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Khan Academy: Trigonometry
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Description
Ximarc Studios Inc is proud to bring you Khan Academy Trigonometry. Khan Academy Trigonometry allows students to learn Trigonometry through various
Ximarc Studios will continue to bring you great video lessons from the Khan Academy.
Khan Academy is a non-profit organization with over 60,000 students world wide and is a library of videos numbering over 1100. Students are able to leverage a wide range of topics in Mathematics, Statistics, Physics, Chemistry and Biology. Khan Academy videos work through a numerous sets of problems providing the most comprehensive set of teaching videos available. Khan Academy also has sample standardized tests SAT and GMAT.
To learn more about the Khan Academy visit the website
For more information on Ximarc Studios visit the website
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Customer Reviews
3.8 out of 5
11 Ratings
11 Ratings | 677.169 | 1 |
Note: Citations are based on reference standards. However, formatting rules can vary widely between applications and fields of interest or study. The specific requirements or preferences of your reviewing publisher, classroom teacher, institution or organization should be applied.
Curves and surfaces for computer graphics
".Read more...
Abstract:
Requires only a basic knowledge of mathematics and is geared toward the general educated specialists.Includes a gallery of color images and Mathematica code listings.Read more...
Reviews
Editorial reviews
Publisher Synopsis
From the reviews of the first edition:"The book gives an introduction to curve and surface representation and modelling techniques that are used in the related fields of geometric modelling, computer aided geometric design (CAGD), and computer graphics. It mainly introduces newcomers to these domains as well as those interested in the mathematical background of graphics applications. It is written in a mathematically very accessible style ... . Historical remarks, citations from the literature adapted to the context as well as numerous examples contribute to a pleasant and readable style." (Gudrun Albrecht, Mathematical Reviews, 2006 i)"This book is concerned with the computation of surfaces. ... The book also deals with curves ... . after reading and understanding a topic, the reader should be able to design and implement the concepts discussed. ... The notions are clearly introduced and are ... illustrated by figures and examples. ... The book has many examples, which are important for a better understanding of the presented concepts ... . Many exercises are sprinkled throughout the text. " (Marian Ioan Munteanu, Zentralblatt MATH, Vol. 1083, 2006)Read more...
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schema:Review ; schema:itemReviewed < ; # Curves and surfaces for computer graphics schema:reviewBody ""." ; . | 677.169 | 1 |
KS4 & 5 Mathematics
KS4
EXAM BOARD: Edexcel
ASSESSMENT: Paper 1 non-calculator one and a half hours
Paper 2 calculator one and a half hours
Paper 3 calculator one and a half hours
KS4 pupils are studying towards their GCSE in Mathematics. During the two years of study all pupils will learn about every aspect of maths including, algebra, statistics, formal methods of calculation, ratio, percentages, area and volume, trigonometry and many others. Problem solving and application of the mathematics they are learning has increased in importance and this is the main focus in paper 3 of their exams.
Paper 2 calculator one and a half hours, 6 questions of varying length
IB mathematical studies concentrates on the application of mathematics at a higher level than GCSE, the aim is to prepare students for the use of math in their future career paths. All bar one topic is an extension of what they learn at GCSE, this being Logic. Students have access to a graphical calculator both during lessons and the exam in order to make their experience as work like as possible. | 677.169 | 1 |
I've always wanted to learn graphing calculator program for list of factors, it seems like there's a lot that can be done with it that I can't do otherwise. I've browsed the internet for some good learning tools , and consulted the local library for some books, but all the data seems to be targeted at people who already understand the subject. Is there any tool that can help new students as well?
Kids can't seem to think of anything beyond tutoring. Why don't you try something yourself? There are numerous resources for graphing calculator program for list of factors which are a lot better than tutoring. Try Algebrator, and you will never need a tutor.
graphing function, percentages and angle-angle similarity were a nightmare for me until I found Algebrator, which is really the best math program that I have come across. I have used it through several algebra classes – Basic Math, Intermediate algebra and Pre Algebra. Just typing in the math problem and clicking on Solve, Algebrator generates step-by-step solution to the problem, and my math homework would be ready. I highly recommend the program. | 677.169 | 1 |
Most effective math challenge solver That could Cause you to a greater Student
In a few classes, all it requires to go an test is take note using, memorization, and remember. Nevertheless, exceeding inside of a math class takes a different style of effort. You can't simply just demonstrate up for any lecture and check out your instructor "talk" about algebra and . You study it by doing: paying attention at school, actively researching, and solving math issues – even if your instructor has not assigned you any. For those who find yourself struggling to try and do effectively with your math class, then stop by very best web page for resolving math difficulties to understand the way you may become an improved math student.
Affordable math specialists on-line
Math programs observe a purely natural development – each builds on the know-how you've acquired and mastered within the earlier system. If you are getting it tough to comply with new ideas at school, pull out your old math notes and overview past materials to refresh on your own. Make certain that you satisfy the stipulations just before signing up for a class.
Assessment Notes The Night Before Class
Dislike every time a teacher phone calls on you and you have forgotten the way to clear up a selected difficulty? Keep away from this moment by examining your math notes. This could help you figure out which concepts or queries you'd like to go around at school the following day.
The considered accomplishing research each night time could appear bothersome, but if you desire to reach , it can be important that you continually apply and learn the problem-solving approaches. Use your textbook or on the internet guides to work via prime math issues with a weekly basis – even though you have no research assigned.
Utilize the Dietary supplements That include Your Textbook
Textbook publishers have enriched modern publications with added substance (for example CD-ROMs or on-line modules) which can be accustomed to help pupils attain further observe in . A few of these elements may additionally incorporate an answer or clarification guide, which may help you with doing the job as a result of math issues all by yourself.
Read Ahead To stay Forward
If you would like to minimize your in-class workload or perhaps the time you devote on research, use your free time after university or on the weekends to go through ahead to your chapters and ideas which will be lined another time you might be in class.
Review Previous Tests and Classroom Examples
The perform you are doing in class, for homework, and on quizzes can give clues to what your midterm or final examination will seem like. Make use of your outdated checks and classwork to make a individual study guideline for the upcoming examination. Seem at the way your trainer frames inquiries – this really is most likely how they can surface on your test.
Figure out how to Get the job done By the Clock
This is the well known research suggestion for people taking timed tests; primarily standardized exams. For those who only have forty minutes to get a 100-point exam, then you can certainly optimally shell out 4 minutes on each 10-point dilemma. Get data regarding how very long the check will be and which sorts of thoughts are going to be on it. Then plan to attack the easier inquiries very first, leaving on your own enough time and energy to devote within the much more hard kinds.
Improve your Assets to have math research aid
If you're obtaining a tough time knowing ideas in class, then make sure you get assist outside of course. Inquire your folks to produce a analyze group and go to your instructor's business hrs to go over challenging challenges one-on-one. Attend study and review classes once your teacher announces them, or retain the services of a private tutor if you need 1.
Talk To By yourself
After you are reviewing complications for an examination, consider to elucidate out loud what method and methods you used to get your options. These verbal declarations will occur in helpful throughout a take a look at whenever you ought to remember the methods you ought to consider to find a resolution. Get added follow by attempting this tactic by using a good friend.
Use Research Guides For Further Practice
Are your textbook or class notes not helping you recognize that which you ought to be mastering in school? Use review guides for standardized exams, such as the ACT, SAT, or DSST, to brush up on old materials, or . Review guides ordinarily occur equipped with comprehensive explanations of how to address a sample issue, , and you also can frequently discover where would be the much better obtain mathtroubles. | 677.169 | 1 |
Patterns, Relations and Algebra
1. Patterns, Functions, and Algebra
Click on the following benchmarks for more information and for links to
annotated OGT items.
a.
Benchmark A: Generalize and explain patterns and sequences in order
to find the next and the nth term.
At this level students should be able to identify a pattern or a relationship
between the terms in a sequence. They should be able to use the relationship
to find the next term. Finally, they should be able to express a pattern in
a table, graph or equation to find the nth term in the
sequence.
Click here for
an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.
b.
Benchmark B: Identify and classify functions as linear or nonlinear,
and contrast their properties using tables, graphs or equations.
This benchmark culminates with a description and comparison of nonlinear
functions. However, before students can successfully describe and compare
the characteristics of nonlinear functions, they should be able to define
a function in terms of domain and range and using f(x) notation.
They should also be able to distinguish between linear and nonlinear functions
that are displayed in tables, graphs or equations Translate information from one representation (words, table,
graph or equation) to another representation of a relation or function.
Students should be able to express linear, quadratic and exponential relationships
in words, a table, a graph or in symbolic form. They should also be able to
explain how these four representations relate and convert from one representation
to another.
Click here for
an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.
d.
Benchmark D: Use algebraic representations, such as tables, graphs,
expressions, functions and inequalities, to model and solve problem situations.
Algebraic representations are most useful when modeling and solving real
world problems. Students should not learn these representations without understanding
their applications. They should understand situations where one variable depends
on another. They should be able to combine monomials and polynomials with
and without using physical models and be able to simplify rational expressions.
Finally, they should be able to represent and solve problem situations using
appropriate equations, inequalities or algebraic representations.
Click here for
an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.
e.
Benchmark E: Analyze and compare functions and their graphs using attributes,
such as rates of change, intercepts and zeros.
In this benchmark, students explore the practical applications of functions
and their graphs. They should be able to relate rates of change, intercepts
and zeros to graphs of real world situations. They should be able to describe
and compare linear, quadratic, and exponential functions using their attributes.
Click here for
an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.
f.
Benchmark F: Solve and graph linear equations and inequalities.
The focus of this benchmark is on linear equations and inequalities. Students
should be able to convert problem situations to appropriate equations, inequalities
or algebraic representations. They should be able to write linear equations
using a variety of information, and they should know several ways to write
and solve linear equations and inequalities. Students should be able to combine
all these skills to solve real world problems.
Click here for
an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.
In this benchmark, students explore various ways to solve quadratic equations.
Students move from graphing quadratics to using factoring, the quadratic formula
and technology to solve them. Finally, they graph circles and solve problems
that can be modeled with linear, quadratic, exponential or square root functions.
In this benchmark, students study several methods of solving systems of
linear equations. These methods include using graphs, substitution and elimination.
Students also learn how the solution relates to intersection of the two lines.
Finally, students solve problems using systems of linear equations and inequalities.
Click here for
an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.
In this benchmark, students study direct and inverse variation. First,
they should be able to distinguish between several types of changes in mathematical
relationships. Second, students should be able to apply direct and inverse
variation to appropriate problem-solving situations. Finally, students should
be able to interpret the graphs of direct and inverse variation.
Click here for
an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.
j.
Benchmark J: Describe and interpret rates of change from graphical and
numerical data.
The focus of this benchmark is on the rate of change of a data set. Rate
of change is a relationship such as distance over time, often described by
using a slope. Students start their explorations within this benchmark by
learning to find slope, midpoint and distance. Next they study how graphs
of equations change when the numbers in the equation are changed. Finally,
they study the relationships between the slopes of parallel and perpendicular
lines.
Click here for
an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.
ABOUT THE MATH
Graphing a function
To graph a function such as y = 5x -
4, follow these steps:
Create a table. In the left column, list some x-values.
Substitute these values for x in the equation and solve
for y. Record these values in the right column next to
their corresponding x-values.
Write the x- and y-values
in each row as coordinate pairs:
Plot the coordinate pairs in the coordinate plane. Each row of the
table becomes a point in the plane, and together these points suggest the
shape of the graph of the function.
The coordinate plane
Students should have a strong understanding of the coordinate plane. The x-axis
(the horizontal number line) and the y-axis (the vertical
number line) are perpendicular lines that cross at the point (0, 0), called
the origin. The two axes of the coordinate system divide the plane into four
separate sections known as quadrants. These are identified as the first, second,
third and fourth quadrants.
Spend time with your students discussing what the points in each quadrant
have in common, and what the points located on the axes have in common. The
diagram below helps to describe these patterns.
Direct and inverse variation
Direct variation occurs when the values of two variables maintain a constant
ratio. The relationship can be expressed as an equation in the form .
The graph of such an equation is a line with an x-intercept
of 0 and a slope of k.
Inverse variation occurs when the variables x and y vary
inversely. For a constant k, or
.
When k and x are greater than 0,
the graph of an inverse variation shows y decreasing
as x increases. When x is less than
0, the graph shows y increasing as x decreases.
Strategies
Help With Fundamentals
Here are a few of the difficulties students might have with this topic,
along with suggestions for addressing these difficulties.
Additional Instruction and Practice
If your students need additional practice, supplement your instruction
with the following activities:Advanced Work
Solving Quadratic Equations by Completing the Square
Completing the square requires students to factor and multiply polynomials.
Problem 1Answer Explanation | 677.169 | 1 |
Get In Touch
A Level
Maths
Mathematics is a universal part of human culture. It is the tool and language of commerce, engineering and other sciences. It helps us recognise patterns and to understand the world around us. The generic nature of mathematics means that almost all industries require mathematicians.
• Mathematics A level students and graduates are in high demand by employers.
• Maths is the essential transferable component across all science, engineering, technology and maths subjects.
• Currently 59% of employers state they are having difficulty recruiting people with Mathematics skills.
• Those with maths A level earn on average around 10% more than those without.
This programme delivered by dedicated tutors at The City of Liverpool College gives you a solid grounding in Mathematics. AS Maths is made up of 3 modules: Core 1 and Core 2 which are Pure Maths modules and then you will have a choice of either Mechanics or Statistics for your Applied Maths module.
If you study Physics you should take Mechanics as your Applied module as a great deal of the content overlaps. Also you will develop excellent skills in the manipulation of formula which will support the study of all of the Sciences.
Psychology and Sociology can also be greatly supported by AS Statistics.
What can I do now? For more information: • • • Edexcel AS/A Level Mathematics; AS/A Level Further Mathematics.Qualification(s)
Edexcel AS/A Level Mathematics; AS/A Level Further Mathematics.
Entry RequirementsClarence Street L3 5TP accept any liability for inaccuracies or omissions which arise due to such changes. | 677.169 | 1 |
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DeMoivre's Theorem and The Complex Plane, Complex Numbers in Polar Form
This lesson is designed for PreCalculus or Trigonometry. It includes a 35 slide animated PowerPoint along with a Task Cards activity and an additional HW assignment. As you know this is very difficult unit. Activity Based Learning with Task Cards really does work to help reinforce your lessons. Task and station cards get your students engaged and keep them motivated. Use all at once or as many as you like. Directions for using Task Cards included in this resource | 677.169 | 1 |
This course equips students with the mathematical knowledge and skills they will need in
many college programs. Students will use statistical methods to analyse problems; solve problems
involving the application of principles of geometry and measurement to the design and
construction of physical models; solve problems involving trigonometry in triangles; and consolidate
their skills in analysing and interpreting mathematical models. | 677.169 | 1 |
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Apprenticeship and Workplace Math 10
Prerequisites
The prerequisite for this course is Mathematics 9.
Course Description
This course focuses on math topics that are useful in our daily tasks. Topics include unit conversion, money and currency, measurement and some trigonometry. This course is intended for students not intending to pursue studies not strongly related to math. | 677.169 | 1 |
Complex variables is a precise, elegant, and captivating subject. Presented from the point of view of modern work in the field, this new book addresses advanced topics in complex analysis that verge on current areas of research, including invariant geometry, the Bergman metric, the automorphism groups of domains, harmonic measure, boundary regularity of conformal maps, the Poisson kernel, the Hilbert transform, the boundary behavior of harmonic and holomorphic functions, the inhomogeneous Cauchy–Riemann equations, and the corona problem.
The author adroitly weaves these varied topics to reveal a number of delightful interactions. Perhaps more importantly, the topics are presented with an understanding and explanation of their interrelations with other important parts of mathematics: harmonic analysis, differential geometry, partial differential equations, potential theory, abstract algebra, and invariant theory. Although the book examines complex analysis from many different points of view, it uses geometric analysis as its unifying theme.
This methodically designed book contains a rich collection of exercises, examples, and illustrations within each individual chapter, concluding with an extensive bibliography of monographs, research papers, and a thorough index. Seeking to capture the imagination of advanced undergraduate and graduate students with a basic background in complex analysis—and also to spark the interest of seasoned workers in the field—the book imparts a solid education both in complex analysis and in how modern mathematics works.
