation direction. In contrast, relying solely on velocity addition (whether Galilean or Einsteinian) suggests that acceleration does not break the symmetry between the accelerated and initial inertial frames. The error in the last argument arises from the incorrect assumption that an inertial observer and an accelerated observer share a common 3-space. According to special relativity, an inertial observer and an accelerated observer (relative to the fixed stars) have distinct 3-spaces. The classical kinematics method used to compute the aberration of light effect is flawed because it treats the speed of light as a classical velocity, disregarding relativistic effects. In the classical approach, no fundamental distinction is made between the aberration of light and the aberration of raindrops. It is particularly interesting to observe that geometric effects in our ordinary space are closely linked to the relativity of simultaneity. In Fig. 15, the transmitted light beam propagates at an angle of , resulting in the phenomenon of light aberration. This naturally raises the question: relative to what does a light beam propagate within an accelerated frame when it undergoes an angular displacement of ? Consider an observer in the accelerated frame measuring the direction of the light beam. If the plane wavefront of the transmitted beam is focused by a lens, it will form a diffraction spot in the focal plane along the optical axis. Determining the direction of the optical axis with respect to the frame’s axes is equivalent to measuring the angular displacement. To detect the aberration of light within the accelerated frame, it is essential to establish a coordinate system with a well-defined reference direction. This necessitates a detailed examination of the method by which coordinates are assigned—a process that inherently involves a physical procedure. Ultimately, this method must provide a consistent framework for defining coordinates in both inertial and accelerated reference frames. In ordinary space, an accelerated frame moves relative to an inertial frame along the line of motion, and conversely, the inertial frame moves relative to the accelerated frame along the line of motion. The angle between the observer’s coordinate system axis and the line of motion is a fundamental geometric parameter in ordinary space. By using the line of motion as the reference axis, the accelerated observer can define a second reference axis. To achieve this, we must provide a practical and operational method for assigning an axis perpendicular to the motion line axis. One possible approach is to use a light beam as a reference direction. We define the second reference direction as follows: Suppose the aberration direction within the accelerated frame is determined relative to the fixed direction of a light beam emitted from a ”plane-wave” source at rest in the accelerated frame. In other words, the coordinate system is established using the electromagnetic axis and the line of motion. For simplicity, we assume that the motion of the aberrated light beam lies in the same plane, with its angular position described by a single angle. When the accelerated system transitions to uniform motion at a constant velocity, Einstein’s standard clock synchronization procedure can be applied. This synchronization method relies on light signals emitted by a source at rest, under the assumption that light propagates isotropically with constant speed . In this framework, the accelerated observer analyzes the light beam from the stationary emitter using the standard Maxwell equations. According to classical electrodynamics, light is emitted perpendicular to the wavefront of the radiation. We consider a specific configuration in which the component of the reference light beam’s velocity along the -axis lies parallel to the wavefront of the reference light beam. The aberration of the light beam transmitted through an aperture can be characterized by the angle between the directions of the reference and transmitted beams, which quantifies the aberration increment. Let us examine the relationship between reference directions in the inertial frame and those in the accelerated frame. In the context of our study on the aberration of light, this approach relies on the use of local reference light sources. To define reference directions perpendicular to the motion in both frames, we employ two local light sources: one remains at rest in the accelerated frame, while the other is stationary in the inertial frame. In essence, the reference electromagnetic axis in each frame is established by an individual light beam. However, reference axes can also be defined using more practical methods than light beams. For example, in an Earth-based frame, the reference axis may be aligned with the gravitational field vector, such as the standard direction of a plumb line. The theory of relativity is founded on the equivalence of all local physical frames of reference. Consequently, when the aberration angle arises, it manifests identically across all local reference systems. Now, let us return to the observations of an accelerated observer as viewed from an inertial frame. In Chapter 3, we discussed how absolute time coordinatization can be transformed into Lorentz coordinatization. Within the inertial frame, this transformation can be understood as a change of the time variable according to . The combined effect of the Galilean transformation and this variable change results in the Lorentz transformation when using absolute time coordinatization in the inertial frame. Consequently, in this framework, the speed of light remains independent of the motion of the source. Simultaneity of events is relative for both inertial and accelerated frames. Events occurring at different locations along the direction of relative motion cannot be simultaneous in both frames when using Lorentz coordinatization—an idea analogous to Einstein’s train-embankment thought experiment. On one hand, the wave equation remains invariant under Lorentz transformations. On the other hand, employing Lorentz coordinatization introduces a time transformation, , which results in a rotation of the radiation phase front by an angle . From the preceding discussion, it follows that in an accelerated system, as viewed by an inertial observer, the electromagnetic axis assigned by the accelerated observer will not be parallel to the -axis of the inertial frame (see Fig. 30). We conclude that, in the accelerated frame, the transmitted light beam propagates along the -axis of the accelerated frame with an angular displacement of . We determine the reference directions in both the inertial and accelerated frames using two local light sources, leading to intriguing consequences. First, both the accelerated and inertial observers can directly measure the angular displacement of a transmitted light beam relative to the reference light beam in the inertial frame. Consider an observer in the inertial frame measuring the direction of the transmitted light relative to a reference light beam from an emitter at rest in that frame. The inertial observer finds that the angular displacement is positive and equal to . Now, we examine the angular displacement of the transmitted light relative to the reference beam within the accelerated frame. On one hand, the angular displacement of the transmitted light with respect to the reference beam from an emitter at rest in the accelerated frame is . On the other hand, the angular displacement of the inertial reference light beam is (see Fig. 16). This implies that the accelerated observer can directly measure the angular displacement of the transmitted light with respect to the inertial frame’s reference light beam. Specifically, the relative angular displacement is . This result is consistent with the measurements of the inertial observer, as expected 262626We have been discussing the electromagnetic axis. If we instead determine the reference directions in both frames using two relativistic electron sources, an important distinction arises. Unlike a reference light beam, a reference relativistic electron beam would not allow for an exact measurement of the transmitted light’s angular displacement within our experimental setup.. 7.3 The Non-Existence of Instantaneous Three-Dimensional Space We derived the results for a non-inertial observer’s observations using Lorentz transformations. Let us follow the standard textbook assumption that the linear motion of observers remains the same in both the accelerated and inertial frames. At first glance, if the reference axes are orthogonal to the direction of motion, they should remain parallel to each other. However, due to the angular displacement , this leads to an apparent paradox. The key to resolving this paradox, as noted above, lies in recognizing that the theory of special relativity eliminates the concept of absolute simultaneity. In the case of the relativity of simultaneity, space and time become intertwined—an observer’s spatial measurements inherently include a subtle influence of time as perceived by another observer. A closer examination reveals that the commonly accepted assumption of a shared line motion among observers is based on the mistaken belief that they share a common 3-space. This is a misconception. The correct approach is to use the light beam as the reference direction. We have already established that accelerated observer can directly measure the angular displacement between the inertial and accelerated electromagnetic reference directions, and this measurement aligns with the inertial observer’s perspective. In other words, the displacement between electromagnetic axes is a physical reality. From the inertial observer’s viewpoint, the moving frame’s axes rotate relative to the inertial frame axes, as illustrated in Fig. 31. This rotation is a relativistic kinematic effect. One can directly verify that the accelerated frame’s axes are rotated by an angle (to first order in ) relative to the inertial frame axes (see the next section for further details). 7.4 Acceleration of a Rigid Body in Special Relativity Now, let us explore further consequences of the relativity of simultaneity. The concept of rigid motion, as understood in Newtonian mechanics, cannot be directly carried over to the special theory of relativity. First, let us study the relativistic kinematics of rotation of the moving frame axes with respect to the inertial frame axes. Consider two emitter-detector setups in an initial inertial frame. Suppose one setup is accelerated from rest to velocity along the -axis, as shown in Fig. 32. The explanation of aberration relies on absolute time coordinatization 272727An alternative explanation involves a clock re-synchronization procedure in the initial inertial frame. Under Einstein’s synchronization, the time for the moving source is obtained by introducing an offset factor in the first-order approximation. This time shift results in a rotation of the radiation wavefront by an angle for both sources. In this synchronization scheme, the radiation of the moving source is described using Maxwell’s equations.. A key distinction between relativistic and Newtonian kinematics is that, in the inertial frame, the accelerated emitter-detector setup undergoes a shearing motion. While counterintuitive, this effect is not paradoxical—it arises naturally from the relativity of simultaneity and the geometric structure of spacetime in special relativity. We have demonstrated that the aberration of light in an accelerated frame, as observed from an inertial frame, can be derived from the Lorentz transformations. In other words, the aberration effect in the accelerated frame, viewed from the initial inertial frame, is a kinematic consequence of special relativity (Fig. 33). It is shown that the relativity of simultaneity is responsible for aberrations to first order in . Furthermore, we identified a fundamental asymmetry between inertial and accelerated frames: acceleration (relative to the inertial frame) influences the propagation of light in the accelerated frame. Specifically, in an accelerated system, an accelerated detector does not receive light emitted by an accelerated source (Fig. 33). In contrast, a detector at rest in the inertial frame continues to receive light (Fig. 32, left). Common textbook presentations of special relativity typically follow the (3+1) approach, which relies on the implicit assumption that distant clocks are synchronized according to the absolute simultaneity convention. According to these textbooks, when a reference frame at rest is set into motion, all points in the three-dimensional reference grid are assumed to move simultaneously. The standard derivation of the aberration of light effect is based on the hidden assumption that the axes of the accelerated frame remain parallel to the axes of the inertial frame. In other words, it is incorrectly presumed that both accelerated and inertial observers share a common three-dimensional space (see Fig. 34 - Fig. 35). According to conventional theory, acceleration does not disrupt the motional symmetry between the accelerated and inertial reference frames. However, our findings (Fig. 32 - Fig. 33) contradict the textbook predictions. A critical flaw in the widely accepted derivation of the aberration of light effect is the omission of Wigner rotation, which, in our analysis, is fundamentally linked to the relativity of simultaneity. In the next chapter, we will explore the theory of Wigner rotation in detail. 7.5 Discussion Let us return to the observations of an inertial observer. According to the special theory of relativity, an orthogonal emitter-detector setup undergoes a shearing motion during acceleration, as illustrated in Fig. 32. This effect is purely kinematic and does not involve any forces. How can this be? The answer is not straightforward, but one way to approach the issue is as follows: Our ordinary intuition often fails when dealing with two space-like separated events. The commonly accepted assumption of rigidity in an accelerated frame is based on the belief that the simultaneous acceleration of the reference space grid has a direct physical meaning. However, this is a misconception. When considering two distant events, we must account for the conventionality of distant simultaneity over a time interval of , where is their spatial separation. The position of the accelerated emitter along the -axis relative to the accelerated detector lacks an exact objective meaning. Due to the finite speed of light, no experimental method can precisely determine this position. Instead, there is an inherent uncertainty (blurring) in the relative position along the -direction, quantified as , which arises from uncertainties in the timing of acceleration. A relevant example involves a light clock—a rod with mirrors at each end. Textbooks typically describe how, from the perspective of an inertial observer, the light inside the orthogonal light clock follows a zigzag path. However, this explanation implicitly assumes that the coordinate axes of the moving observer remain parallel to the axes of the inertial observer. This assumption is incorrect. To properly analyze the behavior of light clocks, we must consider both the phase and initialization of the system. The term ”phase” refers to the zigzag pattern of the light path, which cannot be directly measured in the lab frame. Moreover, the relative position of the accelerated bottom mirror with respect to the accelerated top mirror also lacks an exact objective meaning. Here, too, an uncertainty of order arises, where is the rod length. 8 Kinematics of Wigner Rotation The focus of this chapter is the Wigner rotation. 282828It is well known that the composition of non-collinear Lorentz boosts does not result in a simple boost but rather in a Lorentz transformation that includes both a boost and a spatial rotation. This rotation is known as the Wigner rotation [39, 40]. It is sometimes referred to as the Thomas rotation (see, e.g., [13, 41]). We previously introduced the Wigner rotation in the last chapter, where our discussion primarily centered on the aberration of light. However, the Wigner rotation is also closely related to the aberration of particles. As a relativistic kinematic effect, the Wigner rotation describes how the coordinate axes of a reference frame, moving along a curvilinear trajectory, rotate with respect to the axes of an inertial Lorentz frame. The primary objective of this chapter is to derive an expression for this effect. Throughout, we have aimed to minimize mathematical complexity while maintaining clarity. 8.1 Composition of Lorentz Boosts Let us now consider a relativistic particle accelerating in an initial inertial frame and analyze its evolution within Lorentz coordinate systems. Since the particle’s permanent rest frame is not inertial, we introduce an infinite sequence of comoving frames to address this difficulty. At each instant, the rest frame is a Lorentz frame centered on the particle, moving alongside it. As the particle’s velocity changes at an infinitesimally later instant, a new Lorentz frame, centered on the particle and moving at the updated velocity, is adopted. Let us denote the three inertial frames by . The initial inertial frame is , while is the rest frame of the particle, which has a velocity relative to , and represents the particle’s rest frame at the next instant of proper time , moving with an infinitesimal velocity relative to . All three inertial frames are assumed to be Lorentz frames. To clarify velocity measurements in different inertial frames: an observer in measures the velocity of frame as , while an observer in measures the velocity of frame as . Now, let us examine the transformation of velocity in the theory of relativity. For a rectilinear motion along the axis, the velocity transformation follows the equation: . Just as with Galilean transformations, collinear Lorentz boosts commute—the order in which collinear boosts are applied does not affect the final result. This means that the composition of two Lorentz boosts in the same direction yields another Lorentz boost in that direction, with the total velocity given by the formula above. However, when Lorentz boosts are applied in non-collinear directions, they no longer commute. While the composition of two Galilean boosts or two spatial rotations yields another transformation of the same kind (i.e., a boost or a rotation), the combination of two non-parallel Lorentz boosts results in a transformation that is not purely a boost—it includes a rotation known as the Wigner rotation. To illustrate this, suppose frame moves with velocity along the -axis relative to and in , the particle experiences an infinitesimal acceleration in the -direction, perpendicular to the trajectory of the initial inertial frame . That is, frame moves relative to with a small velocity along the -axis. From the perspective of an inertial observer, time dilation occurs in the Lorentz frame moving with velocity relative to the initial inertial frame, given by , where . Due to this time dilation, the velocity increment in the initial inertial frame corresponds to a velocity in . 8.2 Wigner Rotation Numerous incorrect expressions for the Wigner rotation appear in the literature, underscoring the need for a careful and transparent treatment of this topic. Rather than deriving all transformation matrices for four-vector components, we adopt a geometric perspective that offers clear and intuitive insight into the phenomenon. The mathematics involved is notably straightforward, relying only on basic algebra. The key challenge lies in bridging the conceptual gap that arises from the fact that different Lorentz frames cannot be described within a shared three-dimensional space. Consider a succession of inertial frames . From the perspective of the initial inertial frame, an observer in the frame sees frame moving with velocity and velocity increment . The corresponding rotation of the velocity direction in the frame is given by (Fig. 36). In the frame, the velocity rotation angle is , with both rotations occurring in the same direction. If the proper and lab observers were to share a common three-dimensional space, this discrepancy would lead to a paradox — an apparent inconsistency in the theory. However, in Minkowski spacetime, no such paradox arises: the lab’s 3-space is rotated relative to the proper 3-space by an angle in the same direction as the velocity vector’s rotation in the proper frame. This angular difference is the Wigner rotation — the rotation of the original frame’s axes as seen from the proper frame . In vector form, this can be expressed as . Several key insights can be drawn regarding the theory of Wigner rotation. Notably, the expression for the Wigner rotation angle in the proper frame, can be reformulated as , where is the velocity of the original frame relative to the proper frame, and is its differential change. This alternative form emphasizes the expression of the Wigner rotation in terms of quantities measured in the proper frame. Transforming this expression back to the lab frame yields representing the Wigner rotation angle of the proper frame’s axes as observed in the lab frame. According to the principles of special relativity, this quantity must remain form-invariant under Lorentz transformations. 292929The correct expression for the Thomas (Wigner) rotation was first obtained by V. Ritus . In deriving expressions for the Thomas (Wigner) rotation, the majority of authors (see e.g. ) was supposedly guided by the incorrect expression for Thomas (Wigner) rotation from Moeller’s monograph . The expression obtained by Moeller is given by (and subsequently expression for Thomas precession ). It should be noted that, in his monograph, Moeller stated several times that this expression is valid in the lab Lorentz frame. Clearly, this expression and correct result differ both in sign and magnitude. An analysis of why Moeller obtained an incorrect expression for the Thomas (Wigner) rotation in the lab frame is the focus of Ritus paper . As analyzed in Ritus’s work , Moeller’s error is not computational but rather conceptual. The correct expression, derived decades earlier by several researchers, remained largely unnoticed amid the proliferation of inaccurate results, as discussed in the review . We note that owing to the relativistic effect of time dilation in the reference frame that moves to the lab frame, the Wigner rotation angle in the proper frame is always time higher than in the lab frame. 303030In 1986, M. Stranberg obtained an expression for the Thomas (Wigner) rotation correct both in the lab inertial frame and the comoving reference frame . His paper is one of the few that explicitly states that the angle of Thomas (Wigner) rotation in the comoving frame is times higher than in the lab frame . We derived the exact relation using only rudimentary knowledge of special relativity. In standard textbooks on relativity, the spatial rotation that arises from the composition of two non-parallel Lorentz boosts is typically introduced through an algebraic approach. This reliance on algebra, without sufficient geometrical interpretation, is one reason why many authors arrive at an incorrect expression for the Wigner rotation. These treatments often describe the rotation of a moving object without addressing its geometrical significance, leading to serious difficulties in both the interpretation of the computations and the physical meaning of the results. 8.3 Measuring Wigner rotation The Wigner rotation is a relativistic kinematic effect that arises from the geometry of pseudo-Euclidean space-time. It manifests as a rotation of the spatial coordinate axes of a particle’s proper frame when the particle follows a curvilinear trajectory in the Lorentz lab frame. In this context, we derive the expression for the infinitesimal rotation resulting from the acceleration of a relativistic particle: \| \| \| --- \| \| \| (20) \| Here represents the infinitesimal change in velocity due to acceleration, is the Wigner rotation angle describing the rotation of the particle’s proper frame relative to the Lorentz lab frame, and is the infinitesimal orbital angle. A natural question arises: how can this rotation be observed or measured? Since a moving coordinate system changes its position over time, determining its orientation becomes nontrivial. To explore the physical meaning and possible experimental interpretation of this rotation, we illustrate the problem of defining the orientation of a moving frame using a simple example. The Wigner rotation, when interpreted in the context of the original inertial frame, is intimately related to length contraction. 313131The length of a moving object depends not only on the structure of space-time but also on our choice of synchronization convention. As with the orientation of radiation phase fronts, such quantities do not possess an exact objective meaning. Under the standard clock synchronization, they are typically regarded as ”relativistic kinematic effects.” The Wigner rotation similarly arises as a coordinate effect, lacking an absolute, observer-independent significance. Suppose that, after performing a Lorentz boost along the -axis with velocity , an observer in the lab frame rotates the coordinate system by an angle , such that the new -axis is orthogonal to the vector , and the -axis is aligned with it. Now, consider a rod oriented along the -axis in the comoving frame. The motion takes place in the -plane, and the rod is initially perpendicular to the velocity . After rotating the lab frame axes, the projection of the rod onto the -axis becomes simply , where is the length of the rod in the comoving frame, and also in the lab frame following the first boost along the -axis. After a second infinitesimal Lorentz boost with velocity , this projection undergoes Lorentz contraction and becomes (Fig. 37). Assume now that the lab observer fixes the orientation of the comoving frame axes. In the ultrarelativistic limit , the comoving frame axes become parallel to the rotated lab frame axes , and thus the projection of the rod onto the -axis vanishes. However, for an arbitrary (finite) velocity, the comoving frame axes are not aligned with the rotated lab frame axes. Due to the contracted projection, the angular deviation is . Consequently, one can directly verify that the comoving frame axes are rotated relative to the original lab frame axes by an angle , which is precisely the Wigner rotation angle, as given by Eq.(20). This example illustrates how the Wigner rotation arises naturally from the composition of Lorentz transformations. 8.4 Discussion Numerous incorrect expressions for the Wigner (Thomas) rotation can be found in the literature. Here, we examine the underlying reasons behind these errors. In 1959, Bargmann, Michel, and Telegdi (BMT) formulated a consistent relativistic theory for spin dynamics as observed in the laboratory frame, which was later confirmed experimentally . A common misconception is that the BMT equation inherently includes the standard (but incorrect) expression for the Wigner rotation in the Lorentz lab frame. Many physicists, having learned about Wigner (Thomas) rotation from well-known textbooks, argue: ”The highly precise measurements of the electron’s magnetic-moment anomaly, conducted on relativistic electrons, rely on the BMT equation—where the Wigner rotation is an integral part—thus serving as an experimental confirmation of the standard expression for the Wigner rotation.” This belief, however, is both widespread and misleading. It is important to clarify that the results in the Bargmann-Michel-Telegdi paper were derived using a semiclassical approximation of the Dirac equation. The Wigner rotation was not explicitly considered, as the Dirac equation allows for a complete description of spin dynamics without separating it into Larmor and Wigner components. It should be noted that the application of Wigner (Thomas) rotation theory in relativistic spin dynamics is complex. Bargmann, Michel, and Telegdi began their analysis in the particle’s rest frame, deriving the equation of motion for angular momentum. They then generalized it to the Lorentz lab frame before transforming it back to the rest frame. This raises an interesting question: Why is it convenient to transform this equation to the rest frame at that instant? In experimental practice, working with the proper spin is preferred due to the clear physical interpretation of the three-dimensional spin pseudo-vector . Unlike the three-dimensional momentum , which has well-defined components in every reference frame, angular momentum is specific to a single frame and does not transform. Thus, any meaningful statement about spin implicitly refers to the particle’s instantaneous rest frame. For example, when we say that a particle’s spin in the lab frame makes an angle with its velocity, we actually mean that in the particle’s rest frame, the spin vector forms the same angle with the direction of the lab frame’s motion. Physical phenomena in non-inertial frames can still be analyzed within the framework of standard special relativity by applying well-established relativistic kinematic effects. In the context of spin dynamics, this eliminates the need for measurements using non-inertial devices. The conventional approach to explaining spin dynamics in the lab frame is quite unusual. As previously noted, the laws of physics in the lab frame account for all physical phenomena, including observations made by non-inertial observers. In particular, spin orientation measurements with respect to the lab frame axes can be determined within the proper frame. From the lab frame perspective, the interpretation of spin rotation experiments is straightforward and can be derived from the predictions of the proper observer regarding the measurements made by the lab observer. Thus, the spin rotation measurements by a polarimeter in the lab frame is interpreted from the viewpoint of the proper observer as viewing of the lab observer. This inherent complexity has led textbook authors to derive incorrect expressions for Wigner rotation—an issue that will be discussed further in Chapter 18. Additionally, it is crucial to examine why errors in Wigner rotation theory went undetected for so long. Wigner (or Thomas) rotation is typically presented within the context of spin kinematics as a peculiar effect of special relativity. This limited perspective likely explains why, until recently, researchers failed to recognize the significant discrepancies between different works. The Wigner rotation is a fundamental relativistic phenomenon, on par with time dilation and length contraction. Special relativity demonstrates that the aberration effect in the rotating Earth-based frame, as observed from the inertial Sun-based frame (in Lorentz coordinatization), is a kinematic consequence of the theory. In this context, the shift in aberration images within the Earth-based frame, when viewed from the Sun-based frame, is governed by the Wigner rotation. According to Wigner rotation theory, observers following different trajectories experience different 3-spaces. However, previous literature has incorrectly assumed that an Earth-based observer and an inertial (e.g., Sun-based) observer share a common 3-space—an assumption that has long been taken for granted. 9 Aberration of Relativistic Particles We now examine the phenomenon of aberration for relativistic particles. Consider a particle source initially at rest in an inertial frame, emitting particles that travel with a constant velocity along the -axis. Suppose both the observer (along with their instruments) and the particle source are uniformly accelerated from rest to a velocity along the -axis, under the condition that . Our goal is to understand how the aberration of the particle beam is perceived by the observer in this accelerated frame. To approach this question, we begin by summarizing the key result. The central question is: How should the aberration of a particle beam be described when the source is at rest in the accelerated frame? To predict the outcome of such a measurement, the observer must refer to the Langevin metric, given in Eq. (15). Within this framework, the aberration angle is related to the system parameters by the expression: , where . is the Lorentz factor of the particles. This relativistic expression smoothly reduces to the classical result in the limit . It is crucial to note that a correct description of aberration in the accelerated frame requires incorporating the full metric tensor, even to first order. This is because the cross term in the Langevin metric—representing the leading-order deviation from the Minkowski metric—plays an essential role in capturing the non-inertial kinematics of relativistic particle motion. In the ultrarelativistic limit , the aberration increment approaches the simple and intuitive form: . 9.1 Explanation of Electron Aberration Based on Electrodynamics The aberration of particle trajectories can be effectively addressed using dynamical arguments. Electromagnetic forces, which determine the behavior of an emitted electron beam, are influenced by the acceleration of the source in such a way that they cause a deviation in the direction of electron transport. To analyze these effects, we can derive the electromagnetic fields in an accelerated frame through a Galilean transformation. For readers already familiar with special relativity from standard textbooks, it may be advantageous to approach the aberration of particles—both theoretical and experimental—using a microscopic perspective. From this standpoint, one can calculate relativistic quantities directly from the fundamental theories of matter, without explicitly invoking relativistic kinematics. The aberration of particles associated with the transformation from an inertial to an accelerated frame can be understood as the result of a force acting on the system. Importantly, this issue is unrelated to the concept of the ether and is fully contained within the framework of special relativity. Let us consider a specific case in which an electron gun, serving as a source of relativistic electrons, is initially at rest in an inertial frame and then accelerated to a velocity along the -axis. The electromagnetic forces governing the properties of the emitted electron beam are modified by this acceleration, resulting in a change in the beam’s transport direction. The underlying mechanism behind the change in electron momentum , when both the source and the observer (along with their measuring instruments) are at rest in the accelerated frame, can be fully explained within the framework of electrodynamics. We can derive the electrodynamics equations in an accelerated reference frame by applying a Galilean transformation. When analyzing Maxwell’s equations under such a transformation, a key question arises: What are the transformation laws for the electromagnetic fields and ? The electric and magnetic fields and observed in the accelerated frame—where the particle source is at rest—differ from the fields and observed in the original inertial frame, prior to the Galilean boost. Our goal is to determine the relationship between these sets of fields. To proceed, we transform the coordinates of the inertial observer , moving with velocity relative to the accelerated observer , using the Galilean transformation: , . Substituting these into the Minkowski metric, , we obtain the so-called Langevin metric in the accelerated frame Eq. (15): . This transformed metric allows us to study electrodynamics equations in a non-inertial frame using the Galilean transformation with velocity . In the specific case where , the transformed fields to first order in are: , . Here, is the velocity of the source in the original inertial frame. This result captures the leading-order relativistic correction to the magnetic field observed in the accelerated frame. The accelerated observer experimentally determines the Lorentz force acting on a charge moving with velocity in the region of the electron gun, where electric and magnetic fields and are present. The force is given by: . In the special case where and , the magnetic term simplifies to: . Using the relativistically correct equation of motion, , we expect that after a short time interval , the emitted electron will acquire a transverse momentum in the accelerated frame given by: . The differential change in kinetic energy of the accelerated electrons is: . Upon integration, this yields: . Given our condition , the aberration angle increment becomes: , where the total momentum of the accelerated electron is: . Substituting, we find: . Expressed in vector form, the aberration increment is: . The aberration of particles resulting from the transformation between an inertial and an accelerated coordinate system can be interpreted as a manifestation of the Lorentz force. The electromagnetic forces that govern the behavior of an emitted electron beam are influenced by acceleration relative to the fixed stars, leading to a deviation in the beam’s transport direction, as illustrated in Fig. 38. 9.2 Magnetic Field Measurements in an Accelerated Electron Source We now turn to an apparent paradox. Special relativity introduces concepts that may seem counterintuitive at first. Yet it remains a rigorously mathematical theory, and it leads to a surprising conclusion: when we consider an accelerated electron source, the electromagnetic fields in the accelerated frame are not static—the problem becomes time-dependent. This is a direct implication of special relativity. Specifically, if both the electron source and the observer are at rest in the accelerated frame, one might intuitively expect that everything should remain unchanged over time—that is, the electric field should not vary. However, special relativity tells us otherwise. The time derivative of the electric field, , is nonzero because of the spatial variation of the field, , and the relation . This seemingly paradoxical result is simply a manifestation of the four-dimensional formalism of relativity. It carries no physical contradiction or special consequence—just a shift in perspective due to the geometry of spacetime. To clarify this, it may help to draw an analogy with a familiar concept in classical electromagnetism: the use of gauge transformations. For example, when we work in the Coulomb gauge, the scalar potential appears to propagate instantaneously. However, this is merely a mathematical artifact of the gauge choice. In a complete and self-consistent calculation, all unphysical instantaneous effects cancel out, preserving causality and agreement with physical observation. We derived the electrodynamics equations in an accelerated frame by applying the inverse Galilean transformation (with velocity ) of Maxwell’s equations. Under a Galilean transformation, and assuming in the original frame, the transformed fields become , . The electric field and the induced magnetic field are both static, meaning they do not vary with time—the system is entirely time-independent. Many physicists familiar with standard electrodynamics might argue, based on textbook knowledge, that a static magnetic field must arise from electric currents, which in turn require moving charges. Since no charges are in motion in this case, the presence of a magnetic field appears paradoxical. This represents a fundamentally new kind of situation, distinct from traditional electrodynamics in an initial inertial frame. How can that be? Even when the electron source and the observer are at rest in the accelerated frame, the time derivative is generally nonzero. This can be readily demonstrated by transforming from the coordinates of an inertial observer —moving with velocity relative to the non-inertial observer —to the coordinates of . Using a Galilean transformation, the inertial coordinates can be related to the non-inertial ones as follows: . This allows all fields to be re-expressed in terms of the variables . Taking partial derivatives accordingly, we find: , . Therefore, if the fields are time-independent in the inertial frame (i.e., ), it follows that: . Thus, is not necessarily zero, even though the source and observer are at rest in the non-inertial frame. A natural starting point is to assume that the time derivative originates from a Galilean boost. To generalize, we now consider the case of Lorentzian coordinatization. When the reference frame begins moving at a constant velocity, Einstein’s standard procedure for clock synchronization can be applied. In this context, the time coordinate in the frame —as defined by Einstein synchronization—is obtained by introducing an offset term and applying the first-order approximation: . This transformation allows us to reinterpret the new time coordinate in the accelerated frame in such a way that Maxwell’s equations remain valid, particularly in describing the aberration of electrons. Taking partial derivatives with respect to the transformed coordinates yields: , . Hence, to first order in , we find: neglecting terms of order . It is important to note that changing the (four-dimensional) coordinate system does not give rise to new physical phenomena. The underlying physics remains invariant under such transformations. When the situation is described as we have done here, the paradox seems to disappear entirely; it becomes quite natural that, in general, space and time can no longer be treated as separate entities. The argument for a paradox typically goes like this: If we could measure the magnetic field using an ordinary magnetometer, then a paradox would indeed arise. However, such a measurement is not possible. As discussed earlier, the magnetic field induces a change in the transverse momentum of test particles, described by . For non-relativistic particle velocities, applying a binomial expansion yields , indicating that the momentum perturbation is a relativistic effect. According to special relativity, as , the change in particle momentum vanishes. Thus, a classical magnetometer using test particles would detect no trace of the magnetic field. Moreover, it is not just that this particular classical magnetometer cannot detect ; if the theory of relativity holds, any classical magnetometer—regardless of its operating principle—would similarly fail to reveal any evidence of the field’s effect. To understand magnetic field measurements in an accelerated electron source, we begin by examining how a classical magnetometer behaves when observed from an accelerated frame. Since a full treatment is complex, we consider a simplified, conventional type of magnetometer. One practical approach to observing nuclear magnetism is through the phenomenon of nuclear magnetic resonance (NMR), where a proton resonance apparatus can serve effectively as a proton resonance magnetometer. We present a general analysis of this classical type of magnetometer, independent of its experimental precision. In the inertial lab frame, a current loop possessing a magnetic moment acquires an electric dipole moment when it moves with velocity relative to the lab. This dipole moment is given by: . Since the magnetic properties of materials arise from atomic-scale current loops, a magnetized body undergoing acceleration relative to the lab frame should exhibit an induced electric polarization in that frame. As a result, the potential energy of a moving particle with intrinsic magnetic moment , in the presence of a magnetic field and electric field , is given by: . To analyze this in an accelerated frame, we apply the Galilean transformation (with velocity ) to Maxwell’s equations. Under this transformation, assuming an initial zero electric dipole moment (), the magnetic and electric dipole moments in the accelerated frame become: , . From vector algebra, we use the identity to evaluate terms in the potential energy expression: , . Thus, the additional contribution to the potential energy evaluates to zero, as expected. Consequently, a conventional magnetometer cannot detect the magnetic field in the accelerated frame. 9.3 Aberration of Particles from a Moving Source Let us examine what occurs when the electron gun—serving as the source of a relativistic electron beam—is accelerated from rest to a velocity along the -axis in the initial inertial frame (see Fig. 39). In other words, we consider an active (physical) Lorentz boost. In an active boost, we are observing the evolution of the same physical system over time, analyzed from the perspective of a single reference frame. The simplest method of clock synchronization in this context is to retain the same set of uniformly synchronized clocks used when the particle source was at rest. In other words, we continue to use the synchronization method based on clock transport or Einstein synchronization, which relies on light signals emitted by a stationary dipole source. This approach is typically the most convenient, as it allows for a smooth and consistent definition of the metric tensor. It also preserves simultaneity and follows the convention of absolute time—or absolute simultaneity. We can determine the electric field and magnetic field observed in the inertial frame where the source particle is moving with velocity by applying the Galilean transformation. When analyzing the equations of electrodynamics under such a transformation, a natural question arises: What are the transformation laws for the electromagnetic fields? The equivalence between the active and passive viewpoints stems from the fact that moving a physical system in one direction is equivalent to moving the coordinate system in the opposite direction by the same amount. Consequently, the fields in the inertial frame before an active Galilean boost are identical to the fields and in the comoving coordinate system. Under a Galilean transformation, and assuming in the comoving coordinate system, the transformed fields in the lab frame are given by: , . This relation captures only first-order effects in , where is the velocity of the source relative to the lab frame. An inertial observer experimentally measures the Lorentz force acting on a charge moving with velocity in the region of an electron gun, where both electric and magnetic fields, and , are present. The force is given by: , where is the electric field in the coordinate system moving with velocity relative to the inertial frame. In the special case where and , the magnetic component of the force simplifies to: . The differential change in the kinetic energy of the accelerated electrons is: . Integrating this expression yields the total kinetic energy: . The total transverse momentum of the accelerated electron beam, to first order in , is given by: , where is the initial transverse momentum of the emitted electron. Since we have chosen , the aberration angle becomes: , where is the momentum of the accelerated electron. Substituting, we obtain: . This can be written in vector form as: . It is important to note that this result holds for an arbitrary Lorentz factor . We analyzed the effect of particle aberration by considering terms only up to first order in . A basic explanation of this effect is well-known: in the inertial laboratory frame, particle aberration can be readily understood through the Galilean transformation of velocities between reference frames. It is important to emphasize that the dynamical reasoning presented here clarifies the physical meaning of particle aberration in the inertial frame. The kinematics effects are only an interpretation of the behavior of the electromagnetic fields. Let us now return to the magnetic field measurements in the accelerated electron source, as shown in Fig. 39. In the lab frame, the electron source is in motion and its evolution is analyzed from the perspective of an inertial reference system. A key feature of this situation is that the velocity of the source carries real physical significance. Indeed, the problem is explicitly time-dependent. For a moving electron source, special relativity tells us that a magnetic field arises according to , leading to a change in the transverse momentum of test particles. This change can be quantified as , a result that remains valid even in the non-relativistic limit as . This brings us to an interesting question: what does an inertial observer measure when using a conventional magnetometer? Given that charges in the source are moving, the resulting current produces a magnetic field described by . This field is indeed detectable with a standard magnetometer. To understand this more concretely, consider the operation of an NMR magnetometer. When at rest in the lab frame, it measures the additional potential energy experienced by a proton with magnetic moment in the field , given by . We come to the conclusion that the magnetic field generated by a moving source in an inertial frame is a real field in the sense we have defined it. 9.4 Inertial Frame View of Observations by a Non-Inertial Observer Earlier, we demonstrated that the aberration of light observed in an accelerated frame, when viewed from the initial inertial frame, is a well-known kinematic effect described by the special theory of relativity. Can we not interpret the aberration of particles in an accelerated frame in a similar fashion? As we shall see, the effect of particle aberration in non-inertial frames can indeed be understood within an inertial frame using the framework of standard special relativity—specifically, by employing the theory of Wigner rotation. The Wigner rotation is a relativistic kinematic effect in which the coordinate axes of a particle’s proper reference frame appear to rotate relative to an initial (Lorentz) inertial frame. This rotation arises due to successive Lorentz boosts in different directions. The Wigner rotation angle in the inertial frame can be expressed in vector form as , where is the vector of particle velocity in the initial Lorentz frame before acceleration, is the vector of small velocity change () due to acceleration, is the Lorentz factor associated with , and is the Wigner rotation angle of the proper frame axes as observed from the (Lorentz) inertial frame. 323232This expression for the Wigner rotation is derived in Chapter 8 for the limiting case of an infinitesimal rotation angle, , corresponding to a small change in the particle’s velocity vector. The derivation is based on the composition of Lorentz boosts in the form . In the context of an accelerated electron source, the relevant velocity composition arises from . A careful analysis shows that both orderings yield the same Wigner rotation expression, due to the infinitesimal nature of the boost . We now examine how this formula applies in the specific case where both the observer and the particle source are accelerated to a velocity in the inertial frame. To find the rotation magnitude in the stated problem, we introduce a composition of Lorentz boosts. Let be the initial inertial frame of reference, an accelerated frame with velocity relative to the inertial frame, and is a frame which moves relative to the with velocity along the -axis. Two sequential boosts from the inertial frame to and then to are equivalent to the boost from to and the subsequent rotation. The frame is rotated through the angle with respect to the inertial frame . Now we must be careful about the direction of rotation. There is a good mnemonic rule. The rule says that the direction of the Wigner rotation in the Lorentz lab frame is the same as the direction of the velocity rotation in the Lorentz lab frame. We can easily understand that the coordinate axes of the frame will be parallel to the coordinate axes of the accelerated frame . As viewed from the inertial frame, the coordinate axes of the accelerated frame are also rotated through the angle with respect to the coordinate axes of the inertial frame . Our results show that we cannot remain with the framework of parallel axes when considering the inertial frame view of particle beam observations of the accelerated (up to the velocity ) observer in perpendicular geometry. Consider an electron gun–detector setup initially at rest in an inertial frame of reference. Suppose this system is then uniformly accelerated along the -axis until it attains a velocity . Now, from the perspective of an observer who remains at rest in the original inertial frame, the direction of the electron beam’s propagation appears to shift. This observer measures an angular displacement of the beam given by , where is the velocity of the particles emitted by the gun. This effect, known as particle aberration, is a well-established phenomenon. A key distinction between relativistic and Newtonian kinematics emerges when we examine the behavior of the emitter–detector system under acceleration. From the inertial frame’s viewpoint, the accelerated setup undergoes a shearing motion: the detector no longer remains aligned with the electron beam after acceleration. For instance, if the gun–detector assembly was originally aligned along the -axis, the detector fails to intercept the beam once the acceleration ends. In the accelerated frame, this misalignment is reflected in the angular deviation, and we obtain an expression for the aberration increment: . In the special case of light this expression simplifies to: . 333333It’s straightforward to verify that the infinitesimal Wigner rotation angle is always less than regardless of the particle velocity . Specifically, , where and function is strictly less than 1 for . As discussed in Chapter 5, there is an inherent uncertainty in the relative position along the -axis in the -direction, amounting to , due to the ambiguity in the timing of acceleration. This uncertainty arises because the relevant events are space-like separated. Consequently, the orientation of the axis relative to the inertial frame’s -axis carries an angular uncertainty of . The Wigner inclination angle always lies within this uncertainty bound. The formula matches the result previously derived for observations made by a non-inertial observer within the framework of electrodynamics. According to special relativity, the aberration of particle trajectories observed in an accelerated frame—when viewed from an initial inertial frame (using Lorentz coordinates)—is a purely kinematic effect. In such cases, the apparent shift in particle directions (i.e., aberration) within the accelerated frame is governed by the Wigner rotation. The theory of Wigner rotation reveals that observers following different worldlines possess distinct notions of 3-space. This insight challenges a widespread but incorrect assumption in earlier literature: that a non-inertial observer (e.g., Earth-based) and an inertial observer (e.g., Sun-based) share a common 3-space. It is important to emphasize that the dynamical arguments presented above clarify the physical meaning of the Wigner rotation in a rotating frame. The relativistic kinematic effects observed in the rotating frame—when interpreted from the perspective of an inertial observer—are simply manifestations of how electromagnetic fields transform under relativistic motion. 10 Earth-Based Setups to Detect the Aberration Phenomena 10.1 The Potential of Earth-Based Electron Microscopes Above, we discussed the new relativistic particle kinematics in rotating frames of reference. Since we live on a rotating Earth, the distinction between relativistic kinematics in rotating and non-rotating frames holds both theoretical and practical importance. We propose an aberration-of-particle experiment using an Earth-based particle source. The results of such experiments can be interpreted within the framework of special relativity. However, obtaining an accurate solution in the Earth-based frame requires the use of the metric tensor, even for first-order approximations. We propose an experiment using commercial 200-kV scanning transmission electron microscope (STEM) as a particle source. One could use a vertically oriented optical setup. We suppose that the electron beam is imaged by a lens to a spot that lies in the image plane. Measurement of the spot shift with respect to the optical axis (which is parallel to the zenith-nadir axis) is equivalent to the determination of the angular displacement. Due to the high stability, the aberration increment could be observed. 