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Advanced Placement Program* Summer Institute
AP* Calculus AB, July 22-25, 2013
Lead Consultant: Joe Milliet
What to Bring
Much work is done on the graphing calculator. For our purposes, a TI-83/84 (any edition) will be used in our demonstrations. Please bring one of these calculators or be proficient in doing comparable work on a calculator of your choosing without needing instruction.
If you have a tablet with a digital pen stylus for note-taking using math symbols, your laptop/tablet could be useful in the institute. Other than that, use of a laptop or other digital device is probably a distraction to the institute and your fellow participants.
Sample texts will be ordered for you. How many we receive will depend on publisher generosity.
Course Description
This course will present a detailed analysis of the current Calculus AB curriculum. Features of the institute will include: strategies for teaching various topics; resources available for the AP* teacher; discussion of content from an advanced viewpoint; substantial activities with graphing calculators and their role on the AP* exam; suggested timetables; suggestions for types of assignments; and suggestions for sample test questions. The format of the AP* exam together with an analysis of scoring standards used will be discussed. Sample questions and the grading process from the perspective of an AP* Reader will be addressed. Participants will have opportunities to work "hands-on" with graphing calculators and will receive numerous materials.
About Joe Milliet
Joe Milliet is the mathematics department chair at St. Mark's School of Texas and teaches both middle and high school mathematics. Starting in 1976, Joe taught high school mathematics at Beaumont Central High School in Beaumont, Texas, for 22 years. During that time, he served the AP* Calculus community as a College Board consultant, a workshop leader, exam reader, and member of the AP* Calculus Test Development Committee. From June 1998 to June 1999, Joe served as Associate Director for the AP* Program at the College Board Southwest Region Office (SWRO) in Austin, Texas. From June 1999 to July 2000, Joe worked as a consultant to promote a partnership between the College Board SWRO and the University of Texas, while also developing AP* course support material in conjunction with the Dana Center for Math and Science Education at the University of Texas. Joe returned to teaching in August 2000, joining the faculty at St. Mark's. Having taught mathematics for 35 years total, including AP* Calculus AB and/or BC for 31 years, he currently teaches AP* Calculus along with other mathematics courses. He also continues to serve as an AP* Calculus consultant since his return to teaching. He is co-author of the test preparation book "Be Prepared for the AP* Calculus Exam" published in 2005. In 2008-2009, Joe was the state of Texas' recipient of the Siemens Award for Advanced Placement.
* Advanced Placement Program and AP are registered trademarks of the College Board and have been used with permission. | 677.169 | 1 |
Algebra for Everyone? by John E. Bonfadini, Ed.D., Contributing ColumnistProfessor, George
Mason University
Some people think public education is sick and needs a good dose of antibiotics.
Others are hoping to find a vaccine that will protect every student from any bad
experiences associated with public education. Yes, we have no shortage of Jonas Salks who
think they have the cure for today's educational problems. I guess some readers may
even consider this column in that category. I certainly don't think I have the cure,
only the desire to share with my readers some "Food for Thought." So here is a
little cookie.
Who in the
world ever came up with the idea that everyone needs a shot of algebra? We have placed
some mystic aura around this mathematical offering and hypothesize that forcing all
students to take it will better prepare them for life's future challenges. I just
happen to disagree. In high school I took Algebra I and II. Don't know why, someone
just said these courses were part of some diploma requirement. I believe it was called a
scientific academic diploma. I can remember a student who sat next to me in algebra class,
his name was Joel and he was a math whiz. He could calculate when the hands on a clock
would pass one another or when a boat traveling upstream against given water current would
arrive at the dock. I certainly had difficulty with working those kinds of problems or
maybe I saw no need to know those things. You guessed it; my grades in the two courses
weren't very good.
I liked math back then and still like it today. The math I enjoyed learning had some
real tangible purpose. I've taught electronics in high school and now teach basic
statistics at the college level. I still haven't found a good reason to factor a
trinomial. During my high school teaching days many of the guidance counselors insisted on
telling students they needed algebra as a prerequisite to taking my electronics class. I
kept trying to inform the counselors that a good fundamental math course was all a student
needed if they weren't planning on becoming an engineer. All students need good
fundamental math skills. Somehow we have lost all common sense when it comes to
determining what math most of us need. Take time and list the mathematical-related things
you did during the last three months. How much algebra did you use? Many educator friends
will respond with the "it makes you think" argument. Tell me something that you
do that doesn't require thought. I do far more creative thinking in the bathroom than
I ever did in algebra class.
Preach (and Teach) What You Practice
What's
Your View?
Obviously, there are
at least two sides to every issue. Do you have a different view? This column is meant to
provoke thought, so keep sending comments. Each one is read with the utmost interest. Send
e-mail to: jbonfadi@gmu.edu, or send written
responses to the editor or to John Bonfadini, 7500 Forrester Lane, Manassas, VA
20109.
The music industry has composers who provide the music to singers and musicians. Many
great singers aren't composers. They use music in a practical way — they sing.
Why can't we accept some students who use math in this same practical way? Somehow
education has gotten snobbish. We look down on everyone who doesn't fit the mold of a
liberal arts college graduate. Shame on us. The state is now designing different diplomas
to further classify students. If you take courses like algebra, you somehow get a better
diploma than if you take electronics, auto mechanics, business, or other more practical
courses. One has to ask why two different diplomas are needed. If your answer is that
colleges need to know, well — colleges get a transcript, not a diploma. I can't
imagine that employers who have "HIRING" or "HELP WANTED" signs all
over the place are looking for algebra on prospective employees' transcripts. I only
hope that someone has taught them how to count change so I can get through the line a
little faster.
Last month I had the opportunity to attend the State
Board of Education meeting. The Association of Mathematics Supervisors made a statement
supporting algebra for everyone and condemning the teaching of consumer, general or
technical math. I couldn't believe what I was hearing. I hope they are not reflecting
the sentiments of the state's math teachers. Somehow all disciplines are fighting to
look more academic. Even in my own area, vocational education, schools see more value in
offering courses related to computer technology than the practical arts like masonry, auto
mechanics, carpentry, electrician, air conditioning, electronics, and etc. My only
question is, "Hasn't anyone tried to get their automobile or air conditioning
fixed?" We just need to take a little time to look at what really makes a difference
in the quality of life. I know it's not another course in algebra for everyone. I
recently heard of a vocational administrator who justified supporting every student taking
algebra because he was afraid vocational students who didn't take the course would be
considered second rate (I don't want to use the "D" word). That's a
great educational strategy; sacrifice the child for an image.
I've seen this educational strategy used
many times before. Just shuffle the required educational deck of courses. Can anyone
remember the Sputnik days? Every child had to have more math and foreign language or the
USA was doomed to fall to the evil empire. The "more algebra for everyone"
strategy will do little to bring about real educational change. Compare adding an algebra
course to making changes like reducing class sizes, providing teachers with clerical help,
increasing funding for technology management tools, hiring more and better teachers,
providing teachers with more professional development, increasing parent involvement with
their child's education, and the list goes on.
Let's Fix the Real Problems
Requiring algebra of all students as a means of improving education is like calling in
the real estate agent when your home's plumbing is leaking, the roof needs repair,
the paint is peeling off, the air conditioning has quit, and your dog has ruined all the
carpet. It's a perceived quick fix that doesn't touch the real problems, just
passes them on to another generation. Finally, whatever happened to the idea of designing
programs for the individual child? I guess it's just too difficult, so we'll
recommend cloning everyone to be someone's educational ideal. | 677.169 | 1 |
Modify Your Results
This traditional text offers a balanced approach that combines the theoretical instruction of calculus with the best aspects of reform, including creative teaching and learning techniques such as the integration of technology, the use of real data, real-life applications, and mathematical models. The Calculus with Analytic Geometry Alternate, 6/e, offers a late approach to trigonometry for those instructors who wish to introduce it later in their courses.
Advanced Placement Calculus is aligned with the NCTM Standards and the latest AP guidelines and retains the solid, carefully motivated mathematics that characterize previous editions. The Seventh Edition embraces the best aspects of reform by integrating the latest technological tools and by emphasizing real-life data, practical applications, and mathematical models.
Points, lines, and planes are the basic figures of geometry. In algebra, you used ordered pairs of numbers to locate points on a coordinate plane. You also used tables of ordered pairs and linear equations to locate and draw straight lines on a coordinate plane. Geometry on the coordinate plane is called coordinate geometry.
George Washington is often called the father of our country. He beat the odds to lead the Continental Army to victory in the Revolutionary War, served as first president of the United States, and then retired from public life to farm Virginia's largest plantation. In this groundbreaking new series, DK brings together fresh voices and DK design values to give readers the most information-packed, visually exciting biographies on the market today. Full-color photographs of people, places, and artifacts, and sidebars on related subjects add dimension and relevance to stories of famous lives that students will love to read. Modern scholarship and a variety of narrative approaches give today's reader a chance to explore the extraordinary worlds of George Washington and Abraham Lincoln. This new way of looking at classic subjects creates a unique reading experience that breathes life into the book-report and summer-reading repertoire.
In this thoughtful and incisive biography, the strengths and weaknesses of Washington's presidential leadership are dissected, from his lasting foreign and economic policies to his polarizing denunciation of political parties and his public silence about slavery. The result is a surprising portrait of the multidimensional man behind the myth he so assiduously crafted | 677.169 | 1 |
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MathModelica: Design, Simulate, Create
Jan Brugård
This example-driven screencast from the Wolfram Mathematica Virtual Conference 2011 shows how to develop models of complex systems using drag-and-drop with the Wolfram MathModelica software, and illustrates how to take the models seamlessly into Mathematica for doing your simulation analysis as well as model design.
Channels: Virtual Events
See the first public viewing of the revolutionary Wolfram Calculator. This Wolfram Technology for STEM Education: Virtual Conference for Education talk demonstrates the basic functionality as well as the predictive ...
Learn how to design interactive digital material for the classroom. This Wolfram Technology for STEM Education: Virtual Conference for Education talk highlights the Wolfram Demonstrations Project and how teachers can ...
The Wolfram Cloud Platform provides a convenient way to use Mathematica, create and share resources, and collaborate with students and colleagues. This Wolfram Technology for STEM Education: Virtual ...
computerbasedmath.org has been engaged in a fundamental rethink of math education. This Wolfram Technology for STEM Education: Virtual Conference for Education talk shares some of the thinking behind the ...
Learn how to incorporate engaging digital content into physics classrooms with Mathematica, Wolfram|Alpha, and other Wolfram technologies. This Wolfram Technology for STEM Education: Virtual Conference for Education ...
Wolfram technologies are the tools for providing interactive and engaging materials for STEM education. In this video, Conrad Wolfram shares examples and explains why Wolfram is uniquely positioned to be ...
This video explains the principles of volume rendering and the art of constructing the right transfer functions. Markus van Almsick explores the drawbacks and extravagant possibilities of this new visualization ... | 677.169 | 1 |
Description
This app introduces concepts and develops techniques to enable the user to correctly perform and interpret calculations with data. This content is found in GCSE Mathematics courses, GCSE Statistics courses and in some first applied modules of A-Level Mathematics.
With an inexpensive in-app purchase, the full nine sections of teaching, questions and worked solutions become available. Four useful introductory sections are immediately accessible from the initial download. This app makes an ideal companion for learning, revision, examination practice, and checking knowledge.
* Suitable for students, parents, tutors and teachers.
* Visit to sign up for 1600 questions and solutions from 188 A-Level papers, spanning over 2000 pages and including video demonstrations for over 60 of the subsections. There are 1000+ pages of maths e-text covering Statistics and Pure Mathematics micro Mathematics, you can not only perform mathematical calculations in naturally readable form, but can also create and manage your own collection of interactive formulas! It is free of any charges and does not contain any adds.
Micro Mathematics"Scientific Calculator 3D FreeThis is an app for those who use the linear convolution theorem. It will directly give you the result of the problem as you enter your X and Y values. The data entered is in a very easy and interactive way, and even calculations for the element at the zeroth position are made.
This app is essential for any math student. Draw, chart, graph and plot linear, quadratic, trigonometry and custom equations. Get quadratic roots. Learn and understand math through graphs and equations. This app It is unique because it allows the user to interact with the graphic layout, enabling drag vertices and intersection points, and get in real-time the changes that occur in the equation coefficients.
## Key Features ##
- Several examples of mathematical equations with charts of the most dif, ferent ways; - Module for the assembly of equations f (x). Allows the use of parentheses, basic mathematical operators (+, -, *, /) and many mathematical functions such as sin, cos, tan, ln, sqrt, cube, abs, acos, asin, atan; - Module for the construction and analysis of linear equations, enabling the drag of all the points of intersection between the line and the graphics axes; - Module for the construction and analysis of quadratic equations, enabling drag the vertex of the parabola, and the parable of the intersection points with the graphics axes; - Module for trigonometric analysis.
With micro Mathematics Plus, you can not only perform mathematical calculations in naturally readable form, but can also create and manage your own collection of interactive formulas!
Micro Mathematics PlusThe micro Mathematics Plus has exactly the same user interface as the micro Mathematics free version, but implements more mathematical functionality: functions of many arguments, plots for several functions, contour and 3D plots, summation and product operations, derivative and definite integrals, if-function and logical operators. This version has following mathematical limitations: it does not support complex numbers, special functions, vectors, matrices and many other things from high-level mathematicsFinger Painter is a painting tool, a coloring book that is designed to bring out your child's inner artist. A simple to use interface, easy to use navigation based around the alphabet so your kids also learn their A, B, Cs (it will speak as you touch).
Finger Painter carries levels of difficulty, ranging from 1 star to 3 stars and backed up with a 3D image to aid with colouring in. All illustrations are original, there are no re-purposed or stolen copyrighted images in this product.
Suitable for all children, it will teach them how to paint whilst subconsciously learn about light and shadow.
If there is anything you would like to see, why not leave a comment (in the title) and you *might* just get what you wish for ;)
THERE ARE NO HIDDEN COSTS, NO IN-APP PURCHASES OR ADS (AND NEVER WILL).
Want to see how it's put together?
Let's see what people have done in the Finger Painter gallery:
Please email images to with your "name" and "age" in the subject: images@fp.pixelised.co.uk
MathsTools is free app that allows you to perform various mathematical operations. MathsTools include following features: **Function Plotter-----Looking for an efficient plotter????you need have this. Plot and evalute functions of y(x),parametric and polar. Plot derivates of functions of x. Trace and calculate roots of any functions. Pinch Zoom to get accurate result. Save image of current plot.
**Base Converter----Convert numbers of any length between decimal,Binary,Ternary, Quaternery,Octal,and Hexadecimal.
**Definite Integration---A powerful feature that calculate definite integration of function of x between given limits.
**Factorization---Factorize number upto length of 50 digits into prime factors. You can also calculate LCM and GCD of n numbers.
This sensor fusion app is intended as an illustration of what sensor capabilities your smartphone or tablet have. You can watch graphs of the main sensors in real time, except for video, microphones and radio signals. You can log data to file or stream data to a computer. The app is bundled with a a Matlab interface which allows for on-line processing and filtering for prototyping and demo purposes.
The first versions of this app was developed in collaboration with HiQ (hiq.se), funded by a SAAB award in the name of former CEO Åke Svensson.
The second version of this app, featuring a considerable rewrite of the code base as well as extended functionality and Matlab support, was developed by Gustaf Hendeby as part of introducing the app as part of a lab in the Sensor Fusion course at University of Linköping the spring of 2013.
PTC-Circle: Smart School Information Cum SMS System is a powerful and effective tool for "School/College/Institute Automation". It provides effective tools to manage the various aspects of schools/colleges/institutes.
'Franciscan e-Care' mobile app is a smart tool, a complete and comprehensive package which brings all e-Care facilities at one place at fingertips for all users (School management, Teachers, Parents and Students). Responding to the growing use of mobile technology in the modern world and the increased adoption of smart phones, e-Care has extended its multi-platform capabilities to include support for tablets and smart phones. e-Care now enables user to carry their school world in their pocket with this mobile application.
* Visit to sign up for 1600 more questions and solutions from 188 A-Level papers, spanning over 2000 pages and including video demonstrations for over 60 of the subsections. There are 1000+ pages of maths e-text covering Statistics and Pure Mathematics. | 677.169 | 1 |
Calculus (1,437 Books)
Calculus (Latin, calculus, a small stone used for counting) is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus. | 677.169 | 1 |
The Everything Guide to Calculus 1: A step-by-step guide to the basics of calculus - in plain English!
Paperback
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Meet the Author
Greg Hill has more than twenty-five years of experience teaching AP Calculus and other advanced math classes. He is a two-time Illinois state finalist for the Presidential Award for Excellence in Mathematics and Science Teaching, and is a member of the Illinois and National Councils of Teachers of Mathematics. Hill has been a College Board consultant and AP Calculus exam grader for the past ten years. He currently teaches at Hinsdale Central High School, and also conducts day- and week-long professional development seminars for AP Calculus teachers. He is the author of CLEP Calculus, a test prep book for the College Board's College Level Entrance Exam. | 677.169 | 1 |
Curriculum Pathways has launched an Algebra add-on to its science, technology, engineering and maths (Stemocusing on core Stem subjects, SAS Curriculum Pathways is free for secondary school teachers and students. Designed for both GCSE and A-level students, the online resource includes interactive tools, web lessons, enquiries and audio tutorials. More than 150 UK schools have signed up for SAS Curriculum Pathways since it was launched in March 2012.
An estimated 15 million adults in the UK have numeracy skills equivalent to or below those of an 11-year-old, according to a study by KPMG. The lack of numeracy skills in this country is costing the UK up to £2.4bn a year, the survey revealed.
Geoffrey Taylor, academic programme manager at SAS UK and Ireland, said the figures were concerning.
"With the growing amount of big data being generated, a strong numerical understanding is vital for providing the basis for many careers within the growing business analytics sector, an area where SAS is a market leader," he said.
"Algebra 1 is providing students with the building blocks to develop a key understanding of maths, which may help to produce the analysts of the future for SAS and its clients | 677.169 | 1 |
Viewpoints by P-PrincetonUniversi
An undergraduate textbook devoted exclusively to relationships between mathematics and art, Viewpoints is ideally suited for math-for-liberal-arts courses and mathematics courses for fine arts majors. The textbook contains a wide variety of classroom-tested activities and problems, a series of essays by contemporary artists written especially for the book, and a plethora of pedagogical and learning opportunities for instructors and students.Viewpoints focuses on two mathematical areas: perspective related to drawing man-made forms and fractal geometry related to drawing natural forms. Investigating facets of the three-dimensional world in order to understand mathematical concepts behind the art, the textbook explores art topics including comic, anamorphic, and classical art, as well as photography, while presenting such mathematical ideas as proportion, ratio, self-similarity, exponents, and logarithms. Straightforward problems and rewarding solutions empower students to make accurate, sophisticated drawings. Personal essays and short biographies by contemporary artists are interspersed between chapters and are accompanied by images of their work. These fine artists—who include mathematicians and scientists—examine how mathematics influences their art. Accessible to students of all levels, Viewpoints encourages experimentation and collaboration, and captures the essence of artistic and mathematical creation and discovery.Classroom-tested activities and problem solvingAccessible problems that move beyond regular art school curriculumMultiple solutions of varying difficulty and applicabilityAppropriate for students of all mathematics and art levelsOriginal and exclusive essays by contemporary artistsSolutions manual (available only to teachers) | 677.169 | 1 |
Synopses & Reviews
Publisher Comments:
This book is a revised and up-dated fourth edition of a textbook designed for upper division courses in linear algebra. It includes the basic results on vector spaces over fields, determinants, the theory of a single linear transformation, and inner product spaces. While it does not presuppose an earlier course, many connections between linear algebra and calculus are worked into the discussion, making it best suited for students who have completed the calculus sequence. A special feature of the book is the inclusion of sections devoted to applications of linear algebra, which can either be part of a course, or used for independent study. The topics covered in these sections are the geometric interpretation of systems of linear equations, the classification of finite symmetry groups in two and three dimensions, the exponential of a matrix and its application to solving systems of first order linear differential equations with constant coefficients, and Hurwitz's theorem on the composition of quadratic forms. This revised fourth edition contains a new section on analytic methods in matrix theory, with applications to Markov chains in probability theory. Proofs of all the main theorems are included, and are presented on an equal footing with methods for solving numerical problems. Worked examples are included in almost every section, to bring out the meaning of the theorems, and to illustrate techniques for solving problems. Many numerical exercises are included, which use all the ideas, and develop computational skills. There are also exercises of a theoretical nature, which provide opportunities for students to discover interesting things for themselves.
Synopsis:
"Synopsis"
by Springer, | 677.169 | 1 |
Designed to prepare students for entry into basic calculus. Precalculus I together with Precalculus II is designed to prepare students for entry into the calculus sequence: MATH& 151, MATH& 152, MATH& 153, and MATH& 254. The topics include: absolute value, complex numbers, linear and quadratic equations, rational, polynomial, exponential and logarithmic functions, inverse functions, theory of equations, and sequences and series. Prerequisite: grade of 2.0 or better in MATH 095 or COMPASS test placement. Students completing MATH& 141 may not receive graduation credit for MATH& 144. | 677.169 | 1 |
MATLAB for Engineers hands-on approach and focus on problem solving, this introduction to Matlab uses examples drawn from a range of engineering disciplines to demonstrate Matlabrs"s applications to a broad variety of problems.Encourages readers to type in examples as they go for immediate application of techniques presented. Includes numerous broad-based examples embedded in the text, practice exercises with solutions, and hints related to commonly encountered problems. Introduces m-files early in the text to make it easier for readers to save their work and develop a consistent programming strategy.For those interested in learning Matlab. | 677.169 | 1 |
...
Show More collaborative learning, require student observation, and integrate technology for gathering, recording, and synthesizing data. The third edition provides even more guidance and resources for students to support their active learning of statistics and includes updated real data sets with everyday applications in order to promote statistical literacy. This version of the text isrecommended for use with graphing calculator technology and includes integrated instructions for using a Texas Instruments calculator (specifically a TI-83 or TI-84111.95
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Can you also answer the "General Problems"?
It would be enormously time consuming to also answer all the "General Problems" so I'm limiting coverage only to the regular "Problems", of which there are still more than 1700. If you need help with a "General Problem", my suggestion would be to try and find a regular "Problem" that is similar to the "General Problem", and see if you can apply the same problem solving technique.
Will you answer the question I left below a video?
It depends. I certainly want to, but I need to be time efficient since it would be enormously time consuming to give personal email help to all subscribers. However, often I do respond to comments on the site, although sometimes I leave them for other students to pick up. Here are the things that I'm thinking about when deciding whether I'll be able to get to your question:
Is the question also useful for other students?
Is there a specific time mentioned for the video? (example: "at 2:10, you mentioned a = b^2, so...") Having a specific time to look at helps me know right away exactly what you're asking about.
How much time do I have? (I always wish I had more!)
If you're pointing out an error in a solution, I always respond and thank you for mentioning that. | 677.169 | 1 |
Understanding these topics requires mathematics through calculus and mental models of underlying physical processes. Working with the student we can develop the skills necessary for new areas of interest. Computer software often draws from multiple disciplines and understanding basic concepts is essential. | 677.169 | 1 |
Introduction to Multimedia is designed to help you become a skilled and creative user of current multimedia technology. The purpose of this course is to increase your understanding of multimedia concepts and techniques. You are the reason that this textbook was written. Multimedia is a vast and exciting domain.
This book focuses on 3 critical areas: applying equations in one and two variables;understanding the concept of a function and using functions to describe quantitative relationships; applying the Pythagorean Theorem and the concepts of similarity and congruenceGlencoe MathinteractiveStudent Editionsallow students to work problems, tear out pages, take notes, and even complete homework sheets right in the book. Pages are 3-hole punched so they fit neatly into class binders.
The Glencoe Math interactive Student Editions allow students to work problems, tear out pages, take notes, and even complete homework sheets right in the book. Pages are 3-hole punched so they fit neatly into class binders.
To help chart their journeys, wise travelers consult a map before they begin. Just as maps lead travelers to their destinations, the script on the next five pages points out the ways that you use the mathematics in this text in you daily lives.
Mathematics: Applications and Conceptsis a three-text Middle School series intended to bridge the gap from Elementary Mathematics to High School Mathematics. The program is designed to motivate middle school students, enable them to see the usefulness of mathematics in the world around them, enhance their fluency in the language of mathematics, and prepare them for success in Algebra and Geometry | 677.169 | 1 |
05344967Single Variable Calculus, Volume 1
Stewart's clear, direct writing style in SINGLE VARIABLE CALCULUS guides you through key ideas, theorems, and problem-solving steps. Every concept is supported by thoughtfully worked examples and carefully chosen exercises. Many of the detailed examples display solutions that are presented graphically, analytically, or numerically to provide further insight into mathematical concepts. Margin notes expand on and clarify the steps of the solution | 677.169 | 1 |
Elementary and Intermediate Algebra - Text - 6th editionsosolving skills necessary for success in algebra and beyond | 677.169 | 1 |
I Agol's Recommendations
Very thorough explanations of math and how the concepts we learn are applicable and useful to life outside the classroom. Also, writes everything important on the board, so makes it easy for you to go back and forth between taking notes and what he's saying | 677.169 | 1 |
a visual, graphical approach that emphasizes connections among concepts, this text helps students make the most of their study time. The authors show how different mathematical ideas are tied together through their zeros, solutions, and x-intercepts theme; side-by-side algebraic and graphical solutions; calculator screens; and examples and exercises. By continually reinforcing the connections among various mathematical concepts as well as different solution methods, the authors lead students to the ultimate goal of mastery and success in class. | 677.169 | 1 |
Search Results
This lesson from Illuminations asks students to look at different classes of polynomial functions by exploring the graphs of the functions. Students should already have a grasp of linear functions, quadratic functions,...
This lesson from Illuminations teaches students to use a computer algebra system to determine the square root of 2 to a given number of decimal places. Students will learn how utilizing technology makes an algorithm...
This math unit from Illuminations introduces students to the concepts of cryptology and coding. It includes two lessons, which cover the Caesar Cipher and the Vignere Cipher. Students will learn to encode and decode asks students to use matrix multiplication to transform digital images. Students will use matrix multiplication skills, look at the connections between geometric transformations and matrix... | 677.169 | 1 |
Precise Calculator has arbitrary precision and can calculate with complex numbers, fractions, vectors and matrices. Has more than 150 mathematical functions and statistical functions and is programmable (if, goto, print, return, for).Pc Calculator is a clever note and formula editor combined with an advanced and strong scientific calculator. Being an editor it is extremely user-friendly allowing all possible typing and other errors to be easily corrected and fast recalculated | 677.169 | 1 |
Basic Math Skills Courses
If you are an incoming first-year or transfer student and do not have adequate placement for the mathematics course required by your desired program of study, it may be a good idea for you to take MATH 005 Basic Math Skills and/or MATH 006 Basic Math Skills II the summer or J-Term before you begin at UST. STEM majors (science, technology, engineering, and mathematics) are especially encouraged to take the remedial course(s) when necessary; it is less crucial for business, humanities, and education majors.
These courses are non-credit, but will appear on your UST transcript.
Your math placement score along with your required Mathematics course will determine if you take MATH 005, MATH 006 or both. If you wish take one or both of these courses, please be advised of the following:
1. Academic Counseling & Support can help you enroll. The Academic Counseling & Support phone number is (651)-962-6300, or email Amy Ledin at ledi0009@stthomas.edu.
2. The cost of MATH 005 and MATH 006 is $291 per course during academic year 2015-2016 ( You will be directly billed if you enroll for one or both courses ($291 for one course, $582 for both MATH 005-006). When the bill is ready, you will receive an e-mail notification (via your UST e-mail account) about payment procedures and deadlines.
3. The textbook for Summer 2015 MATH 005 and MATH 006 will be Algebra for College Students, 10th Edition, by Kaufman and Schwitters. This text can be purchased at the UST bookstore. It is not required to purchase a hard copy of the textbook, but purchase of the eBook with access to WebAssign (an online homework system) is required. You will receive more information on the first day of classes from your instructor about purchasing and using WebAssign.
4. MATH 005 will be held beginning on July 22 and ending on August 5, on Mondays through Thursdays from 9 AM to noon. MATH 006 will be held beginning on August 6 and ending on August 20 on Mondays through Thursdays from 9 AM to noon. Class meetings for both MATH 005 and MATH 006 will be held in OWS 257.
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Proofs and Fundamentals : A First Course in Abstract Mathematics - 00 editionpa
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2. Strategies for Proofs 2.1 Mathematical Proofs-What They are and Why We Need Them 2.2 Direct Proofs 2.3 Proofs by Contrapositive and Contradiction 2.4 Cases, and If and Only If 2.5 Quantifiers in Theorems 2.6 Writing Mathematics
8. Number Systems 8.1 Back to the Beginning 8.2 The Natural Numbers 8.3 Further Properties of the Natural Numbers 8.4 The Integers 8.5 The Rational Numbers 8.6 The Real Numbers and the Complex Numbers 8.7 Appendix: Proof of Theorem 8.2.1 | 677.169 | 1 |
this course, you will be Introduced to several methods of numerical approximation such as error analysis, root finding,...
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In this book, there are five chapters: The Laplace Transform, Systems of Homogeneous Linear Differential Equations (HLDE), Methods of First and Higher Orders Differential Equations, Extended Methods of First and Higher Orders Differential Equations, and Applications of Differential Equations. In addition, there are a very useful for college students who studied Calculus II, and other students who want to review some concepts of differential equations before studying courses such as partial differential equations, applied mathematics, and electric circuits II Friendly Introduction to Differential Equations to your Bookmark Collection or Course ePortfolio
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'This review book, used in conjunction with free online YouTube videos, is designed to help students prepare for exams, or...
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'This review book, used in conjunction with free online YouTube videos, is designed to help students prepare for exams, or for self-study. The topics covered here are most of the standard topics covered in a first course in differential equations.The chapters and sections of this review book, organized by topics, can be read independently. Each chapter or section consists of three parts: (1) Theory; (2) YouTube Example; and (3) Additional Practice. In Theory, a summary of the topic and associated solution method is given. It is assumed that the student has seen the material before in lecture or in a standard textbook so that the presentation is concise. In YouTube Example, an online YouTube video illustrates how to solve an example problem given in the review book. Students are encouraged to view the video before proceeding to Additional Practice, which provides additional practice exercises similar to the YouTube example. The solutions to all of the practice exercises are given in this review book's Appendix.For students who self-study, or desire additional explanatory materials, a complete set of free lecture notes by the author entitled An Introduction to Differential Equations can be downloaded by clicking HERE. This set of lecture notes also contains links to additional YouTube tutorials. The lecture notes and tutorials have been extensively used by the author over several years when teaching an introductory differential equations course at the Hong Kong University of Science and Technology with YouTube Examples to your Bookmark Collection or Course ePortfolio
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PUBLISHED BY
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CHAPMAN & HALL, Limited, LONDON
MATHEMATICAL MONOGRAPHS.
EDITED BY
MANSFIELD MERRIMAN AND ROBERT S. WOODWARD.
No. 9.
DIFFERENTIAL EQUATIONS.
BY
W. WOOLSEY JOHNSON,
PROFESSOR OF MATHEMATICS IN THE UNITED STATES NAVAL ACADEMY.
FOURTH EDITION.
FIRST THOUSAND.
NEW YORK:
JOHN WILEY & SONS.
LONDON: CHAPMAN & HALL, LIMITED.
1906.
COPYRIGHT, 1896,
BY
MANSFIELD MERRIMAN AND ROBERT S. WOODWARD
UNDER THE TlTLE
HIGHER MATHEMATICS.
First Edition, September, 1896.
Second Edition, January, 1898.
Third Edition, August, 1900.
Fourth Edition, January, 1906.
FOHRT DRUMMOND, PRINTRR, NRW YORIC.
EDITORS' PREFACE.
THE volume called Higher Mathematics, the first edition
of which was published in 1896, contained eleven chapters by
eleven authors, each chapter being independent of the others,
but all supposing the reader to have at least a mathematical
training equivalent to that given in classical and engineering
colleges. The publication of that volume is now discontinued
and the chapters are issued in separate form. In these reissues
it will generally be found that the monographs are enlarged
by additional articles or appendices which either amplify the
former presentation or record recent advances. This plan of
publication has been arranged in order to meet the demand of
teachers and the convenience of classes, but it is also thought
that it may prove advantageous to readers in special lines of
mathematical literature.
It is the intention of the publishers and editors to add other
monographs to the series from time to time, if the call for the
same seems to warrant it. Among the topics which are under
consideration are those of elliptic functions, the theory of num-
bers, the group theory, the calculus of variations, and non-
Euclidean geometry; possibly also monographs on branches of
astronomy, mechanics, and mathematical physics may be included..
It is the hope of the editors that this form of publication may
tend to promote mathematical study and research over a wider
field than that which the former volume has occupied.
December, 1905.
AUTHOR'S PREFACE.
IT is customary to divide the Infinitesimal Calculus, or Calcu-
lus of Continuous Functions, into three parts, under the heads
Differential Calculus, Integral Calculus, and Differential Equa-
tions. The first corresponds, in the language of Newton, to the
"direct method of tangents, " the other two to the " inverse method
of tangents"; while the questions which come under this last
head he further -divided into those involving the two fluxions and
one fluent, and those involving the fluxions and both fluents.
On account of the inverse character which thus attaches to
the present subject, the differential equation must necessarily
at first be viewed in connection with a "primitive," from which
it might have been obtained by the direct process, and the solu-
tion consists in the discovery, by tentative and more or less arti-
ficial methods, of such a primitive, when it exists; that is to
say, when it is expressible in the elementary functions which
constitute the original field with which the Differential Calculus
has to do.
It is the nature of an inverse process to enlarge the field of
its operations, and the present is no exception; but the adequate
handling of the new functions with which the field is thus enlarged
requires the introduction of the complex variable, and is beyond
the scope of a work of this size.
But the theory of the nature and meaning of a differential
equation between real variables possesses a great deal of interest.
To this part of the subject I have endeavored to give a full treat-
ment by means of extensive use of graphic representations in
AUTHOR S PREFACE. V
rectangular coordinates. If we ask what it is that satisfies an
ordinary differential equation of the first order, the answer must
be certain sets of simultaneous values of x, y, and p. The geo-
metrical representation of such a set is a point in a plane asso-
ciated with a direction, so to speak, an infinitesimal stroke, and
the "solution" consists of the grouping together of these strokes
into curves of which they form elements. The treatment of
singular solutions, following Cayley, and a comparison with the
methods previously in use, illustrates the great utility of this point
of view.
Again, in partial differential equations, the set of simultaneous
values of x, y, z, p, and q which satisfies an equation of the first
order is represented by a point in space associated with the direc-
tion of a plane, so to speak by a flake, and the mode in which
these coalesce so as to form linear surface elements and con-
tinuous surfaces throws light upon the nature of general and
complete integrals and of the characteristics.
The expeditious symbolic methods of integration applicable
to some forms of linear equations, and the subject of development
of integrals in convergent series, have been treated as fully as space
would allow.
Examples selected to illustrate the principles developed in
each section will be found at its close, and a full index of subjects
at the end of the volume.
W. W. J.
ANNAPOLIS, MD., December, 1905.
CONTENTS.
ART. i. EQUATIONS OF FIRST ORDER AND DEGREE Page i
2. GEOMETRICAL REPRESENTATION 3
3. PRIMITIVE OF A DIFFERENTIAL EQUATION 5
4. EXACT DIFFERENTLY EQUATIONS 6
5. HOMOGENEOUS EQUATIONS 9
6. THE LINEAR EQUATION 10
7. FIRST ORDER AND SECOND DEGREE 12
8. SINGULAR SOLUTIONS 15
9. SINGULAR SOLUTION FROM THE COMPLETE INTEGRAL . . . 18
10. SOLUTION BY DIFFERENTIATION 20
11. GEOMETRIC APPLICATIONS; TRAJECTORIES 23
12. SIMULTANEOUS DIFFERENTIAL EQUATIONS 25
13. EQUATIONS OF THE SECOND ORDER 28
14. THE Two FIRST INTEGRALS 31
15. LINEAR EQUATIONS 34
1 6. LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS 36
17. HOMOGENEOUS LINEAR EQUATIONS 40
18. SOLUTIONS IN INFINITE SERIES 42
19. SYSTEMS OF DIFFERENTIAL EQUATIONS 47
20. FIRST ORDER AND DEGREE WITH THREE VARIABLES ... .50
21. PARTIAL DIFFERENTIAL EQUATIONS OF FIRST ORDER AND DEGREE 53
22. COMPLETE AND GENERAL INTEGRALS 57
23. COMPLETE INTEGRAL FOR SPECIAL FORMS 60
24. PARTIAL EQUATIONS OF SECOND ORDER 63
25. LINEAR PARTIAL DIFFERENTIAL EQUATIONS 66
INDEX 72
DIFFERENTIAL EQUATIONS.