"The geometric point of view is the unifying theme in this fine textbook in complex function theory. But the author also studies byways that come from analysis and algebra.... Altogether, the author treats advanced topics that lead the reader to modern areas of research. And what is important, the topics are presented with an explanation of their interaction with other important parts of mathematics. The presentations of the topics are clear and the text makes [for] very good reading; basic ideas of many concepts and proofs are carefully described, non-formal introductions to each chapter are very helpful, a rich collection of exercises is well composed and helps the student to understand the subject. The book under review leads the student to see what complex function theory has to offer and thereby gives him or her a taste of some of the areas of current research. As such it is a welcome addition to the existing literature in complex function theory.... In this reviewer's opinion, the book can warmly be recommended both to experts and to a new generation of mathematicians." —Zentralblatt MATH
"This book provides a very good and deep point of view of modern and advanced topics in complex analysis. … Each chapter contains a rich collection of exercises of different level, examples and illustrations. … The book is very clearly written, with rigorous proofs, in a pleasant and accessible style. It is warmly recommended to advanced undergraduate and graduate students with a basic background in complex analysis, as well as to all researchers that are interested in modern and advanced topics in complex analysis." —Studia Universitatis Babes-Bolyai Mathematica
"This book is an exploration in Complex Analysis as a synthesis of many different areas; the prejudice in the subject is geometric, but the reader may [need] basic information from analysis…, partial differential equations, algebra, and other parts of mathematics. This synthesis is addressed to the students; it gives them the possibility of writing their thesis on the subject and introduces them to some research problems…This captivat[ing] book also contains a collection of exercises, examples and illustrations, as well as an extensive bibliography and a thorough index." —Analele Stiintifice ale Universitatii "Al. I. Cuza" din Iasi | 677.169 | 1 |
At the end of the semester, students will have an
understanding of the concepts and techniques listed below.This understanding will be enhanced through directed group and individual
computer exercises and group and individual projects.The computer algebra system Derive, the spreadsheet Excel and the
graphing calculator make it possible to explore non-routine problems and
applications.In some instances,
students will be allowed to write their own code in the programming language of
their choice instead of, or in conjunction with, the other software.
COURSE CONTENT:
At
the end of the semester, the student will be able to:
.. use Derive to solve equations and systems of equations, graph functions and perform calculus operations
.. use Microsoft Excel to numerically investigate data
.. understand and compute simple interest
.. understand and compute compound interest
.. apply compound interest to population and inflation applications
.. find effective (annual) yield
.. understand what stocks, mutual funds are and find average annual returns
.. understand annuities and find present and future value
.. understand and apply the concepts of mean and median
.. understand and apply the concepts of variance and standard deviation
.. understand and apply the concept of the line of least regression (complete with correlation coefficients, standard error and prediction intervals)
.. understand the concepts of average and instantaneous rates of change | 677.169 | 1 |
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What is Algebra? (Historically)
At the beginning of my second year, I thought it would be interesting to explore definitions of Algebra in historical texts: both how the author described Algebra and also what topics were included in Algebra. Unfortunately, I have not had time to do a more comprehensive search, but I found five British and American texts that were interesting and spanned about a century. (I chose to focus on these countries because the texts are readily available in digital form and also because I am limited in my language).
Please see below for additional details, but I list their definitions here:
Hutton (1831)
In his "Course of Mathematics," he included many Algebra topics, starting from basic operations, moving through roots and powers, to arithmetic and geometric progression, to some equations, and finishing with interest formulas.
Because his definition of Algebra depended on his definition of Mathematics, I include both here. HeJ. M. Taylor, A.M.
James Morford Taylor served 57 years at Colgate University as a professor of Mathematics.George A. Wentworth | 677.169 | 1 |
Welcome to Math 1500
Praxis Math Test Prep: State Math Licensure Exams
This course is a resource to help you prepare exceptionally well for your state's high-school mathematics teachers' licensure exam.
Each state requires its new teachers to demonstrate the necessary content knowledge and practice skills to be an effective secondary mathmematics educator. The content required varies to some degree from state to state, but all demand a proficiency at least at the level provided by beginning mathematics baccalaureate courses. Many states use the Praxis Mathematics: Content Knowledge Test (test code 0061, Mathematics CK) developed by the Educational Testing Service ( The remaining states administer their own version of this test.
Visit your state government's Department of Education website to determine which test your state administers. (Searching under "[state name], secondary mathematics licensure" usually does the trick.
This course covers the entire content knowledge required by all fifty states and the District of Columbia.
How to Use This Course
All state exams have a significant multiple-choice component, and some, including Praxis Mathematics: Content Knowledge Test (0061), are purely multiple choice in their structure. However, a number of states do include one or two "open response" or essay questions to be completed as part of their exam. (See your state's department of education website for details and examples of what might be required.) We will briefly discuss open response questions later in this section. A full and complete understanding of the content covered in this course will help prepare you well for any essay question.
The Questions in This Course
Each question is presented as either a multiple-choice question or a short-answer question. The choice to not present all questions in the multiple-choice format is deliberate: having a selection of answers to review offers the crutch of being able to eliminate incorrect answers. As a result, many of the questions in this course are a tad harder than what shall appear on your exam.
Each page of the course consists of 2–4 questions from one particular theme. Have pencil and paper in hand and work through each question on your own. After you have attempted each question, click to the next page. Not only are the answers revealed, but full explanations of the mathematics behind the topics at hand appear as well. In your notebook, write a summary of the mathematics. If you find you need deeper explanation than offered in this course, search the internet or look at your mathematics textbook for further discussion. We have assumed that the user already has at least at basic introduction to all the topics discussed but, of course, one might have areas that need more thorough review.
Ideally you should start work on this course several months before the date of the exam. A good practice is to work through the material of 10–15 questions each day.
Most states allow the use of a calculator when taking the exam. (Check this on your state's Department of Education website.) However, try to do as much as you can on each question without a calculator. Working to exercise your mental faculties now will serve you well on the test.
Practice in Combination
The styles of the questions in this course mimic the questions that appear in all secondary mathematics licensure exams. However, each state's exam has its own feel and it is important to understand the feel of the exam you will be taking. Download the practice materials from your state's licensure exam website (or the ETS Praxis website) and work through them too. After doing this course, you will find the topics, content, and style familiar. It is nonetheless important to focus on the particular style of exam you will face.
General Study Tips
Set aside, as best you can, a regular time each day to study. Work through 10–15 questions per day, taking notes along the way.
Reserve a regular study place with few distractions. Turn off your cell phone during your study time.
Write summary sheets of your notes. The act of rewriting material forces you to examine content slowly and with purpose and focus. The physical act of writing aids in memory. (You may well find a moment in the exam of remembering on which part of the page you wrote a particular idea, and that will aid the recall of the idea itself.)
Periodically review the material you have covered. Take a few moments at different parts of the day to read over one of your summary sheets.
Consider organizing a study group if you have colleagues also taking the exam. | 677.169 | 1 |
Ashley Logan, AZ
I was quite frustrated with handling complex numbers. After using this software, I am quite comfortable with it. Complex numbers are no more 'COMPLEX' to me. Jeff Kasten, MI
The Algebrator is awesome and out of this world! Thanks for making my life so much easier28:
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of dividing monomials what would 4n to the 6th over 20 n to the 4th be | 677.169 | 1 |
5 IMPORTANT STEPS TO SUCCESS IN CALCULUS III
Having a good graphing calculator can make you Calculus III experience 10 times easier. Being able to graph functions, check homework answers, and check test/quiz questions makes all the difference when taking Calculus III. So invest in a good common calculator. Texas Instruments calculators are standard for most students.
Two great calculators. The Ti-84 on the left offers a more simple approach. While the Ti-89 offers a more advanced approach with a LOT more features. I high recommend the Ti-89.
2: Buy a Good Pencil and Eraser
Two of the most important things for any math class. A nice mechanical pencil will allow you to draw detailed graphs in 3 dimensions. (Yes graphs with 3 axises will appear is Calculus III a lot!). So don't use that dull yellow pencil that you found in the bottem of your backpack unless you are ready to have very sloppy hard to read notes.
The Best Pencil and Eraser Combo I have Ever Used!
3. Have Tons of Paper!
When it comes to Calculus III always use as much paper space as you need to make clear, accurate, and descriptive notes! Use a new page every time you start a problem and make sure you write your own notes along side your professors notes to help you understand the content. Remember you could keep these notes for the rest of your college years and career.
4. Buy a Smart Pen
A smart pen allows you to write your lecture notes in pen, transfer them to your computer to create an exact digital copy, and be able to record your lecture while you write! Smart pens are a must have tool for studying out of class.
Top Recommended Smart Pen: The Echo
5. Get Sleep
Anyone who is looking to succeed in calculus III will preform their best when they are fully rested. Make a point to get the right amount of sleep before the day of class to ensure you are in a good mindset. | 677.169 | 1 |
Math 203C - Algebraic Geometry
Course description:
This course provides an introduction to algebraic geometry. Algebraic geometry
is a central subject in modern mathematics, and an active area of research.
It has connections with number theory, differential geometry, symplectic
geometry, mathematical physics, string theory, representation theory,
combinatorics and others.
Math 203 is a three-quarter sequence. Math 203A covered affine and
projective varieties and the basics of scheme theory.
Math 203B
focused more heavily on the theory of schemes and the modern
language of algebraic geometry, including sheaves, sheaf cohomology, and properties of morphisms.
Math 203C will continue where 203B left off. Planned topics include:
Prerequisites: Math 203B, preferably taken last quarter. If you do not
meet this prerequisite, please contact me as soon as possible!
Grading: 100% homework. Problem sets will be assigned weekly (see below); please do them! It is effectively impossible to learn this subject passively.
(If you are a graduate student, I will only grade your homework if you are
registered for the course. Please register!) | 677.169 | 1 |
GeoGebra - Free download and software reviews
GeoGebra - Graphing Software Mathematics
GeoGebra is a mathematics software provided useful and completely free. This is really a tool for effectively supporting those who have done the research or work with arithmetic, geometry, spreadsheets, graphics, statistics, algebra and calculus. Also , GeoGebra can be shared and used by everyone at
GeoGebra, the free, open-source mathematics software is designed for math classrooms in secondary schools, but anyone who uses geometry, algebra, or calculus should check it out. It combines a flexible, easy-to-use geometry tool with direct input of equations and coordinates. It can create points, vectors, lines, segments, conic sections, and more using preconfigured tools and handle variables for vectors, numbers, and points. It's available in many languages and is supported by a community of users and developers as well as a useful Web-based Help file, a forum, and a wiki. It requires the Java Runtime Environment.
GeoGebra's default interface displays a toolbar full of unique icons for adding a range of objects, including Points, Lines Through Two Points, Polygons, Ellipses, Angles, Reflect Objects, and Sliders. Clicking any object and then clicking on the main two-axis view opened small properties boxes that let us customize and configure each item. As we clicked to add points or other objects, the program added them to either the Free Objects or Dependent Objects lists. Once we'd placed an object, we could easily move it around. For example, we clicked the tool to add a Circle Through Three Points. We added the first two points, which drew the circle. As we moved the cursor around for the third point, the circle moved position, expanded, and contracted to follow, with the changing value displayed in the Free Objects list in the left-hand navigation console as well as in small parentheses next to the cursor. We entered some simple equations in the Input field, and GeoGebra displayed them in the main view. We could also customize much of the program's look and functions on the Options menu. The Tools menu let us create and manage new tools via a simple wizard, a great extra.
We barely scratched the surface of the many ways users can customize GeoGebra and use it to teach, learn, and perform math. It's flexible, very easy to use, and very well supported, too. For anyone who can use it, we recommend it.
You can download GeoGebra for other platforms here:
Although at first glance, this seems to be a complex application but its advantages compared to other similar applications such as: providing multiple objects are closely linked. The purpose of the design that is supported GeoGebra connection geometry, algebra and other math elements in an interactive way and tighter.
This can be done by using the points, vectors, lines, triangles, cones, etc. Besides, GeoGebra also allows users to directly import and manipulate mathematical equations and coordinates.
With all these characteristics, GeoGebra is currently one of the mathematical software is most popular in the world and has received many prestigious awards. It has brought improvements and advances in the process of teaching and learning of students worldwide. So what else are you waiting for, download GeoGebra on your computer to facilitate learning algebra itself, analytic geometry and better.
Mathematical software is completely free and efficient support academic work, teaching and assessment
The interface is easy to use and full interoperability with many powerful features
Access multiple resources in
Available in multiple languages, including: Vietnamese
Provides a fun way to see and experience the math and science
Adapt well to any program or project
Used by millions of people around the world
Disadvantages of GeoGebra
The only downside of GeoGebra : a bit complicated for beginners.
In addition, users also need special attention are: can only install programs in the Java environment. Therefore, before starting to install it, set up a Java Runtime Environment .
Tip switch the interface language programs into Vietnamese
To help users in Vietnam manipulate this software easier, use the Vietnamese language options for the program's main interface. Simply install it after successful programs on your computer, you will see the interface as below.
Then, click the button Options and select Language -> Vietnamese . As a result, you can manipulate the program much simpler.
GeoGebra is dynamic mathematics software for education in secondary schools that joins geometry, algebra, and calculus. On the one hand, GeoGebra is a dynamic geometry system. You can do constructions with points, vectors, segments, lines, conic sections as well as functions and change them dynamically afterwards. On the other hand, equations and coordinates can be entered directly. Thus, GeoGebra has the ability to deal with variables for numbers, vectors and points, finds derivatives and integrals of functions, and offers commands like Root or Extremum. It is a free and open source software
Thus, the use of mathematics useful software becomes easier than ever. | 677.169 | 1 |
50693
ISBN: 0387950699
Publication Date: 2001
Publisher: Springer Verlag
AUTHOR
Gamelin, Theodore W., Gehring, F. W., Halmos, P. R.
SUMMARY
The book provides an introduction to complex analysis for students with some familiarity with complex numbers from high school. The first part comprises the basic core of a course in complex analysis for junior and senior undergraduates. The second part includes various more specialized topics as the argument principle the Poisson integral, and the Riemann mapping theorem. The third part consists of a selection of topics designed to complete the coverage of all background necessary for passing Ph.D. qualifying exams in complex analysis.Gamelin, Theodore W. is the author of 'Complex Analysis', published 2001 under ISBN 9780387950693 and ISBN 0387950699 | 677.169 | 1 |
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This is aligned to the Virginia State Standards (Reporting Category 1), but it can be used for any Algebra 1 course. It contains 13 multiple choice questions covering evaluating expressions, solving equations and inequalities, and evaluating graphs. 3 of the 13 questions are word problems. | 677.169 | 1 |
The mechanical approach to math problem solving relies heavily on manipulation of terms using low level mathematical constructs without using the problem solving abilities of the student. In fact, if students follow only this approach of solving problems, they may tend to become used to mechanical and procedural thinking suppressing their inherent creative and innovative out-of-the-box thinking abilities...
This is the second solution set of 10 practice problem exercise for SSC CGL exam on topic Trigonometry. Students must complete the corresponding question set in prescribed time first and then only refer to this solution set...
This is the second question set of 10 practice problem exercise for SSC CGL exam on topic Trigonometry. Students must complete this question set in prescribed time first and then only refer to the corresponding solution set... | 677.169 | 1 |
Course Content:
Geometric combinatorics refers to a growing body of mathematics concerned
with understanding the combinatorics associated with discrete
geometric objects desribed by a finite set of building blocks.
One primary example we will study in this course are
polytopes, which are bounded polyhedra and
the convex hull of a finite sets of points. We will also study
objects built up from polytopes, such as triangulations and cell
complexes, and other objects from the land of
discrete geometry, such as the arrangements of points, lines, and hyperplanes.
There are many connections to linear algebra, discrete mathematics,
geometry, and topology--- and there are many exciting applications to
other fields such as economics, robotics, and biology.
A tentative sample of topics include:
Course materials:
Taking notes in class will be essential, but I will also provide and
outline of the main theorems on downloadable notes on the course
webpage. Also, for some inexpensive model-building material later on,
please buy some gumdrops (I recommend DOTS) and a box of
toothpicks (available at, e.g., Target).
Coursework:
Homeworks, assigned weekly and due Tuesdays in class,
will be announced on the course webpage:
These will be
worth 30 percent of the course grade.
There will be a midterm, tentatively handed out on March 9, worth 30 percent.
There will be a final project and presentation, worth 40 percent.
Honor Code:
All are expected to abide by the HMC honor code.
Cooperation is ENCOURAGED in this class, but write
up all solutions individually and be sure to credit any collaborators. | 677.169 | 1 |
Normal 0 false false false MicrosoftInternetExplorer4 Finite Mathematics, Eleventh Edition is a comprehensive, yet flexible, text for students majoring in business, economics, life science, or social sciences. The authors delve into greater mathematical depth than other texts, while motivating students through relevant, up-to-date applications drawn from students' major fields of study. Every chapter includes a large quantity of exceptional exercises—a hallmark of this text—that address skills, applications, concepts, and technology. The Eleventh Edition includes updated applications, exercises, and technology coverage. In addition, modern and relevant topics such as health statistics have been added. The authors have also added more study tools, including a prerequisite skills diagnostic test and a greatly improved end-of-chapter summary, and made content improvements based on user reviews.
Many teenagers leave home for college but don't take their faith with them. Popular writer and speaker Sean McDowell offers a solution for this problem: a new way of approaching faith that addresses t... | 677.169 | 1 |
Finest math problem solver That can Make you an improved University student "talk" about algebra and You discover it by carrying out: paying attention in class, actively researching, and fixing math troubles – even though your instructor has not assigned you any. In the event you find yourself battling to try and do properly in your math class, then check out most effective web-site for fixing math complications to understand the way you may become an even better math student.