343434The dedicated STEM with high stability was made It was developed based on a 200-kV commercial instrument. The electron source was installed in the anti-seismic room. Temperature fluctuations in the room were within 0.2 . The magnetic field in the room was less than near the source column. To prevent thermal drift, a shield box was placed over the high-voltage generator. The column was covered with rubber sheets to reduce the temperature fluctuations. The image drift measured during a period of 61 minutes was . The drift speed and direction during this period were approximately constant. The average drift speed was 0.12 nm/min. It is substantially smaller than that conventional commercial microscope, in which the drift speed is about 2 nm/min. Consider the inertial sun-based reference system. In this system, there is the earth which rotates around the sun with orbital velocity . Developing into powers of we can classify effects for velocities as of the first order, of the second order, or higher orders. Clearly, in the case of the rotating (around the sun) earth-based frame, we consider the small expansion parameter neglecting terms of order of . Suppose that an earth-based observer performs an aberration of particle measurement. The aberration angle varies with a one-year temporal period. An approximate formula to express the aberration angle can be found by using the Langevin metric in the earth-based frame or by using the Wigner rotation theory in the sun-based frame. In the aberration of particles, the aberration angle is the apparent angular deviation of the position of the particle source relative to the real location of the source. The reference axis in the earth-based frame can be formed by a plumb line, which is the most fundamental local earth-based coordinate system. When measured with an earth-based particle source, the aberration angle is , where is the orbital velocity of the earth relative to the sun, is the velocity of the accelerated electron, and is the particle relativistic factor. This means that the angle of Wigner rotation depends on . It should be noted that the presented formula can be considered only as a first-order approximation in . In the theory of (infinitesimal) Wigner rotation, we must also consider the relativistic parameter . If a particle motion velocity is non-relativistic, the binomial expansion yields . Let us calculate the parameters for the STEM setup. The Wigner rotation angle parameter (constant of aberration) defines the scale of the aberration increment and is equal to . Here , , . Earth moves around the sun, with a consequent change of the direction of . Therefore the angle of aberration changes. The total change of the order of . It is oscillatory with a period of one year. A specific aspect of our case study needs further investigation. The proposed method of measuring the angle of aberration involves the use of earth-based sources and has a big advantage. The rotation of the earth on its axis should produce a corresponding shift of the image. Obviously, it is important that observation should be recorded in the shortest possible time. In principle, records could be taken over a period of one day. The aberration shift depends only upon the value of , the component of the orbital velocity perpendicular to the earth’s rotation axis 353535 The constant of diurnal aberration defines the scale of the aberration increment and is equal to . Here , , where km/s is the linear velocity of the earth rotation at equator.. In practice, the measurement procedure is complicated by a number of factors, Fig. 40. A correction has to be made for the observer’s position on the earth’s surface and the obliquity of the ecliptic . The image appears to move in an ellipse. The shape of the aberration ellipse depends on the observer’s latitude . For an observer on the equator, the ellipse degenerates into a line segment, and for an observer at the pole of the earth, the ellipse is a circle. At the value mm of the focal length, the major axis of the aberration ellipse of the order of 80 ( nm. 363636Electron-optical system of the STEM contains an electron gun and several magnetic lenses, stacked vertically to form a lens column. The illumination system of the instrument comprises the electron gun, together with two condenser lenses that focus the electrons onto the spacemen. Electrons entering the lens column appear to come from a virtual source. The first condenser lens (C1) is a strong magnetic lens, with a focal length of typically 2 mm. Using a virtual electron source as its object, C1 produces a real image. Because the lens is located cm below the object (virtual source), its object distance is 15 cm and the image distance . The second condenser lens (C2) is a weak magnetic lens ( cm) that provides little magnification but allows the diameter of illumination at the specimen to vary continuously over a wide range. The distance between lens centers is typically 10 cm. Spacemen is located cm below the C2. The value of the minor axis depends on the observer’s position and is given by 80 ( nm. When the latitude is , the value of the scale of image shift per hour depends on the local time and varies from a value of about 10 nm/60 min to about 20 nm/60 min through the day. 373737Experimental results (see Fig. 44 p.167 in ) shows that the image shift is quite close to the theoretical prediction . It is important to note that we used the typical focal length (2 mm) in our estimations. The details of the optical setup may influence the image shift. 10.2 A Thought Experiment with a Point-Like Source We now turn to a simple scaling model for stellar aberration, illustrated in Fig. 41. Consider a scale transformation involving both the linear dimensions of the source and the distance between the source and the observer. This approach allows us to derive a condition for optical similarity between the aberration of light from a distant star and that from an Earth-based source. Such a transformation can assist in the preliminary design of an Earth-based experimental setup, where an incandescent lamp is used in place of a star. Scaling model for stellar aberration using an Earth-based setup with an incandescent lamp. Based on astronomical observations, we know what to expect even without invoking special relativity. Remarkably, by measuring the aberration of light alone—without referencing anything external to Earth—it is possible to determine the Earth’s orbital speed around the Sun. A star ( or an incandescent lamp) is a system of elementary statistically independent point sources. Point-source field diffraction can be divided into categories - the Fresnel (near-zone) diffraction and Fraunhofer ( or telescopic) diffraction. If the Fraunhofer (or equivalently, far zone) approximation is satisfied, then the quadratic phase factor is approximately unity over the detection aperture. Here is the aperture width, is the wavelength of the radiation, and is the distance between the source and observer. In the far zone, a point source produces in front of pupil detection effectively a plane wave. The plane wavefronts of starlight entering the telescope are imaged by the lens to a diffraction spot that lies in the focal plane. It is worth noting that we consider the divergence of the transmitted radiation, , that is relatively small compared to the aberration angle, . An incandescent lamp is a completely incoherent source. The character of the mutual intensity function produced by an incoherent source is fully described by the Van Cittert-Zernike theorem . In our case of interest, imaging system is situated in the far zone . It is the source linear dimension (source diameter) that determines the coherent area of the observed radiation . The condition for neglecting the transverse size of the source, , formulated as the requirement that the change of the correlation function along the aperture be less than unity, that is . 383838 In all cases of practical interest, telescopes are situated in the far zone of the distant stars. If the far zone approximation is satisfied, then the Van Cittert-Zernike theorem takes its simplest form. For example, the coherence area of the light emitted by the circular incoherent source is . The minimal coherent area of light observed on the earth’s surface from the nearest star has a diameter of about 10 m. This diameter is larger than a typical telescope aperture . In this situation, a star can be considered as a point source and a star image is a point spread function in the image plane of a telescope. Our scaling model is based on the far zone approximation. We have only to change the distance and the source diameter. Assuming small parameter conditions , , the scaling approach yields the following requirements: , . Let us estimate the parameters for an Earth-based setup using an incandescent lamp. We aim to describe a series of thought experiments—hypothetical, yet physically plausible. At optical frequencies, the conditions required for the Fraunhofer approximation can be quite stringent. For instance, with a wavelength of m and an aperture width of cm, the observation distance must satisfy m. Obviously, the required conditions are met in a number of important problems. No one has ever done such an experiment, but we know what would happen from the astronomical observations. 10.3 Practical Setup Using an Incoherent Optical Source It should be noted that Fraunhofer diffraction patterns can be observed at distances much closer than implied by equation provided the aperture is illuminated by a spherical wave converging towards the observer, or if a positive lens is properly situated between the observer and the aperture. The operational principle of the Earth-based optical setup designed to detect light aberrations is schematically illustrated in Fig. 42. This configuration follows the well-known two-lens imaging scheme, which enables magnification by adjusting the focal length of the second lens. However, for simplicity, we will assume equal focal lengths for both lenses (i.e., 1:1 imaging) in the following discussion. The two-lens setup is commonly used in image processing applications, as it provides more flexibility for modifying images compared to a single-lens system. In the context of this two-lens arrangement, we now turn to the relatively straightforward task of characterizing the point source radiation in the image plane, assuming that the lenses depicted in Fig. 42 are identical. To perform an aberration of light measurement, the source linear dimension should be radically decreased. We assume that the order of magnitude of the dimensions of the ”point” source is about , where is the optical radiation wavelength. Within an elementary source volume () there is an enormous number of atoms. In optics, the emission from such a ”point” source is described classically using Maxwell’s equations, while the emitting medium (considered as an ensemble of atoms) is treated quantum mechanically. Thanks to modern lithography techniques, it is relatively straightforward to create an unresolved point source at optical wavelengths, allowing for the generation of sufficiently bright radiation. 393939It may be possible to use the masked spatially incoherent UV light source. One could use the light source onto a (micron diameter) hole of a specially designed mask. The mask should be placed at the distance , where is the transverse size of the spatially incoherent UV light source. Consider a diffracting aperture that is circular, and let the radius of the aperture be . The physical properties of the lens can be combined in a single number called the focal length. We assume that the image is produced by a diffraction-limited optical system (i.e. a system that is free from aberrations). Once the wavelength is fixed, the resolution only depends on the numerical aperture of the system. The response of the system at point of the image plane to a function input at coordinates of the object plane is called the point spread function of the system. In our case of interest, one obtains the following amplitude point spread function: , which is the diffraction pattern of a circular aperture, where , , . The first zero of the Airy’s pattern is at . Let us estimate the parameters for the optical setup shown in Fig.42. Suppose cm and m. We will use . The resolution analysis is reduced to the theoretical framework of standard imaging theory. One can take advantage of well-known resolution criteria like Rayleigh criteria i.e. m. One could use a vertically oriented optical setup. The constant of aberration defines the scale of the aberration increment, and is equal to rad. Measurement of the spot shift with respect to the optical axis (which is parallel to the nadir direction) is equivalent to the determination of the angular displacement. As already pointed out, the proposed method of measuring the angle of aberration involves the use of earth-based sources and has a big advantage. The rotation of the earth on its axis should produce a corresponding shift of the image. In principle, records could be taken over a period of one day. The image appears to move in an ellipse. The major axis of the ellipse is about 14 m. The value of the minor axis depends on the observer’s position and is given by m. 404040The small image size—approximately 2 m in the numerical case considered—places significant limitations on the use of a CCD as a detector. To address this, one promising alternative is photon detection using a microchannel plate photomultiplier (MCP-PMT). A feasible approach involves using a commercially available precision 2D translation stage with a resolution of 0.5 m to control the position of a submicron aperture in an opaque screen. For instance, the aperture diameter could be set to 1 m or smaller. The screen is mounted horizontally on the translation stage, which is positioned in the image plane. The MCP-PMT can then detect photons transmitted through the aperture. This setup offers the advantage of compactness, with the overall height of the optical configuration being approximately one meter. We derived our results under the assumption that the dimensions of the source are on the order of the wavelength, . It is important to note that the resolution of the proposed optical techniques is not fundamentally limited by the size of the incoherent source. The radiation field generated by an incoherent source can be modeled as a linear superposition of the fields from individual elementary point sources. The image of each elementary point source corresponds to a point spread function (PSF). An optical system illuminated with incoherent light can be fully described by a convolution integral that relates the object distribution to the resulting image distribution. In this framework, the optical system is characterized by its impulse response, namely, the point image. The rounding or softening of the source’s edges is a characteristic effect of the convolution (”smoothing”) process. It should be remarked that any linear superposition of radiation fields from elementary point sources conserves single-point source characteristics like oscillatory image shift within a period of one day. This argument gives the reason why our technique can be extended to allow for the reconstruction of the angle of aberration from a measurement of image shift of a finite size, incoherent source. The edge object has become useful in aberration shift detection. An alternative application of a ”point” source is an incoherent source with (physically) sharp edges. The edge width of the source is strictly related to the practical resolution achievable. For our particular example, the edge width should be about m. 11 The Aberration of Light from a Laser Source 11.1 Theory of a Laser with an Optical Resonator Suppose an Earth-based observer attempts to measure the aberration of light using a laser as the light source. It is important to emphasize that the Earth’s orbital motion cannot be detected using a laser in this manner. To understand why, consider a simplified scenario in which the laser resonator consists of plane mirrors. For oscillation to occur within the resonator, the total power loss due to diffraction and reflection must be less than the power gained as light passes through the active medium. Diffraction losses, therefore, play a crucial role—not only in determining the threshold for oscillation but also in shaping the distribution of energy within the interferometer during operation. The aim of this study is to examine how diffraction affects the electromagnetic field in a Fabry–Pérot interferometer operating in free space. The results of this analysis are equally applicable to gas lasers, as long as the interferometer is immersed in the active medium. The electric field of an electromagnetic wave in a passive resonator satisfies the equation . For simplicity, we shall assume that the field in the resonator is circularly polarized. Such an assumption does not violate the generality of the analysis because the polarization degeneracy takes place in the system under study. The electric field vector in the plane resonator is presented as an oscillation superposition with different longitudinal wave numbers, , where , , , is the (complex) refractive index of the mirror matter (, is the distance between the mirrors, and is an integer number (). It is evident that this expression satisfies Leontovich’s boundary conduction on the mirror surface when and . 414141An example of introducing approximate boundary conditions is presented here. These conditions, originally derived by Leontovich, apply to surfaces of materials with a high refractive index modulus. A more comprehensive theoretical treatment can be found in the monograph . We assume the field change per one resonator pass to be small i.e. , which means that the laser frequency is close to the natural frequencies of the longitudinal resonator modes. Let us find a solution for the field amplitude in the form . Substituting the ansatz in the wave equation we get . In a plane resonator, the diffraction effects at the edges of the mirrors can be taken into account if one considers the space between the mirrors as a waveguide, and uses diffraction theory at the open end of the waveguide. If almost an integer number of half waves is laid along the length between the mirrors then at the open resonator end we may put boundary conditions on the amplitude function , equivalent by its effect to the open waveguide end action : , where is the direction of the normal to the imaginary side surface of the resonator, and the parameter . 424242The analytical method is based on the introduction of specific, complex boundary conditions known as impedance boundary conditions of the resonance type . Interestingly, is related to one of the most renowned mathematical functions—the Riemann zeta function . Specifically, it is given by ; see . The boundary conditions do not depend on the polarization which means that polarization degeneracy occurs in the system under study. Therefore, the problem concerning the open plane resonator excitation using equivalent boundary conditions can be reduced to the more usual classical formulation of the problem of closed resonator excitation. Let us consider a resonator equipped with plane circular mirrors with radius . The distance between the mirrors is equal to . The system is azimuthally symmetric relative to the resonator axis which coincides with the axis of the polar coordinates. We shall seek the solution of the wave equation in the following form: , where is an integer.The eigenfunctions of the resonator are the solution of the homogeneous equation , satisfying the impedance boundary conditions. The eigenfunctions are orthogonal and form a complete system. In the first approximation for small parameter , where can be referred to as the Fresnel number, the functions have the form , where is theth root of the Bessel function of order , , . Each mode for is four times degenerate. Firstly, for each value of the azimuthal number there are two linear independent functions and . Secondly, because the boundary conditions do not depend on the polarization, two different field polarizations exist for each linear independent eigenfunction. The rigorous results of the three-dimensional theory of a plane Fabry-Perot resonator demonstrate that the radiation within the resonator can be represented as a set of discrete modes. Each mode is characterized by a specific decay rate (decrement) and an eigenfunction that describes the field amplitude distribution along the mirror surface. For the eigenmode associated with the transverse wave number , the corresponding eigenvalue is given by: . Due to the exponential dependence on , only the lower order modes tend to survive in the passive open resonator. We denote a normal mode supported by circular plane mirrors as a mode, with denoting the order of angular variations ( the angular variation are sinusoidal in form) and denoting the order of radial variation. The fundamental mode, , is identified as the dominant mode for resonators with circular plane mirrors. 11.2 Absent of Light Aberration from an Earth-Based Laser We have shown that acceleration influences the field equations. Within the resonator, the electric field of an electromagnetic wave satisfies the wave equation, which transforms into Eq. (17) under a Galilean transformation of space and time coordinates. At first glance, measuring the diffraction losses of the dominant mode may seem equivalent to determining the angular displacement. However, it is important to recognize that the difference between the Langevin and Minkowski metrics does not affect the parameters of a laser incorporating an optical resonator. In our earlier discussion of the Langevin metric, we demonstrated that applying a Galilean transformation to the wave equation yields a more complex, anisotropic form—namely, Eq. (17). This raises a fundamental question: how can we solve this form of the electrodynamics equation, particularly in the presence of boundary conditions? The key lies in recognizing that the underlying electrodynamics theory for optical resonators remains Lorentz-covariant. To address the anisotropic terms and simplify the field equations, we employ a mathematical transformation. By introducing a suitable change of variables, these anisotropic terms can be eliminated from the transformed equations, effectively restoring the Minkowski metric in the accelerated frame. It is important to stress that this transformation is purely a mathematical convenience, not a reflection of any underlying physical reality. The variables used carry no intrinsic physical meaning; they serve only to streamline the problem-solving process. Crucially, the physical content of the theory remains unchanged—results expressed in the transformed coordinates are entirely equivalent to those in the original frame. Changing the four-dimensional coordinate system introduces no new physical effects. A comprehensive analysis of experiments performed in an accelerated frame confirms that optical phenomena within a resonator are unaffected by acceleration relative to the fixed stars. This invariance arises because all radiation measurements in such systems are based on standing-wave phenomena—specifically, round-trip or two-beam interference methods. Let us now consider the aberration of light emitted from a laser resonator at rest in an accelerated frame. Upon analyzing the data, the accelerated observer detects no shift in the direction of energy propagation, as illustrated in Fig. 43. This absence of aberration indicates that the cross-term in the Langevin metric, which would normally cause such an effect, is effectively averaged out over the complete round-trip trajectory of the radiation within the resonator. The distinction between a ”plane wave” emitter and a laser source leads to a striking prediction in the theory of light aberration. Specifically, if a ”plane wave” emitter initially at rest with respect to the fixed stars begins to move, the apparent angular position of the emitter—as observed from the accelerated frame—undergoes a sudden shift by an angle of (see Fig. 44). Understanding this phenomenon requires the framework of special relativity; classical theory is insufficient. But how can such a difference between a ”plane wave” emitter and a laser source arise? To answer this, we must consider: What happens to an accelerated optical resonator? In previous discussions of optical resonators, we applied electrodynamic boundary conditions in 3-space, which define the direction of energy propagation along the optical axis—perpendicular to the mirror surfaces. According to special relativity, acceleration does not alter the orientation of the mirrors relative to the axes of the accelerated frame (i.e., the 3-space of the observer). Thus, an observer co-moving with an accelerating laser source will find that the direction of energy transport (along the optical axis) remains unchanged relative to the observer’s own frame, regardless of the motion with respect to the fixed stars (see Fig. 43). Special relativity tells us that both accelerated and inertial observers share the same absolute space-time but differ in their respective metrics, and therefore in their associated 3-spaces. In the case of an optical resonator, the direction of energy propagation is dictated by boundary conditions within 3-space. In contrast, the emission from an incoherent source—such as a ”plane wave” emitter—is influenced by initial conditions. For an accelerated observer, these conditions involve a nontrivial mixture of space and time coordinates, a consequence of the pseudo-gravitational field experienced during acceleration. The solution lies in using the absolute time coordinate. After acceleration, the wavefront of the light emitted by the incoherent source is perpendicular to the direction . Applying this initial condition, together with the Langevin metric in an accelerating reference frame, allows us to fully explain the observed aberration of light from a ”plane wave” emitter (see Fig. 44). 11.3 A Moving Laser Source in an Inertial Frame of Reference Let us consider the radiation emitted by an accelerated laser source in the inertial frame. The light aberration effect can be described within the framework of standard special relativity, utilizing the relativity of simultaneity. When a laser moves transversely (i.e., parallel to the mirrors), there is a deviation in the energy transport of the light emitted from the optical resonator. In the first-order approximation in the inertial observer would observe an angular displacement of , Fig. 45. When discussing light aberration, it is important to distinguish between the effect from a ”plane wave” emitter and that from the laser source. Consider a ”plane wave” emitter that is accelerated from rest to velocity in a direction perpendicular to its optical axis. 434343It is assumed that after acceleration, the laser moves at constant velocity and operates in a steady-state regime. Suppose an observer, at rest with respect to the inertial frame of reference, measures the direction of light transport. How does the light beam from the moving ”plane wave” emitter appear to this observer? The inertial observer would find an angular displacement of , as illustrated in Fig. 46. The first point to make regarding measurements in an inertial frame is that there is an intuitively plausible way to understand the aberration of light from a moving ”plane wave” emitter. An elementary explanation of this effect is well-known and can be understood in terms of the transformation of velocities between different reference frames. The aberration of light in the initial inertial frame can also be readily explained within the framework of the corpuscular theory of light. This is plausible when considering that a light signal represents a certain amount of electromagnetic energy. Like mass, energy is conserved, so a light signal shares several characteristics with material particles. Consequently, we should expect the group velocities of light signals to follow the same addition theorem as particle velocities. A more detailed treatment based on the wave theory of light confirms this expectation. In the case of a moving laser source, intuition suggests that the aberration increment would be given by . However, special relativity introduces an additional factor of 2. The standard derivation of the aberration of light effect does not account for the mixing of positions and time in an accelerated frame. It incorrectly assumes that accelerated and inertial observers share a common 3-space. In contrast, from the perspective of the initial inertial frame, the coordinate axes of the accelerated frame are rotated by an angle relative to those of the inertial frame. 12 Special Relativity and Reciprocal Symmetry 12.1 Pseudo-Gravitation Fields and the Langevin Metric This chapter explores the nature of special relativistic effects. The dynamics, governed by field equations, are often obscured within the framework of relativistic kinematics. The metric structure of space-time is best understood as an interpretation of how dynamical matter fields are perceived by different observers. A notable concern in the field is the absence of a dynamical explanation for the Langevin metric within special relativity, which has unsettled some physicists. The asymmetry between inertial and accelerated frames can be interpreted as a manifestation of pseudo-gravity experienced by an accelerated observer. The Langevin metric, which governs the transformation from an inertial coordinate system to an accelerated one (relative to the fixed stars), can be understood as arising from the action of a specific force—an issue typically addressed within the framework of general relativity. However, it would be erroneous to suggest that this provides a direct explanation for the effects discussed in earlier chapters. A central aim of this chapter is to show that non-inertial frames in pseudo-Euclidean space-time can be effectively described using the language of general relativity. Suppose that the reference frame is at rest with respect to the fixed stars. Let represent the Lorentz coordinates in the inertial frame , while is a system initially at rest relative to that undergoes uniform acceleration along the -axis, reaching a velocity . The inverse Galilean transformation , defines a new reference frame, , which is uniformly accelerated in the -direction, attaining a velocity relative to . After the acceleration, the frame is, reciprocally, moving along the -axis with velocity relative to . When the system begins moving with constant velocity, it becomes an inertial frame of reference. At first glance, there appears to be symmetry (or reciprocity) between the inertial frame and the inertial frame . According to the principle of relativity (also known as the principle of reciprocity), an observer accelerated relative to the fixed stars should measure the Minkowski metric as . This metric is reciprocal to the one measured by the inertial observer: . So, where does the asymmetry arise? The analysis of the equivalence principle reveals that no additional complexities are required to explain the asymmetry between an inertial frame and an accelerated frame. The resolution of this paradox lies in identifying a pseudo-gravitational potential within the system , which acts as the source of the asymmetry. A uniformly accelerating frame can be treated as if it were an inertial frame, with the addition of a uniform pseudo-gravitational field. By ”pseudo-gravitational field,” we refer to an apparent field (not a true gravitational field) that exerts a force on all objects proportional to their mass. ”Uniform” means that the force experienced by each object is constant, regardless of its position. This concept lies at the heart of the equivalence principle. The principle of equivalence allows us to apply dynamical methods to solve problems involving non-inertial reference frames. To minimize the mathematical complexity of the discussion, we will restrict our analysis to terms up to second order in . Consider a reference frame , in which each point undergoes constant acceleration in the negative -direction over a time interval . In this configuration, frame is effectively in free fall within a pseudo-gravitational field experienced in the accelerating frame . It becomes evident that the pseudo-gravitational acceleration is given by the gradient of a scalar potential, i.e., . A straightforward calculation shows that at time , the coordinate transforms as at . Once the system transitions to uniform motion (i.e. constant velocity), the pseudo-gravitational field vanishes. To determine the pseudo-gravitational potential difference between the two frames, we compute , where we have chosen the potential to be zero at , i.e., at that point. The analysis of an accelerated frame using the equivalence principle does not involve a real gravitational field and therefore does not require the full framework of general relativity. Nevertheless, the insights provided by general relativity about real gravitational fields apply in a limited sense to pseudo-gravitational fields as well. One key concept we rely on is that time runs more slowly as one moves deeper into the potential well of a pseudo-force field. This effect can be used to our advantage when analyzing the metric in a non-inertial frame. The equivalence principle implies that gravity can alter the frequency of an electromagnetic wave and cause clocks to run at different rates. Specifically, the frequency of light increases with the absolute value of the gravitational potential. If a pulse of light is emitted at a point where the gravitational potential is with frequency , then upon reaching a point where the potential is , its frequency—measured in proper time at that location—will be: . Frequency is inversely proportional to the rate of local proper time; therefore, the gravitational frequency shift can be reformulated as a time dilation effect. The rate at which a coordinate clock ticks depends on the gravitational potential at its location. The relationship between coordinate time and proper (physical) time is given by: . This implies that physical time elapses more slowly in regions with lower gravitational potential—i.e., where the absolute value of the potential is higher. If two identical clocks are placed in a gravitational field for some duration, the clock that remains deeper in the field will lag behind the other, appearing to run slower. Let us now consider the situation where a reference frame, initially located at a point with gravitational potential , undergoes free fall toward a region of lower potential . Importantly, during fall, the frame retains the gravitational potential of its starting point, . It is worth exploring what it means for the potential to remain ”unchanged” in this context. Let us first examine the behavior of a clock in free fall. From the perspective of an observer located at , the clock in the freely falling frame appears to tick faster, since it remains associated with the higher potential , regardless of its instantaneous position. Radiation from a standard source within the free-falling frame will therefore appear blue-shifted relative to radiation from a standard source at rest at . 444444 To understand the gravitational blue-shift of a light source in free fall, one must analyze the source’s emission mechanism and its behavior while in motion. The frequency of dipole radiation is proportional to the electron mass. In a gravitational field, mass varies with gravitational potential, according to the relation . As a result, the electron mass is greater at points with lower gravitational potential. A similar time dilation effect occurs in accelerated frames. For a clock at rest in a uniformly accelerated frame, the relation between proper time and coordinate time is also: , where is coordinate time in the accelerated frame , but corresponds to proper time in the inertial frame . From the viewpoint of the accelerated observer, clocks in the inertial frame appear to run faster due to their association with a higher effective (pseudo) potential. According to the equivalence principle, the consistent aging of the inertial observer compared to the accelerated one arises naturally. The slower passage of time in the pseudo-gravitational field explains this outcome directly, requiring only basic algebraic reasoning. Furthermore, it is important to note that the phenomenon of the relativity of simultaneity can be understood from a dynamical perspective. Specifically, the time-offset relationship between two spatially separated clocks can be interpreted as the accumulated time difference resulting from the pseudo-gravitational effects experienced by an accelerated observer. This interpretation implies that a clock positioned at a higher gravitational potential—i.e., farther along the direction of acceleration—runs faster than one at a lower potential. The pseudo-gravitational potential difference between two points along the -axis is given by . When the system transitions to uniform motion with constant velocity, the potential gradient in the comoving system becomes zero. The accumulated time difference between two spatially separated clocks is then . The next aspect we must examine is the origin of the Lorentz deformation. Specifically, we observe that this deformation—referring to changes in relative position along the direction—is closely connected to shifts in the moments of acceleration. By employing the time-offset relation and integrating with respect to an infinitesimal change we arrive at the expression , which characterizes the spatial geometry within the accelerated frame. This equation relates the length of the rod in the inertial frame to the corresponding length in the accelerated frame . Our analysis, which relies on standard measuring rods in the accelerated frame, demonstrates that the coordinate distance acquires an effective length of . This is a purely kinematic effect, arising without the involvement of any forces. We therefore conclude that the various relativistic effects are not independent phenomena, but are instead interconnected and can be comprehensively understood through a generalized principle of equivalence. We now have all the quantities we set out to determine. Let us bring them together into the expression for the new space-time variables. The new independent variables can be expressed in terms of the original independent variables as follows: , . Above we made a simplification in calculating time and space transformations by considering only low velocities. Despite this approximation, the physical principles underlying the space-time transformation were clearly illustrated. The next natural question is: What is the form of the transformation for arbitrary velocity? The answer is straightforward: , . The quantity represents physical time, which is independent on the choice of coordinate time. It is important to note that in the expression for corresponds to the coordinate time in the accelerated frame. Physical time governs the actual flow of events in physical processes. Likewise, the quantity represents the physical distance between two points in three-dimensional space. The space-time interval, when expressed in terms of physically measurable differentials, becomes a diagonal quantity. These measurable differentials, and , are linear combinations of the coordinate time differential and the spatial differential , with coefficients determined by components of the metric tensor. In the new physical coordinates, the space-time interval takes the form: - . The derivation of the Langevin metric in an accelerated frame—based on the concept of a pseudo-gravitational field—highlights a subtle but crucial distinction: the metric determines the form of physical laws, not the physical phenomena themselves. We interpret the Langevin metric to mean that the laws of electrodynamics, in the accelerated frame, take the form given by Eq. (17). By physical facts in this context, we refer to observable phenomena such as the aberration of light emitted by a single plane-wave source in the accelerated frame. To fully describe this aberration effect, the electrodynamics equation must be solved with an appropriate initial condition for the radiation wavefront. After applying an inverse Galilean boost—where the coordinates transform as —the wavefront is found to be perpendicular to the vertical direction , as depicted in Fig. 14. Together, the Langevin metric and this initial condition account for the observed aberration of light in the accelerated frame. 12.2 The Equivalence Principle and Gravitational Physics We should add a further remark concerning the equivalence principle.454545For a general discussion of the equivalence principle, we recommend . It is commonly believed that the equivalence principle lies at the foundation of general relativity. However, this perspective is not entirely accurate. Gravity, as understood through Einstein’s insight, is not merely a universal force acting identically on all bodies—it is fundamentally tied to the geometry of space-time itself. To appreciate this idea, one must consider the role of the Riemann curvature tensor, which, in a precise mathematical sense, represents the gravitational field. The vanishing of this tensor is the necessary and sufficient condition for the absence of gravity. This fact alone shows that the traditional formulation of the equivalence principle has only limited applicability. While the equivalence principle holds for mechanical processes, it breaks down when applied to electrodynamical phenomena. This distinction implies that internal measurements can, in principle, distinguish between an inertial frame and a frame in free fall within a homogeneous gravitational field. For instance, consider a free electric charge in such a gravitational field. Relative to an inertial frame, it experiences uniform acceleration and consequently emits radiation—regardless of whether the gravitational field is weak or strong. In contrast, in a uniformly accelerated frame in flat space-time (i.e., in the absence of gravity), a freely falling charge does not accelerate relative to the inertial frame and, according to classical electrodynamics, does not radiate. Thus, in this context, the equivalence principle is explicitly violated. In the context of special and general relativity, we can examine earth-based experiments where pseudo-gravity and real gravity are intricately combined. With this foundation, we now need to describe the gravitational fields near the Sun, arising from the Sun’s own mass. Consider the simplest case of a transformation from the initial inertial frame, in which spacetime is Minkowskian, to the Earth-based frame. Newton’s theory of gravity is known to hold when gravitational fields are weak enough that velocities do not approach the speed of light. Specifically, this means the gravitational potential energy of a particle is much smaller than its rest mass energy. In such scenarios, general relativity reduces to Newtonian gravity. For weak gravitational fields, the Newtonian potential fully determines the metric, which takes the form: . Here, is chosen to be negative so that, at large distances from the mass , . The condition for a weak field is that , which ensures that . This metric is only accurate to the first order in . At large distances from the source of mass, we say that spacetime asymptotically becomes flat. For a rotating reference frame, such as the Earth revolving around the Sun, the Langevin metric can be applied: , where is the velocity of the Earth, and represents the square of the coordinate distance. The lack of equivalence between equations in an accelerated reference frame and those in an inertial frame with a static gravitational field becomes clear. In the former case, the non-diagonal metric coefficients depend on time, whereas in the latter, the metric remains diagonal and time-independent. This discrepancy explains why a charge at rest in an accelerated frame will radiate, while a charge at rest in a static gravitational field will not. This difference demonstrates why we cannot conclusively deduce the validity of the equivalence principle from observational data alone. Now, returning to the perspective of an Earth-based observer, an additional expression for the invariant interval is required: . At the order of approximation used, this result is simply the superposition of the Langevin metric and the Sun-centered metric in the weak gravitational field of the Sun. Both the Earth’s orbital speed and the Sun’s gravitational field contribute to time dilation and spatial contraction. However, the effects are small, making the Langevin metric an excellent approximation for the metric in the rotating Earth-based reference frame. 12.3 Time Dilation It is commonly believed that special relativity is a reciprocal theory. However, an instructive counterexample to this reciprocity is provided by the time dilation effect. Consider the passage of time in two inertial reference frames. One frame, denoted by , is considered to be at rest relative to the fixed stars. The second frame, , undergoes acceleration relative to until it reaches a velocity along the -axis. As previously discussed, the transformation from an inertial to an accelerated frame was apparently first introduced by Langevin. A central mathematical approach in the analysis of time dilation is the smooth matching—or tailoring—of the metric tensor. The transformation to the accelerated frame, with coordinates denoted by subscript ”n”, takes a Galilean form: , . Following this transformation, the metric in the inertial frame transitions continuously into the Langevin metric of the accelerated inertial frame (see Chapter 5 for details). In these new coordinates, the invariant interval given by Eq. (1) becomes Eq. (15). It is important to note that although coordinate clocks are fixed in the accelerated frame, they register the same time as clocks in the inertial frame. However, due to the presence of the cross-term , the clocks in the accelerated frame are not synchronized in the conventional (Einstein) sense—that is, by exchanging light signals between clocks and accounting for round-trip travel time. Suppose we have two identical physical clocks located at the same point in an inertial reference frame , and let their readings coincide at the initial moment . The first clock remains at rest in frame , while the second clock, at , begins to move with constant velocity along the axis. Now, consider the situation from the perspective of the accelerated reference frame in which the second clock is always at rest. The two frames are not equivalent, as is accelerated with respect to the fixed stars. In this accelerated frame, the first clock moves with velocity . Taking into account the Langevin metric Eq.(15): , we find that the proper time interval for the first (moving) clock is . This implies that the time shown by the first physical clock (which is moving in the accelerated frame) coincides with the coordinate time in that frame. Since the second clock is at rest in the accelerated frame, its proper time evolves according to . This result demonstrates the time dilation experienced by a physical clock at rest in the accelerated frame when compared to clocks in the inertial frame. Importantly, we observe that the slowdown of the accelerated clock does not depend on the choice of reference frame in which the effect is measured. Let us now consider a different and very practical scenario. Suppose we split the accelerated frame into two parts. Frames and both accelerate relative to the inertial frame , reaching the same speed , but in opposite directions. From the perspective of the inertial frame , the situation is perfectly symmetrical. Therefore, no difference in time dilation should exist between physical clocks at rest in and , as compared to a clock at rest in . The following simple analysis confirms this expectation. Assume we have two physical clocks: the first is at rest in frame , while the second is at rest in frame . From the point of view of , the second clock appears to move with velocity . Using the Langevin metric, , we calculate the proper time interval for the second clock as: . This result shows that the time measured by the first clock, , is equal to the proper time of the second clock. This outcome may appear paradoxical at first. After all, and are moving relative to each other at a non-zero speed. Yet we observe an interesting and important fact: The mere presence of relative velocity does not necessarily lead to time dilation. As discussed above, non-inertial frames can be effectively described using the framework of general relativity. The time dilation observed when transitioning from an inertial to an accelerated frame can be interpreted as the consequence of a specific force acting within the system. A resolution to the symmetry paradox reveals that a pseudo-gravitational field, present within the systems and , serves as the agent responsible for the time dilation effect. Clocks measuring proper time and sharing identical histories of acceleration run at the same rates and produce consistent measurements, even when at rest in distinct inertial frames. Let us now consider a different scenario. Suppose that frame accelerates relative to the inertial frame until it reaches a velocity . Then, two new frames, and , each accelerate relative to to the same speed , but in opposite directions. From the perspective of , this setup is not symmetric, and the resulting time dilation experienced in the accelerated frames and differs. In frame , frame moves with velocity , so the proper time in this frame is simply . Meanwhile, frame moves with velocity relative to . Using the Langevin metric, , we find that the proper time in frame is given by . This analysis has been carried out from the point of view of an observer at rest in frame . It is natural to ask: how would the situation appear to an observer in the inertial frame ? From the perspective of frame the frame moves with velocity . Applying the Minkowski metric, we again obtain: , which is consistent with our previous result. It is worth noting that the mathematics involved in relating proper times is remarkably straightforward in this case. This simplicity arises because we are using Galilean boosts and the Galilean addition of velocities. 12.4 The Light-Clock and Observations by a Non-Inertial Observer To understand how a clock slows down in an accelerated system, we need to examine the internal workings of the clock and observe what happens as it moves. We’ll explore this effect using a simplified example—a device known as a light clock. This clock consists of a meter stick with mirrors at both ends; a ”tick” corresponds to a pulse of light bouncing between the mirrors. Now, let us consider two configurations of the light clock: one with its length parallel to the direction of motion and the other oriented orthogonally. According to the principle of relativity (i.e. pseudo-Euclidean spacetime geometry), all clocks—regardless of their construction—must measure time consistently in all inertial frames. We will demonstrate that, regardless of the orientation of the light clock relative to its motion, the elapsed time measured remains the same. Two light clocks illustrating these configurations are shown in Fig. 47. Now, let us examine the workings of a physical light clock under acceleration and observe what happens to it in such a scenario. We analyze the behavior of the clock from the perspective of an observer co-moving with the accelerated frame in which the clock resides. To begin, we configure the light clock so that the light beam travels between two mirrors aligned parallel to the direction of motion relative to the fixed stars. Within the context of the Langevin metric, the speed of light emitted by an accelerated source is no longer equal to . Considering an infinitesimal displacement along the worldline of a light beam, and using the condition , we derive from Eq. (15) that: . This implies that, in the accelerated coordinate system , the velocity of light emitted in the direction parallel to the -axis is: in the positive direction, and in the negative direction. Let represent the (as yet undetermined) length of the rod or meter stick in this frame. The time it takes for light to travel from the left mirror to the right mirror is then: , and for the return path, from the right mirror to the left: . Therefore, the total time interval between the emission and reception of the light signal is: . To begin, we must determine the length of the physical measuring rods. For measuring distances between fixed points in the accelerated frame, we use physical rods identical to those employed in the inertial frame, but now considered at rest relative to the accelerated frame. Regarding spatial distances, the guiding interpretive principle is that represents the length of an infinitesimal rod whose endpoints are simultaneous according to standard simultaneity in the rod’s rest frame. Using the Langevin metric Eq.(15), we find that for , it follows that . This implies that the length of a physical rod at rest in the inertial frame matches the coordinate distance in the accelerated frame. However, when we apply this principle to rods at rest in the accelerated frame, we face a complication: the condition does not necessarily correspond to standard simultaneity in the accelerated frame. As a result, the length of a rod at rest in the accelerated frame generally differs from that of a rod at rest in the inertial frame. The spatial geometry in an accelerated frame is characterized by the differential spatial line element defined within that frame. Although spatial geometry is inherently coordinate-dependent and thus exhibits a degree of conventionality, the proper spatial line element, denoted by , holds particular significance. It defines a coordinate-independent notion of spatial distance within the accelerated frame. By contrast, the spatial line element , obtained by setting in Eq. (15) such that , differs from . Now, how do physical notions of length and time manifest in the coordinate system ? In an accelerated frame, consider the spacetime interval given by . We can decompose this expression as follows: . Here, the quantity has a time-like character and represents the square of physical time in the accelerated frame, while is space-like and corresponds to the square of physical length. These quantities are directly tied to the spacetime interval in a manner analogous to the Minkowski metric in special relativity . They are thus physically meaningful and invariant under coordinate transformations within the accelerated frame. The relation establishes the connection between the three-dimensional spatial line element and the four-dimensional spacetime metric. Using Eq. (15), we obtain the relation , which defines the spatial geometry in the accelerated frame. This equation is asymmetric with respect to the lengths and , as it connects the physical length of a rod measured in the inertial frame with the physical length in the accelerated frame . Our analysis, based on the use of standard measuring rods within the accelerated frame to probe its geometrical properties, shows that a coordinate distance corresponds to a physical length of . Thus, we can directly determine the asymmetry in the length of the meter stick from the metric. This can be interpreted as an effect of acceleration with respect to the fixed stars, suggesting that space in an accelerated frame is inherently non-Euclidean. It is important to emphasize that the Langevin metric characterizes the measurements of an accelerated observer as interpreted by an inertial observer who employs the simplest form of clock synchronization—namely, maintaining the same set of synchronized clocks used when the light clock was at rest. This method preserves Einstein synchronization, established via light signals emitted from a stationary dipole source. Within this framework of absolute simultaneity, the inertial observer perceives the accelerated observer as measuring the speed of light along the -axis as in the positive direction and in the negative direction. Furthermore, the physical meter stick (with mirrors mounted on it), as measured by the accelerated observer, appears compressed along the direction of motion relative to the fixed stars. It is crucial to note that the Langevin metric, in fact, reflects measurements performed by an inertial observer using the Minkowski metric, subsequently re-expressed via the transformation , . The coordinate length in the accelerated frame corresponds to a length measured by the inertial observer in the moving (accelerated) frame under the assumption of absolute time synchronization in the inertial frame. The length of a moving object thus depends not only on the spacetime structure but also on the synchronization convention adopted. Consequently, it is unsurprising that the measured length of a moving physical rod differs from its proper length when at rest. As a result, the time interval between the emission and reception of the light signal in the parallel light clock within the accelerated frame is given by: . Since and represent the time indicated by a clock at rest in the inertial frame, this implies that physical clocks at rest in the accelerated frame tick more slowly compared to those in the original inertial frame (see the Section 13.1 for further details). Let us consider a light clock that is accelerated to a velocity , perpendicular to the direction of the optical pulse. In other words, we orient the light clock in such a way that the effects of length contraction—such as those observed in the case of a meter stick aligned with the direction of motion—do not arise. We analyze this orthogonal light clock from the perspective of an observer co-moving with the accelerated clock If is the infinitesimal displacement along the worldline of a light beam, then . Applying Eq. (15), we find that . Hence, the time interval between the emission and reception of the light signal is . Since and represent the time indicated by a physical clock at rest in the inertial frame, this result shows that clocks at rest in the accelerated frame run slower compared to those in the original inertial frame. 12.5 Metrics in Accelerated Frames In this section, we continue our exploration of non-reciprocity in special relativity. We begin by examining the implications of assuming that the inertial frame has no history of acceleration and that the space-time metric, as perceived by an observer at rest, is diagonal. Consider two emitters within this inertial frame: the first remains stationary, while the second moves with velocity . A Galilean boost transforms the metric associated with the moving emitter, resulting in the form given in Eq. (11). In the coordinates of the inertial frame, the non-accelerated emitter is described by the standard diagonal Minkowski metric: . Now, let us shift our attention to the accelerated frame in which the second emitter is at rest, and the first appears to move with velocity . This raises a fundamental question: Does the principle of reciprocity still apply in this context? Conventional theory holds that acceleration does not intrinsically violate the symmetry of motion between inertial and non-inertial frames. Therefore, an observer in the accelerated frame should also describe the metric of the stationary (second) emitter as diagonal. By reciprocity, the moving (first) emitter should then be described in this frame by a metric analogous to Eq. (11), differing only by the sign of . However, this leads to a deeper inquiry into the principle of reciprocity itself. The principle of reciprocity states that all inertial frames are equivalent. However, this relativity principle does not hold and the Lorentz covariance of all physical equations does not dictate the reciprocal symmetry of nature. A geometric analysis of space-time in the accelerated frame reveals this asymmetry. Specifically, within the accelerated frame, the Langevin metric—given in Eq. (15)—emerges as the correct description of the second emitter’s space-time. This metric governs electrodynamics in the accelerated frame and departs from the simple diagonal form of inertial frames. This departure has significant consequences: Maxwell’s equations, valid in inertial frames, no longer retain their form for an observer at rest in the accelerated frame. This breakdown signals a deeper structural difference between inertial and accelerated reference frames. Notably, because we adopt an absolute time coordinatization, the initial conditions—such as the orientation of radiation wavefronts—remain identical in both frames. We now arrive at a paradox. The equivalence of all inertial frames is often supported by the claim that, after an appropriate coordinate transformation (e.g., diagonalization of the metric tensor), both the inertial and accelerated frames yield diagonal metrics. This seems to imply symmetry: both frames appear to be related by a Lorentz transformation and should be physically equivalent. 