ART. 1. EQUATIONS -OF FIRST ORDER AND DEGREE.
In the Integral Calculus, supposing y to denote an unknown-
function of the independent variable x, the derivative of y with
respect to x is given in the form of a function of x, and it is
required to find the value of y as a function of x. In other
words, given an equation of the form
g=/O), or dy = flx)dx t (I)
of which the general solution is written in the form
y = //(*X*, (2)
it is the object of the Integral Calculus to reduce the expres-
sion in the second member of equation (2) to the form of a
known function of x. When such reduction is not possible,
the equation serves to define a new function of x.
In the extension of the processes of integration of which
the following pages give a sketch the given expression for the
derivative may involve not only x, but the unknown function
y ; or, to write the equation in a form analogous to equation
(i), it may be
Mdx + Ndy = o, (3)
in which J/and TV are functions of x and y. This equation is
in fact the general form of the differential equation of the first
order and degree ; either variable being taken as the independ-
ent variable, it gives the first derivative of the other variable
2 DIFFERENTIAL EQUATIONS.
in terms of x and y. So also the solution is not necessarily an
expression of either variable as a function of the other, but is
generally a relation between x and y which makes either an
implicit function of the other.
When we recognize the left member of equation (3) as an
" exact differential," that is, the differential of some function of
x and y, the solution is obvious. For example, given the equa-
tion
xdy-\-ydx = o, (4)
the solution xy = C, (5)
where C is an arbitrary constant, is obtained by " direct inte-
gration." When a particular value is attributed to (7, the result
is a " particular integral ; " thus^ = x~ l is a particular integral
of equation (4), while the more general relation expressed by
equation (5) is known as the " complete integral."
In general, the given expression Mdx -j- Ndy is not an ex-
act differential, and it is necessary to find some less direct
method of solution.
The most obvious method of solving a differential equation
of the first order and degree is, when practicable, to " separate
the variables," so that the coefficient of dx shall contain x
only, and that of dy, y only. For example, given the equation
( i y)dx -f- (i -{- x}dy = o, (6)
the variables are separated by dividing by (\-\-x](\ y).
dx dy
Thus i h =0,
I + x ' I y
Each term is now directly integrable, and hence
log (I +*) log (i y) = c.
The solution here presents itself in a transcendental form,
but it is readily reduced to an algebraic form. For, taking the
exponential of each member, we find
t ^ = c, whence i -J- x = C(i y) t (7)
where C is put for the constant ^.
GEOMETRICAL REPRESENTATION. 3
To verify the result in this form we notice that differentia-
tion gives dx = Cdy, and substituting in equation (6) we find
- C(i -y) + i +* = o,
which is true by equation (7).
Prob. i. Solve the equation dy -\- y tan x dx = o.
Ans. y=C cos x.
Prob. 2. Solve ^ + '/ = a\ Ans.
.
dx by a
-, , ^ / + 1
Prob. 3. Solve -j- = =VT Ans. y =
ex
Prob. 4. Helrnholtz's equation for the strength of an electric
current C at the time / is
C = - -
~ R R dt*
where , R, and L are given constants. Find the value of C, de-
termining the constant of integration by the condition that its initial
value shall be zero.
ART. 2. GEOMETRICAL REPRESENTATION.
The meani'ng of a differential equation may be graphically
illustrated by supposing simultaneous values of x and y to be
the rectangular coordinates of a variable point. It is conven-
ient to put/ for the value of the ratio dy-.dx. Then P being
the moving point (x, y) and denoting the inclination of its
path to the axis of x, we have
dy
p = -j- = tan 0.
dx
The given differential equation of the first order is a relation
between/, x, andy, and, being of the first degree with respect
to/, determines in general a single value of/ for any assumed
values of x and y. Suppose in the first place that, in addition
to the differential equation, we were given one pair of simul-
taneous' values of x and y, that is, one position of the point P.
Now let P start from this fixed initial point and begin to move
in either direction along the straight line whose inclination
DIFFERENTIAL EQUATIONS.
is determined by the value of p corresponding to the initial
values of x and y. We thus have a moving point satisfying
the given differential equation. As the point P moves the
values of x and y vary, and we must suppose the direction of
its motion to vary in such a way that the simultaneous values
of x, y, and/ continue to satisfy the differential equation. In
that case, the path of the moving point is said to satisfy the
differential equation. The point P may return to its initial
position, thus describing a closed curve, or it may pass to infin-
ity in each direction from the initial point describing an infinite
branch of a curve.* The ordinary cartesian equation of the
path of P is a particular integral of the differential equation.
If no pair of associated values of x and y be known, P may
be assumed to start from any initial point, so that there is an
unlimited number of curves representing particular integrals
of the equation. These form a "system of curves," and the
complete integral is the equation of the system in the usual
form of a relation between x, y, and an arbitrary " parameter."
This parameter is of course the constant of integration. It is
constant for any one curve of the system, and different values
of it determine different members of the system of curves, or
different particular integrals.
As an illustration, let us take equation (4) of Art. I, which
may be written
d y___y_
dx x'
Denoting by the inclination to
the axis of x of the line joining P
with the origin, the equation is
equivalent to tan = tan 6, and.
therefore expresses that P moves
in a direction inclined equally with
OP to either axis, but on the other
* When the form of the functions M and N is unrestricted, there is no
reason why either of these cases should exist, but they commonly occur among
such differential equations as admit of solution.
PRIMITIVE OF A DIFFERENTIAL EQUATION. 5
side. Starting from any position in the plane, the point P
thus moving must describe a branch of an hyperbola having
the two axes as its asymptotes ; accordingly, the complete
integral xy = C is the equation of the system consisting of
these hyperbolas.
ff^\ Lc-
Prob. 5. Write the differential equation which requires P to move
in a direction always perpendicular to OP, and thence derive the
equation of the system of curves described.
Ans ^-_*.*' + v'-C
';<** ,'* + *
* Prob. 6. What is the system described when is the comple-
* . ment of ? 6 - Z^ Ans. x 9 -/ = C.
oJ^jv^TProb. 7. If "^OsJ^, show geometrically that the system described
5 ,^^consists of circles, and find the differential equation.
^ 6 4(|F Ans. 2xydx = (x* y*)dy.
ART. 3. PRIMITIVE OF A DIFFERENTIAL EQUATION.
Let us now suppose an ordinary relation between x and y,
which may be represented by a curve, to be given. By differ-
entiation we may obtain an equation of which the given equa-
, tion is of course a solution or particular integral. But by
combining this with the given equation any number of differ-
ential equations of which the given equation is a solution may
be found. For example, from
y = m(x a) (l)
we obtain directly
2ydy = mdx, (2)
of which equation (i) is an integral; again, dividing (2) by (i)
we have
***
y x-a'
and of this equation also (i) is an integral.
If in equation (i) m be regarded as an arbitrary parameter,
it is the equation of a system of parabolas having a common
axis and vertex. The differential equation (3), which does not
contain m, is satisfied by every member of this system of curves,
DIFFERENTIAL EQUATIONS.
Hence equation (i) thus regarded is the complete integral of
equation (3), as will be found by solving the equation in which
the variables are already separated.
Now equation (3) is obviously the only differential equation
independent of m which could be derived from (i) and (2), since
it is the result of eliminating ;;/. It is therefore the " differ-
ential equation of the system ; " and in this point of view the
integral equation (i) is said to be its "primitive."
Again, if in equation (i) a be regarded as the arbitrary con-
stant, it is the equation of a system of equal parabolas having
a common axis. Now equation (2) which does not contain a
is satisfied by every member of this system of curves; hence it
is the differential equation of the system, and its primitive is
equation (i) with a regarded as the arbitrary constant.
Thus, a primitive is an equation containing as well as x and"
y an arbitrary constant, which we may denote by C, and the
corresponding differential equation is a relation between x, y,
and/, which is found by differentiation, and elimination of C if
necessary. This is therefore also a method of verifying the com-
plete integral of a given differential equation. For example, in
verifying the complete integral (7) in Art. i we obtain by differ-
entiation i = Cp. If we use this to eliminate C from equa-
tion (7) the result is equation (6); whereas the process before
employed was equivalent to eliminating / from equation (6),
thereby reproducing equation (7).
Prob. 8. Write the equation of the system of circles in Prob. 7,
Art. 2, and derive the differential equation from it as a primitive.
Prob. 9. Write the equation of the system of circles passing
through the points (o, b} and (o, b], and derive from it the differ-
ential equation of the system.
ART. 4. EXACT DIFFERENTIAL EQUATIONS.
In Art. I the case is mentioned in which Mdx -(- Ndy is an
" exact differential," that is, the differential of a function of x
andj. Let u denote this function; then
(i)
EXACT DIFFERENTIAL EQUATIONS.
and in the notation of partial derivatives
M=* t N=^.
'dx -dy
3*
Then, since by a theorem of partial derivatives r-
-dy -dx
This condition rnjj^t therefore be fulfilled by M and N in
order that equation (i) may be possible. When it is fulfilled
Mdx -\- Ndy o is said to be an " exact differential equation,"
and its complete integral is
u = C. (3)
For example, given the equation
x(x -\- 2y]dx -f- (x* y*}dy = o,
= 2x, and - = 2*; the
condition (2) is fulfilled, and the equation is exact. To find the
function ?/, we may integrate Mdx, treating y as a constant; thus,
&+Sy = Y,
in which the constant of integration Y may be a function of y.
The result of differentiating this is
;*
- x*dx -\- 2xy dx -f- x*dy = dY,
which should be identical with the given equation ; therefore,
dY = y*dy, whence Y = $y a -}- C, and substituting, the com-
plete integral may be written
The result is more readily obtained if we notice that all |
terms containing x and dx only, or y and dy only, are exact w
differentials; hence it is only necessary to examine the termsj
containing both x and y. In the present case, these are
2xy dx -\- x*dy, which obviously form the differential of x*y ;
whence, integrating and multiplying by 3, we obtain the result
above.
The complete integral of any equation, in whatever way it
'8 DIFFERENTIAL EQUATIONS.
was found, can be put in the form u = C, by solving for C.
Hence an exact differential equation du = o can be obtained,
which must be equivalent to the given equation
Mdx -f- Ndy o, (4)
here supposed not to be exact. The exact equation du = o
must therefore be of the form
}ji(Mdx + Ndy) = o, (5)
where /* is a factor containing at least one of the variables x
and/. Such a factor is called an " integrating factor" of the
given equation. For example,- the result of differentiating
equation (7), Art. I, when put in the form u = C, is
(i -y)dx + (\ 4- x)dy _
(i - yf
so that (i y)~* is an integrating factor of equation (6). It
is to be noticed that the factor by which we separated the
variables, namely, (i y)~\i x)~ l , is also an integrating
factor.
It follows that if an integrating factor can be discovered,
the given differential equation can at once be solved.* Such
a factor is sometimes suggested by the form of the equation.
Thus, given (y x)dy -\-ydx = o,
the terms ydx xdy, which contain both x and y, are not ex-
act, but become so when divided by either x 1 or y*\ and be-
cause the remaining term contains^ only, j/~ 2 is an integrating
factor of the whole expression. The resulting integral is
ig.y + - = c.
Prob. 10. Show from the integral equation in Prob. 9, Art. 3, that
x~* is an integrating factor of the differential equation.
Prob. n. Solve the equation x(x' 1 + 37*)^ -\-y(y* -f 3* a )^ = o.
Ans. x*
* Since f*M and uN in the exact equation (5) must satisfy the condition (2),
we have a partial differential equation for //; but as a general method of finding
ft this simply comes back to the solution of the original equation.
HOMOGENEOUS EQUATION.
, xdyydx
Prob. 12. Solve the equation ydy -\-xdx -j -- . , . = o.
*
(Ans.
2 X
Prob. 13. If w = c is a form of the complete integral and p the
corresponding integrating factor, show that l*f(u) is the general
expression for the integrating factors.
Prob. 14. Show that the expression x a }P(mydx + nxdy) has the
integrating factor jp**- 1 -*^**- 1 .-^; an d by means of such a factor
solve the equation y(y* + 2x*)dx -(- jc(^ 4 2y)</v = o.
Ans. 2x*y y* = ex*. V
Prob. 15. Solve (x 1 -{- y'}dx 2xydy o. Ans. x* y* = ex.
ART. 5. HOMOGENEOUS EQUATION.
The differential equation Mdx -f- Ndy = o is said to be
homogeneous when M and N are homogeneous functions of
v and y of the same degree ; or, what is the same thing, when
dy y
Hr is expressible as a function of . If in such an equation
the variables are changed from x and y to x and v, where
y
v = whence y = xv and dy = xdv -f- vdx,
the variables x and # will be separable. For example, the
equation
(x 2y)dx -f- ydy = o
is homogeneous; making the substitutions indicated and
dividing by x,
(i 2v)dx -f- v(xdv -|- ^^f) = O,
dx vdv
whence -- \- -, -. = o.
x ' (y i/
Integrating, log x + log (v - i) = C\
and restoring^,
x
log ( _y *) -- - = C.
' y x
The equation Mdx -f- -/VWy = o can always be solved when
10 DIFFERENTIAL EQUATIONS.
M and N are functions of the first degree, that is, when it is
of the form
(ax + fy + c]dx + (a'x -f b'y + c'}dy = o.
For, assuming x = x' -f- //, y = y' -f- ^> it becomes
(*'+ b'y'+ ah + bk+ c)dx'+(a'x'+ b'y'+ a'h +b'k+c')dy' =o,
which, by properly determining h and k, becomes
(ax' + /X*' + (a'x' -f- '/>//,
a homogeneous equation.
This method fails when a : b == a' : b', that is, when the
aquation takes the form
(ax -\-by-\- c)dx -\- \m(ax -f- by) -\- c'~\dy = o ;
but in this case if we put z = ax -f- by, and eliminate y, it will
be found that the variables x and 2 can be separated.
Prob. 16. Show that a homogeneous differential equation repre-
sents a system of similar and similarly situated curves, the origin
being the center of similitude, and hence that the complete integral
may be written in a form homogeneous in x, y, and c.
Prob. 17. Solve xdy ydx y(x* -f y*)dx = o.
Ans. x* = * 2cy.
Prob. 1 8. Solve (3^ 7* + i)dx + (jy 3* + $)dy = o.
Ans. (y x + r)'(7 + x i) 5 = r.
Prob. 19. Solve (x*.-{-y*)dx zxydy = o. Ans. .r a y 9 = ex. *
Prob. 20. Solve (i + xy)ydx + (i tfy).*^ = o by introducing
_
the new variable z = xy. Ans. x = Cye*y.
Prob. 21. Solve ~ ax -\-by-\-c. Ans. ata+^-i- -!-& = <?**.
ART. 6. THE LINEAR EQUATION.
A differential equation is said to be "linear" when (one of
the variables, say x, being regarded as independent,) it is of
the first degree with respect to y, and its derivatives. The
linear equation of the first order may therefore be written in
the form
THE LINEAR EQUATION. 11
where P and Q are functions of x only. Since the second
member is a function of x, an integrating factor of the first
member will be an integrating factor of the equation provided
it contains x only. To find such a factor, we solve the equation
dy
which is done by separating the variables ; thus, = Pdx \
whence log y = c I Pdx or
y = Ce ~' M "- (3)
Putting this equation in the form u = c, the corresponding
exact equation is
e SPd \dy + Pydx} = o,
whence tr ' ' is the integrating factor required. Using this
factor, the general solution of equation (i) is
Qdx + C. (4)
In a given example the integrating factor should of course
be simplified in form if possible. Thus
(i -(- x*}dy = (m -\- xy)dx
is a linear equation for/; reduced to the form (i), it is
dy x m
dx i -\-a? i -f- x"
from which
The integrating factor is, therefore,
e fpdx =
whence the exact equation is
dy xy dx mdx
r
J
I
12 DIFFERENTIAL EQUATIONS.
Integrating, there is found
y w* ~
v(i+O =: i/(i + *') + C)
or
y = mx -\- C \/(l + X*).
ri r An equation is sometimes obviously linear, not for^, but
for some function of y.' For example, the equation
dy .
-j \- tan y = x sec y
when multiplied by cos y takes a form linear for sin y ; the
integrating factor is e*, and the complete integral
sin y = x i -|- ce~ x .
dy
In particular, the equation ^- -j- Py = Qy* t which is known as
" the extension of the linear equation," is readily put in a form
linear for y l ~ M .
d y , .
rob. 22. Solve ar-r- + (i 2#)y = ^ a - Ans. y = x*(\ + ce*).
dy
Prob. 23. Solve cos x - -{-y i -(- sin ^r = o.
Ans. Xsec * + tan *) = * + ^
A
Prob. 24. Solve cos x + y sin ^ = i.
rf^
Ans. y = sin * + c cos ^c.
Prob. 25. Solve -^ = *V xy. Ans. 4 = ** + i +^.
Prob. 26. Solve ^ = - ,' , 3 . Ans. - = 2 -/+
dx yy + 0:7 a:
.
y
ART. 7. FIRST ORDER AND SECOND DEGREE.
If the given differential equation of the first order, or re-
lation between x, y, and /, is a quadratic for p, the first step
in the solution is usually to solve for /. The resulting value
of / will generally involve an irrational function of x and y\
>so that an equation expressing such a value of /, like some of
those solved in the preceding pages, is not properly to be re-
FIRST ORDER AND SECOND DEGREE. 13
garded as an equation of the first degree. In the exceptional
case when the expression whose root is to be extracted is a
perfect square, the equation is decomposable into two equa-
tions properly of the first degree. For example, the equation
y x
when solved for / gives 2p = -, or 2p = ; it may therefore
be written in the form
(2px - y)(2py - x) = Q,
and is satisfied by putting either
dy _ y dy _ x
~T~ - OF ~7~ - -
ax 2x ax 2y
The integrals of these equations are
y* = ex and 2y* x 1 = C,
and these form two entirely distinct solutions of the given
equation.
As an illustration of the general case, let us take the equation
Separating the variables and integrating,
Vic Vy = Vc, (2)
and this equation rationalized become^
(x- y y-2c(x+y)+c*=o. (3)
There is thus a single complete integral containing one arbi-
trary constant and representing a single system of curves;
namely, in this case, a system of parabolas touching each axis
at the same distance c from the origin. The separate equa-
tions given in the form (2) are merely branches of the same
parabola.
Recurring now to the geometrical interpretation of a differ-
ential equation, as given in Art. 2, it was stated that an equa-
tion of the first degree determines, in general, for assumed
values of x and y, that is, at a selected point in the plane, a
single value of p. The equation was, of course, then supposed
14 DIFFERENTIAL EQUATIONS.
rational in x and y* The only exceptions occur at points for
which the value of p takes the indeterminate form ; that is,
the equation being Mdx -\- Ndy = o, at points (if any exist)
for which M = o and N = o. It follows that, except at such
points, no two curves of the system representing the complete
integral intersect, while through such points an unlimited num-
ber of the curves may pass, forming a "pencil of curves." f
On the other hand, in the case of an equation of the second
cdegree, there will in general be two values of / for any given
point. Thus from equation (i) above we find for the point
(4, i), p = ; there are therefore two directions in which a
point starting from the position (4, i) may move while satis-
fying the differential equation. The curves thus described
represent two of the particular integrals. If the same values
of x and y be substituted in the complete integral (3), the re-
sult is a quadratic for c, giving c = 9 and c = i, and these
determine the two particular integral curves, Vx -j- Vy = 3,
and Vx Vy = i.
In like manner the general equation of the second degree,
which may be written in the form
where L, M, and A 7 " are one-valued functions of x and y, repre-
sents a system of curves of which two intersect in any given
point for which p is found to have two real values. For these
points, therefore, the complete integral should generally give
two real values of c. Accordingly we may assume, as the
standard form of its equation,
PC*
* In fact / was supposed to be a one-valued function of x and y; thus,
/ = sin" 1 * would not in this connection be regarded as an equation of the first
degree.
f In Prob. 6, Art. 3, the integral equation represents the pencil of circles pass-
ing through the points (o, b) and (o, b)\ accordingly/ in the differential equa-
tion is indeterminate at these points. In some cases, however, such a point is
merely a node of one particular integral. Thus in the illustration given in Art. 2,
/ is indeterminate at the origin, and this point is a node of the only particular
integral, xy = o, which passes through it.
SINGULAR SOLUTIONS. 15
where P, Q, and R are also one-valued functions of x and y.
If there are points which make / imaginary in the differential
equation, they will also make c imaginary in the integral.
Prob. 27. Solve the equation /" +y = i and reduce the inte- ./
gral to the standard form.
Ans. (y -f- cos x)^ 2C sin x -f- y cos x = o.
Prob. 28. Solve yp* + zxp y = o, and show that the intersect-
ing curves at any given point cut at right angles. ,
Prob. 29. Solve (x 9 + i)/ = i. Ans. <rV r - text? = i.
ART. 8. SINGULAR SOLUTIONS.
A differential equation not of the first degree sometimes
admits of what is called a " singular solution ; " that is to say, a
solution which is not included in the complete integral. For
suppose that the system of curves representing the complete
integral has an envelope. Every point A of this envelope
is a point of contact with a particular curve of the complete in-
tegral system ; therefore a point moving in the envelope when
passing through A has the same values of x, y, and /as if it
were moving through A in the particular integral curve. Hence
such a point satisfies the differential equation and will continue
to satisfy it as long as it moves in the envelope. The equation
of the envelope is therefore a solution of the equation.
As an illustration, let us take the system of straight lines
whose equation is
i a
y = cx + -, (I)
C
where c is the arbitrary parameter. The differential equation
derived from this primitive is
y=px + - p * (2)
of which therefore (i) is the complete integral.
Now the lines represented by equation (i), for different
values of c, are the tangents to the parabola
/ = V**- (3)
1G
DIFFERENTIAL EQUATIONS.
A point moving in this parabola has the same value of/ as if it
were moving in one of the tan-
gents, and accordingly equation
(3) will be found to satisfy the
differential equation (2).
It will be noticed that for
any point on the convex side of
the parabola there are two real
values of p ; for a point on the
other side the values of / are
imaginary, and for a point on
the curve they are equal. Thus
its equation (3) expresses the
relation between x and y which must exist in order that (2)
regarded as a quadratic for p may have'equal roots, as will be
seen on solving that equation.
In general, writing the differential equation in the form
the condition of equal roots is
o. (5)
The first member of this equation, which is the " discrimi-
nant " of equation (4), frequently admits of separation into
factors rational in x and_y. Hence, if there be a singular solu-
tion, its equation will be found by putting the discriminant of
the differential equation, or one of its factors, equal to zero.
It does not follow that every such equation represents a solu-
tion of the differential equation. It can only be inferred that
it is a locus of points for which the two values of / become
equal. Now suppose that two distinct particular integral
curves touch each other. At the point "of contact, the two
values of/, usually distinct, become equal. The locus of such
points is called a "tac-locus." Its equation plainly satisfies the
discriminant, but does not satisfy the differential equation. An
illustration is afforded by the equation
SINGULAR SOLUTIONS. 17
of which the complete integral isy 1 -f- (x c)* = a*, and the
discriminant, see equation (5), is j"(y tf 2 ) = o.
This is satisfied by y = a, y a, and y = o, the first two
of which satisfy the differential equation, while U = o does not.
The complete integral represents in this case all circles of radius
a with center on the axis of x. Two of these circles touch at
every point of the axis of x, which is thus a tac-locus, while
y = a and y = a constitute the envelope.
The discriminant is the quantity which appears under the
radical sign when the general equation (4) is solved for/, and
therefore it changes sign as we cross the envelope. But the
values of / remain .real as we cross the tac-locus, so that the
discriminant cannot change sign. Accordingly the factor which
indicates a tac-locus appears with an even exponent (as y 1 in
the example above), whereas the factor indicating the singular
solution appears as a simple factor, or with an odd exponent.
A simple factor of the discriminant, or one with an odd ex-
ponent, gives in fact always the boundary between a region of
the plane in which / is real and one in which p is imaginary ;
nevertheless it may not give a singular solution. For the two
arcs of particular integral curves which intersect in a point on
the real side of the boundary may, as the point is brought up
to the boundary, become tangent to each other, but not to the
boundary curve. In that case, since they cannot cross the
boundary, they become branches of the same particular inte-
gral forming a cusp. A boundary curve of this character is
called a "cusp-locus" ; the value of / for a point moving in it
is of course different from the equal values of/ at the cusp, and
therefore its equation does not satisfy the differential equation.*
Prob. 30. To what curve is the line y = mx -\- a |/(i m*)
always tangent ? Ans. y 1 x* = a*.
Prob. 31. Show that the discriminant of a decomposable differ-
* Since there is no reason why the values of/ referred to should be identical,
we conclude that the equation Z/ 9 -f Mp + N = o has not in general a singular
solution, its discriminant representing a.cusp-locus except when a certain con-
dition is fulfilled.
18 DIFFERENTIAL EQUATIONS.
cntial equation cannot be negative. Interpret the result of equating
it to zero in the illustrative example at the beginning of Art. 7.
Prob. 32. Show that the singular solutions of a homogeneous dif-
ferential equation represent straight lines passing through the origin.
Prob. 33. Solve the equation xp* zyp -\- ax o.
Ans. x* 2cy -f- af = o ; singular solution y = ax*.
Prob. 34. Show that the equation /" -f- zxp y = o has no sin-
gular solution, but has a cusp-locus, and that the tangent at every
cusp passes through the origin.
ART. 9. SINGULAR SOLUTION FROM COMPLETE INTEGRAL.
When the complete integral of a differential equation of
the second degree has been found in the standard form
PS+Qc + R = o (i)
(see the end of Art. 7), the substitution of special values of x
and y in the functions P, Q, and R gives a quadratic for c whose
roots determine the two particular curves of the system which
pass through a given point. If there is a singular solution,
that is, if the system of curves has an envelope, the two
curves which usually intersect become identical when the given
point is moved up to the envelope. Every point on the en-
velope therefore satisfies the condition of equal roots for equa-
tion (i), which is
Q- 4/^ = 0; (2)
and, reasoning exactly as in Art. 8, we infer that the equation
of the singular solution will be found by equating to zero the
discriminant of the equation in c or one of its factors. Thus
the discriminant of equation (i), Art. 8, or "^-discriminant," is
the same as the "/-discriminant," namely, y ^ax, which
equated to zero is the equation of the envelope of the system of
straight lines.
But, as in the case of the /-discriminant, it must not be
inferred that every factor gives a singular solution. For ex-
ample, suppose a squared factor appears in the ^-discriminant.
The locus on which this factor vanishes is not a curve in cross-
ing which c and/ become imaginary. At any point of it there
SINGULAR SOLUTION FROM COMPLETE INTEGRAL. 19
\vill be two distinct values of p, corresponding to arcs of par-
ticular integral curves passing through that point ; but, since
there is but one value of c, these arcs belong to the same par-
ticular integral, hence the point is a double point or node.
The locus is therefore called a " node-locus." The factor repre-
senting it does not appear in the /-discriminant, just as that
representing a tac-locus does not appear in the <r-discriminant.
Again, at any point of a cusp-locus, as shown at the end of
Art. 8, the two branches of particular integrals become arcs ot
the same particular integral ; the values of c become equal, so
that a cusp-locus also makes the ^-discriminant vanish.
The conclusions established above obviously apply also to
equations of a degree higher than the second. In the case of
the <r-equation the general method of obtaining the condition
for equal roots, which is to eliminate c between the original and
the derived equation, is the same as the process of finding the
envelope or "locus of the ultimate intersections" of a, system
of curves in which c is the arbitrary parameter.
Now suppose the system of curves to have for all values of
* a double point, it is obvious that among the intersections
of two neighboring curves there are two in the neighborhood
of the nodes, and that ultimately they coincide with the node,
which accounts for the node-locus appearing twice in the dis-
criminant or locus of ultimate intersections. In like manner,
* It is noticed in the second foot-note to Art. 7 that for an equation of the
first degree p takes the indeterminate form, not only at a point through which all
curves of the system pass (where the value of c would also be found indeter-
minate), but at a node of a particular integral. So also when the equation is of
the th degree, if there is a node for a particular value of c, the n values of e at
the point (which is not on a node-locus where two values of c are equal) deter-
mine n -f- i arcs of particular integrals passing through the point ; and there-
fore there are n -\- i distinct values of / at the point, which can only happen
when p takes the indeterminate form, that is to say, when all the coefficients of
the/-eguation (which is of the wth decree} vanish. See Cayley on Singular So-
lutions' in the Messenger of Mathematics. New Series, Vol. II, p. 10 (Collected
Mathematical Works, Vol. VIII, p. 529). The present t-neory of Singular Solu-
tions was established by Cayley in this paper and its continuation, Vol. VI, p. 23.
See also a paper by Dr. Glaisher, Vol. XII, p. I.
20 DIFFERENTIAL EQUATIONS.
if there is a cusp for all values of c, there are three intersections
of neighboring curves (all of which may be real) which ulti-
mately coincide with the cusp ; therefore a cusp-locus will
appear as a cubed factor in the discriminant.*
Prob. 35. Show that the singular solutions of a homogeneous
equation must be straight lines passing through the origin.
Prob. 36. Solve 3/V 2xyp + 47* x* = o, and show that there
is a singular solution and a tac-focus.
Prob. 37. Solve yp*-\- 2Xp y = o t and show that there is an
imaginary singular solution. Ans. y* = zcx + c 1 .
Prob. 38. Show that the equation (i x^p' = i y* represents
a system of conies touching the four sides of a square.
Prob. 39. Solve yp* ^xp -\-y = o ; examine and interpret both
discriminants. Ans. c 1 + 2cx($y' 8.x 2 ) 3* a y +/* = o.
ART. 10. SOLUTION BY DIFFERENTIATION.
The result of differentiating a given differential equation o
the first order is an equation of the second order, that is, it
d*y
contains the derivative -r-^ ; but, if it does not contain y ex-
plicitly, it may be regarded as an equation of the first order for
the variables x and/. If the integral of such an equation can
be obtained it will be a relation between x, p, and a constant
of integration c, by means of which / can be eliminated from
the original equation, thus giving the relation between x, y>
and c which constitutes the complete integral. For example,
the equation
(i)
* The discriminant of PC* -\- Qc + R = o represents in general an envelope,
no further condition requiring to be fulfilled as in the case of the discriminant
of Z/ 2 -f- Mp -f- -A/ = o. Compare the foot-note to Art. 8. Therefore where
there is an integral of this form there is generally a singular solution, although
Z/ 9 -f- Mp -}- .A 7 = o has not in general a singular solution. We conclude, there-
fore, that this equation (in which Z, M, and N are one-valued functions of x
and y) has not in general an integral of the above form in which P, Q, and R
are one-valued functions of x and y. Cayley, Messenger of Mathematics, New
Series, Vol. VI, p. 23.
SOLUTION BY DIFFERENTIATION. 21
when solved for^, becomes
y = x + ^p\ (2)
whence by differentiation
The variables can be separated in this equation, and its inte-
gral is
Substituting in equation (2), we find
which is the complete integral of equation (i).
This method sometimes succeeds with equations of a higher
degree when the solution with respect to p is impossible or
leads to a form which cannot be integrated. A differential
equation between p and one of the two variables will be ob-
tained by direct integration when only one of the variables is
explicitly present in the equation, and also when the equation
is of the first degree with respect to x and y. In the latter
case after dividing by the coefficient of y, the result of differ-
entiation will be a linear equation for x as a function of p, so
that an expression for x in terms of p can be found, and then
by substitution in the given equation an expression for y in
terms of p. Hence, in this case, any number of simultaneous
values of x and y can be found, although the elimination of p
may be impracticable.
In particular, a homogeneous equation which cannot be
solved for p may be soluble for the ratio y : x, so as to assume
the form y = x(f>(p). The result of differentiation is
in which the variables x and p can be separated.
Another special case is of the form
y = P* +/<, (0
22 DIFFERENTIAL EQUATIONS.
which is known as Clairaut's equation. The result of differ-
entiation is
which implies either
=o, or
The elimination of p from equation (i) by means of the
first of these equations * gives a solution containing no arbi-
trary constant, that is, a singular solution. The second is a
differential equation for/; its integral is p = c, which in
equation (i) gives the complete integral
y = c X +f(c\ (2}
This complete integral represents a system of straight lines,
the singular solution representing the curve to which they are
all tangent. An example has already been given in Art. 8.
A differential equation is sometimes reducible to Clairaut's
form by means of a more or less obvious transformation of the
variables. It may be noticed in particular that an equation of
the form
y = nx p _}_ (j)( x , p)
is sometimes so reducible by transformation to the independent
variable z, where x = z n ; and an equation of the form
by transformation to the new dependent variable v y n . A
double transformation of the form indicated may succeed
when the last term is a function of both x and y as well as of/.
Prob. 40. Solve the equation $y = 2/ 3 + 3/ 2 ; find a singular
solution and a cusp-locus. Ans. (x -j- y -\- c i) a == ( xJ t~ f Y'
a
Prob. 41. Solve zy = xp -\ , and find a cusp-locus.
Ans. aV \2acxy -f- Scy 3 i2x*y* -\- i6ax* == o.
* The equation is in fact the same that arises in the general method for the
condition of equal roots. See Art. 9.
GEOMETRIC APPLICATIONS ; TRAJECTORIES. 23
Prob. 42. Solve (x* (?}p* 2xyp + / a* = o.
Ans. The circle x* + / = a\ and its tangents.
Prob. 43. Solve y = xp + *y.
Ans. ^ -j- <r .ay = o, and i + $x*y = o.
Prob. 44. Solve / s $xyp + By 1 = o.
Ans. y = c(x c}* 2jy = 43? and y = o are singulat
solutions \y = o is also a particular integral.
Prob. 45. Solve x\y px] = yp\ Ans. / = ex* + c\
ART. 11. GEOMETRIC APPLICATIONS ; TRAJECTORIES.
Every property of a curve which involves the direction of
its tangents admits of statement in the form of a differential
equation. The solution of such an equation therefore deter-
mines the curve having the given property. Thus, let it be
required to determine the curve in which the angle between
the radius vector and the tangent is n times* the vectorial
angle. Using the expression for the trigonometric tangent of
that angle, the expression of the property in polar coordi-
nates is
rdB
= tan nO.
dr
Separating the variables and integrating, the complete
integral is
r" = C H sin nB.
The mode in which the constant of integration enters here
shows that the property in question is shared by all the mem-
bers of a system of similar curves.
The solution of a question of this nature will thus in gen-
eral be a system of curves, the complete integral of a differential
equation, but it may be a singular solution. Thus, if we ex-
press the property that the sum of the intercepts on the axes
made by the tangent to a curve is equal to the constant a, the
straight lines making such intercepts will themselves consti-
tute the complete integral system, and the curve required is
the singular solution, which, in accordance with Art. 8, is the
24 DIFFERENTIAL EQUATIONS.
envelope of these lines. The result in this case will be found
to be the parabola Vx ~\- \^y = Va.
An important application is the determination of the
"orthogonal trajectories" of a given system of curves, that is
to say, the curves which cut at right angles every\curve of the
given system. The differential equation of the trajectory is
readily derived from that of the given system ; for at every
point of the trajectory the value of p is the negative reciprocal
of its value in the given differential equation. We have there-
fore only to substitute /"' for p to obtain the differential
equation of the trajectory. For example, let it be required to
determine the orthogonal trajectories of the system of pa-
rabolas
having a common axis and vertex. The differential equation
of the system found by eliminating a is
2 xdy = y dx.
Putting -- in place of -7-, the differential equation oi
dy dx
the system of trajectories is
2.x dx -\- ydy = o,
Whence, integrating,
, The trajectories are therefore a system of similar ellipses
with axes coinciding with the coordinate axes.
Prob. 46. Show that when the differential equation of a system
is of the second degree, its discriminant and that of its trajectory
system will be identical ; but if it represents a singular solution in
one system, it will constitute a cusp locus of the other.
Prob. 47. Determine the curve whose subtangent is constant and
equal to a. Ans. ce*=y*.
Prob. 48. Show that the orthogonal trajectories of the curves
r n =c" sin## are the same system turned through the angle about
2n
the pole. Examine the cases n = i, n = 2, and n = .
Prob. 49. Show that the orthogonal trajectories of a system of
SIMULTANEOUS DIFFERENTIAL EQUATIONS. 25
circles passing through two given points is another system of circles
having a common radical axis.
Prob. 50. Determine the curve such that the area inclosed
by any two ordinates, the curve and the axis of x, is equal to
the product of the arc and the constant line a. Interpret the
singular solution.
Ans. The catenary ^ = \a(e a e a ).