Low-cost math specialists on the web
Math programs follow a purely natural development – each one builds on the awareness you've obtained and mastered through the prior course. When you are obtaining it challenging to stick to new concepts in school, pull out your outdated math notes and evaluation earlier product to refresh on your own. Ensure that you meet the conditions in advance of signing up to get a course.
Evaluation Notes The Evening Ahead of Class
Detest every time a instructor phone calls on you and you have forgotten tips on how to remedy a specific issue? Stay clear of this minute by examining your math notes. This may make it easier to establish which ideas or issues you'd wish to go over in school another working day.
The considered doing homework each individual night time may seem annoying, but if you need to succeed in , it really is essential that you repeatedly practice and grasp the problem-solving procedures. Make use of your textbook or on the web guides to operate through leading math difficulties over a weekly basis – regardless if you've got no research assigned.
Make use of the Dietary supplements That come with Your Textbook
Textbook publishers have enriched present day publications with excess material (for instance CD-ROMs or on the web modules) that will be utilized to assistance learners achieve extra follow in . A few of these materials can also include things like a solution or rationalization guideline, which might help you with performing as a result of math troubles by yourself.
Read through In advance To remain In advance
In order for you to lessen your in-class workload or maybe the time you commit on homework, make use of your free time just after school or to the weekends to browse in advance to your chapters and ideas which will be covered the subsequent time you're in school.
Assessment Aged Checks and Classroom Examples
The perform you need to do in class, for research, and on quizzes can offer clues to what your midterm or ultimate test will search like. Use your old exams and classwork to make a personalized analyze guideline for your upcoming examination. Seem for the way your instructor frames issues – this is certainly possibly how they're going to appear on your own take a look at.
Learn how to Get the job done Because of the Clock
This can be a popular review tip for persons using timed exams; primarily standardized checks. In case you only have forty minutes for any 100-point check, then you can certainly optimally commit four minutes on every single 10-point issue. Get information and facts regarding how extensive the take a look at will be and which varieties of concerns will likely be on it. Then prepare to attack the easier questions initially, leaving you plenty of time for you to shell out around the far more difficult types.
Maximize your Sources to get math homework help
If you're getting a hard time being familiar with ideas at school, then be sure to get enable outside of class. Question your buddies to create a examine team and visit your instructor's office environment hours to go about rough complications one-on-one. Attend study and critique sessions whenever your instructor announces them, or seek the services of a non-public tutor if you want one.
Discuss To Oneself
Once you are reviewing troubles for an exam, check out to clarify out loud what method and methods you used to obtain your remedies. These verbal declarations will occur in helpful for the duration of a take a look at once you need to remember the steps you need to just take to locate a alternative. Get extra follow by hoping this tactic by using a buddy.
Use Analyze Guides For Excess Observe
Are your textbook or course notes not supporting you realize everything you really should be discovering in school? Use research guides for standardized examinations, including the ACT, SAT, or DSST, to brush up on outdated materials, or . Research guides normally come equipped with comprehensive explanations of the way to fix a sample trouble, , and you can often uncover where could be the better get mathtroubles. | 677.169 | 1 |
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Hi everyone ! I need some urgent help! I have had a lot of problems with algebra lately. I mostly have problems with middle school math dilation lesson plan. I can't solve it at all, no matter how much I try. I would be very relieved if anyone would give me any kind of help on this matter .
That is true , there are programs that can assist you with homework. I think there are several ones that help you solve algebra problems, but I heard that Algebrator stands out amongst them. I used the program when I was a student in Algebra 1 for helping me with middle school math dilation lesson plan, and it always helped me out since then. In time I understood all the topics, and then I was able to solve the hardest of the tests without the program. Don't worry; you won't have any problem using it. It was made for students, so it's simple to use. Actually you just have to type in the keyword that's all .Of course you should use it to learn math , not just copy the results, because you won't improve that way.
Some teachers really don't know how to explain that well. Luckily, there are programs like Algebrator that makes a great substitute teacher for algebra subjects. It might even be better than a real teacher because it's more accurate and quicker!
Algebrator is a easy to use software and is definitely worth a try. You will find several exciting stuff there. I use it as reference software for my math problems and can say that it has made learning math much more enjoyable.
Thanks a lot for the detailed information. We will definitely try this out. Hope we get our assignments completed with the aid of Algebrator. If we have any technical queries with respect to its use, we would definitely get back to you again. | 677.169 | 1 |
Through the visual world of geometry, students experience probability as a tangible issue, allowing them to more easily grasp this important, but challenging concept. In this unit, you will find the probabilites for situations such as a meteor striking the United States, your meeting a friend at a mall, hearing a favorite song on the radio, and many others. Instead of using algebraic formulas to solve these probability problems, you will use geometric figures. The geometric solutions do not require any memorization of formulas or terms. Instead, you will be able to use well-known geometric relationships to understand the problem situations and then to solve them. | 677.169 | 1 |
Welcome to My Parent Page!
My algebra and pre-algebra pages on this web site contain several sites to provide help and practice on concepts. There are websites on the math and pre-algebra web pages under resources that also provide explanations of concepts in our textbooks. IXL can be used to practice concepts. Please refer to the parent letter for aides that are provided to assist students. Explore my web site to see what is available.
Algebra Class News
Students are working on chapter 7 which covers ratios, rates, proportions, fractional equations and percents. There will be a vocabulary quiz at the beginning of each week.
Algebra 2 Class News
Students are working on chapter 5 which covers systems of linear equations. There will be a vocabulary quiz at the beginning of each week. The test for chapter 4 is on Wednesday.
Pre-algebra Class News
Students are working on chapter 9 which covers ratios, rates, proportions, and percents. There will be a vocabulary quiz at the beginning of each week. Students are working on programming using Tech Basic. | 677.169 | 1 |
Abstract algebra
Boolean algebraringsvector spaces ... algebras It is normal to build a theory on one kind of structure, like group theory or category theory The purpose of each theory is to organize in a simple-to-complex model the precise definition of a concept, examples, its substructures, the relations between them: morphisms and its applications, inside the own theory as well outside. During history, different fields of mathematics have used algebras . Algebras are about finding or specifying rules on how to calculate with certain mathematical formulas and expressions. Another algebra (which is not abstract ) is elementary algebra , for example.
3. Abstract Algebra - Definition Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. The term abstract algebra is used to distinguish the
Abstract algebra - Definition
Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groupsrings and fields . The term "abstract algebra" is used to distinguish the field from " elementary algebra " or "high school algebra" which teaches the correct rules for manipulating formulas and algebraic expressions involving real and complex numbers Historically, algebraic structures usually arose first in some other field of mathematics, were specified axiomatically, and were then studied in their own right in abstract algebra. Because of this, abstract algebra has numerous fruitful connections to all other branches of mathematics. Examples of algebraic structures with a single binary operation are:Related books
Information for contributors
6. BEACHY / BLAIR: ABSTRACT ALGEBRA An abstract algebra course at the junior/senior level, whether for one or two semesters, has been a wellestablished part of the curriculum for mathematics
INTRODUCTION
Some of the strengths of this undergraduate/graduate level textbook are the gentle introduction to proof in a concrete setting, the introduction of abstract concepts only after a careful study of important examples, and the gradual increase of the level of sophistication as the student progresses through the book. A number theory thread runs throughout several optional sections, and there is an overview of techniques for computing Galois groups. The book offers an extensive set of exercises that help to build skills in writing proofs. Chapter introductions, together with notes at the ends of certain chapters, provide motivation and historical context, while relating the subject matter to the broader mathematical picture.
7. BEACHY: ABSTRACT ALGEBRA II A third edition of Abstract Algebra by John A. Beachy and Bill Blair published by Waveland Press. Includes table of contents, preface and supplements.
Abstract Algebra II
by John A. Beachy
These notes served as a companion volume to the book Abstract Algebra Second Edition (written jointly with Bill Blair, and published by Waveland Press in 1995). The notes became the preliminary version of the book Introductory Lectures on Rings and Modules published by Cambridge University Press, and so it has become necessary to discontinue publication of the notes on the Web. Click here for further information about the book. If you need additional information, please contact John Beachy, Dept. of Mathematical Sciences, Northern Illinois University, DeKalb, Illinois 60115, Tel. 815 / 753-6753, email: beachy@math.niu.edu This site was opened in 12/1995, and last modified on 5/21/2002. This page has been accessed 179,681 times since 8/96. Author's homepageHomepage for Abstract Algebra Second edition
ABSTRACT ALGEBRA
ONLINE STUDY GUIDE
These online notes are intended for students who are working through the textbook Abstract Algebra . The notes are focused on solved problems, and will help students learn how to do proofs as well as computations. There are also some "lab" questions on groups, based on a Java applet written by John Wavrik of UCSD. The pages are in html format, optimized for To work through the problems and related pages, click on the
and William D. Blair If you would like a printed copy of the solved problems, rather than printing the html pages you should download and print one of these supplements, which are typeset using LaTex and are available in pdf format.
ONLINE STUDY GUIDE
These online notes are for students who are working through our textbook Abstract Algebra . The notes are focused on solved problems, which are numbered consecutively, beginning with the exercises in the text. Our intention is to help students learn how to do proofs as well as computations. There are also some "lab" questions on groups, based on a Java applet written by John Wavrik of UCSD.
ABSTRACT ALGEBRA:
A STUDY GUIDE FOR BEGINNERS
John A. Beachy Department of Mathematical Sciences Northern Illinois University
This study guide is intended for students who are working through the Third Edition of our textbook Abstract Algebra (co-authored with William D. Blair). The guide is focused on solved problems, and my goal is to help students learn how to do proofs, as well as computations. The number of problems can be quite overwhelming, so if you don't have enough time to try all of the solved problems, you can at least read the solutions. The files are in pdf format, suitable for viewing with Adobe Acrobat Reader . Please note that this study guide differs substantially from the Online Study Guide The study guide is currently still under revision. The latest draft 166 pages, pdf format In this draft (posted 3/1/2010), the study guide is very close to final form. There are more than 500 problems, and over half of these problems have full solutions in the study guide. I still hope to add answers for many of the computational problems that do not have solutions. Please send me email if you find typos or mathematical mistakes.
12. Contemporary Abstract Algebra Abstract Algebra With GAP This GAP Manual accompanies Contemporary Abstract Algebra, Fifth Edition, and provides instruction and exercises for students who use the GAP software Abstract Algebra/Rings
Introduction to Rings Rings are algebraic structures designed to model and abstract the structure of the integers ( ), so that we can duplicate some of the processes in which integers are used, but in a more general setting. It will be helpful if you have familiarity with the concepts and theorems for groups, because we'll be using many of the same ideas and theorems. Definition: A ring is a set R with two binary operations and that satisfies the following properties: For all
The ring forms an abelian group under the addition operation. R is closed under is associative) R contains an additive identitiy) R contains additive inverses) is commutative) R is closed under is associative) and is distributive over
If you're already familiar with the concepts of groups and semigroups, we can compress the conditions above to:
Abstract algebra
2008/9 Schools Wikipedia Selection
. Related subjects: MathematicsAbstract algebra is the subject area of mathematics that studies algebraic structures , such as groups rings, fields, modules, vector spaces , and algebras. Most authors nowadays simply write algebra instead of abstract algebra The term abstract algebra now refers to the study of all algebraic structures, as distinct from the elementary algebra ordinarily taught to children, which teaches the correct rules for manipulating formulas and algebraic expressions involving real and complex numbers , and unknowns. Elementary algebra can be taken as an informal introduction to the structures known as the real field and commutative algebra. Contemporary mathematics and mathematical physics make intensive use of abstract algebra; for example, theoretical physics draws on Lie algebras. Subject areas such as algebraic number theory, algebraic topology, and algebraic geometry apply algebraic methods to other areas of mathematics. model theory. Two mathematical subject areas that study the properties of algebraic structures viewed as a whole are universal algebra and category theory. Algebraic structures, together with the associated
19. Abstract Algebra Rather than looking for the solutions to a particular problem, abstract algebra is interested in such questions as When does a solution exist? | 677.169 | 1 |
Elementary Numerical Analysis: An Algorithmic Approach
by Samuel Daniel Conte and Carl de Boor
This is a very nice introduction to numerical methods. The topics are presented in a logical and a pedagogical method.
If your looking to learn numerical techniques this book will well serve your purpose.
Readers unfamiliar with this book can see what others have said here.
To learn this material as well as possible I'm working through the book's exercises and wrote up solutions as I went along.
This is a work in progress that currently includes solutions to some questions in Chapters 3.
You can find the computational exercises (without explanations) for free in the links below. In my spare time
I'm currently working on a text solution manual for this book. | 677.169 | 1 |
mathematics
Mathematics is all around us, from the in-depth calculations and programming that make up the apps on your phone to calculating your weekly shop. Maths is for everyone. It is diverse, engaging and essential.
The Mathematics department at Les Quennevais are committed to ensuring that every learner is equipped with the right skills to reach their future destination, whatever that may be, and provide an environment where students can learn and become competent users of mathematics and mathematical application, helping students transition from concrete to abstract concepts and to develop reasoning and problem solving skills.
Throughout their time at Les Quennevais, students will be following a 5 year Mastery in Maths plan. This allows students to acquire a deep, long-term, secure and adaptable understanding of the subject by building on prior knowledge, acquiring new skills and re-evaluating the application of their knowledge as an ongoing process.
At Key Stage 4 all learners will be following the AQA GCSE specification covering all areas of Mathematics; Number, Algebra, Geometry and Measures, Probability, Statistics, Ratio, Proportion and rates of change. This then culminates with three final exams in year 11, where ongoing support will be provided throughout using a combination of extra revision classes, provided during out of school hours, past papers, specimen papers and exemplar student answers with examiner commentaries. | 677.169 | 1 |
Gary Sterns, CA
Wow! The new interface is fantastic and the added functionality takes it to a new level. Bob Albert, CA
Algebra Professor is truly an educational software. My students feel at ease while using it. Its like having an expert sit next to you. Lucy, GA
Algebra homework has always given me sleepless nights but once I started using Algebra Professor it has been fun. Its made my life easy and study enjoyable. Jeff Kasten, MI
The Algebra Professor software helped me very much. I thought the step by step solving of equations was the most helpful. It was easy to use and easy to understand. I would definitely recommend this to anyone. Thanks, Annie Hines01-29:
number pattern helpers
is there a difference between solving a system of equations by the algebraic methos and the graphical method. Why or why not? | 677.169 | 1 |
Kevin Porter, TX
I think it's an awesome program. Richard Penn, DE.
My daughter is dyslexic and has always struggled with math. Your program gave her the necessary explanations and step-by-step instructions to not only survive grade 11 math but to thrive in it. Thanks. Candida Barny, MT Sam Willis, MD06:
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Math Sense
Don't Let a Fear of Math Keep You From a College Degree
Math Sense is a FREE basic math course designed for individuals needing a refresher in math.
Whether you need to prepare for college entry math classes or simply wish to improve your computation skills, we can help you!
Math Sense is open to curriculum students who did not qualify for developmental math classes or other individuals that possess a high school diploma or high school equivalency and need a refresher course in basic math skills. | 677.169 | 1 |
Scientific Calculator
So there are many calculators out there, but I teach chemistry and there are certain things I tell my students to look for when they are getting a calculator.
1. a 2 line display so you can see the order of operations and have an easier time manipulating it with parentheses and such.
2. log and ln buttons so they can do problems that have to do with pH later in the year.
3. sin, cos, and tan so they can use the calculator for most of the math classes they will take.
4. a scientific notation button. This looks like a x10^x, exp, or EE. These are the most common ways that this appears. This way I can have students easily put in numbers like 6.02 x 10^23 and use them in a calculation.
The scientific calculator from silentmatt.com fulfills all of my requirements.
When you open up the calculator if you click on the "Sidebar" option, the display is nice and simple. you can then type in what you want and click on the necessary buttons.
If you have other requirements or other calculators that you have used that work, please let me know. | 677.169 | 1 |
Basic Linear Algebra is a textual content for first 12 months scholars, operating from concrete examples in the direction of summary theorems, through tutorial-type workouts. The publication explains the algebra of matrices with functions to analytic geometry, platforms of linear equations, distinction equations, and complicated numbers. Linear equations are handled through Hermite common kinds, which gives a winning and urban clarification of the inspiration of linear independence. one other spotlight is the relationship among linear mappings and matrices, resulting in the switch of foundation theorem which opens the door to the idea of similarity. The authors are renowned algebraists with substantial event of educating introductory classes on linear algebra to scholars at St Andrews. This e-book is predicated on one formerly released through Chapman and corridor, however it has been generally up to date to incorporate extra explanatory textual content and entirely labored ideas to the workouts that each one 1st 12 months scholars may be capable of resolution.
This publication is a compilation of numerous works from well-recognized figures within the box of illustration conception. The presentation of the subject is exclusive in providing a number of varied issues of view, which should still makethe e-book very helpful to scholars and specialists alike.