464646Many physicists argue that the universality of Maxwell’s equations across inertial frames rules out the existence of any privileged frame. As Dieks notes , “As we already mentioned, it is a basic principle of the special theory of relativity that the line element supplies all information about the physics of the situation, as described in the given coordinates.” However, a key asymmetry becomes evident when we consider the physical history of the frames. While the inertial frame remains unaccelerated, the accelerated frame retains a memory of its dynamical past. This memory is encoded in the metric structure. After diagonalization, the metric tensor in the accelerated frame undergoes a discontinuous shift—from a form such as to the standard inertial values: . This discontinuity highlights the fact that the Lorentz transformation does not preserve the smoothness of the metric across such transitions. Hence, the apparent symmetry between frames breaks down when physical considerations—specifically, the history of acceleration—are taken into account. This leads us to a subtle but important conclusion: although the mathematical form of the metric may appear similar in both frames, the physical context—particularly the presence or absence of past acceleration—introduces observable differences. The accelerated frame thus retains a “memory” of its non-inertial history, breaking the naive equivalence suggested by purely kinematic symmetry arguments. 13 Relativistic Measurements 13.1 The Concept of Time in Special Relativity The theory of relativity reveals that our intuitive understanding of time does not align with its true nature. To clarify the concept of time, we adopt the principle that ”each physical quantity is defined by the method used to measure it.” Let us explore how this principle applies. Consider the case of length. We have a standard unit of length, and we can measure the length of any object accordingly. However, this measurement refers to objective physical length only when the object is at rest. For moving objects—such as a rod in motion—length becomes a convention-dependent quantity. It relies on the chosen method of synchronization and lacks an exact, objective meaning. With this in mind, we can now turn to the problem of defining time within the framework of special relativity. Here is a simple example that illustrates the concept clearly. Suppose we know the law governing muon decay in its rest frame. When we apply a Lorentz transformation to this decay law, we find that, in the laboratory frame, the characteristic lifetime of the particle appears to increase from to . This result can be interpreted as follows: after traveling a distance of , the number of muons observed in the lab frame is reduced to half of the original population. In this context, the time measured corresponds to a ”length”. This leads to an intriguing question: could this interpretation apply to all time measurements within the framework of special relativity? We argue that the answer is yes. Our next example concerns frequency measurements. Consider a Fabry-Perot interferometer. In this setup, the frequency of light is effectively measured through the ”length” of the standing wave it produces. Frequency measurements in such cases inherently follow the principles of interference. Another example that illustrates the connection between frequency and interference is the grating spectrometer. Here, the frequency is inferred from the position of the light spot along the dispersion direction. Although their designs differ, there is no fundamental physical distinction between a diffraction grating and an interferometer—both rely on the interference of incident and reflected light waves to perform frequency measurements. Let us now explore how to compare the clock rates between the initial inertial frame and the accelerated frame . It is generally understood that when comparing clock rates between two frames and in relative motion, one cannot simply compare the readings of a single clock in each frame. This is because clocks from different frames coincide at the same spatial point only once. Therefore, in at least one of the frames, multiple clocks—assumed to be synchronized—must be used for meaningful comparison. We have already pointed out that, within the framework of special relativity, all time measurements ultimately reflect what is essentially a measurement of ”length.” When it comes to measuring the length of objects at rest, we are dealing with an objective physical quantity—one that is independent of any particular convention. Since our empirical access is limited to such length measurements, it is difficult to accept the standard textbook claim that one can only compare the reading of a single clock in one frame with the readings of multiple synchronized clocks in another. Let us now apply our operational interpretation of time measurements to the example of a light clock, in order to see how this approach functions in practice. Let us consider a specific thought experiment. We examine a situation in which time marks can be imprinted on a moving object (a screen) by a clock. To begin, we describe an orthogonal light clock from the perspective of an observer in the same accelerated frame as the clock. The screen is at rest in the initial inertial frame. Suppose the top mirror of the light clock is semitransparent, allowing time marks to be recorded on the (moving) screen, thereby measuring the time intervals between clock ”clicks.” In the case of the Langevin metric the speed of light emitted transversely by the accelerated source is given by . Accordingly, the time interval between the emission and return of the light signal is . During this interval, the object (i.e., the screen in the initial inertial frame) falls a distance , which corresponds to the spacing between consecutive time marks. Taking into account the Langevin metric (Eq. 15), we find that at , it follows that , i.e. the length of the physical rod in the inertial frame coincidence with the coordinate distance in the accelerated frame. Since and represent the time indicated by a physical clock at rest in the initial inertial frame, it follows that clocks at rest in the accelerated frame run slower compared to those in the inertial frame. In fact, the rate of the accelerated clock as measured by the inertial observer corresponds to the spacing between time marks on the screen in the initial inertial frame. Second, we describe the orthogonal light clock from the perspective of an observer who is initially at rest in the same inertial frame as the clock. According to this observer, the time interval between the emission and reception of the light signal is given by . During this interval, the screen in the accelerated frame falls a distance , which corresponds to the spacing between the time marks. An analysis using standard measuring rods in an accelerated frame to examine its geometrical properties reveals that the coordinate distance corresponds to a physical length of . This relationship directly reflects the asymmetry in the length of the meter stick, as derived from the Langevin metric. This result indicates that the physical spacing between time marks, as measured in the accelerated frame, experiences compression in the direction of motion relative to the fixed stars. Since , and represents the physical time rate shown by a clock at rest in the inertial frame, it follows that the accelerated observer perceives a shortening of the mark spacing, given by . Here denotes the physical spacing of time marks in the accelerated frame. This thought experiment clearly demonstrates the absence of reciprocity in time dilation. 13.2 Interference Phenomena We now turn our attention to a specific aspect of the phenomenon of interference. To explore this, let’s examine the processes of reflection and refraction within the framework of special relativity. At this point, we are prepared to analyze what happens when light emitted by a moving source passes through a stationary medium, such as glass. The key task is to calculate the interference between the incident and reflected radiation. At first glance, one might assume that calculating both refraction and reflection in a single inertial frame would require accounting for differences in the metrics, and consequently the electrodynamics equations, for a moving emitter and a stationary glass. This suggests that the possibility of calculating interference effects is not immediately apparent. If the mirror is stationary and the emitter is in motion, it is evident that the equations governing electrodynamics must be the same for all electromagnetic waves. In other words, a consistent metric must be applied to both the incident and scattered waves. However, this raises a paradox: how can the electrodynamics equations be identical for both the incident and reflected waves, given that they are governed by two different metrics? In a previous discussion, we addressed the issue of two distinct metrics in the context of light aberration. In Chapter 4, we focused on the first-order approximation and highlighted that the apparent paradox is resolved. One aspect of the aberration setup can be easily understood based on concepts we have already explored. It is advantageous to begin with the simpler case of a normally incident plane wave, and only later in Section 5.10 do we introduce the theory that includes tilted incoming plane waves. This approach is beneficial because the unique geometry of light aberration ensures that, even after applying a Galilean transformation, the electrodynamics equations in an absolute time coordinate system will retain Maxwell’s form for a normally incident plane wave. Now, let us examine the situation in a collinear geometry, as illustrated in Figure 48. We begin by summarizing the central idea. The apparent paradox is resolved by recognizing that both the Minkowski metric and the metric given by Eq. (11) yield identical predictions. The key point is that all methods used to measure interference—specifically, those involving standing waves—are effectively round-trip measurements. In contrast, the deviation in the direction of energy transport arises from a geometric effect. Since empirical measurements grant access only to the round-trip (two-way) average speed of light, the one-way speed of light remains a matter of convention without intrinsic physical meaning. The two-way speed of light, however, is directly measurable and therefore possesses physical significance. We now turn to an important aspect of interference phenomena. It is commonly believed that, in a setup involving a moving light source and a stationary glass medium, the incident wave and the wave scattered by the dipoles of the glass cannot interfere—as required by the electron theory of dispersion—because they travel at different velocities in an absolute time framework. This notion, however, is incorrect. Physically, an incident wave of a given frequency will excite electrons in the glass to oscillate at that same frequency, regardless of the wave’s velocity. These oscillating electrons then re-emit radiation at the same frequency. Thus, the incident and scattered waves at any given location share the same frequency and can interfere. The difference in their velocities manifests as a relative phase shift that varies spatially. This phase variation influences both the velocity and amplitude envelope of the resulting wave formed by the superposition of the incident and scattered components. The following simple analysis supports these ideas. We consider the case where the metric is non-diagonal, as given in Eq. 11. Let the incoming wave be represented by , where the phase velocity is . Similarly, let the scattered (outgoing) wave be represented by , having the same amplitude and frequency, but a different velocity , and a phase offset . The superposition of these two waves is given by: . Here, the cosine term acts as an amplitude envelope that is stationary in space. Its periodicity is inversely proportional to the difference in the wave numbers and of the two components. This expression can be further simplified by substituting the wave numbers in terms of , , and , leading to: . Consider a source at rest emitting waves with a natural frequency . In the laboratory frame, after applying a Galilean transformation, the velocity of the incoming wave becomes . Given this, the frequency observed in the lab frame is: . This frequency shift corresponds to the well-known Doppler effect. Incorporating this into the expression for the superposition of two waves, we obtain: . Suppose an observer in the laboratory conducts a standing wave measurement. It is important to analyze which aspects of the measured data depend on the chosen synchronization convention and which remain invariant. We emphasize that time oscillation lacks intrinsic physical meaning—its interpretation arises solely from a chosen convention. In particular, consider the time shift appearing in the phase term . This shift is closely tied to the issue of synchronizing distant clocks. Furthermore, from a physical standpoint, the absolute scale of time—or equivalently, frequency—cannot be directly identified. Suppose the laboratory observer measures the wavelength of a standing wave. The relation describes the connection between the spatial line element and the metric. Using Eq. (11), we find , which defines the spatial geometry in the context of the non-diagonal metric. This result reflects the Lorentz contraction of measuring rods. When analyzing the measured data, the laboratory observer finds that the wave number is given by . Notably, this is the same factor one obtains under the assumption of a diagonal metric. In other words, the laboratory observer will measure the same two-way speed of light regardless of the choice of metric representation. The reason a laboratory observer measures the same wavelength lies in the underlying principle of interference. All interference phenomena are governed by a fundamental law known as the relativistic invariance of phase. This principle states that a specific quantity—the phase—remains unchanged under transformations between reference frames. The invariance of phase is a powerful tool for analysis, as it allows us to draw meaningful conclusions (as will be illustrated in the next section) without requiring detailed knowledge of the full set of transformation formulas. However, a puzzle remains in the scenario under discussion. Let us consider a dipole source that accelerates in the laboratory inertial frame to a velocity along the -axis. After acceleration, assuming synchronization in the lab frame remains unchanged, we encounter a complex situation involving the dynamics and electrodynamics of moving charges. In conventional (non-covariant) particle tracking, the particle’s motion can be modeled as a sequence of infinitesimal Galilean transformations, with a corrected form of Newton’s second law applied at each step. Interestingly, this approach does not invoke Lorentz transformations. Within the inertial lab frame, the particle’s motion appears exactly as classical mechanics would predict—following absolute time, without invoking the relativity of simultaneity. That is, time and position remain distinct and do not mix in this description. In contrast, Maxwell’s equations are valid in the inertial frame only when Lorentz coordinates are used. This raises a fundamental question: How can classical mechanics (with Galilean kinematics) be used alongside electrodynamics governed by Einstein’s kinematics? This apparent inconsistency calls for explanation. The resolution lies in the nature of the coupling between particles and fields in collinear geometries. In such configurations—where both the particle motion and the emitted radiation are aligned along the same axis—the standard Maxwell equations and the corrected Newtonian dynamics are sufficient to explain interference phenomena within a single inertial frame. In this context, the dynamical evolution of the particle can be alternatively interpreted as a sequence of infinitesimal Lorentz transformations, rather than Galilean ones. But how is this possible? As discussed in Section 3.6, the key insight lies in the fact that Galilean boosts form a commutative group—just as collinear Lorentz boosts do. This commutative property plays a critical role in the analysis. In the case of collinear geometry, where the particle moves along the same line as the emitted radiation, the velocity vector is perpendicular to the radiation’s wavefront (or the plane of simultaneity). In this configuration, the motion of the source influences only higher-order kinematic terms—those proportional to powers of . At first glance, Newtonian particle dynamics in collinear geometry appears to be entirely non-relativistic. However, relativistic effects do manifest, albeit subtly, through the implicit assumption that the mass of the moving particle corresponds to its relativistic mass. 14 A Preferred Inertial Frame and Second Order Experiments 14.1 Second-Order Optical Experiments and Special Relativity An illustrative example of time dilation is observed in the resonance absorption of gamma rays in Mössbauer rotating disk experiments. When measuring the resonance absorption for the same relative velocity between the source and the observer, a blue shift is detected when the source is positioned at the center of the disk, while a red shift appears when the source is located at the rim. This result reveals an asymmetry in the transverse Doppler effect.474747The first experimental confirmation of time dilation via resonance absorption of gamma rays was reported by Champeney, Isaak, and Khan in 1965 . Notably, when both the source and observer are situated at the rim, no frequency shift is observed. In such a configuration, a pulse emitted by one accelerated observer on the rotating disk, which appears redshifted at the center due to time dilation, is received by a diametrically opposite observer on the rim without any frequency change. This occurs despite both observers moving at the same speed in opposite directions in the laboratory frame. Their mutual time dilation effects effectively cancel the frequency shift in this scenario. At first glance, the transverse Doppler effect in an accelerated frame reveals a fundamental asymmetry between accelerated and inertial observers. This leads to the question: is it experimentally possible to determine the state of motion of an inertial frame from the perspective of an accelerated frame , using the second-order Doppler redshift and blueshift of moving sources? Specifically, can this effect be used to determine the motion of the Earth relative to a Sun-centered inertial frame? Let us examine the resonance absorption of gamma rays in a Mössbauer rotating disk experiment. Consider a setup where the absorber is located at the center of a disk, which remains at rest relative to the Earth-based frame, while the emitter is positioned on the edge of the disk. The emitter thus moves in a circular path around the absorber with a constant tangential velocity . To keep the mathematical complexity manageable, we restrict our analysis to terms up to second order in and , where is the orbital velocity of the Earth. The frequency of the emitter changes with the velocity in a manner as it suggests itself from the transverse Doppler effect. Indeed, the inner frequency of the emitter depends on its velocity , so that (using Langevin metric in the Earth-based frame) . Expanding this expression and neglecting higher-order terms, we obtain . Since the physical clock is at rest in the earth-based frame, the reading of its proper time is . We therefore expect that radiation emitted with a constant intrinsic frequency will arrive at the absorber with a time-dependent frequency, given by . Because the emitter moves in a circular path, the scalar product varies periodically, leading to periodic fluctuations in the observed frequency at the disk center. Consequently, the resonance absorption measured at the absorber will also vary periodically with the emitter’s position along the disk’s circumference. In the actual experiment, no effect on absorption was observed when the disk was set into rotation. This negative result can be explained by considering that only a portion of the total frequency fluctuation is initially taken into account. A more refined analysis reveals the necessity of including the frequency shift caused by the so-called radial Doppler effect. We analyze a configuration based on perpendicular geometry, where, at first glance, the radial Doppler effect would appear to be absent. However, as discussed in previous chapters, the aberration of light in the Earth-based reference frame implies that the radial component of the emitter’s velocity is given by . As a result, the emitted radiation reaching the center of the disk exhibits a periodically varying frequency shift , which exactly matches the periodic variation in the internal resonance frequency. Thus, the two effects — the intrinsic frequency variation and the radial Doppler shift — cancel each other out, leading to no net observable effect. The second-order Doppler redshift observed in moving atoms is a key prediction of special relativity. The first experimental verification of the transverse Doppler effect was conducted in 1938 by Ives and Stilwell . In their experiment, ionized hydrogen molecules were accelerated in a cathode tube to energies of up to 28 keV. The frequencies of light emitted in both the forward (parallel) and backward (antiparallel) directions relative to the motion of the particles were measured using a stationary spectrograph. Let us now analyze this experiment. We discuss a setup based on a parallel geometry. Radiation emitted by the moving atoms, exhibiting Doppler-shifted spectral lines, was observed alongside radiation from stationary atoms present in the same working volume, which produced an unshifted reference line. By including this reference, Ives and Stilwell effectively circumvented the challenging task of directly measuring the asymmetry between the red- and blue-shifted lines. Instead, they compared both to the stationary reference line, simplifying the analysis. At first glance, the Earth’s orbital motion relative to the Sun can be examined through this type of experiment.484848In the Ives and Stilwell experiment, the effect arising from the Earth’s orbital motion relative to the Sun was within the limits of experimental accuracy. This limitation is due to the high velocity of the emitter (). In this discussion, we present a general analysis of such experiments, independent of specific experimental limitations. Specifically, if the internal frequency of an atom depends on its velocity , then (using the Langevin metric in the Earth-based frame) we have , which implies that the frequency of radiation recorded by a spectrograph should, in principle, depend on the Earth’s orbital velocity . In rotating disk experiments, the observable effect due to the transverse Doppler shift is counteracted by another relativistic effect. For the case of perpendicular geometry, we have previously explained the experimental results by accounting for both the aberration of light and the radial (i.e., classical) Doppler effect. In contrast, in the collinear geometry, the aberration of light is absent, and only the radial Doppler effect needs to be considered. By analyzing the geometry of this configuration, we find that the observed frequency (i.e., the frequency of the radiation reaching the detector) is given by , where, according to the Langevin metric, the coordinate velocity of light in the Earth-based frame is . Thus, just as in the case of the rotating disk experiment, we find that in measurements of the second-order Doppler redshift for moving atoms, the two effects caused by the Earth’s orbital motion effectively cancel each other. As a result, no net observable effect remains. Such cancellations are typically understood to arise from deep underlying principles. Given the series of negative results—like the one obtained by Michelson and Morley—it seems reasonable to assume that the consistent failure of various optical experiments to detect the Earth’s orbital velocity is no coincidence. This has led to the widely accepted view that a fundamental law of nature prevents the determination of through laboratory-based experiments. However, in the present case, no such profound implication appears to be at play. This outcome reflects a specific limitation of second-order optical experiments. As discussed in earlier chapters, we already know that it is possible to detect the Earth’s orbital velocity—without referencing anything external—by using the phenomenon of light aberration. 14.2 Features of Second Order Optical Experiments The principle of relativity has been experimentally confirmed, though only to a limited extent. In second-order experiments, the effects caused by Earth’s orbital motion cancel each other out, leaving no observable impact. This suggests that second-order experiments inherently prevent us from determining Earth’s velocity relative to the Sun through laboratory measurements. Let us examine this claim more closely. The key effects under consideration involve the interference of two light beams. A light source is split into two beams using a suitable apparatus, and their overlap produces an interference pattern. Since both beams originate from the same source, second-order interference experiments cannot detect Earth’s orbital velocity. This is because phase is a four-dimensional invariant—independent of the chosen inertial frame and coordinate system—due to the fundamental geometry of space-time. This discussion shows that special relativity correctly predicts the observed null fringe shift in Earth-based frames. One of the most well-known interference experiments that demonstrates this is the Michelson-Morley experiment. Next, we turn to the second-order Doppler shift in moving atoms. The frequencies of emitted light were measured using a grating spectrograph. Notably, there is no fundamental distinction between a diffraction grating and an interferometer in this context. An intriguing example of this effect is the measurement of resonance absorption of gamma rays in a Mössbauer rotating disk experiment. To illustrate this effect, consider a source positioned at a great distance from a thin absorber plate. We aim to determine the resulting field at a distant point on the opposite side of the absorber. According to electrodynamics, the total electric field is the sum of contributions from the external source and from the fields generated by charges within the absorber. The absorber consists of atoms with nuclei, and when the external field acts on these nuclei, it induces motion in their charges. Moving charges generate additional fields, acting as new radiators. Consequently, the two plane waves that interfere propagate in the same direction, leading to interference between the incident and coherently forward-scattered waves. Our model of the nuclear oscillator includes a damping force. With damping taken into account, the refractive index becomes complex, resulting in destructive interference. Thus, this too is an interference phenomenon. 14.3 Wigner Rotation and the Trouton-Noble Experiment Next, we will explore additional second-order experiments that are not directly related to light propagation but can be understood through general dynamical principles. One of the most significant of these is the Trouton-Noble experiment . The Trouton-Noble experiment aims to detect the rotational motion of a charged parallel-plate capacitor suspended at rest in the Earth’s frame. This motion is used as a means to measure the Earth’s velocity relative to the Sun (i.e., relative to the fixed stars). The fundamental concept behind the experiment can be described as follows. Consider two opposite point charges and in the Earth-based frame; the radius-vector pointing from to be denoted by . If the charges are at rest the force acting upon can be written . As the force acts in the direction the moment of force produced by the pair of charges vanishes, . If the pair of charges is made to move with a velocity in the Sun-centered frame then the positive charge will be under the action of the Coulomb attraction of and also under the influence of the magnetic field . Thus the total force acting upon is given by . Since we find that the moment of force produced by the pair of charges is equal to . Denoting the angle between and by , we find for the absolute value of moment of force . In the actual experiment, a charged condenser was suspended from an elastic string. It was positioned such that the angle ; that is, the line perpendicular to the surface of the condenser plates formed an angle of with the assumed direction of the Earth’s orbital velocity, denoted by the vector . The experiment yielded negative results—no effect proportional to was observed, contrary to expectations. The absence of torque on the capacitor in the Earth-based frame is fully consistent with special relativity and can be explained in a straightforward manner. The electrodynamic equations in the Earth-based frame can be derived by applying a Galilean transformation (with velocity ) to Maxwell’s equations. Assuming the magnetic field in the original frame, the transformed fields are given by , , where is the velocity of the Earth relative to the Sun-based frame. Consequently, the induced magnetic field in the Earth-based frame does not exert any torque on the capacitor, which is at rest in that frame. The presence of a magnetic field gives rise to a magnetic force, which, in turn, exerts a torque on a charge in the Sun-based inertial frame. This leads to an apparent paradox: the mechanical equations governing force and torque appear to differ between inertial frames. Specifically, a torque—and thus a time-dependent change in three-dimensional angular momentum—exists in one inertial frame but not in another that is in relative motion. 494949In , Jackson discusses the apparent paradox that different mechanical equations for force and torque govern the motion of a charged particle in different inertial frames. A similar paradox arises in the standard explanation of the Trouton–Noble experiment. In the usual approach , this paradox stems from the hidden assumption that all inertial frames share a common three-dimensional space. However, we demonstrate that a relativistically accurate description of the Trouton–Noble experiment can be achieved only by taking into account the Wigner rotation effect. We argue that the root of this paradox lies in a flawed assumption: that an Earth-based observer and a Sun-based observer share a common three-dimensional space. According to special relativity, observers following different worldlines (i.e., with different trajectories through spacetime) possess different three-dimensional hypersurfaces of simultaneity—different 3-spaces. The standard treatment of the Trouton-Noble experiment implicitly assumes that the coordinate axes of the moving Earth-based observer are parallel to the axes of the stationary Sun-based observer. We demonstrate that a consistent explanation, fully aligned with special relativity, emerges only when the equations of motion are expressed in covariant form. This approach properly accounts for the relativistic transformation of coordinate systems and resolves the apparent contradiction. The key to resolving the paradox commonly associated with explanations of the Trouton–Noble experiment lies in recognizing that spatial measurements made by an Earth-based observer involve a mixing of space and time coordinates, as viewed from the Sun-based (inertial) frame. While it may seem intuitive to assume that the angle between the velocity vector and the radius vector in the Earth frame should match the angle between and in the Sun frame, this is not actually the case. The apparent rotation of the radius vector about the velocity vector is intrinsically linked to the phenomenon of length contraction. Consider a rod aligned along the -axis in the Earth-based frame. Under a Lorentz boost along the -axis with velocity , the rod appears contracted in the Sun-based frame to a length of , where is the rod’s proper length in the Earth frame and is the Lorentz factor. The relationship between the angle in the Earth frame and in the Sun frame is given by . Assuming , this leads to the expression , where represents the rotation angle. This angle is further related to the system parameters by . Consequently, the torque (moment of force) exerted by a pair of charges in the Earth-based frame is found to be . But we’re still not done! We need to apply an additional correction to the moment of force in the Earth-based frame. A more careful analysis reveals that we must account for the change in the perpendicular component of the velocity, , due to the Wigner rotation. Since the force acts along the direction of , there exists a component of the charge’s acceleration that is perpendicular to the velocity . This component results in a rotation of the Earth-based (proper) frame axes as seen from the Sun-based frame. The rate at which the Wigner rotation occurs is given by: . The direction of this rotation of the Earth-based frame (in the Sun-based frame) is the same as the direction of the rotation of the velocity vector in the Sun-based frame. We already have all the expressions we need. For the rate of change of the angle , we find: , where is the mass of the charge. Thus, the second correction to the moment of force is: . As a result, the net moment of force produced by a pair of charges in the Earth-based frame vanishes. 14.4 Time Dilation in Moving Atomic Clock The next topic we need to address is the time dilation effect. We have already discussed this subject in Chapter 12. Here’s a brief summary of our findings: We established that the slowing down of a physical clock that is accelerated with respect to the fixed stars does not depend on the choice of the reference frame in which the effect is measured. In other words, the time dilation effect is absolute, not relative. Due to the asymmetry between the initial inertial and accelerated frames, special relativity makes a striking prediction. Suppose frame is at rest relative to the fixed stars, and frame is accelerated with respect to until it reaches a speed . Now consider two clocks in frame both symmetrically accelerated to speed , but moving in opposite directions. Surprisingly, the time dilation experienced by these two clocks in is not the same. This discrepancy arises because the clocks have different acceleration histories relative to the fixed stars. A natural question arises: is it possible to experimentally determine the state of motion of the inertial frame from the perspective of the accelerated frame , using the time dilation effect observed in a moving (atomic) reference clock? At first glance, it may seem that the Earth’s orbital motion could be detected through time dilation experiments. While this is theoretically possible, a more detailed analysis is required to properly address the question. The proper time interval can be associated with phenomena such as particle lifetimes, atomic transition periods, and nuclear half-lives. The key point, however, is that this interval is entirely determined by the motion of a clock within its initial inertial frame. In 1971, Hafele and Keating conducted a landmark experiment in which four atomic clocks were synchronized with a laboratory reference clock and then flown around the Earth aboard commercial aircraft—two traveling eastward and two westward. 505050This experiment provided the first experimental confirmation of time dilation as predicted by special relativity, using atomic clocks in circular motion. An important follow-up came from the Global Positioning System (GPS), which also involves relativistic effects. GPS satellites, numbering around 80, orbit the Earth at velocities of approximately 4 km/s. The system demonstrates that satellite clocks tick more slowly due to their relative motion, with a measured discrepancy of about 7.1 microseconds per day. Interestingly, while there is a noticeable time difference between the clocks on Earth and those in orbit, no asymmetries are observed among the satellites themselves. This suggests that relative velocity alone does not account for time dilation. Furthermore, the experiment revealed no effect of Earth’s orbital velocity on clock rates, a result attributed to the fact that both the satellites and Earth-based clocks share a common orbital path around the Sun, rendering orbital effects unobservable in this context [56, 57]. Upon return, the airborne clocks were compared with the laboratory reference clock. The results showed that the clocks carried on the planes had run slower than those left stationary on Earth. The observed discrepancies matched the predictions of special relativity, confirming the role of relative acceleration in time dilation. Notably, the experiment found no measurable influence of Earth’s orbital motion on clock rates, reinforcing the idea that time dilation effects are tied to local motion relative to the chosen inertial frame. The null result of the experiment described above can be understood by considering that the setup involves a circular path geometry. A more careful analysis reveals that it is essential to account for the fact that the moving clocks are ultimately compared to a reference clock located at the same point and at rest relative to the Earth. The moving clock travels around the reference clock along a circular path with velocity . We will analyze the effect by expanding terms up to second order in and , where is the Earth’s orbital velocity. The inner rate of the moving clock depends on velocity , so that . Since the reference clock is at rest in the Earth-based frame, the reading of its proper time is . Analyzing the geometry of the situation, we see that for motion along a closed circular path, the cross term averages out over a complete cycle. Specifically, the integral of the mixed term vanishes: . Therefore, no observable effect from the Earth’s orbital velocity remains, which explains the null result. 15 The Principle of Relativity and Modern Cosmology In the preceding discussion, we explored the fundamental concepts necessary for understanding the phenomenon of the aberration of light. At first glance, the source-observer asymmetry associated with stellar aberration appears to contradict the principle of relativity. This principle, first introduced by Poincaré—who coined the term—states that an observer in uniform motion relative to the fixed stars cannot, by any internal measurement, detect this motion, provided they do not observe external celestial objects. Now, let us examine the stellar aberration experiment more closely. One might argue that in the case of stellar aberration, the key distinction is that the accelerated (Earth-based) observer does, in fact, observe external celestial objects—the stars. However, the motion of the stars relative to Earth is never accompanied by any aberration. The aberration shift, as inferred from astronomical observations, remains even when a star moves with the same velocity as Earth. Thus, when considering Earth-based observations of stellar aberration, we cannot rely solely on the notion of ”looking outside to the stars.” In Chapter 10, we introduced a simple scaling model for stellar aberration and derived a condition for optical similarity between the aberration of light from a distant star and that from an incoherent Earth-based source. This analysis revealed that even without observing external celestial objects, it is possible to determine Earth’s orbital velocity around the Sun through aberration of light measurements. There is no fundamental conflict between the structure of special relativity and the aberration of light phenomena. As discussed in previous chapters, special relativity does not require the principle of the irrelevancy of velocity formulated earlier. The principle of special relativity applies to physical laws, not to physical facts. It asserts that the same laws must hold in all inertial frames, which we interpret to mean that these laws should be expressed by equations that retain the same form in all inertial frames. In special relativity, the transformations preserving this form-invariance between inertial frames are the Lorentz transformations. 515151In the special theory of relativity the form-invariant transformations between inertial frames are Lorentz transformations . For a general discussion about the Principle of Relativity we suggest reading the paper . According to the special principle of relativity, all inertial frames are equivalent concerning physical laws, but not necessarily concerning physical facts. The existence of absolute acceleration in nature implies the existence of absolute velocity. A closer examination of the problem of time dilation in an accelerated frame reveals the fundamental asymmetry between an accelerated frame and an initial inertial frame (i.e., one without a history of acceleration). The proper time of any particle moving arbitrarily within the initial inertial frame always runs slower than the physical time of that frame. This is a fundamental characteristic of pseudo-Euclidean space-time. Uniform motion is not relative in this framework, allowing us to determine a particle’s absolute velocity within the initial inertial frame. The formalism of classical physics relies on the structure of absolute space, which, in turn, must be associated with some underlying ”substance”—the ether—which serves as an absolute rest frame. Could space-time itself be the new ether? It shares similar properties, notably providing an ”absolute rest reference frame.” The key distinction between the initial inertial frame and Newtonian absolute space lies in the shift introduced by special relativity: we now conceive of a unified space-time model, without an inherent separation between space and time. While the initial inertial frame is not physically identical to absolute space, it represents the concept of absolute space-time. The principle of relativity has a long history in physics. In Newtonian mechanics, physical laws remain invariant under Galilean transformations. However, from a mathematical perspective, the Lorentz transformation is fundamentally different from the Galilean transformation. Notably, Lorentz boosts alone do not form a group, whereas the set of Galilean boosts does. This reflects the essence of Galilean relativity: all inertial frames are equivalent, meaning that no one frame is preferred over another. The underlying reason for this symmetry is that the equations of motion in Newtonian mechanics do not explicitly depend on velocity, ensuring that a system’s internal dynamics remain unchanged under a Galilean boost. Historically, Galileo’s relativity principle was mistakenly assumed to apply universally across all of physics. However, the principle of relativity does not hold across the entire domain of Lorentz-covariant physical laws, and Lorentz covariance itself is not a fundamental symmetry of nature. Since the space-time metric tensor is a continuous quantity, an accelerated inertial frame is connected to its initial state by a Galilean transformation rather than a Lorentz transformation. This distinction is particularly evident in electrodynamics: Galilean transformations do not preserve the form-invariance of Maxwell’s equations under a change of inertial frames. The reason is that electrodynamic equations explicitly depend on velocity, leading to the breakdown of Galilean relativity in this context. According to special relativity, a fundamental difference exists between an accelerated inertial frame (relative to the fixed stars) and an inertial frame without an acceleration history. This distinction is closely linked to inertial forces and forms the core idea of the equivalence principle. Understanding the relationship between inertia and gravity remains an open question in fundamental physics. Now we want to discuss a serious difficulty associated with the ideas of modern cosmology. The difficulty we speak of is associated with the concept of source-observer asymmetry when applied to the motion relative to the cosmic microwave background (CMB). One may say that perhaps there is no use worrying about these difficulties since there are so many things about the universe that we still don’t understand. We live in a universe governed by the unknown. Dark energy and dark matter together make up nearly 96 percent of the universe’s total energy-mass, yet their true nature remains a mystery. However, it is essential to recognize that the foundational assumptions of modern cosmology are only approximations. As our understanding deepens, these assumptions will inevitably evolve. We will not delve into the principles of modern cosmology at this time but will assume their validity and focus on discussing some of their implications. According to the cosmological principle, the universe should appear isotropic — lacking any preferred direction — to a comoving observer with no peculiar motion relative to the cosmic fluid of the expanding universe. However, if such an observer has a peculiar motion, it can introduce a dipole anisotropy in the observed properties of certain objects. This anisotropy can, in turn, be used to infer the observer’s peculiar velocity. A well-known example of this phenomenon is the anisotropy observed in the cosmic microwave background (CMB) radiation. The CMB dipole is almost universally attributed to kinematic effects, specifically the Doppler shift caused by the relative motion between the Earth (the observer) and the reference frame in which the CMB appears nearly isotropic. By subtracting this dipole component, the CMB rest frame is established as the fundamental reference frame of the universe. The observed dipole indicates that the solar system is moving at 370 km/s relative to the observed universe in the direction of galactic longitude and latitude . This is quite far from the galactic rotation direction of 250 km/s towards and . The motion relative to the cosmic microwave background results from the sum of many components of velocity due to the gravitational attraction of various mass concentrations. The existence of clusters and super clusters of galaxies and our motion is a natural consequence of the large-scale organization of matter. The peculiar velocity consists of the five vector contributions: the motion of the earth around the sun (30 km/s), the hypothetical circular motion around our galaxy (250 km/s), the motion of our galaxy in the local group, and the motion of the local group with respect to the cosmic microwave background. The origin of the velocity of the local group is still uncertain and has been under discussion over the past two decades. This peculiar velocity is believed to be generated by the spatial inhomogeneities of mass (mainly dark matter) distribution in nearly large-scale structures. The velocity of the local group with respect to the cosmic microwave background is estimated to be 500 km/s. Now let us see how aberration of light varies with the speed relative to the rest frame of the universe. Consider first the textbook explanation. It is very easy to understand how the aberration of light effect comes about from the point of view of the conventional aberration of light theory. The existence of an aberration of the line of sight by motion can be recognized by considering an observer in a car driving through a rainstorm: raindrops falling vertically appear to be oblique. An important application of the Galilean velocity transformation law is provided by stellar aberration, the change in the apparent direction of a star caused by the Earth’s motion around the Sun. The apparent positions of all fixed stars are thus always a little displaced in the direction of the Earth’s motion at that moment, and hence describe a small elliptical figure during the annual revolution of the Earth around the Sun. What about the motion relative to the CMB rest frame? According to textbooks, if the Earth’s motion were uniform, the aberration effect would be undetectable since the ”true” direction of the star is unknown. Indeed, who can say where a given star should be? On the other hand, the true direction of an Earth-based source is known. But, according to conventional theory, the aberration of light phenomenon does not exist in an aberration of light experiments using an Earth-based light source. We are now in a position to understand the nature of our difficulty with the motion relative to the CMB. It is believed that the dipole anisotropy is produced by the Doppler effect due to acceleration (during the billions of years) of the solar system with respect to the rest frame of the universe. This acceleration is believed to be generated by the spatial inhomogeneities of (Dark matter) mass distribution in nearly large-scale structures. A correct solution of the aberration problem in the Earth-based frame requires the use of metric tensor even in first-order experiments since the crossed term in the Langevin metric plays a fundamental role in the non-inertial kinematics of a light ( or relativistic particle ) beam produced by the Earth-based source. According to the relativistic theory of aberration presented in this book, the proper rotation of the Earth on its axis should produce a corresponding shift of the image. The aberration shift will depend also upon the value of , the component of the solar system velocity (relative to the CMB) perpendicular to the Earth’s rotation axis. The rotation of the Earth produces aberration in an amount larger enough to be taken into account in precise observation work using an electron microscope as an Earth-based particle source. Experimental results show (see Chapter 10) that the image shift is quite close to the theoretical prediction for the (30 km/s) orbital velocity and clearly indicate that the signal associated with motion (370 km/s) relative to the CMB does not exist. The simplest explanation is that the CMB dipole might be of non-kinematical origin. More recent observations by astronomers also cast doubt on the CMB dipole, being the ultimate representative of the solar peculiar velocity. In recent years observations have emerged hinting at an anisotropic universe. The discovery of the preferred direction in the universe was serendipitous. Until very recently the velocity of the solar system in the rest frame of the universe is inferred from CMB temperature dipole anisotropy. Obviously, an independent measurement of this velocity is needed to fully establish the kinematical origin of the dipole. Another such quantity that could be employed to look for departures from isotropy is the angular distribution of distant radio sources in the sky. This could provide an independent check on the interpretation of CMB dipole. The radio data clearly indicate that significantly larger dipoles exist in the rest frame of the radio galaxies. While the velocity of the solar system inferred from the CMB temperature dipole anisotropy is 370 km/s, the radio dipole measurements find the speed of motion to be around 1000 km/s (i.e. to be around three times larger than that of CMB). From the Hubble diagram of quasars, the motion of the solar system is derived, which is out to be the largest value ever found, km/s. 525252Recent observations of the cosmic dipole anisotropies is revolutionized our understanding of the universe. The various dipoles, including CMB dipole, all pointing along the same direction, suggest a preferred direction in the universe, raising thereby uncomfortable questions about the cosmological principle, the basis of the standard model in modern cosmology [60, 61, 62]. On the other hand, a common direction for all these dipoles, determined from completely independent surveys by different groups employing different techniques, indicates that these dipoles are not resulting from some systematics in the observations or the data analysis, but could instead suggest an inherent anisotropy. This is totally unexpected in a standard model of the universe. We have a preferred direction, aligned with the CMB dipole, in the universe. That is, going to the CMB rest frame, we see an anisotropic background. There is a difficulty with such some sort of an ”axis” of the universe which, in turn, would be against the cosmological principle. Three independent dipole vectors pointing along the same particular direction could imply an anisotropic universe, violating the cosmological principle, a cornerstone of modern cosmology. Let us be conservative and say that there are two kinds of the Earth velocity. The total velocity with respect to the initial (i.e. privilege ) inertial frame could be the sum of the orbital velocity and the velocity of the Sun. The velocity of the Sun consists the two contributions: the circular motion around our galaxy and the motion of our galaxy. In Earth-based experiments where we measure the Earth’s velocity with respect to the privileged frame by seeing the aberration of light (or particles), we are measuring the recorded (in the accelerational history) velocity. So the velocity with respect to the privilege frame consists of two contributions: a recorded motion plus an unrecorded motion. We know that there is definitely a recorded (orbital) motion. It is therefore impossible to get all the Earth’s velocity to be recorded in the accelerational history in the way we hoped. Clearly something else has to be added. Let’s take a closer look at why the Earth moves relative to the initial inertial frame. For the Earth to be in motion, a force must have acted upon it at some point. To deepen our understanding, it is useful to examine the origins of these forces in greater detail. We must say immediately that orbital motion is the result of non-gravitational forces. It is well known that a full 98 percent of all the angular momentum in the solar system is concentrated in the planets, yet a staggering 99.8 percent of all the mass in our solar system is in our sun. Perhaps the first scientifically respectable theory of the origin of the solar system was given by Hoyle (1960) . He invoked the action of the magnetic field to transfer angular momentum from the central body, the Sun, to the ejected matter which eventually formed the planets. In contrast, the unrecorded motion around our galactic and the motion of our galactic is the result of the action of gravity. We began by discussing the gravitational interaction between the Earth and the Sun. Our confidence in the theory of gravity is so strong that we use it to describe the forces acting between entire galaxies. However, we may be making too great an extrapolation, applying our limited understanding of gravity to cosmic scales. Perhaps the entire difficulty is that a modification of gravity may be responsible for unrecorded velocities. Given the many unresolved mysteries surrounding dark matter, dark energy, and cosmic anisotropy, a deeper exploration of fundamental physics is essential to uncovering the true nature of the universe. 16 Relativistic Particle Dynamics 16.1 Manifestly Covariant Formulation The equations of dynamics can be formulated as tensor equations within Minkowski space-time. Upon selecting a coordinate system, these equations may be expressed in terms of components rather than abstract geometric objects. Leveraging the geometric structure of Minkowski space-time, one can identify the class of inertial frames and, for any such frame, adopt a Lorentz frame with orthonormal basis vectors. In any Lorentz coordinate system, the law of motion takes the form: \| \| \| --- \| \| \| (21) \| where here the particle’s mass and charge are denoted by and respectively. The electromagnetic field is described by a second-rank, antisymmetric tensor with components . The coordinate-independent proper time is a parameter describing the evolution of the physical system under the relativistic laws of motion, Eq. (21). The covariant equation of motion for a relativistic charged particle, subject to the four-force , as given in Eq. (21), represents a relativistic extension of Newton’s second law. While the classical form remain valid in the particle’s instantaneous comoving Lorentz frame, the relativistic formulation embeds these three spatial equations into the four-dimensional Minkowski spacetime. 535353The notion of embedding relies on the fact that Newton’s second law can always be applied in any Lorentz frame in which the particle is instantaneously at rest . That is, if a comoving Lorentz frame is defined at a given instant, the particle’s evolution can be accurately predicted in this frame over an infinitesimal time interval. Geometrically, Newton’s law holds rigorously on a hyperplane orthogonal to the particle’s world line. As the particle moves, the hyperplane—and its normal vector —tilts accordingly along the world line. To formalize this embedding, one introduces a projection operator that continuously projects vectors in Minkowski space onto hyperplanes orthogonal to the world line. This operator is given by . The presence of highlights that only three of the four components of the covariant equation are independent. The immediate generalization of to an arbitrary Lorentz frame is given by Eq. (21), as can be verified by reducing the equation to the particle’s rest frame. In Lorentz coordinates, the four-velocity satisfies the kinematic constraint . Due to this condition, the four-dimensional equation of motion, Eq. (21), contains only three independent components. Using the explicit form of the Lorentz force, one can show that Eq. (21) automatically preserves the constraint , as required. To demonstrate this, we take the scalar product of both sides of the equation of motion with . Exploiting the antisymmetry of the electromagnetic field tensor, , we find . This result implies that the quantity is conserved, since . 16.2 Conventional Particle Tracking Having expressed the equation of motion in 4-vector form, Eq. (21), and determined the components of the 4-force, we have, on the one hand, ensured compliance with the principle of relativity and, on the other hand, obtained a complete set of four equations describing particle motion. This represents a covariant, relativistic generalization of Newton’s three-dimensional equation of motion, formulated with the particle’s proper time as the evolution parameter. We now aim to describe the motion of a particle in the Lorentz laboratory frame, using the lab time as the evolution parameter. To begin, we consider the standard treatment found in textbooks. Let us determine the spatial components (i.e., the first three components) of the four-force. For this purpose, we examine the spatial part of the dynamical equation, Eq. (21): . The prefactor appears due to the change in the evolution parameter from the proper time —natural for covariant formulations—to the lab frame time , which is necessary for introducing the conventional three-force vector . In this context, the spatial part of the four-force is related to the three-force by . Explicitly, the relativistic form of the three-force is given by \| \| \| --- \| \| \| (22) \| The time component of the four-force equation is \| \| \| --- \| \| \| (23) \| The evolution of the particle is governed by these four equations, together with the invariant constraint \| \| \| --- \| \| \| (24) \| According to the non-covariant (3+1) approach we seek the initial value solution to these equations. Using the explicit expression for Lorentz force we find that the three equations Eq.