Prob. 51. Show that a system of confocal conies is self-orthog-
onal.
ART. 12. SIMULTANEOUS DIFFERENTIAL EQUATIONS.
A system of equations between n -j- I variables and their
differentials is a " determinate" differential system, because it
serves to determine the n ratios of the differentials ; so that,
taking any one of the variables as independent, the others vary
in a determinate manner, and may be regarded as functions of
the single independent variable. Denoting the variables by .#,
y, 2, etc., the system may be written in the symmetrical form
dx _ dy _ dz _
~X~~^Y~~Z~- "'
where X, Y, Z . . . may be any functions of the variables.
If any one of the several equations involving two differen-
tials contains only the two corresponding variables, it is an
ordinary differential equation ; and its integral, giving a re-
lation between these two variables, may enable us by elimina-
tion to obtain another equation containing two variables only,
and so on until n integral equations have been obtained.
Given, for example, the system
dx_d^ _dz^ ,.
x~ z " y'
The relation between dy and dz above contains the varia-
bles y and z only, and its integral is
7' s? = a. (2)
Employing this to eliminate z from the relation between
dx and dy it becomes
dx _ dy
~'
20 DIFFERENTIAL EQUATIONS.
of which the integral is
*) = *>*' (3)
The integral equations (2) and (3), involving two constants
of integration, constitute the complete solution. It is in like
manner obvious that the complete solution of a system of n
equations should contain n arbitrary constants.
Confining ourselves now to the case of three variables, an
extension of the geometrical interpretation given in Art. 2
presents itself. Let x, y, and z be rectangular coordinates of
P referred to three planes. Then, if P starts from any given
position A, the given system of equations, determining the
ratios dx : dy : dz, determines the direction in space in which P
moves. As P moves, the ratios of the differentials (as deter-
mined by the given equations) will vary, and if we suppose P
to move in such a way as to continue to satisfy the differential
equations, it will describe in general a curve of double curva-
ture which will represent a particular solution. The complete
solution is represented by the system of lines which may be
thus obtained by varying the position of the initial point A.
This system is a " doubly infinite " one ; for the two relations
between x, y, and z which define it analytically must contain
two arbitrary parameters, by properly determining which we
can make the line pass through any assumed initial point.*
Each of the relations between x, y and z, or integral equa-
tions, represents by itself a surface, the intersection of the two
surfaces being a particular line of the doubly infinite system.
An equation like (2) in the example above, which contains only
one of the constants of integration, is called an integral of the
differential system, in contradistinction to an " integral equa-
* It is assumed in the explanation that X, V, and Zare one-valued functions
of x, y, and 2. There is then but one direction in which P can move when
passing a given point, and the system is a non-intersecting system of lines. But
if this is not the case, as for example when one of the equations giving the ratio
of the differentials is of higher degree the lines may form an intersecting sys-
tem, and there would be a theory of singular solutions, into which we do not
here enter.
SIMULTANEOUS DIFFERENTIAL EQUATIONS. 27
tion " like (3), which contains both constants. An integral
represents a surface which contains a singly infinite system of
lines representing particular solutions selected from the doubly
infinite system. Thus equation (2} above gives a surface on
which lie all those lines for which a has a given value, while b
may have any value whatever ; in other words, a surface which
passes through an infinite number of the particular solution
lines.
The integral of the system which corresponds to the con-
stant b might be found by eliminating a between equations (2)
and (3). It might also be derived directly from equation (i) ;
thus we may write
dx^ _ dy _dz _ dy -J- dz _du
x z y y ' -f- z u'
in which a new variable u =. y -f- z is introduced. The rela-
tion between dx and du now contains but two variables, and
its integral,
y + z = bx, (4)
is the required integral of the system ; and this, together with
the integral (2\ presents the solution of equations (i) in its
standard form. The form of the two integrals shows that in
this case the doubly infinite system of lines consists of hyper-
bolas, namely, the sections of the system of hyperbolic cylinders
represented by (2) made by the system of planes represented
by (4).
A system of equations of which the members possess a cer-
tain symmetry may sometimes be solved in the following
manner. Since
dx _ dy _ dz _ \dx -f- pdy -f- vdz
'
if we take multipliers A, /v, v such that
we shall have \dx -f- pdy -f- y dz = o.
If the expression in the first member is an exact differential,
28 DIFFERENTIAL EQUATIONS.
direct integration gives an integral of the given system. For
example, let the given equations be
dx dy dz
mz ny nx Iz ly mx '
/, m and n form such a set of multipliers, and so also do x, y
and z. Hence we have
Idx -j- mdy -f- ndz = o,
and also xdx -\-ydy-\- zdz = o.
Each of these is ai\ exact equation, and their integrals
Ix -J- my -\- nz = a
a nd x* -J- y* -j- z* = b*
constitute the complete solution. The doubly infinite system
of lines consists in this case of circles which have a common
axis, namely, the line passing through the origin and whose
direction cosines are proportional to /, m, and n.
dx dy dz
Prob. 132. Solve the equations -5 5 . = ^ = , and
x y z 2xy 2xz
interpret the result geometrically. (Ans. y=az, x*-\-y* -^-z^bz.}
dx dy dz
^Prob. 53. Solve
x z
Prob. 54. Solve ,, ~ = , ^ = , ^- .
(b c}yz (c a)zx (a b)xy
Ans. x* +/ + 2 a = A, ax' -f b? + c# - B.
ART. 13. EQUATIONS OF THE SECOND ORDER.
A relation between two variables and the successive deriva-
tives of one of them with respect to the other as independent
variable is called a differential equation of the order indicated
by the highest derivative that occurs. For example,
is an equation of the second order, in which x is the independent
EQUATIONS OF THE SECOND ORDER. 29
variable. Denoting as heretofore the first derivative by/, this
equation may be written
/ , v\^P_\ j. i _ / \
and this, in connection with
which defines /, forms a pair of equations of the first order,
connecting the variables x, y, and /. Thus any equation of the
second order is equivalent to a pair of simultaneous equations
of the first order.
When, as in this example, the given equation "does not con-
tain^ explicitly, the first of the pair of equations involves only
the two variables x and/ ; and it is further to be noticed that,
when the derivatives occur only in the first degree, it is a linear
equation for/. Integrating equation (i) as such, we find
and then using this value of/ in equation (2), its integral is
y = c, - mx + c t log \x + y(i + *')], (4)
in which, as in every case of two simultaneous equations of the
first order, we have introduced two constants of integration.
An equation of the first order is readily obtained also
when the independent variable is not explicitly contained in
the equation. The general equation of rectilinear motion in
ffs
dynamics affords an illustration. This equation is = f(s),
where s denotes the distance measured from a fixed center of
d*v
force upon the line of motion. It may be written = f(s), in
U-f
connection with = v, which defines the velocity. Eliminat-
dt
ing dt from these equations, we have vdv = f(s)ds, whose
integral is $v* = I f(s)ds -\- c, the "equation of energy" for
the unit mass. The substitution of the value found for v in the
30 DIFFERENTIAL EQUATIONS.
second equation gives an equation from which t is found in
terms of s by direct integration.
The result of the first integration, such as equation (3) above,
is called a "first integral" of the given equation of the second
order ; it contains one constant of integration, and its complete
integral, which contains a second constant, is also the "com-
plete integral " of the given equation.
A differential equation of the second order is " exact " when,
all its terms being transposed to the first member, that member
is the derivative with respect to x of an expression of the first
order, that is, a function of x, y and p. It is obvious that the
terms containing the second derivative, in such an exact differ-
ential, arise solely from the differentiation of the terms con-
taining/ in the function of x, y and/. For example, let it be
required to ascertain whether
is an exact equation. The terms in question are (i x*}-f-,
ax
which can arise only -from the differentiation of (i x^p.
Now subtract from the given expression the complete deriva-
tive of (i x*}p, which is
, cTy dy
/T _ y- \ - 2 X *
\ / J a _/
ax ax
the remainder is x -\- y, which is an exact derivative, namely,
ax
that of xy. Hence the given expression is an exact differ-
ential, and
(i-^ + ^y^ (6)
is the first integral of the given equation. Solving thi-s linear
-equation for y, we find the complete integral
y = Cl x + c t tf(i *?). (7)
,.
\Prob. 55- Solve (i - * 2 )
Ans. y = (sin* 1 #)" + c, sin" 1 x + c v
THE TWO FIRST INTEGRALS. 31
* Prob. 56. Solve = . Ans. y = - + cjc\
dx x x
v Prob. 57. Solve -^ = cfx tfy.
Ans. c?x b*y A sin for -f- .5 cos for.
vprob. 58. Solve y + ' = i. Ans. / = ** + ^ + c v
ART. 14. THE Two FIRST INTEGRALS.
We have seen in the preceding article that the complete
integral of an equation of the second order is a relation be-
tween x, y and two constants c t and c 9 . Conversely, any rela-
tion between x, y and two arbitrary constants may be regarded
as a primitive, from which a differential equation free from both
arbitrary constants can be obtained. The process consists in
first. obtaining, as in Art. 3, a differential equation of the first
order independent of one of the constants, say c 9 , that is, a rela-
tion between x, y,p and <:, , and then in like manner eliminating
r, from the derivative of this equation. The result is the equa-
tion of the second order or relation between x, y, p and q (q
denoting the second derivative), of which the original equation
is the complete primitive, the equation of the first order being
the first integral in which c l is the constant of integration. It
is obvious that we can, in like manner, obtain from the primi-
tive a relation between x, y, p and c 9 , which will also be a first
integral of the differential equation. Thus, to a given form of
the primitive or complete integral there corresponds two first
integrals.
Geometrically the complete integral represents a doubly
infinite system of curves, obtained by varying the values of c t
and of independently. If we regard c t as fixed and c t as
arbitrary, we select ffom that system a certain singly infinite
system ; the first integral containing c, is the differential equa-
tion of this system, which, as explained in Art. 2, is a relation
between the coordinates of a moving point and the direction
of its motion common to all the curves of the system. But
DIFFERENTIAL EQUATIONS.
the equation of the second order expresses a property involv-
ing curvature as well as direction of path, and this property
being independent of c l is common to all the systems corre-
sponding to different values of c lt that is, to the entire doubly
infinite system. A moving point, satisfying this equation,
may have any position and move in any direction, provided its
path has the proper curvature as determined by the value of q
derived from the equation, when the selected values of x, y
and/ have been substituted therein.*
For example, equation (7) of the preceding article repre-
sents an ellipse having its center at the origin and touching
the lines x = I, as in the diagram ; c 1 is the ordinate of the
point -of contact with x = i, and c 9 that of the point in which
the ellipse cuts the axis of y. If we regard ^, as fixed and c,
as arbitrary, the equation represents the system of ellipses
touching the two lines at fixed points, and equation (6) is the
differential equation of this system. In
like manner, if , is fixed and c 1 arbitrary,
equation (7) represents a system of ellipses
cutting the axis of y in fixed points
and touching the lines x= i. The
corresponding differential equation will be
found to be
Finally, the equation of the second order, independent of ,
and [(5) of the preceding article] is the equation of the
doubly infinite system of conies f with center at the origin,
and touching the fixed lines x i,.
* If the equation is of the second or higher degree in q, the condition for
equal roots is a relation between x, y and/, which may be found to satisfy the
given equation. If it does, it represents a system of singular solutions; each
of the curves of this system, at each of its points, not only touches but osculates
with a particular integral curve. It is to be remembered that a singular solu-
tion of a first integral is not generally a solution of the given differential equa-
tion; for it represents a curve which simply touches but does not osculate a set
of curves belonging to the doubly infinite system.
f Including hyperbolas corresponding to imaginary values of c*.
THE TWO FIRST INTEGRALS. 33
But, starting from the differential equation of second order,
we may find other first integrals than those above which corre-
spond to , and a . For instance, if equation (5) be multiplied
by/, it becomes
which is also an exact equation, giving the first integral
in which c t is a new constant of integration.
Whenever two first integrals have thus been found inde-
pendently, the elimination of / between them gives the com-
plete integral without further integration.* Thus the result
of eliminating p between this last equation and the first inte-
gral containing c l [equation (6), Art. 13] is
/ -2c,xy + cfx 1 = C? - c t \
which is therefore another form of the complete integral. It
is obvious from the first integral above that c t is the maximum
value of y, so that it is the differential equation of the system
of ellipse inscribed in the rectangle drawn in the diagram. A
comparison of the two forms of the complete integral shows
that the relation between the constants is c* = c? -j- c*.
If a first integral be solved for the constant, that is, put in
the form <j>(x, y, p) = c, the constant will disappear on differ-
entiation, and the result will be the given equation of second
order multiplied, in general, by an integrating factor. We can
thus find any number of integrating factors of an equation
already solved, and these may suggest the integrating factors
of more general equations, as illustrated in Prob. 59 below.
* The principle of this method has already been applied in Art. 10 to the
solution of certain equations of the first order; the process consisted of forming
the equation of the second order of which the given equation is a first integral
(but with a particular value of the constant), then finding another first integral
and deriving the complete integral by elimination of /.
31 DIFFERENTIAL EQUATIONS.
Prob. 59. Solve the equation y + c?y = o in the form
y = A cos ax + B sin 0.x;
and show that the corresponding integrating factors are also inte-
grating factors of the equation
where X is any function of x; and thence derive the integral of this
equation.
/* /*
Ans. <y sin ax I cos ax . Xdx cos ax I sin fl.r . Xdx.
Prob. 60. Find the rectangular and also the polar differential
^equation of all circles passing through the origin.
ART. 15. LINEAR EQUATIONS.
A linear differential equation of any order is an equation of
the first degree with respect to the dependent variable y and
each of its derivatives, that is, an equation of the form
where the coefficients />,... P n and the second member X are
functions of the independent variable only.
The solution of a linear equation is always supposed to be
in the f orm y =f(x)\ and if j, is a function which satisfies the
equation, it is customary to speak of the function j,, rather than
of the equation y = jj/,, as an "integral" of the linear equa-
tion. The general solution of the linear equation of the first
order has been given in Art. 6. For orders higher than the
first the general expression for the integrals cannot be effected
by means of the ordinary functional symbols and the integral
sign, as was done for the first order in Art. 6.
The solution of equation (i) depends upon that of
LINEAR EQUATIONS. 36
The complete integral of this equation will contain arbi-
trary constants, and the mode in which these enter the expres-
sion for y is readily inferred from the form of the equation.
For let y l be an integral, and c t an arbitrary constant ; the re-
sult of putting y = cjf 1 in equation (2) is ^, times the result of
putting y = y l ; that is, it is zero ; therefore c l y l is an integral.
So too, if y t is an integral, j/ a is an integral ; and obviously
also c l y l -\- c^y t is an integral. Thus, if n distinct integrals/,,
y t ,. . . y n can be found,
y = Wi + w* + - + c y (3)
will satisfy the equation, and, containing, as it does, the proper
number of constants, will be the complete integral.
Consider now equation (i); let Fbe a particular integral of
it, and denote by u the second member of equation (3), which
is the complete integral when X = o. If
y=Y-\-u (4)
be substituted in equation (i), the result will be the sum of the
results of putting y = Fand of putting y = u ; the first of
these results will be X, because Fis an integral of equation (i),
and the second will be zero because u is an integral of equa-
tion (2). Hence equation (4) expresses an integral of (i); and
since it contains the n arbitrary constants of equation (3), it
is the complete integral of equation (i). With reference to
this equation F is called " the particular integral," and u is
called "the complementary function." The particular integral
contains no arbitrary constant, and any two particular integrals
may differ by any multiple of a term belonging to the comple-
mentary function.
If one term of the complementary function of a linear
equation of the second order be known, the complete solution
can be found. For let y l be the known term ; then, if y = yp
be substituted in the first member, the coefficient of v in the
result will be the same as if v were a constant : it will there-
fore be zero, and v being absent, the result will be a linear equa-
tion of the first order for v', the first derivative of v. Under
36 DIFFERENTIAL EQUATIONS.
the same circumstances the order of any linear equation can
in like manner be reduced by unity.
A very simple relation exists between the coefficients of an
exact linear equation. Taking, for example, the equation of
the second order, and indicating derivatives by accents, if
is exact, the first term of the integral will be P^y' Subtracting
the derivative of this from the first member, the remainder is
(/>, /V)y + P,y. The second term of the integral must
therefore be (P l P ')y ; subtracting the derivative of this ex-
pression, the remainder, (P t />/ -j- P<>"}y, must vanish. Hence
P 9 PI -j- P " = o is the criterion for the exactness of the
given equation. A similar result obviously extends to equa-
tions of higher orders.
V Prob. 61. Solve x (3 + x) -\- $y = o, noticing that e* is
an integral. Ans. y c^ + c a (x a + 3^ -f 6.v + 6.
v Prob. 62. Solve (x* x)-^-. -4- z(2x 4- i)-f- + 2y = o.
dx ax
Ans. (^ *) b y = f i(- a;4 ~~ 6' ra 4~ 2Jf -J 4x 3 log ^) + <r a ^: 3 .
d*y . O d*y a dv
v Prob. 63. Solve^rn + cos "^z ~ 2 sin P-^ y cos c/ = sin 2#.
ac/ f at/
Ans. y e- sin '^ ^ sin *(^ +
ART. 16. LINEAR EQUATIONS WITH CONSTANT
COEFFICIENTS.
The linear equation with constant coefficients and second
member zero may be written in the form
J*-*y + . . . + A n y = o, (i)
7 Jt
in which D stands for the operator -j-, D* for ^-,, etc., so that
D" indicates that the operator is to be applied n times. Then,
since ZV* = me mx , D*e mx = m*e mx , etc., it is evident that if
LINEAR EQUATIONS, CONSTANT COEFFICIENTS. 37
y e mx be substituted in equation (l), the result after rejecting
the factor e* will be
Aftf + A^"- 1 + . . . + A H = o. (2)
Hence, if m satisfies equation (2), e mx is an integral of equation
(i) ; and if m lt m^ . . . m n are n distinct roots of equation (2),
the complete integral of equation (i) will be
y = c^ x + /"* + . . . + c n e" n *. (3)
For example, if the given equation is
dy
the equation to determine m is
n? m 2 = o,
of which the roots are m l = 2, m t = i; therefore the in-
tegral is
y = cS* + c,e-\
The general equation (i) may be written in the symbolic
form f(D) .y = o, in which / denotes a rational integral func-
tion. Then equation (2) is f(m) = o, and, just as this last
equation is equivalent to
(m m^(m m t ) . . . (m m n ) = o, (4)
so the symbolic equation f(D) . y = o may be written
(D - m t )(D - m,) ..."(/>- m n }y = o. (5)
This form of the equation shows that it is satisfied by each of
the quantities which satisfy the separate equations
(D - m,}y = o, (D m^y = o...(D m n )y = o ; (6)
that is to say, by the separate terms of the complete integral.
If two of the roots of equation (2) are equal, say to m lt two
of the equations (6) become identical, and to obtain the full
number of integrals we must find two terms corresponding to
the equation
(D-m l )y = o; (7)
in other words, the complete integral of this equation of which
^ = e n ^ is known to be one integral. For this purpose we
38 DIFFERENTIAL EQUATIONS.
put, as explained in the preceding article, y =7,^. By differen-
tiation, Dy = De m \ x v = e m ^ x (m.y -J- Dv] ; therefore
(D m^f^v = e m *Dv. (8)
In like manner we find
(D m^e m ^ x v = e m i*D*v. (9)
Thus equation (7) is transformed to D*v = o, of which the
complete integral is v = c,x -{-c t ; hence that of equation (7) is
y = ev(c l x+Ct). (io>
These are therefore the two terms corresponding to the squared
factor (D m,Y in f(D}y = o.
It is evident that, in a similar manner, the three terms
corresponding to a case of three equal roots can be shown to-
be c m ^(c^ -\- CyX -(- c a ), and so on.
The pair of terms corresponding to a pair of imaginary-
roots, say m l = a -\- ifi, m t a i/3, take the imaginary form
Separating the real and imaginary parts of &* and e-#*, and
changing the constants, the expression becomes
e ax (A cos fix-\-B sin fix}. (\ i)
For a multiple' pair of imaginary roots the constants A and
B must be replaced by polynomials as above shown in the case
of real roots.
When the second member of the equation with constant
coefficients is a function of X, the particular integral can also
be made to depend upon the solution of linear equations of
the first order. In accordance with the symbolic notation
introduced above, the solution of the equation
JL_ ay = X , or (D - a}y = x (12)
is denoted by y = (D a)~ l X, so that, solving equation (12),
we have
D^=-a X = r/-T-r (13)
as the value of the inverse symbol whose meaning is " that
LINEAR EQUATIONS, CONSTANT COEFFICIENTS. - 39
function of x which is converted to X by the direct operation
expressed by the symbol D a" Taking the most convenient
special value of the indefinite integral in equation (13), it gives
the particular integral of equation (12). In like manner, the par-
ticular integral of f(D)y = X is denoted by the inverse symbol
fi-fi-X. Now, with the notation employed above, the symbolic
J\ i
fraction may be decomposed into partial fractions with constant
numerators thus:
N
in which each term is to be evaluated by equation (13), and
may be regarded (by virtue of the constant involved in the
indefinite integral) as containing one term of the complement-
ary function. For example, the complete solution of the
equation
is thus found to be
y =
When X is a power of x the particular integral may be
found as follows, more expeditiously than by the evaluation of
the integrals in the general solution. For example, if X = x*
the particular integral in this example may be evaluated by
development of the inverse symbol, thus :
_ i _ _ _!
y ~ D*-D~2 X ~ 2
* The validity of this equation depends upon the fact that the operations
expressed in the second member of
f(D) = (D - mi )(D _,) + ...+(/>_,,)
are commutative, hence the process of verification is the same as if the equation
were an algebraic identity. This general solution was published by Boole in
the Cambridge Math. Journal, First Series, vol. n, p. 114. It had, however,
been previously published by Lobatto, Theorie des Characteristiques, Amster-
dam, 1837.
40 DIFFERENTIAL EQUATIONS.
The form of the operand shows that, in this case, it is only
necessary to carry the development as far as the term contain-
ing D\
For other symbolic methods applicable to special forms of
X we must refer to the standard treatises on this subject.
d*y dy
Prob. 64. Solve JH-< 3-7 + y .
^dx dx
Ans. y = <**(Ax + ) + c<r*.
' Prob. 65. Show that
and that ~ sin (ax + ^) = T sin (ax + ft).
Prob. 66. Solve (Z> 5 + i )y = e* + sin zx + sin x. (Compare
Prob. 59, Art. 14.)
Ans. y = A sin x -f- B cos x -\- %e* -J sin zx $x cos x.
ART. 17. HOMOGENEOUS LINEAR EQUATIONS.
The linear differential equation
in which A , A^ etc., are constants, is called the "homogene-
ous linear equation." It bears the same relation to x m that
the equation with constant coefficients does to e mx . Thus, if
yx m be substituted in this equation, the factor x m will divide
out from the result, giving an equation for determining m,
and the n roots of this equation will in general determine the
n terms of the complete integral. For example, if in the
equation
jd*y . dy
*'-4 + 2*/- - 2y - o
da? dx
we put y = x m , the result is m(m i) + 2m 2 o, or
(m i)(m -}- 2) = o. , ^
The roots of this equation are m 1 = I and m 9 = 2.
Hence y = c^x -j- CyX~*
is the complete integral.
Equation (i) might in fact have been reduced to the form
with constant coefficients by changing the independent vari-
HOMOGENEOUS LINEAR EQUATIONS. 41
able to 0, where x = e , or 6 = log x. We may therefore at
once infer from the results established in the preceding article
that the terms corresponding to a pair of equal roots are of the
form
(c l + c t log x)x m , (2)
and also that the terms corresponding to a pair of imaginary
roots, a ifi, are
x*[A cos (/3 log x) + B sin (ft log *)]. (3)
The analogy between the two classes of linear equations
considered in this and the preceding article is more clearly
seen when a single symbol $= xD is used for the operation of
taking the derivative and then multiplying by x, so that
$x m = mx m . It is to be noticed that the operation x^D 1 is not
the same as & or xDxD, because the operations of taking the
derivative and multiplying by a variable are not "commu-
tative," that is, their order is not indifferent. We have, on the
contrary, x^D* = 8(8 i) ; then the equation given above,
which is
(x*D* + 2xD 2)y = o,
becomes
[8(8 i) + 28 2]j = o, or (8-1X8 + 2)^ = 0,
the function of 8 produced being the same as the function of
m which is equated to o in finding the values of m.
A linear equation of which the first member is homoge-
neous and the second member a function of x may be reduced
to the form
A$).y = x- (4)
The particular integral may, as in the preceding article (see
eq. (14)), be separated into parts each of which depends upon
the solution of a linear equation of the first order. Thus,
solving the equation
-ay = X, or (8 - a)y = X, (5)
we find
X=x" Cx- a - l Xdx. (6)
a v
8
The more expeditious method which may be employed
DIFFERENTIAL EQUATIONS.
when X is a power of x is illustrated in the following example :
d, v /z"i/
Given x* 2-f- = *'. The first member becomes homo-
dx ax
geneous when multiplied by x, and the reduced equation is
( 8 _ 3$'jjy X \
The roots of /($) =o are 3 and the double root zero, hence
the complementary function is cj? -f- c ., -f- c 3 log x. Since in
general f($)x r f(r)x r , we infer that in operating upon x* we
may put $ = 3. This gives for the particular integral
i i ^ _ i i ^
but fails with respect to the factor $ 3.* We therefore
now fall back upon equation (6), which gives
JT- x 3 = x* / x~ l dx = x* log x.
The complete integral therefore is
y =
dy
Prob. 67. Solve zx -~ 4- 3^-7 $y = x*.
ax ax
Ans. y = c 1 x +
Prob. 68. Solve (*'>' + 3 ^Z>' + D)y = -.
oc
Ans. y = c, + <r, log x + <r,(log ^) a
ART. 18. SOLUTIONS IN INFINITE SERIES.
We proceed in this article to illustrate the method by
which the integrals of a linear equation whose coefficients are
algebraic functions of x may be developed in series whose
terms are powers of x. For this purpose let us take the
equation
' '
* The failure occurs because x 3 is a term of the complementary function
having an indeterminate coefficient; accordingly the new term is of the same
form as the second term necessary when 3 is a double root, but of course with
a determinate coefficient.
SOLUTIONS IN INFINITE SERIES. 43-
which is known as " Bessel's Equation," and serves to define-
the "Besselian Functions."
If in the first member of this equation we substitute (or y
the single term Ax m the result is
A(m* - ri l }x m + Ax m +\ (2)
the first term coming from the homogeneous terms of the
equation and the second from the term x*y which is of higher
degree. If this last term did not exist the equation would be
satisfied by the assumed value of y, if m were determined so as
to make the first term vanish, that is, in this case, by Ax n or
Bx~ n . Now these are the first terms of two series each of
which satisfies the equation. For, if we add to the value of y
a term containing x m+2 , thus/ = A Q x m -j- A^" 1 ^ 2 , the new term-
will give rise, in the result of substitution, to terms containing.
x m+2 and x m+4 respectively, and it will be possible so to take
A i that the entire coefficient of x m+ * shall vanish*. In like
manner the proper determination of a third term makes the
coefficient of x mJr * in the result of substitution vanish, and so
on. We therefore at once assume
= A,x' + A, x m + 2 -\- A t x m + 4 + . . . , (3)
in which r has all integral values from o to oo. Substituting
in equation (i)
2[{(m + 2/) 9 - n 9 }A^ m + 2r -\- ^X* +2(r+1) ] = o. (4)
The coefficient of each power of x in this equation must sep-
arately vanish ; hence, taking the coefficient of x m+2r , we have
[(m + 2 ry-n>]A r +A r _ I =o. (5)
When r = o, this reduces to m* n* = o, which determines
the values of m, and for other values of r it gives
~ (m + 2r + n)(m -\-2r- n) Ar ~ 1 '
the relation between any two successive coefficients.
For the first value of m, namely n, this relation becomes
A __ !_ . A
"'- ' -"
4-i DIFFERENTIAL EQUATIONS.
whence, determining the successive coefficients in equation (3),
the first integral of the equation is
- -^ -, + wlw
In like manner, the other integral is found to be
r-jj - .. .J. (7)
. . -, (8)
and the complete integral is 7 = A y l -J-
This example illustrates a special case which may arise in
this form of solution. If n is a positive integer, the second
series will contain infinite coefficients. For example, if n == 2,
the third coefficient, or B v is infinite, unless we take B = o, in
which case B^ is indeterminate and we have a repetition of the
solution y r This will always occur when the same powers of
x occur in the two series, including, of course, the case in which
m has equal roots. For the mode of obtaining a new integral
in such cases the complete treatises must be referred to.f
It will be noticed that the simplicity of the relation between
consecutive coefficients in this example is due to the fact that
equation (i) contained but two groups of terms producing
different powers of x, when Ax m is substituted for y as in ex-
pression (2). The group containing the second derivative
necessarily gives rise to a coefficient of the second degree in
in, and from it we obtained two values of m. Moreover, be-
cause the other group was of a degree higher by two units, the
assumed series was an ascending one, proceeding by powers
of x\
* The Besselian function of the wth order usually denoted byy is the value
, of y\ above, divided by 2"! if n is a positive integer, or generally by 2 n r(n-\-i).
For a complete discussion of these functions see Lommel's Studien liber die
Bessel'schen Functionen, Leipzig, 1868; Todhunter's Treatise on Laplace's,
Lame's and Bessel's Functions, London, 1875, etc -
f A solution of the kind referred to contains as one term the product of the
regular solution and log x, and is sometimes called a " logarithmic solution."
See also American Journal of Mathematics, Vol. XI, p. 37. In the case of
Bessel's equation, the logarithmic solution is the "Besselian Function of the
iecond kind."
SOLUTIONS IN INFINITE SERIES. 45
In the following example,
there are also two such groups of terms, and their difference
of degree shows that the series must ascend by simple powers.
We assume therefore at once
The result of substitution is
- l ']= o.
Equating to zero the coefficient of x" lJrr ~ 2 ,
(m -\- r + i)(m + r 2)A r + a(m + r i)A r . t = o, (12)
which, when r = o, gives
(;//+!)(* 2)/4.=o, (13)
and when r > o,
m-\-r I
A r ^7 - i - i - \7 - i --- f^*r 1" (14)
(in -\- r -{- i)(m -\-r~-2)
The roots of equation (13) are m-=.2 and m = i; taking
nt=2, the relation (14) becomes
A i r+l A
(r+*Y
vh< nee the first integral is
Taking the second value w = i, equation (14) gives
r ~ 2
\-
3)
, =
gral is the finite expression
whence B, = -- ^ , and ^, = o*; therefore the second inte~
* Bt would take the indeterminate form, and if we suppose it to have a finite
value, the rest of the series is equivalent to B^y\, reproducing the first integral.
46 DIFFERENTIAL EQUATIONS.
When the coefficient of the term of highest degree in the
result of substitution, such as equation (11), contains m, it is
possible to obtain a solution in descending powers of x. In
this case, m occurring only in the first degree, but one such
solution can be found; it would be identical with the finite
integral (16). In the general case there will be two such solu-
tions, and they will be convergent for values of x greater than
unity, while the ascending series will converge for values less
than unity.*
When the second member of the equation is a power of x,
the particular integral can be determined in the form of a series
in a similar manner. For example, suppose the second mem-
ber of equation (9) to have been x*. Then, making the sub-
stitution as before, we have the same relation between consecu-
tive coefficients; but when r = o, instead of equation (13) we
have
(m 4- i)(m 2)A x m ~ 2 = x
to determine the initial term of the series. This gives m = 2$
and A -f ; hence, putting m = in equation (14), we find for
the particular integral f
7 9.3 9.11.3.5
A linear equation remains linear for two important classes
of transformations ; first, when the independent variable is
changed to any function of x, and second, when for y we put
vf(x). As an example of the latter, let y = e~ ax v be substituted
in equation (9) above. After rejecting the factor e'**, the
result is
d*v dv 2v _
dx* dx x*
Since this differs from the given equation only in the sign
*When there are two groups of terms, the integrals are expressible in terms
of Gauss's " Hypergeometric Series."
f If the second member is a term of the complementary function (for ex-
ample, in this case, if it is any integral power of x), the particular integral will
take the logarithmic form referred to in the foot-note on p. 346.
SYSTEMS OF DIFFERENTIAL EQUATIONS. 47
of a, we infer from equation (16) that it has the finite integral
v = - -J . Hence the complete integral of equation (9) can
X
be written in the form
xy <r,(2 ax) -f c^e- ax (2 -f- ax).
<Ty
Prob. 69. Integrate in series the equation -T-J + ^ry = o.
Ans. ^(.-I^+Lj**-- . . .)-HB(_J^+'^'_ . . .).
^y . </y
Prob. 70. Integrate in series x*. -, + x* + (x 2)y = o.
Prob. 71. Derive for the equation of Prob. 70 the integral
y 9 = e~*(x~ l + i + *)> an d find its relation to those found above.
ART. 19. SYSTEMS OF DIFFERENTIAL EQUATIONS.
It is shown in Art/ 12 that a determinate system of n differ-
ential equations of the first order connecting n -J- I variables
has for its complete solution as many integral equations con-
necting the variables and also involving n constants of inte-
gration. The result of eliminating n I variables would be a
single relation between the remaining two variables containing
in general the n constants. But the elimination may also be
effected in the differential system, the result being in general
an equation of the wth order of which the equation just men-
tioned is the complete integral. For example, if there were
two equations of the first order connecting the variables x and
y with the independent variable /, by differentiating each we
should have four equations from which to eliminate one vari-
able, say y, and its two derivatives * with respect to /, leaving
a single equation of the second order between x and /.
It is easy to see that the same conclusions hold if some of
the given equations are of higher order, except that the order
of the result will be correspondingly higher, its index being in
* In general, there would be n* equations from which to eliminate i
variables and n derivatives of each, that is, (n i)( -f- i) = n* I quantities
leaving a single equation of the wth order.
48 DIFFERENTIAL EQUATIONS.
general the sum of the indices of the orders of the given equa-
tions. The method is particularly applicable to linear equa-
tions with constant coefficients, since we have a general method
of solution for the final result. Using the symbolic notation,
the differentiations are performed simply by multiplying by
the symbol D, and therefore the whole elimination is of exactly
the same form as if the equations were algebraic. For ex-
ample, the system
cFy dx _ dx dy
when written symbolically, is
(2D* 4)y Dx = 2t,
whence, eliminating x,
2t
2 D*-4 D
2D 4D~ y ~
o
which reduces to
(D-i)
Integrating,
y = (A + Bty + Ce~* - \t,
the particular integral being found by symbolic development,
as explained at the end of Art. 16.
The value of x found in like manner is
x = (A' + B'ty + C'e-V - f
The complementary function, depending solely upon the deter-
minant of the first members,* is necessarily of the same form
as that for y, but involves a new set of constants. The re-
lations betv/een the constants is found by substituting the
values of x and y in one of the given equations, and equating
to zero in the resulting identity the coefficients of the several
terms of the complementary function. In the present ex-
ample we should thus find the value of x, in terms of A, B,
and C, to be
* The index of the degree in D of this determinant is that of the order of
the final equation ; it is not necessarily the sum of the indices of the orders of
the given equations, but cannot exceed this sum.
SYSTEMS OF DIFFERENTIAL EQUATIONS. 49"
In general, the solution of a system of differential equations
depends upon our ability to combine them in such a way as
to form exact equations. For example, from the dynamical
system
~dJ- ' d?~ ' d? -
where X, Y, Z are functions of x, y, and 2, but not of t t
we form the equation
dx ,dx . dy jdz . dz ,dz , , ,,, ,
- d -- r- -^-d-~ + -rd Xdx -4- Ydy -\-Zdz.
dt dt '<#<#' dt dt
The first member is an exact differential, and we know that for
a conservative field of force the second member is also exact,
that is, it is the differential of a function U of x> y, and z.
The integral
is that first integral of the system (i) which is known as the
equation of energy for the unit mass.
Just as in Art. 13 an equation of the second order was re-
garded as equivalent to two equations of the first order, so the
system (i) in connection with the equation defining the resolved
velocities forms a system of six equations of the first order, of
which system equation (2) is an " integral " in the sense ex-
plained in Art. 12.
Prob. 72. Solve the equations - = = dt as a system im-
my mx
ear in /. Ans. x A cosmt-\-smmt,y=A sinmtJB cosmt.