This can be an abridged version of the author's prior two-volume paintings, Ring conception, which concentrates on crucial fabric for a basic ring thought path whereas ommitting a lot of the cloth meant for ring thought experts. it's been praised via reviewers:**"As a textbook for graduate scholars, Ring thought joins the easiest. | 677.169 | 1 |
Don Copeland Margaret Thomas, NY
I was just fascinated to see human-like steps to all the problems I entered. Remarkable! Alex Martin, NH02-07:
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GED Mathematics
If you are math-phobic, or just need a quick refresher then this prgram will help you prepare for even the most challenging problems on the GED mathematics test. Work along as the math professor guides you through many practice examples on the chalkboard while offering step-by-step solutions and clear explanations along the way. This easy-to-follow review course includes sample problems in all of these areas: number operations, probability, statistics, data analysis, algebra, and coordinate geometry. You will also learn about the structure of the exam and the rules for filling in the grids, how to judge when to use the calculator and when not to. | 677.169 | 1 |
This book demonstrates how some of the fundamental ideas of algebra can be introduced, developed, and extended. It focuses on repeating and growing patterns, introduces the concepts of variable and equality, and examines relations and functions. Its activities are designed to capture the interest of small children as they investigate growing patterns, use pictures of dogs with varying numbers of spots to solve for missing addends, and use spinners to identify and explore functions. The supplemental CD-ROM features interactive electronic activities, master copies of activity pages for students, and additional readings for teachers.
Navigating through Algebra in Prekindergarten-Grade 2 is the first of four grade-band books that demonstrate how some of the fundamental ideas of algebra can be introduced, developed, and extended. The introduction to this book is an overview of the development of algebraic reasoning from prekindergarten through grade 12. Each of the three chapters that follow the introduction focuses on a basic idea of algebra. Chapter 1 deals with repeating and growing patterns, chapter 2 introduces the concepts of variable and equality, and chapter 3 introduces relations and functions.
Each chapter begins with a discussion of the foundational ideas and the expectations for students' accomplishments by the end of grade 2. This discussion is followed by student activities that introduce and promote familiarity with the basic ideas. At the beginning of each activity, the recommended grades are identified in the margin and a summary of the activity is presented. The goals to be achieved, the prerequisite skills and knowledge, and the materials necessary for conducting the activities are presented. Some of the activities have blackline masters, which are signaled by an icon and identified in the materials list and can be found in the appendix. They can also be printed from the CD-ROM that accompanies the book. The CD, also signaled by an icon, contains two applets for students to manipulate and resources for professional development.
All the activities have the same format. Each consists of three sections: "Engage," "Explore," and "Extend." The "Engage" section presents tasks designed to capture students' interest. "Explore" presents the core investigation that all students should be able to do. "Extend" offers additional activities for students who demonstrate continued interest and want to do some challenging mathematics. Throughout the activities, questions are posed to stimulate students to think more deeply about the mathematical ideas. After some questions, possible responses are shown in parentheses. Margin notes include teaching tips, anticipated student responses to some of the questions or activities, students' work, copies of blackline masters and solutions to problems on them, and quotations from Principles and Standards for School Mathematics (National Council of Teachers of Mathematics 2000). The discussion for each activity identifies connections with other content strands in the curriculum and to process strands, offers insights about students' performance, and suggests ways to modify the activities for students who are experiencing difficulty or who are in need of enrichment. Although grade levels are recommended, most of the activities can be modified for use by students at other levels in the pre-K-2 grade band. In order to make modifications that will most enhance students' learning, teachers are urged to observe students' performance by taking note of the appropriateness of their mathematical vocabulary, the clarity of their explanations, the robustness of the rationale for their solutions, and the complexity of their creations.
Book Description National Council of Teachers of Mathematics36513
Book Description National Council of Teachers of Mathematics. Condition: New. Paperback. Worldwide shipping. FREE fast shipping inside USA (express 2-3 day delivery also available). Tracking service included. Ships from United States of America. Seller Inventory # 0873534999 2001. PAP. Condition: New. New Book. Shipped from UK in 4 to 14 days. Established seller since 2000. Seller Inventory # CE-9780873534994 | 677.169 | 1 |
Math
At McAuliffe Manual, our units and lessons are in complete alignment with the new Common Core State Standards that Colorado has adopted and is a mastery-based curriculum. Each lesson develops students' problem solving abilities by building their conceptual understandings, skills, mathematical processes, attitudes towards math, application of math in real-world contexts, and self-awareness. Additional resources are utilized to differentiate for below- and above-grade level students.
As an IB Middle Years Programme candidate school, McAuliffe Manual provides both Standard Mathematics and Accelerated Mathematics courses. The Standard Mathematics course sequence teaches Common Core Standards at the pace recommended by the state of Colorado. Upon successful completion of the standard pathway, students will be prepared for High School Algebra 1 during their 9th grade year and will be on track for completing Pre-Calculus math during high school.
The Accelerated course sequence concentrates 7th grade Pre-Algebra, 8th grade Algebra, and High School Algebra I into two years. Accelerated courses include the same Common Core State Standards as the Standard courses, but the pacing of the courses is faster. Upon successful completion of the accelerated pathway, students will be ready for Geometry during their 9th grade year and will be prepared for taking AP math or HL (Higher Level) IB Math during high school. Decisions to accelerate students into high school mathematics before ninth grade are based on solid evidence of student learning, work ethic, and commitment. Students who are not placed in the accelerated math sequence at McAuliffe may still take an AP math or HL IB Math course in high school by successfully "doubling up" on math courses during their 10th or 11th grade year, as DPS recommends. | 677.169 | 1 |
Basic calculator functions explained
Topics covered include mean and standard deviation, data entry, and calculating. Probability for Middle School - Before the scientific calculator can be whipped out of the desk drawer, it's important to understand how probability works. History, the earliest known version of a calculator was a device called an abacus. script Now we will need to setup a few variables. You can name them whatever you like.I have tried using a simply math formula single player commands 1.4 4 mac which was 23 as you seen below by using ripting : using System; using neric; using nq; using System.Functions, modern scientific calculators generally have many more features than a standard four or five-function calculator, and the feature set differs between manufacturers and models; however, the defining features of a scientific calculator include: In addition, high-end scientific calculators generally include: While most scientific models.A basic calculator covers most general math functions - addition, subtraction, multiplication, and division.
If they do, then it calls another function 'DoSum which does the calculation. The first electronic calculators were developed in 1961, and the first scientific calculator was developed and released in 1968. This is a follow up question.Tasks; using ripting; using ripting; using th; using parser; namespace ConsoleApplication4 class Program static void Main(string args) Expression eh new Expression 28 String result lculate Int32(result adLine).Text Based Basic autocad 2015 serial number mac Formula Calculator Function/Class.There is also some overlap with the financial calculator market.RedCrab is a portable tool that can be used immediately after extraction.The value given to the sign variable inside this function is the value of the key pressed by the user.However before understanding each function, first you need to setup your html page. | 677.169 | 1 |
... routines. SimplexCalc is an indispensable calculator designed for math teachers, scientists, engineers, university and college faculty and students, financiers and other professionals. Features include: * Ease of use * Scientific ...
... your calculating routines. It contains more than hundred mathematical functions and physical constants to satisfy your needs ... (cut, copy and paste operations) - All functions, mathematical and physical constants can also be used in ...
AstroGrav is a full-featured, high precision solar system simulator that calculates the gravitational interactions between all astronomical bodies, so that the motions of asteroids and comets are simulated much more accurately ...
Math Practice is an easy to use software addressed ... make their first steps into the world of math, providing fairly simple teaching tools and a basic ... division. Once you click on one of these, math exercises are automatically launched, pick difficulty by choosing ...
... creates graphs, as the name implies. It targets mathematical students and other similar users. The interface is based on a clean and well-organized window. FX Graph supports multiple tabs, allowing ...
GeoGebra portable is free and multi-platform dynamic mathematics software for all levels of education that joins geometry, algebra, tables, graphing, statistics and calculus in one easy-to-use package. Features includes Graphics, ...
RekenTest is open source educational software to practice arithmetic skills. It supports basic arithmetic operations like addition and subtraction, the muliplication tables and so on, as well as more advanced arithmetic ...
... mosaic maker Artensoft Photo Mosaic Wizard does complex math to fit cell images so that they form ... the colors, photomosaic maker will solve a complex mathematical task of matching and fitting cell images without ... | 677.169 | 1 |
MATH 1007 (Ideas in Math) Homework
NOTE: All homework is due before start of class on the day it's due.
Due Friday, January 20, 2017:
Read the syllabus carefully
Read "Polya's Problem Solving Techniques" handout
Read "How to Learn From a Math Book" handout
Solve the puzzle of the day (on Canvas)
Purchase access to Top Hat
Purchase Geometer's SketchPad
Please write on your notecard: your name, major, why you're taking this class, what "math" means to you, your previous experience(s) in math and how you feel about mathematics, and as much as or as little as you'd like to tell me about yourself - maybe just something interesting or unique to you.
Due Monday, January 23, 2017:
Do the homework due last Friday if you haven't already
Read the "How to Read Mathematics" handout. As you read about the birthday paradox, try and really read and understand it, practicing what you just learned about reading math as you go.
Solve the puzzle of the day (on Canvas)
Download and install Sokoban++
On Top Hat, contribute to the discussion on reading math (it is mandatory)
Due Wednesday, January 25, 2017:
Do the homework due last Friday and Monday if you haven't already and turn it in ASAP
Solve the first Sokoban puzzle (on Canvas) - Sokoban is the puzzle of the day for Wednesdays
Pick your favorites of the first twenty projects (on Canvas)
Finish the in-class handout
On Top Hat, contribute to the discussion on reading math if you haven't already (it is mandatory)
Due Friday, January 27, 2017:
Do the previous homework if you haven't already and turn it in ASAP
Do the puzzle of the day (on Canvas)
Do the Magic Square Homefun
Answer the "Sums of Numbers 2" question on Top Hat
Due Monday, January 30, 2017:
Do the previous homework if you haven't already and turn it in ASAP
Do the puzzle of the day (on Canvas)
Contribute to the Top Hat discussion on magic squares (it is mandatory)
Pick your favorites of the next twenty projects (21-40, on Canvas)
Due Wednesday, February 1, 2017:
Solve Sokoban level 3 (on Canvas)
Pick your favorites of the next twenty projects (41-60, on Canvas)
Work on the Tilings and Fibonacci Numbers handout
Due Friday, February 3, 2017:
Do the puzzle of the day (on Canvas)
Finish the Tilings and Fibonacci Numbers handout
Due Monday, February 6, 2017:
Do the puzzle of the day (on Canvas)
Answer the Top Hat question about converting miles to km
Watch the videos on Fibonacci numbers and take the quiz (on Canvas)
Pick your favorites of the last twenty projects (61-80)
Due Wednesday, February 8, 2017:
Solve Sokoban level 4 (on Canvas)
Finish the handout on Skyscrapers and Battleship
Read the handout on Complexity Theory given on Friday
Due Friday, February 10, 2017:
Do the puzzle of the day (on Canvas)
Finish the handout on Skyscrapers and Battleship if you haven't yet
Reread the handout on Complexity Theory, writing out as many relationships between the different classes of problems as you can. Be sure to note the relationships you're unsure of so you can ask in class.
Due Monday, February 13, 2017:
Do the puzzle of the day (on Canvas)
Figure out what projects would be most interesting to you and which group you want to work with, partially based on that decision.
Due Wednesday, February 15, 2017:
Solve Sokoban level 5 (on Canvas)
If you haven't already, figure out what projects would be most interesting to you and which group you want to work with, partially based on that decision.
Due Friday, February 17, 2017:
Solve the puzzle of the day (on Canvas)
Get together with your new group to study Complexity Theory and work on the Group Contract (due Wednesday, February 22)
Prove that "Subset Sums" is in NP (adding two numbers is one operation)
Due Monday, February 20, 2017:
Solve the puzzle of the day (on Canvas)
Get together with your new group to work on Graph Theory and work on the Group Contract (due Wednesday, February 22)
Finish the Graph Theory handout given in class
Due Wednesday, February 22, 2017:
Solve Sokoban level 6 (on Canvas)
Get together with your new group to work on Graph Theory and FINISH the Group Contract.
The Group Contract is due, no exceptions.
Finish the Graph Theory handout given in class
Due Friday, February 24, 2017:
Solve the puzzle of the day (on Canvas)
Study for Midterm #1
Due Monday, February 27, 2017:
Solve the puzzle of the day (on Canvas)
Schedule an appointment with Barbara Mento and find good, credible, understandable sources to use for your project.
Due Friday, March 3, 2017:
Solve the puzzle of the day (on Canvas)
Finish the Graph Theory 3 handout
Turn in the Colorings and Bipartite Graphs worksheet
Find good, credible, understandable sources to use for your project and start reading them
Due Monday, March 13, 2017:
Solve the puzzle of the day (on Canvas)
Finish the Geometer's SketchPad coloring exercise (on Canvas)
Finish the Graph Theory 4 handout
Finish the Graph Theory 3 homefun (i.e. the 3-Coloring, graph isomorphisms, and planar graphs exercise packet) and turn that in
Work on your group project and get ready to turn in an outline on Wednesday the 15th
Due Friday, March 31, 2017:
Solve the riddle of the day (on Canvas)
Complete the Geometer's SketchPad assignment on the Nine-point Circle (on Canvas)
Turn in the rough draft of your presentation if you haven't already
Due Monday, April 3, 2017:
Solve the riddle of the day (on Canvas)
Complete the Geometer's SketchPad assignment on the Gergonne Point (on Canvas)
Get ready for your presentation, and work on your individual papers for the project
Wednesday, April 5, 2017: Midterm Exam #2
Friday, April 7, 2017:
Read the Triangle Geometry Part I handout
Complete the "Complements and Anticomplements" SketchPad assignment on Canvas
Solve the puzzle of the day (on Canvas)
Due Monday, April 10, 2017:
Solve the puzzle of the day (on Canvas)
Read the Triangle Geometry Part II handout; try and read it all, but definitely at least read up through the end of the Isotomic Conjugates section
Do the Isotomic Conjugates SketchPad assigment on Canvas
Due Wednesday, April 12, 2017:
Solve Sokoban level 17 (on Canvas)
First group presentation takes place
Turn in a rough draft of your project paper in class
Due Wednesday, April 19, 2017:
Solve Sokoban level 20 (on Canvas)
Finish reading the Triangle Geometry Part II handout, and as you read it, do the "Symmedian Point and Kosnita Point" and then "Isogonal and Cyclocevian Conjugates" SketchPad assignments on Canvas.
Second group presentation takes place | 677.169 | 1 |
Hey MysticCat! check this out and tell me what you think. I know nothing about it but can vouch for the person who referred me to it. I'll come back and post the isbn here as soon as she sends it to me.
Visual Mathematics: A Step by Step Guide (title is what she quoted, so best to wait for the isbn, right?)
And to be clear, obviously I know there are many professions and occupations where you would need to know higher forms of math. But I think he's like me: I knew I disliked math enough that any career that required lots of it was automatically excluded from consideration. I guess what I'm looking for are examples from anyone not in a math-oriented field as to how they've used what they learned in algebra.
When I was in nursing, I used basic ratio/proportion algebra to figure out medication dosages nearly everyday while at the hospital. But that was about the only time I used it. Higher forms of math was never used.
I'm still trying to figure out why elementary calculus & finite math was necessary for me to complete my business degree. A lot of economics, accounting, & finance formulas come out of calculus, but one does not need to know how to do calculus to find the answers, just basic algebra. Even then in the real world, computer programs run those formulas for you. I saw some of the finite math stuff in my statistics class, but outside of that I've never seen it again.
Ok, I win at worst HS math situation (I remember a board of education member once saying to me, "you never did quite have luck with math classes, did you?"). But I still see the value to a basic understanding of algebra. I can't think of anything specific that algebra taught me that, say, basic math didn't (especially because I somehow managed to get myself into a math class in my junior year called "Finite Math." It was basic math all the way to introductory algebra. We only got to do it that way because the first semester was statistics. I don't know how this managed to come after Algebra II, but it was SOOO cool that it did!), but I can say that, imo, some of these annoying classes (things like Algebra or World History) just teach useful skills.
My math classes taught me that some problems only one answer. They taught me how to collaborate with classmates when a particular teacher was being unreasonable, they taught me how to verbalize my concerns over a grade (with a less tangible subject like, say, English, where an essay is incredibly subjective, math tends to be more objective, and if a teacher marked an answer wrong but it was clearly right, you can easily win that one!), and how to attack one challenge multiple ways (because algebra is that kind of subject: there is [usually] only one answer, but there are [usually] multiple paths to that one answer).
Personally? I still hate math, and I avoid using anything beyond basic math when I can. Nonetheless, I still love solving ratios, calculating my grades with fancy formulas, and the occasional multiplication/division question. But algebra? I hardly remember it-I just remember that I learned about things other than math in my high school math classes.