(22) automatically imply the constraint Eq.(24), once this is satisfied initially at . In this (3+1) formalism, the four-dimensional equations of motion are decomposed into three spatial and one temporal component, effectively eliminating any mixing between space and time in the dynamical equation Eq.(21). This approach to relativistic particle dynamics relies on the use of three independent equations of motion Eq.(22) for three independent coordinates and velocities, ”independent” meaning that equation Eq.(23) (and constraint Eq.(24)) are automatically satisfied. One might expect that the particle’s trajectory in the lab frame, denoted as and derived from the previous reasoning, should correspond to the covariant trajectory . However, identifying them in this way leads to paradoxical results. Specifically, the trajectory fails to account for relativistic kinematic effects. In the non-covariant (3+1) approach, the dynamics in the lab frame are formulated without reference to Lorentz transformations. As a result, the motion of a particle in a constant magnetic field, when described from the lab frame, appears identical to its Newtonian counterpart—relativistic effects are absent in this formulation. In conventional particle tracking, the lab-frame trajectory can be interpreted as arising from a sequence of Galilean boosts, which follow the motion of the particle as it accelerates under the influence of the magnetic field. The standard Galilean velocity addition rule is applied to determine the boosts at each instant, tracing the particle’s progression along its curved path. The old kinematics is especially surprising, because we are based on the use of the covariant approach. Where does it come from? The previous commonly accepted derivation of the equations for the particle motion in the three-dimensional space from the covariant equation Eq.(21) includes one delicate point. In Eq.(22) and Eq.(23) the restriction has already been imposed. One might well wonder why because in the accepted covariant approach, the solution of the dynamics problem for the momentum in the lab frame makes no reference to the three-dimensional velocity. Equation Eq.(21) tells us that the force is the rate of change of the momentum , but does not tell us how momentum varies with speed. The four-velocity cannot be decomposed into when we deal with a particle accelerating along a curved trajectory in the Lorentz lab frame. Actually, the decomposition comes from the relation . In other words, the presentation of the time component as the relation between the and lab time is based on the hidden assumption that the type of clock synchronization, which provides the time coordinate in the lab frame, is based on the use of the absolute time convention. It is important to stress at this point that the situation when only one clock in the comoving frame is involved in dynamics cannot be realized. The Newton law can be written down in the proper frame only when a space-time coordinate system has been specified. The type of clock synchronization which provides the time coordinate in the Newton equation has never been discussed in textbooks. It is clear that without an answer to the question about the method of distant clock synchronization used, not only the concept of acceleration but also the dynamics law has no physical meaning. A proper frame approach to relativistic particle dynamics is forcefully based on a definite (Einstein) coordinatization assumption. After this, the dynamics theory states that the equation of motion in the proper frame is . It should be stressed that in the case of velocity increment , we also deal with the distant events. It can be said with some abuse of language that a 3-velocity vector is always a ”spatially extended” object. Let us return to our consideration of the relation between the and lab time . The calculation carried out in the case of a spatially extended object shows that the temporal coincidence of two distant events has absolute character: implies . It should be noted that usually the notation ”” arose from the proper time. Now let us recall the standard concept of the object’s proper time. Let a point-like object move uniformly and rectilinearly relative to the lab frame . The proper frame can be fixed to the moving object. The object is at rest in this frame, so that events happening with this object are registered by one clock. This clock counts the proper time at the point where the object is located. As we already know, the proper time on moving object flows slower than the time . This phenomenon was called time dilation. The proper time can also be introduced for a point particle moving with acceleration. This standard interpretation of the proper time has nothing to do with dynamics and one may wonder where this contradiction existing between the name and the content of time in dynamic law comes from. To avoid being overly abstract for too long, we have introduced a concrete example. Consider the case where the velocity increment is not aligned with the direction of uniform motion. For instance, imagine a particle moving ”upward” with a velocity increment of in the frame , while the frame itself is moving ”horizontally” with velocity in the lab frame. According to standard textbook treatments, one would apply the relation , leading to the familiar result: . However, this approach can lead to paradoxical outcomes. Specifically, it implies a trajectory that does not appropriately mix spatial and temporal components. One of the key conclusions drawn from the discussion in Chapter 7 is that special relativity does not allow for an absolute notion of simultaneity. Consequently, there is no well-defined concept of an instantaneous three-dimensional space. The standard expression for the transformation of a velocity increment, , rests on a hidden assumption—that the -axis and -axis remain parallel. In reality, the Lorentz transformation induces a Wigner rotation, which causes these axes to become oblique. This insight immediately reveals a fundamental distinction between non-covariant and covariant descriptions of point particle trajectories. (We will explore velocity transformations in more detail in Section 16.5.) 16.3 Incorrect Expansion of the Relation for Arbitrary Motion Textbook authors are fundamentally mistaken in their widespread belief about the general validity of the standard momentum–velocity relation. According to the theory of relativity, the equation does not hold for particles moving along curved trajectories in the Lorentz lab frame. Many experts who have studied relativity through conventional textbooks may find this claim counterintuitive or even unsettling at first. But how can such an unconventional momentum–velocity relation arise? We know that the components of the four-momentum transform between Lorentz frames in the same way as the components of the four-position vector . However, when translating from four-vector notation back to the familiar three-dimensional velocity vector , defined via , subtleties emerge. Surprises are to be expected in this transition, especially when dealing with non-rectilinear motion. It is well established that for rectilinear accelerated motion, the standard momentum–velocity relation does hold. In such cases, the combination of the usual relation and the covariant velocity transformation law (Einstein’s velocity addition) is consistent with the covariant momentum transformation. Both the non-covariant and covariant approaches yield the same trajectory in the Lorentz lab frame. We can see why by examining the transformation of the 3-velocity in the theory of relativity. For a rectilinear motion, this transformation is performed as . The relativistic factor is given by: . The new momentum is then simply times the above expression. But we want to express the new momentum in terms of the primed momentum and energy, and we note that . Thus, for a rectilinear motion, the combination of Einstein’s addition law for parallel velocities and the usual momentum-velocity relation is consistent with the covariant momentum transformation. It is well known that collinear Lorentz boosts, like their Galilean counterparts, form a commutative group. This implies that the result of applying successive Lorentz boosts in the same direction is independent of the order in which they are applied. However, Lorentz boosts in different directions do not commute and do not form a group. In contrast, Galilean boosts always form a group, regardless of direction. A key relativistic effect arises when composing non-collinear Lorentz boosts: the result is not simply another boost but a combination of a boost and a spatial rotation, known as the Wigner rotation. This phenomenon has no analogue in non-covariant physics. One of its consequences is the emergence of a non-trivial relationship between momentum and velocity: , which also lacks a non-covariant counterpart. Relativity further reveals that this unusual momentum-velocity relationship is tied to motion along curved trajectories. In such cases, the concept of a single, shared ”ordinary space” breaks down, leading to a fundamental distinction between covariant and non-covariant particle trajectories. 16.4 Standard Integration of the 4D Covariant Equation of Motion Attempts to solve the dynamical equation Eq. (21) in a manifestly covariant form can be found in the literature. However, the resulting trajectory does not account for relativistic kinematic effects. Consequently, it cannot be identified with , even though it may initially appear to follow a covariant prescription. We begin by analyzing the textbook’s treatment of the integration of the covariant equation of motion. Consider, for example, the motion of a charged particle in a given electromagnetic field. Of particular practical importance is the simplest case: a uniform electromagnetic field, meaning that the field tensor remains constant throughout the entire space-time region of interest. Specifically, we focus on the motion of a particle in a constant, homogeneous magnetic field, represented by the tensor components where and are orthonormal space like basis vectors , . In the lab frame of reference where is taken as the time axis, and and are space vectors the field is indeed purely magnetic, of magnitude and parallel to the axis. Let us set the initial four-velocity , where is the initial particle’s velocity relative to the lab observer along the axis at the instant , and . The components of the equation of motion are then , , , . We seek the initial value solution to these equations as done in the existing literature 545454Textbook authors, such as those in [5, 64, 65], conclude from the covariant equations that a charged particle in a constant magnetic field moves in a uniform circular trajectory. This analysis does not account for relativistic kinematics, and instead, the Galilean vector law of velocity addition is applied. The non-relativistic kinematics are derived from the relation . It is only after making this substitution for hat the authors obtain the familiar formula for the non-covariant trajectory.. A distinctive aspect of relativistic dynamics is the presence of constraints. The evolution of the particle is subject to , but also to the constraint . However, such a condition can be weakened requiring its validity at certain values of only, let us say initially, at . Therefore, if vanishes initially, i.e. , then at any . In other words, the differential Lorentz-force equation implies the constraint once this is satisfied initially. Integrating with respect to the proper time we have where . We see that is constant with time, meaning that the energy of a charged particle moving in a constant magnetic field is constant. After two successive integrations we have where . This enables us to find the time dependence of the particle’s position since . From this solution, we conclude that the motion of a charged particle in a constant magnetic field is indeed uniform circular motion, consistent with classical expectations, yet derived within the fully covariant relativistic framework. One might expect that the particle’s trajectory in the lab frame, derived from the previous reasoning , should be identified with . However, the trajectory does not include relativistic kinematics effects. We have found that the standard integration of the four-dimensional covariant equation of motion, as shown in Eq. (21), yields a particle trajectory that is mathematically identical to the one derived from Newtonian kinematics and dynamics. Specifically, the trajectory of the electron fails to incorporate relativistic effects, and instead, the Galilean law of velocity addition is applied. It seems that we must have made a mistake. Notice that we are using textbooks to solve covariant equations of motion. While we did not make a computational error in our integrations, we did make a conceptual one. First, we should clarify that there is no issue with the initial integration of Eq. (21) with respect to proper time from the initial conditions. This yields the four-momentum, which has a well-defined, objective meaning—it is convention-invariant. The key point to recognize is that the concept of velocity is only introduced in the second step of the integration. However, in the accepted covariant approach, the solution to the dynamics problem for momentum in the lab frame makes no reference to three-dimensional velocity. The initial condition we used is , which includes and , not as velocities but as the time and space components of the initial four-momentum. The three-dimensional trajectory and velocity, which are convention-dependent, emerge only after the second integration step. So, where does the old kinematics come from? The second integration is performed using the relation . It is only after substituting for that we recover the conventional (non-covariant) expression for the trajectory of an electron in a constant magnetic field. We should expect results similar to those obtained in the case of the (3+1) non-covariant particle tracking. In fact, based on the structure of the four components of the equation of motion (Eq. 21), we can derive an equivalent mathematical formulation of the dynamical problem. As previously mentioned, the evolution of the particle in the lab frame is subject to a constraint. This implies that the dynamics are described by only three independent equations of motion. It is easy to see from the initial set of four equations, , , , , that the presentation of the time component simply as the relation between proper time and coordinate time is just a simple parametrization that yields the corrected Newton’s equation Eq.(22) as another equivalent form of these four equations in terms of absolute time instead of proper time of the particle. This approach to solving the dynamical equations from the initial conditions relies on three independent spatial coordinates and velocities, without constraints, and is closely tied to classical kinematics. The presentation of the time component simply as the relation between proper time and coordinate time is based on the hidden assumption that the type of clock synchronization, which provides the time coordinate in the lab frame, is based on the use of the absolute time convention. 16.5 Convention-Invariant Particle Tracking So far, we have described the motion of a particle in three-dimensional space using the vector-valued function . This function defines a prescribed curve (or path) along which the particle moves. The motion along this path is parameterized by , where represents a particular parameter—specifically, the arc length in the case of interest. It is important to distinguish between the concepts of path and trajectory. A trajectory provides a complete description of a particle’s motion by specifying its position as a function of time. In contrast, a path is a purely geometric construct, independent of time. Paths can take various forms, such as straight lines, circular arcs, or helical curves. If we connect the origin of a Cartesian coordinate system to a point on the path, the resulting vector is called the position vector, denoted by . The derivative of this vector with respect to arc length, , yields the tangent vector to the curve at that point. The direction of this tangent vector is determined by the orientation of the curve as traced by increasing arc length. As established in Chapter 2, the path has an exact, objective meaning—it is invariant under changes in convention. Similarly, the components of the four-momentum vector are also convention-invariant. In contrast, the trajectory —as measured in a laboratory frame—is convention-dependent, reflecting the conventionality inherent in the definition of velocity. As such, it lacks the same level of objective meaning as the path . We now describe how to determine the position vector in the convention-invariant particle tracking approach. Consider a particle moving in a uniform magnetic field, with no electric field present. Using Eq. (21), we obtain: \| \| \| --- \| \| \| (25) \| Since and the constraint holds, it follows that , where . The unit vector satisfies: , where is the differential path length. From this we derive: \| \| \| --- \| \| \| (26) \| These equations are identical to those obtained using the non-covariant particle tracking approach. Consequently, , as expected. Both formulations describe the same physical reality. Since the curvature radius of the trajectory in a magnetic field—and therefore the three-momentum—has an objective (i.e., convention-invariant) meaning, both approaches yield the same physical results. 16.6 Einstein’s Velocities Addition Vs. Covariant Addition In conventional particle tracking, the trajectory of a particle, denoted by , is often described from the laboratory frame as the result of successive Galilean boosts. These boosts account for the motion of a particle accelerating in a constant magnetic field. The classical Galilean rule for the addition of velocities is used to determine the appropriate boost at each instant, effectively tracking the particle’s motion along its curved trajectory. However, we cannot rely on Newtonian kinematics for mechanics while simultaneously applying Einsteinian kinematics to electrodynamics. A consistent and accurate description requires solving the dynamical equations in their covariant form. Only this approach ensures the correct coupling between Maxwell’s equations and the particle trajectories observed in the laboratory frame. In this covariant framework, the trajectory is understood, from the lab frame, as the result of successive Lorentz transformations. This introduces relativistic kinematic effects that are absent in the classical treatment. Let us apply our algorithm for reconstructing to a few examples to observe its performance in practice. To begin, we revisit the standard textbook derivation of velocity transformations between reference frames. Assume a frame moves relative to the system with velocity along the axis. Let be the vector of the particle velocity in the system and the velocity vector of the same particle in the system. From Lorentz transformation we have , , , , where . Dividing the spatial components by the time component yields the relativistic velocity transformation formulas: , , . These expressions describe the relativistic law of composition of velocities and form the basis of velocity transformations in special relativity, as presented in standard texts. We begin with the simple case of relativistically combining perpendicular velocities. While there is no length contraction in directions perpendicular to motion, time dilation still affects the observed velocities. As a result, in the frame , the particle’s velocity components are , . It is important to clarify what is meant by ”perpendicular” in this context. Two velocity vectors can be meaningfully said to be perpendicular only if they are defined within the same reference frame. Therefore, when textbook authors assert that and are perpendicular, what they actually rely on—often implicitly—is the assumption that the spatial axes and are parallel. In other words, they assume that the observers in frames and share a common 3-space. This is a subtle but important misconception. This is a good point to make some general remarks about Einstein’s velocity addition. It should be noted that presented above the commonly accepted derivation of velocity composition does not follow from the composition of Lorentz boosts. In particular, second boost from to particle proper frame does not Lorentz boost. Einstein’s velocity addition is based on the relation , which implicitly assumes an absolute time convention for second boost. However, it does not account for the interplay between transverse positions and time , which is inherent in Lorentz boost matrices. In contrast, every Lorentz boost represents a transformation between reference frames within a Lorentzian framework (i.e., under Lorentz coordinatization including also transverse direction). While Lorentz boosts intertwine space and time coordinates, Einstein’s velocity addition relies on the assumption of a shared three-dimensional space across all inertial frames. The consequence of this difference is the Wigner rotation accompanying the composition of non-parallel 3D velocities. One of the consequences of the non-commutativity of non-collinear Lorentz boosts is the absence of a shared, common notion of ordinary space. Suppose that in the Lorentz frame , a traveler observes a particle moving along the axis. In other words, the particle has a velocity component in this frame, while the frame itself is moving with velocity along the axis relative to initial Lorentz frame . The proper frame is rotated by an angle relative to frame . The coordinate axes of the proper frame remain parallel to those of frame . From the perspective of frame , the coordinate axes of frame are also rotated by the angle with respect to those of frame . Let us consider the simple case where the boost velocity approaches the speed of light (i.e., the ultrarelativistic limit), while the transverse velocity remains small. This means we are taking the limit where , treating as a small parameter and keeping terms up to second order. In this regime, the axes of the frame appear rotated with respect to those of by an angle . Notably, this rotation is in the same direction as the rotation of the particle’s velocity vector in . We now consider the physical consequences of this rotation. First, note that the transverse velocity component acquires a projection onto the -axis when viewed from frame . As a result, the component of the particle’s velocity along the -axis in frame is slightly reduced from . Our goal is to calculate the decrement in the horizontal velocity in frame . This relativistic correction to the horizontal velocity appears only at second order in . The particle’s trajectory, as viewed from , can be considered the result of a sequence of infinitesimal Lorentz transformations. By integrating over infinitesimal transverse velocity increments , we find that the corrected horizontal velocity is , where is the transverse component of the particle’s velocity as seen in frame . So that the Lorentz transformation effectively rotates the particle’s velocity vector by an angle in ultrarelativistic asymptotic. This result stands in contrast with the textbook prediction, which states that the total speed of the particle in frame increases from to . Our result contradicts this claim. The standard derivation of velocity addition neglects the mixing of transverse spatial coordinate with time in Lorentz transformations. This omission leads to an incorrect conclusion about the magnitude of the resulting velocity in initial inertial frame . It is important to highlight another effective approach to covariant particle tracking. Here, we revisit the motion of an accelerated relativistic particle from the perspective of an inertial lab observer—without changing reference frames. Specifically, we consider a particle undergoing acceleration within the lab frame, representing an example of an active boost. In this scenario, we track the system’s evolution entirely within a single (lab) reference frame. The simplest synchronization method maintains the same set of uniformly synchronized clocks used before the acceleration, without any adjustments. This approach is based on the absolute time (or absolute simultaneity) convention, which preserves simultaneity across space. Under this convention, the standard Galilean velocity addition rule can be applied instant by instant to track the particle’s motion as it accelerates. As discussed in Chapter 3, absolute time coordinatization can be transformed into Lorentz coordinatization. By combining Galilean transformations with appropriate changes of variables, we can derive the Lorentz transformation within the framework of absolute time. To further illustrate this idea, we now examine how distant lab clocks resynchronize during the particle’s acceleration. This analysis reveals a direct connection between the transformation of particle velocity (in Lorentz coordinates) and time dilation. As previously emphasized, the Lorentz coordinate system is a conceptual construct. The notion of synchronized clocks in this system is purely hypothetical, used primarily for the application of Maxwell’s equations in radiation calculations. Suppose that, before acceleration, we choose a Lorentz coordinate system in the lab frame. Immediately after the acceleration, particle velocity changes by an infinitesimal amount along the -axis. If clock synchronization is fixed, this corresponds to adopting the absolute time convention. To maintain Lorentz coordinates in the lab frame—as discussed previously—we must perform a clock resynchronization by introducing an infinitesimal time shift. The simplest case arises when the is very small, allowing us to work up to second-order terms in . This restriction greatly simplifies the calculations for two reasons. First, relativistic corrections due to the composition of non-collinear velocity increments only appear at order and can be neglected. Second, time dilation effects also enter only at higher orders. To maintain a Lorentz coordinate system in the lab frame after the acceleration, a resynchronization of clocks is required. In this context, operations involving the rule clock structure in the lab frame can be interpreted as a change of variables governed by the transformation in Eq. 13: , , where and the particle’s displacement is given by . This transformation effectively rescales the time variable—adjusting the synchronization of all clocks from to , with . As a result, we see that the particle’s speed after acceleration is unaffected by its transverse motion. Although no second-order relativistic correction appears in the transverse (-axis) velocity component, the longitudinal component is modified. Specifically, the longitudinal velocity changes to with . Thus, after clock resynchronization, the total electron speed in the lab frame remains —a result that aligns perfectly with the outcome derived earlier using a sequence of Lorentz transformations in changing frames. All pieces fit together seamlessly. We will explore the topic of relativistic velocity composition in greater detail in the next chapter 17 Mathematical Analysis of Relativistic Velocity Composition In this chapter, we shall explore the subject of relativistic velocity addition more mathematically. Textbook authors often assume that the principle of relativity requires Einstein’s velocity addition law to obey group composition. However, the non-associativity of non-collinear velocity addition is not widely recognized. Many physicists, having studied relativity from standard textbooks, might find this claim surprising, since Einstein’s velocity addition is commonly assumed to be associative. Yet, a straightforward algebraic exercise reveals its non-associative nature. The root cause of this issue lies in the presence of the Wigner rotation. In fact, the breakdown of commutativity and associativity in Einstein’s velocity addition law naturally manifests as Wigner rotation, offering deeper insight into the structure of relativistic transformations. 17.1 Parametrization of Lorentz Transformations Einstein formally defined the special theory of relativity through two explicit postulates. Minkowski, in 1908, recognized that these postulates could be reformulated as a single geometric axiom: spacetime possesses a pseudo-Euclidean geometry. As a consequence of this spacetime geometry, Lorentz coordinate systems are related via Lorentz transformations. Consider a four-vector, , with squared-length where is the Minkowski space-time metric. Since Lorentz transformations preserve the space-time interval, they satisfy such that . This condition defines the most general Lorentz transformation matrix . The set of all such matrices forms the symmetry group of the Minkowski metric under matrix multiplication, denoted by . In particular, is a Lie group known as the Lorentz group. It is widely accepted that the relationship between inertial frames is fully determined by their relative velocities and orientations. Consequently, a general Lorentz transformation, denoted , is parameterized by a velocity vector and a rotation vector , representing a combined boost and spatial rotation. Here, is a vector in three-dimensional Euclidean space, while belongs to the Lie algebra of the rotation group , which consists of all possible spatial rotations in three dimensions. This parameterization of the Lorentz group is standard in the literature. Now, we turn to a significant conceptual and structural challenge associated with this parametrization of the Lorentz group. A comparison with the Galilean group provides valuable insight. It is well known that the Galilean transformation , which satisfies the group composition law, is isomorphic to the semi-direct product group , where represents the group of Galilean boosts and denotes the rotation group. In this structure, acts as a normal subgroup, and the composition of transformations remains straightforward. In contrast, the composition of Lorentz transformations is significantly more intricate. The essential distinction between the Galilean and Lorentz groups lies in their group-theoretic structure: the Lorentz group is a simple Lie group, meaning it does not admit a nontrivial decomposition into a direct or semi-direct product of boosts and rotations. In particular, is not a normal subgroup of the Lorentz group . As a result, the combined action of Lorentz boosts and spatial rotations does not obey the same algebraic structure as in the Galilean case. This leads to an important conclusion: standard textbook analyses of inertial frames often overlook the fundamental geometric difference between Newtonian space and time, and the pseudo-Euclidean geometry of Minkowski spacetime. In classical mechanics, the components of the velocity three-vector transform under Galilean transformations in the same way as the components of the position vector . This correspondence arises because, in Newtonian physics, all inertial observers share a common, absolute three-dimensional space. Time is also absolute, so the spatial position and velocity vectors can be consistently defined across all inertial frames. However, in special relativity, this intuition fails. When transitioning from the four-dimensional spacetime formalism back to the three-dimensional velocity vector , defined as , subtle but profound issues emerge. The Lorentz group acts on four-dimensional spacetime, and all of its subgroups also act on four-vectors, not directly on spatial vectors. Therefore, there is no concept of a universal or common instantaneous three-dimensional spatial frame among all inertial observers within the framework of the Lorentz group. This absence of a shared simultaneity structure underlies many of the non-intuitive features of relativistic kinematics, including the non-associativity and non-commutativity of velocity addition. Every Lorentz transformation that relates inertial observers corresponds to an isometry of the Minkowski metric. However, when two reference frames are related by a general Lorentz transformation—i.e., an element of the Lorentz group —the concept of relative velocity between them is not immediately well-defined. To extract the relative velocity, one typically attempts to factor the Lorentz transformation into a pure boost (parameterized by a velocity vector) and a spatial rotation. However, such a factorization is not unique: there are many ways to choose the rotation subgroup within , and different choices can lead to different definitions of the relative velocity . Mathematically, the space of all boosts (i.e., all Lorentz transformations modulo spatial rotations) is the coset space . Since is not a normal subgroup of , different embeddings of yield different decompositions and thus different velocities. The commonly used decomposition, often referred to in the literature as the ”standard choice,” implicitly assumes a privileged reference frame—a specific frame in which the rotation subgroup is defined. In the absence of such a preferred frame, the Lorentz transformation between two systems remains well-defined as an isometry of spacetime, but the notion of a unique relative velocity becomes ambiguous. To define a single, well-determined velocity, one must fix a specific reference frame as distinguished. Physically, this can be modeled by introducing a preferred timelike vector field on spacetime, which selects a particular congruence of worldlines—effectively, a material or ”ether-like” reference frame. However, this dependence on a distinguished timelike direction violates the relativity principle, which asserts the equivalence of all inertial frames. Thus, within the geometry of pseudo-Euclidean spacetime, the equivalence of all reference systems cannot be maintained. 17.2 Observer-Independence and Lorentz Covariance A standard approach to representing Lorentz transformations is via transformation matrices acting on spacetime four-vectors. Any Lorentz transformation can, in principle, be decomposed into a product of a rotation and a boost. However, the explicit form of this decomposition depends on the choice of reference frame, which in the matrix formalism is determined by the selection of a distinguished timelike vector, typically denoted by , in a chosen basis. For example, in his widely used textbook, Barut defines a Lorentz boost (see Eq. (1.28) and the accompanying discussion in ) as: \| \| \| \| where and . This boost corresponds to motion along the -axis and is defined relative to a specific basis. However, the Lorentz boost is not fully determined by the velocity parameter alone. The very notion of velocity presupposes a distinction between space and time—i.e., a decomposition of spacetime—which requires the selection of a specific timelike direction. This decomposition identifies the observer through the choice of a basis vector , which represents the time direction in that observer’s rest frame. As made explicit in Barut and other standard references , the Lorentz boost is expressed as a matrix relative to a given orthonormal basis. In the “standard” formulation, the preferred inertial frame is associated with the basis vector . Rotations then occur within the three-dimensional spatial subspace orthogonal to , and any boost involving a velocity vector orthogonal to implicitly defines the boost with respect to this preferred frame. Thus, even though Lorentz transformations are covariant under changes of inertial observers, their explicit matrix representations—and the definition of velocity itself—rely on a choice of observer. Decompositions of spacetime based on different choices of the timelike basis vector are mutually incomparable, as they correspond to distinct three-dimensional spatial slices for which no canonical identification exists. Successive Lorentz boosts are composed through matrix multiplication, assuming both boosts are expressed in the same basis. Specifically, if is the boost from frame to frame , and is the boost from to , then their composition is given by: . Here, is the velocity of frame relative to the preferred frame , and is the velocity of frame relative to , as observed from . It is crucial to distinguish between observer-independence and Lorentz covariance. Although Lorentz covariance refers to the invariance of physical laws under Lorentz transformations, it does not imply that all inertial frames are physically equivalent in a pseudo-Euclidean spacetime. The geometry of Minkowski space allows for Lorentz covariance without guaranteeing the equivalence of all inertial observers. This subtle distinction opens the door to the possibility of a physically preferred inertial frame, thereby challenging the traditional interpretation of the principle of relativity. In this context, relative velocity alone does not fully characterize the relationship between inertial frames. 555555In particular, a non-zero relative velocity does not, by itself, cause time dilation. See Section 12.3 for further discussion. A deeper tension emerges between the principle of relativity—which asserts the equivalence of all inertial frames—and the geometric structure of Minkowski spacetime. In fact, only one of these principles can hold unconditionally. 565656For a more detailed analysis, see “How Do You Add Relative Velocities?” by Oziewicz in . Oziewicz argues that within pseudo-Euclidean spacetime, the equivalence of all reference frames is not possible. Given the strength of experimental evidence supporting Lorentz invariance, we adopt Lorentz covariance as the foundational principle of the theory. As discussed in earlier chapters, further mathematical considerations support the view that pseudo-Euclidean geometry does not entail the complete equivalence of all inertial frames. One key insight comes from analyzing the continuity of the metric tensor under coordinate transformations. The Langevin metric, for instance, arises from a continuous deformation of the Minkowski metric when transitioning between an inertial frame and an accelerated frame. This smooth transition can be modeled via a Galilean boost relating the coordinate systems of the inertial and accelerated observers. However, restoring the diagonal Minkowski form in the accelerated frame often requires an abrupt coordinate transformation, resulting in a discontinuity in the metric tensor. This discontinuity signals a breakdown in the symmetry between inertial frames: if a Lorentz transformation fails to preserve the smoothness of the metric, then the equivalence of the frames involved is compromised. This raises a natural and important question: How can observers identify a preferred inertial frame? Within the structure of the Lorentz group, relative velocity between frames enters exclusively through the boost transformations. However, a Lorentz boost is always defined relative to a specific reference frame—typically one associated with a distinguished timelike vector. Special relativity, in its standard formulation, does not specify such a frame; yet physical observations suggest the existence of a unique inertial frame—one not subjected to prior accelerations, such as relative to the fixed stars. There is, however, no intrinsic contradiction between the mathematical framework of special relativity and the physical asymmetry introduced by selecting a preferred frame. The theory is built on the requirement that physical laws remain invariant under Lorentz transformations. But special relativity is not a self-contained theory in the sense of fully determining physical phenomena from dynamical laws alone—initial conditions must also be specified. The principle of relativity asserts that the form of physical laws is independent of velocity, but not necessarily that initial conditions must be velocity-independent. While Lorentz coordinatization appears to establish symmetry among inertial frames, the asymmetry emerges from the history of acceleration. Once a frame has undergone acceleration, information about that acceleration is not encoded in the transformation laws themselves but in the initial conditions from which they operate. Consider, for instance, the relativistic aberration of particles emitted from a source at rest in an accelerated frame. As discussed in Chapter 9, special relativity predicts that—alongside a static electric field—a magnetic field also arises in the accelerated frame, leading to deviations in the observed trajectory of particles. This deviation is a measurable consequence of the frame’s prior acceleration. The mathematical framework of special relativity remains entirely self-consistent when formulated relative to an initial inertial frame—one that has not undergone prior acceleration with respect to the distant stars. Such a frame functions as a privileged reference system. As demonstrated in Chapter 7, when the laws of physics are formulated relative to this frame, they accurately account for all observed phenomena, including those reported by accelerated observers. In particular, phenomena like the aberration of light in an accelerated frame can be fully derived using standard tools of special relativity—specifically, Lorentz transformations and Wigner rotations—when analyzed from the standpoint of the initial inertial frame. 17.3 Einstein Velocity Addition and Conventional Particle Tracking Composition of relativistic velocities, also named as ”Einstein’s addition of velocities” is somewhat confusing due to fact that this addition is nonassociative. This is contrast with associativity of the Lorentz group. Most texts on special relativity present the relativistic velocity addition formula only for parallel velocities. In this simplified case, Einstein’s velocity addition is both commutative and associative, making it a commutative group operation analogous to the Galilean velocities addition . However, in the general case, Einstein’s velocity addition does not form a group. The relativistic reciprocity principle states that if an inertial frame moves with velocity relative to another inertial frame , then, reciprocally, the velocity of relative to is . This principle suggests that the inverse of a velocity is simply its negation: , analogous to the Galilean addition of velocities. This is often presented in textbooks as a self-evident fact requiring no further discussion. For composite velocities, , reciprocity implies . However, since Einstein’s velocity addition is generally non-commutative, this leads to a contradiction, known as the reciprocity paradox. This paradox is closely related to the Mocanu paradox, as discussed by Mocanu and was latter resolved by Ungar who demonstrated the crucial role of the Wigner rotation in the transformation. 575757It is sometimes referred to as the Thomas rotation Furthermore, Ungar also discovered that Einstein’s velocity addition is non-associative meaning that . Being non-associative, the Einstein’s addition is not a group operation. Non-associativity is paradoxical: for a system of four bodies the of three non-parallel velocities gives two distinct velocities between two bodies. Let us illustrate non-commutativity of Einstein’s velocity addition using only simple case of relativistically combining perpendicular velocities. Consider a coordinate system moving with respect to with a velocity and a frame moving with respect to with a velocity . It is important to note that represents the velocity of frame as measured by an observer at rest in , while represents the velocity of as measured by an observer in . The question then arises: what is the velocity transformation from to ? Now, consider the case, where and are perpendicular. The ”perpendicular” means that velocity is perpendicular to line motion of frame in . The velocity of frame as observed from is given by . According to reciprocity principle, accepted in all textbooks, an observer in sees frame moving with velocity , while an observer in sees frame moving with velocity . Similarly, applying Einstein’s velocity addition, we find that the velocity of frame as observed from is: . According to textbooks, should be reciprocal of , but they point in different directions. This reciprocity discrepancy leads to the Mocanu contradiction and highlights the non-associativity of Einstein’s velocity addition. Under Lorentz coordinatization, moving frames inherently possess different simultaneity relations. As a result, the velocity and its inverse are each tangent to a different instantaneous space and these two spaces are not parallel. Consequently, the inverse velocity cannot be truly reciprocal. The reciprocity of frames contradicts relativity of simultaneity. The commonly accepted derivation of velocity composition does not follow from the composition of Lorentz boosts and based on absolute time convention. In Enstein’s velocity addition all velocities , , , are implicitly assumed to lie within the same preferred 3-space as seen by a preferred observer. It is often assumed that the acceleration of a rigid reference frame grid has a direct physical meaning. However, this is a misconception. Despite this, Einstein’s velocity addition is widely used in relativistic dynamics raising a reasonable question: How can this be? It is important to emphasize that Einstein’s velocity addition is fully compatible with conventional particle tracking within the initial inertial frame, which itself relies on the absolute time convention. Crucially, Einstein’s velocity addition inherently incorporates relativistic time dilation. Similarly, conventional particle tracking also accounts for relativistic time dilation, albeit implicitly, through the use of relativistic mass for moving particles. Non-covariant approach presents no fundamental challenges in mechanics. The preference for a non-covariant formulation within the framework of dynamics arises from its simplicity and practicality. What is essential is the consistent application of the chosen convention across both dynamics and electrodynamics. In an absolute-time coordinatization, the equations of electrodynamics are not isotropic. This poses a challenge, as electrodynamics involves partial differential equations, whereas dynamics deals only with ordinary differential equations. Only by adopting a kinematic framework based on Lorentz coordinatization can we ensure the correct coupling between the isotropic Maxwell equations and particle trajectories in the laboratory frame. 17.4 Velocity Addition and the Wigner Rotation While the coordinate transformation formula for Lorentz boosts and the velocity addition formula may appear formally similar, they exhibit significant differences in structure and properties. Specifically, the composition of Lorentz boosts is associative, as it corresponds to the matrix product of Lorentz transformation matrices. However, this composition does not form a group under velocity addition alone. In contrast, Einstein velocity addition is not associative. The consequence of this difference is the Wigner rotation accompanying the composition of non-parallel velocities. For completeness, it is worth noting earlier attempts to address the limitations of Einstein velocity composition. In a series of mathematically rigorous works, Ungar proposed a refined framework to resolve the apparent paradox of non-associativity in relativistic velocity addition (see [70, 71] and references therein). In this refined framework, the composition of velocities is not solely represented by , but rather by the pair , where denotes the Wigner rotation arising from the non-commutative nature of Lorentz boosts. Notably, the binary operation remains the standard Einstein velocity addition; what changes is the recognition that a rotation must accompany it to fully capture the transformation. It should be noted that the Wigner rotation leaves invariant preferred observer. A common textbook derivation of the Wigner rotation begins by decomposing the product of two Lorentz boosts, and , as follows: , where is identified as the Wigner rotation, and the resultant boost corresponds to the Einstein velocity addition . However, this standard approach is mathematically incomplete, as it treats the rotation as a secondary effect rather than an intrinsic component of the velocity composition itself. As shown in Section 16.6, the Wigner rotation is not a mere additive correction—it is fundamentally embedded in the structure of relativistic velocity addition. Consider, for example, the composition of two perpendicular velocities and as discussed earlier. Suppose approaches the speed of light, while remains small. In this regime, the Lorentz factor becomes very large, i.e., . Treating as a small parameter and retaining terms up to second order, we observe that the intermediate frame appears to rotate with respect to . This effect is a manifestation of the Wigner rotation. Ungar’s approach—based on the Einstein velocity addition within an improved composition law—suggests that the total speed of in the Lorentz frame increases from to . However, this standard velocities addition overlooks a critical point. In the Section 16.6 we found that in Lorentz coordinatization, the total velocity vector of frame in , after a boost with velocity in , undergoes a simple rotation by an angle without any change in its magnitude, . This is intuitively reasonable, since we are using Lorentz coordinatization and in our relativistic asymptotic approaching the speed of light. Let us now re-examine, from a covariant perspective, why Einstein’s velocity addition formula proves inadequate—even when only two velocities are involved. This topic was already introduced in the previous chapter, where we provided a general overview. That earlier discussion covers what is typically sufficient for most purposes. However, we will now revisit the topic in greater detail to develop a more complete understanding. The boost from frame to , as observed from , can be understood as a sequence of infinitesimal Lorentz transformations, each involving a small velocity increment . To provide a complete analysis, we must also account for the Wigner rotation which occurs in the same direction as the rotation of the velocity vector of as seen from . The coordinate axes of the frame remain parallel to those of frame . From the perspective of frame , the coordinate axes of frame are also rotated by the same angle with respect to those of frame . The infinitesimal velocity increment of as observed from can be decomposed into two physically meaningful components: 1) a transverse component of , perpendicular to the original velocity , and 2) a tangential component of which is aligned with the direction of , and arises due to the Wigner rotation. Here denotes the Wigner rotation angle. In the ultra-relativistic limit we approximate it as . Notably, analyzing the transverse component is considerably more straightforward than handling the tangential one. In fact, to second order in , no relativistic corrections affect the transverse velocity component. We now proceed to compute the net effect of accumulating these infinitesimal increments. This process is equivalent to integrating over the transverse velocity: . Evaluating this from to gives the total tangential correction due to the Wigner rotation. This results in a tangential velocity component of . Thus, to second order in , the total speed of frame with respect to remains . This analysis is carried out in the limit of a small Wigner rotation angle, . Even within this simplified regime, we can clearly demonstrate the fundamental difference between covariant velocity addition and Einstein’s velocity addition. To properly interpret the physical meaning of covariant velocity addition as presented here, it is essential to recognize that Einstein’s velocity addition is fully compatible with conventional particle tracking methods, which are based on the absolute time convention. However, the only rigorous way to maintain consistency between Maxwell’s equations and particle trajectories is to formulate and solve the equations of motion in a fully covariant manner. Experimental results support our correction to the conventional theory of relativistic velocity addition; further details are provided in the final two chapters. 17.5 Lorentz-Covariant Particle Tracking In covariant particle tracking, a particle’s trajectory is described from the perspective of an initial inertial frame as a result of successive Lorentz boosts. Let us now apply this approach to a concrete example and assess its practical implications. Based on experimental evidence, we adopt the Sun-centered frame as the preferred inertial frame (see Chapter 15 for further discussion). Once such a preferred frame is established, a unique and well-defined relative velocity exists between any two inertial frames. Now, let us consider the Earth-based frame. For an observer on Earth, the velocity relative to the initial inertial frame is approximately . As previously demonstrated, current experimental techniques usually lack the sensitivity required to detect the influence of orbital velocity on the relativistic particle dynamics. Consequently, in accelerator physics, the Earth-based frame can be treated with good accuracy as the preferred inertial frame. Suppose an observer in the initial inertial frame establishes a coordinate system using Einstein’s synchronization procedure, which relies on light signals emitted by a dipole source at rest and assumes that light propagates isotropically at speed . This procedure enables the construction of a Lorentz coordinate system in . Now, consider two additional inertial frames, and , both moving relative to . The simplest synchronization method is to retain the same set of synchronized clocks across all frames. This preserves simultaneity and leads to an absolute-time coordinatization based on . To describe physical phenomena across these frames, the preferred observer uses a metric tensor to derive the (electro)dynamical equations. According to the equivalence between passive and active Lorentz boosts within a single inertial frame, the metric in Eq. 11 characterizes the measurements of the preferred observer. Let us now examine velocity composition within the absolute-time coordinatization framework. Assume that frame moves with velocity relative to , and frame moves with velocity relative to . Importantly, both and are defined from the perspective of the observer in . 585858The velocity used here denotes the relative velocity as measured by a preferred observer, and should not be confused with the in the preceding section, which refers to the relative velocity measured in frame when observing this observer from frame . In this setup, velocity transformation is particularly straightforward: under absolute-time coordinatization, Galilean velocity addition applies. Thus, the velocity of as observed from is simply . This observation highlights a key advantage: when the preferred observer adopts the absolute-time coordinatization, relativistic kinematics becomes extremely simple. In this formulation, all the complexity of relativistic particle tracking is absorbed into the frame metrics. Consider now two electrons, both moving with the same velocity relative to the preferred frame. The first electron is accelerated directly from rest to within the preferred frame. The second reaches the same final velocity through two steps: first accelerated to , then from to . Despite having the same final velocity, the two electrons have different acceleration histories. As a result, the electrodynamics of these two electrons will differ due to differences in their respective metrics. The contrast between these seemingly equivalent cases is both subtle and intriguing. This paradoxical asymmetry between the two setups is most naturally understood from the perspective of electrodynamics. Our everyday intuition often breaks down when confronted with the counterintuitive nature of special relativity. The dynamics governed by electromagnetic fields are, in this context, encoded in the language of pseudo-Euclidean geometry, which can obscure their physical implications. The idea that each electron ”remembers” the acceleration it experienced relative to a preferred frame may seem surprising at first. However, this is entirely consistent with our understanding of relativistic electrodynamics. Let us consider the case where the velocity approaches the speed of light, while the velocity is perpendicular to and remains small. More specifically, we are interested in the limit where . In this setup, the first electron is accelerated in parallel geometry, reaching a final velocity . The second electron, in contrast, is finally accelerated in a direction perpendicular to the ultrarelativistic velocity . It is well known that transverse acceleration in the ultrarelativistic regime results in radiation that is enhanced by a factor of approximately compared to the case of parallel acceleration. Thus, in the ultrarelativistic limit, the first electron (undergoing parallel acceleration) emits negligible radiation. In contrast, the second electron emits significant synchrotron radiation due to its transverse acceleration. From the standpoint of electrodynamics, these two configurations are fundamentally different. In particular, the radiation emitted by the second electron carries information about its acceleration history. Now let us consider a scenario where the deflection angle of the second electron is smaller than . In this regime, synchrotron radiation is effectively absent. This leads to a compelling question: in the absence of radiation, where—within the framework of absolute time coordinatization—is the information about the second electron’s acceleration physically located? According to conventional particle tracking, both electrons end up with the same final velocity after acceleration. However, the second electron underwent a qualitatively different acceleration process, raising subtle questions about how this information is encoded in the system when no radiation is emitted. One might be tempted to assert: “Since an electron is a structureless particle, this situation seems paradoxical.” However, this is not accurate. An ultrarelativistic electron cannot be regarded as structureless in the conventional sense. The electromagnetic field associated with such an electron exhibits a macroscopic transverse extent. The spatial distribution of this virtual radiation field is well described by the Ginzburg–Frank formula (see Appendix A4 for further details). For a rapidly moving electron, the transverse components of the electric and magnetic fields are nearly equal in magnitude and orthogonal to one another—effectively indistinguishable from those of a real radiation beam. On a microscopic level, when the electron experiences transverse acceleration, its electromagnetic field is perturbed in a way that reflects this acceleration. 