Prob. 73. Solve the system -5 -- 1- *y = e*. -f- -4- z = o.
dx dx
Ans. y - Ae nx + Be~ nx + -r . z nAe nx -\- nEe'** -- ^ .
n t n i
Prob. 74. Find for the system - = x<t>(x,y), -^ = y<f>(x,y}
a first integral independent of the function <f>.
dy dx
50 DIFFERENTIAL EQUATIONS.
Prob. 75. The approximate equations for the horizontal motion
of a pendulum, when the earth's rotation is taken into account, are
d*x dy , gx d*y . dx , gy
2f-> r i +i r = > i + * r is+i = <
show that both x and y are of the form
A cos ,/ -f- -B sin ,/ + C cos nj -f- D sin #,
ART. 20. FIRST ORDER AND DEGREE WITH THREE
VARIABLES.
The equation of the first order and degree between three
variables x, y and z may be written
Pdx + Qdy + Rdz = o, (i)
where P, Q and R are functions of x, y and z. When this
equation is exact, P, Q and R are the partial derivatives of
some function u, of x, y and z ; and we derive, as in Art. 4,
'dP = 3Q dQ = d_ -dR = -dP_ ,.
-dy d*' dz ~ 'dy' d* dz
for the conditions of exactness. In the case of two variables,
when the equation is not exact integrating factors always exist;
but in this case, there is not always a factor // such that j*P,
pQ and pR (put in place of P, Q, and R) will satisfy all three
of the conditions (2). It is easily shown that for this purpose
the relation
must exist between the given values of P, Q, and R. This is
therefore the " condition of integrability " of equation (i).*
When this condition is fulfilled equation (i) may be inte-
grated by first supposing one variable, say z, to be constant.
Thus, integrating Pdx -f- Qdy = o, and supposing the constant
of integration C to be a function of z, we obtain the integral, so
* When there are more than three variables such a condition of integra-
bility exists for each group of three variables, but these conditions are not all
independent. Thus with four variables there are but three independent con-
ditions.
FIRST ORDER AND DEGREE, THREE VARIABLES. 51
far as it depends upon x and y. Finally, by comparing the
total differential of this result with the given equation we de-
termine dC in terms of z and dz, and thence by integration the
value of C.
It may be noticed that when certain terms of an exact
equation forms an exact differential, the remaining terms must
also be exact. It follows that if one of the variables, say z
can be completely separated from the other two (so that in
equation (i) ^becomes a function of z only and P and Q func-
tions of x and y, but not of z) the terms Pdx -f- Qdy must be
thus rendered exact if the equation is integrable.* For example,
zydx zxdy y*dz = o.
is an integrable equation. Accordingly, dividing by y*2. which
we notice separates the variable z from x and y, puts it in the
exact form
ydx xdy dz
y
of which the integral is x = y log cz.
Regarding x, y and z as coordinates of a moving point,
an integrable equation restricts the point to motion upon one
of the surfaces belonging to the system of surfaces represented
by the integral ; in other words, the point (x, y, z) moves in an
arbitrary curve drawn on such a surface. Let us now consider
in what way equation (i) restricts the motion of a point when
it is not integrable. The direction cosines of a moving point
are proportional to dx, dy, and dz; hence, denoting them by
/, m and , the direction of motion of the point satisfying
equation (i) must satisfy the condition
Pl+Qm+Rn = o. (4)
It is convenient to considernn this connection an auxiliary
system of lines represented, as explained in Art. 12, by the
simultaneous equations
dx dy dz
= = fe\
P Q R'
*In fact for this case the condition (3) reduces to its last term, which ex-
presses the exactness of Pdx -f- Qdy,
52 DIFFERENTIAL EQUATIONS.
The direction cosines of a point moving in one of the lines
of this system are proportional to P, Q and R. Hence, de-
noting them by A, ju, v, equation (4) gives
A/ -j- pin -(- vn = o (6)
for the relation between the directions of two moving points,
whose paths intersect, subject respectively to equation (i) and
to equations (5). The paths in question therefore intersect at
right angles; therefore equation (i) simply restricts a point to
move in a path which cuts orthogonally the lines of the auxili-
ary system.
Now, if there be a system of surfaces which cut the auxiliary
lines orthogonally, the restriction just mentioned is completely
expressed by the requirement that the line shall lie on one of
these surfaces, the line being otherwise entirely arbitrary..
This is the case in which equation (i) is integrable.*
On the other hand, when the equation is not integrable, the
restriction can only be expressed by two equations involving
an arbitrary function. Thus if we assume in advance one such
relation, we know from Art. 12 that the given equation (i)
together with the first derivative of the assumed relation forms
a system admitting of solution in the form of two integrals-
Both of these integrals will involve the assumed. function. For
any particular value of that function we have a system of lines
satisfying equation (i), and the arbitrary character of the func-
tion makes the solution sufficiently general to include all lines
which satisfy the equation.f
Prob. 76. Show that the equation
(mz ny)dx + (nx lz]dy + (ly mx}dz = o
is integrable, and infer from the integral the character of the auxil-
*It follows that, with respect to the system of lines represented by equations
(5), equation (3) is the condition that the system shall admit of surfaces cutting
them orthogonally. The lines of force in any field of conservative forces form
such a system, the orthogonal surfaces being the equipotential surfaces.
f So too there is an arbitrary element about the path of a point when the
single equation to which it is subject is integrable, but this enters only into one
of the two equations necessary to define the path.
PARTIAL EQUATIONS, FIRST ORDEX OO
iary lines. (Compare the illustrative example at the end of Art. 12.)
Ans. nx Iz = C(ny mz).
Prob. 77. Solve yz'dx z*dy e*dz o. Ans. yz = e*(i-\-cz).
Prob. 78. Find the equation which in connection with^ .f(x)
forms the solution of dz = aydx -f- bdy.
Prob. 79. Show that a general solution of
ydx (x z)(dy dz)
is given by the equations
yz=(t>(x), y = (x z) (/>'(*).
(This is an example of " Monge's Solution.")
ART. 21. PARTIAL DIFFERENTIAL EQUATIONS OF FIRST
ORDER AND DEGREE.
Let x denote an unknown function of the two independent
variables x and y, and let
denote its partial derivatives : a relation between one or both
of these derivatives and the variables is called a " partial dif-
ferential equation " of the first order! A value of z in terms of
x and y which with its derivatives satisfies the equation, or a
relation between x, y and z which makes z implicitly such a
function, is a " particular integral." The most general equation
of this kind is called the " general integral."
If only one of the derivatives, say/, occurs, the equation
may be solved as an ordinary differential equation. For if y is
considered as a constant,/ becomes the ordinary derivative of
z with respect to x\ therefore, if in the complete integral of
the equation thus regarded we replace the constant of integra-
tion by an arbitrary function of j, we shall have a relation
which includes all particular integrals and has the greatest pos-
sible generality. It will be found that, in like manner, when
both p and q are present, the general integral involves an arbi-
trary function,
We proceed to give Lagrange's solution of the equation of
54r DIFFERENTIAL EQUATIONS.
the first order and degree, or " linear equation," which may be
written in the form
Pp+Qq = R, (I)
P, Q and R denoting functions of x, y and z. Let u = a, in
which u is a function of x, y and z, and #, a constant, be an
integral of equation (i). Taking derivatives with respect to x
and y respectively, we have
and substitution of the values of / and q in equation (i) gives
the symmetrical relation
Consider now the system of simultaneous ordinary differ-
ential equations
dx_ _ dy__ dz_
~J>~~Q^~R (3)
Let u = a be one of the integrals (see Art. 12) of this sys-
tem. Taking its total differential,
"du 3 9
-^+-^ + --^=0:
and since by equations (3) dx, dy and dz are proportional to />,
<2 and 7?, we obtain by substitution
which is identical with equation (2). It follows that every
integral of the system (3) satisfies equation (i), and conversely,
so that the general expression for the integrals of (3) will be
the general integral of equation (i).
Now let v == b be another integral of equations (3), so that
v is also a function which satisfies equation (2). As explained
in Art. 12, each of the equations u a, v = b is the equation
of a surface passing through a singly infinite system of lines
belonging to the doubly infinite system represented by equa-
tions (3). What we require is the general expression for any
PARTIAL EQUATIONS, FIRST ORDER.
55
surface passing through lines of the system (and intersecting
none of them). It is evident that f(u, v) = f(a, b) = C is such
an equation,* and accordingly f(u, v), where f is an arbitrary
function, will be found to satisfy equation (2). Therefore, to
solve equation (i), we find two independent integrals u = a,
v = b of the auxiliary system (3), (sometimes called Lagrange's
equations,) and then put
= 00)> (4)
an equation which is evidently equally general with/(#, v) = o.
Conversely, it may be shown that any equation of the form
(4), regarded as a primitive, gives rise to a definite partial
differential equation of Lagrange's linear form. For, taking
partial derivatives with respect to the independent variables
x and j, we have
3*^3-3-
and eliminating (f>'(v) from these equations, the term contain-
ing/^ vanishes, giving the result
3 3;
37
P +
(5)
which is of the form Pp -f- Qq R.\
* Each line of the system is characterized by special values of a and b which
we may call its coordinates, and the surface passes through those lines whose
coordinates are connected by the perfectly arbitrary relation f(a, 6) = C.
f These values of P, Q and R are known as the " Jacobians " of the pair
of functions u, v with respect to the pairs of variables y, z ; z, x ; and x, y re-
spectively. Owing to their analogy to the derivatives of a single function they
are sometimes denoted thus :
_ 3(, z/) 3(,
3(.v,
g _ d(u, v)
The Jacobian vanishes if the functions u and v are not independent, that is
to say, if can be expressed identically as a function of v. In like manner,
56 DIFFERENTIAL EQUATIONS.
As an illustration, let the given partial differential equa-
tion be
(mz ny]p -f- (nx lz)q = ly mx, (6)
.for which Lagrange's Equations are
dx dy dz
_____^^__ * __ 17 >
mz ny ~ nx Iz ~ ly mx' *''
These equations were solved at the end of Art. 12, the two
integrals there found being
Ix -j- my -\- nz = a and x* -\- y* -j- ^ = b*> (8)
Hence in this case the system of " Lagrangean lines" con-
sists of the entire system of circles having the straight line
for axis. The general integral of equation (6) is then
Ix -j- my + nz = (t>(x* -\- y* -f- z*), (10)
which represents any surface passing through the circles just
mentioned, that is, any surface of revolution of which (9) is the
axis.*
Lagrange's solution extends to the linear equation contain-
ing n independent variables. Thus the equation being
the auxiliary equations are
dx\ dx^ _ _ dx n _ dz
~I\'-~-^\~- : ^ = ^~'
! - = o is the condition that (a function of x, y and 2) is expressible
<H*, >, 2)
identically as a function of u and v, that is to say, that = o shall be an in-
tegral of Pp + Qq= R.
* When the equation Pdx -f- Qdy + Rdz = o is integrable (as it is in the
above example; see Prob. 76, Art. 20), its integral, which may be put in the form
V = C, represents a singly infinite system of surfaces which the Lagrangean
lines cut orthogonally ; therefore, in this case, the general integral may be de-
fined as the general equation of the surfaces which cut orthogonally the system
V = C. Conversely, starting with a given system V = C, u = J\v) is the gen-
eral equation of the orthogonal surfaces, if u = a and v = b are integrals of
COMPLETE AND GENERAL INTEGRALS. 5?
and if u l = c lt u y = c a , . . . u n = c n are independent integrals,
the most general solution is
/(,, . . . ) = o,
where /is an arbitrary function.
^ 2 ^ 2T / y\
Prob. 80. Solve xz- \-yz~~ = xy. Ans. xy z* =/[ ).
dx dy \y)
Prob. 81. Solve (y + z)p + (z + x}q = x + y.
Prob. 82. Solve (x +^)(/ q) z.
Ans. (x-\-y) log 2 x =/(x-\-y).
Prob. 83. Solve x(y z)p -\-y(z x)q = z(x y).
Ans. x -\- y-\- z= f(xyz).
ART. 22. COMPLETE AND GENERAL INTEGRALS.
We have seen in the preceding article that an equation be-
tween three variables containing an arbitrary function gives
rise to a partial differential equation of the linear form. It
follows that, when the equation is not linear in / and q, the
general integral cannot be expressed by a single equation of
the f^rm 0(, v) = o; it will, however, still be f^und to- depend
upon a single arbitrary function. X .
It therefore becomes necessary to consider an integral hav-
ing as much generality as can be given by the presence of arbi-
trary constants. Such an equation is called a " complete in-
tegral " ; it contains two arbitrary constants (n arbitrary con-
stants in the general case of n independent variables), because
this is the number which can be eliminated from such an equa-
tion, considered as a primitive, and its two derived equations.
For example, if
(*-a)' + o/-)' + *' = /P,
a and b being regarded as arbitrary, be taken as the primitive,
the derived equations are
x a -\- zp = o, y b -f- zq = O,
and the elimination of a and b gives the differential equation
A/ +V +!)*.
of which therefore the given equation is a complete integral.
58 DIFFERENTIAL EQUATIONS.
Geometrically, the complete integral represents a doubly in-
finite system of surfaces ; in this case they are spherical sur-
faces having a given radius and centers in the plane of xy.
In general, a partial differential equation of the first order
with two independent variables is of the form
F(x, y, z, p, q) = O, (l)
and a complete integral is of the form
f(x, y, 2, a, b) = o. (2)
In equation (i) suppose x, y and z to have special values,
namely, the coordinates of a special point A ; the equation
becomes a relation between p and q. Now consider any sur-
face passing through A of which the equation is an integral of
(i), or, as we may call it, a given "integral surface " passing
through A. The tangent plane to this surface at A determines
values of / and q which must satisfy the relation just men-
tioned. Consider also those of the complete integral surfaces
[equation (2)] which pass through A. They form a singly in-
finite system whose-tangent planes at A have values of p and
q which also satisfy the relation. There is obviously among
them one which has the same value of /, and therefore also
the same value of q, as the given integral. Thus there is one
of the complete integral surfaces which touches at A the given
integral surface. It follows that every integral surface (not in-
cluded in the complete integral) must at every one of its points
touch a surface included in the complete integral.*
It is hence evident that every integral surface is the en-
velope of a singly infinite system selected from the complete
integral system. Thus, in the example at the beginning of
this article, a right cylinder whose radius is k and whose axis
lies in the plane of xy is an integral, because it is the envelope
* Values of x, y, and z, determining a point, together with values of/ and q,
determining the direction of a surface at that point, are said to constitute an
"element of surface." The theorem shows that the complete integral is ' com-
plete " in the sense of including all the surface elements which satisfy the differ-
ential equation. The method of grouping the "consecutive" elements to form
an integral surface is to a certain extent arbitrary.
COMPLETE AND GENERAL INTEGRALS. 59
of those among the spheres represented by the complete in-
tegral whose centers are on the axis of the cylinder. If we
make the center of the sphere describe an arbitrary curve in
the plane of xy we shall have the general integral in this ex-
ample.
In general, if in equation (2) an arbitrary relation between
a and b, such as b = </>(a), be established, the envelope of the
singly infinite system of surfaces thus defined will represent
the general integral. By the usual process, the equation of
the envelope is the result of eliminating a between the two
equations
o
f(x, y, z, a, 0(rt) ) = o, ~Tf( x i y> z > a > 0(*) ) = - (3)
These two equations together determine a line, namely, the
" ultimate intersection of two consecutive surfaces." Such
lines are called the " characteristics " of the differential equa-
tion. They are independent of any particular form of the
complete integral, being in fact lines along which all integral
surfaces which pass through them touch one another. In the
illustrative example above they are equal circles with centers
in the plane of xy and planes perpendicular to it.*
The example also furnishes an instance of a " singular so-
lution " analogous to those of ordinary differential equations.
*The characteristics are to be regarded not merely as lines, but as " linear
elements of surface," since they determine at each of their points the direction
of the surfaces passing through them. Thus, in the illustration, they are cir-
cles regarded as great-circle elements of a sphere, or as elements of a right
cylinder, and may be likened to narrow hoops. They constitute in all cases a
triply infinite system. The surfaces of a complete integral system contain them
all, but they are differently grouped in different integral surfaces.
If we arbitrarily select a curve in space there will in general be at each of
its points but one characteristic through which the selected curve passes; that
is, whose tangent plane contains the tangent to the selected curve. These char-
acteristics (for all points of the curve) form an integral surface passing through
the selected curve ; and it is the only one which passes through it unless it be
itself a characteristic. Integral surfaces of a special kind result when the se-
lected curve is reduced to a point. In the illustration these are the results of
rotating the circle about a line parallel to the axis of z.
60 DIFFERENTIAL EQUATIONS.
For the planes z = k envelop the whole system of spheres
represented by the complete integral, and indeed all the sur-
faces included in the general integral. When a singular solu-
tion exists it is included in the result of eliminating a and b
from equation (2) and its derivatives with respect to a and b,
that is, from
but, as in the case of ordinary equations, this result may in-
clude relations which are not solutions.
Prob. 84. Derive a differential equation from the primitive
2x + my -f- nz = a, where /, m, n are connected by the relation
f + m* + *' = i.
Prob. 85. Show that the singular solution of the equation
found in Prob. 84 represents a sphere, that the characteristics con-
sist of all the straight lines which touch this sphere, and that the
general integral therefore represents all developable surfaces which
touch the sphere.
Prob. 86. Find the integral which results from taking in the
general integral above /' -\-m* = cos" (a constant) for the arbitrary
relation between the parameters.
ART, 23. COMPLETE INTEGRAL FOR SPECIAL FORMS.
A complete integral of the partial differential equation
F(x,y,z,p,q) = o (i)
contains two constants, a and b. If a be regarded as fixed and
b as an arbitrary parameter, it is the equation of a singly in-
finite system of surfaces, of which one can be found passing
through any given point. The ordinary differential equation
of this system, which will be independent of b, may be put in
the form
dz = pdx -\- qdy, (2)
in which the coefficients/ and q are functions of the variables
and the constant a. Now the form of equation (2) shows that
these quantities are the partial derivatives of z, in an integral
of equation (i); therefore they are values of p and q which
COMPLETE INTEGRAL FOR SPECIAL FORMS. 61
satisfy equation (i). Conversely, if values of/ and q in terms
of the variables and a constant a which satisfy equation (i) are
such as to make equation (2) the differential equation of a sys-
tem of surfaces, these surfaces will be integrals. In other
words, if we can find values of/ and q containing a constant a
which satisfy equation (i) and make dz = pdx -f- qdy inte-
grable, we can obtain by direct integration a complete inte-
gral, the integration introducing a second constant.
There are certain forms of equations for which such values
of / and q are easily found. In particular there are forms in
which / and q admit of constant values, and these obviously
make equation (2) integrable. Thus, if the equation contains
/ and q only, being of the form
F(P* 4) = 0, (3)
we may put p = a and q = b, provided
F(a,t)=o. (4)
Equation (2) thus becomes
dz = adx -f- bdy,
whence we have the complete integral
z = ax+by-\ r c, (5)
in which a and b are connected by the relation (4) so that a, b
and c are equivalent to two arbitrary constants.
In the next place, if the equation is of the form
*=P* + W+f(P,q\ (6)
which is analogous to Clairaut's form, Art. 10, constant values
of p and q are again admissible if they satisfy
z = ax + by+f(a,b\ (7)
and this is itself the complete integral. For this equation is
of the form z ax + by + c, and expresses in itself the rela-
tions between the three constants. Problem 84 of the preced-
ing article is an example of this form.
In the third place, suppose the equation to be of the form
F(z,p,q} = o, (g)
62 DIFFERENTIAL EQUATIONS.
in which neither x nor y appears explicitly. If we assume
q = ap, p will be a function of z determined from
F(z, p, ap] o, say / = (f>(z), (9)
Then dz = pdx -\- qdy = o becomes dz (f)(z)(dx -|- ady), which is
integrable, giving the complete integral
A fourth case is that in which, while z does not explicitly
occur, it is possible to separate x and / from y and q, thus put-
ting the equation in the form
/.(*>/) =/,(* ?) (II)
If we assume each member of this equation equal to a con-
stant #, we may determine/ and q in the forms
/ = <&(>> a \ q = fad', a). (12)
and dz = pdx + qdy takes an integrable form giving
(13)
It is frequently possible to reduce a given equation by trans-
formation of the variables to one of the four forms considered
in this article.* For example, the equation x*p* + y q* = z*
may be written
*The general method, due to Charpit, of finding a proper value olp consists
of establishing, by means of the condition of integrability, a linear partial dif-
ferential equation for/, of which we need only a particular integral. This may
be any value of / taken from the auxiliary equations employed in Lagrange's
process. See Boole, Differential Equations (London 1865), p. 336 ; also For-
syth, Differential Equations (London 1885), p. 316, in which the auxiliary equa-
tions are deduced in a more general and symmetrical form, involving both /
and q. These equations are in fact the equations of the characteristics regarded
as in the concluding note to the preceding article. Denoting the partial deriva-
tives of F(x, y, z,p, q) by X, Y, Z, P, Q, they are
dx _ dy dz dt> >/,/
' ~ ~
IP - Q Pp+Qq ~ X+ty ~ Y+Zq'
See Jordan's Cours d'Analyse (Paris, 1887), vol. in, p. 318 ; Johnson's Differ-
ential Equations (New York, 1889), p. 300. Any relation involving one or both
the quantities /and q, combined with f=o, will furnish proper values of/
PARTIAL EQUATIONS, SECOND ORDER. 63
whence, putting x' = log x, y' = log y, z' = log 2, it becomes
/'* + ^' a == l > which is of the form F(p', q'} = o, equation (3).
Hence the integral is given by equation (5) when a? -j- &* = i;
it may therefore be written
z' = x' cos a-\-y' sin a -f- c,
and restoring x, y, and #, that of the given equation is
z = cx co * a y* Q ".
Prob. 87. Find a complete integral for/ 1 q* = i.
Ans. 2 = .# sec or + j> tan a + .
Prob. 88. Find the singular solution of z = px + gy +/?
Ans. 2 = xy.
Prob. 89. Solve by transformation q = zyp*.
Ans. z = ax -\~ ofy* + ^
Prob. 90. Solve z(p*q*} = x y.
Ans. 2? = (# + a)i + (y + )1 + b.
Prob. 91. Show that the solution given for the form F(z,p, q ) o
represents cylindrical surfaces, and that F(z, o, o) = o is a singular
solution.
Prob. 92. Deduce by the method quoted in the foot-note two
complete integrals of pq = px -j- qy.
f x Y
Ans. 22 = f -f~ a y) + A an( i z== xy-\-y ^(x'' -}- a) -\- b.
ART. 24. PARTIAL EQUATIONS OF SECOND ORDER.
We have seen in the preceding articles that the general
solution of a partial differential equation of the first order de-
pends upon an arbitrary function ; although it is only when
the equation is linear in / and q that it is expressible by a
single equation. But in the case of higher orders no general
account can be given of the nature of a solution. Moreover,
when we consider the equations derivable from a primitive con-
taining arbitrary functions, there is no correspondence between
their number and the order of the equation. For example, if
and q. Sometimes several such relations are readily found ; for example, for
the equation z = pq we thus obtain the two complete integrals
and 4* =/
64- DIFFERENTIAL EQUATIONS.
the primitive with two independent variables contains two ar-
bitrary functions, it is not generally possible to eliminate them
and their derivatives from the primitive and its two derived
equations of the first and three of the second order.
Instead of a primitive containing two arbitrary functions,
let us take an equation of the first order containing a single
arbitrary function. This may be put in the form u = $(v\
u and v now denoting known functions of x, y, z, p, and q.
(f>'(v) may now be eliminated from the two derived equations
as in Art. 21. Denoting the second derivatives of z by
9V 9** ,_9!f
= a*" ~ a-ra/ ~ ay
the result is found to be of the form
Rr + Ss-\-Tt+ U(rt - /) = V, (i)
in which R, S, T, U, and V are functions of x, y, z, p, and,^.
With reference to the differential equation of the second order
the equation u = </>(v) is called an " intermediate equation of
the first order " : it is analogous to the first integral of an ordi-
nary equation of the second order. It follows that an inter-
mediate equation cannot exist unless the equation is of the
form (i); moreover, there are two other conditions which
must exist between the functions R, S, T, and U.
In some simple cases an intermediate equation can be ob-
tained by direct integration. Thus, if the equation contains
derivatives with respect to one only of the variables, it may be
treated as an ordinary differential equation of the second order,
the constants being replaced by arbitrary functions of the
other variable. Given, for example, the equation xr p = xy,
which may be written
xdp pdx = xy dx.
This becomes exact with reference to x when divided by x*,
and gives the intermediate equation
A second integration (and change in the form of the arbitrary
function) gives the general integral
z = $yx* log x -f
PARTIAL EQUATIONS, SECOND ORDER. 65
Again, the equation p-\-r-\-s=i is already exact, and
gives the intermediate equation
which is of Lagrange's form. The auxiliary equations* are
dz
dx = dy = - -,
x - z -f <t>(y)
of which the first gives x y = a, and eliminating x from the
second, its integral is of the form
z = a -}- 0( y] 4" e ~ y b.
Hence, putting b = $(a), we have for the final integral
in which a further change is made in the form of the arbitrary
function 0.
Prob. 93. Solve / q = e* + e y .
Ans. z = ye y c* + <p
Prob. 94. Solve r+p* = y\
Ans. z = log [^00 - e-**] +
Prob. 95. Solve /(j /) = x.
Ans. 2 = (x +y) logj> + 0(*) + #(*
Prob. 96. Solve /^ qr = o. Ans. ^c = 0(.y) + ^( 2 )-
Prob. 97. Show that Monge's equations (see foot-note) give for
Prob. 96 the intermediate integral p = (p(z) and hence derive the
solution.
* In Monge's method (for which the reader must be referred to the complete
treatises) of finding an intermediate integral of
Rr + Ss + Tt = V
when one exists, the auxiliary equations
Rdy* - Sdy dx + Tdo* = o, Rdp dy + Tdq dx = Vdx dy
are established. These, in connection with
dz = pdx -f- qdy,
form an incomplete system of ordinary differential equations, between the five
variables x, y, z, p, and q. But if it is possible to obtain two integrals of the
system in the form = a, v = b t u = 0(z>) will be the intermediate integral.
The first of the auxiliary equations is a quadratic giving two values for the ratio
dy.dx. If these are distinct, and an intermediate integral can be found, for
each, the values of p and q determined from them will make dz = pdx -(- qdy
jntegrable, and give the general integral at once.
6G DIFFERENTIAL EQUATIONS.
Prob. 98. Derive by Monge's method for q*r 2pqs -\- p*t = o
the intermediate integral/ = q <fi(z), and thence the general integral.
Ans. y + x<f>(z) =
ART. 25. LINEAR PARTIAL DIFFERENTIAL EQUATIONS.
Equations which are linear with respect to the dependent
variable and its partial derivatives may be treated by a method
analogous to that employed in the case of ordinary differential
equations. We shall consider only the case of two independ-
ent variables x and y, and put
D- D' -
" a*' ~ a/
so that the higher derivatives are denoted by the symbols D*,
DD', D rt , D 3 , etc. Supposing further that the coefficients are
constants, the equation may be written in the form
f(D, D')z =. F(x, y\ (i)
in which f denotes an algebraic function, or polynomial, of
which the degree corresponds to the order of the differential
equation. Understanding by an " integral" of this equation
an explicit value of z in terms of x and y, it is obvious, as in
Art. 1 5, that the sum of a particular integral and the general
integral of
AD, D'}z = o (2)
will constitute an equally general solution of equation (i). It
is, however, only when f(D, D') is a homogeneous function of D
and D' that we can obtain a solution of equation (2) containing
n arbitrary functions,* which solution is also the "comple-
mentary function" for equation (i).
Suppose then the equation to be of the form
.
and let us assume 2 = <p(y -f- mx), (4)
* It is assumed that such a solution constitutes the general integral of an
equation of the th order; for a primitive containing more than independent
arbitrary functions cannot give rise by their elimination to an equation of the
th order.
LINEAR PARTIAL EQUATIONS. 67
where m is a constant to be determined. From equation (4),
Dz = m(p'(y -f- mx) and D 'z = <j>\y -j- mx}, so that Dz = mD'z,
D*z = m*D'*z, DD'z = mD'*y, etc. Substituting in equation (3)
and rejecting the factor D' n z or (f> (n) (y -\- mx), we have
Ajn* + A,m n - 1 -f . . . + A n = o (5)
for the determination of m. If ;,, t 9 , . . . m n are distinct roots
of this equation,
* = <t>i(y + m **) +<P*( y + m **} + + <i>n(y + m n x) (6)
is the general integral of equation (3).
For example, the general integral of ; --., = o is thus
QX oy
found to be z = (f)(y -\- x] -j- $(y *) Any expression of the
form Axy + Bx -f- Cy -\- D is a particular integral ; accordingly
it is expressible as the sum of certain functions of x -\- y and
x y respectively.
The homogeneous equation (3) may now be written sym-
bolically in the form
(D - mjy)(D - m,D'} ...(D- m n D'}z = o, (7)
in which the several factors correspond to the several terms of
the general integral. If- two of the roots of equation (5) are
equal, say, to m lt the corresponding terms in equation (6) are
equivalent to a single arbitrary function. To form the general
integral we need an integral of
(D - m.DJz - o (8)
in addition to 0(j -)- m } x}. This will in fact be the solution of
(D - m,D'}z = <f>(y + ,*); (9)
for, if we operate with D m^D' upon both members of this
equation, we obtain equation (8). Writing equation (9) in the
form
p m,q = (f>(y + mx\
Lagrange's equations are
-V . f . k j
1 0(7 + *!*)
giving the integrals y -f- #*,.* = #, ^ = x<p(a) -}- b. Hence the
integral of equation (9) is
z = x(f)(y -|- m^x) -\- if)(y -\- ntji), (10)
63 DIFFERENTIAL EQUATIONS.
and regarding also as arbitrary, these are the two independ-
ent terms corresponding to the pair of equal roots.
If equation (5) has a pair of imaginary roots m = ^ zV,
the corresponding terms of the integral take the form
which when and ip are real functions contain imaginary
terms. If we restrict ourselves to real integrals we cannot
now say that there are two radically distinct classes of inte-
grals ; but if any real function of y -(- JJLX ~\- ivx be put in the
form X-\-iY, either of the real functions X or Y will be an
integral of the equation. Given, for example, the equation
of which the general integral is
s = 0(7 + "0
to obtain a real integral take either the real or the coefficient
of the imaginary part of any real form of (f>(y -j- ix]. Thus, if
0(/) = # we find e y cosx and e y sin;r, each of which is an
integral (see Chap. VI, p. 245).
As in the corresponding case of ordinary equations, the
particular integral of equation (i) may be made to depend
upon the solution of linear equations of the first order. The
inverse symbol -=? ^^(x, y) in the equation corresponding
to equation (14), Art. 16, denotes the value of z in
(D mD'}z F(x, y) or p mq = F(x, y). (l 1}
For this equation Lagrange's auxiliary equations give
y -j- mx = a, z = / F(x, a mx)dx -f- b = F^(x, a) -f- b,
and the general integral is
mx). (12)
The first term, which is the particular integral, may there-
fore be found by subtracting mx from y in F(x, y}, inte-
LINEAR PARTIAL EQUATIONS. 69
grating with respect to x, and then adding mx to y in the
result.*
For certain forms of F(x, y) there exist more expeditious
methods, of which we shall here only notice that which applies
to the form F(ax -\- by). Since DF(ax -f- by) = aF'(ax -f by)
and D'F(ax -f- by) = bF'(ax -\- by), it is readily inferred that,
when f(D, D') is a homogeneous function of the wth degree,
f(D,D')F(ax + by) = f(a, b)F*\ax + by). (13)
That is, if t ax -\- by, the operation of f(D, D') on F(t) is
equivalent to multiplication by/(a,b) and taking the th de-
rivative, the final result being still a function of /. It follows
that, conversely, the operation of the inverse symbol upon a
function of t is equivalent to dividing by f(a, b) and integrating
n times. Thus,
ff. . .fPWT. (H)
^ y
When ax -(- by is a multiple of y -f- w,-^, where m^ is a root of
equation (5), this method fails with respect to the correspond-
ing symbolic factor, giving rise to an equation of the form (9),
of which the solution is given in equation (10). Given, for ex-
ample, the equation
6*2 d*Z 6*2
or (D D') (D + 2D')z = sin (x y) -f sin (x + y).
The complementary function is <p(y + *") + $(y 2x). The
part of the particular integral arising from sin (x y), in which
a = i, b = i, is - - / / sin tdf = ~ sin (x y). That aris-
* The symbolic form of this theorem is
corresponding to equation (13), Art. 16. The symbol e^xD 1 here indicates the
addition of mx to y in the operand. Accordingly, using the expanded form
of the symbol,
f\ 22 X2
ffttxIJ f ( lA ~~" 1 1 | fttX ** ' ~~i ~^~ ~T~ ) f\ V) "^ f\ V ^~ fttX\
8y 2 ! <5y*
the symbolic expression of Taylor's Theorem.
70 DIFFERENTIAL EQUATIONS.
ing from sin (x -\-y) which is of the form of a term in the com-
plementary function is ^ jr, cos (x + j), which by equa-
o
tion (10) is \x cos (x -\-y). Hence the general integral of
the given equation is
* = 0( y + *) + $(y - 2*}+% sin (x -y)-\x cos (* + j)..
If in the equation f(D, D')z = o the symbol f(D, D'), though
not homogeneous with respect to D and D', can be separated
into factors, the integral is still the sum of those corresponding
to the several symbolic factors. The integral of a factor of
the first degree is found by Lagrange's process; thus that of
(D mD' a)z = o (15)
is z = ^"0( y -\- mx). (16)
But in the general case it is not possible to express the
solution in a form involving arbitrary functions. Let us, how-
ever, assume
*=<***+*, (17)
where c, h, and k are constants. Since De hx+ky = he kx + k:r
and >'<**+**= k<* x + ky , substitution in f'(D,D'}z = o gives
cf(fi, k)e hx ~ lrky = o. Hence we have a solution of the form (17)
whenever h and k satisfy the relation
Ak>k} = o, (18)
c being altogether arbitrary. It is obvious that we may also
write the more general solution
z=2ce** +F <*, (19)
where k = F(h) is derived from equation (18), and c and h admit
of an infinite variety of arbitrary values.
Again, since the difference of any two terms of the form
fkx + FWy w jth different values of h is included in expression
(19), we infer that the derivative of this expression with respect
to h is also an integral, and in like manner the second and
higher derivatives are integrals.
For example, if the equation is
LINEAR PARTIAL EQUATIONS. 71
for which equation (18) is f? k o, we have classes of in-
tegrals of the forms
27)], e** + k (x -f 2/^) s + 6y(x -f-
In particular, putting // = o we obtain the algebraic integrals
Cl x y c&? + 27), ,(*' + 6*y), etc.
The solution of a linear partial differential equation with
variable coefficients may sometimes be effected by a change of
the independent variables as illustrated in some of the exam-
ples below.
Prob. 99. Show that if m l is a triple root the corresponding.
terms of the integral are o?<t>(y + m t x) -\- x$(y-{- w,#)
B u 9 ' 2 9>j5 9 ^
Prob. 100. Solve 2^-^ -- 3^^ -- 2^ a = o.
d* dxdy dy
Prob. ici. Solve ^^ + 2 ^ + - 3 = ^.
Ans. 2 = 0(^) -f- $(x +y)+ xx(x +y) y log *.
Prob. 102. Solve (D 1 + $DD' + 6D n )z = (y - 2 X y\
Ans. z = (f>(y 2 x) + ip(y 3^) + x log (y 2x).
tfz &z $z
Prob. 103. Solve ^-; ^-^- -f ^ -- 2 = 0.
Prob. 104. Show that for an equation of the form (15) the solu-
tion given by equation (19) is equivalent to equation (16).
i tfz i fiz i d*z i dz .
Prob. 105. Solve ^-5 -- 1^- = ^1 -- ;^~ by transpose
x dx x dx y dy y 9 dy
tion to the independent variables x* an
Prob.,06. So,ve,'l
INDEX.
Auxiliary system of lines, 51, 55.
Besselian functions, 44 note.
Bessel's equation, 43.
Boundary (of real solutions), 17.
Characteristics, 59.
Clairaut's equation, 22, 61.
Complementary function, 35, 48.
Complete integral, 2, 30, 57, 60.
Condition of integrability, 50.
Cusp-locus, 17, 20.
Decomposable equations, 13.
Differentiation, solution by, 20.
Direct integration, 2.
Discriminant, 16, 18.
Doubly infinite systems, 26, 31.
Envelope, 15.
Equation of energy, 29, 49.
Equipotential surfaces, 52 note.
Exact differentials, 2, 51.
Exact equations, 6, 27, 30, 36.
Extension of the linear equation, 12.
Finite solutions, 45
First integral, 30, 31.
First order and degree, i, 50.
First order and second degree, 12.
General integral, 53, 57.
Geometrical applications, 23, 31.
Geometrical representation of a differ
ential equation, 3, 13, 15, 26.