I use algebra on a regular basis, especially to figure out complex percentages. I have trouble visualizing where the numbers go in the equation, so I'll write an equation fully out and then do the calculations in my head.
People use basic algebra all the time without thinking about it. Once you have the concepts in mind, you forget that it's "algebra" and you just think of it as working out an equation.
I currently teach Algebra I in the same state you live, MC, and in a county not very far away. Let me know if you decide you want an algebra tutor. (;
This is very true and in the state you live, it is a result of standardized testing. There are "more creative" things that can be done with Algebra but because Algebra I is and always will be an EOC class, teachers are less likely to do more fun things. Their goal is to teach it exactly how it will look on the EOC so that the students can pass it. (As an special ed teacher teaching Algebra, my approach is a bit different, but gets the same result.)
This is certainly part of the problem, but I came along long before teaching for the test, and it was a problem then as well. Teaching for the test exacerbates the problem, I think.
And thanks for the tutoring offer. We actually have a really great tutor and who is making some good progress. She's a friend whose family he's known all his life, and she has a son a year older than ours with similar spectrum challenges, so she has a really good sense of how to approach things with him. She is seeing promise.
Quote:
Originally Posted by psusue
The other point I have is that he has no idea yet what will truly interest him-- many of the topics that he may go on to love haven't yet been covered.
Exactly! We keep reminding him of this, along with the basic "you need this to get into college to study what you want to study." Your whole post was very helpful and encouraging. Thanks!
Quote:
Originally Posted by southbymidwest
MysticCat, I think your son has to look at this at a macro versus micro level also. School is about learning about all sorts of stuff, even that which you don't have an affinity for, or like. I'd bet big money that the vast majority of us took classes in something that we despised, and thought that they were "stupid" to take.Thanks everyone for the suggestions. There's so much that I don't think that I can respond to everything everyone has said, but I really appreciate it all!
ETA:
Quote:
Originally Posted by psusue
For example, I HATED math with the burning passion of a thousand fiery suns . . . .I've also taught older HS students how different gambling games have different odds.... and how to calculate them so should they ever visit a casino at some point in their lives, they can maximize winning potentialMC, you are so clever, I am sure that you have tried every which way to get him to understand the importance of algebra on mathematical and non-mathematical levels. But I do find this to be a very interesting and educational discussion, and I am glad that you brought it up.
Algebra is needed for construction, such as if you are building a table or laying tile on a new floor. Algebra is needed in all types of projects that require building, that is if you want to do it right.
Algebra is also important if you ever want to get good with money and understanding dividends and interest rates and if you are making a decent return on an investment.
Algebra is also important for programming as well as just basic counting, say if you need to do fractions and split a certain amount of money between multiple people at different rates.
How sad is it that I read this three times and still thought "I would have no clue how to do that"?
If it hasn't already been obvious, clearly part of the problem is that our son has two parents for whom math was always something of a foreign language. So, for example, when he was struggling with linear equations, I had to look them up and try to remember/learn again what they are before I could even think of helping him -- I was basically trying to learn them along with him.
Seriously. thanks for the suggestion. Once I wrap my head around it -- and I'll do my best to do that -- I'll try it on him. And yes, I wish he had you for a teacher, too.
Well that was a few sentence explanation of a larger idea. Let's say your monthly plan is $50. You only get 150 texts a month with your plan.
your equation would be y=.10x+50
(x would be number of texts a month over the 150 you get included in plan. y would be the total amount you have to spend)
If you use less than 150 texts in the month, you only pay the $50. So your first point on the graph would be (0,50).
Now let's say you have a super social teenager, and he decides that "texts are pretty cheap!" and uses 1,000 texts in that one month. That's 850 more texts that you get in your plan. Each additional text over your plan costs .10 (I think that's what I used as a random example in previous post). so 850 texts at .10 each comes out to be $85 in texts. (craziness!)
So if you wanted to graph that point, it would be (850, 135) (the 850 is the number of texts beyond what you get. the 135 is the total cost for the month, the $50 monthly charge + the $85 in texts.
If you plot those two points, connect them with a ruler, and you have your line graphed.
Did that make any more sense or just cause more confusion? (it's hard to explain without writing down or speaking) | 677.169 | 1 |
Discrete Mathematics: An Open Introduction
Publisher: University of Northern Colorado2017 ISBN/ASIN: 1534970746 Number of pages: 345
Description: This book was written to be used as the primary text for a transitions course (introduction to proof), as well as an introduction to topics in discrete mathematics. Topics: Counting; Sequences; Symbolic Logic and Proofs; Graph Theory; Generating Functions; Introduction to Number Theory.
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Topics in Discrete Mathematics by A.F. Pixley - Harvey Mudd College This text is an introduction to a selection of topics in discrete mathematics: Combinatorics; The Integers; The Discrete Calculus; Order and Algebra; Finite State Machines. The prerequisites include linear algebra and computer programming. (3767 views)
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Book Title: "An Introduction to Mathematics" by Alfred North Whitehead
Title
An Introduction to Mathematics
Author(s)
Alfred North Whitehead
Description
"The object of the following chapters is not to teach mathematics, but to enable students from the very beginning of their course to know what the science is about, and why it is necessarily the foundation of exact thought as applied to natural phenomena." Thus begins this volume by the prominent English philosopher and mathematician Alfred North Whitehead, a concise statement on the nature and meaning of mathematics for the general student. Expertly written and abounding in insights, the book presents a lively exposition of mathematical concepts, the history of their development, and their applications to the physical world.
Whitehead explains in broad terms what mathematics is about, what it does, and how mathematicians do it. Generations of readers who have stayed with the philosopher from the beginning to the end have found themselves amply rewarded for taking this journey. As The New York Times observed decades ago, "Whitehead doesn't popularize or make palatable; he is simply lucid and cogent ... A finely balanced mixture of knowledge and urbanity .... Should delight you."
Dover republication of the edition originally published by Henry Holt & Co., New York, 1911.
Library of Congress Cataloging-in-Publication Data
???
CIP Number
???
LC Control Number
2016058718
LC Call Number
QA7.W53 2017
DDC Call Number
510—dc23
PICTURES
Book Review (Posted on 08-14-2017)
For being one of the most disliked science among the rest, books like these need to flourish into many academic institutions. Yeah, I know most of you are going to say, "but Kris, I already know what Math is anyway LOL. It's all just calculating numbers!" Just calculating numbers? That's what happens when you don't do research.
Author Alfred North Whitehead, who happens to be both a Mathematician and a Philosopher, wrote this book welcoming those into the realm and beauty of Mathematics. Although known for his linguistic style, which may overwhelm the reader a bit, Whitehead did a wonderful job explaining what the true goal of Math, its philosophy, its disciplines and what makes the science applicable to the real world. While the book tries to attract the layperson, those even with knowledge in basic arithmetic should be able to grasp this book.
The longest section of the book is his take on Imaginary Numbers. Whitehead explains some brief history as to why it came to be as some solutions could not be solved, nor could mathematicians back then could arrive at a final answer (square root of 2, for example). It came to the point where I was curious about digging in to Number Theory, a disciplinary study in finding numerical patterns and theoretical primes among, well, numbers (best area if you're studying Cryptography). Besides that, due to Whitehead's explanations, he got me very excited about Geometry, Trigonometry and Differential Calculus. The explanations and introduction to those subjects were gentle to me, making me want to dive deep into them (I'm an Algebra and Mathematical Logic guy). Not only that, but Whitehead made the explanations clear enough to where the reader's eye's won't go bonkers when they see funky, complex and "cool" Math symbols. In other words, all those subscripts and Greek symbols won't seem like you're studying a foreign language with Math; It's all about what its function is and the problem its trying to solve.
Saying that means that I strongly recommend that schools introduce books like these in the classrooms early on. We can't just do Math if we don't know what it's truly all about and what its underlying philosophy is (we could say the same for many other subjects). One my beliefs is "Know Before You Do," and I really wished I knew all about Math before I was a teenager, even though I was very good at Algebra. Nevertheless, perhaps Math won't be so hated and crazy if we knew what it was about and learn about its workings and symbolic notations before taking classes on it. Agree to disagree, but that's how I feel.
Though he passed away during the World War II era, wherever you are, author Whitehead, this review is to thank you for writing this book and your amazing contributions to the wonderful science: Mathematics. I enjoyed your book and look forward to more. Highly recommended, folks!
(NOTE: This review marks as the very first book by Dover Publications! Thank you Dover for your amazing science books. I personally am a fan, and here's to many more coming!)
Get $10 off at the #1 digital bookstore, eBooks.com! Use the coupon codes to redeem the discount: | 677.169 | 1 |
Online Math Classes for Home School Students
Monthly Archives: January 2015
Having taught an SAT prep class 3 times a year I've become very familiar with the content on the current SAT. They've made a few changes since I took the test in high school (mainly adding a 3rd writing section), but overall the math content has been pretty stable. Here is a snapshot of the basic breakdown of math content taken from Gruber's Complete Guide to the SAT, the prep book I use in my course.
Basic Math: this is a review of Middle School concepts and includes fractions, decimals, percents, and rates problems (these can be pretty tricky).
Algebra: It starts with concepts as simple as equation, inequalities, and graphs but continues into topics like exponents and roots.
Geometry: This is an area where many of my current students struggle. Questions include area, perimeter, and volume as well as the properties of shapes. Students are expected to have memorized the properties of all kinds of triangles, quadrilaterals, and polygons.
Miscellaneous: Finally, there is a category that contains all sorts of misfit problems. Many of these types of problems come from probability and statistics and others are pretty random. Set theory is one example of this, a topic many students don't cover in high school.
For the New SAT I've created a chart to show you the main topics, what types of questions it contains, and most importantly how many questions are included in this category. All of the main topics are hyperlinks to the College Board website if you'd like to read about it in detail.
Analyzing and fluently solving equations and systems of equationsCreating expressions, equations, and inequalities to represent relationships between quantities and to solve problemsRearranging and interpreting formulas
Making area and volume calculations in contextInvestigating lines, angles, triangles, and circles using theoremsWorking with trigonometric functions
6
The first thing to notice is that "additional topics" which is mainly geometry comprises only 6 questions on the whole test. This reflects a movement in the math education world away from teaching geometry. Their argument is that it is not essential for college-level mathematics. I wouldn't overreact and cancel all plans for a geometry class just yet, but it is something to keep in mind.
The other takeaway is that the content isn't as broad, but it is deep. So it is going to be more important for your child to master a few skills very well than for them to make it to the final chapter of their math textbook.
But there's more to come on how to prepare for this new test in an upcoming post… | 677.169 | 1 |
This is an example of the use of the Manipulate function in Mathematica 6 to animate a 3D plot. The function is sin(mx) x^p y + b where m, b, and p are being manipulated.
Mathematica is a general computing environment, organizing many algorithmic, visualization, and user interface capabilities within a document-like user interface paradigm. It was originally conceived by Stephen Wolfram, developed by a team of mathematicians and programmers that he assembled and led, and it is sold by his company Wolfram Research of Champaign, Illinois.
Since version 1.0 in 1988, Mathematica has steadily expanded into more and more general computational capabilities. Besides addressing nearly every field of mathematics, it provides cross-platform support for a wide range of tasks such as giving computationally interactive presentations, a multifaceted language for data integration, graphics editing, and symbolic user interface construction. An organized index of its functionality can be found here.
Many major educational and research organizations have Mathematica site licenses, and individual licenses are also sold. With Mathematica 6, a free interactive player is provided for running Mathematica programs which have been converted using a free web service or published on the Wolfram Demonstrations Project website. | 677.169 | 1 |
Kaseberg/Cripe/Wildman's respected INTERMEDIATE ALGEBRA is known for an informal, interactive style that makes algebra more accessible to students while maintaining a high level of mathematical accuracy. This new edition introduces two new co-authors, Greg Cripe and Peter Wildman. The three authors have created a new textbook that introduces new pedagogy to teach students how to be better prepared to succeed in math and then life by strengthening their ability to solve critical-thinking problems. This text's popularity is attributable to the author's use of guided discovery, explorations, and problem solving, all of which help students learn new concepts and strengthen their skill retention Intermediate Algebra: Everyday Explor | 677.169 | 1 |
"Every prospective teacher should read it. In particular, graduate students will find it invaluable. The traditional mathematics professor who reads a paper before one of the Mathematical Societies might also learn something from the book: 'He writes a, he says b, he means c; but it should be d.' "--E. T. Bell, Mathematical Monthly "[This] elementary textbook on heuristic reasoning, shows anew how keen its author is on questions of method and the formulation of methodological principles. Exposition and illustrative material are of a disarmingly elementary character, but very carefully thought out and selected."--Herman Weyl, Mathematical Review "I recommend it highly to any person who is seriously interested in finding out methods of solving problems, and who does not object to being entertained while he does it."--Scientific Monthly "Any young person seeking a career in the sciences would do well to ponder this important contribution to the teacher's art."--A. C. Schaeffer, American Journal of Psychology "Every mathematics student should experience and live this book"--Mathematics Magazine "In an age that all solutions should be provided with the least possible effort, this book brings a very important message: mathematics and problem solving in general needs a lot of practice and experience obtained by challenging creative thinking, and certainly not by copying predefined recipes provided by others. Let's hope this classic will remain a source of inspiration for several generations to come."--A. Bultheel, European Mathematical Society
Reseña del editor:-from building a bridge to winning a game of anagrams. Generations of readers have relished Polya's deft-indeed, brilliant-instructions on stripping away irrelevancies and going straight to the heart of the problem. This book usually ship within 10-15 business days and we will endeavor to dispatch orders quicker than this where possible. BTE9780691164076 | 677.169 | 1 |
introduction
Competitive exams are meant for real-men and women. This is no country for crybabies, kids, college teens and no0bs. So first of all, you must get rid of the following "loser" mindsets:
Yaar this maths is so hard, I can't do it.
I'm not from science/engineering background hence this is not my cup of tea.
I'm poor in maths and I cannot improve.
Thik hai, dekh lenge. (alright, I'll see).
Maths is not difficult. All it requires is concept clarity + lot of practice. In SSC-CGL exam, you've to face Mathematics at two stages
Stage
Maths-Questions
Penalty
Tier-I (Prelims)
50 Qs
Negative 0.25
Tier-II (Mains) Paper I: Arithmetical Ability
100 Qs worth 200 marks
Negative 0.50
The Approach for Maths, stands on two pillars.
Pillars
How?
Conceptual clarity
NCERTs (Free download links @bottom)
Mrunal.org/aptitude
For some topics, directly Quantitative aptitude books.
Lot of practice
From Quantitative aptitude books.
There are lot of books in market, the question is, which one to refer? It is explained at the bottom of this article.
#1: Getting the conceptual clarity
We'll divide Maths or Quantitative Aptitude, into topics and further into subtopics.
Your task is to cover one topic at a time, first get conceptual-clarity and then solve maximum questions at home.
Whenever you learn any shortcut technique, you note it down in your diary.
Similarly, whenever you make any mistake while solving sums, you also note that down in your diary. Night before the exam, you review that diary of mistakes. (why do this? Because it is the "Art of Aptitude" (Click ME)
Topic
Subtopics
How to approach
Number theory
Divisibility, remainders
LCM and HCF
Unknown numbers from given conditiofor
Fractions-comparisions.
NCERT Class 7 Chap 2, 9 (fraction)
NCERT Class 10 Chap 1 (divisibility)
Finally your Quantitative aptitude book.
Basic Maths
Simplification (BODMAS)
NCERT Class 8 Chap 1
Surds, indices
NCERT Class 8 Chap 12
Then NCERT Class 9 Chap 1
Roots, squares, Cubes
Basics from NCERT Class 8 Chap 6 and 7.
Algebra
Linear equation
"Mother's age was x and daughters age.."
"3 mangos and 5 bananas purchased for…"
X+1/2x+3=3/8 then find X.
^This type of stuff. Just practice and you'll get a hang of it.
Basics given in NCERT Class 8 Chap 2 and 9.
Then NCERT Class 9 Chap 4
Lastly NCERT Class 10 Chap 3.
Quadratic equations, Polynomials
Factorization and roots. Heavily asked in Tier-II.
NCERT Class 8 Chap 14
And then NCERT Class 9 Chap 2
Lastly NCERT Class 10 Chap 4
Avg and Ratios
Wine-Water mixture (Alligations)
Can be solved without formula. Go through
%
Basic % (increase, decrease in consumption, population)
Also do NCERT Class 8 Chap 8.
Data-interpretation cases.
Mere extention of % concept. Just practice.
For long division, use this approximation method:
Breakup: SSC-CGL Tier II (2010, 2011, 2012)
Here too, Geometry+Trigonometry have been given emphasis like never before.
Almost 65% of the paper is made up of Geometry, Trig, Percentage and Algebra (and in that too, mostly Quadratic equations.)
#2: Practice
Merely knowing the concepts or formulas won't help. Because unless you practice different variety of questions, you won't become proficient in applying those concepts flawlessly in the actual-exam.