595959Certain assumptions are necessary when applying relativistic kinematics in practice. If the electron undergoes a sudden acceleration, there is a minimum distance—known as the formation length—required for the development of the (virtual laser-like) self-field of an ultrarelativistic electron. Only when the distance traveled after the first acceleration exceeds this formation length can a second acceleration (or ’boost’) be considered kinematically independent. If this condition is not met, the two accelerations effectively combine into a single boost from rest to a final velocity and the electron’s acceleration history will appear nearly identical from the perspective of relativistic electrodynamics. Under the framework of classical (Galilean) kinematics, the orientation of the virtual radiation phase front remains unchanged. However, since Maxwell’s equations are not invariant under Galilean transformations—as discussed throughout this book—the adoption of such kinematics necessitates the use of anisotropic field equations. Consequently, although the phase front remains planar, the direction of propagation is no longer perpendicular to it. In other words, the (virtual) radiation beam’s direction of motion diverges from the normal to its phase front. Within the absolute time coordinatization, electrodynamics predicts that the virtual radiation beam of the second electron propagates along the direction of , with a phase front tilt . This tilt is the crucial clue to resolving the paradox. The key insight is this: The information about the electron’s acceleration is not lost—it is embedded in the perturbation of the electron’s self-fields. Mathematically, this information is encoded in the difference between metrics of these two electrons This rises a reasonable question: why does there is no virtual phase front perturbation in the situation with first electron acceleration? The key distinction is in collinear geometry - the first electron accelerates perpendicular to the plane of the virtual radiation wavefront (i.e., the plane of simultaneity). In the case of the boost with velocity the second electron undergoes acceleration along the virtual radiation wavefront. Absolute time coordinatization in the preferred frame can be transformed into Lorentz coordinatization. In the Chapter 3 we consider a general method for metric diagonalization in the case when particle accelerated from rest to velocity in the preferred frame. The change variables according to Eq. 13 completed by Galilean transformation is mathematically equivalent to the Lorentz transformation with the same velocity. As a result, covariant particle tracking in the case of the first electron is straightforward. Similar there is no problem with the first boost of the second electron. To maintain a Lorentz coordinate system in the preferred frame after the second boost with velocity , additional changes in rule-clock structure is required as described in Section 16.6. Thus, after addition variable change, the total speed of the second electron decries in the Lorentz coordinatization from to . This result stands in contrast with the speed of the first electron which remains . In covariant particle tracking, information about difference in acceleration history between two setups is recorded in the covariant velocities difference. An equally valid interpretation exists within the non-covariant framework, which tracks particle motion using absolute time and applies Galilean transformations to field equations. The velocity of an electron and the speed of light are convention-dependent. In absolute-time coordinatization, the speed of light differs from the electrodynamics constant because Maxwell’s equations are not invariant under Galilean transformations. However, the dimensionless ratio of electron velocity to light velocity remains convention-independent, meaning it is unaffected by distant clock synchronization or variations in clock rhythms. In electrodynamics, parameters related to synchrotron radiation, which have direct physical significance, depend on this ratio. According to non-covariant particle tracking, the second electron’s velocity is equal . However, Maxwell’s equations are not invariant under Galilean transformations, leading to a change in the speed of light. Specifically, the velocity of light increases from , before the second boost with velocity , to after the second boost. This variation arises because, under the absolute time convention, clocks are not resynchronized after the boost. As a result, the speed of light differs from the electrodynamical constant after the second boost. Nevertheless, the ratio of the electron velocity to the speed of light remains invariant across different synchronization conventions—that is, it does not depend on the method of distant clock synchronization or the rate of the clocks. Our calculations demonstrate that both covariant and non-covariant treatments—when the correct coupling between fields and particles is used—yield the same prediction for the relativistic electrodynamics phenomena. This prediction is convention-invariant and depends solely on the dimensionless parameter , where is the coordinate velocity of the second electron, and is the coordinate velocity of light. This is illustrated that all relativistic effects can, in principle, be derived directly from underlying physical laws - without invoking relativity as an independent framework. Further details are provided in the final two chapters. We have shown that covariant particle tracking in the preferred frame can be achieved without switching reference frames. Initially, we employed conventional particle tracking—i.e., absolute time coordinatization—within the preferred frame. Then, by applying Galilean transformations alongside appropriate redefinitions of space and time variables, we derived the covariant particle trajectory, . From the standpoint of electrodynamics involving relativistically moving charges, this approach ensures consistency between Maxwell’s equations and particle trajectories within the preferred frame. There is, however, one important case where tracking requires a change of reference frame: the description of a particle’s angular momentum (spin) relative to the preferred frame. In the non-relativistic regime, spin is represented as a three-dimensional pseudovector . For a relativistically moving particle, by contrast, angular momentum must be described using the antisymmetric four-tensor . In experimental practice, however, the spin vector remains the preferred object due to its intuitive physical interpretation as a three-dimensional pseudovector. As a result, any meaningful statement about is understood to refer implicitly to the particle’s instantaneous rest frame—that is, its proper frame. As previously noted in Chapter 8, spin orientation measurements with respect to the preferred frame axes can be determined within the proper frame. In other words, spin rotation is interpreted from the viewpoint of the proper observer as the apparent rotation of the preferred frame axes. Successive Lorentz boosts to the proper frame are composed via matrix multiplication, assuming all transformations are expressed in the same (preferred) basis. However, under Lorentz coordinatization, moving frames inherently adopt different simultaneity conventions. As a result, the instantaneous spatial hypersurfaces of the proper and preferred frames differ and are not parallel. From a purely kinematic standpoint, spin rotation within the proper frame is absent. However, the preferred frame undergoes relativistic motion relative to the proper frame, which gives rise to the Wigner (Thomas) rotation of the preferred frame axes as seen from the proper frame. This rotation is given by , where is the orbital deflection angle of the particle in the preferred frame. The direction of the preferred frame’s rotation, as observed in the proper frame, aligns with the direction of velocity rotation—both in the preferred and the proper frames. Accordingly, the spin vector rotates with respect to the preferred frame axes as . In the next chapter, we will continue our discussion of relativistic spin rotation. 18 Relativistic Spin Dynamics We aim to demonstrate that while studying relativistic particle dynamics, we simultaneously explored a wide range of related topics. As an example, consider the motion of spin in a given field. The equations governing spin dynamics can be formulated as tensor equations in Minkowski space-time. Here, we focus on the case of a particle with a magnetic moment in a microscopically homogeneous electromagnetic field. Notably, the torque influences only the spin, while the force affects only the momentum. Consequently, the overall motion of the system in any reference frame is determined solely by its charge, independent of its magnetic dipole moment—a topic addressed in the previous chapter. Our current focus is on the dynamics of spin motion. 18.1 Magnetic Dipole at Rest in an Electromagnetic Field Let us first consider the spin precession for a nonrelativistic charged particle. The proportionality of magnetic moment and angular momentum has been confirmed in many ”gyromagnetic” experiments on many different systems. The constant of proportionality is one of the parameters charactering a particular system. It is normally specified by giving the gyromagnetic ratio or factor, defined by . This formula says that the magnetic moment is parallel to the angular momentum and can have any magnitude. For an electron is very nearly 2. Suppose that a particle is at rest in an external magnetic field . The equation of motion for the angular momentum in its rest frame is . In other words, the spin precesses around the direction of the magnetic field with the frequency . In the same nonrelativistic limit the velocity processes around the direction of with the frequency : . Thus, for spin and velocity precess with the same frequency, so that the angle between them is conserved. 18.2 Derivation of the Covariant (BMT) Equation of Spin Motion To derive the equation governing spin motion, we start from the well-established spin dynamics in the particle’s rest frame and apply known relativistic transformation laws. Since spin is inherently defined in the rest frame, constructing covariant expressions requires introducing a four-dimensional quantity associated with spin that transforms appropriately under Lorentz transformations. There are two general ways to generalize the spin vector —and, by extension, the Larmor equation —to arbitrary frames: one approach uses an axial four-vector, while the other employs an antisymmetric rank-2 tensor. The most commonly used representation is the four-(pseudo)vector , often referred to as the four-spin vector. It is defined such that, in the rest frame, its spatial components coincide with the components of , while its time-like component vanishes. When normalized by its invariant magnitude, becomes the polarization four-vector. Being space-like, its spatial components remain non-zero in all inertial frames. 606060An arguably more physically natural representation is the antisymmetric four-tensor. The motion of a spinning point particle with mass and charge in Minkowski spacetime is then described by its polarization tensor —an antisymmetric four-tensor that incorporates both intrinsic magnetic and electric dipole moments. It is defined as , where is the intrinsic angular momentum tensor. Let the spin of the particle be represented in the rest frame by . The four-vector is by definition required to be purely spatial at time in an instantaneous Lorentz rest frame of the particle and to coincide at this time with the spin of the particle; that is . At a later instant in an instantaneous inertial rest frame , we have similarly . The BMT equation is the manifestly covariant equation of motion for a four-vector spin in an electromagnetic field : \| \| \| --- \| \| \| (27) \| where is the four-dimensional particle velocity vector. With Eq.(21), one has \| \| \| --- \| \| \| (28) \| The BMT equation is valid in any Lorentz frame and, together with the covariant force law, consistently describes the motion of a charged particle with spin and magnetic moment. The covariant equation of spin motion for a relativistic particle subjected to the four-force (as expressed in the Lorentz lab frame, Eq. (27)) serves as a relativistic generalization of the classical spin dynamics equation in the particle’s rest frame. This generalization embeds the three Larmor spin precession equations into the four-dimensional structure of Minkowski space-time. In Lorentz coordinates, there exists a kinematic constraint , which expresses the orthogonality between the four-spin and the four-velocity vectors. Due to this condition, the four-dimensional dynamical equation, Eq. (27), effectively contains only three independent equations of motion. By substituting the explicit form of the Lorentz force, one finds that Eq. (27) inherently satisfies the constraint , as required. To demonstrate this, note that in any Lorentz frame, the time component of the spin satisfies . Although vanishes in the particle’s rest frame, its derivative with respect to proper time, , does not necessarily vanish. In fact, the condition leads directly to . The immediate generalization of and to arbitrary Lorentz frames is Eq.(27) as can be checked by reducing to the rest frame. A methodological parallel can thus be drawn between this formulation and the relativistic generalization of Newton’s second law. To fully grasp the significance of embedding the spin dynamics law within Minkowski space-time, it is important to recall that the spin dynamics equation, as presented earlier in the Lorentz comoving frame, is characterized as a phenomenological law. Its formulation does not provide a microscopic interpretation of a particle’s magnetic moment. In other words, the spin dynamics law is generally accepted as phenomenological, with the magnetic moment introduced in an ad hoc manner. The coordinate system in which the classical equations of motion for a particle’s angular momentum are valid is referred to as the Lorentz rest frame. The relativistic generalization of the three-dimensional equation to an arbitrary Lorentz frame enables accurate predictions of spin behavior across different reference frames. 18.3 Changing Spin Variables When Bargmann, Michel, and Telegdi first formulated the correct laws of spin dynamics, they derived a manifestly covariant equation in Minkowski space-time, Eq. (28), describing the motion of the four-spin . Their approach closely resembled the four-tensor equations already established in relativistic particle dynamics. To apply Eq. (28) to concrete physical scenarios, it is necessary to express the standard three-dimensional spin vector in terms of the four-vector . The relationship between and follows directly from the Lorentz transformation: . With the help of this relation, one can work out the equation of motion for . Let us restrict our treatment of spinning particle dynamics to purely transverse magnetic fields. This means that the magnetic field vector is oriented normally to the particle line motion. In this practically important case one has, after somewhat lengthy calculations : \| \| \| --- \| \| \| (29) \| What must be recognized is that in the accepted covariant approach (indeed, Eq.(28) is manifestly covariant), the solution of the dynamics problem for the spin in the lab frame makes no reference to the three-dimensional velocity. In fact, the Eq.(29) includes relativistic factor and vector , which are actually notations: , . All quantities are defined in the lab frame and possess exact, objective meaning—that is, they are independent of any choice of convention. The evolution parameter is likewise measured in the lab frame and has a similarly objective interpretation. For example, it is straightforward to show that , where denotes the differential path length. We now have the equation in a form that is convenient for solving. Consider a charged, spinning particle moving through a bending magnet of length in the lab frame. The orbital deflection angle of the particle is given by . Additionally, the proper time interval is . From Eq. (29), we can express the spin rotation angle relative to the lab frame axes as , which becomes: . The spin vector directly represents the spin as perceived in a comoving system. In the lab frame, if we say that a particle’s spin makes an angle with its velocity, this means that in the particle’s rest frame, the spin makes the same angle with the direction of motion of the lab frame. This leads us to conclude that the conventional approach used to describe spin dynamics in the lab frame is rather unconventional. Specifically, the measurement of spin rotation in the lab frame is interpreted, from the proper observer’s perspective, as viewing this of the lab observer. 18.4 An Alternative Approach to BMT Theory Above we described the BMT equation, Eq.(29), in the standard manner. It uses a spin quantity defined in the proper frame but observed with respect to the lab frame axes. That means that we know the orientation of the proper spin with respect to the lab coordinate system which is moving with velocity and acceleration in the proper frame. Let’s look at what the equation Eq.(29) says in a little more detail. It will be more convenient if we rewrite this equation as \| \| \| --- \| \| \| (30) \| Now let’s see how we can write the right-hand side of Eq.(30). The first term is that we would expect the spin rotation due to a torque with respect to the proper frame axes . Here is the angle of the velocity rotation in the lab frame. It has also been made evident by our analysis in Chapter 8 that the angle of rotation corresponds to the Wigner rotation of the lab frame axes with respect to the proper frame axes. With these definitions, we have . We now present a new approach to the BMT theory, offering an alternative method for addressing this complex problem. It is important to note that and represent rotations with respect to the axes of the proper frame. However, our primary goal is to determine the spin motion relative to the laboratory frame axes. In doing so, we must pay close attention to the signs of the rotations, as they play a critical role in the analysis. A helpful mnemonic can be used to remember the signs of various rotational effects. It consists of three parts: The direction of velocity rotation in the proper frame is the same as the direction of velocity rotation in the lab frame. The direction of lab frame rotation as observed in the proper frame is the same as the direction of velocity rotation in the proper frame. When (which is the case for an electron, where is positive and approximately equal to 2), the direction of spin rotation due to a torque in the proper frame matches the direction of velocity rotation in the proper frame. We now turn to the question of determining the proper spin rotation relative to the lab frame axes. This is straightforward: the relevant rotation angle is given by the difference . With this, we begin to grasp the fundamental framework of spin dynamics. It becomes clear why the Wigner rotation of the lab frame axes, as seen from the proper frame, must be accounted for when analyzing spin dynamics relative to the lab frame. Why is the new derivation of the BMT equation so simple? The key reason is that the decomposition of the particle’s spin motion relative to the lab frame axes into dynamic and kinematic components can only be properly realized in the particle’s proper frame. In this frame, there is no need to consider a relativistic ”generalization” of the phenomenological classical equation of motion for angular momentum. This approach allows the spin dynamics problem to be separated into two distinct parts: the trivial (Larmor) dynamic problem and the kinematic problem of the Wigner rotation of the lab frame axes in the proper frame. 18.5 Spin Tracking By expressing the spin motion equation in four-vector form, Eq. (28), and determining the components of the four-force, we have not only ensured compliance with the principle of relativity but also obtained the four-component formulation of the spin motion equation. This represents a covariant relativistic generalization of the conventional three-dimensional equation for the motion of a magnetic moment, where the particle’s proper time serves as the evolution parameter. Next, we aim to describe the spin motion relative to the Lorentz lab frame, using the lab time as the evolution parameter. When going from the proper time to the lab time , the frequency of spin precession with respect to the lab frame can be obtained using the well-known formula . We then find \| \| \| --- \| \| \| (31) \| The frequency of spin precession can be written in the form , where is the particle revolution frequency. The old kinematics comes from the relation . The presentation of the time component simply as the relation between proper time and coordinate time is based on the hidden assumption that the type of clock synchronization that provides the time coordinate in the lab frame is based on the use of the absolute time convention. In the previous chapter, we saw that the particle path has an exact objective meaning i.e. it is convention-invariant. The spin orientation at each point of the particle path has also exact objective meaning. In contrast to this, and consistently with the conventionality of the three-velocity, the function describing the spinning particle in the lab frame has no exact objective meaning. We now describe how to determine the spin orientation along the path in convention-invariant spin tracking. Starting with the covariant equation (Eq. 27), we obtain Eq. (Eq. 29). By using the relation , we arrive at the convention-invariant equation of spin motion: \| \| \| --- \| \| \| (32) \| where is the path length, used as the evolution parameter. These three equations precisely correspond to the equations for components of the proper spin vector derived from the non-covariant spin tracking equation (Eq. 31). It is important to note that there are two distinct approaches—covariant and non-covariant—that yield the same spin orientation along the path. Both approaches describe the same physical reality correctly, meaning that the orientation of the proper spin at any point along the particle’s path in the magnetic field is objective and convention-invariant. 19 Relativity and Electrodynamics The differential form of Maxwell’s equations describing electromagnetic phenomena in the Lorentz lab frame is given by Eq.(8). Now let us use these equations to discuss the phenomena called radiation. To evaluate radiation fields arising from external sources in Eq. (8), we need to know the velocity and the position as a function of the lab frame time . As discussed above, it is generally accepted that one should solve the Maxwell’s equations in the lab frame with current and charge density created by particles moving along non-covariant trajectory like . The trajectory , which follows from the solution of the corrected Newton’s second law Eq. (4) under the absolute time convention, does not include, however, relativistic kinematics effects. We now turn to the motion of a particle in a given magnetic field. Maxwell’s equations, in their standard form, are valid only within Lorentz reference frames. According to the proper coupling between fields and particles, Eq. 10 implies the following expressions for the charge and current densities: \| \| \| --- \| \| \| (33) \| \| \| (34) \| where . Here, the covariant trajectory of the particle, as observed in the laboratory frame, results from a sequence of Lorentz transformations. In this book, we emphasize the significant role of the Wigner rotation as a regulator of the velocity addition law. As previously discussed, the four-velocity of an accelerating particle moving along a curved path in the Lorentz lab frame cannot, in general, be decomposed as . 616161Consider, for instance, a typical textbook treatment , which discusses the projection of an arbitrary world line onto the Lorentz lab frame basis: ”The particle, which is assumed to curry the charge , creates the current density . […] Furthermore, in any frame of reference , one recovers the expected expressions for the charge and current densities by integrating over by means of relation between proper time and coordinate time and using the formula , , , ”. The presentation of the time component simply as the relation between proper time and coordinate time is based on the hidden assumption that the type of clock synchronization, which provides the time coordinate in the lab frame, is based on the use of the absolute time convention. 19.1 The Dipole Approximation The distinction between covariant and non-covariant particle trajectories was never fully understood. As a result, physicists failed to recognize the contribution of relativistic kinematic effects to radiation. This oversight naturally raises an important question: why did this error in radiation theory remain undetected for so long? To address this question, we will analyze the subject more mathematically. For an arbitrary velocity parameter , performing covariant calculations of the radiation process is highly challenging. However, in certain cases, significant simplifications arise. One such case is the non-relativistic radiation regime. The non-relativistic asymptotic limit offers a fundamental simplification for covariant calculations. This is because the non-relativistic assumption justifies the dipole approximation, which is of great practical importance. When considering only the dipole component of radiation, all details of the electron trajectory are effectively ignored. Consequently, dipole radiation is completely insensitive to the distinction between covariant and non-covariant particle trajectories. 626262Similarly, the ultrarelativistic asymptotic limit also simplifies covariant calculations. This is due to the paraxial approximation, which naturally follows from the ultrarelativistic assumption. More details on this effect will be discussed in the next chapter. We want now to solve electrodynamics equations mathematically in a general way and consider the radiation associated with the succeeding terms in (multi-pole) expansion of the field in powers of the ratio . Radiation theory is naturally developed in the space-frequency domain, as one is usually interested in radiation properties at a given position in space and at a certain frequency.636363In this book we define the relation between temporal and frequency domain via the following definition of Fourier transform pair: . Suppose we are interested in the radiation generated by an electron and observed far away from it. In this case, it is possible to find a relatively simple expression for the electric field. We indicate the electron velocity in units of with , the electron trajectory in three dimensions with and the observation position with . Finally, we introduce the unit vector \| \| \| --- \| \| \| (35) \| pointing from the retarded position of the electron to the observer. In the far zone, by definition, the unit vector is nearly constant in time. If the position of the observer is far away enough from the charge, one can make the expansion . We then obtain the following approximate expression for the the radiation field in the space-frequency domain: \| \| \| \| \| --- --- \| \| \| \| \| (36) \| where is the frequency, is the negative electron charge and we make use of Gaussian units. A different constant of proportionality in Eq.(LABEL:revwied) and the well-known textbooks is to be ascribed to the use of different units and the definition of the Fourier transform. Let us now examine in greater detail how the dipole radiation term arises. In the integrands of the expression for the radiation field amplitude, Eq. (LABEL:revwied), the time argument can be neglected if the trajectory of the charge changes insignificantly during this interval. Determining the conditions under which this approximation holds is straightforward. Let represent the characteristic size of the system. Then, the time delay can be estimated as . To ensure that the charge distribution remains essentially unchanged over this duration, it is necessary that , where is the wavelength of the emitted radiation. This condition can also be expressed in an alternative form: , where denotes the characteristic velocity of the charges in the system. We consider the radiation corresponding to the zeroth-order term in the expansion of Eq. (LABEL:revwied) in powers of . In this approximation, all details of the electron trajectory are neglected. This is the essence of the dipole approximation, where the spatial scale of the electron’s motion is assumed to be much smaller than the wavelength of the emitted radiation. Under this condition, Eq. (LABEL:revwied) yields fields that closely resemble those predicted by instantaneous, non-retarded theories. Therefore, it is appropriate to use a non-covariant approach when analyzing dipole radiation. However, this is only the first and most practically significant term. The remaining terms indicate that there are higher-order corrections to the dipole radiation approximation. Calculating these corrections requires detailed knowledge of the electron’s trajectory. In particular, to determine the correction to dipole radiation, we must use the covariant form of the trajectory rather than relying on a non-covariant approach. Nevertheless, corrections to multipole radiation are generally expected to be small. For instance, the covariant correction to quadrupole radiation is typically regarded as a contribution of higher order than the quadrupole term itself. 19.2 An Illustrative Example It is beneficial to illustrate errors in standard coupling fields and particles in accelerator and plasma physics using a relatively simple example, where the core physical concepts remain clear and unobscured by unnecessary mathematical complexities. This example is primarily intended for readers with limited knowledge of accelerator and synchrotron radiation physics. Fortunately, the error in standard coupling fields and particle interactions can be explained in a straightforward manner. An electron kicker setup serves as a practical example for illustrating the distinction between covariant and non-covariant trajectories. Consider a simple case where an ultrarelativistic electron, moving with velocity along the -axis in the inertial (lab) frame, is subjected to a weak dipole magnetic field directed along the -axis. For simplicity, we assume that the kick angle is small compared to , where is the Lorentz factor. This corresponds to the limit . We begin with non-covariant particle tracking calculations. The electron’s trajectory, derived from the corrected form of Newton’s second law under the absolute time convention, does not account for relativistic effects. Consequently, as in classical Newtonian kinematics, the Galilean vector addition of velocities is employed. Non-covariant particle dynamics predicts that the electron’s direction changes as a result of the kick, while its speed remains constant (see Fig. 49). According to this model, the magnetic field affects only the direction of motion and not the magnitude of the velocity. After the kick, the components of the beam’s velocity are , where . In the ultrarelativistic limit, where , we can use a second-order approximation to obtain . In contrast, covariant particle tracking—which relies on Lorentz coordinates—yields different results for the electron’s velocity. To explore this, consider a sequence of passive Lorentz transformations used to track the motion of a relativistic electron accelerated by a kicker field. Let be the lab frame and a frame comoving with the electron at velocity relative to . Upstream of the kicker, the electron is at rest in the frame. This implies that is related to by a Lorentz boost , with aligned along the axis. The boost transforms a four-vector event in space-time according to . Now, let us analyze the particle’s dynamics from the perspective of the frame. In this frame, the electron remains at rest, while the kicker moves toward it with velocity . Due to its motion, the kicker’s magnetic field generates an electric field perpendicular to it. As the kicker interacts with the stationary electron in , the particle experiences a combination of perpendicular electric and magnetic fields. We consider a small expansion parameter, , and retain terms up to second order—specifically, terms of order —while neglecting higher-order contributions such as . This corresponds to employing the second-order kick angle approximation. It is straightforward to show that the acceleration in crossed fields leads to a particle velocity along the -axis and along the -axis. Within this second-order approximation, relativistic corrections to velocity composition—typically arising at third order—do not appear. We begin by analyzing the textbook treatment of the composition of motions. Let be a reference frame fixed with respect to a particle located downstream of the kicker. As is well known, non-collinear Lorentz boosts do not commute. However, in our second-order approximation, we can neglect the difference between and , where . Here and in the ultrarelativistic regime of interest. Therefore, we can approximate the transformation from coordinates in frame to coordinates in using a sequence of two commuting non-collinear Lorentz boosts: , where and are unit vectors along the and axis respectively. It is important to note that, as seen by an observer in , the axes of frame remain parallel to those of . The full transformation from the lab frame to the downstream particle frame can thus be written as: . In the special case where the velocities are collinear, the composition of boosts simplifies: , leading to the overall transformation: . Textbooks typically emphasize that, within the Lorentz framework, a magnetic field can only alter the direction of an electron’s motion, but not its speed. According to standard textbooks, in relativistic kinematics, the composition of perpendicular velocities is analyzed using Einstein’s velocity addition theorem. We previously discussed the law of velocity composition in Section 16.6, where it was shown that, within the Lorentz coordinatization, velocity addition is governed by the Wigner rotation. In the ultrarelativistic approximation, a simple result emerges: . This implies that a Lorentz transformation induces a rotation of the particle’s velocity by an angle approximately equal to (Fig. 50). This result highlights that the apparent contradiction is rooted in the presence of the Wigner rotation. Now let us revisit the kinematics of a relativistic electron accelerated by the kicker field, analyzing it from the perspective of an inertial lab observer without changing reference frames. Specifically, we consider an electron in the lab frame undergoing acceleration due to the kicker field. The simplest synchronization method involves maintaining the same set of uniformly synchronized clocks as in the pre-kick state, without modification. This approach, based on the absolute time (or absolute simultaneity) convention, preserves simultaneity. The standard Galilean velocity addition rule is then applied to track the electron’s motion instant by instant as it moves through a constant magnetic field. By combining Galilean transformations with variable changes, we ultimately derive the Lorentz transformation within the framework of absolute time coordinatization in the lab frame. This analysis establishes a direct link between the electron’s speed reduction after the kick (in Lorentz coordinates) and time dilation. Suppose that, upstream of the kicker, we choose a Lorentz coordinate system in the lab frame. Immediately after the electron enters the magnetic field, its velocity changes by an infinitesimal amount along the -axis. At this initial stage, Eq. (22) allows us to express the differential in terms of the differential within the Lorentz coordinate system defined upstream of the kicker. If clock synchronization is fixed, this corresponds to adopting the absolute time convention. To maintain Lorentz coordinates in the lab frame—as discussed previously—we must perform a clock resynchronization by introducing an infinitesimal time shift. The simplest case arises when the kick angle is very small, allowing us to work up to second-order terms in . This restriction greatly simplifies the calculations for two reasons. First, relativistic corrections due to the composition of non-collinear velocity increments only appear at order and can therefore be neglected. Second, time dilation effects also enter only at higher orders. Thus, Eq. (22) enables us to express the small velocity change due to the kick in the initial Lorentz coordinate system and defer clock resynchronization until after the kicker. Consequently, the post-kick motion can be described as a composition of two Lorentz boosts in the perpendicular and directions. The first boost imparts a velocity along the -axis, while the second applies an additional velocity along the axis. The second-order approximation ensures that the two boosts commute. To maintain a Lorentz coordinate system in the lab frame after the electron receives a transverse kick, a resynchronization of clocks is required. In this context, operations involving the rule clock structure in the lab frame can be interpreted as a change of variables governed by the transformation in Eq. 13: , , where and the electron’s displacement is given by . This transformation effectively rescales the time variable—adjusting the synchronization of all clocks from to , with . As a result, it becomes evident that the electron’s downstream speed is no longer independent of its transverse motion in the magnetic field (see Fig. 50). Although no second-order relativistic correction appears in the transverse (-axis) velocity component, the longitudinal component is modified. Specifically, the longitudinal velocity changes to with , leading to . As a result, the total electron speed in the lab frame, after clock resynchronization downstream of the kicker, decreases from to . 19.3 Electron Motion Accelerated by a Kicker in a Bending Magnet In our relativistic, yet non-covariant, analysis of electron motion in a magnetic field, the electron maintains the same velocity—and therefore the same relativistic factor —both upstream and downstream of the kicker. Suppose the electron then enters a bending magnet, i.e., a region with a uniform magnetic field directed along the -axis (see Fig. 51). The resulting motion within the bending magnet closely resembles that predicted by non-relativistic dynamics, with the only difference being the presence of the relativistic factor in the expression for the cyclotron frequency: . The curvature radius of the trajectory follows from the relation , where is the component of the velocity perpendicular to the magnetic field . According to non-covariant particle tracking, the correction to the radius after the kick is of order . At first glance, one might expect that in covariant particle tracking, the total speed of an electron in the lab frame—after passing through the kicker—would decrease from to . This, in turn, would suggest a corresponding decrease in the magnitude of the three-momentum from to , based on our approximation. However, such a change in momentum would imply a correction to the bending radius on the order of , leading to a stark contradiction with the radius calculated using noncovariant tracking. Since the curvature radius in the bending magnet has an objective, convention-invariant meaning, this apparent discrepancy poses a paradox. The resolution lies in recognizing that, in Lorentz coordinates, the momentum three-vector transforms as the spatial part of the four-momentum under Lorentz boosts. To analyze the motion of the relativistic electron affected by the kicker, we consider a composition of Lorentz boosts that follow the particle’s trajectory. Under such a transformation, the longitudinal component of the momentum remains effectively unchanged—accurate to order . Let us verify the validity of this assertion. The four-momentum is given by . Consider the Lorentz frame , which is at rest with respect to the electron upstream of the kicker. In the special case where the electron is initially at rest in this frame, its four-momentum is . We now focus on the dynamics in frame . Acceleration due to the crossed electric and magnetic fields of the kicker induces a velocity component along the -axis, and a second-order longitudinal component along the -axis. After the interaction with the kicker, the four-momentum in becomes: , where the expression is evaluated to second order in , consistent with our approximation scheme. We observe that the transverse acceleration results in an increase in the time-like component of the four-momentum—that is, the energy of the electron. Specifically, the energy increases from to . Recall that frame is related to the lab frame via a Lorentz boost. Under a boost to a frame moving with velocity , the longitudinal component of the momentum (which is normal to the magnetic field of the bending magnet) transforms as: . This shows that, within our approximation, the momentum component along the -axis remains unchanged. Furthermore, applying the Lorentz transformation to the time component yields: , consistent with our earlier result. It is intriguing to examine the implications of having two distinct approaches—covariant and noncovariant—that yield the same particle three-momentum. The key insight is that both frameworks accurately describe the same physical reality. The curvature radius of the trajectory in a magnetic field, and consequently the three-momentum, possesses an objective meaning, making it invariant under different conventions. 19.4 Redshift of the Synchrotron Radiation Critical Frequency Next, we explore the intriguing problem of synchrotron radiation emission in a bending magnet, both with and without the influence of a preceding kick. We will focus on the physical interpretation without delving into computational details (see the next chapter for a more thorough analysis). Consider the setup illustrated in Fig. 51. An ultrarelativistic electron is initially moving along the -axis in the laboratory frame. Before entering a uniform magnetic field directed along the -axis (i.e., a bending magnet), the electron receives a small transverse kick from a weak dipole field oriented along the -axis. An electron undergoing acceleration along a curved trajectory emits electromagnetic radiation. At relativistic speeds, this radiation is concentrated into a narrow cone, tangent to the electron’s path. Furthermore, the radiation amplitude is significantly enhanced in this direction—a phenomenon known as Doppler boosting. Synchrotron radiation arises specifically when a relativistic electron is accelerated by a bending magnet. In the ultrarelativistic limit, the key features of the emitted spectrum can be intuitively understood to depend primarily on the small difference between the speed of the electron and the speed of light. We now turn to the radiation emitted by an ultrarelativistic electron traversing a bending magnet. Consider the case in which the electron is moving toward the observer (Fig. 52). The electromagnetic source follows a trajectory over time, where denotes the emission time of the radiation. However, because electromagnetic signals propagate at a finite speed (the speed of light), a signal emitted at time from position reaches the observer at a later time . Consequently, the observer perceives the motion of the source as a function of , not . It is important to note that the prime notation here indicates retarded time and should not be confused with primes denoting quantities in a Lorentz-transformed frame, as used in subsequent sections. Let the coordinates of the electron be denoted by , where the -axis is aligned with the direction of observation. We continue to assume that the detector is located far from the radiation source. At a given moment , the electron’s position components are , , and . The distance to the observer is approximately . If we denote the observation time by , then is not equal to ; rather, it is delayed by the time it takes light to travel from the electron to the observer. Disregarding a constant delay—equivalent to shifting the time origin—we find the relation . Our goal is to express as a function of the observation time , rather than the emission time . Assuming , we can use the approximation: , where is the observation angle (Fig. 52). This relation implies that the observer perceives a time-compressed motion of the electron as it travels from point to point , corresponding to an apparent spatial interval of . Let us assume (this assumption will be justified in a moment) . In this case, one has . One can distinguish between radiation emitted at point and radiation emitted at point only when compressed distance , i.e. for , where is the filtered radiation wavelength. This means that, as concerns the radiative process, we cannot distinguish between point and point on the bend such that . It does not make sense at all to talk about the position where electromagnetic signals are emitted within (here we assume that the bend is longer than ). This characteristic length is called the formation length for the bend. Physically, the formation length can be interpreted as the effective longitudinal size of the single-electron radiation source in the space-frequency domain. Importantly, radiation from a single electron is always diffraction-limited. The spatial coherence condition is expressed through the space-angle product: , where being the transverse size and the divergence of the source. Since it follows that the divergence angle is strictly related to and : . One may check that using , one obtains as it must be. In particular, at one obtains the characteristic wavelength , which is the familiar result for radiation from a bending magnet (see Fig. 53). It is evident from the preceding discussion that, according to conventional synchrotron radiation theory, introducing a kick to the electron’s motion results merely in a rigid rotation of the angular distribution of the emitted radiation, aligning it with the new direction of motion. This conclusion is intuitive: the electron retains its speed after the kick and, due to the Doppler effect, emits radiation along the new trajectory. However, a more rigorous treatment—based on the proper coupling between fields and particles—reveals a noteworthy prediction of synchrotron radiation theory for the setup described above. Specifically, there is a red shift in the critical frequency of the synchrotron radiation observed in the kicked direction. To demonstrate this effect, we turn to a covariant formulation that explicitly incorporates Lorentz transformations. When the kick is applied, covariant particle tracking predicts a nonzero red shift in the critical frequency. This shift arises because, in Lorentz coordinates, the electron’s velocity decreases from to , while the speed of light remains unchanged at the electrodynamic constant . The resulting red shift in the critical frequency can be approximated by the relation . Here, we observe a second-order correction in that is, however, multiplied by a large factor . To validate the predictions of our synchrotron radiation theory, we propose an experimental test using third-generation synchrotron radiation sources. Synchrotron radiation from bending magnets spans a broad frequency range; however, employing narrow-bandwidth sources offers a more promising approach for studying redshifts in the radiation spectrum. Narrower bandwidths enhance the sensitivity of output intensity to redshift effects and reduce the demands on beam kicker strength and photon beamline aperture. Undulators, which generate quasi-monochromatic synchrotron radiation, are particularly well-suited for this purpose. By forcing the electron beam into a periodic undulating path, they induce interference effects that significantly narrow the emitted bandwidth. Since undulators typically consist of many periods, the resulting bandwidth scales inversely with the number of periods. Consequently, insertion devices at third-generation facilities provide an effective means to boost sensitivity to redshift signals, even at small kick angles, . In essence, we argue that, despite decades of experimental work, synchrotron radiation theory remains insufficiently confirmed, and a more precise test is both timely and necessary. 20 Synchrotron Radiation 20.1 Introductory Remarks Accelerator physics has traditionally been framed within the context of classical (Newtonian) kinematics, which is fundamentally incompatible with Maxwell’s equations. Here, we aim to revisit this perspective by applying modern insights from dynamics and electrodynamics to explore a central question in greater detail: Why did the error in synchrotron radiation theory remain undetected for so long? The electromagnetic radiation phenomena under consideration are inherently complex. In a general setup, covariant calculations of the radiation process can be highly intricate. However, certain configurations allow for significant simplifications. One such case is the synchrotron radiation setup. Much like the non-relativistic limit, the ultrarelativistic limit introduces essential simplifications into covariant calculations. This is because the ultrarelativistic regime justifies the paraxial approximation: the formation length of the radiation is much longer than its wavelength, and as a result, radiation is emitted within a narrow cone of angles on the order or smaller. Consequently, the small-angle approximation becomes valid. In this regime, the transverse velocity is small compared to the speed of light, allowing us to use a second-order relativistic approximation for the transverse motion. Rather than using a small total velocity parameter as in the non-relativistic case, we now deal with a small transverse velocity parameter . The next step is to analyze the longitudinal motion using the same approach. Notably, in synchrotron radiation, the longitudinal dynamics are particularly straightforward. When expanding transformations up to second order in , relativistic corrections to the longitudinal motion are absent within this approximation. In the covariant framework, relativistic kinematic effects associated with synchrotron radiation appear as successive corrections at different orders of approximation: First order : Effects such as the relativity of simultaneity and the Wigner rotation arise. In the ultrarelativistic limit, the Wigner rotation manifests already at first order and is a direct consequence of the relativity of simultaneity. Second order : Effects such as time dilation and the relativistic correction to the velocity addition law emerge. Relativistic correction in the law of composition of velocities, which already appears in the second order, results directly from the time dilation. The first-order kinematic term plays a significant role primarily in the description of coherent radiation from a modulated electron beam. In a storage ring, however, the longitudinal positions of electrons within a bunch are essentially uncorrelated. As a result, the radiation emitted by different electrons is also uncorrelated. In this case, the total radiated power is simply the sum of the individual contributions—intensities are added rather than electric fields. For incoherent synchrotron radiation, where one considers the motion of a single ultrarelativistic electron in a constant magnetic field, relativistic effects influence only the second-order kinematic terms, specifically . 20.2 Paraxial Approximation of the Radiation Field The general method to derive the frequency spectrum is to transform the electric field from the time domain to the frequency domain by the use of the Fourier transform. First, let us rewrite Eq. (LABEL:revwied) as follows \| \| \| \| \| --- --- \| \| \| \| \| (38) \| Eq. (LABEL:revwied) and Eq. (LABEL:rrevwied) are equivalent but include different integrands. This is no mistake, as different integrands can lead to the same integral (see Appendix A1). We call the observation distance along the optical axis of the system, while fixes the transverse position of the observer. Using the complex notation, in this and in the following sections we assume that the temporal dependence of fields with a certain frequency is of the form: \| \| \| --- \| \| \| (40) \| With this choice for the temporal dependence we can describe a plane wave traveling along the positive -axis with \| \| \| --- \| \| \| (41) \| In the following, we will always assume that the ultra-relativistic approximation is satisfied, which is the case for SR setups. As a consequence, the paraxial approximation applies too. The paraxial approximation implies a slowly varying envelope of the field with respect to the wavelength. It is therefore convenient to introduce the slowly varying envelope of the transverse field components as \| \| \| --- \| \| \| (42) \| We will now replace all vectors by their components to obtain directional dependency of the synchrotron radiation. The emission angle is taking with respect to the -axis. Here is the observation angle projected onto the plane, is the observation angle projected onto the plane. The components of the unit vector can be approximated by , , , so . We consider the motion in a static magnetic field. According to conventional particle tracking the magnitude of the velocity is constant and is equal , where is the longitudinal coordinate along the path. The transverse components of the envelope of the field in Eq. (LABEL:rrevwied) in the far zone and paraxial approximation finally becomes \| \| \| \| \| --- --- \| \| \| \| \| (43) \| where the total phase is \| \| \| --- \| \| \| (44) \| \| \| (45) \| Here and are the horizontal and the vertical components of the transverse velocity of the electron, and specify the transverse position of the electron as a function of the longitudinal position, and are unit vectors along the transverse coordinate axis. 20.3 Undulator Radiation Traditionally, courses on synchrotron radiation theory begin by following the historical development of the field, starting with radiation from bending magnets. In contrast, we will begin this chapter by exploring a more advanced topic: the theory of synchrotron radiation from undulator setups, where covariant calculations of the radiation process become notably straightforward. To generate specific characteristics of synchrotron radiation, special insertion devices known as undulators are often employed. In such setups, the resonance approximation—which is always applicable—significantly simplifies the theoretical treatment. This approximation complements, rather than replaces, the paraxial one, and leverages another large parameter: the number of undulator periods . The frequency of the radiation emitted by a particle traversing an undulator can be derived by analyzing the interference between radiation produced at successive undulator periods (see Fig. 54). This frequency is subject to Doppler shifting, and the shortest wavelength is observed along the undulator axis. Under the resonance approximation, radiation at this shortest wavelength is emitted within an angular cone much narrower than (see Fig. 55). This sharply defines the relevant observation angles. Outside the diffraction angle, the intensity falls to zero with an accuracy . This leads us to an important question: Why did the error in insertion device theory remain undetected for so long? We address this in detail below, focusing specifically on radiation within the central cone (see Fig. 55). In this approximation, the electron trajectory is effectively neglected. Regardless of the strength of the undulator (i.e., the undulator parameter), the amplitude of the electron’s oscillation remains much smaller than the diffraction-limited size of the radiation at the undulator exit—again, a consequence of . As a result, undulator radiation theory, in this context, produces fields closely resembling those predicted by dipole-like instantaneous emission models. For practical purposes, especially when describing radiation into the central cone, the conventional (non-covariant) approach has therefore been considered adequate. However, there is a specific scenario in which the conventional theory breaks down. The covariant formulation predicts a non-zero red shift in the resonance frequency when the electron motion experiences perturbations—specifically, transverse kicks with respect to the longitudinal axis. Experimental observations support this correction for spontaneous undulator emission, validating the covariant approach in such cases. 20.3.1 Conventional Theory Equation (43) provides a general framework for characterizing the far-field radiation from an electron following an arbitrary trajectory. In this section, we present a straightforward derivation of the frequency-domain representation of the radiated field produced by an electron traversing a planar undulator. The magnetic field along the undulator axis is given by \| \| \| --- \| \| \| (46) \| Here , and is the undulator period. The Lorentz force is used to derive the equation of motion of the electron in the presence of a magnetic field. Integration of this equation gives \| \| \| --- \| \| \| (47) \| Here , where is the deflection parameter defined as \| \| \| --- \| \| \| (48) \| being the electron mass at rest and being the maximal magnetic field of the undulator on the axis. In this case, the electron path is given by \| \| \| --- \| \| \| (49) \| where is the oscillation amplitude. We write the undulator length as , where is the number of undulator periods. With the help of Eq. (43) we obtain an expression, valid in the far zone: \| \| \| --- \| \| \| (50) \| \| (51) \| Here \| \| \| --- \| \| \| (52) \| \| (53) \| where the average longitudinal Lorentz factor is defined as \| \| \| --- \| \| \| (54) \| The choice of the integration limits in Eq. (51) implies that the reference system has its origin in the center of the undulator. Typically, calculating the intensity distribution using Eq. (51) alone is insufficient, as it neglects additional contributions—both interfering and non-interfering—from other segments of the electron’s trajectory. To obtain the total field in a given setup, one must account for the full electron path and include these extra terms alongside Eq. (51). However, there are specific scenarios where the contribution from Eq. (51) dominates over the others. In such cases, Eq. (51) can be considered to have an independent physical significance. One such situation arises when the resonance approximation is applicable. This approximation does not replace the paraxial approximation, which is based on the condition , but is instead used in conjunction with it. It exploits another typically large parameter—the number of undulator periods, . Under these conditions, the integral over in Eq. (51) simplifies significantly. This simplification occurs regardless of the frequency of interest, due to the long integration range compared to the characteristic scale of the undulator period. A well-known expression for the angular distribution of the first harmonic field in the far zone (see Appendix A2 for a detailed derivation) can be derived from Eq. (51). This expression is axisymmetric and can therefore be represented as a function of a single observation angle, , where . The resulting distribution for the slowly varying envelope of the electric field is given by: \| \| \| --- \| \| \| (55) \| \| (56) \| Here , and \| \| \| --- \| \| \| (57) \| is the fundamental resonance frequency. Finally is defined as \| \| \| --- \| \| \| (58) \| being the -th order Bessel function of the first kind. The integration over longitudinal coordinate can be carried out leading to the well-known final result: \| \| \| --- \| \| \| (59) \| \| (60) \| where . Therefore, the field is horizontally polarized and azimuthal symmetric. Eq. (60) describes a field with a spherical wavefront centered in the middle of the undulator. 