Homogeneous equations, 9.
Homogeneous linear equations, 40
Integral, 26, 34, 49, 66.
Integral equation, 26.
Integral surface, 58.
Integrating factors, 8, n, 33.
Jacobians, 55 note.
Lagrange's lines, 55.
Lagrange's solution, 53, 56.
Linear elements of surface, 59 note.
Linear equations, 10, 34, 36.
Linear partial differential equation, 66.
Logarithmic solutions, 44 note, 46 note.
Monge's method, 65, Prob. 97 note.
Monge's solution, 53, Prob. 79
Node-locus, 19.
Non-integrable equation, 51.
Operative symbols, 36, 41.
Order, equations of first, i.
of second, 28.
Orthogonal surfaces, 52 note,
Orthogonal trajectories, 24.
Parameters or arbitrary constants, 4,
15, 26, 31, 60.
Partial differential equations.
first order and degree, 53.
linear, 66. /
second order, 63
Particular integral. 2
determined in series, 46.
of linear equation, 35, 38. 41
Pencil of curves, 14.
Primitive 5 55
Separation of variables. 2, 51.
Series, solutions in, 42,
Simultaneous equations, 25, 47
Singular solutions, 15, 18, 26 note, 59.
Symbolic solutions, 37 et seq., 41, 67.
Systems of curves, 4, 26, 31.
Systems of differential equations, 47.
Tac locus, 1 6
Trajectories, 23
Transcendental and algebraic forms of
solution, 2.
Transformation of linear equations, 46.
Ultimate intersections, 19, 59.
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The treatise on conic sections by the Hellenistic mathematician Apollonius from Perga is regarded as a supreme achievement of Greek mathematics and maintained its authority right up to the 18th century. This new edition is the first to consider all Greek and Arabic sources, with the Arabic texts being presented in the first ever critical... more...
Designed for those seeking help studying calculus in school - also valuable for adults attempting to learn/re-learn calculus. A resource for instructors supplementing their instruction. 501 Calculus Problems helps users prepare for academic exams and build problem-solving skills. Unlike textbooks, full answer explanations are provided for all problems.... more...
Features an introduction to advanced calculus and highlights its inherent concepts from linear algebra Advanced Calculus reflects the unifying role of linear algebra in an effort to smooth readers' transition to advanced mathematics. The book fosters the development of complete theorem-proving skills through abundant exercises while also promoting... more...
Sequences and Their Limits Computing the Limits Definition of the Limit Properties of Limits Monotone Sequences The Number e Cauchy Sequences Limit Superior and Limit Inferior Computing the Limits-Part II
Real Numbers The Axioms of the Set R Consequences of the Completeness Axiom Bolzano-Weierstrass Theorem Some Thoughts about... more...
Your INTEGRAL tool for mastering ADVANCED CALCULUS
Interested in going further in calculus but don't where to begin? No problem! With Advanced Calculus Demystified , there's no limit to how much you will learn.
Beginning with an overview of functions of multiple variables and their graphs, this book covers the fundamentals, without spending... more...
Advanced Calculus of Several Variables provides a conceptual treatment of multivariable calculus. This book emphasizes the interplay of geometry, analysis through linear algebra, and approximation of nonlinear mappings by linear ones. The classical applications and computational methods that are responsible for much of the interest and importance... more...
Designed to help motivate the learning of advanced calculus by demonstrating its relevance in the field of statistics, this successful text features detailed coverage of optimization techniques and their applications in statistics while introducing the reader to approximation theory. The Second Edition provides substantial new coverage of the material,... more... | 677.169 | 1 |
...Since then, I have taken related math classes which expand upon my knowledge, such as abstract algebra and Fourier analysis, although most of my expertise comes from physics. Classes such as quantum mechanics or even classical mechanics make extensive use of the mathematical tools and concepts s | 677.169 | 1 |
An Introduction to the History of Mathematics (Saunders Series)
9780030295584
ISBN:
0030295580
Edition: 6 Pub Date: 1990 Publisher: Cengage Learning
Summary: This classic best-seller by a well-known author introduces mathematics history to math and math education majors. Suggested essay topics and problem studies challenge students. CULTURAL CONNECTIONS sections explain the time and culture in which mathematics developed and evolved. Portraits of mathematicians and material on women in mathematics are of special interest.
Howard Eves is the author of An Introduct...ion to the History of Mathematics (Saunders Series), published 1990 under ISBN 9780030295584 and 0030295580. One hundred forty An Introduction to the History of Mathematics (Saunders Series) textbooks are available for sale on ValoreBooks.com, twelve used from the cheapest price of $27.85, or buy new starting at $311.80 | 677.169 | 1 |
MERLOT Search - category=2545&sort.property=overallRating
A search of MERLOT materialsCopyright 1997-2015 MERLOT. All rights reserved.Tue, 30 Jun 2015 08:18:45 PDTTue, 30 Jun 2015 08:18:45 PDTMERLOT Search - category=2545&sort.property=overallRating
4434MathPagesMi tarea
This is an exceptional site for locating all types of content material in creating modules. The are links to the following disciplines: Sciences, History, Art and Culture, Humanities, and general resources. On several of the links, one can find audio files, e.g., Christmas carols. While this site is designed for native speakers in middle school or a secondary level, it is quite appropriate for Spanish language students having an intermediate language proficiency or higher in secondary or college courses.Applied Discrete Structures
Applied Discrete Structures by Al Doerr and Ken Levasseur is a free open content textbook in discrete mathematics. Originally published in 1984 & 1989 by Pearson, the book has been updated to include references to Mathematica and Sage, the open source computer algebra system. Contents:Front Matter: Contents and IntroductionChapter 1: Set Theory I Chapter 2: Combinatorics Chapter 3: Logic Chapter 4: More on Sets Chapter 5: Introduction to Matrix Algebra Chapter 6: Relations and Graphs Chapter 7: Functions Chapter 8: Recursion and Recurrence Relations Chapter 9: Graph Theory Chapter 10: Trees Chapter 11: Algebraic Systems Chapter 12: More Matrix Algebra Chapter 13: Boolean Algebra Chapter 14: Monoids and Automata Chapter 15: Group Theory and Applications Chapter 16: An Introduction to Rings and Fields Solutions to Odd-Numbered Exercises Combinatorial Math: How to Count Without Counting
A collection of JavaScript for computing permutations and combinations counting with or without repetitions.Diamond Theory
Symmetry properties of the 4x4 array. The invariance of symmetry displayed in the author's Diamond 16 Puzzle (online) suggests insights into finite geometry, group theory, and combinatorics.Erdös-Ko-Rado theorems
This video was recorded at 6th Slovenian International Conference on Graph Theory, Bled 2007. I will show that this theorem has a natural proof using linear algebra, and that this approach also applies to situations where sets are replaced by objects such as subspaces, permutations or partitions.Minimax Policies for Combinatorial Prediction Games
This video was recorded at 24th Annual Conference on Learning Theory (COLT), Budapest 2011. We address the online linear optimization problem when the actions of the forecaster are represented by binary vectors. Our goal is to understand the magnitude of the minimax regret for the worst possible set of actions. We study the problem under three different assumptions for the feedback: full information, and the partial information models of the so-called "semi-bandit״, and "bandit" problems. We consider both L∞-, and L2-type of restrictions for the losses assigned by the adversary. We formulate a general strategy using Bregman projections on top of a potential-based gradient descent, which generalizes the ones studied in the series of papers György et al. (2007), Dani et al. (2008), Abernethy et al. (2008), Cesa-Bianchi and Lugosi (2009), Helmbold and Warmuth (2009), Koolen et al. (2010), Uchiya et al. (2010), Kale et al. (2010) and Audibert and Bubeck (2010). We provide simple proofs that recover most of the previous results. We propose new upper bounds for the semi-bandit game. Moreover we derive lower bounds for all three feedback assumptions. With the only exception of the bandit game, the upper and lower bounds are tight, up to a constant factor. Finally, we answer a question asked by Koolen et al. (2010) by showing that the exponentially weighted average forecaster is suboptimal against L∞ adversaries.Mudd Math Fun Facts
This archive is designed as a resource for enriching your courses with mathematical Fun Facts! It is designed to pique the interest of students in different areas of mathematics. The fun facts were originally conceived as five minute warm ups at the beginning of lectures so that non mathematics majors would not think math was just calculus. Presentation suggestions are also given.Polyforms
Background and description of polyform sets as a new puzzle genre.The Diamond 16 Puzzle
In solving this puzzle, you permute rows, columns, and quadrants in a 4x4 array of 2-color tiles to make a variety of symmetric designs. A link to underlying theory is provided. | 677.169 | 1 |
Mathematical Problem Solving
Solving problems is an important role in mathematics. Mathematics teaching and learning is aimed at developing children's ability to solve various mathematics problems.
There are various techniques for solving mathematical problems and their application vary from one mathematical problem to the other. The key to successfully solving mathematical problems is flexibility and creativity. One should be able to change methodology if it is not effective for a mathematical problem.
Brainstorming techniques are quite useful in mathematical problem solving. Important information relevant for finding a solution to a problem has to be separated from the rest. Knowledge of techniques in algebra and geometry must be gained, as well as acquaintance with statistical ideas, logic and logical concepts, number theory and arithmetic basic operations. A deep understanding of binary number systems, counting techniques, linear programming techniques and matrix algebra are also crucial for solving mathematical problems. Important techniques are also probability laws, calculus ideas such as differentiation, integrals and different equation solving techniques, complex numbers, probability distributions and solid geometrical concepts and ideas.
After gaining knowledge, suitable approaches to problem solving must be selected, along with techniques which can help solve the problem fast and save time. Different techniques have not the same effectiveness and one must find the most effective. The intensive practice of problem solving makes a person able to quickly choose the technique in case of an examination. The exercises speed up the cognitive retrieval and one is able to quickly choose the most suitable technique.
The ability to solve mathematical problems can be developed only though the continual practice of problem solving. Some textbooks in the United States offer linear models for solving mathematical problems. These models involve reading, deciding, solving and examining the result. However, such models are often inconsistent and cannot be practically used. Linear models do not encourage students to think. Such traditional models present problem solving as a series of steps and a process that has to be remembered. One has to be able to pose or formulate a problem in order to solve it. This area has been subject to studies in the United States in recent years.
Garofola and Lester suggest in their article, Metacognition, cognitive monitoring, and mathematical performance,that problem solving is a series of processes of which students are unaware. Students with a good knowledge base are more able to use heuristics in solving geometry problems. The successful problem solvers are using mathematical structure similarities to categorize math problems.
Algorithms are important in solving math problems and they must be developed. However, using an algorithm is still not problem solving. The process of creating an algorithm is problem solving. Problem solving can be taught by teaching children to create their algorithms.
A useful method in mathematics is also heuristics. These are types of information that help students find a solution to a math problem. Heuristics are similar to strategies, techniques, and rules-of-thumb. These techniques are not very valuable out of the problem context, but in doing mathematics they are very helpful. Heuristics are considered very important in various theories in mathematics. However, despite the use of explicit instructions and heuristics, problem solving is not that simple as such a simple analysis is limited. Theories must include classroom contexts, past knowledge and experience. Several studies have been focused on the research of heuristic processes. These studies found that task specific heuristic instruction was more effective than general heuristic instruction. The heuristic of subgoal generation also enables students to make problem solving plans. Thinking aloud, acting as a teacher and direct instruction enhanced the development of students' abilities to generate subgoals.
In the past, problem solving research involving technology has often dealt with programming as a major focus. This research has often provided inconclusive results. Indeed, the development of a computer program to perform a mathematical task can be a challenging mathematical problem and can enhance the programmer's understanding of the mathematics being used. Too often, however, the focus is on programming skills rather than on using programming to solve mathematics problems. There is a place for programming within mathematics study, but the focus ought to be on the mathematics problems and the use of the computer as a tool for mathematics problem solving. Further basic mathematical problem solving techniques are the use of graphs, patterns and logic concepts. Counting strategies such as tree diagram, binomial expansion and pascal triangle can also enhance problem solving. Trigonometry is used for finding relationships between angles and sides, along with properties waves and simple harmonic motion, as well as projectile motion and circular motion and elliptical motions. | 677.169 | 1 |
ISBN-10:
0131444425
Publication Year:
2004
ISBN-13:
9780131444423
Language:
English
Author:
K. Elayn Martin-Gay
Product Type:
Textbook
Format:
Trade Cloth
ISBN:
9780131444423
Detailed item info
Synopsis
Elayn Martin-Gay's success as a developmental math author starts with a strong focus on mastering the basics through well-written explanations, innovative pedagogy and a meaningful, integrated program of learning resources. The revisions to this edition provide new pedagogy and resources to build reader confidence and help readers develop basic skills and understand concepts. Martin-Gay's 4-step problem solving process-Understand, Translate, Solve and Interpret-is integrated throughout. Also includes new features such as Study Skills Reminders, "Integrated Reviews", and "Concept Checks." For readers interested in learning or revisiting essential skills in beginning and intermediate algebra through the use of lively and up-to-date applications | 677.169 | 1 |
Use this savings calculator to see how a consistent approach to investing can make your money grow. Whether saving for a house, a car, or other special purchase, the savings calculator will help you d... More: lessons, discussions, ratings, reviews,...
This is a full lesson that gives students experience with exponential functions in an application format. There is a movie, an applet for gathering information, and a graphing applet. There is a les... More: lessons, discussions, ratings, reviews,...
The "Exponential Decay" Java applet was used on the discussion board with an online Algebra II course. Students were asked to use the applet, and then go to the discussion board and discuss the questi... More: lessons, discussions, ratings, reviews,...
Explore examples of exponential growth and decay, bounded growth, and carrying capacity in this simulation of the effects of birth and death rates and food supply on a deer population. Includes questi... More: lessons, discussions, ratings, reviews,...
The applet below is a simulation of the law of radioactive decay. The red circles of this simulation symbolize 1000 atomic nuclei of a radioactive substance whose half-life period T1/2 amounts to 20 s... More: lessons, discussions, ratings, reviews,...
Americans today owe more money than ever before. The fact that 'interest never sleeps' means that the situation will continue to worsen unless steps are taken at the individual level to reduce or elim... More: lessons, discussions, ratings, reviews,...
This set of problems consists of a handful of randomly selected graphs for which the user is to find a Hamiltonian circuit or trail (i.e., a walk in the graph that uses every vertex exactly once) if i... More: lessons, discussions, ratings, reviews,...
The user will create their own set of data and then see how changes can occur in the mean, median, and mode and the shape of the histogram when changes are made in the data set in this activity. More: lessons, discussions, ratings, reviews,...
This flash allows the student to load and modify six existing datasets. The flash calculates the mean, median, and quartiles, and plots a histogram. Students are encouraged to examine the effects of...Explore how the parameters in a rational equation affect the graph of the equation. Dynamically change the parameters and immediately see how the graph changes. Undefined points are clearly visible....This has a full explanation of how to graph a hyperbola from a given equation. If you scroll down to the middle of the page, there is a flash movie that takes you through the graphing process step by... More: lessons, discussions, ratings, reviews,...
Flash introduction to finding the equation of an hyperbola centered on (0,0) and with its major axis on the x-axis. With step-by-step instructions and an illustrated glossary, students can learn how t... More: lessons, discussions, ratings, reviews,...
Flash introduction to finding the equation of an hyperbola centered on (0,0) and with its major axis on the y-axis. With step-by-step instruction and an illustrated glossary, students can learn how to | 677.169 | 1 |
High School Algebra & Functions Worksheets and Printables
These high school algebra & functions worksheets help make learning engaging for your high schooler! Covering a wide range of topics, we have many high school algebra & functions worksheets available to help supplement your child's education.
These high school algebra & functions worksheets help make learning engaging for your high schooler! Covering a wide range of topics, we have many high school algebra & functions worksheets available to help supplement your child's education. | 677.169 | 1 |
Included: (Click to preview)
About this Series
Lessons: 3
Total Time: 0h 19m
Use: Watch Online & Download
Access Period: Unlimited
Created At: 11/20/2009
Last Updated At: 11/20/2009
These three algebra lessons will teach you how to approach and apply exponential functions. First, you'll learn to use properties of exponents to solve exponential equations. Then, you'll learn how to find present value and future value, two common problems involving exponents. Last, we'll look at finding interest rates to match given value goals. These types of problems also involve work with exponentsLessons Included
A very clear, concise and understandable explanation of the problem. If only my textbook offered examples this digestible!
Below are the descriptions for each of the lessons included in the
series:
College Algebra: Using Exponent Properties
In this lesson, we'll examine how to solve exponential equations. These are equations in which the unknown variable (usually x) is found in the exponent (like 2^x = 4). One approach to this involves making the bases equivalent on both sides of the equation (given that this will mandate that the exponents are equivalent). Other equations we'll solve in this lesson include 8^x = 2, 8^x = 4, (1/3)^x = 27, 3^(-x)=27, and x^(1/3)=27.
This lesson is perfect for review for a CLEP test, mid-term, final, summer school, or personal growth!
Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, College Algebra. This course and others are available from Thinkwell, Inc. The full course can be found at Present Value & Future Value Find Interest Rate to Match GoalsSupplementary Files:
Once you purchase this series you will have access to these files:
Buy Now and Start Learning
Buy this series to watch it immediately. View it as many times as you
need and download it to your computer or iPod (most lessons).
Satisfaction's guaranteed, so go get started! | 677.169 | 1 |
Dr. Chapin is a professor of mathematics education at Boston University, where she teaches graduate- and undergraduate-level courses in mathematics curriculum, mathematics content and methodology, mathematics for special-needs students (gifted and learning disabled), and educational reform. Dr. Chapin has been the principal investigator on a number of projects, including Project Challenge (Jacob K. Javits Gifted & Talented Students Education Program) and Partners in Change Project (U.S. DOE), and has worked with many colleagues as an investigator or consultant on grant-related research. Dr. Chapin is author or co-author of many mathematics textbooks and programs, including the Prentice Hall textbook series Middle Grade Mathematics: Tools for Success for students in grades 6-8; MEGA Projects: Math Explorations and Group Activities for students in grades 1-8; Algebra and Advanced Algebra for students in grades 8-12; and Classroom Discussions: Using Math Talk To Help Students Learn. She is also the senior author on Math Matters: Understanding the Math You Teach, Grades K-6. Dr. Chapin is a frequent speaker at national meetings of mathematicians, mathematics educators, researchers, and policy makers. | 677.169 | 1 |
6: Real Numbers to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Pre-Algebra Module 6: Real Numbers
Select this link to open drop down to add material Pre-Algebra Module 6: Real Numbers to your Bookmark Collection or Course ePortfolio
Module 7 of Pre-Algebra Course. Students will demonstrate proficiency in using the metric system of measurement and will...
see more
Module 7 of Pre-Algebra Course. Students will demonstrate proficiency in using the metric system of measurement and will correctly solve applications using proportions, conversions, and direct and indirect variation 7: Measurement and Proportion to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Pre-Algebra Module 7: Measurement and Proportion
Select this link to open drop down to add material Pre-Algebra Module 7: Measurement and Proportion 8: Percent to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Pre-Algebra Module 8: Percent
Select this link to open drop down to add material Pre-Algebra Module 8: Percent Physics to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Thermal Physics
Select this link to open drop down to add material Thermal Static Electricity to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Static Electricity
Select this link to open drop down to add material Static Electricity Electricity to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Current Electricity
Select this link to open drop down to add material Current Electricity to your Bookmark Collection or Course ePortfolio
This is a tutorial about the wavelike behaviors of light. The topics included the following:Lesson 1 - How Do We Know Light...
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This is a tutorial about the wavelike behaviors of light. The topics included the following:Lesson 1 - How Do We Know Light is a Wave?Lesson 2 - Color and VisionLesson 3 - Mathematics of Two-Point Source Interference Waves and Color to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Light Waves and Color
Select this link to open drop down to add material Light Waves and Color to your Bookmark Collection or Course ePortfolio | 677.169 | 1 |
More on fractions, percents, decimals, probability and graphs--plus statistics, inequalities and geometry. Also includes "real-world" math such as following recipes and paying taxes.
Students watch 115 animated lectures on CD-ROM, do problems in the 588-page workbook (consumable) and enter their answers on your computer--which records their grade automatically. When they need help or review, you've got a printed Answer Key plus audiovisual step-by-step solutions to every homework and quiz problem.
Teaching Textbooks offers a money-back guarantee for thirty (30) days from the date of purchase. | 677.169 | 1 |
Quick Review Math Handbook hot words hot topics
9780078601262
ISBN:
0078601266
Pub Date: 2004 Publisher: McGraw-Hill Higher Education
Summary: "Quick Review Math Handbook: Hot Words, Hot Topics" (available in English and Spanish) provides students and parents with a comprehensive reference of important mathematical terms and concepts to help them build their mathematics literacy. This handbook also includes short-instruction and practice of key standards for Middle School and High School success.
Glencoe McGraw-Hill Staff is the author of Quick Rev...iew Math Handbook hot words hot topics, published 2004 under ISBN 9780078601262 and 0078601266. Three hundred forty six Quick Review Math Handbook hot words hot topics textbooks are available for sale on ValoreBooks.com, two hundred eighty seven used from the cheapest price of $0.01, or buy new starting at $2 | 677.169 | 1 |
Sharing great ideas and resources with maths teachers around the world
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"What's the point of algebra?"
How many times have you been asked "what's the point of algebra?"? I feel like I have never had a good answer to this question that is relavent to the kids and their lives. Let's find out what Great Maths Teaching Ideas' readers think! Do you have a good answer to this question that will satisfy the kids? If so, please share it with us in the comments section below!
Great question William! I think it's something that every maths teacher faces. Once the language of algebra is understood, students can begin to apply to a host of interesting problems e.g. comparisons of phone bills, look at rates of change, in geometry, etc.
A little while ago I wrote a blog contemplating the issue. In the end, I think that we should not limit what our students are learning because it is difficult for some. Learning is not intended to be easy!
For me, algebra is all about equations, and equations are about relationships.
If you want to look at the relationship between quantities, cause and effect, variables, parameters or change, you're going to use algebra.
If you've ever wondered how something would change if you fiddled with this or that, you're going to use algebra.
What if it's on sale? What if I increase my data usage? What if I talk less and text more?…all algebra.
One simple, relatable answer I've been giving kids lately is that they might need to be a spreadsheet manipulator at some point. I tell my classes, "I know that only about half of you will end up majoring in math." (The top level kids, who will major in math/science, laugh; the lower-level kids don't get my veiled sarcasm and truly believe that half of them will be math majors – the half that does NOT include them as individuals, of course. "But MANY of you will be responsible for working with formulas at some point, and it's much better to be the person who enters a formula correctly OR recognizes a bad formula than to be the person who enters a bad formula." They seem to understand that, and in fact a lot of them say that their parents are database or spreadsheet users at work.
I generally talk about the fact that we are trying to find out something we don't know ,similar to Laurence. we don't know if Mars can accommodate life, we don't know if global warming will flood the planet, we don't know if there will be an Ice age ever again, and we don't know the value of x, or do we….?
I like this answer & Laurence's. At age 40 something I'm having to learn basic algebra in prep for a psychometric test & I began to wonder, whats the point of algebra. The idea that its a method to quantify the unknown intrigues me & gives me that bit of extra motivation to wrestle with it!
This is an ongoing theme with my y11 set 5.
They ask at least once a lesson when studying any algebraic topic, it has become like a 'read us a story' request to settle them before the 'bedtime' of something they wont really enjoy.
My response is threefold:
Some students will actually need to manipulate expressions and equations in spreadaheets, cashflow, planning for unknown situation where you dont know quantities in advance, some element of abstract rather than concrete thinking etc
Most employers will expect staff to be able to learn and follow routines and procedures in a logical manner, not with numbers or algebra, but these are the skills that we are really learning when we learn algebra.
Thirdly, the one that I am less pleased about rekating to them, as I believe the first two should be sufficient, is that you can get a better GCSE grade if you can understand and do more algebra!
I think we should teach algebra through projects and questions that are relevant to our students today — not years in the future. What are your students interested in? I've come up with a couple of ideas/projects/questions that provide a context for teaching algebra. Plot the growth of YouTube use. Then create an equation from the data and make a prediction about YouTube use in the future. Create an equation representing Justin Bieber's number of twitter followers and how fast that number is increasing. Is he going to over take Lady Gaga as the twitter account with the most followers? Create a project about owning/buying a car (lots of teenagers are interested in driving). I don't think it's enough to tell students that they might use algebra in the future. Many of them won't have to use algebra for their careers. Use algebra now, in class, to study problems and questions that the kids are interested in TODAY.
I always tell them it's like, "I'm thinking of a number," and then I give them a clue as to what the number is, like, "If I double the number and subtract 1, I get the same thing as if I just added 3 to the original number." We start by guessing, but then things get difficult because I start "thinking of a number" like 2.5, which they might not guess right away. The point is that whenever faced with a problem, you don't have all the information, and you have to use the information you do have in order to figure it out.
There's a lot of answers to this:
1) it's a good place to practice reasoning and problem solving
2) the content is powerful and has become the language of quantitative thought. It's the generalization of what we know about numbers and how they relate to each other. It has become a language in which we can solve problems in other areas of math. How big, how far, how fast, how long, how many… and why.
3) because of that, those who learn it have academic power and can choose from a wider variety of careers and vocations. I want my students to have that power.
4) it's beautiful. We started by asking concrete questions about number and wound up with a discipline filled with symmetry and amazement, structures that surprise and confound us. As a bonus, after we find these things in algebra, sometimes we get to recognize these same structures in nature. It's beautiful AND real!
My son has Diabetes Type 1. We use Algebra every day. The insulin producing cells in his pancreas were attacked and killed by his own immune system. When he eats we use algebra. If 30 weight grams of goldfish crackers has 20 grams of carbs, but there's only 20 weight grams of crackers left in the bag, then how many grams of carbs is that? 30/20 = 20/x 30x=20×20 30x=400 30x/30=400/30 x=13.3 There are about 13 grams of carbs in that serving of crackers. Now that he has an insulin pump, the pump does other calculations for us, for instance, how many units of insulin he needs if he gets 1 unit for 50 grams of carbs. For those crackers: 1unit/50carbs=Xunits/13 carbs Can you do the math?
I tell mine that they do Algebra all the time in their heads. Ie Coles has a deal that 3 chocolates cost $2.50, how much is each chocolate (3x=2.50). Algebra let's us solve harder problems that we can't do in our heads. It is exercise for our brains!! | 677.169 | 1 |
Elementary Education
Mathematics for Elementary Teaching I
Class Level: Junior
Credits: 2
Department: Education
Term:
Description: This course is the first foundational course in the mathematics content area for elementary education majors. It includes problem solving, sets, functions, exploration of our number system including properties, place value, basic operations and algorithms, and basic concepts of algebra. Problem solving is stressed in each unit. The NCTM Principles and Standards and Indiana's Academic Standards for Mathematics are introduced. Prerequisite: LA 103. Taken concurrently with EDE 336, EDE 344, EDE 365, and EFE 384. Additional prerequisite: 2.50 GPA (A = 4.0) and admission to the teacher education program. Fall, junior year | 677.169 | 1 |
Details
ITEM#: 13456810
W. Michael Kelley is an award-winning former math teacher and author of seven math books. His method of making intimidating math topics very approachable, even humorous, has helped students and adults alike conquer their fear of numbers.
Soon math problems will be no problem at all...
Most math and study guides are as dry and difficult as the professors that write them. In The Humongous Book of Basic Math and Pre-Algebra Problems, author W. Michael Kelley enjoys being the exception. It is full of solved problems, but along the margin Kelley makes notes, adding missing steps and simplifying concepts. In this way questions that would normally baffle students suddenly become crystal clear. His unique method fully prepares students to solve those difficult, obscure problems that were never covered in class but always seem to find their way onto exams.
Annotated notes throughout the book to clarify each problem
An expert author on the topic with a great track record for helping students and math enthusiasts
Author's website calculus-help.com reaches thousands of students every month | 677.169 | 1 |
Practice makes perfect! Get perfect with a thousand and one practice problems! 1,001 Geometry Practice Problems For Dummies gives you 1,001 opportunities to practice solving problems that deal with core geometry topics, such as points, lines, angles, and planes, as well as area and volume of shapes. You'll also find practice problems on more... more...
This first complete English language edition of Euclides vindicatus presents a corrected and revised edition of the classical English translation of Saccheri's text by G.B. Halsted. It is complemented with a historical introduction on the geometrical environment of the time and a detailed commentary that helps to understand the aims and subtleties... more...
First summary of research in the field of applications of hyperbolic geometry to solve theoretical physics problems Clearly written and well presented Provides an extensive list of relevant literature more...
This volume presents an accessible, self-contained survey of topics in Euclidean and non-Euclidean geometry. It includes plentiful illustrations and exercises in support of the thoroughly worked-out proofs. The author's emphasis on the connections between Euclidean and non-Euclidean geometry unifies the range of topics covered.
The text opens with... more...
A Programmer's Geometry provides a guide in programming geometric shapes. The book presents formulas and examples of computer representation and coding of geometry. Each of the nine chapters of the text deals with the representation and solution of a specific geometrical problem, such as areas, vectors, and volumes. The last chapter provides a brief... more...
Geometric Transformations, Volume 2: Projective Transformations focuses on collinearity-preserving transformations of the projective plane. The book first offers information on projective transformations, as well as the concept of a projective plane, definition of a projective mapping, fundamental theorems on projective transformations, cross ratio,... more...
Generalized Functions, Volume 5: Integral Geometry and Representation Theory is devoted to the theory of representations, focusing on the group of two-dimensional complex matrices of determinant one. This book emphasizes that the theory of representations is a good example of the use of algebraic and geometric methods in functional analysis, in which... more...
Computational Geometry: Curve and Surface Modeling provides information pertinent to the fundamental aspects of computational geometry. This book discusses the geometric properties of parametric polynomial curves by using the theory of affine invariants for algebraic curves. Organized into eight chapters, this book begins with an overview of the objects... more... | 677.169 | 1 |
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ECE 3250 Mathematics of Signal and System Analysis
Course description
Course aims to deepen students⿿ working knowledge of mathematical tools relevant to ECE applications. While the course emphasizes fundamentals, it also provides an ECE context for the topics it covers, which include foundational material about sets and functions; linear algebra; inner products and orthogonal representations; basic ideas from multivariable calculus; and elementary convex analysis.
Outcome 1: Deepen their understanding of fundamental concepts from real analysis and linear algebra to which they have been exposed in their calculus and differential equation courses by putting them to work in an engineering context.
Outcome 2: Achieve a sophisticated understanding of fundamental signals and systems concepts, a few of which they have been exposed to on an elementary level in ECE 2200, by learning the mathematics behind them.
Outcome 3: Attain an appreciation of the central role that advanced mathematics plays in modeling, analysis, and design of engineering systems.
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The Cornell University School of Continuing Education and Summer Sessions strives to offer valuable educational opportunities in many formats for any person, in any study, at any time, and in any place. | 677.169 | 1 |
"...a very appealing and valuable book for independent study or as a companion book in advanced probability theory courses. Highly recommended." Choice "This is an excellent book, which should be in every library, … see full wiki
An excellent tool for self-study for the student with the proper background
This set of solved problems involves measure theory and probability and the level of difficulty is that of the Ph. D. student. The problems delve deeply into the theory of probability, independence, Gaussian variables, distributed computations and random processes. There are approximately 100 problems and nearly complete solutions to all of them are included. There is a statement on the back cover that many of the problems can lead the student on to research topics in probability and I fully agree with that. The chapter headings are: | 677.169 | 1 |
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Department of Mathematics
Laurier's Department of Mathematics offers a wide range of courses mathematics, applied mathematics, financial mathematics and statistics. These topics represent dynamic, evolving areas of study with many fascinating applications. Mathematics and statistics are often used to solve problems from the sciences, arts, finance, economics and business. Once put within a mathematical framework, real-world problems and processes can be studied and solved in an efficient and reliable manner. To understand our modern world, a solid grasp of the fundamental methods of mathematics is essential!
Mathematics Research
With facilities in the Bricker Academic and Science Research Buildings, our faculty, graduate students and postdoctoral fellows conduct research on:
Mathematical modeling
Financial mathematics
Mathematics education
Lie algebras
Game theory
Geometry and Topology
Optimization and Graph theory
Number theory
Dynamical systems and Differential equations
Mathematical biology and physics
Mathematical and applied statistics
Environmetrics
Monte Carlo methods and Stochastic processes
The department offers an Honours Seminar course, MA489, where undergraduate students have the opportunity to be directly involved in a research project with faculty members.
New Math Building
A new, $103-million building, scheduled to open in 2015, will meet the growing demand of Laurier's business and math programs, and expand our ability to deliver you integrated and engaged learning opportunities at the local and global levels.
Housing Laurier's School of Business and Economics together with the Department of Mathematics will enhance the synergies between Laurier's business and applied financial math programs, and serve as an iconic representation of the leadership role Laurier plays in Canadian business and Waterloo's technology industry.
Mathematics Assistance Centre
The Centre provides two categories of services: diagnostic and review, and course support. The first is an individual program that refreshes your knowledge in fundamental math skills, prerequisite to a university course. The second is supplemental instruction for particular courses, and includes support such as homework sessions, mock tests and exam reviews.
The MS2Discovery Interdisciplinary Research Institute
The Department is an active player in the research program and activities of the MS2Discovery Interdisciplinary Research Institute. Through the Institute, our students have access to a number of interdisciplinary graduate programs, linking mathematics with other disciplines.
Professor Roderick Melnik, who holds a Tier 1 NSERC Canada Research Chair in Mathematical Modeling, leads the Institute. He and his colleagues also organize seminar and conference series that provide a forum for academics, students, and business and industry professionals to exchange ideas and develop interdisciplinary collaboration. Melnik is head of the Laboratory of Mathematical Modelling, which hosts a number of postdoctoral fellows working in the department.
Business and Financial Mathematics Association
Laurier's Business and Financial Mathematics Association hosts a number of workshops, seminars and events to engage you in learning from the experience of others. The association's flagship event is a Financial Careers Night, which features panelists from the financial and insurance sectors.
Calculus Preparation Evaluation
Are you required to take calculus as part of your program at Laurier? Are you thinking of taking a calculus course as an elective?
Our Calculus Preparation Evaluation is compulsory for all students who intend to take an introductory-level calculus course, including MA100, MA103, MA110* and MA129.
The purpose is to maximize your chance for successful completion of your calculus requirement, by:
Deciding which course is best suited to both your skills and your future goals.
Preparing you for introductory calculus by reviewing key prerequisite material. | 677.169 | 1 |
״At first blush one might think that of all areas of mathematics certainly arithmetic should be the simplest, but it is a...
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״At first blush one might think that of all areas of mathematics certainly arithmetic should be the simplest, but it is a surprisingly deep subject. We assume that students have some familiarity with basic set theory, and calculus. But very little of this nature will be needed. To a great extent the book is self-contained. It requires only a certain amount of mathematical maturity. And, hopefully, the student's level of mathematical maturity will increase as the course progresses. Before the course is over students will be introduced to the symbolic programming language Maple which is an excellent tool for exploring number theoretic questions Number Theory to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Elementary Number Theory
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Euclid's Elements form one of the most beautiful and influential works of science in the history of humankind. Its beauty...
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Euclid's Elements form one of the most beautiful and influential works of science in the history of humankind. Its beauty lies in its logical development of geometry and other branches of mathematics. It has influenced all branches of science but none so much as mathematics and the exact sciences. The Elements have been studied 24 centuries in many languages starting, of course, in the original Greek, then in Arabic, Latin, and many modern languages.The text of all 13 Books is complete, and all of the figures are illustrated using the Geometry Applet, even those in the last three books on solid geometry that are three-dimensional's Elements to your Bookmark Collection or Course ePortfolio
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The materials here form a textbook for a course in mathematical probability and statistics for computer science students.״Why...
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The materials here form a textbook for a course in mathematical probability and statistics for computer science students.״Why is this course different from all other courses?״ * Computer science examples are used throughout, in areas such as: computer networks; data and text mining; computer security; remote sensing; computer performance evaluation; software engineering; data management; etc. * The R statistical/data manipulation language is used throughout. Since this is a computer science audience, a greater sophitication in programming can be assumed. It is recommended that my R tutorial, R for Programmers, be used as a supplement. * Throughout the units, mathematical theory and applications are interwoven, with a strong emphasis on modeling: What do probabilistic models really mean, in real-life terms? How does one choose a model? How do we assess the practical usefulness of models? * There is considerable discussion of the intuition involving probabilistic concepts. However, all models and so on are described precisely in terms of random variables and distributions.For topical coverage, see the book's detailed table of contents Algorithms to Z-Scores: Probabilistic and Statistical Modeling in Computer Science to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material From Algorithms to Z-Scores: Probabilistic and Statistical Modeling in Computer Science
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This OPEN TEXTBOOK is for Politically-Oriented Web-Enhanced Research Methods for Undergraduates Topics and Tools: Resources...