Second, despite knowing concept and formulas, people make silly mistakes either in calculation or in pluging the values.
Third reason- Tier I has 200 questions in 120 limits. =not even 2 minutes per question. Plus, questions reasoning and comprehension might take more than 5 minutes! Therefore speed is essential. Since there is negative marking system, accuracy also matters.
So it is beyond doubt that you have to practice excimer number of questions at home.
The question is where to get the practice? Which book should be used for SSC exam?
Choice of Quantitative Aptitude Book?
In all competitive exams, "uncertainity factor" is involved. Despite your best preparation, you might lose the success-train by 2-3 marks.
Therefore you must never put all eggs in one basket.
While you are preparing for SSC, you should also keep open mind and apply for other competitive exams, such as IBPS, ACIO, ONGC, Railways, LIC, CDS, Coast Guard etc. (Depending on your career-taste).
Publication houses will come up with new books for each and every of ^these exams, but we have neither the time nor the money to buy a new book for every new exam.
Such readymade books are only skimmed down version of original topicbooks. For example, if there is SSC-FCI exam, or ACIO exam, these people will combine a few topics of GK, maths, reasoning and english. And present you a book.
Problem= you don't get comprehensive understanding or coverage. Besides, given the population of India, competition level is always high, irrespective of exam. So half-hearted preparations with readymade "condensed" books don't help much.
Almost all of these exams follow same structure:
General awareness
Maths
Reasoning (Verbal, Non-Verbal)
English vocabularly, grammar and comprehension.
How do they differ from each other?= number of questions, difficulty level and inclusion / exclusion of particular subtopics.
So when you're picking up books for the first time, you should choose the books, that have universal usefulness for similar exams. That way your time, effort and money will be saved.
Books for Maths/Quantitative Aptitude
DONOT use Quantitative Aptitude by R.S.Agarwal for SSC-CGL.
Problems with RS Agarwal's Math book:
The way SSC-CGL question pattern is transforming, R.S.Agarwal's book on Quantitative aptitude, is just not 'upto the mark' to match this changing environment.
Its chapter on Trigonometry (Height and Distance) is simply insufficient to handle SSC-CGL level bombarding.
Similarly coverage of algebra, quadratic equations and number theory is either absent or just for namesake.
The printing and presentation is very " cluttered". He has written the book assuming that you were already good at maths from school level.
If you're already good at basic concepts, use this book for practicing and improving your speed, else don't bother, there are better books in market.
Pricing factor
Author
Quantitative Aptitude For Competitive Examinations (S.Chand)
R.S.Agarwal
Quantum CAT (Arihant)
Sarvesh Kumar
Fast Track Objective Arithmetic (Arihant)
Rajesh Verma
The point is, both books of Arihant Publication (Sarvesh or Rajesh) are way better than R.S.Agarwal, in terms of content, presentation, language and coverage, without being too expensive than R.S.Agarwal's book.
And both of them have universal application for almost all of the competitive exams in India (for maths segment).
My advice, go with either Rajesh Verma or Sarvesh Kumar. Then the question, which one to pick up?
Fast Track Objective Arithmetic by Rajesh Verma.
Quantam CAT by Sarvesh Kumar
Algebra, quadratic equation and Trigonometry specific chapters are given for exclusively for SSC.
If you solve all the sums of this book, then mathematics portion of SSC-CGL (Tier-I and II) will be as easy as a walk in the park.
Although book is written for CAT and Management exams, he starts explaining everything from basics. Then exercises are divided into "Introductory<level 1<Level 2<Final round", based on difficulty level.
Thus it becomes ideal choice for any aptitude exam.
So for lower level exams (SSC/IBPS), you should solve all his solved examples, then introductory exercises, finally level 1.
That'll be quite sufficient.
While it is excellent for SSC, IBPS, UGC, LIC, CDS etc. level exams, its utility starts diminishing as you move towards higher-end exams. | 677.169 | 1 |
Analysis of the real line
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The purpose of this report is to describe the course, Analysis of the Real Line, taught at The University of Texas at Austin. Course materials are presented using the inquiry based learning method. Students work a series of warm up problems before being presented rigorous problems in calculus, including topics on integration, exponential functions, and real number line analysis. Additionally, students consider aspects of these problems that could be incorporated into a high school curriculum. Typical problems in several major areas are summarized along with warm up problems that introduce or extend the topics. | 677.169 | 1 |
Mathematics (M)
This course is for students not ready for college level mathematics and covers the pre-algebra through intermediate algebra mathematics skills needed for college level mathematics courses. The course is delivered in a lab setting allowing students to progress at their own level with the aid of an on-site instructor. The class is organized into three distinct levels of Arithmetic, Beginning Algebra, and Intermediate Algebra with the student required to complete each segment in sequence. Arithmetic topics include concepts and topics of the real number system: including numeric operations, decimals, exponents, radicals, integers, ratios, proportions, fractions, factors, prime numbers, and numeric story problem applications. Beginning Algebra topics include: Power numbers, radicals, logarithms, rational expressions, linear properties, graphs, ordered pairs, relations, polynomial factoring, functions, solutions to linear and systems of two equations. Intermediate Algebra topics include determinants, complex distance and slope, relating data to equation type, application formulas, and application story problems. This course may be repeated as necessary. Formerly MATH 093.
M 105. Contemporary Mathematics. 4 Credits.
This course is desidned to meet the general education mathematics requirement for the liberal arts major. It surveys some of the important ideas and practical applications in matematics and uses a variety of mathematical skills and technology to solve real problems. Topics include problem solving, financial math, mathematical modeling (linear and quadratic), and elementary statistics.
M 109. Col Alg with Sci Applications. 3 Credits.
M 111. Technical Mathematics. 3 Credits.
This course is intended for AAS-degree students enrolled in vocational programs who are not planning to transfer to other degree programs or institutions. This course is a basic mathematics course for developing mathematics skills through introductory algebra as they relate to technical programs. This course includes measurement systems, use of measuring tools, as well as development of area and volume concepts with respect to technical applications. Formerly MAAS 106 or MATH 106.
This course surveys a wide variety of topics including: properties and theorems of the real and complex number systems, the function concept including inverse functions, graphing techniques, linear, quadratic, polynomial, exponential and logarithmic functions, solving systems of equations in two or more variables using matrices and matrix algebra. The development of problem-solving skills is emphasized. Formerly MATH 112.
M 130. Math for Elementary Teachers I. 3 Credits.
The topics included in this course are directly related to elementary mathematics education. The specific number topics included in this course include: numeral system, problem solving, set theory foundation of the real number system, arithmetic algorithms, statistics, probability, and algebra notations. The specific geometry topics include: plane and solid shape classification and properties, congruence, similarity, symmetry, trigonometry, measurement, and transformations. Prerequisite: M 095 or ACT score of 20 or higher or university placement examination. Course Fee: $5.00. Formerly MATH 120.
This course surveys a wide variety of topics including sets and logic, mathematical patterns, number systems, number theory, algebra, geometry, probability, and statistics. The development of problem-solving skills is emphasized. Prerequisite: M111 or M 095, ACT scores 20 to 22, or university placement examination. Formerly MATH 110.
The topics included in this course are: differentiation and integration with positive reinforcement of concepts in algebra, trigonometry and analytic geometry. Prerequisite: ACT scores 25-26 or M 121 or M 151 or university placement examination. Formerly MATH 133.
M 171. Calculus I. 5 Credits.
Developing the concepts of calculus and analytic geometry including rates of change, limits, derivatives and anti-derivatives, concepts of integration, and the application of integration. Prerequisite: M 151 or both M 121 and M 112. Formerly MATH 220.
M 172. Calculus II. 5 Credits.
Further development of the concepts of integration and applications, work with infinite series, plane curves, and parametric vectors and vector valued functions, and partial differentiation. Prerequisite: M 171. Formerly MATH 221.
Use of computers in the classroom focusing on software systems in current use in University and public school situations. The software systems studied are used primarily in science and mathematics but are also adapted for use in developing communication skills. Formerly MATH 320. | 677.169 | 1 |
Math Assignment Help
Introduction
Struggling with your math problems? Are you tensed about the complicated and mind twisting theorems? Is math assignment creating a chaos in your mind? Don't worry; we have solutions to all your unsolved and complex doubts. Go for our Mathematics Homework Help and Math Project Help and get all your queries solved on time.
Mathematics is a branch of science that widens our knowledge on shapes, arrangements and questions that helps in getting logical answers to every practical issues. Studying Mathematics help us to report all the financial matters like expenses, incomes, profit and loss etc. of personal as well as professional life. Apart from this, Mathematical Knowledge is extensively used in findings, business and research. This is the reason that this subject is co-related with all other areas of studies i.e., Science, Finance, Research, Engineering, Medical, Economics, Arts, and many more.
The development of human civilization from the very beginning generated the need for providing coherent solution for number of complex questions and this is the reason that mathematics is introduced in the real world. Though only simple calculation were used in early days, but the formation of complex society and growing population added to the extensive study of every topic in Mathematics. This is the reason that presently Mathematics is one of the major subject in the curriculum of school as well as college students.
If you are finding problem in any section of Mathematics; get a professional help to clear your doubts. Whether its theorems in geometry or any Algebraic calculation, the team of skillful tutors from onlineassignmenthelp.com.au will help you get an accurate answer. Don't keep waiting for the right time, once your doubts get accumulated you might not be able to proceed to next section of your Mathematics course. Our online Tutors are from best college and Universities of Australia; they will simplify any lengthy procedure for you, so that you can understand each step and solve any similar equations in future. You can never achieve brilliancy in Mathematics by mugging up solutions or practicing it without getting the reasons for every step. So, our competent Faculty of Online Math tutors will provide you trick to remember and solve any question of the questionable topic or chapter. If you have any queries, seek our Math Homework Help and Math Assignment help service from active tutors and get the answer on time without piling up your work.
Mathematics can be categorized into two broad area: Pure Mathematics and Applied mathematics. Pure mathematics provides key knowledge of this subject whereas applied mathematics include those subjects that can be applied in the tangible world.
Areas categorized under pure Mathematics are:
1. Analysis: The area of analysis mainly includes the study, knowledge and idea on calculus. Calculus is the result of development in modern science that help us to construct simple model of change and hence infer the significance. The field of Calculus can be further classified into two major sub-field: Differential Calculus and Integral Calculus. Differential calculus describes about the ways of change in Differential quantities whereas integral calculus teaches the method of discovering indefinite integral to determine volume, length and area.
2. Algebra: It is the study of one or more operation that involves variables along with numbers, also termed as Algebraic expressions. Hence, this area of mathematics is mostly concerned with solving equations that involves the polynomial operation of one or more variables. It can be further divided into elementary algebra, abstract algebra and linear algebra. The recording of natural number along with integer and arithmetic procedure is sectioned under elementary algebra whereas the process of solving the given equation is included under abstract algebra. Linear algebra encompasses the ideas and notion on vectors.
3. Geometry: Geometry is the study of shapes, its properties, and configuration. It is also an axiomatic study of associated geometrical objects. It can be further sectioned into conventional Geometry, Algebraic Geometry, Discrete Geometry and many more.
4. Arithmetic: Arithmetic is study of number. It can also be called as the branch of Mathematics that deals with numbers.
Areas categorized under applied Mathematics are:
Physical science: Use of mathematics to provide a logical explanation to findings and research in Physics.
Computational science: Use of mathematical knowledge in computational field for demonstration and simulation.
Statistics: The Mathematical representation and explanation of Statistical data or collected information.
Probability: The study of event that is likely to happen is termed as mathematical study of probability. | 677.169 | 1 |
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Summary
Demystified is your solution for tricky subjects like trigonometry If you think a Cartesian coordinate is something from science fiction or a hyperbolic tangent is an extremeexaggeration, you need Trigonometry DeMYSTiFieD, Second Edition, to unravel this topic's fundamental concepts and theories at your own pace. This practical guide eases you into "trig," startingwith angles and triangles. As you progress, you willmaster essential concepts such as mapping, functions,vectors, and more. You will learn to transform polar coordinates as well as apply trigonometry in the real world. Detailed examples make it easy to understand the material, and end-of-chapter quizzes and a final exam help reinforce key ideas. It's a no-brainer! You'll learn about: Right triangles Circular functions Hyperbolic functions Inverse functions Geometrical optics Infinite-series expansions Trigonometry on a sphere Simple enough for a beginner, but challenging enough for an advanced student, Trigonometry DeMYSTiFieD,Second Edition, helps you master this essential subject. | 677.169 | 1 |
Creditsifier p 14-16 from: '''Rev. Graeme MacNeil''' (for all his support and for getting me started with internet communication) * '''Robert Fant''' (for his continuing collaboration and57 10-13 from:():In particular, my thanks always to 16 from:
* Markus Hohenwarter (geogebra designer Markus Hohenwarter (geogebra author497 from:Such an inspiration to meSuch an inspiration to me - a new lease on life! in45 1-19:
(:notitle:) !! Creditsand to the following persons, * Markus Hohenwarter (geogebra designer) and Bernadette Gotthardt * Cynthia Lainus and so many others from the * Math Forum @ Drexe * Pavel Safronov (geoboard) and George Reese (mste of uiuc | 677.169 | 1 |
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I originally bought Algebrator for my wife because she was struggling with her algebra homework. Now only did it help with each problem, it also explained the steps for each. Now my wife uses the program to check her answers | 677.169 | 1 |
Please note that every semester we covered slightly different material, in a slightly different order. In years 2000-2012, we were using either Stewart's 5th or 6th edition Calculus text. You might have to look around and check a few different exams to find one covering the material you've studied. | 677.169 | 1 |
The new edition of INTERMEDIATE ALGEBRA is an exciting and innovative revision that takes an already successful text and makes it more compelling for ... more »today's instructor and student. The new edition has been thoroughly updated with a new interior design and other pedagogical features that make the text both easier to read and easier to use. Known for its clear writing and an engaging, accessible approach that makes algebra relevant, INTERMEDIATE ALGEBRA helps students to develop problem-solving skills and strategies that they can use in their everyday lives. The new edition welcomes two new co-authors Rosemary Karr and Marilyn Massey who along with David Gustafson have developed a learning plan to help students succeed in Intermediate Algebra and transition to the next level in their coursework. The authors have developed an acute awareness of students' approach to homework and present a learning plan keyed to new Learning Objectives and supported by a comprehensive range of exercise sets that reinforces the material that students have learned setting the stage for their successBook Seller: textbook tycoon Bookstore Rating: 4.1(of 5) Book Location: Kentucky Quantity: 1Book Seller: Good Deals On Used Books Shipping Expedited: 3.00 Book Location: USA Textbook Description: Brooks Cole. Hardcover. 04958314255831425/9780495831426 (0-495-83142-5/978-0-495-831425831425/9780495831426 (0-495-83142-5/978-0-495-83142-6)
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In constructing ordinary differential models it is important to know the laws of the branch of science relating to the nature of the problem being studied. in practical life we often have to deal with cases where the flaws that en-
. Of course.Preface Differential equations belong to one of the main mathematical concepts. It must be emphasized that there are different types of differential models. For instance. in the theory of electric circuits Kirchhoff's laws. in mechanics these may be Newton's laws. one characteristic of which is that the unknown functions in these equations depend on a single variable. It is clear that differential models constitute a particular case of the numerous mathematical models that can be built as a result of studies of the world that surrounds us. in the theory of chemical reaction rates the law of mass action. This book considers none but models described by what is known as ordinary differential equations. The differential equations arrived at in the process of studying a real phenomenon or process are called the differential model of this phenomenon or process. They are equations for finding functions whose derivatives (or differentials) satisfy"given conditions.
rich in content rather than purely illustrative.P. Tikhonov and D. They give an idea of the role that ordinary differential equations play in solving practical problems. he can turn to the fascinating books by A. the variables.8 Preface able building a differentia lequation (or several differential equations) are not known.
. The first was to use examples. Passage to the limit will then lead to a differential equation. Moiseev. 1979) (nn Russian). and we must resort to various assumptions (hypotheses) concerning the course of the process at small variations of the parameters. this will mean that the hypothesis underlying the model reflects the true situation. Mathematigs Stages an Experiment (Moscow: Nauka. and if it so happens that the results of investigation of the differential equation as the mathematical model agree with the experimental data. The second goal was to acquaint the read* If the reader wishes to know more about mathematical models. Of course. Kostomarov. Stories About Applied Mathematics (Moscow: Nauka. 1979) (in Russian). and N .N. the examples far from exhaust the scope of problems solvable by ordinary differential equations. I had two goals in mind.N.* When working on this book. taken from various fields of knowledge so as to demonstrate the possibilities of using ordinary differential equations in gaining an understanding of the world about us.
then the scope of solvable equations broadens considerably. it often happens that the most essential and interesting properties of the solutions cannot be revealed by
. we can say that the great variety of solutions to differential equations is such that for their representation in closed form a finite number of analytical operations is insufficient. The fact is that only in rare cases are we able to solve a differential equation in the so-called closed form. To return to differential equations.and second-order algebraic equations the solutions can always be easily expressed in terms of radicals and for third. A similar situation exists in the theory of algebraic equations: while for first. If an infinite series of this or that form is used to represent the solutions. for a general algebraic equation of an order higher than the fourth the solution cannot be expressed in radicals. that is. Unfortunately. even when it is known that the differential equation has a solution.Preface 9 er with the simplest tools and methods used in studying ordinary differential equations and characteristic of the qualitative theory of diyjerential equations.and fourth-order equations the solutions can still be expressed in terms of radicals (although the formulas become very complicated). represent the solution as a formula that employs a finite number of the simplest operations involving elementary functions. In other words.