20.3.2 Why Did the Error in Insertion Device Theory Remain Undetected so Long? We have seen that, in full generality, the expression for the undulator field in the far zone under the ultrarelativistic (i.e., paraxial) approximation can be written as Eq. (141). Within the resonance approximation (), and for frequencies near the first harmonic, this expression simplifies to the well-known form given by Eq. (60), where the field is horizontally polarized and exhibits azimuthal symmetry. The angular divergence of the radiation is significantly smaller than the characteristic angle . This narrow divergence arises from the resonance nature of undulator radiation, mathematically expressed through the sinc-like term . To characterize the angular width of the radiation peak around the forward direction (), we introduce a small angular displacement . By identifying the first zero of the function at , we can define the natural angular width of the first harmonic radiation, denoted . The resulting cone of radiation with aperture is commonly referred to as the central cone. It can be shown that the angular width satisfies . We now aim to determine the characteristic transverse size of the radiation field at the exit of the undulator. Radiation emitted by the magnetic poles interferes coherently along the undulator axis, with constructive interference occurring within an angle of approximately . This corresponds to a transverse interference size at the undulator exit on the order of . Meanwhile, the amplitude of the electron’s transverse oscillation is given by . Comparing this with the interference size, we find , where we used the relation . This inequality holds regardless of the value of , since . Therefore, the electron’s oscillation amplitude is always much smaller than the diffraction-limited size of the radiation field at the undulator exit. We analyze the radiation associated with the first-order term in the expansion of Eq. (147) in powers of the small parameter (see Eq. (150)). However, this approximation neglects the term , and as a result, all information about the transverse electron trajectory in the phase factor of Eq. (53) is lost. Under this approximation, the scale of the electron’s orbit is much smaller than the radiation’s diffraction size, and Eq. (60) yields fields that closely match those predicted by dipole radiation theory. Therefore, we consider the non-covariant approach sufficient for describing transverse electron motion in this context. Several important points can be made regarding the result discussed above. As previously explained, by considering only the radiation emitted within the central cone, we overlook critical information about the electron’s transverse motion. To present a complete analysis, we must also account for the effects of acceleration along the -direction—that is, along the undulator axis. We assume that the transverse velocity is small compared to the speed of light , and we introduce as a small expansion parameter. In this framework, we neglect terms of order , corresponding to a second-order relativistic approximation for the transverse dynamics. It is worth noting that, under the ultrarelativistic approximation, analyzing the longitudinal motion is considerably simpler than analyzing the transverse motion. In a constant magnetic field, the electron acquires a transverse velocity , and its longitudinal velocity is reduced by , where is the total velocity. When we apply the relevant transformations consistent with a second-order approximation in , no relativistic correction to the longitudinal motion arises. Consequently, within this level of approximation, the distinction between covariant and non-covariant constrained electron trajectories has no impact on the undulator radiation observed in the central cone. 20.3.3 Influence of the Kick According to Conventional Theory Equation (60) can be extended to describe a particle with an initial offset and deflection angle relative to the longitudinal axis, under the assumption that the magnetic field in the undulator is independent of the particle’s transverse position (see Appendix A3). Although this generalization can be derived directly from Eq. (161), it is often more efficient to apply certain intuitive geometrical arguments that are consistent with a rigorous mathematical treatment. Consider first the effect of a transverse offset with respect to the longitudinal axis . Since the particle experiences the same magnetic field, the far-field radiation pattern is simply shifted by . As a result, Eq. (60) can be generalized by replacing the transverse observation coordinate with . Equivalently, the observation angle should be substituted with , leading to: \| \| \| \| \| --- --- \| \| \| \| \| (61) \| Let us now examine the effect of a deflection angle . Assuming the magnetic field experienced by the electron remains independent of its transverse position, the electron’s trajectory continues to follow a sinusoidal path. However, the effective undulator period is modified and becomes , due to the projection of the undulator axis onto the new trajectory. This leads to a relative red shift in the resonant wavelength given by . In typical scenarios of interest, the deflection angle can be approximated as . Consequently, the relative red shift scales as . This should be compared with the intrinsic relative bandwidth of the resonance, which is approximately , where denotes the number of undulator periods. For instance, if , the red shift induced by the deflection angle becomes negligible in all practical situations. Therefore, introducing a deflection angle effectively corresponds to a rigid rotation of the entire system. When performing this rotation, it is important to note that the phase factor in Eq. (61)—which represents a spherical wavefront emanating from —remains invariant under rotation. In contrast, the argument of the function in Eq. (61) is altered by the rotation, as the point is mapped to . As a result, after the rotation, Eq. (61) transforms accordingly: \| \| \| --- \| \| \| (62) \| \| (63) \| Finally, in the far-zone regime, we can take the limit , which justifies neglecting the term in the argument of the function, as well as the quadratic phase term . As a result, Eq. (63) simplifies further, yielding a generalized form of Eq. (60) in its final expression: \| \| \| \| \| --- --- \| \| \| \| \| (64) \| It is evident from the above that, within the framework of conventional synchrotron radiation theory, considering the radiation from a single electron at a detuning from resonance, the effect of a kick is merely a rigid rotation of the angular distribution in the direction of the new electron trajectory. This is intuitively reasonable, since after the kick, the electron maintains its velocity and emits radiation in the new, kicked direction due to the Doppler effect. Following this rotation, Eq. (60) transforms into Eq. (64). 20.3.4 Influence of the Kick According to Correct Coupling of Fields and Particles According to the correct coupling between fields and particles, undulator radiation theory makes a striking prediction regarding radiation from a single electron, both with and without a transverse kick. Specifically, when a kick is introduced, the theory predicts a red shift in the resonance wavelength of the undulator radiation along the direction of the electron’s velocity. To demonstrate this effect, we consider a covariant treatment that explicitly employs Lorentz transformations. Within this framework, introducing a kick leads to a non-zero red shift in the resonance frequency. This shift arises because, in Lorentz coordinates, the electron’s longitudinal velocity is reduced—from to -while the speed of light remains constant at its electrodynamical value, . Consequently, the expression given in Eq. (168) requires correction: it should use the modified electron velocity rather than the unperturbed velocity . The resulting shift in the total phase , appearing under the integral in Eq. (161), is given by , where we have used the ultrarelativistic approximation. Now, consider the case where the electron moves along a straight, constrained trajectory parallel to the undulator axis, without any kick. The corresponding radiation field in the far zone is described by Eq. (60). Referring back to Eq. (64), we find that the conventional undulator radiation theory provides the following expression for the radiation field after the kick: \| \| \| --- \| \| \| (65) \| \| (66) \| The covariant equations indicate that, when the kick is applied, the resulting radiation field is described by the following formula: \| \| \| --- \| \| \| (67) \| \| (68) \| This formula is nearly, but not exactly, of the same form as Eq. (66). The key difference lies in the additional term appearing in the argument of the function. Special attention should be paid to the shift in resonance frequency between the undulator radiation setups with and without a kick. Recalling the definition of the detuning parameter, , we can express the redshift in resonance frequency as . Alternatively, this redshift can also be written in terms of the relativistic factor as . This reveals a second-order correction in , which, however, is amplified by the large factor . We are now prepared to more generally examine how the field expression is modified by the introduction of a kick. Suppose that, in the absence of a kick, the electron moves along a trajectory that makes an angle with respect to the undulator axis. The corresponding field is given by Eq. (64). Let denote the kick angle of the electron relative to its initial trajectory. According to the conventional approach, the expression for the field after the kick becomes: \| \| \| --- \| \| \| (69) \| \| (70) \| In contrast, the covariant approach gives \| \| \| --- \| \| \| (71) \| \| (72) \| This brings us to an intriguing situation. According to conventional theory, the resonance wavelength depends solely on the observation angle relative to the electron’s velocity direction. However, Eq. (70) shows that for any kick angle and for any angle between the undulator axis and the initial electron velocity, the radiation emitted along the velocity direction experiences no redshift. This highlights a critical distinction between the conventional and covariant formulations. In contrast, the result from the covariant approach—Eq. (72)—explicitly depends on the magnitude of the kick angle . In this case, radiation along the velocity direction exhibits a redshift only when the kick angle is nonzero. We are thus led to an important conclusion: when the electron is accelerated in the lab frame upstream of the undulator, the covariant trajectory retains this information. 20.3.5 Experimental Test of SR Theory in 3rd Generation Light Source One way to highlight the incompatibility between the standard approach to relativistic electrodynamics, which typically involves Maxwell’s equations, and the particle trajectories derived from non-covariant particle tracking is through a direct laboratory test of synchrotron radiation theory. Let us now explore the potential of synchrotron radiation sources to validate the predictions of a revised synchrotron radiation theory. In modern synchrotron radiation sources, the electron beam emittance is sufficiently small that one can disregard the effects of finite beam size and angular divergence, particularly in the soft X-ray wavelength range. This makes it feasible to model the synchrotron radiation source using the approximation of a filament electron beam, thereby enabling the use of analytical models for single electron synchrotron radiation fields. The basic setup for the test experiment is illustrated in Fig. 56. The soft X-ray undulator beamline should be tuned to a minimum photon energy, typically corresponding to the ”water window” wavelength range. The radiation pulse passes through a monochromator filter , and its energy is then measured by the detector. Precise monochromatization of the undulator radiation is not required in this case; a monochromator line width of is sufficient. To conduct the proposed test experiment, it is essential to control the beam kick, for example, using a corrector magnet. Without the kick, the maximum pulse energy detected will align with the monochromator line tuned to resonance. However, when the kick is applied, conventional synchrotron radiation theory predicts no redshift in the resonance wavelength. In contrast, one of the immediate consequences of the corrected theory is the occurrence of a non-zero redshift. The proposed experimental procedure is relatively simple, as it involves relative measurements in the electron beam’s velocity direction, both with and without the transverse kick. This measurement is crucial because conventional theory predicts the absence of redshift, a result that, to our knowledge, has never been observed at synchrotron radiation facilities. However, an XFEL-based experiment confirms our correction for spontaneous undulator emission—further details can be found in the next chapter. 20.4 Synchrotron Radiation from Bending Magnets Consider a relativistic electron moving along a circular orbit. In the standard treatment, the observer is located in a vertical plane tangent to the electron’s circular trajectory at the origin, positioned at an angle above the orbital plane. In this configuration, the -axis is not fixed but varies with the observer’s position. Notably, the electron’s motion exhibits cylindrical symmetry about the vertical axis through the center of the orbit. Owing to this symmetry, it is sufficient—when calculating the spectral and angular distribution of emitted photons—to consider an observer in this specific geometric arrangement, without resorting to a more general configuration. It is important to emphasize the distinction between the general geometric setup used here and the more restrictive arrangement often assumed in standard synchrotron radiation treatments of bending magnet radiation. To proceed, we begin with a broad overview of the main results. The goal is to solve the electrodynamics problem using Maxwell’s equations in their conventional form. Since the particle undergoes relativistic acceleration, its dynamics must be analyzed within the framework of special relativity. However, defining Lorentz coordinates for the laboratory frame becomes nontrivial in the presence of acceleration. The only consistent method to introduce Lorentz coordinates in this context is to employ individual coordinate systems—so-called ”ruler-clock” structures—at each point along the electron’s trajectory. Several notable effects emerge from the cylindrical symmetry and the use of the paraxial approximation. In our analysis, we demonstrate that the derivation of bending magnet radiation does not require covariant particle tracking. Nonetheless, there is one key instance in which conventional theory fails: the covariant approach predicts a nonzero redshift of the critical frequency when perturbations influence the electron’s motion along the magnetic field—that is, when the trajectory deviates from the nominal orbit. 20.4.1 Conventional Theory Consider a single relativistic electron moving along a circular orbit and an observer. In conventional treatments, the horizontal observation angle is typically assumed to be zero. This simplification arises because most textbooks focus on calculating the intensity radiated by a single electron in the far field—specifically, the square modulus of the field amplitude—without addressing more complex scenarios such as source imaging. Equation (43) can be used to calculate the far-zone radiation field emitted by a relativistic electron moving along a circular arc. Assuming a geometry with a fixed -coordinate, the transverse position of the electron can be expressed as a function of the curvilinear abscissa as follows: \| \| \| --- \| \| \| (73) \| and \| \| \| --- \| \| \| (74) \| where is the bending radius. Since the integral in Eq. (43) is performed along we should invert in Eq. (74) and find the explicit dependence : \| \| \| --- \| \| \| (75) \| so that \| \| \| --- \| \| \| (76) \| where the expansion in Eq. (75) and Eq. (76) is justified, once again, in the framework of the paraxial approximation. With Eq. (43) we obtain the radiation field amplitude in the far zone: \| \| \| --- \| \| \| (77) \| where \| \| \| --- \| \| \| (78) \| \| \| (79) \| One can easily reorganize the terms in Eq. (79) to obtain \| \| \| --- \| \| \| (80) \| \| \| (81) \| With redefinition of as under integral we obtain the final result: \| \| \| --- \| \| \| (82) \| \| \| (83) \| where \| \| \| --- \| \| \| (84) \| and \| \| \| --- \| \| \| (85) \| In standard treatments of bending magnet radiation, the phase term is absent. The horizontal observation angle is always equal to zero. 20.4.2 Why Did the Error in Synchrotron Radiation Remain Undetected so Long? Our case of interest involves an ultrarelativistic electron undergoing circular acceleration. As previously noted, conventional (non-covariant) particle tracking describes the dynamical evolution in the laboratory frame using the absolute time convention. In this framework, simultaneity is treated as absolute, requiring only a single set of synchronized clocks in the lab frame to describe the electron’s accelerated motion. However, adopting the absolute time convention leads to significantly more complex field equations, which vary with the particle’s velocity—that is, they differ at each point along the electron’s trajectory. This complexity is precisely why the covariant approach is preferred when treating both dynamics and electrodynamics. We begin by considering an electron moving along a circular trajectory that lies in the -plane and is tangent to the -axis. Due to the cylindrical symmetry of the problem, it is not necessary to consider a general observation point to calculate the spectral and angular distributions of the emitted photons. Instead, we assume the observer is located in the vertical plane tangent to the trajectory at the origin. Within the ultrarelativistic (paraxial) approximation, we perform transformations accurate up to order , which significantly simplifies the calculations. This second-order approximation allows us to neglect higher-order terms while retaining essential relativistic effects. In the lab frame, manipulations involving the rule-clock structure can be interpreted as a change of variables governed by the transformation in Eq. (13): , , where reflects the second-order approximation. This combination of a Galilean transformation and a change of variables effectively results in a transverse Lorentz transformation. Since this modified Galilean approach is mathematically equivalent to a Lorentz transformation, applying these variable changes naturally leads to the correct form of Maxwell’s equations. To retain Lorentz coordinates in the laboratory frame, as previously discussed, it suffices to apply a time transformation given by . At this order of expansion, the relativistic correction to the particle’s offset does not appear; such corrections arise only at order . Therefore, for our case of interest, . Although this time transformation was initially derived for a specific scenario, the result generalizes to any transverse velocity direction. In general, we can write: . To complete our analysis, we now consider the relativistic correction to longitudinal motion. We emphasize again that when evaluating the transformations up to second order in , no correction to the longitudinal motion appears at this level of approximation. In summary, we have demonstrated a covariant approach applicable to arbitrary trajectories—providing a general framework for analyzing the system directly in the space-frequency domain under the paraxial approximation. Let us now explore how to apply the covariant method to a specific case. We will use our understanding of the relativistically correct approach for calculating synchrotron radiation emission to determine the photon angular-spectral density distributions from a bending magnet. In the ultrarelativistic limit, the electron undergoes uniform transverse acceleration, given by . Under this approximation, the electron’s velocity and displacement in the transverse direction can be expressed as: , . With these expressions, we now have all the quantities needed to evaluate the relativistic time shift: . There is no time difference! This means that covariant particle tracking is not required to derive the radiation from a bending magnet. Such a precise cancellation often hints at a deeper underlying principle. However, in this case, it appears to be a mere coincidence—there is no evident profound significance. This cancellation is not surprising when one analyzes the general expression for the radiation field from a bending magnet in the far zone, given by Eq. (83). In our previous discussion of undulator radiation, we learned that the relativistic correction appears only when the transverse electron trajectory is included in the total phase under the integral Eq.(43). Referring back to Eq.(45) for the phase factor , we see that the term which depends on the transverse position of the electron can be written as . We conclude that the observation angle in the total phase factor under the integral must be related to the contribution of the transverse electron trajectory. Now, examining Eq. (83), we observe that the phase factor includes only the component of the observation angle. This implies that the transverse constrained motion of the electron, in the bending magnet does not contribute to synchrotron radiation. Therefore, it is justified to use a non-covariant approach when analyzing the constrained electron motion along the nominal orbit in the -plane. We emphasize that the cancellation of the relativistic time shift and the independence of the Fraunhofer propagator (more precisely, the paraxial approximation of the Green’s function for the inhomogeneous Helmholtz equation in the space-frequency domain) with respect to the observation angle in the far zone are two aspects of the same phenomenon. Both arise from the cylindrical symmetry inherent in the motion of an electron along a circular arc. Due to this symmetry, calculating the spectral and angular photon distributions in the far field does not require placing the observer at a general position. Instead, it suffices to consider an observer located in the vertical plane tangent to the circular trajectory at the origin. In this configuration, the observation angle , while corresponds to elevation above the orbital plane. In other words, within this specific geometry, the -axis is effectively defined by the observer’s position. However, this approach offers limited insight into the near-field radiation, where the Fresnel propagator must be used. In the near zone, the radiation field is sensitive to the detailed, constrained motion of the electron. Far-field arguments, though, are useful in demonstrating that synchrotron radiation from bending magnets is unaffected by differences between non-covariant and covariant electron trajectories. Te near-zone field can be calculated from the knowledge of the far-zone field. This is made possible through the use of the space-frequency domain and by exploiting the paraxial approximation . The cancellation of the relativistic time shift ensures consistent results in both the far and near zones, as expected 646464It is important to emphasize that the inverse field problem—based solely on far-field data—cannot be solved without invoking the paraxial approximation.. 20.4.3 Influence of the Kick According to Conventional Theory Up to this point, we have examined the case of an electron moving along a circular trajectory confined to the -plane and tangent to the -axis. The resulting phase difference in the emitted fields is influenced by both the observer’s position and the specifics of the electron’s motion. We now shift focus to the radiation emitted by a single electron traversing a bending magnet, allowing for arbitrary angular deflection and spatial offset relative to the nominal orbit. An approximate representation of the electron’s path—given by Eqs. (155) and (158) in Appendix A3—can be used to characterize the emitted field for a general trajectory. By applying Eqs. (75) and (76), we derive an approximate expression for : \| \| \| --- \| \| \| (86) \| so that \| \| \| --- \| \| \| (87) \| and \| \| \| --- \| \| \| (88) \| It is clear that the offsets and are consistently subtracted from and , respectively. This implies that a shift in the particle trajectory on the vertical plane is equivalent to an opposite shift of the observer. Taking this into account, we define the modified angles and in order to obtain \| \| \| --- \| \| \| (89) \| and \| \| \| --- \| \| \| (90) \| \| \| (91) \| One can easily reorganize the terms in Eq. (91) to obtain \| \| \| --- \| \| \| (92) \| \| \| (93) \| \| \| (94) \| \| \| (95) \| Redefinition of as gives the result \| \| \| --- \| \| \| (96) \| \| \| (97) \| where \| \| \| --- \| \| \| (98) \| and \| \| \| --- \| \| \| (99) \| In the far zone we can neglect terms in and , which leads to \| \| \| --- \| \| \| (100) \| \| \| (101) \| where \| \| \| --- \| \| \| (102) \| and \| \| \| --- \| \| \| (103) \| It is evident from the discussion above that the field distribution in the far zone depends solely on the observation angle relative to the direction of the electron’s velocity. According to the conventional (though incorrect) approach to coupling fields and particles, radiation theory predicts the behavior of bending magnet radiation from a single electron, both with and without an applied kick. Specifically, when a kick is introduced, the angular distribution in the far zone is predicted to undergo a rigid rotation. 20.4.4 Influence of the Kick According to Correct Coupling of Fields and Particles Let us now discuss the covariant treatment, which explicitly employs Lorentz transformations. Consider the radiation from a bending magnet, emitted by a single electron that receives a transverse kick with respect to the nominal orbit in the -plane. Due to the kick, there is an additional translation along the -axis with a constant velocity . The corresponding offset of the electron can be written as: . We now incorporate the velocity and offset into the relativistic time shift: . Thus, the resulting shift in the total phase accumulated along the path is given by: . This result is consistent with our redshift calculation in the undulator case when a kick is introduced, as expected. We would like to make a historical note: the distinction between covariant and non-covariant particle trajectories was not fully understood in the past. As a result, accelerator physicists did not recognize that relativistic kinematic effects contribute to synchrotron radiation. This raises an important question: how can storage rings function effectively under such assumptions? The key point is that in this context, the dynamics of the electron beam are primarily governed by synchrotron radiation emitted from bending magnets. Due to the cylindrical symmetry of the system, both covariant and non-covariant descriptions of electron motion along a circular arc produce similar synchrotron radiation characteristics—with one important exception. The covariant framework predicts a small redshift in the critical frequency of the emitted radiation when the electron motion experiences perturbations in the vertical direction. However, since synchrotron radiation from bending magnets spans a broad frequency range, the overall output intensity is largely insensitive to this redshift. 20.5 Problem-Solving: Multiple Trajectory Kicks We now consider the case of arbitrarily spaced kickers, each introducing a distinct rotation angle. Our goal is to explore how covariant particle tracking can be applied in this scenario and to gain insight into the resulting dynamics—particularly when an undulator is placed downstream of the kicker configuration. Formally, calculating the radiation from the undulator requires accounting for all trajectory perturbations experienced by the electron since its generation. This may seem surprising at first, but it aligns with the general expression for the radiation field from a single electron, as given by Eq. (LABEL:revwied). It is important to note that, in principle, the entire history of the electron - from to -must be considered, since the integral in Eq. (LABEL:revwied) extends over this range. However, the physical interpretation of this statement depends on the specific scenario. In practice, the integration should be restricted to the time intervals during which the electron contributes significantly to the radiation field. Ultimately, it is electrodynamics itself that determines which portions of the particle’s trajectory are relevant for calculating the undulator radiation and which may be neglected. The most crucial general statement in this context is that the trajectory must be computed using covariant methods—provided one wishes to employ the standard formulation of Maxwell’s equations. Let us consider the ultrarelativistic assumption , which is commonly valid in typical synchrotron radiation setups. In general, introducing a small parameter into a physical theory leads to significant simplifications. Specifically, the ultrarelativistic approximation corresponds to a paraxial regime, under which Eq. (LABEL:revwied) simplifies to Eq. (43) Let us now consider a scenario in which the deflection angle of the first bending magnet upstream of the undulator is much larger than . In other words, we examine a standard synchrotron radiation setup, where an electron enters the system through a bending magnet, traverses a straight section, passes through an undulator, continues along another straight section, and exits via a second bending magnet. Although the integration in Eq.(43) is performed from to , the only (edge) part of the trajectory into the bending magnets contributing to the integral is of the order of the radiation formation length . Mathematically, this is reflected in the behavior of in Eq.(43), which exhibits increasingly rapid oscillations as exceeds the formation length. At the critical wavelength, the formation length is approximately , where is the bending radius. This corresponds to an angular interval along the orbit of roughly . Typically, the critical wavelength of the radiation from a bending magnet in a synchrotron radiation source is about 0.1 nm. The formation length in this case is only a few millimeters. Note that for ultrarelativistic systems, the formation length is generally much longer than the radiation wavelength. This seemingly counterintuitive result arises because, in ultrarelativistic systems, it is not possible to localize radiation sources within a macroscopic portion of the trajectory. The formation length can be interpreted as the longitudinal size of a single electron source. It is meaningless to specify the exact location where electromagnetic signals are emitted within the formation length. As a result, in the context of the radiative process within a bending magnet, we cannot distinguish between radiation emitted at point and radiation emitted at point if the distance between these two points is shorter than the formation length . Now, consider the case of a straight section of length placed between the bending magnet and the undulator. The same reasoning applied to the bending magnet can be used to define a region of the trajectory in which it is indistinguishable to the observer whether radiation originates from different points. Just as in the bending magnet case, the observer perceives a time-compressed motion of the source. For straight-line motion, the apparent time corresponds to an apparent distance . At the critical wavelength, the bending magnet formation length is of the same order as the straight section formation length . Intuitively, bending magnets can be seen as ”switchers” for the ultrarelativistic electron trajectory. While we focus on bending magnets here, other configurations could also serve as switchers, provided they share a common characteristic: the switching process must depend exponentially on the distance from the starting point. In this context, a characteristic length can be associated with any switcher. Consider, for example, a plasma accelerator, where an electron is accelerated by high-gradient fields. In this case, the accelerator itself acts as a switcher for the relativistic electron trajectory, since acceleration in the GeV range occurs over a distance of only a few millimeters. In the (soft) X-ray range, the acceleration distance is shorter than the formation length for the following straight section. In this particular case length plays the role of the characteristic length of the switcher , which switches on the ultrarelativistic electron trajectory. Let us now return to our examination of the standard synchrotron radiation setup and analyze the radiation process in an insertion device, such as an undulator. The ”creation” of the relativistic electron occurs within a distance on the order of from the start of the straight section upstream of the undulator. It is assumed that the length of the straight section, , is much longer than the formation length, , which is always the case in the X-ray range. When the switching distance , the specifics of the switcher become irrelevant for describing the radiation from the undulator installed within the straight section (see Fig. 57). Downstream of the switcher, the electron moves uniformly. The fields associated with an electron traveling at a constant velocity exhibit interesting behavior as the speed of the charge approaches that of light. Specifically, in the space-frequency domain, the fields of a relativistic electron become equivalent to those of a beam of electromagnetic radiation. For a rapidly moving electron, the transverse electric and magnetic fields are nearly equal in magnitude and mutually perpendicular. These fields are indistinguishable from the radiation fields of a beam. This virtual radiation beam has a macroscopic transverse size on the order of (see Appendix A4). At the exit of the switcher, we have a ”naked” (or ”field-free”) electron, meaning an electron that is not accompanied by any virtual radiation fields. Within a distance on the order of downstream of the switcher, the electron undergoes a process of field formation, becoming a ”field-dressed” electron—one that is now accompanied by the fields generated by its fast motion. The electron’s trajectory can be divided into two fundamentally distinct segments: before and after the switcher. When the electron is accelerated upstream of the switcher in the lab frame, information about this acceleration is embedded in the first segment of its covariant trajectory. However, this pre-acceleration history—along with the associated electromagnetic fields of the ultrarelativistic electron—is effectively ”washed out” during the switching process. As a result, the electron enters the straight section in a ”naked” state, devoid of its previous field configuration. We begin by describing the field formation process along the straight section downstream of the switcher, using a covariant framework. The first step is to synchronize distant clocks within the lab frame, employing the standard Einstein synchronization procedure. Within this Lorentzian lab frame, we assume the electron follows a rectilinear trajectory at constant velocity ; this serves as our initial condition. From here, the evolution of the electromagnetic field can be analyzed using Maxwell’s equations. When one analyzes the process of ”field-dressed” electron formation from the viewpoint of the noncovariant approach, one assumes the same initial conditions (rectilinear trajectory with velocity ) for the electron motion. Then one solves the electrodynamics problem of field formation by using the usual Maxwell’s equations. We already mentioned that the type of clock synchronization that results in the time coordinate in an electron trajectory is never discussed in accelerator physics. However, we know that the usual Maxwell’s equations are only valid in the Lorentz frame. The noncovariant approach is obviously based on a definite synchronization assumption, but this is actually a hidden assumption. In our case of interest, within the lab frame, the Lorentz coordinates are then automatically enforced. So one should not be surprised to find that in this simple case of rectilinear motion, there is no difference between covariant and noncovariant calculations of the initial conditions at the undulator entrance. Due to the nature of undulator radiation, the radiation field within the central cone can be accurately calculated using the instantaneous (dipole-like) approximation. This justifies the use of a noncovariant approach to describe the constrained electron motion through the undulator. In conclusion, for a standard synchrotron radiation setup, it does not matter whether one adopts the covariant approach with Einstein synchronization or the noncovariant approach based on absolute time. Both yield the same results for the radiation field within the central cone. Let us now examine the behavior of a weak dipole magnet (commonly referred to as a kicker) installed in the straight section upstream of the undulator. This kicker is characterized by a small kick angle such that . What should we expect in terms of undulator radiation under these conditions? At first glance, the setup appears similar to the switcher configuration: the electron trajectory is again divided into two segments—before and after the kicker. However, there is a crucial difference. In this case, electrodynamics dictates that both segments of the trajectory must be considered when calculating the resulting undulator radiation. As the electron passes through the kicker, there is no significant synchrotron radiation. More precisely, any emitted radiation is indistinguishable from the electron’s self-fields. As a result, the virtual radiation fields are not washed out as they are in the switcher scenario. The electron emerges from the kicker still ”field-dressed,” but with a perturbed electromagnetic field that now carries information about the acceleration experienced relative to an inertial frame. According to conventional theory—rooted in Newtonian kinematics—the Galilean vector law of velocity addition is applied. Within this non-covariant framework, the electron’s direction changes after the kick, but its speed remains constant. In contrast, covariant particle tracking—based on Lorentz coordinates—leads to a different prediction: the electron’s speed is slightly reduced, from to . This discrepancy arises because the covariant approach properly accounts for the relativistic addition of non-parallel velocities. In standard electrodynamics, the usual algorithm involves solving Maxwell’s equations using particle trajectories derived from non-covariant tracking. According to this method, the undulator radiation emitted along the post-kick velocity direction exhibits no redshift in resonance frequency, regardless of the kick angle . However, when the coupling between fields and particles is treated correctly within the covariant framework, a notable prediction emerges. Specifically, synchrotron radiation theory indicates a redshift in the resonance frequency of the undulator radiation in the kicked direction. This redshift is given by: . This result underscores the importance of using covariant dynamics in accurately modeling radiation phenomena in relativistic beamlines. 20.6 Helical Trajectories and Synchrotron Radiation The observation of a redshift in bending magnet radiation inherently suggests a similar issue within the conventional theory of cyclotron radiation. In the ultrarelativistic regime, well-established analytical expressions describe the spectral and angular distribution of radiation emitted by an electron moving in a uniform magnetic field, where the motion has a non-relativistic component parallel to the field and an ultrarelativistic component perpendicular to it. According to the conventional framework—just as in the case of bending magnet radiation—the angular and spectral distribution of the emitted radiation depends on the total velocity of the particle, a consequence of the Doppler effect. In contrast, the covariant approach predicts a redshift of the critical frequency that arises specifically when there are perturbations in the electron’s motion along the direction of the magnetic field. It is important to note that cyclotron-synchrotron radiation is a fundamental emission process in both plasma physics and astrophysics. Therefore, the corrections we propose have significant implications extending well beyond synchrotron radiation facilities. 20.6.1 Existing Theory Let us now examine the relativistic cyclotron radiation in greater detail. Here, we present only the final results and discuss their connection to the conventional synchrotron radiation theory associated with bending magnets. In the case of uniform translational motion with non-relativistic velocity along the direction of the magnetic field (see Fig. 58), a widely accepted expression in astrophysics describes the angular and spectral distributions of radiation emitted by an ultra-relativistic electron following a helical trajectory656565The angular and spectral distributions of radiation from an ultra-relativistic electron on a helical orbit were derived in [74, 75]. These results are now standard examples in the literature (see, e.g., ) and are not elaborated upon here. \| \| \| \| --- \| \| \| (105) \| \| \| where and are the modified Bessel functions, , , ( is the angle between and and that between and ); the angle is the angular distance between the direction of the electron velocity and the direction of observation . Here the is defined by . We have already discussed the radiation emitted by an ultrarelativistic electron following a helical trajectory in the previous section. Equation (101) presents the result we derived earlier for synchrotron radiation from a single electron undergoing an angular deflection relative to the nominal orbit. At first glance, Eq. (105) appears different from Eq. (101). However, the two become equivalent upon introducing the small deflection angle and the observation angle , under the assumption that the observer lies in the vertical plane tangent to the trajectory (i.e., ). The integrals in Eq. (101) can then be expressed in terms of modified Bessel functions: \| \| \| --- \| \| \| (106) \| \| \| (107) \| Then, making the necessary variable changes, the formula reduces to Eq.( 105). 20.6.2 Methodology of Solving Problems Involving Boosts The derivation leading to Eq. (105) is rather involved, so it is useful to present an independent and more intuitive argument. A particularly simple way to analyze radiation from ultrarelativistic helical motion leverages special relativity and requires minimal calculation. In the case of uniform translational motion, radiation analysis can often be simplified by identifying a reference frame in which the problem is already solved—such as the frame where the particle undergoes circular motion—and then transforming the result back to the original frame. The reference system in which the electron moves in circular motion can be transformed to a reference system in which the electron proceeds following a helical trajectory. Eq. ( 105) holds, indeed, in the frame for a particle whose velocity is . The Lorentz transformation, which leads to the value for the -component of the velocity yields , where , is the velocity of the electron in the frame and the phase angle is invariant. This means that, in order to end up in with a transverse (to the magnetic field direction) velocity , one must start in with . In the ultrarelativistic approximation , and one finds the simple result , so that a Lorentz boost with non-relativistic velocity leads to a rotation of the particle velocity of the angle (if angle is small and , we would write ). If one transforms the radiation field for a particle in a circular motion in the system , one obtains the result that the effect of a boost amounts to a rigid rotation of the angular-spectral distribution of the radiation emitted by the electron moving with velocity on a circle that is, once more, Eq. ( 105) 666666A covariant approach to analyzing radiation from helical motion is presented in . It is generally assumed that , which is why makes no distinction between the covariant and non-covariant treatments of electron motion along a helix downstream of the kicker setup.. Note: the subscript ”⟂” used here to indicate the velocity in the transverse to the magnetic field direction should not be confused with the ”⟂” referring to an acceleration of the electron in the transverse direction in the proceeding sections. It follows quite naturally that the covariant analysis of radiation from helical motion, as considered above, is grounded in the use of Lorentz transformations. In other words, within the laboratory frame, Lorentz coordinates are inherently applied. It is assumed that, in this Lorentz lab frame, the electron follows a helical trajectory with velocity , which serves as an initial condition. In the ultrarelativistic approximation, applying a Lorentz boost along the direction of the magnetic field with a non-relativistic velocity transforms the motion into a circular trajectory, still with velocity . As a result, a single boost along the field direction does not alter the radiation properties. Let us consider synchrotron radiation from a single electron that has received a transverse kick relative to the nominal orbit in the -plane. As a result of this kick, the electron experiences an acceleration that induces a velocity component along the -axis and a corresponding change along the -axis. Limiting the analysis to second-order terms simplifies the calculations significantly. To describe the motion of the particle downstream of the kicker, we can employ a sequence of two commuting, non-collinear Lorentz boosts. When the kick is applied, covariant particle tracking predicts a modification to the initial conditions at the entrance of the synchrotron radiation setup. If we retain the Lorentz coordinate system of the lab frame downstream of the kicker, we find that the covariant velocity of the particle on the resulting helical orbit decreases from to . Applying a covariant analysis of radiation for a helical trajectory with this reduced velocity leads to a redshift in the critical wavelength. It is important to note that the single passive Lorentz boost to the reference frame , discussed above, is merely a mathematical device. One can interpret this transformation as a change of variables, rather than a physical alteration of the system. Under this kinematic transformation, neither the electron’s nor the observer’s motion relative to the fixed stars is affected. True (i.e., physically real) acceleration is a dynamic process and cannot be captured by coordinate transformations alone. Therefore, we conclude that accelerating an electron with respect to the fixed stars in the lab inertial frame—before it enters the uniform magnetic field—is an absolute acceleration. The effects of this acceleration are intrinsically encoded in the covariant particle trajectory. 20.6.3 On the Advanced ”Paradox” Related to the Coupling Fields and Particles We now wish to highlight that two distinct sets of initial conditions can result in the same uniform translation along the magnetic field direction in the Lorentz lab frame. To illustrate this, we first consider an electron moving along a circular trajectory in the -plane. Next, we rotate the magnetic field vector within the -plane by a small angle , under the assumption that . In this configuration, the electron moves uniformly along the direction of the magnetic field with a velocity component . Importantly, this rotation of the magnetic field does not affect the radiation characteristics of the electron in circular motion. This invariance is intuitive: after rotating the bending magnet, the electron retains its velocity, and due to the Doppler effect, it continues to emit radiation along the direction of motion. The change in the curvature radius due to the rotation is of order , and thus can be neglected. We now consider a second scenario. Suppose a kicker is installed in the straight section upstream of the bending magnet, imparting a kick in the -direction with angle . When such a kick is applied, a redshift in the critical wavelength of the emitted radiation occurs. This is a consequence of the relativistic velocity addition law: the longitudinal component of the electron’s velocity is reduced from to after the kick. The relative redshift of the critical frequency is given by: . Here we observe a second-order correction in , but it is enhanced by the large factor . In the covariant framework, the outcome depends on the absolute magnitude of the kick angle . Radiation emitted along the velocity direction experiences a redshift only when the kick angle is nonzero. This result reflects the concept of ”absolute” acceleration—acceleration relative to the fixed stars. The distinction between these two situations—both ultimately resulting in uniform translation along the direction of the magnetic field—is quite intriguing. This difference arises due to the presence of two distinct Lorentz coordinate systems within the lab frame. When we accelerate the electron upstream of the bending magnet, we effectively change its associated Lorentz frame. To maintain a consistent Lorentz coordinate system in the lab frame after the electron receives a transverse kick, a clock resynchronization must be performed. Consequently, we should expect the electron’s velocity to be modified. This explains the difference between the two experimental setups: when the electron’s motion upstream of the bending magnet remains unperturbed (relative to the fixed stars), no clock resynchronization is necessary. However, when we do perturb the electron’s motion, resynchronization is required. This leads us to consider an apparent paradox. The argument is as follows: under absolute time coordinatization in the lab frame, the initial conditions at the entrance to the bending magnet appear to be identical in both cases. Specifically, the electron’s velocity magnitude and its orientation with respect to the magnetic field are the same. Yet, when we accelerate the electron upstream of the magnet, this information is seemingly absent from its noncovariant trajectory. So, where is the information about this acceleration encoded? Since the electron is traditionally considered a structureless particle, the situation seems paradoxical. However, this paradox hinges on a subtle but critical point: an ultrarelativistic electron is not truly structureless in practice. Consider an electron moving uniformly at a constant velocity. As its speed approaches the speed of light, the electromagnetic fields it generates begin to resemble those of a radiation beam. This virtual radiation beam has a macroscopic transverse extent of order , and its field distribution is described by the Ginzburg-Frank formula (see Appendix A4). When the electron is subjected to a transverse kick—such as by a kicker magnet—its self-fields are perturbed, now reflecting information about the acceleration. Under the traditional (Galilean) kinematics, the orientation of the virtual radiation phase front remains unchanged. However, Maxwell’s equations are not invariant under Galilean transformations. As discussed throughout this book, adopting Galilean kinematics leads to anisotropic field equations. As a result, although the phase front of the virtual beam remains planar, the direction of propagation is no longer perpendicular to the phase front. That is, the motion of the virtual radiation beam and the normal to its phase front diverge. Therefore, within the absolute time synchronization convention, electrodynamics predicts that the virtual radiation beam propagates in the kicked direction with a phase front tilt . In this way, the information about the electron’s acceleration is embedded in the perturbation of its self-electromagnetic field. 21 Relativity and X-Ray Free Electron Lasers 21.1 Introductory Remarks In the previous chapter, we examined why the error in radiation theory remained undetected for so long. Within the covariant framework, relativistic kinematic effects in synchrotron radiation emerge in successive orders of approximation. Rather than using the total velocity parameter as in non-relativistic cases, we employ the small transverse velocity parameter . In our earlier discussion of bending magnet radiation, we found that the motion of a single ultrarelativistic electron in a constant magnetic field, according to relativistic theory, affects kinematic terms only at the second order, i.e., It has been shown that, due to a combination of the ultrarelativistic (or paraxial) approximation and the specific symmetry of conventional synchrotron radiation setups, second-order relativistic kinematic effects cancel out—except for a non-zero redshift in the critical frequency. This redshift arises when the electron experiences perturbations in the direction of the bending magnetic field. However, because synchrotron radiation from bending magnets spans a broad frequency range, the output intensity is largely insensitive to this redshift. As a result, spontaneous synchrotron radiation shows no observable sensitivity to the differences between covariant and non-covariant particle trajectories. This situation changes in the 21st century with the advent of X-ray Free Electron Lasers (XFELs). In XFELs, first-order kinematic effects play a crucial role in the description of radiation. In such cases, the covariant coupling between fields and particles predicts outcomes that sharply contrast with those of conventional treatments. In this chapter, we critically reexamine existing XFEL theory, with particular focus on coherent undulator radiation from modulated electron beams. The discussion is primarily intended for readers with limited background in accelerator and XFEL physics. The conventional theory of X-ray Free Electron Lasers (XFELs) typically relies on a hybrid approach: Newtonian kinematics—corrected for relativistic mass—is applied to particle dynamics, while Einsteinian principles underpin electrodynamics. In practice, the relativistic treatment of particle motion often reduces to a modified form of Newton’s second law, with velocity composition still grounded in Galilean transformations. For rectilinear motion of a modulated electron beam, both covariant and non-covariant approaches yield identical particle trajectories. Consequently, Maxwell’s equations remain consistent with conventional particle tracking in such cases. However, the relativity of simultaneity—which involves the intermixing of spatial and temporal coordinates—leads to fundamental differences between covariant and non-covariant descriptions when the beam follows a curved path. According to the theory of relativity, these discrepancies emerge specifically in the presence of acceleration along curved trajectories. First-order relativistic effects in XFELs can arise under various conditions, particularly when the electron beam experiences a trajectory kick676767Angular kicks are commonly used in XFEL diagnostics and experiments. For instance, standard gain length measurements rely on such kicks. They are also employed in beam-splitting techniques, where polarization components are separated through angular deflection of the modulated beam [78, 79].. Among these, one of the most illustrative effects involves the generation of coherent undulator radiation by an ultrarelativistic, modulated electron beam that has been deflected by a weak dipole field prior to entering a downstream undulator. This scenario serves as a critical test for the correct coupling between electromagnetic fields and particle dynamics. To set the stage, it is helpful to begin with a broad overview of key results. Let us now consider the predictions of the standard XFEL theory in the context of non-collinear electron beam motion. A well-established result from conventional particle tracking states that, following a weak transverse kick, the trajectory of the electron beam changes direction while the modulation wavefront retains its original orientation (see Fig. 60). This results in a misalignment between the electron beam direction and the normal to the modulation wavefront—commonly referred to as a wavefront tilt. In the conventional framework, this wavefront tilt is treated as a physical phenomenon. That is, a transverse kick does not alter the orientation of the modulation wavefront, thereby reducing the efficiency of radiation in the direction of the beam motion. 