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This OPEN TEXTBOOK is for Politically-Oriented Web-Enhanced Research Methods for Undergraduates Topics and Tools: Resources for introductory research methods courses in political science and related disciplines The POWERMUTT Project is a cross between an introductory political science research methods textbook and an online resource for teaching and learning such methods. It includes: Topics. Each topic is equivalent to a short chapter in a traditional textbook. Tools. These are brief step-by-step tutorials for carrying out specific techniques. At present, the Tools described are some of those found in SPSS, a leading software package for statistical analysis. Datasets and codebooks. Data, and codebooks describing them, on public opinion, the American states, the U.S. Congress, and the countries of the world. Links to other sites providing additional information about research methods. Compared to traditional textbooks, POWERMUTT offers several important advantages, including: Flexibility. Your instructor may have decided to adopt the entire POWERMUTT site as the main course "textbook," or to use just a small portion of the site's resources as supplementary material. Interactivity. Want to see exactly how a table or graph was generated? With POWERMUTT PUPs (Pop Up Protocols), the answer is just a click away. Just as close is additional information on other resources within POWERMUTT or elsewhere on the Web. Affordability. In fact, it's free! However, your instructor may ask you to purchase hard copy of all or part of POWERMUTT for a nominal cost at your campus copy center. While most of the materials in the project are for reference, some, especially the Topics, need to be studied carefully. A highlighter will really mess up your monitor. You can save money by directly downloading the Topics and printing them at home drop down to add material Introduction to Research Methods in Political Science: to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Introduction to Research Methods in Political Science:
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From the preface: "This is a book on linear algebra and matrix theory. While it is self-contained, it will work best for...
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From the preface: "This is a book on linear algebra and matrix theory. While it is self-contained, it will work best for those who have already had some exposure to linear algebra. It is also assumed that the reader has had calculus. Some optional topics require more analysis than this, however." A solutions manual to the exercises in the textbook is Algebra, Theory and Applications to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Linear Algebra, Theory and Applications
Select this link to open drop down to add material Linear Algebra, Theory and Applications MAPLE-based textbook for probability and statistics to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material MAPLE-based textbook for probability and statistics
Select this link to open drop down to add material MAPLE-based textbook for probability and statistics to your Bookmark Collection or Course ePortfolio
'Mathematical Reasoning: Writing and Proof is designed to be a text for the first course in the college mathematics curriculum...
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'Mathematical Reasoning: Writing and Proof is designed to be a text for the first course in the college mathematics curriculum that introduces students to the pro-cesses of constructing and writing proofs and focuses on the formal development of mathematics. The primary goals of the text are as follows:To help students learn how to read and understand mathematical definitions and proofs;To help students learn how to construct mathematical proofs;To help students learn how to write mathematical proofs according to ac-cepted guidelines so that their work and reasoning can be understood by others; andTo provide students with material that will be needed for their further study of mathematics Mathematical Reasoning: Writing and Proof to your Bookmark Collection or Course ePortfolio
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According to The Orange Grove, "This book covers the following: Foundations of Trigonometry, Angles and their Measure,...
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According to The Orange Grove, "This book covers the following: Foundations of Trigonometry, Angles and their Measure, Applications of Radian Measure, Cosine and Sine, Beyond the Unit Circle, The Six Circular Functions and Fundamental Identities, Beyond the Unit Circle, Trigonometric Identities, Graphs of the Trigonometric Functions, Graphs of the Cosine and Sine Functions, Graphs of the Secant and Cosecant Functions, Graphs of the Tangent and Cotangent Functions, The Inverse Trigonometric Functions, Inverses of Secant and Cosecant: Trigonometry Friendly Approach, Inverses of Secant and Cosecant: Calculus Friendly Approach, Using a Calculator to Approximate Inverse Function Values, Solving Equations Using the Inverse Trigonometric Functions, Trigonometric Equations and Inequalities, Applications of Trigonometry, Applications of Sinusoids, Harmonic Motion, The Law of Sines, The Law of Cosines, Polar Coordinates, Graphs of Polar Equations, Hooked on Conics Again, Rotation of Axes, The Polar Form of Conics, Polar Form of Complex Numbers, Vectors, The Dot Product and Projection, and Parametric Equations. In addition, exercises and answers are provided for the reader.Most questions from this textbook are available in WebAssign. The online questions are identical to the textbook questions except for minor wording changes necessary for Web use. Whenever possible, variables, numbers, or words have been randomized so that each student receives a unique version of the question. This list is updated nightlycalculus to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Precalculus
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This is a free, open textbook that is part of the Connexions collection at Rice University. The book includes the following...
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This is a free, open textbook that is part of the Connexions collection at Rice University. The book includes the following topics: 1. Sampling and Data 2. Descriptive Statistics 3. The Normal Distribution 4. Confidence Interval 5. Hypothesis Testing 6. Linear Regression and Correlation Business Statistics to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Principles of Business Statistics
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'This course introduces students to the basic concepts and logic of statistical reasoning and gives the students...
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'This course introduces students to the basic concepts and logic of statistical reasoning and gives the students introductory-level practical ability to choose, generate, and properly interpret appropriate descriptive and inferential methods. In addition, the course helps students gain an appreciation for the diverse applications of statistics and its relevance to their lives and fields of study. The course does not assume any prior knowledge in statistics and its only prerequisite is basic algebra.'The course is organized in 4 units:'Unit 1 Exploratory Data Analysis. This is organized into two modules – Examining Distributions and Examining Relationships. The general approach is to provide students with a framework that will help them choose the appropriate descriptive methods in various data analysis situations.Unit 2 Producing Data. This unit is organized into two modules – Sampling and Study Design.Unit 3 Probability. The unit is a classical treatment of probability and includes basic probability principles, conditional probability, discrete random variables (including the Binomial distribution) and continuous random variables (with emphasis on the normal distribution).Unit 4 Inference. This unit introduces students to the logic as well as the technical side of the main forms of inference: point estimation, interval estimation and hypothesis testing. The unit covers inferential methods for the population mean and population proportion, Inferential methods for comparing the means of two groups and of more than two groups (ANOVA), the Chi-Square test for independence and linear regression. The unit reinforces the framework that the students were introduced to in the Exploratory Data Analysis for choosing the appropriate, in this case, inferential method in various data analysis scenarios.Throughout the course there are many interactive elements. These include: simulations, "walk-throughs" that integrate voice and graphics to explain an example of a procedure or a difficult concept, and, most prominently, computer tutors in which students practice problem solving, with hints and immediate and targeted feedback. The most elaborate of such activities is the "StatTutor," a tool that supports students as they go through the processes of data analysis while emphasizing the big picture of statistics Probability and Statistics to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Probability and Statistics
Select this link to open drop down to add material Probability and Statistics to your Bookmark Collection or Course ePortfolio | 677.169 | 1 |
Description
This application will find the distance between two points on a Cartesian plane. Simply enter the (x, y) coordinates of the two points and it will quickly calculate the distanceGraphIt is a utility that is used to display on a Cartesian plane any mathematical function, which can be f(x)=2, f(x)=2*x, f(x)=sin(x) ecc... The management function is entrusted to you: you can add more or modify existing onesBest math tool for school and college! If you are a student, it will helps you to learn geometry!
Note: Trigonometric functions are used for computing unknown lengths and angles in triangles (in navigation, engineering and physics). The sine and cosine functions are also commonly used to model periodic function phenomena such as sound and light waves, the position and velocity of harmonic oscillators surfaceElectrical Engineering Formulas helps with complex calculations of single phase, three phase and direct current. You can calculate since apparent power, electrical motor horse power, the interesting of this tool is you can use the prefix in order to simplify the calculations.
Also it has a amazing tool to plot the behavior of voltage, percentage efficient, between another more.
You will not forget math formulas anymore. This app is everything you need. Perfect app for college and highschool students. You will find everything about mathematics in it. This app includes maths topics such like: algebra, analysis, integrals, derivatives, logic and more. Try it now!
Ms Excel allows you to use formulas to perform not only mathematical operations but also a myriad of other complex actions, such as parsing textual values, searching for certain values in a range of data, performing recursive calculations, and much more.
Feature of Excel Formulas
+ Common Mathematical Operations + Manipulating Text with Formulas in microsoft excel 2007 or 2010 + Working with Dates and Times + Performing Conditional Analysis with microsoft excel spreadsheet + Using Lookup Formulas, spreadsheet +Common Business and Financial Formulas
To leverage the full power of Ms Excel formulas, you need to understand how Ms Excel formulas work as well as some of the ground rules for working with formulas. The goal of this chapter, therefore, is for you to get acquainted with the fundamentals of using Microsoft Excel 2010 formulas.
Formula Deck helps you learn and revise formulas quickly, to help you ace your exams. With FormulaDeck you can:
• Get access to all Physics, Chemistry and Mathematics formulas at one place. • Browse Formulas without any internet connection once you have logged in to the app. • No more spending hours on writing important formulas on a paper, just use Formula Deck Cheat Sheets. • Mark formulas which you find hard and revise them all together in the Marked to Memorize list. • Helpful for Students preparing for Class 12 Boards (CBSE, ISC, etc.), IIT JEE and Other Engineering Entrance Exams InstallNeed help with those tricky drawings in Draw Something? This app will generate all playable words in Draw Something based on the letters they give you to solve the word. Simply enter the letters and word length and it will list all the possible drawing answers.
Are you worried about cheating? Well, don't. Yes, the app will give you all the words you can play, but by the nature of this game, you still must correlate the correct word to the drawing. In essence, it's like a playing a guessing game while you win at Draw Something! Win-win. ;)
This app will simplify fractions (mixed, proper, & improper) and tell you if the fraction that you entered is already in simplest form. Also, just added, you can enter any decimal and it will automatically convert it into simplest fraction form. It will also convert rational, repeating decimals for you! - No advertisements! ;)
- Expand your vocabulary with the "Learn A Word" option, exclusive to the pro version.Sound Healing is founded on the premise that all matter is vibrating at specific frequencies. Science has proven that sound, or vibration, has a strong impact upon substance.
This app uses the Solfeggio scale. This ancient scale is part of a 6-tone sequence of electro-magnetic frequencies called the Original Solfeggio Scale. These particular frequencies were rediscovered by Dr. Joseph Puleo in 1974. The 6 tones were then manifested into tuning forks. The hertz are at which the forks vibrate to balance your cells. This app emulates the sound of those tuning forks and aim to stimulate and balance each chakra that correlates with each on of the tones.
This MW3 (Modern Warfare 3) themed app will encode any phrase into the NATO phonetic alphabet (as appears in the Allied Maritime Signal and Maneuvering Book used by all allied navies of NATO). This alphabet is used in the Modern Warfare video game series.
*** (As with all my paid apps, this app contains an ad-free environment so your child can focus on learning, uninterrupted.) ***
This application targets the Common Core Standard for Kindergartners in the area of counting and cardinality. These activities are designed to help know number names and count sequence. The app contains different animals for each set of ten numbers and provides voice-overs so the child can hear the number as they navigate through each game. Check out the in-game screen shots!
This is the first programming experience for Ben (8 years old) and Evan (10 years old)! It's a clean, simple, and fun version of Tic-Tac-Toe! Enjoy and leave some positive feedback for these young scholars! ;)
How good are your reflexes and coordination? The goal of the game is simple; touch as many dolphins in thirty seconds as you can without touching the water or touching a shark! If you touch the water, you'll hear a "splash" sound and you will lost 6 points, but the game will keep going. if you touch a shark, you'll hear a "chomp" sound and the game is over. For every 10 dolphins you touch, you'll gain 1 second on the timer. What's your high score going to be? ;)
Fly your way through space in this endless runner (flyer) game, navigating your ship through countless gates as the gates come at you faster and faster all while getting smaller and smaller every second! Test your coordination and see how far your reflexes will get you! Also available is dart mode, where you try to "thread the needle" between vertically moving gates!
Features:
- Smooth, fast, arcade action
- Select from 6 different ships
- 4 levels of difficulty to test your reflex and coordination, including "Dart Mode", which will test your timing as well.
- Uses leaderboards through Google Play to track your best scores against your friends and gaming community. | 677.169 | 1 |
Mathematics for Class XI is a reference book by R. D. Sharma for students of class XI following the syllabus issued by the Central Board of Secondary Education.
Summary Of The Book
Mathematics for Class XI is a reference book aimed at helping students of Class XI in their efforts to prepare for class exams as well as to provide a good foundation in the preparation for competitive exams.
The chapters in this book provide a description of concepts covered in Class XI, along with a multitude of problems for each chapter. The book has an algorithmic approach and comes with clear explanations of the theory supplemented by illustrations, making concepts easy to understand and remember. The book also contains concise summaries of concepts and formulae at the end of each chapter, making it a good book to have handy for revisions just before the exam.
For students looking to learn by practice, this book is a great source for problems. It comes with many exercises, and the problems listed are graded and come with the solved answers. The book also contains a number of exercises with unsolved problems that the student may use to further sharpen his/her skills.
Mathematics for Class XI addresses the entire syllabus for Class XI. Some topics covered in the book include trigonometric equations, functions, ratios, and graphs of trigonometric functions, sine and cosine formulae and their applications, mathematical induction, and transformation formulae.
With detailed explanations of the theory and a wealth of solved and unsolved problems, this book makes for a reliable source book for Class XI students to help in their efforts of doing well in their competitive exams.
About R. D. Sharma
R. D. Sharma is an Indian author and teacher. He is a widely published author of Mathematics textbooks and reference books.
Some other books by R. D. Sharma include Mathematics for Class XII, Mathematics for Class 12 (Set of 2 Volumes), Objective Mathematics For IIT-JEE, AIEEE and All Other Engineering Entrance Examinations, and Mathematics Class-VII.
R. D. Sharma has a Doctorate in Mathematics, and completed his Bachelor's and Master's degrees, with Honours and double gold medals, from the University of Rajasthan. He is presently with the Directorate of Technical Education, Delhi, serving as the Head of Department, Science and Humanities.
Recent top reviews
BETTER UNDERSTANDING AND CLEAR CONCEPTS PROVIDED BY THIS BOOK REGARDING MATHS TO STD XI STUDENTS.....
★★★★★
★★★★★
SUBIPRA
Mar 25, 2014
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BEST BOOK FOR MATHEMATICS CLASS XI
'I AM A REGULAR CONSUMER OF THIS PRODUCT"RD SHARMA" SINCE CLASS VII TILL XII IT IS A SELF TUTOR AND PROVIDES U WITH THE REAL CONCEPTS WHERE NCERT FAILS.BUY THIS BOOK IF U WANT TO BE MASTER OF LIMITS AND DERIVATIVES.THE MAIN FEATURE IS IT SHOWS THE STEPS LITERALLY OR ALGORITHM FOR EACH AND EVERY … (View complete review)
★★★★★
★★★★★
rasheed haq
Mar 13, 2014
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Good book and good service from flipkart
i bought this book for my sibling who is studying in +1. It wasnt available in local store flipkart gave the book at a good price and delivery was given in time.
★★★★★
★★★★★
ABHINAV ROY
Mar 3, 2014
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KING OF MATHEMATICS BOOKS!!!!!!! BY ABHINAV ROY
YO!!!!!!!! its outstanding!!!!!!!! all the concepts are elaborated very well!!!! A lot of sums are given for practicing which will help you a lot.Once you solve the entire book,NO ONE WILL BE ABLE TO STOP YOU FROM DOING EXCELLENT IN EXAMS!!!! hope it helps BY:- ABHINAV ROY(ROYAL KING)
★★★★★
★★★★★
rupesh prakash
Mar 1, 2014
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best of all......
the best book for mathematics.if you want to excel and get good marks,you must practice out.one problem is that it has many unwanted questions which are not asked for exams.please do believe me.i was very poor in maths.but i brought this book in class 9 and got A1 in maths(u wont believe,but u have … (View complete review)
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Mathematics by R.D.Sharma for class IX, X, XI and XII
… who are finding it difficult with the basics or with class XI and XII math, R.D.Sharma's books are sufficient to score better and foremost important understand better. (Expand)
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Every student of Class XI should go through
Helpful for conceptual clarity on doubts.Problem solving skills would get a boost if you work out all exercises.I would recommend each CBSE Class XI student must possess this mathematical gem.Definitely you would derive pleasure out of it.
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Look no further,'cause you're staring at the BEST.
E…EE or any other competitive exam whatsoever. It spans about some 1200 odd pages and trust me you're gonna get sums that will take you for a ride. The key features of the book are :
1. Algorithmic approach 2. Stress on concepts 3. Tremendously huge number of practice sums(unsolved) as well as hundreds of solved examples from a myriad of topics.
HOWEVER,if you are really serious about acing your mathematics exam or simply enhancing your knowledge and challenging yourself,you shall have to devote a considerable amount of time because not only does this book house a number of sums,believe me some of them are so difficult to get thorugh it will take you days before you can finally brag in fornt of your geeky acquaintances that you were able to solve them while they couldn't .
So if you are hard working and are ready to relentlessly toil to get a really good score in mathematics,look no further because I got on my doubts whether you shall find some other book to suit your needs better .
Last but not the least I would like to applaud the amazing service that flipkart again provided me by way of their quick delivery of my order. I hope they keep up the good work.
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BEST OF ALL
Features 1) Number of examples problem based on each concepts 2) makes concepts easy to understand and have practical and application type of sums 3)Good quality of questions in exercises 4)MCQs at d end of each chapter 5)Summary contains impt formulas,definations and revalant points.
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Best for Boards exams.!!
A very nice book for Boards preparation. The book was very easy to solve as I had the concepts beforehand. But if you are looking for something that gives you an edge over others, well try out some JEE books. But for boards, its enough
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Maths Basic for IIT+School
Firstly,,, i'm a bit unhappy because the the delivery time period was two-three days and 1 got it after four days. But it is not a very big problem. talking about the book,, it is one of a kind,, the most basic book for IIT basics + school exams. The price offered by flipkart is the best, as always. Thank you flipkart.
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The best book of mathematics for class 11
… in achieving our target. The book is an elixir of our career and so without thinking any more , we shall flipkart it! (Expand)
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Only For Extraordinary Students !!
If U Get More Than 90% In Mathematics , This Is The Book For U Man !! Otherwise Stay Away....................................................................................................................... | 677.169 | 1 |
Product Description
For Grades 7-10, this series emphasizes the development of understanding mathematical concepts and their applications, as well as proficiency in problem solving, mathematical reasoning and higher order thinking. Students will work on investigative skills, communication in mathematics, appropriate computation and estimation skills and mental calculation through graded exercises. Exercises are provided for different levels, while chapter reviews, challenging and problem-solving questions, revision exercises for review, the history of math and other elements are also included. Many questions require students to apply knowledge to new situations rather than following a procedure. Textbook 2 is recommended for Grade 8; 444 pages, hardcover. Teacher involvement is generally required.
Excellent book! My wife and I are both engineers, so math is important to us. Our son attends a great school but needed some other approaches to understanding the same material. This book, and the workbook with extra problems, fulfill that role very well | 677.169 | 1 |
Find a Mill Neck have studied the following topics that fall under the description "discrete mathematics': algebraic number theory, fractal geometry, dynamical systems, difference equations, group theory, set theory, combinatorics, game theory etc. These were all at masters or undergraduate level. High School topics in discrete mathematics tend to be derivative and simpler than these subjects. | 677.169 | 1 |
Algebra and Trigonometry
2nd
Algebra and Trigonometry by James Stewart
Book Description
Choose the algebra textbook that's written so you can understand it. ALGEBRA AND TRIGONOMETRY reads simply and clearly so you can grasp the math you need to ace the test. And with Video Skillbuilder CD-ROM, you'll follow video presentations that show you step-by-step how it all works. Plus, this edition comes with iLrn, the online tool that lets you sign on, save time, and get the grade you want. With iLrn, you'll get customized explanations of the material you need to know through explanations you can understand, as well as tons of practice and step-by-step problem-solving help. Make ALGEBRA AND TRIGONOMETRY your choice today.
Buy Algebra and Trigonometry book by James Stewart from Australia's Online Bookstore, Boomerang Books.
This manual contains solutions to odd-numbered Section Exercises, selected Chapter Review Exercises, odd-numbered Discussion Exercises, and all Chapter Test Exercises, giving students a way to check their answers and ensure that they took the correct steps to arrive at an answer.
Books By Author James Stewart
Shows students how calculus relates to biology, with a style that maintains rigor without being overly formal. This book includes topics on calculus with examples drawn from many areas of biology, including genetics, biomechanics, medicine, pharmacology, physiology, ecology, epidemiology, and evolution | 677.169 | 1 |
Eighth Grade Reading Comprehension and Writing Skills 1st edition
1576857115
9781576857113
Details about Eighth Grade Reading Comprehension and Writing Skills:
What makes this newest addition to the Express Review Guides series different than other math practice books is that it contains only math word problems – the kinds encountered at school and on high-stakes tests. Questions are divided into areas such as algebra, geometry, fractions, percents, decimals, and more. Within each chapter, questions move from easy to advanced, giving students the opportunity to gain confidence in each math area. | 677.169 | 1 |
Edexcel and AQA Style GCSE Maths Questions
Arranged by Topic
GCSE Maths Topic Booklets for Revision
FREE booklets written for students working towards Edexcel 1MA0 and also AQA 4360 GCSE Maths exams. High quality model answers can be bought for less than the cost of 1 hour with a private tutor. They are a great alternative to hiring a tutor and they actually will improve your grade if you work through them all. I use them myself in my work as a Maths tutor and my students consistently achieve A* or A grades.
Edexcel Style GCSE Questions by topic These FREE booklets of GCSE 1MA0 style past paper questions arranged by topic make revision and practice much easier. Trying to revise using past papers usually means going from one topic to another and so no topic is covered thoroughly. These topic booklets have questions that are just like those in the Edexcel 1MA0 exams. Great for revision.
AQA Style GCSE Questions by topic FREE booklets for students and teachers of AQA 4360 Maths. The questions are just like the AQA ones. Study and revision is much easier as topics can be done one at a time.
Resources Useful FREE extras such as graph and isometric paper, reflection images and 3D paper models. There is now a FREE collection of 28 cursive
handwriting sheets, using the nonsense poems of Edward Lear.
MATHS
TUTOR
If you live in the area around Knutsford I can offer tuition in Maths from primary to GCSE. I am a highly experienced teacher and tutor. I come to you to teach in your own home. I specialise in helping to greatly boost the grades for people who have a forecasted "C" or "D". My rates are very competitive and I provide all materials needed. There are no hidden extras. For more information please contact me at: peter@bland.in or phone 01565 220 375 or 0790 710 6489
GCE O and A
level papers
A FREE and fascinating collection of JMB Syllabus B past papers from the 1970s, 1980s and 1994. These papers are available nowhere else. There are also 10 examples of Additional Maths papers from JMB and AEB.
Please email me at peter@bland.in if you would like to see them, and you will receive copies of them, free of charge. | 677.169 | 1 |
...
Show More the calculations. By seeing the mathematics and understanding the underlying geometry, students will develop mathematical maturity and learn to think abstractly | 677.169 | 1 |
According to OER Commons, "Crossroads in Mathematics: Standards for Introductory College Mathematics Before Calculus has two...
see more
According to OER Commons, "Crossroads in Mathematics: Standards for Introductory College Mathematics Before Calculus has two major goals: to improve mathematics education at two-year colleges and at the lower division of four-year colleges and universities and to encourage more students to study mathematics. The document presents standards that are intended to revitalize the mathematics curriculum preceding calculus and to stimulate changes in instructional methods so that students will be engaged as active learners in worthwhile mathematical tasks. Preparation of these standards has been guided by the principle that faculty must help their students think critically, learn how to learn, and find motivation for the study of mathematics in appreciation of its power and usefulness' (direct from website). Users can access all chapters of the book as well as the Illinois Mathematics Association of Community Collegesroads in Mathematics: Standards for introductory college mathematics before calculus to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Crossroads in Mathematics: Standards for introductory college mathematics before calculus
Select this link to open drop down to add material Crossroads in Mathematics: Standards for introductory college mathematics before calculusLogic to the Rescue is designed to teach kids critical thinking. A combination of fiction and non-fiction, it weaves examples of logical fallacies into a fictional sword-and-sorcery fantasy. Simple examples for testing a hypothesis and setting up experiments in chemistry and physics are integrated into the plot.Who will it appeal to?Kids ages 10 to 14, though adults who want to brush up on their knowledge of logical fallacies such as post hoc ergo propter hoc, ad hominem, and counting the hits may also enjoy the to the Rescue to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Logic to the Rescue
Select this link to open drop down to add material Logic to the Magnetic Math to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Magnetic Math
Select this link to open drop down to add material Magnetic Math to your Bookmark Collection or Course ePortfolio | 677.169 | 1 |
This article needs rewriting to enhance its relevance to psychologists.. Please help to improve this page yourself if you can..
Calculus is a central branch of mathematics, developed from algebra and geometry. The word stems from the nascent development of mathematics: the early Greeks used pebbles arranged in patterns to learn arithmetic and geometry, and the Latin word for "pebble" is "calculus."
Calculus is built on two major complementary ideas, both of which rely critically on the concept of limits. The first is differential calculus, which is concerned with the instantaneous rate of change of quantities with respect to other quantities, or more precisely, the local behaviour of functions. This can be illustrated by the slope of a function's graph. The second is integral calculus, which studies the accumulation of quantities, such as areas under a curve, linear distance traveled, or volume displaced. These two processes act inversely to each other, as shown by the fundamental theorem of calculus.
Contents
Differential calculus
The derivative measures the sensitivity of one variable to small changes in another variable. Consider the formula:
for an object moving at constant speed. The speed of a car, as measured by the speedometer, is the derivative of the car's distance traveled, as measured by the odometer, as a function of time. Calculus is a mathematical tool for dealing with this complex but natural and familiar situation.
Differential calculus can be used to determine the instantaneous speed at any given instant, while the formula "speed = distance divided by time" only gives the average speed, and cannot be applied to an instant in time because it then gives an undefined quotient zero divided by zero. Calculus avoids division by zero by using the concept of the limit which, roughly speaking, is a method of controlling an otherwise uncontrollable output, such as division by zero or multiplication by infinity. More formally, differential calculus defines the instantaneous rate of change (the derivative) of a mathematical function's value, with respect to changes of the variable. The derivative is defined as a limit of a difference quotient.
The derivative of a function, if it exists, gives information about its graph. It is useful for finding optimum solutions to problems, called maxima and minima of a function. It is proved mathematically that these optimum solutions exist either where the tangent of the graph is flat, so that the slope is zero; or where the graph has a sharp turn (cusp) where the derivative does not exist; or at the endpoints of the graph. Another application of differential calculus is Newton's method, a powerful equation solving algorithm. Differential calculus has been applied to many questions that were first formulated in other areas, such as business or medicine.
Integral calculus
There are two types of integral in calculus, the indefinite and the definite. The indefinite integral is simply the antiderivative. That is, F is an antiderivative of f when f is a derivative of F. (This use of capital letters and lower case letters is common in calculus. The lower case letter represents the derivative of the capital letter.)
The definite integral evaluates the cumulative effect of many small changes in a quantity. The simplest instance is the formula
for calculating the distance a car moves during a period of time when it is traveling at constant speed. The distance moved is the cumulative effect of the small distances moved in each instant. Calculus is also able to deal with the natural situation in which the car moves with changing speed.
Integral calculus determines the exact distance traveled during an interval of time by creating a series of better and better approximations, called Riemann sums, that approach the exact distance as a limit. More formally, we say that the definite integral of a function on an interval is a limit of Riemann sum approximations.
Applications of integral calculus arise whenever the problem is to compute a number that is in principle (approximately) equal to the sum of a large number of small quantities. The classic geometric application is to area computations. In principle, the area of a region can be approximated by chopping it up into many pieces (typically rectangles, or, in polar coordinates, circular sectors), and then adding the areas of those pieces. The length of an arc, the area of a surface, and the volume of a solid can also be expressed as definite integrals. Probability, the basis for statistics, provides another important application of integral calculus.
The symbol of integration is ∫, a stretched s (which stands for "sum"). The precise meanings of expressions involving integrals can be found in the main article Integral. The definite integral, written as:
is read "the integral from a to b of f(x) dx".
Foundations
There is more than one rigorous approach to the foundation of calculus. The usual one is via the concept of limits defined on the continuum of real numbers. An alternative is nonstandard analysis, in which the real number system is augmented with infinitesimal and infinite numbers. Calculus also can be rigorously developed by using explicit uniform estimates. This approach avoids both limits and infinitesimals, it is more in the spirit of constructive mathematics that fits much better with the computational aspect of the subject than the other two approches. It is also much simpler conceptually and technically. The tools of calculus include techniques associated with elementary algebra, and mathematical induction. The foundations of calculus are included in the field of real analysis, which contains all full definitions and proofs of the theorems of calculus as well as generalisations such as measure theory and distribution theory.
Fundamental theorem of calculus
The fundamental theorem of calculus states that differentiation and integration are, in a certain sense, inverse operations. More precisely, if one defines one function as the integral of another continuous function, then differentiating the newly defined function returns the function you started with. Furthermore, if you want to find the value of a definite integral, you usually do so by evaluating an antiderivative.
Here is the mathematical formulation of the Fundamental Theorem of Calculus: If a function f is continuous on the interval [a, b] and if F is a function whose derivative is f on the interval [a, b], then
Also, for every x in the interval [a, b],
This realization, made by both Newton and Leibniz, was key to the massive proliferation of analytic results after their work became known. The fundamental theorem provides an algebraic method of computing many definite integrals—without performing limit processes—by finding formulas for antiderivatives. It is also a prototype solution of a differential equation. Differential equations relate an unknown function to its derivatives, and are ubiquitous in the sciences.
Applications
The development and use of calculus has had wide reaching effects on nearly all areas of modern living. It underlies nearly all of the sciences, especially physics. Virtually all modern developments such as building techniques, aviation, and other technologies make fundamental use of calculus. Many algebraic formulas now used for ballistics, heating and cooling, and other practical sciences were worked out through the use of calculus. In a handbook, an algebraic formula based on calculus methods may be applied without knowing its origins. The success of calculus has been extended over time to differential equations, vector calculus, calculus of variations, complex analysis, and differential topology.
History
The origins of integral calculus are generally regarded as going back no further than to the time of the ancient Greeks, circa 200 BC. The Hellenic mathematician Eudoxus is generally credited with the method of exhaustion, which made it possible to compute the areas of regions and the volumes of solids. Archimedes developed this method further, inventing heuristic methods which resemble integral calculus. After him, the development of calculus did not advance appreciably for over 500 years.1
Leibniz and Newton pulled these ideas together into a coherent whole and they are usually credited with the probably independent and nearly simultaneous "invention" of calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the notation used in calculus today; he often spent days determining appropriate symbols for concepts. The fundamental insight that both Newton and Leibniz had was the fundamental theorem of calculus. Virtually all modern methods of symbolic integration follow from this theorem, and it has proven indispensible in the development of modern mathematics and physics. For example, see Integration by parts and Integration by substitution.
When Newton and Leibniz first published their results, there was some controversy over whether Leibniz's work was independent of Newton's. While Newton derived his results years before Leibniz, it was only some time after Leibniz published in 1684 that Newton published. Later, Newton would claim that Leibniz got the idea from Newton's notes on the subject; however examination of the papers of Leibniz and Newton show they arrived at their results independently, with Leibniz starting first with integration and Newton with differentiation. This controversy between Leibniz and Newton divided English-speaking mathematicians from those in Europe for many years, which slowed the development of mathematical analysis. Today, both Newton and Leibniz are given credit for independently developing calculus. It is Leibniz, however, who is credited with giving the new discipline the name it is known by today:
"calculus". Newton's name for it was "the science of fluxions". Some others who contributed important ideas are Descartes, Barrow, Fermat, Huygens, and Wallis.
Since the time of Leibniz and Newton, many mathematicians have contributed to the continuing development of calculus. In the 18th century, calculus was put on a much more rigorous footing by Cauchy, Riemann, Weierstrass, and others. It was also during this time period that the ideas of calculus were generalized to Euclidean space and the complex plane. Calculus continues to be further generalized, such as the development of the Lebesgue integral in 1900. | 677.169 | 1 |
...I also contribute to the Resources:Answers section, providing on-line homework help to students. Having a good understanding of basic elementary and middle school mathematics is essential to mastering the higher-level math disciplines that a student will encounter in high school and college, and | 677.169 | 1 |
A First Course in Probability and Markov Chains
Provides an introduction to basic structures of probability with a view towards applications in information technology
A First Course in Probability and Markov Chains presents an introduction to the basic elements in probability and focuses on two main areas. The first part explores notions and structures in probability, including combinatorics, probability measures, probability distributions, conditional probability, inclusion-exclusion formulas, random variables, dispersion indexes, independent random variables as well as weak and strong laws of large numbers and central limit theorem. In the second part of the book, focus is given to Discrete Time Discrete Markov Chains which is addressed together with an introduction to Poisson processes and Continuous Time Discrete Markov Chains. This book also looks at making use of measure theory notations that unify all the presentation, in particular avoiding the separate treatment of continuous and discrete distributions.
A First Course in Probability and Markov Chains:
- Presents the basic elements of probability. - Explores elementary probability with combinatorics, uniform probability, the inclusion-exclusion principle,
independence and convergence of random variables. - Features applications of Law of Large Numbers. - Introduces Bernoulli and Poisson processes as well as discrete and continuous time Markov Chains with discrete states. - Includes illustrations and examples throughout, along with solutions to problems featured in this book.
The authors present a unified and comprehensive overview of probability and Markov Chains aimed at educating engineers working with probability and statistics as well as advanced undergraduate students in sciences and engineering with a basic background in mathematical analysis and linear algebra.
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"This is useful not only as review material to mathematics students, but also to students in the engineering and information sciences which may be curious about theoretically understanding the material presented before." (Zentralblatt MATH, 1 August 2013 | 677.169 | 1 |
Essential Mathematics and Statistics for Science, 2nd Edition
Essential Mathematics and Statistics for Science, Second
Edition is a completely revised and updated version of this
invaluable text which allows science students to extend necessary
skills and techniques, with the topics being developed through
examples in science which are easily understood by students from a
range of disciplines. The introductory approach eases students into
the subject, supported by revision material and on-line
self-assessments. The book then progresses to cover topics relevant
to both first and second year level study as well as supporting
data analysis for final year projects. The practical applications
of the theory are developed by numerous worked examples and
practice questions. The revision of the material in the book has
been matched, on the accompanying website, with the extensive use
of video, providing worked answers to all the questions in the book
plus additional tutorial support. The second edition has also
improved the learning approach for key topic areas to make it even
more accessible and user-friendly, making it a perfect resource for
less confident students, as well as an easily accessible resource
for students of all abilities.
Applied approach providing mathematics and statistics from the
first to final years of undergraduate science courses.
Second edition substantially revised to improve the learning
approach to key topics and the organisation of resources for ease
of use in teaching.
Extensive use of video for all worked answers now provides a
very successful style of learning support for students via the
website.
Resource Site Click here to find links to:
Over 200 videos showing step-by-step workings of problems in the book
Additional materials including related topic areas, applications, and tutorials on Excel and Minitab
Interactive multiple-choice questions for self-testing, with step-by-step video feedback for any wrong answers | 677.169 | 1 |
Mathematics for Retail Buying
Synopses & Reviews
Publisher Comments:
New to this Edition - At the end of each of the six units of study, a new section entitled, "The Acaptation of Excel Spreadsheets" is included. - Includes a CD-ROM that contains templates for use with specific computer spread-sheet problems and cases. - Examples of current industry programs on collecting and recording data or the movement of inventory. - All practice problems apply industry terminology used in realistic merchandising situations. - To become familiar with real-world examples, store forms have been added for the computation of appropriate practice problems - Answer Manual provides computations and solutions for all of the practice problems and case studies in the text.
Synopsis:About the Author
Table of Contents
Merchandising for a Profit Defining the Basic Profit Factors Profit and Loss Statements How to Increase Profits Retail Pricing and Repricing of Merchandise Retail Pricing Basic Markup Equations Used in Buying Decisions Repricing of Merchandising The Relationship of Markup to Profit Types of Markup Averaging or Balancing Markup Limitations of the Markup Percentage as a Guide to Profits The Retail Method of Inventory Explanation of the Retail Method of Inventory General Procedures for Implementing the Retail Method of Inventory Shortages and Overages An Evaluation of the Retail Method of Inventory Dollar Planning and Control Six-Month Seasonal Dollar Merchandise Plan Open-to-Buy Control Invoice Mathematics: Terms of Sale Terms of Sale Dating Anticipation Loading Shipping Terms | 677.169 | 1 |
Nested requests
OBJECTIVE: To understand and use the fundamental problem-solving
element, Request-Response-Result, when requests are nested.