Lyapunov laid the foundations at the end of the 19th century. They constitute the essence of the qualitative theory of differential equations. More than that. This shows how important it is to develop methods that make it possible to acquire the data on the various properties of the solutions without solving the differential equations themselves.S.10 Preface studying the form of the series.V. the theory has been intensively developing and its methods are widely used when studying the world about us.M. And indeed.V. I am indebted to Professors Yu. which is based on the general theorems regarding the existence and uniqueness of solutions and the continuous dependence of solutions on the initial data and parameters. ever since J. even if a differential equation can be solved in closed form. . Amefkin
. The role of existence and uniqueness theorems is partially discussed in Section 2. Poincaré and A. Fedoryuk for the constructive remarks and comments expressed in the process of preparing the book for publication. Bogdanov and M. As for the general qualitative theory of ordinary differential equations.H. . such methods do exist. more often than not it is impossible to analyze such a solution since the relationship between the various parameters of the solution often proves to be extremely complicated. l _ V.2.
when Tom returned. they were given both simultaneously and proceeded as follows. Tom poured some of his cream into the coffee. covered the cup with a paper napkin. the amount
.1 Whose Coffee Was Hotter? When Tom and Dick ordered coffee and cream in a lunch room. In accordance with our assumption on the basis of a law of physics. and went to make a phone call. Dick covered his cup with a napkin and poured the same amount of cream into his coffee only after 10 minutes. Whose was hotter? We will solve this problem on the natural assumption that according to the laws of physics heat transfer through the surface of the table and the paper napkin is much less than through the sides of the cups and that the temperature of the vapor above the surface of the coffee in the cups equals the temperature of the coffee.Chapter 1 Construct1on of D1fferent1al Models and Their Solut1ons 1. We start by deriving a relationship indicating the time dependence of the temperature of the coffee in Dick's cup before the cream is added. The two started drinking their coffee at tl1e same time.
(1) where T is the coffee's temperature at time t. can be rewritten as follows: dT __ qs ———-T_6 — —T:. 6 the temperature of the air in the lunch room. we find that T=6—|—(T0—9)exp(—T.dt. (1) and (2) together. 1] the thermal conductivity of the material of the cup. (3) Denoting the initial temperature of the coffee by T0 and integrating the differential equation (3). and m the mass (or amount) of coffee in the cup. after variable separation. The amount of heat given off by the coffee is dQ = —-cm dT.l'%—t). which. l the thickness of the cup.—.12 Differential Equations in Applications of heat transferred to the air from Dick's cup is determined by the formula dQ Z qui}?-Sat. (2) where c is the specific heat capacity of the coffee. and s the area of the cup's lateral surface. (4) This formula is the analytical expression of the law whereby the temperature of the
. If we now consider Eqs. we arrive at the following equation: q J-`{it-dr: -cmdr.
Construction of Differential Models 13 coffee in Dick's cup varied prior to the addition of the cream.Ch... which now assumes the form Cm (T0 " 90) = cimi (90 "" T1): (8)
.. (5) where BD is the temperature of the Dick's coffee with cream at time t. ¤>¤xp( .. and ml the mass of cream added to the coffee. We use the heat balance equation. To derive the law for temperature variation of the coffee in Tom's cup we again employ the heat balance equation. Equation (5) yields i Clml Cm 9D— cm-|—c1m1 Tl-l_ cm—|—c1m1 T` (6) Bearing in mind formula (4). 1. ¢)].. T1 the temperature of the cream. which here can be written as cm (T — Bo) = ¢1mi(6D — Ti).. [Now let us establish the law whereby the temperature of the coffee in Dick's cup changed after Dick poured in cream. C1 the specific heat capacity of the cream. we can rewrite (6) as follows: HD CTn+C1m1 1 + cm+c1m1 .. <7> which constitutes the law whereby the temperature of the coffee in Dick's cup varies after cream is added. >< [a+<T...I|f.
and Z: 2 X 10*3 m._..*lS X €Xp( l(¢m+ Cimi) tl ° (9) Thus. using Eq.1 ><103 J/kg·K.one in which the object's tem-
.... cxé.6 V/m·K and assuming. 0 :20°C. (4) with 0. 1. to answer the question posed in the problem we need only turn to formulas (7) and (9) and carry out the necessary calculations. m=8 >< 10*2 kg. The calculations show that Tom's coffee was hotter. for the sake of definiteness._. we arrive at the law for the temperature (0T) variation of the coffee in Tom's cup as analytically given by the following formula: "_ Clml .14 Differential Equations in Applications where 00 is the temperature of the mixture. bearing in mind that C1 z 3. lf we solve (8) for 00. that ml : 2 >< 10*2 kg.. T1 =20°C. serving as the initial temperature and cm + clml substituted for cm... _.2 Steady-State Heat Flow The reader will recall that a steady-state heat flow is. s = 11 ><10*3 mz. we get c m em 69: cmil-dlml 11*+ cm—|—c1m1 TO' Then. T0 =80°C.9 >< 103 J/kg·K. and nz0.Cm L 9T`"`9+li ¢m—|—c1m1 Ti T ¢m—|—c1m1 To 9] .
made of a homogeneous material.€Ilt· is related to the effects of heat flows.I'€ 1) 20 cm in diameter. at each point of which the temperature is the same. Construction of Differential Models 15 'l\ \§ w \u\ %n\ § it N §·a x n 2 is ni n in 20 Fig. say 95 °C. 1. while the
. an important role is played by the so—called isothermal surfaces. 2 perature at each point does not vary with time. let US COI1Sld€I' 3 h€3l»··COIldllCtlI1g (Flg`U. We assume that the temperature of the pipe is 160 °C and the outer surface of the protective covering has a temperature of 30 °C.Ch. 1 Fig. II'1 pI'Ol'Jl€I1'lS Wll0SB physical COI`ll. The dashed curve in Figure 2 is known as an isotherm. To clarify this statement. designated by a dashed curve in Figure 2. and protected by a layer of magnesium oxide 10 cm thick. It is intuitively clear then that there is a surface.
and sc the radius of the base of the cylindrical surface lying inside the outer cylinder. (IO) where F (x) is the cross-sectional area normal to the direction of heat flow. for one. In general.7 >< 10*4. on the nonsteady-state nature of the heatiflow and on the nonhomogeneity of the material. and lc the thermal conductivity coefficient. depending. isotherms may have various shapes. according to which the amount of heat released per unit time by an object that is in stable thermal state and whose temperature T at each point is solely a function of coordinate x can be found according to the formula dT Q = -—kF (ac) -8-5:.== const. In the case at hand the isotherms (isothermal surfaces) are represented by concentric circles (cylinders).16 Differential Equations in Applications surface corresponding to this curveiis known as an isothermal surface.r•. Then on the basis of (IO)
. To this end we turn to the Fourier law.xl. assuming that the thermal conductivity coefficient lc is equal to 1. The statement of the problem implies that F (:2:) = 2. We wish to derive the law of temperature distribution inside the protective coating and find the amount of heat released by the pipe over a section 1 m long in the course of 24 hours. where l is the length of the pipe (cm).
a the temperature of the air. Inspection showed that the poacher's shot had been precise and the boar had died on the spot. and k a positive proportionality coefficient. (13) where x is the temperature of the object at time t. Reasoning that the poacher should return for the kill. Let us see what reasoning this involved.two forest rangers found the carcass. the rate at which an object cools off in air is proportional to the difference between the temperature of the object and that of the air. Solution of the problem lies in an analysis of the relationship that results from integrating the differential equation (13). heading directly for the dead boar.18 Differential Equations in Applications 1. Here one must bear in mind that after the
. 2%. the rangers decided to wait for him and hid nearby. This they did by using the law of heat emission. Soon two men appeared.3 An Incident in a National Park While patrolling a national park. the men denied having anything to do with the poaching. When confronted by the rangers.: -k (x-G). It only remained to establish the exact time of the kill. But by this time the rangers had collected indirect evidence of their guilt. According to the law of heat emission.of a slain wild boar.
22314 31-21 1 . 1. 2*
. Using these data and Eq. In the first case integration of the differential equation (13) with variables separable leads to 1n%-5%: —kt.Ch. In other words.·—--OE-—@ ln 1. xqéa. x. we can establish when the shot was fired by putting t = O as the time when the strangers were detained. was 31 °C and after an hour 29 °C and if when the shot was fired the temperature of the boar was :1. we get t— 14. Then. (14).6: -2. we find that 31--21 Now.2: = 37 into (14). if at the time when the strangers were confronted by the rangers the temperature of the carcass. roughly 2 hours and 6 minutes passed between the time the boar was killed and the time the strangers were detained. substituting this value of lc and .10630. (14) where xo is the temperature of the object at time t = O. = 37 °C and the temperature of the air a = 21 °C. In " 0. the temperature of the air could have remained constant but also could have varied with time.. Construction of Differential Models 19 boar was killed.
(16). We will also assume that after an hour had passed after the discovery the temperature of the carcass was 25 °C and that of the air was down to ——1° C. where t-= 15* is the time when the rangers discovered the carcass. that is.t —{. To illustrate one of the methods for determining the time when the boar was killed. we get x=(37—t—lc·1)e·"'—|—t* —. let us assume that the temperature of the carcass was 30 °C when the strangers were detained. Let us also suppose that it is known that on the day of the kill the temperature of the air dropped by 1 °C every hour in the afternoon and was 0 °C when the carcass was discovered.k'1. If we now assume that the shot was fired at t=0 and that :1:0 = 37 °C at t = 0. we get a (t) = t* —— t. If we 11ow bear in mind that sc = 30 °C at t=t* and :c=25°C at t-==t*-{-1. Integrating Eq.
.20 Diierential Equations in Applications In the second case. when the temperature of the air varies in time.—}— kx: lca(t). (t) is the temperature of the air at time 13. the cooling off of the carcass is expressed by the following nonhomogeneous linear differential equation $35. (16) where 0.
suppose that at if: 0 the carcass of the slain boar was found. (17) We can arrive at the same equation starting from different assumptions. Then a (t) == ——t and we arrive at the differential equation 13%+ kx: -kz (18) (with the initial data 2:. (17) cannot be solved algebraically for Ic. (37 — t* . (19) Setting sc = 25 and t = 1 in this relationship. 1. (18). Construction of Differential Models 21 the last formula yields (37 —. for one. Indeed.Irl) exp [—k (t* -}. These two equations can be used to derive an equation for Ic.
. we again arrive at Eq. Eq. as is known. Solving Eq. = 30 at t= 0). But it is easily solved by numerical methods for finding the roots of transcendental equations.1)] ""' kl-1 7-' 26. (30 -— Irl)e·'* — 26 + Irl = 0. namely. from which we must {ind as as an explicit function of t.Irl == 30. (17). we get x = (30 —— Irl) exp (——Ict) -—— t + Irl. Indeed.t* ——.Ch. which enables solving the initial problem numerically.Irl) exp (——-Ict*) -l..
22 Differential Equations in Applications by Newton's method of approximation. (22).2M) exp (kx).Vb -1. This method. To show how Newton's method is used.+1. as well as other methods of successive approximations. (22) by qa (sc). cp" (x) = (Wax —{. are of the type (asc —|— b) exp (kx) -i— cx —{— cl = 0. (17) to the form 30k — 1 —|. is a way of using a rough estimate of the true value of a root used to obtain more exact values of the root. C3I1 130 f0U. (20) and Eq. (22) If we denote the left-hand side of Eq. (21) Both equations. -76. to the form (37k — 1 —l. (19). Then according to Newton's method for finding a root of Eq. differentiation with respect to .Hd
. (20) and (21). we transform Eq.kt) exp (kt) — 30k —l— 1 = 0. The process can be continued until the required accuracy is achieved. if for the ith approximation xi we have the inequality (P ($2) (P" (xi) > 0» tht? D€XlL &pPI`0XlIIl&l&l0U.a) exp (M:) + c. setting :1: = 37.2: yields cp' (sc) = (lax —[— Ptb -|.(1 — 26k) exp (lc) = 0.
let us turn to Eq.60768
.859806 -71.525956 -16.1794539 ·—~2. 1) there is a root of the equation cp (k) : 0. The results can be listed in a table of the approximate values of the root and the respective values of the function and its iirst and second derivatives. It can the11 easily be verified that tp (0) = 0. the function cp increases in a small neighborhood of the origin and then dec1·eases to a negative value at lc = 1. This implies that in the interval (0. and q>' (0) > 0.49665 .11805 -82.2281622 -. (20).:) exp (k).(25 -|— 26}.3514252 -8.103379 -67.52333 . Thus.24 Differential Equations in Applications After starting the program enter the requested values of the coefficients of the equation and the initial value of the root. Below we give the protocol of the program: Solution of equation by Newton's method lambda: 1 a=—26 b=1c=30 d=—1 Approximate value of root = 0.5 X f f' f" . To hnd this root we run the program.748013E·—-03 -5.497021 ——-66. _ Employing this general procedure.02506 .784655 -32.Difierentiating its left-hand side cp (k) with respect to lc. cp (1) < 0. we arrive at cp' (lc) = 30 .5 -5.519438E-02 -6.5182 .65141 -105.322836 -1.1884971 -5.
463635 -66. and cp" (kn). which prove to be equal to 5.Ch.1789525 0 -5.5587 . The protocol of the program is given below: Solution of equation by Newton's method lambda = 0. rp (kn). we can find the time when the boar was killed.629395E-06 -5.55884 . The final step in solving the problem consists in substituting the calculated value kn z k z 0.1917861
.621243 c = 0 d = -4. To employ the above scheme.1789525 a =-·-· 0. respectively. Construction of Differential Models 25 .46373 -66.9270549 . (21) and solving the latter for t (the time when the wild boar was killed).1992802 -·-1.368575.368575 Approximate value of root = -1 X f f' f" -1 .463642 -66.506184E-03 .d= -30k-l—1. find the sought time t.621243 and -4. Then.b : 37k-1. selecting -1 for the value of to and bearing in mind that in this case a =k. and f" stand for kn. cp' (kn).188775 3. (21) by g (t).178952 into Eq.1789526 -4. To this end we find the coefficients b and d. and then running the above program.181972 . respectively.768372E-07 -5. f'.c =0. we denote the left—hand side of Eq.178954 -7. f. and ft:-k.9639622 . 1.5587 OK In this protocol X.1789525 b = 5.
We take a vessel (Figure 3) whose horizontal cross section has an area that is a function of the distance from the bottom of the vessel to the cross Section. The Water Clock The two problems that we now discuss illustrate the relationship between the physical content of a problem and geometry. f'.26 Differential Equations in Applications `cs H Fig. These results imply that the boar was killed approximately l hour and 12 minutes before the rangers discovered the carcass. g' (tn). 3 -1. g (tn). respectively.192557 1.430512E-06 . But first let us examine some general theoretical conclusions.9263298 .1916388 -1.1916387 OK In this protocol X. and g" (tn).4 Liquid Flow Out of Vessels. Suppose
. 1.192559 0 . f. and f" stand for tn.9263295 .
where g = 9. d . which causes the level of the liquid to change by -——dx (the "minus" because the level lowers). 1.Ch. The above reasoning leads us to the following differential equation ks dt == —S (:1:) dx. In the course of an infinitesimal time interval dt the outflow of the liquid can be assumed uniform. and k is the rate constant of the outflow process. the rate v at which the liquid flows out of the vessel at the moment when the liquid's level is aiheight ..2: is given by the formula v == kl/2g:c..-..8 m/sz. whereby during dt a column of liquid with a height of v dt and a cross-sectional area of s will flow out of the vessel....`-LY-$*9. 23 ks \/2gx at ( ) Let us now solve the following problem. We will also suppose that the area of the vessel's cross section at height x is denoted by S (:2:) and that the area of the opening in the bottom of the vessel is s. which can be rewritten as at T. As is known. Construction of Differential Models 27 that initially at time t = O the level of the liquid in the vessel is at a height of h meters. The
. A cylindrical vessel with a vertical axis six meters high and four meters in diameter has a circular opening in the bottom..
1916388 -4.26 Differential Equations in Applications `cs H Fig. g (tn). We take a vessel (Figure 3) whose horizontal cross section has an area that is a function of the distance from the bottom of the vessel to the cross section. f'. f.430512E—06 .192559 0 . g' (tn). Suppose
.192557 1. The Water Clock The two problems that we now discuss illustrate the relationship between the physical content of a problem and geometry. But first let us examine some general theoretical conclusions.9263298 . and f" stand for tn.4 Liquid Flow Out of Vessels. and g" (tn). respectively. These results imply that the boar was killed approximately 1 hour and 12 minutes before the rangers discovered the carcass.9263295 .1916387 OK In this protocol X. 1. 3 -1.