686868In standard XFEL operation, the electron beam is guided to remain closely aligned with the undulator axis. Nonetheless, random perturbations in the focusing system can introduce angular deviations or ”kicks.” The resultant misalignment between beam trajectory and wavefront normal has been discussed in the literature. It is traditionally understood that coherent radiation is emitted along the wavefront normal. Consequently, under the conventional field-particle coupling (which we argue is flawed), the mismatch between these directions diminishes radiation efficiency . Studies of trajectory errors in XFEL amplification have shown that undulator magnetic fields must meet stringent tolerances. Interestingly, these tolerances have proven to be more stringent than necessary according to the corrected, four-dimensional (4D) covariant XFEL theory—an insight that helps explain the remarkable success of XFEL technology in recent decades. The covariant approach, applied within the frameworks of both mechanics and electrodynamics, predicts an effect that stands in stark contrast to conventional theory. Specifically, in the ultrarelativistic limit, the modulation wavefront—defined as a plane of simultaneity—is always perpendicular to the velocity of the electron beam (see Fig. 61). Consequently, Maxwell’s equations predict strong coherent undulator radiation emitted in the direction of the kick imparted to the modulated electron beam. Experimental results confirm this prediction. Observations from XFEL facilities have demonstrated that even the direction of coherent undulator radiation emission lies outside the predictive scope of the conventional theory (see Section 20.6 for further discussion). It is worth noting that the lack of a dynamical explanation for the readjustment of the modulation wavefront under Lorentz coordinatization has been a point of concern for some XFEL experts. However, we suggest that a useful way to conceptualize this readjustment is to view it as a consequence of transforming to a new time variable within the Galilean (i.e., single-frame) formulation of electrodynamics. 21.2 Modulation Wavefront Orientation Let us consider a modulated electron beam propagating along the -axis of a Cartesian coordinate system in the laboratory frame. For example, assume that the modulation wavefront is perpendicular to the beam velocity . How can we determine this orientation? As the electron bunch moves, its position changes over time. A natural way to characterize the orientation of the modulation wavefront is to ask: at what time does each electron cross the -axis of the reference frame? If a synchronization convention has been adopted—i.e., a method for timing distant events—we can use it to define the orientation of the modulation wavefront. Specifically, if electrons corresponding to the region of maximum density cross the -axis simultaneously at a given position , then the wavefront is perpendicular to the -axis. In other words, the modulation wavefront can be identified as a plane of simultaneous events—where the events are the arrivals of electrons at the point of maximum density. In short, it is a plane of simultaneity. We now examine the case where the electron beam is accelerated in the lab frame to acquire a small transverse velocity component along the -axis. This raises an important question: how should synchronization be defined in the lab frame after this acceleration? Prior to the kicker setup, we chose a Lorentz coordinate system for the lab frame. 696969Conventional XFEL theory adopts the absolute time convention to describe particle dynamics. Notably, employing the standard coupling of Maxwell’s equations with the corrected Newtonian equations to compute radiation from an ultrarelativistic electron beam in an undulator does not inherently introduce inaccuracies. In the case of straight-line motion—whether for the electron beam or the emitted undulator radiation—both covariant and non-covariant approaches produce identical trajectories. This ensures internal consistency between Maxwell’s equations and conventional particle tracking methods. However, this consistency brings to the forefront a critical question: how should synchronization be defined in the laboratory frame for a modulated electron beam upstream of the kicker setup? This is particularly relevant considering that the absolute time coordinatization was originally adopted in alignment with FEL theory. The most straightforward synchronization method employs the same set of synchronized clocks used in conventional particle tracking. This approach defines simultaneity based on light signals emitted from a stationary dipole source, assuming isotropic light propagation at speed in all directions. After the beam acquires a small transverse velocity , maintaining the original synchronization leads to complications in the electrodynamics of moving charges. As a result of such a boost, the transformation of time and spatial coordinates has the form of a Galilean transformation. To maintain a Lorentz coordinate system in the lab frame following the acceleration, it is necessary to perform a clock resynchronization. Specifically, this involves a time shift of the form given in Eq. 13: . This adjustment is valid in the first-order approximation, where is so small that can be neglected and one arrives at the coordinate transformation , . This differs from a pure Galilean transformation by the inclusion of the relativity of simultaneity—the only first-order relativistic correction in . This resynchronization introduces a time shift for electrons located at different transverse positions. For instance, electrons at a transverse position with maximum density will cross the lab frame’s -axis at a later time compared to electrons at . The resulting time shift is given by . This time shift results in an effective rotation of the modulation wavefront by an angle: in the first order approximation. In ultrarelativistic limits, , and the modulation wavefront rotates exactly as the velocity vector . What does this readjustment of the wavefront mean in terms of measurements? In the framework of absolute time coordination, simultaneity between two events is considered absolute. This absolute nature of temporal coincidence stems from the convention of absolute time synchronization. In this classical kinematics view, the modulation wavefront remains unchanged. However, within the covariant approach, we adopt a criterion for simultaneity based on the invariance of the speed of light. It becomes evident that, due to the motion of electrons along the -axis (i.e., along the plane of simultaneity prior to the boost), with a velocity , the simultaneity of events is no longer absolute. Instead, it becomes dependent on the kick angle . This reasoning mirrors the principles of Einstein’s train-embankment thought experiment. The orientation of the wavefront lacks an exact objective meaning due to the relativity of simultaneity. However, the statement that the wavefront orientation has an objective meaning within a certain accuracy can be illustrated by considering the wavefront in its proper orientation, with the angle’s uncertainty (or “blurring”) given by . This relationship defines the limits within which nonrelativistic theory remains applicable. For a very nonrelativistic electron beam, where is negligible, the angle of ”blurring” becomes extremely small. In this case, the wavefront tilt angle is almost perfectly sharp, with . This represents the limiting case of nonrelativistic kinematics. The angle ”blurring” is a characteristic feature of relativistic beam motion. In the ultrarelativistic limit the wavefront tilt has no exact objective meaning. This is because, due to the finite speed of light, no experimental method exists by which we could measure this tilt with certainty. 21.3 XFEL Radiation Configuration One of the most fundamental effects that serve as a crucial test for the proper coupling of fields and particles is the production of coherent undulator radiation by a modulated ultrarelativistic electron beam, which is deflected by a weak dipole field before entering a downstream undulator. Our goal is to study the emission of coherent undulator radiation from such a system. At the heart of an XFEL source is the undulator, which compels electrons to follow curved, periodic trajectories. There are two primary undulator configurations: helical and planar. To understand the basic principles of undulator operation, let us begin by examining the helical undulator. The magnetic field along the axis of the helical undulator is given by \| \| \| --- \| \| \| (108) \| where is the undulator wavenumber and are unit vectors directed along the and axes. We neglected the transverse variation of the magnetic field. It is necessary to mention that in XFEL engineering we deal with very high-quality undulator systems, which have a sufficiently wide good-field-region, so that our studies, which refer to a simple model of undulator field nevertheless yield a correct quantitative description in a large variety of practical problems. The Lorentz force is used to derive the equation of motion of electrons with charge and mass in the presence of magnetic field \| \| \| --- \| \| \| (109) \| \| \| (110) \| Introducing , we obtain \| \| \| --- \| \| \| (111) \| Integration of the latter equation gives \| \| \| --- \| \| \| (112) \| where and is the undulator parameter. The explicit expression for the electron velocity in the field of the helical undulator has the form \| \| \| --- \| \| \| (113) \| This means that the reference electron in the undulator moves along the constrained helical trajectory parallel to the axis. As a rule, the electron rotation angle is small and the longitudinal electron velocity is close to the velocity of light, . Let us consider a modulated ultrarelativistic electron beam moving along the -axis within the field of a helical undulator. In this study, we make the following assumptions: First, in the absence of any deflection, the electrons follow constrained helical trajectories that are parallel to the -axis. Second, the electron beam density at the entrance of the undulator is given by \| \| \| --- \| \| \| (114) \| where In other words we consider the case in which there are no variations in amplitude and phase of the density modulation in the transverse plane. Under these assumptions, the transverse current density may be written in the form \| \| \| --- \| \| \| (115) \| Even though the measured quantities are real, it is generally more convenient to use complex representation, starting with real , one defines the complex transverse current density: \| \| \| --- \| \| \| (116) \| The transverse current density has an angular frequency and two waves traveling in the same direction with variations and will add to give a total current proportional to . The factor indicates a fast wave, while the factor indicates a slow wave. The use of the word ”fast” (”slow”) here implies a wave with a phase velocity faster (slower) than the beam velocity. Having defined the sources, we now turn to the electrodynamics problem. Maxwell’s equations can be manipulated in various ways to derive forms that are more suitable for specific applications. For instance, from Maxwell’s equations (Eq. 8), we can derive an equation that depends solely on the electric field vector (in Gaussian units): \| \| \| --- \| \| \| (117) \| With the help of the identity \| \| \| --- \| \| \| (118) \| and Poisson equation \| \| \| --- \| \| \| (119) \| we obtain the inhomogeneous wave equation for \| \| \| --- \| \| \| (120) \| Once the charge and current densities and are specified as a function of time and position, this equation allows one to calculate the electric field at each point of space and time. This nonhomogeneous wave equation thus serves as the complete and accurate formulation for describing radiation. However, we aim to apply it to a simplified scenario in which the second term on the right-hand side—associated with the current density—dominates the contribution to the radiation field. It is important to recall that our focus is on coherent undulator radiation, which exhibits a divergence much smaller than the angle . Under this condition, it can be shown that the gradient term, , on the right-hand side of the wave equation becomes negligible. As a result, we simplify the wave equation to the following form: \| \| \| --- \| \| \| (121) \| We now consider the case in which the phase velocity of the current wave is close to the speed of light. This condition can be satisfied under the resonance condition . This is the condition for synchronism between the transverse electromagnetic wave and the fast transverse current wave with the propagation constant . When the phase velocity of the current wave matches that of the electromagnetic wave, a spatial resonance can occur between the electromagnetic field and the electrons. This resonance enables a cumulative interaction between the modulated electron beam and the transverse electromagnetic wave, even in free space. Under this synchronism condition—and provided the undulator has a sufficiently large number of periods—the contributions of all other (non-synchronous) waves become negligible. Thus, it is justified to focus solely on the resonant wave in analyzing the beam–field interaction. Here follows an explanation of the resonance condition which is elementary in the sense that we can see what is happening physically. The field of electromagnetic wave has only transverse components, so the energy exchange between the electron and electromagnetic wave is due to a transverse component of the electron velocity. For effective energy exchange between the electron and the wave, the scalar product should be kept nearly constant along the whole undulator length. We see that required synchronism takes place when the wave advances the electron beam by the wavelength at one undulator period , where is the radiation wavelength. This tells us that the angle between the transverse velocity of the particle and the vector of the electric field remains nearly constant. Since this resonance condition may be written as . We will employ an adiabatic approximation, which is applicable in all practical XFEL scenarios where the modulation wavelength is much shorter than the electron bunch length , i.e. . Since our focus is on coherent emission near the modulation wavelength, the theory of coherent undulator radiation is most naturally formulated in the space-frequency domain. This is because, in such cases, one is typically interested in the radiation properties at a fixed modulation frequency. We first apply a temporal Fourier transformation to the inhomogeneous wave equation to obtain the inhomogeneous Helmholtz equation \| \| \| --- \| \| \| (122) \| where is the Fourier transform of the current density . The solution can be represented as a weighted superposition of solutions corresponding to a unit point source located at . The Green function for the inhomogeneous Helmholtz equation is given by (for unbounded space and outgoing waves) \| \| \| --- \| \| \| (123) \| with . With the help of this Green function we can write a formal solution for the field equation as: \| \| \| --- \| \| \| (124) \| This is simply a mathematical description of the concept of Huygens’ secondary sources and wave propagation, which is, of course, well-known. However, it is worth recalling how this directly follows from Maxwell’s equations. We can consider the amplitude of the radiation emitted by the plane of oscillating electrons as the resultant of radiated spherical wavelets. This is because Maxwell’s theory exhibits no intrinsic anisotropy. The electrons situated on the plane of simultaneity each generate spherical wavelets, which, according to Huygens’ principle, combine to form the resulting radiated wave. If the plane of simultaneity is the -plane (i.e., the beam modulation wavefront is perpendicular to the -axis), Huygens’ construction reveals that plane wavefronts will be emitted along the -axis. In summary, according to Maxwell’s electrodynamics, coherent radiation is always emitted in the direction normal to the modulation wavefront. We have already emphasized that Maxwell’s equations are valid only within a Lorentz reference frame, i.e., in an inertial frame where the Einstein synchronization procedure is applied to assign values to the time coordinates. It is crucial to apply Einstein’s time order consistently, both in dynamics and electrodynamics. Our previous discussion naturally leads to the conclusion that Maxwell’s equations in the lab frame are compatible only with covariant trajectories, , which are calculated using Lorentz coordinates and, therefore, include relativistic kinematic effects. Let us revisit the modulated electron beam that was kicked transversely with respect to its direction of motion, as discussed earlier. Conventional particle tracking indicates that, while the direction of the electron beam changes after the kick, the orientation of the modulation wavefront remains unchanged. In other words, the direction of the electron’s motion and the normal to the wavefront are not aligned. Therefore, according to the conventional coupling of fields and particles, which we consider incorrect, the coherent undulator radiation produced in the kicked direction downstream in the undulator is expected to be significantly suppressed once the kick angle exceeds the divergence of the output coherent radiation. To estimate the loss in radiation efficiency in the kicked direction using the conventional coupling of fields and particles, we assume that the spatial profile of the modulation closely follows the electron beam’s transverse distribution, modeled as a Gaussian with standard deviation . A modulated electron beam in an undulator can be viewed as a sequence of periodically spaced radiating oscillators. These oscillators emit radiation that interferes constructively in the forward direction () when they are all in phase, leading to strong on-axis emission. To understand the angular distribution, consider a triangle formed by a radiation path at a small angle , with altitude and base . The diagonal path length exceeds the base by an amount . When equals one radiation wavelength, destructive interference occurs, as the phase contributions from different oscillators become uniformly distributed over to . In the limit of a small electron beam size (), constructive interference occurs within an angle , where is the undulator length. In the limit for the large size of the electron beam, the angle of coherence is about instead. The boundary between these two asymptotes is for sizes of about . The parameter can be referred to as the electron beam Fresnel number. For XFELs, the transverse beam size typically exceeds , indicating a large Fresnel number. Consequently, the angular distribution of radiation in the far zone is approximately Gaussian with standard deviation . However, under the conventional framework, a discrepancy arises after the beam is kicked: the direction of electron motion no longer aligns with the modulation wavefront. As a result, the radiation intensity in the kicked direction is suppressed and can be approximated as , where is the on-axis intensity without kick and is the kick angle. This exponential suppression reflects the misalignment between the wavefront and the new direction of electron motion. We presented a study of the very idealized situation to illustrate the difference between the conventional and covariant coupling of fields and particles. We solved the dynamics problem of the motion of relativistic electrons in the prescribed force field of a weak kicker magnet by working only up to the order of . This approximation is of particular theoretical interest because it is relatively simple and at the same time forms the basis for understanding relativistic kinematic effects such as relativity of simultaneity. Let us discuss the region of validity of our small kick angle approximation . Since in XFELs, the Fresnel number is rather large, we can always consider a kick angle that is relatively large compared to the divergence of the output coherent radiation, and, at the same time, it is relatively small compared to the angle . In fact, from , with some rearranging, we obtain . Then we recall that . Therefore, the first-order approximation used to analyze the kicker setup in this chapter is of practical significance for XFEL engineering. One of the goals of this chapter is to demonstrate the experimental predictions that we expect from our corrected radiation theory. To illustrate the essential physical principles clearly, we worked out a simple case. Surprisingly, the first-order approximation used to analyze the kicker setup in this chapter also has important practical applications. As shown above, our corrected coupling of fields and particles predicts an effect that is in complete contrast to the conventional treatment. Specifically, in the ultrarelativistic limit, the plane of simultaneity—the wavefront orientation of the modulation—is always perpendicular to the electron beam velocity. As a result, we predict strong emission of coherent undulator radiation from the modulated electron beam in the direction of the kick, as shown in Fig. 62. XFEL experts have indeed observed an apparent wavefront readjustment due to relativistic kinematics effects, though this conclusion was never drawn. In this book, we are the first to consider the idea that the results of the conventional theory of radiation from relativistically moving charges are inconsistent with special relativity. In previous literature, identification of the trajectories in the source part of the usual Maxwell’s equations with the trajectories calculated by conventional particle tracking in the (”single”) lab frame has always been taken as self-evident. 21.4 Modulation Wavefront Tilt and Maxwell’s Theory In the existing literature, the theoretical analysis of XFELs driven by electron beams with wavefront tilt is typically based on standard simulation codes and conventional applications of Maxwell’s equations. However, even when using only a kicker setup—without the inclusion of undulator radiation—we can demonstrate that the traditional coupling between fields and particles in XFEL theory is fundamentally flawed. This conventional XFEL theory relies on the absolute time convention (i.e., old kinematics) for describing particle dynamics. In this work, we present a straightforward demonstration of the inconsistency between standard particle tracking methods and Maxwell’s electrodynamics. Our goal is to show, in a clear and simple manner, that conventional XFEL theory fails to accurately describe the electromagnetic field distribution produced by a fast-moving, modulated electron beam downstream of a kicker. Under Maxwell’s electrodynamics, the fields associated with a modulated electron beam moving at constant velocity exhibit intriguing behavior as the beam velocity approaches the speed of light. Specifically, in the space-time domain, these fields increasingly resemble those of a laser beam (see Appendix A4). For a rapidly moving, modulated electron beam, the electric and magnetic fields are nearly equal in magnitude, transverse, and mutually perpendicular. In the limit , they become virtually indistinguishable from the fields of a laser beam. According to Maxwell’s equations, the wavefront of such a beam is always orthogonal to its direction of propagation. This is indeed the case for virtual laser-like radiation beam in the region upstream the kicker. Let us now examine the impact of a transverse kick on the modulation wavefront of the electron beam. Conventional particle tracking predicts that the kick causes a misalignment between the direction of electron motion and the normal to the modulation wavefront, effectively tilting the wavefront. This leads to an inherent contradiction: within the conventional ”single-frame” approach, the post-kick propagation direction of the radiation beam is no longer perpendicular to its wavefront. In other words, the direction of motion of the virtual radiation beam and the normal to its wavefront diverge. As a result, the beam appears to propagate along the kicked trajectory while maintaining a tilted wavefront—a scenario indistinguishable from real radiation in the ultrarelativistic limit, yet fundamentally inconsistent with Maxwell’s electrodynamics. The existing literature includes theoretical treatments of XFELs driven by electron beams with tilted modulation wavefronts. These analyses typically rely on Maxwell’s equations and standard simulation tools. However, by isolating the kicker setup—excluding the undulator—we demonstrate that the conventional XFEL framework fails to correctly model the coupling between fields and particles. This discrepancy underscores a persistent issue in XFEL physics. Initially emerging in the context of coherent undulator radiation from ultrarelativistic, modulated electron beams, the problem has now shifted focus to the tilt of the self-electromagnetic fields of the modulated electron beam. 21.5 Discussion Finally, we offer some remarks on an alternative perspective for analyzing our complex problem. While we have already obtained results using Lorentz coordinatization, it is instructive to understand why coherent undulator radiation is still emitted in the kicked direction when described within the framework of absolute time synchronization. The underlying physics is, in fact, quite simple. If we interpret the emission of coherent undulator radiation from a kicked, modulated electron beam in terms of the aberration of light, we can gain a remarkably intuitive understanding of the phenomena occurring in the undulator. Within the framework of Galilean-transformed electrodynamics, a measurement of the coherent radiation will detect emission only along the kicked direction. This is directly analogous to the classical aberration effect—i.e., the apparent deviation in the direction of energy transport—associated with light emitted by a moving source in an inertial frame (see Fig. 1). There is an intuitively appealing way to grasp the aberration of light in this context. As shown in Fig. 39, the aberration effect in an inertial frame can be readily understood within the corpuscular (particle-like) theory of light. This phenomenon arises naturally from the transformation of velocities between reference frames. Applying this analogy to a transversely kicked electron beam in an undulator, it is reasonable to expect a similar aberration effect. This becomes especially clear when one considers that a pulse of undulator radiation carries a quantifiable amount of electromagnetic energy. Like mass, energy is a conserved quantity, and in many respects, a coherent X-ray pulse behaves analogously to a stream of particles. Accordingly, we expect that the group velocity of undulator radiation pulses transforms according to the same velocity addition rules that apply to particles in an inertial frame. A more rigorous analysis based on the wave theory of light fully supports this expectation. Let us now demonstrate how Galilean-transformed electrodynamics predicts a deviation in the direction of energy transport for a radiated X-ray pulse. A modulated electron beam with finite transverse size, when traversing an undulator, effectively acts as an active medium. This medium diffracts the emitted radiation into multiple plane-wave components, each corresponding to a Fourier component of the spatial modulation of the beam. Consider an ultrarelativistic electron beam propagating along the -axis and kicked transversely along the -axis. The transverse electron density is assumed to vary according to , where is the transverse wavenumber associated with a particular Fourier component. By applying the Galilean transformation and performing the necessary partial differentiations, one arrives at the modified wave equation, Eq. (12). The additional terms introduced by the Galilean transformation give rise to a predicted Doppler effect. One significant consequence of this effect, in the context of absolute time coordinatization, is an angular frequency dispersion in the light emitted from the kicked, modulated electron beam with finite transverse size. Specifically, the Doppler shift of the radiated wave, to first order, is given by , where is the transverse velocity component of the kicked beam. This relation implies that a coherent X-ray beam with finite transverse extent propagates along the -axis with a group velocity given by . Thus, according to the corrected coupling between conventional particle tracking and electrodynamics, the direction of X-ray beam propagation is identified with the direction of energy transport, rather than with the orientation of the wavefront. This distinction arises because, under Galilean transformation, the energy transport direction and the wave normal transform differently. 21.6 Experimental Test: Electrodynamics - Dynamics Coupling The fact that our theory aligns well with experimental observations is clearly illustrated by its agreement with results from an ”X-ray beam-splitting” experiment. This technique separates a circularly polarized XFEL pulse from the linearly polarized XFEL background, thereby maximizing the degree of circular polarization. In this experiment, it was demonstrated that when a modulated electron beam is kicked by an angle significantly larger than the divergence of the XFEL radiation, the modulation wavefront realigns itself along the new direction of beam motion (see Fig. 62). This realignment is essential to explain the emission of coherent radiation from a short undulator placed downstream of the kicker and oriented along the kicked direction—see, for example, Fig. 14 in . Although these results were unexpected at the time, they had immediate practical implications. The observed ”apparent wavefront readjustment” made it possible to eliminate the unwanted linearly polarized background radiation without any additional hardware—an important breakthrough for XFEL polarization control. In the existing literature, theoretical treatments of XFELs driven by electron beams with tilted modulation wavefronts have been presented (e.g., [82, 83]), relying on Maxwell’s equations and implemented through standard simulation codes using Maxwell solvers. However, we assert that this approach is fundamentally flawed. In the ultrarelativistic limit relevant to XFELs, a tilted modulation wavefront is inconsistent with Maxwell’s electrodynamics. Specifically, within the Lorentz lab frame—where Maxwell’s equations are valid and Einstein synchronization is enforced—the concept of a tilted modulation wavefront contradicts the principles of special relativity. When the beam is treated properly in Lorentz coordinates, the modulation wavefront must remain perpendicular to the electron beam velocity. Therefore, the notion of a persistent wavefront tilt under these conditions is not physically meaningful. It is important to note that the results of the beam-splitting experiment at LCLS support our revised understanding of spontaneous undulator emission . The experiment clearly demonstrated that when a modulated electron beam is kicked by an angle much larger than the divergence of the XFEL radiation, the modulation wavefront realigns itself along the new direction of beam motion. As a result, coherent radiation emitted from an undulator placed downstream of the kicker is observed along the kicked direction, with practically no suppression. Within the framework of conventional XFEL theory, this leads to a second major puzzle. According to the standard model of undulator radiation, if a modulated electron beam is in perfect resonance before the kick, then after the kick it should remain in resonance along the new velocity direction. However, experimental observations contradict this expectation. Specifically, the data show a redshift in the resonance wavelength following the kick. Maximum radiation power is achieved only when the undulator is detuned to match the reduced longitudinal velocity of the beam after the kick . It is worth emphasizing that any linear superposition of radiation fields from individual electrons preserves the fundamental characteristics of single-particle emission—such as its parametric dependence on undulator settings and polarization. Consider a modulated beam deflected by a weak dipole field before entering a downstream undulator. The resulting radiation field is simply the coherent sum of emissions from individual electrons. Since the observed coherent undulator radiation exhibits a redshift after the kick, it follows that the radiation from each individual electron must also experience a redshift under the same conditions. This line of reasoning reinforces the conclusion that the results of the beam-splitting experiment in validate our correction to the conventional theory of spontaneous undulator emission. 21.7 Wavefront Tilt and Degradation of Electron Beam Modulation In conventional XFEL theory, wavefront tilt is often treated as a physically meaningful and measurable quantity. However, a common misconception in accelerator physics concerns the interpretation of this tilt. In the ultrarelativistic regime, wavefront tilt lacks a unique, objective definition. Its value depends on the choice of clock synchronization convention in the laboratory frame and can therefore be assigned arbitrarily within the interval . For example, when the evolution of a modulated electron beam is described using Lorentz coordinates, the orientation of the modulation wavefront is always perpendicular to the electron beam velocity—i.e., . This illustrates that wavefront tilt is a coordinate-dependent artifact rather than a physical observable. As a general principle, no genuine physical effect can depend on an arbitrary constant or an arbitrary function. Let us consider a specific example. Some papers (see, e.g., ) claim that wavefront tilt leads to significant degradation of electron beam modulation in XFELs. To analyze this, suppose the modulation wavefront is initially perpendicular to the beam velocity. One effect that can influence XFEL performance is the presence of betatron oscillations, which introduce an additional spread in longitudinal velocity. Even for particles with identical energies, differing betatron angles lead to different longitudinal velocities. Thus, beyond the velocity spread caused by energy dispersion, betatron motion introduces another source of longitudinal velocity spread. To assess the impact of this effect, we consider the total dispersion in longitudinal velocity. The deviation from the nominal longitudinal velocity is given by . The finite angular spread of the electron beam results in a difference in arrival times at a given longitudinal position, which is the well-known normal debunching effect. From the standpoint of conventional XFEL theory, this time difference is said to be amplified by the kick angle . According to conventional (non-covariant) particle tracking, the wavefront is tilted by an angle . It is commonly believed that this tilt has physical significance, and that the deviation of the longitudinal velocity component—defined as the component perpendicular to the wavefront under Galilean kinematics—is given by . If this description were accurate, the cross term would indeed lead to a significant degradation of the modulation amplitude. This phenomenon, referred to as modulation smearing, is proposed as a separate mechanism from normal debunching (see ). However, this interpretation overlooks a critical point. Many experts assume that all forms of debunching are physically meaningful. According to the theory of relativity, while normal debunching is an objective effect, the so-called smearing mechanism is not. The proposed smearing is a coordinate-dependent artifact—it arises solely from the choice of a particular reference frame and synchronization convention in four-dimensional spacetime, and thus lacks physical meaning. Let us now examine, from a physical perspective, why the smearing mechanism is not valid even within Galilean kinematics. In this classical framework, the cross term would appear to cause modulation degradation in the forward direction. However, Galilean-transformed electrodynamics tells us that coherent radiation is observable only in the kicked direction. In that direction, the cross term does not appear in the expression for the deviation of the longitudinal velocity component. It follows naturally that the smearing effect is not a real physical phenomenon. 22 Appendix 22.1 A1. Radiation from Moving Charges We start with the solution of Maxwell’s equation in the space-time domain, the well-known Lienard-Wiechert expression, and we subsequently apply a Fourier transformation. The Lienard-Wiechert expression for the electric field of a point charge reads (see, e.g. ): \| \| \| \| \| --- --- \| \| \| \| \| (125) \| denotes the displacement vector from the retarded position of the charge to the point where the fields are calculated. Moreover, , , while and denote the retarded velocity and acceleration of the electron. Finally, the observation time is linked with the retarded time by the retardation condition . As is well-known, Eq. (125) serves as a basis for the decomposition of the electric field into a sum of two quantities. The first term on the right-hand side of Eq. (125) is independent of acceleration, while the second term linearly depends on it. For this reason, the first term is called ”velocity field”, and the second ”acceleration field” . The velocity field differs from the acceleration field in several respects, one of which is the behavior in the limit for a very large distance from the electron. There one finds that the velocity field decreases like , while the acceleration field only decreases as . Let us apply a Fourier transformation: \| \| \| \| \| --- --- \| \| \| \| \| (127) \| \| \| As in Eq. (125) one may formally recognize a velocity and an acceleration term in Eq. (127) as well. Since Eq. (127) follows directly from Eq. (125), that is valid in the time domain, the magnitude of the velocity and acceleration parts in Eq. (127), that include terms in and respectively, do not depend on the wavelength . It is instructive to take advantage of integration by parts. With the help of \| \| \| --- \| \| \| (128) \| Eq. (127) can be written as \| \| \| \| \| --- --- \| \| \| \| \| (130) \| \| \| Eq. (LABEL:revtrasfbiss) may now be integrated by parts. When edge terms can be dropped one obtains \| \| \| \| \| --- --- \| \| \| \| \| (133) \| \| \| The only assumption made going from Eq. (127) to Eq. (133) is that edge terms in the integration by parts can be dropped. This assumption can be justified by means of physical arguments in the most general situation accounting for the fact that the integral in has to be performed over the entire history of the particle and that at and , the electron does not contribute to the field anymore. Let us give a concrete example for an ultra-relativistic electron. Imagine that bending magnets are placed at the beginning and at the end of a given setup, such that they deflect the electron trajectory of an angle much larger than the maximal observation angle of interest for radiation from a bending magnet. This means that the magnets would be longer than the formation length associated with the bends, i.e. , where is the bending radius. In this way, intuitively, the magnets act like switches: the first magnet switches the radiation on, and the second switches it off. Then, what precedes the upstream bend and what follows the downstream bend does not contribute to the field detected at the screen position. With this caveat Eq. (133) is completely equivalent to Eq. (127). The derivation of Eq. (133) is particularly instructive because shows that each term in Eq. (133) is due to a combination of velocity and acceleration terms in Eq. (127). In other words the terms in and in in Eq. (133) appear as a combination of the terms in (acceleration term) and (velocity term) in Eq. (127). As a result, one can say that there exist contributions to the radiation from the velocity part in Eq. (127). The presentation in Eq. (133) is more interesting with respect to that in Eq. (127) (although equivalent to it) because of the magnitude of the -term in Eq. (133) can directly be compared with the magnitude of the -term inside the integral sign. The bottom line is that physical sense can be ascribed only to the integral in Eq. (127) or Eq. (133). The integrand is, in fact, an artificial construction. In this regard, it is interesting to note that the integration by parts giving Eq. (133) is not unique. First, we find that \| \| \| \| \| --- --- \| \| \| \| \| (135) \| \| \| Note that Eq. (135) accounts for the fact that is not a constant in time. Using Eq. (135) in the integration by parts, we obtain \| \| \| \| \| --- --- \| \| \| \| \| (137) \| \| \| Similarly as before, the edge terms have been dropped. Eq. (127), Eq. (133) and Eq. (137) are equivalent but include different integrands. This is no mistake, as different integrands can lead to the same integral. If the position of the observer is far away enough from the charge, one can make the expansion . Using Eq. (137), we obtain Eq. (LABEL:revwied). 22.2 A2. Undulator Radiation in the Far Zone Calculations pertaining undulator radiation are well established see e.g. . In all generality, the field in Eq. (51) can be written as \| \| \| --- \| \| \| (138) \| \| \| (139) \| \| \| (140) \| \| (141) \| Here , and \| \| \| --- \| \| \| (142) \| is the fundamental resonance frequency. Using the Anger-Jacobi expansion: \| \| \| --- \| \| \| (143) \| where indicates the Bessel function of the first kind of order , to write the integral in Eq. (141) in a different way: \| \| \| --- \| \| \| (144) \| \| \| (145) \| \| \| (146) \| \| (147) \| where707070Here the parameter should not be confused with the velocity. \| \| \| --- \| \| \| (148) \| Up to now we just re-wrote Eq. (51) in a different way. Eq. (51) and Eq. (147) are equivalent. Of course, definition of is suited to investigate frequencies around the fundamental harmonic but no approximation is taken besides the paraxial approximation. \| \| \| --- \| \| \| (149) \| the first phase term in under the integral sign in Eq. (147) is varying slowly on the scale of the undulator period . As a result, simplifications arise when , because fast oscillating terms in powers of effectively average to zero. When these simplifications are taken, resonance approximation is applied, in the sense that one exploits the large parameter . This is possible under condition (149). Note that (149) restricts the range of frequencies for positive values of independently of the observation angle , but for any value (i.e. for wavelengths longer than ) there is always some range of such that Eq. (149) can be applied. Altogether, application of the resonance approximation is possible for frequencies around and lower than . Once any frequency is fixed, (149) poses constraints on the observation region where the resonance approximation applies. Similar reasonings can be done for frequencies around higher harmonics with a more convenient definition of the detuning parameter . Within the resonance approximation we further select frequencies such that (i.e. ). Note that this condition on frequencies automatically selects observation angles of interest . In fact, if one considers observation angles outside the range , condition (149) is not fulfilled, and the integrand in Eq. (147) exhibits fast oscillations on the integration scale . As a result, one obtains zero transverse field, , with accuracy . Under the constraint imposed by , independently of the value of and for observation angles of interest , we have \| \| \| --- \| \| \| (150) \| This means that, independently of , and we may expand in Eq. (147) according to , being the Euler gamma function Similar reasonings can be done for frequencies around higher harmonics with a different definition of the detuning parameter . However, around odd harmonics, the before-mentioned expansion, together with the application of the resonance approximation for (fast oscillating terms in powers of effectively average to zero), yields extra-simplifications. Here we are dealing specifically with the first harmonic. Therefore, these extra-simplifications apply. We neglect both the term in in the phase of Eq. (141) and the term in in Eq. (141). First, non-negligible terms in the expansion of are those for small values of , since , with . The value gives a non-negligible contribution . Then, since the integration in is performed over a large number of undulator periods , all terms of the expansion in Eq. (147) but those for and average to zero due to resonance approximation. Note that surviving contributions are proportional to , and can be traced back to the term in only, while the term in in Eq. (147) averages to zero for . Values already give negligible contributions. In fact, . Then, the term in in Eq. (147) is times the term with and is immediately negligible, regardless of the values of . The term in would survive averaging when and when . However, it scales as . Now, using condition we see that, for observation angles of interest , . Therefore, the term in is negligible with respect to the term in for , that scales as . All terms corresponding to larger values of are negligible. Summing up, all terms of the expansion in Eq. (143) but those for and or give negligible contribution. After definition of \| \| \| --- \| \| \| (151) \| that can be calculated at since , we have \| \| \| --- \| \| \| (152) \| \| (153) \| 22.3 A3. Approximating the Electron Path Let us now discuss the case of the radiation from a single electron with an arbitrary angular deflection and an arbitrary offset with respect to a reference orbit defined as the path through the origin of the coordinate system, that is . If the magnetic field in the setup does not depend on the transverse coordinates, i.e. , an initial offset , shifts the path of an electron of . Similarly, an angular deflection at tilts the path without modifying it. Cases when the magnetic field of SR sources include focusing elements (or the natural focusing of insertion devices) are out of the scope of this paper. Assuming further that and , which is typically justified for ultrarelativistic electron beams, one obtains the following approximation for the electron path: \| \| \| --- \| \| \| (154) \| \| (155) \| where the subscript ‘r’ refers to the reference path. This gives a parametric description of the path of a single electron with offset and deflection . The curvilinear abscissa on the path can then be written as \| \| \| --- \| \| \| (156) \| \| \| (157) \| \| \| (158) \| where we expanded the square root around unity in the first passage, we made use of Eq. (155), and of the fact that the curvilinear abscissa along the reference path is . We now substitute Eq. (155) and Eq. (158) into Eq. (43) to obtain: \| \| \| --- \| \| \| (159) \| \| \| (160) \| \| (161) \| where the total phase is \| \| \| --- \| \| \| (162) \| \| \| (163) \| \| \| (164) \| which can be rearranged as \| \| \| --- \| \| \| (165) \| \| \| (166) \| \| \| (167) \| \| (168) \| 22.4 A4. Self-Fields of a Modulated Electron Beam In general, the electrodynamical theory is based on the exploitation, for the ultra-relativistic particles, of the small parameter . By this, Maxwell’s equations are reduced to much simpler equations with the help of paraxial approximation. Whatever the method used to present results, one needs to solve Maxwell’s equations in unbounded space. We introduce a cartesian coordinate system, where a point in space is identified by a longitudinal coordinate and transverse position . Accounting for electromagnetic sources, i.e. in a region of space where current and charge densities are present, the following equation for the field in the space-frequency domain holds in all generality: \| \| \| --- \| \| \| (169) \| where and are the Fourier transforms of the charge density and of the current density . Eq. (169) is the well-known Helmholtz equation. Here indicates the Fourier transform of the electric field in the space-time domain. Eq. (169) can be solved with the help of an appropriate Green’s function yielding \| \| \| \| \| --- --- \| \| \| \| \| (170) \| the integration in being performed over the entire transverse plane. An explicit expression for the Green’s function to be used in Eq. (170) is given by \| \| \| \| \| --- --- \| \| \| \| \| (171) \| that automatically includes the proper boundary conditions at infinity. The transverse field can be treated in terms of paraxial Maxwell’s equations in the space-frequency domain (see e.g. ). From the paraxial approximation follows that the electric field envelope does not vary much along on the scale of the reduced wavelength . As a result, the following field equation holds: \| \| \| --- \| \| \| (172) \| where the differential operator is defined by \| \| \| --- \| \| \| (173) \| being the Laplacian operator over transverse cartesian coordinates. Eq. (172) is Maxwell’s equation in paraxial approximation. Eq. (169), which is an elliptic partial differential equation, has thus been transformed into Eq. (172), which is of parabolic type. Note that the applicability of the paraxial approximation depends on the ultra-relativistic assumption but not on the choice of the axis. If, for a certain choice of the longitudinal direction, part of the trajectory is such that , the formation length is very short (), and the radiated field is practically zero. As a result, Eq. (169) can always be applied, i.e. the paraxial approximation can always be applied, whenever . Complementarily, it should also be remarked here that the status of the paraxial equation Eq. (172) in Synchrotron Radiation theory is different from that of the paraxial equation in Physical Optics. In the latter case, the paraxial approximation is satisfied only by small observation angles. For example, one may think of a setup where a thermal source is studied by an observer positioned at a long distance from the source and behind a limiting aperture. Only if a small-angle acceptance is considered the paraxial approximation can be applied. On the contrary, due to the ultra-relativistic nature of the emitting electrons, contributions to the SR field from parts of the trajectory with formation length (the only non-negligible) are highly collimated. As a result, the paraxial equation can be applied at any angle of interest because it practically returns zero field at angles where it should not be applied. The source-term vector is specified by the trajectory of the source electrons, and can be written in terms of the Fourier transform of the transverse current density, , and of the charge density, , as \| \| \| \| --- \| \| \| (174) \| and are regarded as given data. We will treat and as macroscopic quantities, without investigating individual electron contributions. In the time domain, we may write the charge density and the current density as \| \| \| --- \| \| \| (175) \| and \| \| \| \| \| --- --- \| \| \| \| \| (176) \| where denote the velocity of the electron. The quantity has the meaning of transverse electron beam distribution, while is the longitudinal charge density distribution. In the space-frequency domain, Eq. (175) and Eq. (LABEL:curr) transform to: \| \| \| --- \| \| \| (178) \| and \| \| \| --- \| \| \| (179) \| It should be remarked that and satisfy the continuity equation. In other words, one can find . We find an exact solution of Eq. (172) without any other assumption about the parameters of the problem. A Green’s function for Eq. (172), namely the solution corresponding to the unit point source can be written as (see e.g. ): \| \| \| \| \| --- --- \| \| \| \| \| (180) \| assuming . When the paraxial approximation does not hold, and the paraxial wave equation Eq. (172) should be substituted, in the space-frequency domain, by a more general Helmholtz equation. Yet, the radiation formation length for is very short with respect to the case , i.e. we can neglect contributions from sources located at . Since it is assumed that electrons are moving along the -axis, we have . Thus, after integration by parts, we obtain the solution \| \| \| \| \| --- --- \| \| \| \| \| (182) \| \| \| Eq. (LABEL:ggeneralfin) describes the field at any position . First, we make a change in the integration variable from to . In the limit for , corresponding to the condition , we can write for the transverse field \| \| \| \| \| --- --- \| \| \| \| \| (185) \| \| \| We now use the fact that, for any real number : \| \| \| --- \| \| \| (186) \| where is the zero order modified Bessel function of the second kind. Using Eq. (186) we can write Eq. (185) as \| \| \| --- \| \| \| (187) \| \| \| (188) \| where is the modified Bessel function of the first order. Let us assume a Gaussian transverse charge density distribution of the electron bunch with rms size i.e. . Within the deep asymptotic region when the transverse size of the modulated electron beam the Ginzburg-Frank formula can be applied \| \| \| --- \| \| \| (189) \| Analysis of Eq.(189) shows a typical scale related to the transverse field distribution of order in dimensional units. Here is the modulation wavelength. 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2894	https://math.stackexchange.com/questions/594003/figuring-out-an-angle-in-an-isosceles-triangle	geometry - Figuring out an angle in an isosceles triangle - Mathematics Stack Exchange Join Mathematics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Loading… Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about Stack Overflow the company, and our products current community Mathematics helpchat Mathematics Meta your communities Sign up or log in to customize your list. more stack exchange communities company blog Log in Sign up Home Questions Unanswered AI Assist Labs Tags Chat Users Teams Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Try Teams for freeExplore Teams 3. Teams 4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams Hang on, you can't upvote just yet. You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Figuring out an angle in an isosceles triangle Ask Question Asked 11 years, 10 months ago Modified11 years, 8 months ago Viewed 199 times This question shows research effort; it is useful and clear 0 Save this question. Show activity on this post. A problem from BdMO 2013: Let A B C A B C be an isoscles triangle with A B=A C A B=A C.The bisector of ∠B∠B meets A C A C at D D.Given that B C=B D+A D B C=B D+A D,we need to figure out ∠A∠A. If we consider ∠B=∠C=∠2 x∠B=∠C=∠2 x,then after bisecting angle B,we get a triangle with angles equal to x x,2 x 2 x,and 180−3 x 180−3 x.But that does not get us any further except that ∠A D B=3 x∠A D B=3 x.I also tried extending BD to A′A′ such that A′D=A D A′D=A D but that does not help at all.Finally,I tried to utilize the Angle Bisector Theorem but that yielded nothing good as well.A prod in the correct direction would be appreciated. NOTE: I am looking for a hint,not the whole solution. geometry euclidean-geometry contest-math Share Share a link to this question Copy linkCC BY-SA 3.0 Cite Follow Follow this question to receive notifications edited Dec 5, 2013 at 14:58 rah4927rah4927 asked Dec 5, 2013 at 13:21 rah4927rah4927 3,994 3 3 gold badges 26 26 silver badges 62 62 bronze badges 2 Here A=C A=C and not B=C B=C Albanian_EAGLE –Albanian_EAGLE 2013-12-05 13:26:21 +00:00 Commented Dec 5, 2013 at 13:26 1 And moreover B C=B D+A D B C=B D+A D is impossible since A D=D C A D=D C and triangle B D C B D C would be degenerate!Albanian_EAGLE –Albanian_EAGLE 2013-12-05 13:27:50 +00:00 Commented Dec 5, 2013 at 13:27 Add a comment\| 1 Answer 1 Sorted by: Reset to default This answer is useful 1 Save this answer. Show activity on this post. We take E E on B C B C such that B E=B D B E=B D; so E C=B C−B E=B C−B D=A D E C=B C−B E=B C−B D=A D. Now we have a theorem that A B B C=A D D C A B B C=A D D C; so in △C E D△C E D and △C A B△C A B we have a common angle C E C D=A D C D=A B C B=C A C B C E C D=A D C D=A B C B=C A C B. So we get Δ C E D Δ C A B Δ C E D Δ C A B so we have ∠C D E=∠D C E=∠A B C=2 x∠C D E=∠D C E=∠A B C=2 x [let] Hence ∠B D E=∠B E D=4 x∠B D E=∠B E D=4 x, so 9 x=180 9 x=180, or x=20 x=20. Thus ∠A=180−4 x=100∠A=180−4 x=100 Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications edited Jan 27, 2014 at 17:15 Sawarnik 7,434 7 7 gold badges 37 37 silver badges 79 79 bronze badges answered Dec 5, 2013 at 17:42 krishan actonkrishan acton 659 4 4 silver badges 15 15 bronze badges 1 1 +1.The theorem A B/B C=A D/D C A B/B C=A D/D C is called angle bisector theoren,just for reference.Btw,I am perplexed at your idea of a hint.rah4927 –rah4927 2013-12-05 18:18:47 +00:00 Commented Dec 5, 2013 at 18:18 Add a comment\| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions geometry euclidean-geometry contest-math See similar questions with these tags. Featured on Meta Introducing a new proactive anti-spam measure Spevacus has joined us as a Community Manager stackoverflow.ai - rebuilt for attribution Community Asks Sprint Announcement - September 2025 Report this ad Related 3geometry - triangle 2Bisector of angle formed at the orthocentre passes through the circumcentre 1ABC is an isosceles triangle -prove AD=BC 0In the following figure, A D A D is the bisector of ∠A∠A.Prove that: ∠C B A=∠D A B∠C B A=∠D A B 1Isosceles triangle with duplicated side of 2 each and base 1+5–√1+5, find the third angle. 0Isosceles triangles + angle chasing 7Suppose ∠B A C=60∘∠B A C=60∘ and ∠A B C=20∘∠A B C=20∘. 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2895	https://www.intmath.com/integration/3-area-under-curve.php	Areas Under Curves Skip to main content Interactive Mathematics Home Tutoring Features Reviews Pricing FAQ Problem Solver More Lessons Forum Interactives Blog Contact About SearchSearch IntMath Search for: Close Login Create Free Account Search IntMath Search: Close Home Tutoring Features Reviews Pricing FAQ Problem Solver More Lessons Forum Interactives Blog Contact About Login Create Free Account Want Better Math Grades? ✅ Unlimited Solutions ✅ Step-by-Step Answers ✅ Available 24/7 ➕Free Bonuses ($1085 value!) Invalid email address Thank you for booking, we will follow up with available time slots and course plans. Home Integration The Area Under a Curve On this page Integration 1. The Differential 2. Antiderivatives and The Indefinite Integral The Area Under a Curve 4. The Definite Integral 5. Trapezoidal Rule 6. Simpson’s Rule 6a. Riemann Sums 6b. Fundamental Theorem of Calculus Applet 7. Integration Mini-lectures 7a. The Differential 7b. Difference Between Differentiation and Integration 7c. Given dy/dx, find y = f(x) 7d. Integration by Substitution 7e. Difference Between Definite and Indefinite Integrals 7f. Area Under a Curve Related Sections Math Tutoring Need help? Chat with a tutor anytime, 24/7. Chat now Online Calculus Solver Solve your calculus problem step by step! Online Calculus Solver IntMath Forum Get help with your math queries: See Forum 3. The Area under a Curve by M. Bourne A building has parabolic archways and we need to supply glass to close in the archways. How much glass is needed? Open image in a new page Parabolic archways. To answer this, we need to know the area under the curve. We'll see howto do this in 2 ways on this page: Using an approximation (finding areas of rectangles) Using integration Before integration was developed, mathematicians could only approximate the answer by dividing the space into rectangles and adding the areas of those rectangles, something like this: Open image in a new page Approximating area under a curve using rectangles. The height of each rectangle is found by calculating the function values, as shown for the typical case x = c, where the rectangle height is f(c). We get a better result if we take more and more rectangles. In the above diagram, we are approxcimating the area using inner rectangles (each rectangle is inside the curve). We could also find the area using the outer rectangles. [This method was known to the Ancient Greeks. See Archimedes and the area of a parabolic segment.] See the Riemann Sums applet where you can interactively explore this concept. Example 1: Approximation using rectangles (a) Find the area under the curve y = 1 − x 2 between x = 0.5 and x = 1, for n = 5, using the sum of areas of rectangles method. Answer The area we are trying to find is shaded in this graph: Open image in a new page Required area. Sincen= 5, the width of each rectangle will be: h=Deltax=(b-a)/n=(1-0.5)/5=0.1 We aim to find the sum of the areas of the following 5 rectangles: Open image in a new page Required area with n=5 (upper rectangles). Now the height of each rectangle is given by the function value for that particular x-value. For example, since y = f(x) = 1 − x^2, the first rectangle has height given by: f(0.5) = 1 − (0.5)2 = 0.75 It has area given by: "Area"_1= 0.75 × 0.1 = 0.075 The second rectangle has height: f(0.6) = 1 − (0.6)2 = 0.64 The 5th rectangle has height f(0.9) = 1 − (0.9)2 = 0.19 Adding the areas together gives us the following. (We are writing it using summation notation, which just means the sum of the 5 rectangles. Also, we are adding the heights first then multiplying by the width, which is the same for each rectangle.) A=sum_(i=1)^5A_i =(0.75+0.64+0.51+ 0.36+ {:0.19)(0.1) =2.45(0.1) =0.245 In the above answer, we are finding the area of the "outer" rectangles. To find a better approximation, we could also find the area of the inner rectangles, and then average the 2 results. The graph for the inner rectangles is as follows: Open image in a new page Required area with n=5 (lower rectangles). And this is the sum of the areas for the inner rectangles (the 5th one has height 0, so area 0): A=sum_(i=1)^5A_i = (0.64+0.51+0.36+ 0.19+ {:0)(0.1) =1.7(0.1) =0.17 The average of the 2 areas is given by: (0.245 + 0.17)/2 = 0.2075. A third way of doing this problem would be to find the mid-point rectangles. The diagram for this would be: Open image in a new page Required area with n=5 (mid-point rectangles). This time our area is (0.6975 + 0.5775 + 0.4375 + 0.2775 + {:0.0975) × 0.1 = 0.20875 (The first one comes from f(0.55) = 1 − (0.55)2 = 0.6975). This answer is slightly above the average of the outer and inner rectangles, and less work! (b) Find the area under the curve given in part (a), but this time use n = 10, using the sum of areas of (upper) rectangles method. Answer Since n = 10, h=Deltax=(1-0.5)/10=0.05 Here are the 10 rectangles we are using this time: Open image in a new page Required area with n=10 (upper rectangles). We take the outer rectangles and find the areas (10 of them) as follows: A=sum_(i=1)^10A_i =(0.6975+0.64+0.19+ ...+0.9755+ {:0)(0.05) =3.7875(0.05) =0.189375 You can play with this concept further 0n the Reimann Sums page. Finding Areas using Definite Integration There must be a better way than finding areas of rectangles! Integration was developed by Newton and Leibniz to save all this "adding areas of rectangles" work. General Case Open image in a new page The curvey = f(x), completely above x-axis. Shows a "typical" rectangle, Δ x wide and y high. [NOTE: The curve is completely ABOVE the x-axis]. When Δ x becomes extremely small, the sum of the areas of the rectangles gets closer and closer to the area under the curve. If it actually goes to 0, we get the exact area. We use integration to evaluate the area we are looking for. We can show in general, the exact area under a curve y = f(x) from x=a\displaystyle{x}={a}x=a to x=b\displaystyle{x}={b}x=b is given by the definite integral: Area=∫a b f(x)d x\displaystyle\text{Area}={\int_{{a}}^{{b}}} f{{\left({x}\right)}}{\left.{d}{x}\right.}Area=∫a bf(x)d x How do we evaluate this expression? If F(x) is the integral of f(x), then ∫a b f(x)d x=[F(x)]a b\displaystyle{\int_{{a}}^{{b}}} f{{\left({x}\right)}}{\left.{d}{x}\right.}={{\left[{F}{\left({x}\right)}\right]}_{{a}}^{{b}}}∫a bf(x)d x=[F(x)]a b=F(b)−F(a)\displaystyle={F}{\left({b}\right)}-{F}{\left({a}\right)}=F(b)−F(a) Mini-Lecture See the mini-lecture on the difference between definite and indefinite integrals. This means: To evaluate a definite integral, follow these steps: integrate the given function (do not include the K) substitute the upper limit (b) into the integral substitute the lower limit (a) into the integral subtract the second value from the first value the answer will be a number This forms part of The Fundamental Theorem of Calculus. Mini-Lecture See the mini-lecture on the area under a curve. Example 2: Evaluation of Definite Integral Evaluate: ∫1 1 0 3 x 2 d x\displaystyle{\int_{{1}}^{{{10}}}}{3}{x}^{2}{\left.{d}{x}\right.}∫1 1 03 x 2 d x Answer int_1^(10)3x^2dx=[3xx(x^3)/3]_1^10 (We integrate) =[x^3]_1^10 =10^3-1^3 (Substitute upper and lower values and substract) =1000-1 =999 Example 3: Arches problem Returning to our arches problem above... Open image in a new page Parabolic arch. If the arch is 2 m wide at the bottom and is 3 m high, (i) find the equation of the parabola (ii) find the area under each arch using integration. Answer (i) We place the parabola so that the left side of the arch passes through (0, 0) and the right side will pass through (2, 0), since the arch is 2 m wide at the bottom. The vertex of the arch is at (1, 3). General form of a parabola: y = ax^2+ bx + c At x = 0, y = 0 and on substituting, we get 0 = 0 + 0 + c. So c = 0. At x = 1, y = 3 and on substituting, we get 3 = a + b. At x = 2, y = 0 and on substituting, we get 0 = 4a + 2b. This gives the simultaneous equations a + b = 3 2a + b = 0 Subtracting the 1st line from the 2nd gives a = -3. And so b = 6. So the required parabola is y = -3x^2+ 6x, with xin metres. Open image in a new page Parabolic arch, 2 m wide and 3 m high.. The process of finding the equation is called modeling. It is a very important skill in science and engineering. (ii) Now for the area: int_0^2(-3x^2+6x)dx =[-x^3+3x^2]_0^2 =[-(2^3)+3(2)^2]-[0+0] =[-8+12] =4\ "m"^2 Example 4 Find the (exact) area under the curve y = x 2 + 1 between x = 0 and x = 4 and the x-axis. Answer This is the area we need to find: Area bounded by a parabola, the axes and x=4.. The area is given by: int_0^4(x^2+1)dx =[x^3/3+x]_0^4 =(4^3/3+4)-(0^3/3+0) =76/3\ "units"^2 ~~25.3\ "units"^2 Mini-Lecture See the mini-lecture on the difference between definite and indefinite integrals. Mini-Lecture See the mini-lecture on the area under a curve. 2. Antiderivatives and The Indefinite Integral 4. The Definite Integral Tips, tricks, lessons, and tutoring to help reduce test anxiety and move to the top of the class. Email Address Sign Up Interactive Mathematics Instant step by step answers to your math homework problems. Improve your math grades today with unlimited math solutions. Instagram TikTok Facebook LinkedIn About Us Tutoring Problem Solver Testimonials Affiliate Disclaimer California Privacy Policy Privacy Policy Terms of Service Contact © 2025 Interactive Mathematics. All rights reserved. top