A response may make one or more requests. In this case the response
becomes the current requesting equation. Responses to a requesting
equation are indented one level to the right of the position of the
requesting equation.
If there is more than one request, that is, sequential variables as
described in Lesson 2, responses to each of the requests are made. When
responses to all the
requests have been completed, the previous requesting equation becomes
the current requesting equation.
The following examples shows a simple case of nested requests in which
each response makes one request.
The nesting can go to considerable depth. The following example
illustrates this for a problem a bit more complex than the previous one.
The example also illustrates an important property of a problem
statement. It can readily be seen that the problem statement is the
source of information with which to respond to a request.
Problems of any consequence will have several nested request structures
in the solution.
The problem statement is one of the sources of information needed to
respond to requests.
A problem statement is not an object to be understood. It is only
a source of information in the same sense as a dictionary or telephone
book. It is a place to look for information to respond to what the
equations ask for. Let the equation tell you what to look for. Equations
talk.
Request-Response-Result
Request-Response-Result is used repeatedly in our daily activities. Have
you students journal use of this for a day. Have them identify situations
in which sequential and nested requests were encountered and how
responses generated from available information led to results. | 677.169 | 1 |
Calculator Tab is a free online scientific calculator which works like your regular calculator. One feature that sets it appart from other calculators is that it's memory bank can store an unlimited amount of numbers and descriptions of these numbers for an indefinite length of time. The stored numbers can be sorted by date, by the number of uses or by name. Your saved numbers and their respective descriptions are saved directly on your computer, never entering the sphere of the internet, so that they are as secure as your workstation.
This calculator follows the standard order of operations. It can handle values from 4.9e-324 to 1.7e+308. To perform a calculation, press the calculator buttons with the mouse as you would write the symbols of the calculation. For example to calculate 2 + 5 * 8 press the buttons . To see the result (which in this case will be 42) and to terminate the calculation press the button. The exception to this approach is when using mathematical functions which only take one value: in these functions, the value always comes before the function. For more details see the section "Functions with one (x) value" below. To be able to use your keyboard instead of the mouse, you must first click anywhere on the calculator with the mouse.
Calculator Tab will not allow you to make ambiguous enteries and will tell you, what is not allowed if you try to make such an entry. If you, for example, try to enter the entry of will generate an error telling you, that you first need to enter a number before continuing with the calculation. The entry of will be ignored and you can continue with your calculation as though you had not entered it. This way you can always be sure about what exactly the calculator is doing.
There are two possibilities to store a value in the memory bank: 1. Using the button you can quickly store the displayed value in the memory bank. You don't need to enter a description. 2. Using the button you can store the displayed value in the memory bank and add your own description, which can be up to 64 characters long.
To open the memory bank press this button:
The memory bank will open and you will see your saved values. You can only see three values at a time. If you have more then tree values saved, you can use the scroll bar to the right of the saved values to scroll through the memory bank. To insert a saved value into the calculator, click on the value or the grey field surrounding it. To erase a value click on the button beside the corresponding value.
There are 2 possibilities to enter a negative number: 1. Press the button before pressing the number button. For example to calculate 2 + (-5) press the buttons . 2. Or press the button after the number button. For the same calculation as above press the buttons .
Making a bracket negative can only be accomplished using the first method, because the button can only change the sign of the last entered value. If you try to change the sign of a bracket using the button, you will get an error message.
Functions which take one value (, , , , , , , , , , , , , , ) are used in the following fashion: [value X] [Function], meaning that the value is always entered before the function. For example to enter sin(30) press the buttons .
Using the percentage buttons and , you can calculate either the percentage of a certain number, or what percentage of a certain number another number constitutes. The labels of the percentage buttons are meant as mnemonics. To calculate 5% of 50 (which is 2.5) enter . Here the mnemonic is clear: 5% (of) 50 is reflected in the label "x % y".
If on the other hand you want to know how many percent of 50 the value of 5 constitues, use the button . Here the mnemonic is: (How many) % (of) 50 (is) 5, which is reflected in the label "% y x". Keep in mind though, that the value of "x" comes before the value of "y" as explained above in the section "Functions with two (x and y) values". Accordingly, in order to calculate how many percent of 50 the value of 5 constitutes (which is 10%), enter .
To select your options for various settings, press the button. To turn to the next page of options press the button. To save your settings press the button. To cancel any changes you have made press the button.
Decimal-separator
You can select, whether a point or a comma should be used as a decimal separator.
Order of saved values
You can select, how your saved values should be ordered: by date (most recent first); by most used (values used more frequently appear first); or by alphabet (values with descriptions first in the alphabet will appear first).
Descriptions on roll-over
You can select, whether a short description of the fuctionality should appear, when you hover over a button. As you get more familiar with
the calculator, you might not need this feature.
The "plus" button is used for adding two values or for specifying a positive exponent in scientific notation (note: since the default exponent is positive, it does not need to be explicitly specified).
The "minus" button is used for subtracting one value from an other, for entering a negative value or for specifying a negative exponent in scientific notation. For information about entering a negative value see the above section "Negative numbers".
The "root" button is used for calculating the root of a value. Although it is theoretically possible to calculate the root of a negative value if the root is an odd number, this calculator only calculates the roots of values which are greater than or equal to zero.
RESTRICTIONS: The value who's root you want to calculate must be greater than or equal to zero, except where the root is a number between -1 and 1 inclusive, which corresponds to raising the vaule to the reciprocal of the root. Otherwise a value smaller than zero will generate an error and terminate the calculation.
The "exponent" button is used for raising a value (base) to the power of another value (exponent).
RESTRICTIONS: Since raising a negative base to a power between -1 and 1 exclusive corresponds to the root reciprocal of the exponent, and trying to calculate a root of a negative number generates an error, this configuration will generate an error and terminate your calculation.
The "logarithm" button is used for calculating the logarithm to base 10 of a value. To calculate the logarithm of a value to a base other than 10 divide the logarithm of the value by the logarithm of the desired base.
RESTRICTIONS: The value must be greater than zero. A value less than or equal to zero will generate an error and terminate your calculation.
The "natural logarithm" button is used for calculating the logarithm to base e of a value. To calculate the logarithm of a value to a base other than e divide the natural logarithm of the value by the natural logarithm of the desired base.
RESTRICTIONS: The value must be greater than zero. A value less than or equal to zero will generate an error and terminate your calculation.
The "arcsine" button is used for calculating the arcsine of a value. Arcsine is the inverse function of sine. For now, degrees are the only unit this function supports. It returns a number between -90 and 90 inclusive.
RESTRICTIONS: The value must be a number between -1 and 1 inclusive.
The "arccosine" button is used for calculating the arccosine of a value. Arccosine is the inverse function of cosine. For now, degrees are the only unit this function supports. It returns a number between -90 and 90 inclusive.
The "arctangent" button is used for calculating the arctangent of a value. Arctangent is the inverse function of tangent. For now, degrees are the only unit this function supports. It returns a number between -90 and 90 inclusive.
The "Pi" button is used for entering (an approximation of) the constant Pi into the calculator. Pi is for example useful in trigonometric functions and for calculating the circumference, area and volume of circular geometrical objects.
The "modulo" button is used for calculating modulo of two values. Modulo is the remainder of one value (dividend) divided by another (divisor). The way it is implemented in this calculator, the result has always the same sign as the dividend.
RESTRICTIONS: The divisor must not equal zero. A zero in the divisor will generate an error and terminate the calculation.
The "scientific notation" button is used for entering values in scientific notation. A value written in scientific notation has the form: a x 10b. It is especially useful for entering very large or very small values. For example instead of entering 123000 you can enter 1.23e+5. And instead of entering 0.000123 you can enter 1.23e-4.
To enter a negative exponent press the minus (-) button after pressing the "scientific notation" button. When entering a positive exponent, you do not need to enter the plus (+) sign, because the exponent is positive by default.
RESTRICTIONS: Trying to place scientific notation in a point of an expression, where it cannot be placed, will generate an error message.
The "toggle positive / negative" button is used for switching the sign of the last entered number. The sign of a bracket cannot be switched. To make a bracket negative use the "minus" button before opening a bracket. For more information about negative vaules see the above section "Negative numbers".
The "bracket" buttons are used for opening and closing brackets. Brackets are useful when the order of calculation should diverge from the standard order of operations. If for example in the expression 5 + 3 / 2 the term 5 + 3 should be evaluated first and then be divided by 2, then you it must be entered as (5 + 3) / 2.
Once a bracket has been opened, numbers will apear beside the bracket symbols on the bracket buttons, which indicate how many brackets have been opened or closed. The number beside the "close bracket" will be red as long as not all opened brackets have been closed. Opened brackets will be closed automatically upon evaluating the expression using the "is equal to" button. Once a bracket has been closed the output will be updated to reflect the value of the closed bracket.
The sign of a bracket cannot be switched. To make a bracket negative use the "minus" button before opening a bracket. For more information about negative vaules see the above section "Negative numbers".
RESTRICTIONS: Trying to open or close a bracket at a point of an expression, where a bracket cannot be placed, will generate an error message.
The "x percent of y" button is used for calculating a percentage of a value. For example to calculate 5% of 50 (which is 2.5), enter .
The "how many percent of y is x?" button is used for calculating how many percent one value of another value constitutes. For example to find out how many percent of 50 the value 5 constitutes (which is 10%), enter .
For more information about percentages see the above section "Percentages".
The "ceiling" button is used for obtaining the ceiling of a value. The ceiling of a value is the smallest integer (whole number or negative) not less than the value. For example the ceiling of 4.2 is 5 and the ceiling of -4.2 is -4.
The "floor" button is used for obtaining the floor of a value. The floor of a value is the highest integer (whole number or negative) less than or equal to the value. For example the floor of 4.2 is 4 and the floor of -4.2 is -5.
The "factorial" button is used for calculating the factorial of a value (for integer values) or for calculating (an aproximation of) the gamma function for the value + 1 for values greater than or equal to zero and non-integer values smaller than zero. The algorithm used for this operation is based on the "StieltjesLnFactorial" algorithm by Peter Luschny.
RESTRICTIONS: The value must be greater than or equal to zero, or non-integer smaller than zero. Integer values smaller than zero will generate an error and terminate the calculation. | 677.169 | 1 |
This fall I will be teaching a new course entitled "Applied Mathematics" which is intended for students who demonstrate a...
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This fall I will be teaching a new course entitled "Applied Mathematics" which is intended for students who demonstrate a need to reduce the Algebra II requirement in the Michigan Merit Curriculum due to academic difficulty in Algebra I and/or Geometry. The course features interwoven strands of algebra and functions, statistics, and probability, with a focus on applications of mathematics. Students will learn to recognize and describe important patterns that relate quantitative variables and develop strategies to make sense of real-world data. The course will develop students' abilities to solve problems involving chance and to approximate solutions to more complex probability problems by using simulation. The goal that will be addressed in this lesson is to review Algebra I fundamentals, more specifically mathematical models (price-demand model, formulas as models, and operations with real numbers) to lay the foundation for Lesson Plan to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Applied Math Lesson Plan
Select this link to open drop down to add material Applied Math Lesson Plan to your Bookmark Collection or Course ePortfolio
This is a lesson plan for solving systems of linear equations to Algebra 1 students. Methods to be taught include...
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This is a lesson plan for solving systems of linear equations to Algebra 1 students. Methods to be taught include substitution, elimination, and graphing. The lesson also includes the use of graphing calculators and spreadsheets to solve systems of equations. The lesson involves practice with real world application problems, as well as creation and presentation of original problems Systems of Equations Lesson Plan to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Systems of Equations Lesson Plan
Select this link to open drop down to add material Systems of Equations Lesson Plan to your Bookmark Collection or Course ePortfolio
The goal for this lesson is to provide students with an understanding of how to find the area of any regular polygon. This is...
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The goal for this lesson is to provide students with an understanding of how to find the area of any regular polygon. This is a discovery-based lesson in which students collaborate with their peers and test ideas using Area of Regular Polygons to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Area of Regular Polygons
Select this link to open drop down to add material Area of Regular Polygons to your Bookmark Collection or Course ePortfolio
The goal that will be addressed during this lesson plan is to provide the concepts, methods, and tools for understanding...
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The goal that will be addressed during this lesson plan is to provide the concepts, methods, and tools for understanding characteristics and transformations of graphs from their parent functions using a goal-directed instructional design Functions- Goal-directed Instructional Design Plan to your Bookmark Collection or Course ePortfolio
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This lesson plan was developed to help early grade school teachers demonstrate to students how to use simple deductive logic...
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This lesson plan was developed to help early grade school teachers demonstrate to students how to use simple deductive logic to solve a simple 4x4 matrix Sudoku puzzle. There is a description of instructional objectives as they apply to the NET-S and to the Michigan Grade level Content Expectations. There is a description of important content for the learner to grasp in order to complete the objectives. There is a PDF of an example puzzle to use as practice or assessment. There is a link to developmentally appropriate puzzle web sites to be used as practice or assessment. Finally, there is a brief description of an instructional a 4x4 Sudoku Puzzle to your Bookmark Collection or Course ePortfolio
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This classroom activity presents College Algebra students with a ConcepTest, a Question of the Day, and a Write-pair-share...
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This classroom activity presents College Algebra students with a ConcepTest, a Question of the Day, and a Write-pair-share activity concerning the effect of the coefficient of x on the vertex of a parabola where a>0, b<0 and a and c are arbitrarily fixed values in f(x)=ax^2+bx+c Coefficient of x on Parabola Vertex (b<0) to your Bookmark Collection or Course ePortfolio
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This series of lectures, created by Salman Khan of the Khan Academy, focuses on topics covered in a first year course in differential equations. A basic understanding of differentiation and integration from Calculus...
This geometry lesson introduces the Power of Points theorem. While it is often taught in three parts (the Chord-Chord Power theorem, the Secant-Secant Power theorem, and the Tangent-secant Power theorem), this lesson... lesson plan involves comparing different methods to determine the shortest route when traveling from Cleveland to Boston. Students will be given the opportunity to interpret data presented in table and graph format... | 677.169 | 1 |
Precise Calculator has arbitrary precision and can calculate with complex numbers, fractions, vectors and matrices. Has more than 150 mathematical functions and statistical functions and is programmable (if, goto, print, return, for). | 677.169 | 1 |
Practice Makes Perfect in Geometry: Three-Dimensional Figures
One of the best ways to succeed in Geometry is to practice taking real test questions. This volume contains 133 problems on Three-Dimensional Figures divided into four chapters: Definitions and Shapes; Rectangular Solids; Cylinders, Cones, Spheres; and Prisms and Pyramids. Try the problems. With a little Practice, Practice, Practice, you'll be Perfect, Perfect, Perfect. Good Luck!!!
About the author: Mr. Parnell holds teaching certification in physics, chemistry, biology, general science, mathematics, and business and distributed education. Over the course of forty years of teaching, he has taught students from the sixth grade through graduate school.
In addition, Mr. Parnell has taught for the New York State Research Foundation preparing students for the SAT exam in mathematics. | 677.169 | 1 |
Without a basic understanding of maths, students of any science discipline are ill-equipped to tackle new problems or to apply themselves to novel situations. In this book, Keith Gregson covers a few essential topics that will help encourage an understanding of mathematics so that the student can build on their understanding and apply it to their... more...
For more than two thousand years a familiarity with mathematics has been regarded as an indispensable part of the intellectual equipment of every cultured person. Today, unfortunately, the traditional place of mathematics in education is in grave danger. The teaching and learning of mathematics has degenerated into the realm of rote memorization, the purpose of this book is to give a comprehensive exposition of the theory of boundary integral equations for single and double layer potentials on curves with exterior and interior cusps. The theory was developed by the authors during the last twenty years and the present volume is based on their results. The first three chapters are devoted to... more...
With the advent of rich Internet applications, the explosion of social media, and the increased use of powerful cloud computing infrastructures, a new generation of attackers has added cunning new techniques to its arsenal. For anyone involved in defending an application or a network of systems, Hacking: The Next Generation is one of the few books... more...
This Wrox Blox is a value-packed resource to help experienced .NETdevelopers learn the new .NET release. It is excerpted from theWrox books: Professional C# 4 and .NET 4, Professional ASP.NET4, and WPF Programmer's Reference by Christian Nagel,Bill Evjen, Scott Hanselman, and Rod Stephens, and includes morethan 100 print book pages drawn from... more... | 677.169 | 1 |
This material is a part of a collection of learning materials at the site of Rio Rancho Math Camp. The site provides ״an...
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This material is a part of a collection of learning materials at the site of Rio Rancho Math Camp. The site provides ״an intellectually stimulating experience for middle school students who like math, and want to experience hands on, challenging math activities." The Topology topic is designed to give a student simple yet insightful introduction on the subsject of Topology. It contains illustrations, videos, and interactive exervises Math Wizards: Topology to your Bookmark Collection or Course ePortfolio
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Forum for Science, Industry and Business
The Aftermath of Calculator Use in College Classrooms
13.11.2012
Students may rely on calculators to bypass a more holistic understanding of mathematics, says Pitt researcher
Math instructors promoting calculator usage in college classrooms may want to rethink their teaching strategies, says Samuel King, postdoctoral student in the University of Pittsburgh's Learning Research & Development Center.
King has proposed the need for further research regarding calculators' role in the classroom after conducting a limited study with undergraduate engineering students published in the British Journal of Educational Technology.
"We really can't assume that calculators are helping students," said King. "The goal is to understand the core concepts during the lecture. What we found is that use of calculators isn't necessarily helping in that regard."
Together with Carol Robinson, coauthor and director of the Mathematics Education Centre at Loughborough University in England, King examined whether the inherent characteristics of the mathematics questions presented to students facilitated a deep or surface approach to learning. Using a limited sample size, they interviewed 10 second-year undergraduate students enrolled in a competitive engineering program. The students were given a number of mathematical questions related to sine waves—a mathematical function that describes a smooth repetitive oscillation—and were allowed to use calculators to answer them. More than half of the students adopted the option of using the calculators to solve the problem.
"Instead of being able to accurately represent or visualize a sine wave, these students adopted a trial-and-error method by entering values into a calculator to determine which of the four answers provided was correct," said King. "It was apparent that the students who adopted this approach had limited understanding of the concept, as none of them attempted to sketch the sine wave after they worked out one or two values."
After completing the problems, the students were interviewed about their process. A student who had used a calculator noted that she struggled with the answer because she couldn't remember the "rules" regarding sine and it was "easier" to use a calculator. In contrast, a student who did not use a calculator was asked why someone might have a problem answering this question. The student said he didn't see a reason for a problem. However, he noted that one may have trouble visualizing a sine wave if he/she is told not to use a calculator.
"The limited evidence we collected about the largely procedural use of calculators as a substitute for the mathematical thinking presented indicates that there might be a need to rethink how and when calculators may be used in classes—especially at the undergraduate level," said King. "Are these tools really helping to prepare students or are the students using the tools as a way to bypass information that is difficult to understand? Our evidence suggests the latter, and we encourage more research be done in this area."
King also suggests that relevant research should be done investigating the correlation between how and why students use calculators to evaluate the types of learning approaches that students adopt toward problem solving in mathematics | 677.169 | 1 |
Careers by Major
For each major that interests you, choose "Information" to find an outline of common career areas, typical employers, and strategies designed to maximize career opportunities. Choose "Links" to find a list of Web sites that provide information about listed majors and related careers.
Keep in mind information sheets and websites are representative of typical career paths associated with each major and not a comprehensive list. You may want to explore information and websites from multiple majors to help you learn about a wide range of career opportunities.
Disclaimer: Please note that the websites listed under "Links" are not maintained by the Powell Resource Center, but are provided as a convenience to students.
Adobe Acrobat Reader is required to view the Areas of Employment, Employers and Strategies Information. You can download this free from the Adobe site.
What is Mathematics?
Mathematics is the study of the measurement, properties, and relationships of quantities and sets, using numbers and symbols. It is a body of related courses concerned with knowledge of measurement, properties, and relations quantities, which can include theoretical or applied studies of arithmetic, algebra, geometry, trigonometry, statistics, and calculus.
Mathematicians have an opportunity to make a lasting contribution to society by helping to solve problems in such diverse fields as medicine, management, economics, government, physics, and psychology. | 677.169 | 1 |
Linear Algebra
9780135367971
ISBN:
0135367972
Edition: 2 Pub Date: 1971 Publisher: Prentice Hall
Summary: This introduction to linear algebra features intuitive introductions and examples to motivate important ideas and to illustrate the use of results of theorems. Linear Equations; Vector Spaces; Linear Transformations; Polynomials; Determinants; Elementary canonical Forms; Rational and Jordan Forms; Inner Product Spaces; Operators on Inner Product Spaces; Bilinear Forms For all readers interested in linear algebra. ...> Hoffman, Kenneth is the author of Linear Algebra, published 1971 under ISBN 9780135367971 and 0135367972. Five hundred thirty five Linear Algebra textbooks are available for sale on ValoreBooks.com, twenty two used from the cheapest price of $40.98, or buy new starting at $179.54 | 677.169 | 1 |
Transcript
1. Interactive Exercises Set 2 <ul><li> </li></ul><ul><li> </li></ul>Exercise 2.1 Exercise 2.2 Exercise 2.3 Exercise 2.4 Exercise 2.5 There are five exercises in this set. Each question is followed by three answers. The best answer is worth ten points; the second best answer is worth five points; the remaining answer is zero. The answers are given following each question. Keep track of your score.
13. Interactive Exercise 2.2 Current Catalog Description Introduction to a broad range of topics in Discrete Mathematics. Textbook Rosen, K., "Discrete Mathematics and Its Applications (Third Edition)", McGraw-Hill; 1995. References Three books on Discrete Mathematics topics are placed on reserve in the campus library. (A) = 0 points Needs to be more specific; topics should be listed. Textbook title should be italicized. Books should be listed.
16. Exercise 2.3 Suppose MATH 2240 Pre-calculus is a prerequisite for CSCI 1301 Programming; CSCI 1301 Programming is a prerequisite for all CSCI courses; and CSCI 3202 Architecture is a prerequisite for both CSCI 4120 Operating Systems and CSCI 4350 Artificial Intelligence. Which page should be used as Appendix H: Prerequisite Structure for Computer Science Courses? Interactive Exercise 2.3 | 677.169 | 1 |
SHARP SCIENTIFIC CALCULATOR
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Foundations of Mathematical & Computational Economics
9780324235838
ISBN:
0324235836
Edition: 1 Pub Date: 2006 Publisher: Thomson Learning
Summary: Economics doesn't have to be a mystery anymore. FOUNDATIONS OF MATHEMATICAL AND COMPUTATION ECONOMICS shows you how mathematics impacts economics and econometrics using easy-to-understand language and plenty of examples. Plus, it goes in-depth into computation and computational economics so you'll know how to handle those situations in your first economics job. Get ready for both the test and the workforce with this ...economics textbook.
Dadkhah, Kamran is the author of Foundations of Mathematical & Computational Economics, published 2006 under ISBN 9780324235838 and 0324235836. Two hundred twenty nine Foundations of Mathematical & Computational Economics textbooks are available for sale on ValoreBooks.com, sixty used from the cheapest price of $4.35, or buy new starting at $22.13.[read more | 677.169 | 1 |
Algebra for College Students
9780495105107
ISBN:
0495105104
Edition: 8 Pub Date: 2006 Publisher: Thomson Learning
Summary: Kaufmann and Schwitters have built this text's reputation on clear and concise exposition, numerous examples, and plentiful problem sets. This traditional text consistently reinforces the following common thread: learn a skill; use the skill to help solve equations; and then apply what you have learned to solve application problems. This simple, straightforward approach has helped many students grasp and apply fundam...ental problem solving skills necessary for future mathematics courses in an easy-to-read format. The new Eighth Edition of ALGEBRA FOR COLLEGE STUDENTS includes new and updated problems, revised content based on reviewer feedback and a new function in iLrn. This enhanced iLrn homework functionality was designed specifically for Kaufmann/Schwitters' users. Textbook-specific practice problems have been added to iLrn to provide additional, algorithmically-generated practice problems, along with useful support and assistance to solve the problems for students.
Kaufmann Schwitters Staff is the author of Algebra for College Students, published 2006 under ISBN 9780495105107 and 0495105104. Thirteen Algebra for College Students textbooks are available for sale on ValoreBooks.com, eleven used from the cheapest price of $0.04, or buy new starting at $86.90.[read more brand new condition. FAST, reliable shipping with free tracking and a 100% money-back guarantee from a socially responsible company with thousands of happy customers. E [more]
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0471867Further Mathematics for the Physical Sciences
Further Mathematics for the Physical Sciences Further Mathematics for the Physical Sciences aims to build upon the readera s knowledge of basic mathematical methods, through a gradual progression to more advanced methods and techniques. Carefully structured as a series of self--paced and self--contained chapters, this text covers the essential and most important techniques needed by physical science students. Starting with complex numbers, the text then moves on to cover vector algebra, determinants, matrices, differentiation, integration, differential equations and finally vector calculus, all within an applied environment. The reader is guided through these different techniques with the help of numerous worked examples, applications, problems, figures and summaries. The authors aim to provide high--quality and thoroughly class--tested material to meet the changing needs of science students. Further Mathematics for the Physical Sciences: aeo Is a carefully structured text, with self--contained chapters. aeo Gradually introduces mathematical techniques within an applied environment. aeo Includes many worked examples, applications, problems and summaries in each chapter. Further Mathematics for the Physical Sciences will be invaluable to all students of physics, chemistry and engineering, needing to develop or refresh their knowledge of basic mathematics. The booka s structure will make it equally valuable for course use, home study or distance learning | 677.169 | 1 |
we try to provide lesson 6 3 solving systems by elimination practice c related manual download for free. if you can't find ebooks that you are looking for, try to use our search form on left top of this page.
Solving Systems of Linear Equations; Row Reduction
1hits
description:
gauss jordan elimination method calculator The method reviewed here can be implemented to solve a linear system a11x1 .... We will use Gaussian Elimination to solve the linear system x1 . In the Exploration, use the Row Reduction Calculator to practice solving systems of linear.
Lesson - Math Problem Solving - cccoe net
1hits
description:
multiplication story problems 3rd grade Students will model the process of multiplication. Students will practice solving story problems. California Content Standards: California 3rd Grade State ...
211 LECTURE 15 Gauss-Jordan Elimination Today we study an
1hits
description:
gauss jordan elimination Gauss-Jordan Elimination. Today we study an efficient method for solution of systems of linear equations. It is called. Gauss-Jordan Elimination. We begin with a ... | 677.169 | 1 |
An 8th grade algebra class, in other words, might be matched with a 7th grade prealgebra class, or a 7th grade general-math class paired with an 8th grade prealgebra or general-math class in which students had similar achievement levels.
Learning Games to Go initiative, which aims to create learning games that support students 'prealgebra and reading skills, the game is aligned with Maryland State Voluntary Curriculum standards as well as national standards.
The game addresses three prealgebra topics-proportions, variables and equations, and numbers and operations-and the students must complete three puzzles, which are designed to enforce the concepts behind the topics, in order to move on.
It is intended for students who (1) have had a previous course in prealgebra, (2) wish to meet the prerequisite of a higher level course such as elementary algebra, and (3) need to review fundamental mathematical concepts and techniques.
Fundamentals of Mathematics is a work text that covers the traditional topics studied in a modern prealgebra course, as well as topics of estimation, elementary analytic geometry, and introductory algebra. | 677.169 | 1 |
Description
Copyright 2012
Dimensions: 6-3/4 X 9-1/2
Pages: 616
Edition: 3rd
Book
ISBN-10: 0-273-75058-5
ISBN-13: 978-0-273-75058-1
Success in today's sophisticated financial markets depends on a firm understanding of key financial concepts and mathematical techniques. Mastering Financial Calculations explains them in a clear, comprehensive way — so even if your mathematical background is limited, you'll thoroughly grasp what you need to know.
Mastering Financial Calculations starts by introducing the fundamentals of financial market arithmetic, including the core concepts of discounting, net present value, effective yields, and cash flow analysis. Next, walk step-by-step through the essential calculations and financial techniques behind money markets and futures, zero-coupon analysis, interest rate and currency swaps, bonds, foreign exchange, options, and more. Making use of many worked examples and practical exercises, the book explains challenging concepts such as forward pricing, duration analysis, swap valuation, and option pricing - all with exceptional clarity.
Whether you are a trader, fund manager, corporate treasurer, programmer, accountant, risk manager, or market student, you'll gain the ability to manipulate and apply these techniques with speed and confidence | 677.169 | 1 |
from 100 providers and counting...
Complex analysis is the study of functions that live in the complex plane, i.e. functions that have complex arguments and complex outputs. In order to study the behavior of such functions we'll need to first understand the basic objects involved, namely the complex numbers. We'll begin with some history: When and why were complex numbers invented? Was it the need for a solution of the equation x^2 = -1 that brought the field of complex analysis into being, or were there other reasons?
Analytic Combinatorics is based on formal methods for deriving functional relationships on generating functions and asymptotic analysis treating those functions as functions in the complex plane. This course covers the symbolic method for defining generating functions immediately from combinatorial constructions, then develops methods for directly deriving asymptotic results from those generating functions, using complex asymptotics, singularity analysis, saddle-point asymptotics, and limit laws. The course teaches the precept "if you can specify it, you can analyze it".
The period of the demise of the Kingdom of Judah at the end of the sixth
century B.C.E., the fall of Jerusalem to the Babylonians, the exile of
the elite to Babylon, and the reshaping of the territory of the new province
of Judah, culminating at the end of the century with the first return of
exiles – all have been subjects of intense scrutiny in modern scholarship.
This course takes into account the biblical textual evidence, the results
of archaeological research, and the reports of the Babylonian and Egyptian | 677.169 | 1 |
Mathcad Prime 3.0 Essentials
In this course, you will learn the basics of Mathcad Prime. You will learn about Mathcad Prime's extensive functionality, such as opening and working with Mathcad files, navigating workspaces, defining variables and expressions, and solving equations. In addition, you will learn how to plot graphs, solve for roots, and manipulate data. At the end of each module, you will complete a set of review questions to reinforce critical topics from that module. At the end of the course, you will complete a course assessment in Pro/FICIENCY intended to evaluate your understanding of the course as a whole.
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PTC University has many options to get you the right training without leaving your office. Learn more about Virtual Classes, On-site training, and eLearning.
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PTC University Training Advisors are here to help you find the right training. | 677.169 | 1 |
Author(s): Musser, Gary L.; Peterson, Blake E.; Burger, William FMathematics for Elementary Teachers: A Contemporary Approach, 10th Edition makes readers motivated to learn mathematics. With new-found confidence, they are better able to appreciate the beauty and excitement of the mathematical world. The new edition of Musser, Burger, and Peterson's best-selling textbook focuses on one primary goal: helping students develop a true understanding of central concepts using solid mathematical content in an accessible and appealing format. The components in this complete learning program work in harmony to help achieve this goal. The Tenth Edition features the Common Core Standards to accompany the NCTM standards that are integrated throughout the text. | 677.169 | 1 |
CH 8 Infinite Sequences and Series
Sequence
Series
The Integral Test and Estimates of Sums
The Comparison Test
Alternating Series
Absolute Convergence and The Ratio and Root Test
Power Series
Taylor and Maclaurin Series | 677.169 | 1 |
Highlights of Calculus is a series of short videos that introduces the basic ideas of calculus — how it works and why it is...
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Highlights of Calculus is a series of short videos that introduces the basic ideas of calculus — how it works and why it is important. The intended audience is high school students, college students, or anyone who might need help understanding the subject.In addition to the videos, there are summary slides and practice problems complete with an audio narration by Professor Strang. You can find these resources to the right of each video.This resource is also available on Highlights for High School.About the InstructorProfessor Gilbert Strang is a renowned mathematics professor who has taught at MIT since 1962. Read more about Prof. StrangAcknowledgementsSpecial thanks to Professor J.C. Nave for his help and advice on the development and recording of this program.The video editing was funded by the Lord Foundation of Massachusetts to your Bookmark Collection or Course ePortfolio
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a free online course offered by the Saylor Foundation.'"Everything is numbers." This phrase was uttered by the lead...
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This is a free online course offered by the Saylor Foundation.'"Everything is numbers." This phrase was uttered by the lead character, Dr. Charlie Epps, on the hit television show "NUMB3RS." If everything has a mathematical underpinning, then it follows that everything is somehow mathematically connected, even if it is only in some odd, "six degrees of separation (or Kevin Bacon)" kind of way.Geometry is the study of space (for now, mainly two-dimensional, with some three-dimensional thrown in) and the relationships of objects contained inside. It is one of the more relatable math courses, because it often answers that age-old question, "When am I ever going to use this in real life?" Look around you right now. Do you see any triangles? Can you spot any circles? Do you see any books that look like they are twice the size of other books? Does your wall have paint on it?In geometry, you will explore the objects that make up our universe. Most people never give a second thought to how things are constructed, but there are geometric rules at play. Most people never think twice about a rocket launch, but if that rocket is not launched at an exact angle, it will miss its target. A football field has to be measured out to be a rectangle; if you used another shape, such as a trapezoid, that would give an unfair advantage to one team, because that one team would have more space to work with.In this course, you will study the relationships between lines and angles. Have you ever looked at a street map? Believe it or not, there is a lot of geometry on a map, as you will see from this course. You will learn to calculate how much space an object covers, which is useful if you ever have to, say, buy some paint. You will learn to determine how much space is inside of a three-dimensional object, which is useful for those times you are trying to fit four suitcases, three kids, two adults, and a dog into the back of your vehicle.These are just some of the topics you will be learning. As you will quickly see, everything is not just numbers; it is also relationships. Even nature itself knows this. What did the little acorn say when it grew up? "Gee, I'm a tree!" to your Bookmark Collection or Course ePortfolio
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This series is one part of UC Irvine's Musicianship 15 ABC sequence for music majors. An understanding of music notation and...
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This series is one part of UC Irvine's Musicianship 15 ABC sequence for music majors. An understanding of music notation and basic musical terms is helpful but not required for these presentations. The math involved is basic. Pitch systems use mathematics to organize audible phenomenon for creative expression. The cognitive processes we develop through exposure to music comprise a kind of applied mathematics; our emotional responses to musical nuance grow out of a largely unconscious mastery of the patterns and structures in music. This series of presentations covers the basic mathematics and cognitive phenomenon found in the tonal system used in Western music and much of the music of the world. Over the course of several presentations we will explore basic concepts of pitch and frequency, the organizing rules of tonal systems, and the mathematical construction of basic scales and chords. The reasoning and purpose of equal temperament, the standard tuning system for tonal music, will be explored in this context. Presentations will include graphics and computer applications designed specifically to illustrate these link to open drop down to add material Introduction to Pitch Systems in Tonal Music to your Bookmark Collection or Course ePortfolio
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This subject provides an introduction to the mechanics of materials and structures. You will be introduced to and become...
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This subject provides an introduction to the mechanics of materials and structures. You will be introduced to and become familiar with all relevant physical properties and fundamental laws governing the behavior of materials and structures and you will learn how to solve a variety of problems of interest to civil and environmental engineers. While there will be a chance for you to put your mathematical skills obtained in 18.01, 18.02, and eventually 18.03 to use in this subject, the emphasis is on the physical understanding of why a material or structure behaves the way it does in the engineering design of materials and structures050 Engineering Mechanics I to your Bookmark Collection or Course ePortfolio
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The course material emphasizes mathematical models for predicting distribution and fate of effluents discharged into lakes,...
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The course material emphasizes mathematical models for predicting distribution and fate of effluents discharged into lakes, reservoirs, rivers, estuaries, and oceans. It also focuses on formulation and structure of models as well as analytical and simple numerical solution techniques. Also discussed are the role of element cycles, such as oxygen, nitrogen, and phosphorus, as water quality indicators; offshore outfalls and diffusion; salinity intrusion in estuaries; and thermal stratification, eutrophication, and sedimentation processes in lakes and reservoirs. This course is a core requirement for the Environmental MEng77 Water Quality Control to your Bookmark Collection or Course ePortfolio
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This course introduces programming languages and techniques used by physical scientists: FORTRAN, C, C++, MATLAB®, and...
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This course introduces programming languages and techniques used by physical scientists: FORTRAN, C, C++, MATLAB®, and Mathematica. Emphasis is placed on program design, algorithm development and verification, and comparative advantages and disadvantages of different010 Computational Methods of Scientific Programming (MIT) to your Bookmark Collection or Course ePortfolio
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The subject introduces the principles of ocean surface waves and their interactions with ships, offshore platforms and...