2896	https://math.stackexchange.com/questions/1312849/matrix-with-zeros-on-diagonal-and-ones-in-other-places-is-invertible	Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams Matrix with zeros on diagonal and ones in other places is invertible Ask Question Asked Modified 9 years, 8 months ago Viewed 51k times $\begingroup$ Hi guys I am working with this and I am trying to prove to myself that n by n matrices of the type zero on the diagonal and 1 everywhere else are invertible. I ran some cases and looked at the determinant and came to the conclusion that we can easily find the determinant by using the following $\det(A)=(-1)^{n+1}(n-1)$. To prove this I do induction n=2 we have the $A=\begin{bmatrix} 0 & 1\ 1 & 0 \end{bmatrix}$ $\det(A)=-1$ and my formula gives me the same thing (-1)(2-1)=-1 Now assume if for $n \times n$ and $\det(A)=(-1)^{n+1}(n-1)$ Now to show for a matrix B of size $n+1 \times n+1$. I am not sure I was thinking to take the determinant of the $n \times n$ minors but I am maybe someone can help me. Also is there an easier way to see this is invertible other than the determinant? I am curious. linear-algebra matrices Share edited Jun 8, 2015 at 5:42 Martin Sleziak 56.3k2020 gold badges211211 silver badges391391 bronze badges asked Jun 5, 2015 at 3:34 KoriKori 1,54011 gold badge1212 silver badges3030 bronze badges $\endgroup$ 3 3 $\begingroup$ Do you know about eigenvalues? $\endgroup$ Ben Grossmann – Ben Grossmann 2015-06-05 03:47:32 +00:00 Commented Jun 5, 2015 at 3:47 1 $\begingroup$ Determimant of such matrix was computed in several posts on this site: math.stackexchange.com/questions/81016/…, math.stackexchange.com/questions/84206/… This post is a bit more general: math.stackexchange.com/questions/86644/… $\endgroup$ Martin Sleziak – Martin Sleziak 2015-06-05 09:30:02 +00:00 Commented Jun 5, 2015 at 9:30 4 $\begingroup$ I am not sure this should have been closed as a duplicate. Calculate the determinant is not the same question as Is the matrix invertible? (And serveral answer posted in this thread show that it is invertible without calculating the matrix determinant.) $\endgroup$ Martin Sleziak – Martin Sleziak 2015-06-07 06:24:25 +00:00 Commented Jun 7, 2015 at 6:24 Add a comment \| 9 Answers 9 Reset to default 38 $\begingroup$ If you have studied eigenvalues and eigenvectors there is a very easy proof. Let $A$ be your $n\times n$ matrix, with $n\ge2$. Then $A+I$ is the matrix consisting entirely of $1$s, which clearly has $n-1$ zero rows after row-reduction. Therefore $A$ has eigenvalue $-1$, repeated (at least) $n-1$ times, and since ${\rm trace}(A)=0$, the other eigenvalue is $n-1$. Since every eigenvalue of $A$ is non-zero, the determinant of $A$ is non-zero, so $A$ is invertible. Share answered Jun 5, 2015 at 3:58 DavidDavid 84.9k99 gold badges9696 silver badges166166 bronze badges $\endgroup$ 3 11 $\begingroup$ Clearly, this also proves the said formula $\det A = (-1)^{n-1}(n-1)$. $\endgroup$ Jeppe Stig Nielsen – Jeppe Stig Nielsen 2015-06-05 14:37:56 +00:00 Commented Jun 5, 2015 at 14:37 $\begingroup$ @JeppeStigNielsen How do we conclude that the multiplicity of the eigenvalue $-1$ is $n-1$? $\endgroup$ ZSMJ – ZSMJ 2019-08-03 15:34:41 +00:00 Commented Aug 3, 2019 at 15:34 $\begingroup$ @AkashGaur The argument of the answer claimed to prove that the $n$ eigenvalues of $A$ are $-1,-1,\ldots,-1$ ($n-1$ times), and $n-1$ (once). That was not my statement. I just commented that if those $n$ numbers are really the eigenvalues, then their product is the determinant. $\endgroup$ Jeppe Stig Nielsen – Jeppe Stig Nielsen 2019-08-04 17:39:16 +00:00 Commented Aug 4, 2019 at 17:39 Add a comment \| 30 $\begingroup$ This is easy to calculate by row reduction: Add all rows to first: $$\det(A) =\det \begin{bmatrix} 0 & 1 & 1 &...&1 \ 1 & 0 & 1 &...&1 \ 1 & 1 & 0 &...&1 \ ... & ... & ... &...&... \ 1 & 1 & 1 &...&0 \ \end{bmatrix}=\det \begin{bmatrix} n-1 & n-1 & n-1 &...&n-1 \ 1 & 0 & 1 &...&1 \ 1 & 1 & 0 &...&1 \ ... & ... & ... &...&... \ 1 & 1 & 1 &...&0 \ \end{bmatrix} \ =(n-1)\det \begin{bmatrix} 1 & 1 & 1 &...&1 \ 1 & 0 & 1 &...&1 \ 1 & 1 & 0 &...&1 \ ... & ... & ... &...&... \ 1 & 1 & 1 &...&0 \ \end{bmatrix}=(n-1)\det \begin{bmatrix} 1 & 1 & 1 &...&1 \ 0 & -1 & 0 &...&0 \ 0 & 0 & -1 &...&0 \ ... & ... & ... &...&... \ 0 & 0 & 0 &...&-1 \ \end{bmatrix}$$ where in the last row operation I subtracted the first row from each other row. This shows $$\det(A)=(n-1)(-1)^{n-1}$$ Share answered Jun 5, 2015 at 4:29 N. S.N. S. 135k1212 gold badges150150 silver badges271271 bronze badges $\endgroup$ Add a comment \| 20 $\begingroup$ Here's an alternative approach. The Woodbury matrix identity says that if we start with an invertible matrix $B$ and update it by adding a product of matrices $UCV$, then the inverse of the sum is, $$(B + U C V)^{-1} = B^{-1} - B^{-1} U (C^{-1} + V B^{-1} U)^{-1} V B^{-1}.$$ This holds whenever all the matrices in the formula have sizes such that the formula makes sense, and all the inverses on the right hand side in the formula exist. The Woodbury formula can be proved by direct verification (multiply it out), and can be derived in a number of straightforward ways - see the wikipedia article linked above for more details. Now in your situation we can take: $B := -I$ $C := 1$ $U = \mathbf{1}$, the column vector of all ones $V = \mathbf{1}^T$, the row vector of all ones, so that $B + U C V = -I + \mathbf{1}\mathbf{1}^T = A$ is the matrix of all ones except on the diagonal that we wish to invert. In this case the Woodbury formula becomes, \begin{align} (-I + \mathbf{1}\mathbf{1}^T)^{-1} &= -I - \mathbf{1}(1 - \mathbf{1}^T I \mathbf{1})^{-1}\mathbf{1}^T \ &= -I - \mathbf{1}(1 - n)^{-1}\mathbf{1}^T \ &= -I + \frac{1}{n-1}\mathbf{1}\mathbf{1}^T. \end{align} So, here we have proved that the matrix is invertible for all $n$-by-$n$ matrices whenever $n > 1$ (Ie., it is not a scalar). Further we have a complete formula for the inverse, $$\boxed{A^{-1} = -I + \frac{1}{n-1}\mathbf{1}\mathbf{1}^T}$$ Share edited Jun 5, 2015 at 5:10 answered Jun 5, 2015 at 4:01 Nick AlgerNick Alger 20.1k1111 gold badges7474 silver badges108108 bronze badges $\endgroup$ Add a comment \| 10 $\begingroup$ Using the Gauss-Jordan method it is possible to think of elementary row operation which would convert such a $n\times n$ matrix to the identity matrix. Thus an inverse exist. Namely if you multiply the first row by $n-2$, subtract the remaining $n-1$ rows from it once, then you get $1-n$ for the first element followed by only zeros for the other elements of the first row. By normalizing this row you get a one followed by zeros, which can then be used to clear the first column of the other rows. This process can be repeated for the remaining rows until the final two rows, which can be swapped to obtain the identity matrix. If you wish to actually calculate the inverse you could make use of symmetry, namely all diagonals will be equal and all other elements will be equal. This information can already be obtained after reducing the first row, namely all diagonals will be equal to $\frac{2-n}{n-1}$ and all other elements will be equal to $\frac{1}{n-1}$. Share edited Jun 5, 2015 at 4:59 answered Jun 5, 2015 at 4:09 Kwin van der VeenKwin van der Veen 12.8k22 gold badges2121 silver badges5151 bronze badges $\endgroup$ 3 $\begingroup$ Clever I like it I think all answers are good but this is clever. $\endgroup$ Kori – Kori 2015-06-05 04:24:57 +00:00 Commented Jun 5, 2015 at 4:24 $\begingroup$ How did you conclude that all the off diagonal elements are equal by symmetry? The diagonal entries are clearly all equal by the inverse matrix formula. $\endgroup$ 19021605 – 19021605 2025-07-15 12:22:50 +00:00 Commented Jul 15 at 12:22 $\begingroup$ @19021605 It is indeed incorrect that the first row shows this, since the inverse will also be symmetric however one row does not define a symmetric matrix. But since all other rows can be transformed in the same way as the first (but the operations are permuted with respect to the element of that row that lies on the diagonal). From this it can be shown via the Gauss-Jordan method that all off diagonals of the inverse of this matrix are the same. $\endgroup$ Kwin van der Veen – Kwin van der Veen 2025-07-15 20:32:43 +00:00 Commented Jul 15 at 20:32 Add a comment \| 7 $\begingroup$ A square matrix is invertible if its columns (or equivalently, rows) are linearly independent. Let $v_i$ be the $i$th column of the matrix. If $$ c_1v_1 + c_2v_2 + \dots + c_n v_n = 0 $$ has only the trivial solution, the columns are independent. The $i$th row in this equation is $$c_1 + c_2 + \dots + c_{i-1} + c_{i+1} + \dots + c_n = 0,$$ which can equivalently be written as $$c_1 + c_2 + \dots + c_n = c_i.$$ So each of the $c_i$s are equal to their sum, so we have $$c_1=c_2=\dots=c_n,$$ and also $c_i=nc_i$, so we must have $$c_1=c_2=\dots=c_n=0.$$ Share answered Jun 5, 2015 at 12:11 JiKJiK 5,9552020 silver badges4141 bronze badges $\endgroup$ Add a comment \| 6 $\begingroup$ Let $n\ge 2$ and let $E$ be a $n\times n$ -matrix consisting only of ones. $$E= \begin{pmatrix} 1 & 1 & \ldots & 1 \ 1 & 1 & \ldots & 1 \ \ldots & \ldots & \ldots & \ldots \ 1 & 1 & \ldots & 1 \end{pmatrix} $$ It is easy to notice that $E^2=nE$, i.e., it is a matrix which has $n$ in each position. We are interested in finding the inverse of the matrix $A=E-I$. We have $$A^2=(E-I)^2=E^2-2E+I=(n-2)E+I$$ For $n=2$ we get $A^2=I$, which means $A^{-1}=A$. What about $n\ge3$? Now we can combine the equalities \begin{align} A&=E-I\ A^2&=(n-2)E+I \end{align} to get $$A^2-(n-2)A=(n-1)I$$ which can be rewritten as $$A\cdot (A-(n-2)I) = A\cdot (E-(n-1)I) = (n-1)I.$$ So we see that $$A^{-1} = \frac1{n-1} (E-(n-1)I) = \frac1{n-1} \begin{pmatrix} -(n-2) & 1 & \ldots & 1 \ 1 & -(n-2) & \ldots & 1 \ \ldots & \ldots & \ldots & \ldots \ 1 & 1 & \ldots & -(n-2) \end{pmatrix} $$ We can easily check that product of this matrix with the matrix $A$ is indeed $I$. Note that a more general approach (which gives the same result for $A^{-1}$) is mentioned in Nick Alger's answer. It might be also worth mentioning that the inverse is precisely the matrix from this question: Find the inverse of a matrix with a variable For a generalization of the question from the original post see: Inverse of constant matrix plus diagonal matrix Inverse of a matrix with $a+1$ on the diagonal and $a$ in other places Share edited Apr 13, 2017 at 12:20 CommunityBot 1 answered Jun 5, 2015 at 20:04 Martin SleziakMartin Sleziak 56.3k2020 gold badges211211 silver badges391391 bronze badges $\endgroup$ Add a comment \| 3 $\begingroup$ You can certainly expand by minors, but you will have to deal with multiple types of minors. It may be easier to reduce the matrix instead: Add the 2nd row to the first, the third to the second, and so on, until you add the last row to the second-to-last. What does the matrix look like now? What do you need to do to the last row to make the matrix upper-triangular? Share answered Jun 5, 2015 at 3:49 user7530user7530 50.7k1111 gold badges9292 silver badges159159 bronze badges $\endgroup$ 1 $\begingroup$ The matrix determinant can be found in the Sequences encyclopedia: a(n) = n(-1)^n. oeis.org/A038608 $\endgroup$ Enrique Pérez Herrero – Enrique Pérez Herrero 2021-06-10 18:56:14 +00:00 Commented Jun 10, 2021 at 18:56 Add a comment \| 2 $\begingroup$ Alternative "approach": we know that $$\det A = \sum_{\sigma \in S_n} \operatorname{sign}(\sigma) A_{1,\sigma(1)} \cdots A_{n, \sigma(n)}.$$ (In fact, some authors take this as the definition of the determinant.) In our case, $$A_{i, \sigma(i)} = \begin{cases} 0 & \text{ if } i = \sigma(i), \ 1 & \text{ otherwise,} \end{cases}$$ so $$A_{1,\sigma(1)} \cdots A_{n, \sigma(n)} = \begin{cases} 1 & \text{ if $\sigma$ is a derangement,} \ 0 & \text{ otherwise.} \end{cases}$$ Therefore, $$\det A = #{\text{even derangements of $n$ elements}}-#{\text{odd derangements of $n$ elements}}.$$ Combinatorialists have formulas for each of the terms above, so you can subtract them. However, apparently these formulas are usually obtained by calculating this determinant using one of the other methods given here! I don't know if there are other ways of getting the same result that (1) avoid this determinant and (2) are easy to understand -- that could be its own question here. Share edited Jun 5, 2015 at 10:42 answered Jun 5, 2015 at 5:00 Daniel McLauryDaniel McLaury 25.4k33 gold badges5353 silver badges119119 bronze badges $\endgroup$ Add a comment \| 1 $\begingroup$ Alternatively, a square matrix is invertible if it's rows are linearly independent. (Of course formally proving this is equal to proving the matrix is invertible for which there are many routes ) I mention this because particularly in this case it is intuitively evident that the rows are independent, e.g. because they have all have one zero in a unique position and you can easily see you get the standard normal basis using the same operation on every row, by taking (1,..,1) minus the rows of your matrix. Formally proving this implies the rows are independent without directly proving the matrix is invertible is probably harder, but interesting on its own. I'm not sure if an affine transformation always retains the rank of a matrix, but you can also get the same result here with a single linear transformation matrix, i.e. a reflection across the plane $x_1=....x_2$ then negative scaling, but I'd have to work out the specifics. Share answered Jun 5, 2015 at 22:46 Thomas BosmanThomas Bosman 18144 bronze badges $\endgroup$ Add a comment \| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions linear-algebra matrices See similar questions with these tags. Featured on Meta Introducing a new proactive anti-spam measure Spevacus has joined us as a Community Manager stackoverflow.ai - rebuilt for attribution Community Asks Sprint Announcement - September 2025 Linked 2 Linear Algebra, determinants Finding $A^{-1}$ using characteristic polynomial. 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2897	https://sud03r.github.io/papers/cccg-16.pdf	CCCG 2016, Vancouver, British Columbia, August 3–5, 2016 Counting Convex k-gons in an Arrangement of Line Segments Martin Fink∗ Neeraj Kumar∗ Subhash Suri∗ Abstract Let A(S) be the arrangement formed by a set of n line segments S in the plane. A subset of arrangement vertices p1, p2, . . . , pk is called a convex k-gon of A(S) if (p1, p2, . . . , pk) forms a convex polygon and each of its sides, namely, (pi, pi+1) is part of an input segment. We want to count the number of distinct convex k-gons in the arrangement A(S), of which there can be Θ(nk) in the worst-case. We present an O(n log n + mn) time algorithm, for any ﬁxed constant k, where m is the number of pairwise segment intersections. We can also report all the convex k-gons in time O(n log n+mn+\|K\|), where K is the output set. We also prove that the k-gon counting problem is 3SUM-hard for k = 3 and k = 4. 1 Introduction We consider the problem of counting, and enumerating, all convex k-gons formed by the arrangement A(S) of a set S of n line segments in the plane. A set of vertices p1, p2, . . . , pk of the arrangement A(S) is called a convex k-gon if (p1, p2, . . . , pk) forms a convex polygon and each of its sides (p1, p2), (p2, p3), . . . , (pk−1, pk), (pk, p1) is part of an input segment. We note that such a k-gon is not necessarily a face of the arrangement, and in general there can be Θ(nk) convex k-gons, for any ﬁxed k. We are interested in the problem of counting these k-gons. That is, given a set of n line segments in the plane, how many convex k-gons exist in their arrangement? Surprisingly, this natural-sounding problem appears not to have been explored in computational geometry. We are motivated by an application in computer vision where the case of counting, and enumerating, convex quadrilaterals arises. Speciﬁcally, the arrangement is the camera image representing linear boundaries of objects in the scene, and the goal is to estimate (the counting problem) the number of “rectangular” objects in the input scene, which may represent important features such as desks, door frames, walls etc. Due to the per-spective transformation, the rectangles in the scene map to convex quadrilaterals in the image. The computational problem then becomes the fol-lowing: can we ﬁnd all convex quadrilaterals in the n-segment arrangement in better than the naive O(n4) ∗Department of Computer Science, University of California, Santa Barbara, {fink\|neeraj\|suri}@cs.ucsb.edu time? Counting convex k-gons is a natural generaliza-tion of this problem. We call this the k-gon reporting problem, which leads to a natural counting version of the problem, where we are just interested in counting the number of k-gons formed by the segments. Un-like the segment intersection problem [2, 3], in which the maximum number of intersecting line segments is O(n2), the number of combinatorially distinct k-gons in an n-segment arrangement can be Ω(nk). See Fig. 1. Therefore, it is desirable to be able to count the k-gons in time much faster than the number of combinatorially distinct k-gons. Figure 1: An arrangement of n segments with Ω(nk) convex k-gons. Each of the k = 5 groups contains ⌊n k ⌋ segments. Our Contribution. We present a sweep-line algorithm for counting the number of k-gons in worst-case time O(n log n + mn), using O(n2) space, for any constant k, where m is the number of pairwise segment intersections. The algorithm works for non-constant values of k as well, but in that case takes O(n log n + mn2) time and O(n3) space. In either case, the running time is independent of k. By maintaining additional information during the counting algorithm, we can also recover all the k-gons, in worst-case time O(n log n + mn + \|K\|) time, using O(mn + n2) space, where K is the output. Finally, we show that counting the number of triangles and the number of quadrilaterals are both 3SUM-hard, suggesting that a running time signiﬁcantly better than O(n2) is unlikely. 28th Canadian Conference on Computational Geometry, 2016 Related Work. The problem of counting convex k-gons in a set of n points has been considered by several re-searchers [9, 8, 7]. The algorithm in achieves a running time of O(nk−2), which was then improved to O(n⌈k/2⌉) in . This bound was improved signiﬁcantly to O(n3) by Mitchell et al. using dynamic programming. In , Eppstein et al. study the related problem of ﬁnding minimum area k-gons for point sets. Some results are also known for the restricted problem of counting faces in line arrangements. For instance, every arrangement of lines (or pseudolines) in the plane results in Ω(n) triangular faces . Another related problem is one of counting and report-ing simple cycles of given length in a graph. In general, the time bound for counting the cycles is exponential in k , however for k ≤7 the problem can be solved in O(n2.376) time. These cycle counting results in graphs, however, do not solve our k-gon problem because of the convexity constraint. Indeed, an arrangement A(S) of segments can be easily viewed as a graph, whose vertices are the segments and whose edges correspond to pairs of intersecting segments. However, cycles in this graph are not necessarily convex polygons. An exception is the case of triangles which are always convex: every 3-cycle correspond to a triangle in the segment arrangement, but only for non-degenerate input, namely, no three segments intersecting in a common point. 2 Counting Convex k-gons Let P be a convex k-gon in the arrangement A(S) formed by the n line segments of S.1 Let L be a vertical line intersecting P. Since P is convex, L can only intersect two sides of P. The span of P with respect to L is the (ordered) pair of segments of P that intersect L. Although the number of k-gons can be exponential in k, the number of distinct spans is only quadratic. Observation 1 There are O(n2) distinct spans among all k-gons of A(S) with respect to a vertical line L. In other words, Observation 1 tells us that although there could be Ω(nk) k-gons, at a given vertical line L, all k-gons intersecting L can be assigned to one of the O(n2) distinct segment pairs. This suggests the existence of a natural sweep line based approach for the counting problem. The key idea is to keep track of convex open polygons with up to k sides as we sweep a vertical line L across the arrangement. When sweeping over an intersection, some open polygons may become closed k-gons, which we must count, while other open polygons can be extended using the intersection vertex, and new open polygons start growing at the intersection. We start by ﬁxing some notation. 1For the rest of the paper, we drop the qualiﬁer “convex” and simply refer to P as a k-gon. a b L Figure 2: Open 5-gons: Σ(a, b, 5) at a given sweep line L; two members of the set are shown by dotted lines. Notation. Observe that when we are sweeping a verti-cal line L across the arrangement, at a given x-coordinate xL, we may have come across two types of potential k-gons: • Closed k-gons: these are the k-gons all of whose sides are to the left of the line L. • Open j-gons: these are j-gons, for j ≤k, whose j sides lie (partially or fully) to the left of L. More precisely, L intersects the two open sides of these convex polygons and their remaining j −2 sides are to the left of L. See Fig. 2. Observe that open j-gons are only potential candidates for closed k-gons; not all of them necessarily become k-gons. At an x-coordinate xL, we can now represent an open j-gon P by the triplet (a, b, j), where a and b are the segments forming the top and bottom sides of P at xL and j is the number of sides we have seen so far. Note that 2 ≤j ≤k and this includes the sides of P formed by the segments a and b. The triplet (a, b, j) succinctly combines all open j-gons for which a and b are the open sides intersecting the line L; we let Σ(a, b, j) denote the set of these open j-gons, and we let σ(a, b, j) = \|Σ(a, b, j)\|. 2.1 Algorithm Our algorithm moves a sweep line L across the arrange-ment A(S) with the segment intersections as key event points. For the sake of simplicity, we assume no degenera-cies for now, that is, every intersection involves exactly two line segments of S. (We will show how to handle degenerate cases later in this section.) We maintain an array of counters σ(a, b, j) which keep track of all open j-gons whose span is (a, b) on line L, for all 2 ≤j ≤k. A global counter keeps track of all the closed k-gons that have been encountered already. In the following, we explain these steps in detail. CCCG 2016, Vancouver, British Columbia, August 3–5, 2016 a b pi c d L Figure 3: After the intersection of a and b, the sweep line L gets new open polygons for Σ(b, a, 2), Σ(c, b, j +1), and Σ(a, d, j + 1) shown in dash dotted; parent polygons in Σ(c, a, j) and Σ(b, d, j) are still active. 1. Set count = 0 and σ(a, b, j) = 0 for all segments a, b and 2 ≤j ≤k. 2. Compute all m intersections of the n line segments using and order them from left to right. 3. Process the intersections one by one from left to right, moving the sweep line L accordingly. When L contains the intersection of segments a and b (where to the left of L, a is above b), we perform the following updates. See Fig. 3. (a) Open k-gons of Σ(a, b, k) become closed k-gons. We update the count: count += σ(a, b, k) (b) An open 2-gon of b and a begins at this inter-section. We initialize the count: σ(b, a, 2) = 1 (c) For all 2 ≤j < k, each open j-gon with a as the lower side can be extended to an open j + 1-gon with b as the lower side. Let S′ L be the set of segments intersecting sweep line L above the intersection point (a, b). We update the count: ∀c ∈S′ L σ(c, b, j + 1) += σ(c, a, j) Similarly, for each 2 ≤j < k, each open j-gon with b as the upper side can be extended to an open j + 1-gon with a as the upper side. Let S′′ L be the set of segments that intersect sweep line L below the intersection point (a, b). ∀d ∈S′′ L σ(a, d, j + 1) += σ(b, d, j) 4. Return count. Correctness. Let SL be the set of all segments that intersect the sweep line L. Let AL be the set of all possible spans for the k-gons intersecting L. That is, AL = {(a, b) \| a, b ∈SL, a above b on L}. As the sweep line moves from left to right we maintain the following invariant: count is the total number of closed k-gons to the left of L; and σ(a, b, j) is the number of open j-gons with span (a, b) on L, for all (a, b) ∈AL and 2 ≤j ≤k. The invariant is trivially satisﬁed before processing the ﬁrst intersection. For the general case, after moving the sweep line L over the intersection (a, b), the segments a and b switch their vertical order. Therefore, 1. AL no longer includes the span (a, b). 2. count now includes the new k-gons that complete at the intersection (Step 3a); these correspond to the open k-gons counted by σ(a, b, k). 3. AL includes the new span (b, a). Right after the crossing, the open 2-gon formed by b and a is the only polygon with that span, covered by σ(b, a, 2) = 1. 4. Open polygons with span (c, b) ∈AL can now also use the vertex (a, b). Any such open j-gon (j ≤k) must consist of the new intersection and an open j− 1-gon with span (c, a) right before the intersection (compare Step 3c). 5. Analogously, open polygons with span (a, d) ∈AL can now use vertex (a, b). Such a new open j-gon (j ≤k) consists of the new intersection and an open j −1-gon with span (b, d). It is easy to see that the algorithm maintains the invariant after processing each intersection. Hence, when eventually the sweep line L is right of all intersections, count is the total number of k-gons. Analysis. Computing and storing all m intersections takes O((n+m) log n) time and O(m+n) space . Since we perform O(n) updates for each of the m events, the total running time for our algorithm is O(n log n + mn). The total space requirement is O(n2) since we store information for all pairs of segments that may intersect the sweep line. Handling Degenerate Cases. We now show how to extend our algorithm so that it can also handle segment arrangement with degeneracies, that is, with three or more segments intersecting in a single point. (For parallel segments, we do not need to do anything special). For an intersection point pi of a set Si of more than two 28th Canadian Conference on Computational Geometry, 2016 segments, we ﬁrst update the number of closed k-gons for every pair of segments in Si. For extending open j-gons (2 ≤j < k), we need to be a bit more careful. Since we do not want degenerate k-gons, we should only extend the j-gons which we have seen before the current intersection. More precisely, we compute the updates for every pair of segments in Si, and apply them collectively. One way to achieve this is to process the updates in Step 3(b) and 3(c) in decreasing order of j as follows: • For j in k −1 down to 3 perform updates in Step 3(c) for every segment pair in Si. • Perform updates in Step 3(b) for every segment pair in Si. Observe that these modiﬁcations do not aﬀect the over-all runtime since m ∈O(n2) is the number of pairwise intersections. 3 Reporting Convex k-gons We now turn to the problem of reporting all the k-gons in an arrangement of n line segments. We solve this problem by extending our algorithm for counting k-gons. The key idea is to keep track of how the values σ(a, b, j) are updated as we move the sweep line across the ar-rangement, and to remember the values that contributed to the total number. Recall that the total number of k-gons formed by n segments can be Ω(nk). Therefore, we would like the total running time to be linear in size of the output. We start by describing a reporting graph that will help us reconstruct all k-gons. Reporting Graph. Our reporting graph is a labeled di-rected acyclic graph G = (V, E, L). Its vertices represent the sets of polygons Σ(a, b, j) and its edges keep track of how these sets grow. The function L: E →N assigns a label L(e) to each edge e ∈E. The label is a timestamp and represents the intersection at which the edge was created. To construct the digraph G, we extend the counting algorithm from Section 2.1 as follows: 1. For every pair (a, b) of segments and 2 ≤j ≤k add a vertex (a, b, j) to G. 2. Deﬁne Q = ∅to be the set that keeps track of closed k-gons grouped by their rightmost vertex. 3. Refer to step 3 of the counting algorithm. Suppose the sweep line L is currently at the ith intersection event (a, b). Recall that S′ L and S′′ L are respectively the sets of segments that intersect L above and below the intersection point (a, b). We modify the reporting graph as follows; see Fig. 4 for an example. (a) If σ(a, b, k) > 0, insert (a, b, k) to Q. (b) For all values 2 ≤j < k and each segment c ∈S′ L with σ(c, a, j) > 0, create an edge (c, b, j + 1), (c, a, j) . (c) Similarly, for all 2 ≤j < k and a segment d ∈S′′ L with σ(b, d, j) > 0, create an edge (a, d, j + 1), (b, d, j) . Label each edge e created for this intersection event with the timestamp L(e) = i. The reporting graph G has the following properties. • G has O(n2) vertices and O(mn) edges. The vertices of the form (a, b, k) have in-degree zero (sources) and vertices of the form (a, b, 2) have out-degree zero (sinks). • An edge of G represent the extension of an open j-gon into a j + 1-gon (2 ≤j < k), with the inter-section point (a, b) being the newly added corner. As a result, we get two new sets of j + 1-gons: one with b as the lower side and another with a as the upper side. • There is exactly one label on every edge since two segments can only intersect once. • Each complete k-gon corresponds to a path that starts at a vertex in Q (a source in G) and ends at a sink of G. Since G is acyclic, every such path has exactly k −2 edges. The intersection points corresponding to these k −2 edges along with the two intersection points for the source and sink will be the k vertices of the output k-gon. Enumerating all k-gons. With the reporting graph G, enumerating all k-gons seems pretty straightforward. We can simply start at vertices in Q one by one and recur-sively explore all distinct paths to sinks. However, there is one small caveat. Since the segments may continue to grow further after we close a k-gon, it is possible that a vertex v of G gets an additional successor w′ after it got a predecessor u; see Fig. 4. Observe that in such a case, the path (u →v →w′ →· · · ) does not correspond to a valid k-gon since the corresponding vertices are not ordered chronologically. In order to ﬁx this we can use the timestamps of the edges: we only recurse using the edges whose timestamp is smaller than that of the parent edge. Because of our construction, we are guaranteed to ﬁnd at least one such edge. Moreover, since the edges are added in the order of their timestamps, we can stop at the ﬁrst edge for which the timestamp is higher than the parent value. This way we spend no extra time on objects that are not a member of our output set. Consequently, we get a running time of O(n log n + mn) for constructing the G, and O(\|K\|) time for reporting the k-gons that form our CCCG 2016, Vancouver, British Columbia, August 3–5, 2016 1 2 3 4 5 6 7 a b c d e (a) Arrangement of ﬁve line segments. Numbers 1 through 7 indicate intersections from left to right events. The two valid 4-gons formed by the segments are shown by red and blue dotted lines. (c, a, 2) (d, a, 2) (b, a, 3) (e, a, 4) (b, d, 4) 2 7 4 3 (b) Reporting graph. The edge shown as dash-dotted was added after its predecessors and therefore does not contribute to a valid 4-gon. The other two valid dotted paths to the sink (c, a, 2) represent the closed 4-gons with vertices (1, 2, 4, 6) and (1, 2, 3, 7). Figure 4: An arrangement and the corresponding reporting graph. output set K. Since G has O(n2) vertices and O(mn) edges, the total space requirement is O(n2 + mn). 4 3SUM-Hardness In this section, we show that counting the number of triangles in an arrangement of straight-line segments is at least as hard as the 3SUM problem. Since it is widely believed that 3SUM cannot be solved in o(n2) time, this also holds for the problem of counting triangles. Theorem 1 Counting the number of triangles in an arrangement of straight-line segments is 3SUM-hard. Proof. We reduce the problem Point-on-3-Lines to counting triangles. Gajentaan and Overmars showed that Point-on-3-Lines is as hard as 3SUM . In Point-on-3-Lines one has to decide whether a given arrangement of straight-lines contains a point in which at least three lines intersect. It is easy to see that the problem remains 3SUM-hard even if no pair of lines is parallel. We transform such an arrangement of lines (with no parallel pairs) to an arrangement of straight-line segments by shortening all lines to segments. We must ensure that all crossings of lines are preserved as crossings of the corresponding segments. To this end, we determine the bounding box of the line arrangement, which is not hard to achieve in O(n log n) time. Consider the resulting arrangement. Since it contains all crossings, each triple of segments forms a triangle unless either the three segments intersect in a single point, or (ii) two of the segments are parallel—which cannot happen since the input lines did not contain parallel pairs. Therefore, the arrangement of segments contains n 3 triangles if and only if there is no point in which three or more lines intersect. We have seen that we can check the existence of a point lying on at least three lines by counting the triangles in the arrangement of segments. Furthermore, transform-ing the instance and determining the number α needed only constant time. Hence, an o(n2)-time algorithm for counting triangles in segment arrangements implies an o(n2)-time algorithm for Point-on-3-Lines. □ We can use almost the same 3SUM-hardness proof for convex quadrilaterals rather than triangles. Observe that any arrangement of four straight-lines (without parallel pairs) forms exactly one quadrilateral face unless three of the lines meet in a point. Hence, the number of quadrilaterals is n 4 if and only if there is no triple of lines meeting in a point. Theorem 2 Counting the number of convex quadrilater-als in an arrangement of straight-line segments is 3SUM-hard. Unfortunately, for larger values of k, e.g., k = 5 the hardness reduction does not seem easy to adjust. The problem is that not every set of ﬁve straight lines forms a 5-gon, even if they are in general position. 5 Conclusion We introduced the problem of counting and reporting k-gons in an arrangement of line segments, and presented an O(n log n+mn) time algorithm for counting all the k-gons, for any ﬁxed constant k, where m is the number of intersecting segment pairs. Our algorithm for reporting all the k-gons runs in time O(n log n + mn + \|K\|), where K is the output set. We also prove that the k-gon counting problem is 3SUM-hard for k = 3 and k = 4. 28th Canadian Conference on Computational Geometry, 2016 References N. Alon, R. Yuster, and U. Zwick. Finding and counting given length cycles. Algorithmica, 17:209–223, 1997. J. L. Bentley and T. A. Ottmann. Algorithms for report-ing and counting geometric intersections. IEEE Transac-tions on Computers, 100(9):643–647, 1979. B. Chazelle. Reporting and counting segment inter-sections. Journal of Computer and System Sciences, 32(2):156–182, 1986. D. Eppstein, M. Overmars, G. Rote, and G. Woeginger. Finding minimum area k-gons. Discrete & Computational Geometry, 7(1):45–58, 1992. S. Felsner and K. Krieger. Triangles in euclidean ar-rangements. In Graph-Theoretic Concepts in Computer Science, pages 137–148. Springer, 1998. A. Gajentaan and M. H. Overmars. On a class of o(n2) problems in computational geometry. Computational Geometry, 5(3):165 – 185, 1995. J. S. Mitchell, G. Rote, G. Sundaram, and G. Woeg-inger. Counting convex polygons in planar point sets. Information Processing Letters, 56(1):45–49, 1995. G. Rote and G. Woeginger. Counting convex k-gons in planar point sets. Information Processing Letters, 41(4):191–194, 1992. G. Rote, G. Woeginger, Z. Binhai, and W. Zhengyan. Counting k-subsets and convex k-gons in the plane. In-formation Processing Letters, 38(3):149–151, 1991.
2898	https://www.purplemath.com/modules/ratio6.htm	Select a Course Below Solving Proportions: Similar Figures RatiosProportionsProportionalitySolvingWord ProblemsSimilar FiguresSun's Rays / Parts Purplemath Proportionality equations can be used to "solve" "similar figures". What are "similar figures"? "Similar" is a geometric term, referring to geometric figures (squares, triangles, etc) that are the same shape, but one of the pair of figures is larger than is the other. Content Continues Below MathHelp.com Solving Proportions For a mental picture of similar figures, think of what happens when you use the "enlarge" or "reduce" setting on a copier, or when you get an eight-by-ten enlargement of a three-by-five picture you really like, or, if you've used a graphics program, think about preserving the "aspect ratio" of an image that you're resizing. Each of these examples results in a pair of similar figures (or similar pictures, anyway). How do you "solve" similar figures? In the context of ratios and proportions, the point of similarity is that the corresponding sides of similar figures are proportional; that is, that the lengths of matching sides are proportional. So, given two similar figures, not all of whose sides are labelled with their lengths, you can create a proportionality equation mixing known and unknown information, and solve for the unknown value. Note: For this process to work, the similar figures must have at least one pair of corresponding sides both of whose lengths is known. This pair will be your reference pair, forming one side of your proportionality equation. For instance, look at the similar triangles ABC and abc below: The "corresponding sides" are the pairs of sides that "match", other than for the enlargement or reduction aspect of their relative sizes. So A corresponds to a, B corresponds to b, and C corresponds to c. Since these triangles are similar, then the pairs of corresponding sides are proportional. That is, A : a = B : b = C : c. In words, this tri-part equation says, "Big-A is to little-A as big-B is to little-B and as big-C is to little-C". This proportionality of corresponding sides can be used to find the length of a side of a figure, given a similar figure for which sufficient measurements are known. I'll set up my proportions, using ratios in the form (big triangle length) ÷ (small triangle length), and then I'll solve the proportions. (By the way, you can set up your proportions as you like; there is no rule that says "bigger goes on top".) Since they've only given me the length of side a for the little triangle, my reference ratio will be A : a. First, I'll find the length of b. Here's my set-up: (There is no requirement that you label things, as I did above with the initial fraction containing the words "big" and "small", but this can be very helpful in reminding yourself how you're wanting to set things up. It's a quick way of keeping yourself out of trouble.) Filling in my known values, I get: b = [21 × 81]/48 b = 1701/48 b = 35.4375 (I'll need to remember to give the rounded-to-whole-number value for this side's length (in other words, I'll need to remember to round to "35") in my hand-in answer.) Now that I've found one length, I will, by the same method, find the length of the remaining side, c. c × 48 = 21 × 68 c = 1428/48 c = 29.75 For my answer, I could just slap down the two numbers I've found, but those numbers won't make much sense without their units. Also, in re-checking the original exercise, I'm remindedthat I'm supposed to round my values to the nearest whole number, so "29.75", with or without units, would be wrong. The right answer is: b = 35 mm, c = 30 mm Affiliate Advertisement While I could have used the length I'd found for b for finding the length of c in the above exercise, I instead returned to the value of a. Why? Because it was a known-good "exact" value. While the decimal value was in this case an exact value (that is, the value for b was not rounded), it's generally better to make a habit of returning to known-good values whenever you can. That way, when a decimal value has been rounded, you're ignoring the rounded derived value and returning to the exact original value. This practice will help you avoid the potential for round-off errors. The photo lab, when enlarging the original picture, will be maintaining the aspect ratio of the original; that is, the rectangles representing the outer edges of the original and enlarged pictures will be similar figures. Using this fact, I can set up a proportion and solve, using "h" to stand for the height value that I'm seeking: h = [(9)(3.5)]/5 31.5 = 5h = 31.5/5 h = 6.3 The height of the picture will be: 6.3 inches In the first exercise above, the ratios were between corresponding sides, and the proportionality was formed from those pairs of sides. The ratios in the proportions contained fractions formed from the original large value divided by the new small value. In the second exercise above, the ratios were between the two different dimensions, and the proportionality was formed from the sets of dimensions. The ratios in the proportion contain fractions formed from the old height and old width, and from the new height and the new width. For many exercises, you will be able to set up your ratios and proportions in more than one way. This is perfectly okay. Just make sure that you label things well, clearly define your variables, and set things up in a sensible and consistent manner. Doing so should help you dependably reach the correct solutions. If you're ever not sure of your solution, remember to plug it back into the original exercise, and verify that it works. Content Continues Below There is another topic, kind of an off-shoot of similar-figures questions, which you may encounter. It is based on the fact that, if two figures (or three-dimensional shapes) are similar, then not only are their lengths proportional, but so also are their squares (being their areas) and their cubes (being their volumes). A "rectangular prism" is just fancy geometrical talk for "a brick", so I know that I'm working with three-dimensional shapes. I am given that the shapes are similar, and I'm provided with two comparative lengths. This gives me my basic ratio: (small)/(large): 15/27 = 5/9 This 5/9 is the simplified (that is, reduced to lowest terms) linear ratio for the two prisms, and it's what I'll use for finding my answers for volume and surface area. (a) To find the volume of the larger prism, I need to cube the linear ratio they gave me (that is, I need to cube the reduced fraction that I'd created when I'd put the two lengths into a ratio, above). Putting the values for the smaller prism in the tops of the ratios, this gives me: (5/9)3 = 125/729 This is the ratio I'll use for setting up my volume proportion: V = [(2000)(729)]/125 V = 11664 Checking my units, I get an answer of: (a) 11,664 cm3 (b) To find the surface area of the one side of the smaller prism, I need to square the linear ratio they gave me (that is, I need to square the reduced fraction created by putting the two lengths into a ratio). Putting the values for the smaller prism in the tops of the ratios, this gives me: (5/9)2 = 25/81 This is the ratio I'll use for setting up my area proportion. A = [(243)(25)]/81 A = 75 Checking my units, I get an answer of: (b) 75 cm2 Affiliate URL: Page 1Page 2Page 3Page 4Page 5Page 6Page 7 Select a Course Below Standardized Test Prep K12 Math College Math Homeschool Math Share This Page Visit Our Profiles © 2024 Purplemath, Inc. All right reserved. Web Design by
2899	https://www.sciencelearn.org.nz/resources/49-refraction-of-light	Article Refraction of light Refraction is the bending of light (it also happens with sound, water and other waves) as it passes from one transparent substance into another. This bending by refraction makes it possible for us to have lenses, magnifying glasses, prisms and rainbows. Even our eyes depend upon this bending of light. Without refraction, we wouldn’t be able to focus light onto our retina. Refraction See more By using the example of spearing a fish, Associate Professor Gordon Sanderson, an ophthalmologist from Otago University, explains the principle of refraction. Rights: University of Waikato. All Rights Reserved. Change of speed causes change of direction Light refracts whenever it travels at an angle into a substance with a different refractive index (optical density). This change of direction is caused by a change in speed. For example, when light travels from air into water, it slows down, causing it to continue to travel at a different angle or direction. How much does light bend? Refraction of light in water See more When light travels from air into water, it slows down, causing it to change direction slightly. This change of direction is called refraction. When light enters a more dense substance (higher refractive index), it ‘bends’ more towards the normal line. Rights: The University of Waikato Te Whare Wānanga o Waikato The amount of bending depends on two things: Change in speed – if a substance causes the light to speed up or slow down more, it will refract (bend) more. Angle of the incident ray – if the light is entering the substance at a greater angle, the amount of refraction will also be more noticeable. On the other hand, if the light is entering the new substance from straight on (at 90° to the surface), the light will still slow down, but it won’t change direction at all. Refractive index of some transparent substances \| Substance \| Refractive index \| Speed of light in substance(x 1,000,000 m/s) \| Angle of refraction ifincident ray enterssubstance at 20º \| --- --- \| \| Air \| 1.00 \| 300 \| 20 \| \| Water \| 1.33 \| 226 \| 14.9 \| \| Glass \| 1.5 \| 200 \| 13.2 \| \| Diamond \| 2.4 \| 125 \| 8.2 \| All angles are measured from an imaginary line drawn at 90° to the surface of the two substances This line is drawn as a dotted line and is called the normal. If light enters any substance with a higher refractive index (such as from air into glass) it slows down. The light bends towards the normal line. If light travels enters into a substance with a lower refractive index (such as from water into air) it speeds up. The light bends away from the normal line. A higher refractive index shows that light will slow down and change direction more as it enters the substance. Lenses A lens is simply a curved block of glass or plastic. There are two kinds of lens. A biconvex lens is thicker at the middle than it is at the edges. This is the kind of lens used for a magnifying glass. Parallel rays of light can be focused in to a focal point. A biconvex lens is called a converging lens. Converging lens See more Each light ray entering a converging (convex) lens refracts inwards as it enters the lens and inwards again as it leaves. These refractions cause parallel light rays to meet (converge) – focussing in on a focal point – before spreading out again. Rights: The University of Waikato Te Whare Wānanga o Waikato A biconcave lens curves is thinner at the middle than it is at the edges. Light rays refract outwards (spread apart) as they enter the lens and again as they leave. Concave lens See more Each light ray entering a diverging (concave) lens refracts outwards as it enters the lens and outwards again as it leaves. These refractions cause parallel light rays to spread out, travelling directly away from an imaginary focal point. Rights: The University of Waikato Te Whare Wānanga o Waikato Refraction can create a spectrum Isaac Newton performed a famous experiment using a triangular block of glass called a prism. He used sunlight shining in through his window to create a spectrum of colours on the opposite side of his room. This experiment showed that white light is actually made of all the colours of the rainbow. These seven colours are remembered by the acronym ROY G BIV – red, orange, yellow, green, blue, indigo and violet. Prism See more When white light shines through a prism, each colour refracts at a slightly different angle. Violet light refracts slightly more than red light. A prism can be used to show the seven colours of the spectrum that make up white light. Rights: Lawrence Lawry/Science Photo Library Newton showed that each of these colours cannot be turned into other colours. He also showed that they can be recombined to make white light again. The explanation for the colours separating out is that the light is made of waves. Red light has a longer wavelength than violet light. The refractive index for red light in glass is slightly different than for violet light. Violet light slows down even more than red light, so it is refracted at a slightly greater angle. The refractive index of red light in glass is 1.513. The refractive index of violet light is 1.532. This slight difference is enough for the shorter wavelengths of light to be refracted more. Rainbows A rainbow is caused because each colour refracts at slightly different angles as it enters, reflects off the inside and then leaves each tiny drop of rain. Rainbow See more A rainbow is formed when light enters each water droplet, and the different colours bend (refract) at slightly different angles. They reflect off the inside of the drop before refracting again as they leave. The shorter wavelengths refract more. Rights: Public domain A rainbow is easy to create using a spray bottle and the sunshine. The centre of the circle of the rainbow will always be the shadow of your head on the ground. The secondary rainbow that can sometimes be seen is caused by each ray of light reflecting twice on the inside of each droplet before it leaves. This second reflection causes the colours on the secondary rainbow to be reversed. Red is at the top for the primary rainbow, but in the secondary rainbow, red is at the bottom. Activity ideas Use these activities with your students to explore refration further: Investigating refraction and spearfishing – students aim spears at a model of a fish in a container of water. When they move their spears towards the fish, they miss! Angle of refraction calculator challenge – students choose two types of transparent substance. They then enter the angle of the incident ray in the spreadsheet calculator, and the angle of the refracted ray is calculated for them. Light and sight: true or false? – students participate in an interactive ‘true or false’ activity that highlights common alternative conceptions about light and sight. This activity can be done individually, in pairs or as a whole class. Useful links Learn more about different types of rainbows, how they are made and other atmospheric optical phenomena with this MetService blog and Science Kids post. Learn more about human lenses, optics, photoreceptors and neural pathways that enable vision through this tutorial from Biology Online. Published: 26 April 2012Updated: 23 May 2020 Referencing Hub media Explore related content #### Article Reflection of light Reflection is when light bounces off an object. If the surface is smooth and shiny, like glass, water or polished ... Read more #### Activity Angle of refraction calculator challenge In this activity, students choose two types of transparent substance. They then enter the angle of the incident ray in ... Read more #### Activity What is refraction? In this activity, students are introduced to an analogy through a role-play that helps them visualise the refraction of light. ... Read more

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