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The subject introduces the principles of ocean surface waves and their interactions with ships, offshore platforms and advanced marine vehicles. Surface wave theory is developed for linear and nonlinear deterministic and random waves excited by the environment, ships, or floating structures. Following the development of the physics and mathematics of surface waves, several applications from the field of naval architecture and offshore engineering are addressed. They include the ship Kelvin wave pattern and wave resistance, the interaction of surface waves with floating bodies, the seakeeping of ships high-speed vessels and offshore platforms, the evaluation of the drift forces and other nonlinear wave effects responsible for the slow-drift responses of compliant offshore platforms and their mooring systems designed for hydrocarbon recovery from large water depths. This course was originally offered in Course 13 (Department of Ocean Engineering) as 13.022. In 2005, ocean engineering subjects became part of Course 2 (Department of Mechanical Engineering), and this course was renumbered 2.24.24 Ocean Wave Interaction with Ships and Offshore Energy Systems (13.022) to your Bookmark Collection or Course ePortfolio
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This course provides an integrated introduction to electrical engineering and computer science, taught using substantial...
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This course provides an integrated introduction to electrical engineering and computer science, taught using substantial laboratory experiments with mobile robots. Our primary goal is for you to learn to appreciate and use the fundamental design principles of modularity and abstraction in a variety of contexts from electrical engineering and computer science. Our second goal is to show you that making mathematical models of real systems can help in the design and analysis of those systems. Finally, we have the more typical goals of teaching exciting and important basic material from electrical engineering and computer science, including modern software engineering, linear systems analysis, electronic circuits01SC Introduction to Electrical Engineering and Computer Science I (MIT) to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material 6.01SC Introduction to Electrical Engineering and Computer Science I (MIT)
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The study of "abstract algebra" grew out of an interest in knowing how attributes of sets of mathematical objects behave when...
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The study of "abstract algebra" grew out of an interest in knowing how attributes of sets of mathematical objects behave when one or more properties we associate with real numbers are restricted. The student will begin this course by reviewing basic set theory, integers, and functions in order to understand how algebraic operations arise and are used. The student then will proceed to the heart of the course, which is an exploration of the fundamentals of groups, rings, and fields. This free course may be completed online at any time. See course site for detailed overview and learning outcomes. (Mathematics 231 I to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Abstract Algebra I
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Accessible to all students with a sound background in high school mathematics, A Concise Introduction to Pure Mathematics, Third Edition presents some of the most fundamental and beautiful ideas in pure mathematics. It covers not only standard material but also many interesting topics not usually encountered at this level, such as the theory of solving cubic equations, the use of Euler's formula to study the five Platonic solids, the use of prime numbers to encode and decode secret information, and the theory of how to compare the sizes of two infinite sets.
New to the Third Edition The third edition of this popular text contains three new chapters that provide an introduction to mathematical analysis. These new chapters introduce the ideas of limits of sequences and continuous functions as well as several interesting applications, such as the use of the intermediate value theorem to prove the existence of nth roots. This edition also includes solutions to all of the odd-numbered exercises.
By carefully explaining various topics in analysis, geometry, number theory, and combinatorics, this textbook illustrates the power and beauty of basic mathematical concepts. Written in a rigorous yet accessible style, it continues to provide a robust bridge between high school and higher level mathematics, enabling students to study further courses in abstract algebra and analysis.
$54.2454.24,"ASIN":"1439835985","moqNum":1,"isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":29.96,"ASIN":"0521675995","moqNum":1,"isPreorder":0}],"shippingId":"1439835985::34aRp7oe1duUJdM3EQphHwj8jf0MOn3536RRaM3MM%2B%2FI0DXBtsvykpsIwDMbO2PfL7lmnxDi%2Ft3DH0DiVeAcZVTvztIlSZCnT%2B2OmRl4641qk2bQTCnykA%3D%3D,0521675995::LsSaftQdXig3UfquwqM91bg5ABL0dD0HUGmpkF3vHQUiTDlIrGXbY5WT5VpnrfOjTDkLkoqXgXbkLbVyA27Zxt3zvSwfvFM6LCt5mqi0Y%2B9AQ6ovQ8myIt would in fact be difficult to find in this excellent book three consecutive pages that do not contain material useful to students or practitioners. … A diligent, active reader of this outstanding book will have the best foundation at minimum cost for making meaningful contributions to mathematics, science, or engineering. ―Computing Reviews, November 2011
Now in an updated and expanded third edition, A Concise Introduction to Pure Mathematics provides an informed and informative presentation into a representative selection of fundamental ideas in mathematics … . Of special note is the inclusion of solutions to all of the odd-numbered exercises. An ideal, accessible, elegant, student-friendly, and highly recommended choice for classroom textbooks for high school and college level mathematics curriculums, A Concise Introduction to Pure Mathematics is further enhanced with a selective bibliography, an index of symbols, and a comprehensive index. ―Library Bookwatch, December 2010
This book displays a unique combination of lightness and rigor, leavened with the right dose of humor. When I used it for a course, students could not get enough, and I have been recommending independent study from it to students wishing to take a core course in analysis without having taken the prerequisite course. The material is very well chosen and arranged, and teaching from Liebeck's book has in many different ways been among my most rewarding teaching experiences during the last decades. ―Boris Hasselblatt, Tufts University, Medford, Massachusetts, USA
In addition to preparing students to go on in mathematics, it is also a wonderful choice for a student who will not necessarily go on in mathematics but wants a gentle but fascinating introduction into the culture of mathematics. … This book will give a student the understanding to go on in further courses in abstract algebra and analysis. The notion of a proof will no longer be foreign, but also mathematics will not be viewed as some abstract black box. At the very least, the student will have an appreciation of mathematics. As usual, Liebeck's writing style is clear and easy to read. This is a book that could be read by a student on his or her own. There is a wide selection of problems ranging from routine to quite challenging. ―From the Foreword by Robert Guralnick, University of Southern California, Los Angeles, USA
Praise for Previous Editions: The book will continue to serve well as a transitional course to rigorous mathematics and as an introduction to the mathematical world … . ―Gerald A. Heuer, Zentralblatt MATH, 2009
…a pleasure to read … a very welcome and highly accessible book. ―Michael Ward, The Mathematical Gazette, March 2007
About the Author
Martin Liebeck is a professor and head of the Pure Mathematics Section in the Department of Mathematics at Imperial College London. He earned his B.A., M.Sc., and D.Phil. in mathematics from the University of Oxford. Dr. Liebeck has published over 100 research articles and seven books. His research interests encompass algebraic groups, finite simple groups, probabilistic group theory, permutation groups, and algebraic combinatoricsNow in an updated and expanded third edition, "A Concise Introduction To Pure Mathematics" by Martin Liebeck provides an informed and informative presentation into a representative selection of fundamental ideas in mathematics including the theory of solving cubic equations, the use of Euler's formula to study the five Platonic solids, the use of prime numbers to encode and decode secret information, the theory of how to compare the sizes of two infinite sets, the limits of sequences and continuous functions, the use of the intermediate value theorem to prove the existence of nth roots, and so much more. Of special note is the inclusion of solutions to all of the odd-numbered exercises. An ideal, accessible, elegant, 'student friendly', and highly recommended choice for classroom textbooks for highschool and college level mathematics curriculums, "A Concise Introduction To Pure Mathematics" is further enhanced with a selective bibliography, and index of symbols, and a comprehensive index.
I have not yet finished the book. However, so far every chapter introduces and reviews a familiar concept, and gradually takes it to a whole new level. A lot of the book deals with proofs, starting with simple statements and using those proofs to go further in depth. There is a wonderful bit of humor in each lesson, and a lot of the questions at the end of each chapter are really thought-provoking (answers are only given to odd numbered problems). Although this probably sounds a bit cheesy, this book has already given me a greater appreciation of the beauties and wonders of math.
Cool book! I enjoy reading so-called pure math, though I must admit to some disappointment whenever I open such a book and find that it's full of . . . numbers! Any mathematician will tell you that math isn't about the objects (of which numbers are a tiny subset) but about the rules for combining those objects. This being said, there's a lot to enjoy in this book. I also recommend George Exner's An Accompaniment to Higher Mathematics. That really does take a more "pure" approach to this world. Maybe most fun (?) of all is The Princeton Companion to Mathematics edited by Timothy Gower, a very well-bound book that interlaces so many different fields of math, with excellent introductions, and a 1,000 page hardcover bargain at $84! | 677.169 | 1 |
Course Objectives: The
primary goal of Math 225
is to prepare you for upper level courses in mathematics, especially
courses that rely on understanding and writing mathematics. The course
includes an introduction
to the format of definition, theorem and proof, the design and writing
of logical arguments and formal proofs,
and techniques of problem solving in higher level mathematics. During
the semester, fundamental concepts in several areas of mathematics are
introduced, partly to show the structure of mathematical
writing in different contexts, but also to lay the foundations for
higher level courses such as abstract algebra and analysis.
Additional Policies:
1. I encourage you to discuss homework with other students, or with me
during office hours. You should be aware that the homework is intended
for you to learn from the course; working on homework at least in
part on your own will help you master the material, keep up with the
course and prepare for tests.
Homework will be graded on a scale of 1-10. Points will be awarded for
amount of homework attempted; selected problems will be graded in
detail.
2. I expect you to read sections of the book around the time of
lectures and homework from those
sections. The book has additional examples and discussion that you will
find helpful. Some test questions
may resemble examples from the book.
3. You are expected to attend all classes on time. Classroom discussion
and questions in class help
clarify issues in this course, so please feel free to participate by
asking questions.
4. Arriving late for a class or leaving early is very disruptive of
class. If you need to leave
early, please let me know at the beginning of class, and sit near the
door so you can slip out
quietly.
5. If you are unavoidably absent from a test, a score for that test
will be assessed at the end of the
semester, based on your performance in homework, the other tests, and
on the final, with an emphasis on
the material of the missed test.
Attendance regulations can be found at
Further Statements: Reasonable accommodations will be made for
students with verifiable
disabilities. In order to take advantage of available
accommodations, students must register with Disability Services for
Students at 1900 Student Health Center, Campus Box 7509,
515-7653. For more information on NC State's policy on working
with students with disabilities, please see the Academic
Accommodations
for Students with Disabilities Regulation (REG02.20.1). | 677.169 | 1 |
A Survey of Mathematics with Applications
Browse related Subjects ...
Read More Important" sections throughout the text. Real-life and up-to-date examples motivate the topics throughout, and a wide range of exercises help students to develop their problem-solving and critical thinking skills. Angel Note: This is a standalone book, if you want the book/access card please order the ISBN listed below: 0321837533 / 9780321837530 A Survey of Mathematics with Applications plus MyMathLab Student Access Kit Package consists of: 0321431308 / 9780321431301 MyMathLab/MyStatLab -- Glue-in Access Card 0321654064 / 9780321654069 MyMathLab Inside Star Sticker 0321759664 / 9780321759665 Survey of Mathematics with Applications, A
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Very good. Hardcover. Has minor wear and/or markings. SKU: 978032175966559665 | 677.169 | 1 |
Navigation
CORE 8 Overview
Our text is Connected Mathematics 3 (also called CMP3). It is a series of consumable paperback texts/workbooks, which will be issued to the students one unit at a time. Students need to bring the text to class each day, as they will be taking notes in it and doing problems from it.
In this course, students will learn to solve complex math problems using a variety of mathematical knowledge and skills, collaborate with others to complete a task, and communicate effectively using the language of mathematics. Students will explore the "big ideas" in mathematics and develop the critical thinking skills needed to apply and communicate concepts in real-world situations.
Students will extend their mathematical knowledge by:
writing algebraic models from a variety of physical, numeric, and verbal descriptions | 677.169 | 1 |
Nonlinear finite elements is a course of about 150 to 200 pages designed by mechanical engineering researcher User:Banerjee. It is the best developed of a number of related mechanical engineering projects created by the same user - other projects include Introduction to Elasticity (about 100 pages) and Waves in composites and metamaterials (a 25-page lecture series). Nonlinear finite elements contains a lecture series, homework assignments and solutions. Course description: "an introductory course on nonlinear finite element analysis of solid mechanics and heat transfer problems. Nonlinearities can be caused by changes in geometry or be due to nonlinear material behavior. Both types of nonlinearities are covered in this course. The course aims to (1) provide the mathematical foundations of the finite element formulation for engineering applications (solids, heat, fluids) and (2) expose students to some of the recent trends and research areas in finite elements." For related materials, see also continuum mechanic, finite element analysis and nonlinear finite elements (category).
The Music Lesson, by Jan Vermeer van Delft. In general, paintings prior to the 20th century are in the public domain due to expiration of copyright, which means you can use images of them freely in educational materials. For works of art created in the 20th century, normally you cannot reuse any copies. Click on the image for a full size version which you can freely re-use and modify. Print it and use it for your lessons, integrate it into your pages on Wikiversity, or use it in other learning resources and websites. Use the links below to find more images like this one. | 677.169 | 1 |
Course in Enumeration
9783540390329
ISBN:
3540390324
Pub Date: 2007 Publisher: Springer
Summary: Combinatorial enumeration is a readily accessible subject full of easily stated, but sometimes tantalizingly difficult problems. This book leads the reader in a leisurely way from the basic notions to a variety of topics, ranging from algebra to statistical physics. Its aim is to introduce the student to a fascinating field, and to be a source of information for the professional mathematician who wants to learn more ...about the subject. The book is organized in three parts: Basics, Methods, and Topics. There are 666 exercises, and as a special feature every chapter ends with a highlight, discussing a particularly beautiful or famous result.
Aigner, Martin is the author of Course in Enumeration, published 2007 under ISBN 9783540390329 and 3540390324. Two hundred six Course in Enumeration textbooks are available for sale on ValoreBooks.com, fifty two used from the cheapest price of $69.88, or buy new starting at $73.45.[read more]
Ships From:Salem, ORShipping:Standard, ExpeditedComments:Almost new condition. SKU:9783540390329-2-0-3 Orders ship the same or next business day. Expedite... [more] | 677.169 | 1 |
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Show More pedagogical features are carefully coordinated to ensure that students are better able to make connections between mathematical concepts and understand the content. With a bright, appealing design, the new Fifth Edition builds on the Larson tradition of guided learning by incorporating a comprehensive range of student success materials to help develop students' proficiency and conceptual understanding of algebra. The text also continues coverage and integration of geometry in examples and exercises.
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College algebra
This book fosters the development of problem solving skills, critical thinking and communication of mathematical ides.Subjects covered include: Equations, Inequalities, and Mathematical Models, Functions and Graphs, Modeling with Polynomial and Rational Functions, Exponential and Logarithmic Functions, Matrices and Linear Systems, and Conic Sections and Nonlinear Systems.
Review: College Algebra [With CDROM]
Review: College Algebra [With CDROM]
User Review - Goodreads
this was an excellent book for when I was going to college after I graduated and got a B on my college alegbra course I gave the books to the library. than they went through them and gave them to another libraryRead full review
About the author (1998)
Bob Blitzer is a native of Manhattan and received a Bachelor of Arts degree with dual majors in mathematics and psychology (minor: English literature) from the City College of New York. His unusual combination of academic interests led him toward a Master of Arts in mathematics from the University of Miami and a doctorate in behavioral sciences from Nova University. Bob is most energized by teaching mathematics and has taught a variety of mathematics courses at Miami-Dade College for nearly 30 years. He has received numerous teaching awards, including Innovator of the Year from the League for Innovations in the Community College, and was among the first group of recipients at Miami-Dade College for an endowed chair based on excellence in the classroom. Bob has written "Intermediate Algebra for College Students, Introductory Algebra for College Students, Essentials of Intermediate Algebra for College Students, Introductory and Intermediate Algebra for College Students, Essentials of Introductory and Intermediate Algebra for College Students, Algebra for College Students, Thinking Mathematically, College Algebra, Algebra and Trigonometry," and "Precalculus," all published by Pearson Prentice Hall. | 677.169 | 1 |
Description
Your hands-on guide to real-world applications of linearalgebra
Does linear algebra leave you feeling lost? No worries--this easy-to-follow guide explains the how and the why ofsolving linear algebra problems in plain English. From matrices tovector spaces to linear transformations, you'll understand the keyconcepts and see how they relate to everything from genetics tonutrition to spotted owl extinction.Line up the basics -- discover several different approachesto organizing numbers and equations, and solve systems of equationsalgebraically or with matrices
Mary Jane SterlingWith its use of multiple variables, functions, and formulas algebra can be confusing and overwhelming to learn and easy to forget. Perfect for students who need to review or reference critical concepts, "Algebra I Essentials For Dummies" provides content focused on key topics only, with discrete explanations of critical concepts taught in a typical Algebra I course, from functions and FOILs to quadratic and linear equations."
Get a grip on all types of word problems by applying them to real life
Are you mystified by math word problems? This easy-to-understand guide shows you how to conquer these tricky questions with a step-by-step plan for finding the right solution each and every time, no matter the kind or level of problem. From learning math lingo and performing operations to calculating formulas and writing equations, you'll get all the skills you need to succeed!
Trigonometry deals with the relationship between the sides and angles of triangles... mostly right triangles. In practical use, trigonometry is a friend to astronomers who use triangulation to measure the distance between stars. Trig also has applications in fields as broad as financial analysis, music theory, biology, medical imaging, cryptology, game development, and seismology.
From sines and cosines to logarithms, conic sections, and polynomials, this friendly guide takes the torture out of trigonometry, explaining basic concepts in plain English and offering lots of easy-to-grasp example problems. It also explains the "why" of trigonometry, using real-world examples that illustrate the value of trigonometry in a variety of careers. Tracks to a typical Trigonometry course at the high school or college level Packed with example trig problems From the author of "Trigonometry Workbook For Dummies"
"Trigonometry For Dummies" is for any student who needs an introduction to, or better understanding of, high-school to college-level trigonometry.
Now, it is easier than ever before to understand complex mathematical concepts and formulas and how they relate to real-world business situations. All you have to do it apply the handy information you will find in "Business Math For Dummies." Featuring practical practice problems to help you expand your skills, this book covers topics like using percents to calculate increases and decreases, applying basic algebra to solve proportions, and working with basic statistics to analyze raw data. Find solutions for finance and payroll applications, including reading financial statements, calculating wages and commissions, and strategic salary planning.
Navigate fractions, decimals, and percents in business and real estate transactions, and take fancy math skills to work. You'll be able to read graphs and tables and apply statistics and data analysis. You'll discover ways you can use math in finance and payroll investments, banking and payroll, goods and services, and business facilities and operations. You'll learn how to calculate discounts and markup, use loans and credit, and understand the ins and outs of math for business facilities and operations. You'll be the company math whiz in no time at all! Find out how to: Read graphs and tables Invest in the future Use loans and credit Navigate bank accounts, insurance, budgets, and payroll Calculate discounts and markup Measure properties and handle mortgages and loans Manage rental and commercial properties
Complete with lists of ten math shortcuts to do in meetings and drive your coworkers nuts and ten tips for reading annual reports, "Business Math""For Dummies" is your one-stop guide to solving math problems in business situations.
Algebra II builds on your Algebra I skills to prepare you for trigonometry, calculus, and a of myriad STEM topics. Working through practice problems helps students better ingest and retain lesson content, creating a solid foundation to build on for future success.
"Algebra II Workbook For Dummies, 2nd Edition" helps you learn Algebra II by doing Algebra II. Author and math professor Mary Jane Sterling walks you through the entire course, showing you how to approach and solve the problems you encounter in class. You'll begin by refreshing your Algebra I skills, because you'll need a strong foundation to build upon. From there, you'll work through practice problems to clarify concepts and improve understanding and retention. Revisit quadratic equations, inequalities, radicals, and basic graphs Master quadratic, exponential, and logarithmic functions Tackle conic sections, as well as linear and nonlinear systems Grasp the concepts of matrices, sequences, and imaginary numbers
"Algebra II Workbook For Dummies, 2nd Edition" includes sections on graphing and special sequences to familiarize you with the key concepts that will follow you to trigonometry and beyond. Don't waste any time getting started. "Algebra II Workbook For Dummies, 2nd Edition" is your complete guide to success. manner. With just enough refresher explanations before each set of problems, you'll sharpen your skills and improve your performance. You'll see how to work with angles, circles, triangles, graphs, functions, the laws of sines and cosines, and more! 100s of Problems! * Step-by-step answer sets clearly identify where you went wrong (or right) with a problem * Get the inside scoop on graphing trig functions * Know where to begin and how to solve the most common equations * Use trig in practical applications with confidencePassing grades in two years of algebra courses are required for high school graduation. "Algebra II Essentials For Dummies" covers key ideas from typical second-year Algebra coursework to help students get up to speed. Free of ramp-up material, "Algebra II Essentials For Dummies"5 total
gerryr1
Linear Algebra Well written for a beginners book. Quick delivery from Overstock.com.
User reviews
gerryr1
Overstock.com
Linear Algebra Well written for a beginners book. Quick delivery from Overstock.com.
SimilarThis text is written for a course in linear algebra at the (U.S.) sophomore undergraduate level, preferably directly following a one-variable calculus course, so that linear algebra can be used in a course on multidimensional calculus. Realizing that students at this level have had little contact with complex numbers or abstract mathematics the book deals almost exclusively with real finite-dimensional vector spaces in a setting and formulation that permits easy generalization to abstract vector spaces. The parallel complex theory is developed in the exercises. The book has as a goal the principal axis theorem for real symmetric transformations, and a more or less direct path is followed. As a consequence there are many subjects that are not developed, and this is intentional. However a wide selection of examples of vector spaces and linear trans formations is developed, in the hope that they will serve as a testing ground for the theory. The book is meant as an introduction to linear algebra and the theory developed contains the essentials for this goal. Students with a need to learn more linear algebra can do so in a course in abstract algebra, which is the appropriate setting. Through this book they will be taken on an excursion to the algebraic/analytic zoo, and introduced to some of the animals for the first time. Further excursions can teach them more about the curious habits of some of these remarkable creatures.
This book provides students with the rudiments of Linear Algebra, a fundamental subject for students in all areas of science and technology. The book would also be good for statistics students studying linear algebra.It is the translation of a successful textbook currently being used in Italy. The author is a mathematician sensitive to the needs of a general audience. In addition to introducing fundamental ideas in Linear Algebra through a wide variety of interesting examples, the book also discusses topics not usually covered in an elementary text (e.g. the 'cost' of operations, generalized inverses, approximate solutions). The challenge is to show why the 'everyone' in the title can find Linear Algebra useful and easy to learn.The translation has been prepared by a native English speaking mathematician, Professor Anthony V. Geramita.
"A logical development of the subject . . . all the important theorems and results are discussed in terms of simple worked examples. The student's understanding . . . is tested by problems at the end of each subsection, and every chapter ends with exercises."
—CURRENT SCIENCE
A cornerstone of undergraduate mathematics, science, and engineering, this clear and rigorous presentation of the fundamentals of linear algebra is unique in its emphasis and integration of computational skills and mathematical abstractions. The power and utility of this beautiful subject is demonstrated, in particular, in its focus on linear recurrence, difference and differential equations that affect applications in physics, computer science, and economics.
• Rich selection of examples and explanations, as well as a wide range of exercises at the end of every section
• Selected answers and hints
• Excellent index
This second edition includes substantial revisions, new material on minimal polynomials and diagonalization, as well as a variety of new applications. The text will serve theoretical and applied courses and is ideal for self-study. With its important approach to linear algebra as a coherent part of mathematics and as a vital component of the natural and social sciences, Linear Algebra, Second Edition will challenge and benefit a broad audience.
Linear Algebra: A Geometric Approach, Second Edition, presents the standard computational aspects of linear algebra and includes a variety of intriguing interesting applications that would be interesting to motivate science and engineering students, as well as help mathematics students make the transition to more abstract advanced courses. The text guides students on how to think about mathematical concepts and write rigorous mathematical arguments.
This is a short text in linear algebra, intended for a one-term course. In the first chapter, Lang discusses the relation between the geometry and the algebra underlying the subject, and gives concrete examples of the notions which appear later in the book. He then starts with a discussion of linear equations, matrices and Gaussian elimination, and proceeds to discuss vector spaces, linear maps, scalar products, determinants, and eigenvalues. The book contains a large number of exercises, some of the routine computational type, while others are conceptual.
CliffsQuickReview Linear Algebra demystifies the topic with straightforward explanations of the fundamentals. This comprehensive guide begins with a close look at vector algebra (including position vectors, the cross product, and the triangle inequality) and matrix algebra (including square matrices, matrix addition, and identity matrices). Once you have those subjects nailed down, you'll be ready to take on topics such asLinear systems, including Gaussian elimination and elementary row operationsReal Euclidean vector spaces, including the nullspace of a matrix, projection into a subspace, and the Rank Plus Nullity TheoremThe determinant, including definitions, methods, and Cramer's RuleLinear transformations, including basis vectors, standard matrix, kernal and range, and compositionEigenvalues and Eigenvectors, including definitions and illustrations, Eigenspaces, and diagonalization
CliffsQuickReview Linear Algebra acts as a supplement to your textbook and to classroom lectures. Use this reference in any way that fits your personal style for study and review — the information is clearly arranged and offered in manageable units. Here are just a few of the features you'll find in this guide:A review of core conceptsClear diagrams and loads of formulasEasy to understand definitions and explanationsPlenty of examples and detailed solutions
With titles available for all the most popular high school and college courses, CliffsQuickReview guides are a comprehensive resource that can help you get the best possible grades.
Elementary Linear Algebra develops and explains in careful detail the computational techniques and fundamental theoretical results central to a first course in linear algebra. This highly acclaimed text focuses on developing the abstract thinking essential for further mathematical study. The authors give early, intensive attention to the skills necessary to make students comfortable with mathematical proofs. The text builds a gradual and smooth transition from computational results to general theory of abstract vector spaces. It also provides flexbile coverage of practical applications, exploring a comprehensive range of topics.
Includes a wide variety of applications, technology tips and exercises, organized in chart format for easy referenceMore than 310 numbered examples in the text at least one for each new concept or applicationExercise sets ordered by increasing difficulty, many with multiple parts for a total of more than 2135 questionsProvides an early introduction to eigenvalues/eigenvectorsA Student solutions manual, containing fully worked out solutions and instructors manual available | 677.169 | 1 |
Ideal for courses that require the use of a graphing calculator, PRECALCULUS: REAL MATHEMATICS, REAL PEOPLE, Alternate Edition...
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Free Delivery Worldwide : Precalculus : Undefined : McGraw-Hill Education - Europe : 9780077349912 : 0077349911 : 01 Feb 2010 : Suitable for either one or two semester college algebra with trigonometry or precalculus courses, this title introduces a unit circle approach to trigonometry and includes a chapter on limits to provide students with a foundation for calculus concepts. It employs a large number of pedagogical devices that guide a student through the course.
Ideal for courses that require the use of a graphing calculator, PRECALCULUS: REAL MATHEMATICS, REAL PEOPLE new subtitle, this...
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Free Delivery Worldwide : Precalculus : Hardback : McGraw-Hill Education - Europe : 9780073312637 : 0073312630 : 09 Feb 2007 : Part of the College Algebra series, this title introduces a unit circle approach to trigonometry and can be used in one or two semester college algebra with trig or precalculus courses. It features Explore-Discuss boxes which encourage students to think critically about mathematical concepts. It also includes a CD with practice problems.
Precalculus, Fifth Edition , by Lial, Hornsby, Schneider, and Daniels, engages and supports students in the learning process by developing both the conceptual understanding and the analytical skills necessary for success in mathematics. With the Fifth Edition , the authors adapt to the new ways in which students are learning, as well as the ever-changing classroom environment.
Free Delivery Worldwide : Precalculus : Loose-leaf : McGraw-Hill Science/Engineeri ng/Math : 9780077366490 : 0077366492 : 17 Jul 2009 : Three components contribute to a theme sustained throughout the Coburn Series: that of laying a firm foundation, building a solid framework, and providing strong connections. Not only does Coburn present a sound problem-solving process to teach students to recognize a problem, organize a procedure, and formulate a solution, the text encourages students to see beyond procedures in an effort to gain a greater understanding of the big ideas behind mathematical concepts. Written in a readable, yet ...
Free Delivery Worldwide : Precalculus: Graphs & Models : Hardback : McGraw-Hill Education - Europe : 9780073519531 : 0073519537 : 04 Mar 2011 : Benefiting from the feedback of hundreds of instructors and students across the country, this title emphasizes connections in order to improve the level of student engagement in mathematics and increase their chances of success in precalculus and calculus.
Free Delivery Worldwide : Precalculus: Graphs and Models : Hardback : McGraw-Hill Education - Europe : 9780077221294 : 007722129X : 10 Apr 2008 : The Barnett Graphs & Models Series in college algebra and precalculus maximizes student comprehension by emphasizing computational skills, real-world data analysis and modeling, and problem solving rather than mathematical theory | 677.169 | 1 |
GED science(
Visual
) 3
editions published
between
2002
and
2004
in
English
and held by
453 WorldCat member
libraries
worldwide
An in-depth preparation for the GED science test, which covers life science, earth and space science and physical science
Basic English English as a second language(
Visual
) 2
editions published
between
1989
and
1995
in
English
and held by
439 WorldCat member
libraries
worldwide
Thirty-five lesson course in basic English. Each lesson contains instruction in listening, speaking, reading, and writing. Designed for native speakers of other languages who want to learn English correctly and easily
GED calculator essentials(
Visual
) 3
editions published
between
2002
and
2004
in
English
and held by
359 WorldCat member
libraries
worldwide
Teaches precisely how to use the Casio fx-260 solar scientific calculator, the official calculator of the GED, by taking the viewer step-by-step through 75 math problems that demonstrate which keys to use and when to use them
GED mathematics(
Visual
) 3
editions published
in
2004
in
English
and held by
351 WorldCat member
libraries
worldwide
Refines skills for taking the GED mathematics test; includes arithmetic, charts and graphs, probability, statistics, algebra, and geometry
Pre-GED mathematics(
Visual
) 4
editions published
between
1999
and
2004
in
English
and held by
330 WorldCat member
libraries
worldwide
Provides a thorough review of algebra and geometry. Covers solving equations, factoring, radicals, angle relationships, similar triangles, the Pythagorean theorem, area, volume, distance, slope, and other topics. Presents strategies for solving word problems and common sense approaches to multiple-choice questions
GED social studies(
Visual
) 3
editions published
in
2004
in
English
and held by
316 WorldCat member
libraries
worldwide
Refines skills for taking the GED social studies test, including history, geography, economics, and political science
GED language arts, reading(
Visual
) 2
editions published
in
2004
in
English
and held by
306 WorldCat member
libraries
worldwide
Tips and techniques for refining the skills needed for the GED language arts reading test, which covers many different types of reading materials such as fiction, non-fiction, poetry and drama
GED language arts, writing(
Visual
) 2
editions published
in
2004
in
English
and held by
295 WorldCat member
libraries
worldwide
Refines skills for taking the GED language arts writing test, including how to organize what you write, tips for improving writing style, test taking tips, and much more
Pre-GED social studies by Karl M. A Weber(
Visual
) 5
editions published
between
1999
and
2004
in
English
and held by
277 WorldCat member
libraries
worldwide
Instructor Karl Weber presents social studies basics including developing reading skills for social studies; using context to define words; distinguishing opinions from facts; interpreting graphic images
Pre-GED language arts, reading by Karl Weber(
Visual
) 4
editions published
between
2002
and
2004
in
English
and held by
259 WorldCat member
libraries
worldwide
Helps students build a solid foundation in basic reading skills, including learning new words by using word roots and context clues, identifying key points and main ideas in a variety of reading materials, and recognizing an author's purpose and message. Also discusses how plot, characters, setting, and tone bring literature to life and how poets use rhythm, rhyme, imagery, and symbols
Pre-GED language arts, writing by Karl Weber(
Visual
) 3
editions published
between
2002
and
2004
in
English
and held by
252 WorldCat member
libraries
worldwide
Reviews basic skills necessary for writing in correct and proper English. Covers topics such as how to use the dictionary to learn new words, understanding subject and predicate, the different types of sentences, how to correct errors in sentence structure, usage, and mechanics, and irregular verbs
Video math review for the GED(
Visual
) 4
editions published
between
1986
and
1993
in
English
and held by
249 WorldCat member
libraries
worldwide
Provides step-by-step solutions, strategies for multiple choice questions, time-saving hints, and review of important concepts for the major areas tested on the exam: arithmetic, charts and graphs, probability and statistics, algebra and geometry
English grammar(
Visual
) 1
edition published
in
2004
in
English
and held by
231 WorldCat member
libraries
worldwide
Instruction in English grammar explains the different roles that words play in sentences, including verbs, nouns, pronouns, adjectives, adverbs, prepositions, conjunctions, articles, and interjections
English grammar(
Visual
) 1
edition published
in
2004
in
English
and held by
228 WorldCat member
libraries
worldwide
Instruction in English grammar which explains the use of commas, colons, semicolons, periods, question marks, exclamation points, quotation marks, parentheses, dashes, and other punctuation marks
Video review for the ASVAB(
Visual
) 1
edition published
in
1986
in
English
and held by
227 WorldCat member
libraries
worldwide
Prepares men and women to take the Armed Services Vocational Aptitude Battery (ASVAB), the entrance examination used by the Army, Navy, Air Force, Marine Corps, Coast Guard and National Guard
English grammar(
Visual
) 1
edition published
in
2004
in
English
and held by
226 WorldCat member
libraries
worldwide
Instruction in English grammar which explains the six essential verb tenses, irregular verbs, tricky verbs such as lie and lay, and the subjunctive mood
English grammar sentence structure(
Visual
) 1
edition published
in
2004
in
English
and held by
225 WorldCat member
libraries
worldwide
Perfect for classroom use or self-study, this program uses easy-to-follow examples and practice exercises from everyday life to help you master even the most complicated English grammar topics. It's alos great for reviewing forgotten grammar or preparing for many verbal standardized test such as the SAT, TOEFL, GED, AND GRE
English grammar(
Visual
) 1
edition published
in
2004
in
English
and held by
218 WorldCat member
libraries
worldwide
Instruction in English grammar which explains the secrets of good spelling; simple spelling rules and their exceptions; mastering plurals, prefixes, and suffixes; and the capitalization of words
English grammar(
Visual
) 1
edition published
in
2004
in
English
and held by
215 WorldCat member
libraries
worldwide
Instruction in English grammar which explains using personal, indefinite, interrogative and relative pronouns; mastering the subjective, objective, and possessive cases; using who and whom; and elliptical clauses and other pronoun problems
Writing a great research paper(
Visual
) 5
editions published
in
2007
in
English
and held by
214 WorldCat member
libraries
worldwide
A step-by-step guide to constructing a paper that is technically correct, original and compelling! The viewer will recognize the value of true scholarship; master essential research techniques; dramatically improve writing; keep motivated, productive, and relaxed through the entire process. The strategies learned will apply to any research paper assignment on any topic - in high school, college, graduate school, and beyond | 677.169 | 1 |
Getting from Arithmetic to Algebra
Balanced Assessments for the Transition
Getting from Arithmetic to Algebra by Judah L. Schwartz
Book Description
The title of this book is Getting from Arithmetic to Algebra and is written by author Judah L. Schwartz. The book Getting from Arithmetic to Algebra is published by Teachers' College Press. The ISBN of this book is 9780807753200 and the format is Paperback. The publisher has not provided a book description for Getting from Arithmetic to Algebra by Judah L. Schwartz.
Algebra: Form and Function offers a fresh approach to algebra that focuses on teaching readers how to truly understand the principles, rather than viewing them merely as tools for other forms of mathematics. Meant for a College Algebra course, Algebra: Form and Function is an introduction to one of the fundamental aspects of modern society.
Books By Author Judah L. Schwartz
This comprehensive, research-based book helps teachers use standards-linked assessments to collect accurate formative data about students' strengths and weaknesses and increase mathematical understandings for all learners.
Brings together leading experts to offer an in-depth examination of how computer technology can play an invaluable part in educational efforts through its unique capacities to support the development of students; understanding of difficult concepts | 677.169 | 1 |
Synopses & Reviews
Publisher Comments:
Whether you need help solving equations or determining the slope of a line, this guide gives you the tools you need to find your answers Beginning with the basics, you will learn and practice all the skills needed to enhance your algebra expertise. This comprehensive guide covers all the key concepts, including:
Variables and expressions
Linear equations and inequalities
Monomials and polynomials
Exponents
Rational expressions
The Pythagorean theorem
Area and perimeter
Graphs and charts
Inside you'll find hundreds of examples to illustrate the basics and plenty of exercises to ensure mastery of these fundamentals. No matter if you're a student looking for a companion to your textbook, or a curious learner who's been away from the classroom too long, this will be your indispensable algebra primer | 677.169 | 1 |
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