Construction of Differential Models 27 that initially at time t = 0 the level of the liquid in the vessel is at a height of h meters. In the course of an infinitesimal time interval dt the outflow of the liquid can be assumed uniform.*H"L. We will also suppose that the area of the vessel's cross section at height ag is denoted by S (ac) and that the area of the opening in the bottom of the vessel is s. where g = 9. and k is the rate constant of the outflow process. d . the rate v at which the liquid flows out of the vessel at the moment when the liquid's level is aiheight x is given by the formula v = kl/2gx. which can be rewritten as dz : ___. 1.8 m/sz.. which causes the level of the liquid to change by —d:c (the "minus" because the level lowers). As is known. The
. The above reasoning leads us to the following difierential equation mi/QE at Z -s ix) ax. ks 1/2g:c x (23) Let us now solve the following problem. whereby during dt a column of liquid with a height of v di and a cross—sectional area of s will flow out of the vessel. A cylindrical vessel with a vertical axis six meters high and four meters in diameter has a circular opening in the bottom.Ch..
If we put x = 0 in the last formula.152 dx dt V. This problem can easily be solved via Eq. We wish to determine the shape of the water clock that would ensure that the water level lowered at a constant rate. S (. An ancient water clock consists of a bowl with a small hole in the bottom through which water flows out of the bowl (Figure 4).304 [ l/6 — l/EL Ogxgfi.6. T By hypothesis. Find how the level of water in the vessel depends on time t and the time it takes all the water to flow out.
. Eq.28 Differential Equations in Applications radius of this opening is 1/12 m. Such clocks were used in ancient Greek and Roman courts to time the lawyers' speeches. (23). (23) assumes the form _ __ 217. Now a second problem. We rewrite this equation in the form · _ __ S (=¤> Q. Since for water k = 0. so as to avoid prolonged speeches. Integrating this differential equation yields t= 434. we find that it takes approximately 18 minutes for all the water to flow out of the vessel.1:) = 4at and s = 1/144. which is the sought dependence of the level of water in the vessel on time t.
The latter means that the sought shape of the water clock is obtained by rotating curve (26) about the 2: axis. we arrive at the equation x = cr".
. (24) yields the following result: 1/ 2% Z —--2 a. (25). 4 Precisely. Squaring both sides of Eq. 1. which is constant by hypothesis. if we suppose that the bowl has the shape of a surface of revolution. Eq.Ch.z· § 'Y Fig. in accordance with the notations used in Figure 4. Construction of Differential Models 29 . (26) with c = aznz/2gk2s". <25> ks ]/2g where a = vx ·—-= da:/dt is the projection on the :1: axis of the rate of motion of the water's free surface.
All further information about the commodities is distributed among the buyers via personal contact between them. we arrive at the following differential equation: %:kx (N.
.5 Effectiveness of Advertising Suppose that a retail chain is selling commodities of a certain type.2:). In this equation lc is a positive proportionality factor. If we suppose that time is reckoned from the moment when the promotion materials and advertisements were released and N /·y persons have learned about the commodities. (27) with the initial condition that sc = N /y at t = O. the rate of change in the number of persons knowing about B is proportional both to the number of buyers knowing about B and to the number of buyers not knowing about B. say B. We may assume with a high probability that after the TV and radio network have released the information about B. Integrating.30 Differential Equations in Applications 1. Let us also assume that to speed up sales the chain has placed promotion materials in the local TV and radio network. we find that '%·' ln + C. about which only rz: buyers out of N potential buyers know at time t.
categories that emerge and function on the market. Such a price establishes an "equilibrium" on the market. and that he does this immediately after the apple harvest has been taken in and then once each following week (i. Let us consider the following problem. the week's supply will depend on both the expected price of apples in the coming week and the expected change in price in the following
.32 Differential Equations in Applications that the problem of dissemination of technological innovations can also be reduced to Eq. while the second represents the demand for commodities on the market. One of the main laws of a market economy is the law of supply and demand.e. with a week's interval). 1. Suppose that in the course of a (relatively long) time interval a farmer sells his produce (say. that is. The farmer's stock of apples being fixed after he has collected his harvest.6 Supply and Demand As is known. The first category represents the commodities that exist on the market or can be delivered to it. supply and demand constitute economic categories in a commodity economy. in the sphere of trade. (27). which can be formulated simply by saying that for each commodity some price must exist that will cause the supply and the demand to be just equal. apples) on the market.
one function is given by a linear dependence expressed mathematically in the form y = ap' + bp —l. the higher the supply of apples on the market in the coming week. and the supply s and demand q are given by the functions s=44p' —}—2p—1. both supply and demand are functions of these quantities. 3-0770
. then supply will be restrained provided that the expected rise in price exceeds storage costs. Construction of Differential Models 33 weeks. q==4p'-—2p—|—39. if the price of apples in the coming week is high and then a fall is expected in the following weeks. the lower the supply of apples on the market. As practice has shown. If it is assumed that in the coming week the price of apples will fall and in the following weeks it will grow. then the greater the expected fall in price in the following weeks. in our example. If we denote the price of apples in the coming week by p and the time derivative of price (the tendency of price formation) by p'. and c are real numbers (constants). in t weeks it was p (t) roubles for 1 kg. For example.c. On the other hand. b. the price of apples in the coming week is taken at 1 rouble for 1 kg. Say. where a. depending on various factors the supply and demand of a commodity may be represented by different functions of price and tendency of price formation. In these conditions the greater the expected rise in price in the following weeks. 1.Ch.
It shows how as a result of the interaction of two molecules of hydrogen and one molecule of oxygen two molecules of water are formed. Take. for the supply and demand to be in equilibrium. 1. if we want the supply and demand to be in equilibrium all the time.7 Chemical Reactions A chemical equation shows how the interaction of substances produces a new substance. the price must vary according to formula (29). the equation 2H2 —{— O2 -—>2H2O. (29) Thus. If we allow for the initial condition that p = 1 at t = 0. we must require that 44p' —l-2p—1=4p' —2p—§—39. This condition leads us to the differential equation dp __ Integration yields p = Cermt -}— 10. the equilibrium price is given by the formula p = —9e·1°' + 10. for instance.
.34 Differential Equations in Applications Then.
M. are molecules of the interacting substances. p.——>mM + nN + pP + where A.bB —l. N. . One of the basic laws of the theory of chemical reaction rates is the law of mass action. b. Construction of Differential Models 35 Generally a chemical reaction can be written in the form aA —i. c.Ch. m. Two liquid chemical substances A and B occupying a volume of 10 and 20 liters.cC —l. The rate at which a new substance is formed in a reaction is called the reaction rate. according to which the rate of a chemical reaction proceeding at a constant temperature is proportional to the product of the concentrations of the substances taking part in the reaction at a given moment. . positive integers that stand for the number of molecules participating in the reaction.. . . Assuming that the temperature does not change in the process of the reaction and that every two volumes of substance A and one vol3* r
. B. respectively. molecules of the reaction products. and the constants a. 1. . and the active mass or concentration of a reacting substance is given by the number of moles of this substance per unit volume. form as a result of a chemical reaction a new liquid chemical substance C. Let us solve the following problem. n. C. P.
By hypothesis. As for the moment t = 1/3. Let us denote by x the volume (in liters) of substance C that has formed by the time t(in hours). x (0): 0. Thus. This means that we are left with 10 -— 2x/3 liters of A and 20 ·—— x/3 liters of B. Thus. by that time 2. the solution of the initial problem has been reduced to the solution of a socalled boundary value problem: ft. We must also bear in mind that since initially (t = 0) there was no substance C. {ind the amount of substance C at an arbitrary moment of time t if it takes 20 min to form 6 liters of C.-:k (15-x) (60-:c).1:/3 liters of substance A and x/3 liters of substance B will have reacted.36 Differential Equations in Applications ume of substanceBform three volumes of substance C. in accordance with the law of mass action. we have sc = 6.
. we may assume that x = 0 at t = 0. we arrive at the following differential equation dx 2::: rc w:K('°"T) (Z0-?) which can be rewritten as dx where k is the proportionality factor (lc = 2K/9).
As a result we arrive at the relationship _?_g_.
. a formal study of the above relationship between x and t shows that for a finite t. $:15 i-(2/3)*** i·—(i/@(2/3)3' l This gives the amount of substance C that has been formed in the reaction by time t. From practical considerations it is clear that only a finite volume of substance C can be formed when 10 liters of A interact chemically with 20 liters of B. 1.'_ Z 4€45m_ Since :1: = 6 at t = 1/3. _T4(€15k)3t:. substituting these values into the relationship yields emh = 3/2.Ch.1: becomes iniinite. However. namely at (2/3).4(3/2)3t. that is. while the chemical reaction is considered only for t nonnegative._. But this fact does not contradict practical considerations because it is realized only for a negative value of t. A remark is in order. Hence. we iirst integrate the differential equation and allow for the initial condition x(0) :0.** = 4. Construction of Differential Models 37 To solve it. the variable .
Spectrum 16. B = bx. No. prey situation.38 Differential Equations in Applications 1. Allowing for (31). (cz-. Murray. 2: 48-54 (1983/84). The basic object in ecology is the evolution of populations. The simplest situation is the one in which A = ax. then there is reason to say that the rate at which x varies in time is given by the formula dx —(F.
.b) as. (31) where a and b are the coefficients of births and deaths of individuals per unit time.* Let x (t) be the number of individuals in a population at time t. respectively." Math. "Some simple mathematical models in ecology.D. (30) The problem consists in finding the dependence of A and B on :1:. Below we describe differential models of populations that deal with their reproduction or extinction and with the coexistence of various species of animals in the predator vs.8 Differential Models in Ecology Ecology studies the interaction of man and. If A is the number of individuals in the population that are born per unit time and B the number of individuals that die off per unit time. (32) * For more detail see J. in general. living organisms with the environment. we can rewrite the differential equation (30) as dx Ti-T -.A—B..
say the equation %—·:f(x) =ax-—bx2. practically all models describing real phenomena and processes are nonlinear. Construction of Differential Models 30 Assuming that at t = to the number of individuals in the population is sc = :1:.Ch. However. tends to
. (32).)l. we get .2:) is a nonlinear function. where f (.2:0 exp [(a —— b) (t — t. Assuming that x = xo at t = to and solving the last equation. and solving Eq. we get :1: (t) = . but if a<b. 1.. it still reflects the actual situation in some cases. : _ 33 "' ll ¤=0+<¤/b—x0>exp1—¢»<¢—¢0>1 ( ) We see that as t —> oo the number of individuals in the population. then a:——>O as t-+00 and the population becomes extinct. where a > 0. then the number of individuals sc tends to infinity as t—> oo. Although the above model is a simplified one.. This implies that if a > b. and instead of the differential equation (32) we are forced to consider an equation of the type dx __ jg--f (iv). b > O. x (t).
Let us study in greater detail the two-species predator vs. 2. which have the same period but differ in phase. n. The difference between the two is clearly seen in Figure 6. big and small {ish.Z'0 6 Fig.. say. first introduced by the Italian mathematician Vito Volterra (1860-1940) to explain the oscillations in catch volumes in the Adriatic Sea. for one thing.120 1-zz--1***-@"""'"*°— 0 b <. 6 a/b. where the small fish serve as prey for the big.
. the populations of fruit pests and some types of bacteria.). Note that formula (33) describes. If we consider several coexisting species. then by setting up differential equations for each species we arrive at a system of equations %?—=f. x. Two cases are possible here: a/b > xo and a/b < xo. prey model. (1:.. i=1.40 Differential Equations in Applications cz J? ·5>.
(35) the term ——d. As a result. (34) dy -— as TH-. (34) for the big iish the term bxy reflects the dependence of the increase in the number of big fish on the number of small {ish. and the cycle is repeated. that is. This.z:y expresses the decrease in the number of small fish as a function of the number of big {ish. a situation will emerge in which there will not be enough food and the number of big Hsh will diminish. prey fish.-cx—dxy. The model constructed by Volterra has the form da: Y: —a:c—|—b:cy. Then the number of predator fish will grow as long as they have sufficient food. 1. and d are positive constants. b. 1:=ct.
. ot=a/c. To make the study of these two equations more convenient we introduce dimensionless variables: db u(t)=-Fx. in turn. whose number we denote by y. v(·•:)=—&—y. starting from a certain moment the number of small {ish begins to increase. In Eq. Construction of Differential Models 41 Let ac be the number of big fish (predators) that feed on small fish (prey). while in Eq.Ch. ( ) where ez. Finally. assists a new growth in the number of the big species. c.
(36) with oc positive and the prime standing for differentiation with respect to 1. decreases. which is followed by a prolonged period of time 1: in the
. as u approaches a value close to unity.42 Differential Equations in Applications As a result the differential equations (34) and (35) assume the form u. Let us assume that at time rc = to the number of individuals of both species is known. that is. Let us now assume that the initial values uo and vt. u (T0) = wo. (v — 1). v' : v (1 -. Then.u). Since un > 1 and vo < 1. We see that the (u.' : ow. Figure 7 depicts the dependence of u on v for different values of H. v)—plane contains only closed curves. where H is a constant determined by the initial conditions (37) and parameter oc. We get ow+u—-lnv°°u =ow0—i-un-—lnvg°u0 EH. are specified by point A on the trajectory that corresponds to the value H = H3. Let us establish the relationship between u and v. The same is true of variable v. the first equation in (36) shows that at first variable u. v (10) = va (37) Note that we are interested only in positive solutions. To this end we divide the first equation in system (36) by the second and integrate the resulting differential equation. v' vanishes.
1. the third group consists of individuals
. that is. then the other two will also become extinct. The first include individuals that are susceptible to a certain disease but are healthy. Clearly. if this is the third population (the source of food). Suppose that a population consisting of N individuals is split into three groups. For example. A typical graph of u and v as functions of time is shown in Figure 8 (for the case where v0>1 and u0<1). that is. We denote the number of such idividuals in the population at time t by I (t).44 Differential Equations in Applications its maximum when v does. it can be demonstrated that one of them will become extinct.9 A Problem from the Mathematical Theory of Epidemics Let us consider a differential model encountered in the theory of epidemics. if two populations fight for the same source of food (the third population). The second group incorporates individuals that are infected. In conclusion we note that the study of communities that interact in a more complex manner provides results that are more interesting from the practical standpoint than those obtained above. they are ill and serve as a source of infection. We denote the number of such individuals at time t by S (t). Finally. oscillations in the populations occur in different phases.
(38) Let us also assume that when the number of infected individuals exceeds a certain fixed number I *. This means that we have allowed for the fact that the infected individuals have been isolated for a certain time interval (as a result of quarantine or because they have been far from individuals susceptible to the disease). they can infect the individuals susceptible to the disease. We denote the number of such individuals at time t by R (t). Construction of Differential Models 45 who are healthy and immune to the disease. therefore.Ch. 1. S (t) —l— I (t) —l— R (t) = N. %:l 0 `fI()·" * (39) I§I. Now. We. arrive at the following differential equation -—ocS if I t > I*. we suppose that when the number of infected individuals I (t) is greater than I *. the rate of change in the number of individuals susceptible to the disease is proportional to the number of such individuals. Thus. As for the rate of change in the number of infected individuals that eventually recover. these assumptions simplify matters and in a number of cases they reflect the real situation. since each individual susceptible to the disease eventually falls ill and be-
. we assume that it is proportional to the number of infected individuals. Because of the first assumption. Clearly.
we must hx the initial conditions. Finally. hence. and that initially the number of infected individuals was I (0). on = B (the reader is advised to study the case where ot =. the rate of change of the number of recovering individuals is given by the equation dR/dt -·= BI. For the solutions of the respective equations to be unique. the rate of change in the number of infected individuals is the difference. For the sake of simplicity we assume that at time t = O no individuals in the population are immune to the disease.& B). Case 1. per unit time. R (O) = O. we find ourselves in a situation in which we must consider two cases. 40 dt --BI if 1(t) { 1*. between the newly infected individuals and those that are getting well. and. dj {ocS——B1 if 1(t) > 1*. Next we assume that the illness and recovery coefficients are equal. With the passage of time the individuals in the population will not become infected because in this case dS/dt = O. we have an equation valid for all values
. Hence. that is.46 Differential Equations in Applications comes a source of infection. 1 (O)< 1*. in accordance with Eq. that is. (38) and the condition R (O) == O. Hence. ( ) We will call the proportionality factors ot and B the coefyicients of illness and recovery.
(40) then leads us to the following differential equation dI 7]-. The case considered here corresponds to the situation when a fairly large number of infected individuals are placed in quarantine. 1.——otI. hence. ----——--—--—————-———. Equation.Ch. R(t)=N—S(t)——I(t) = I (O) [1 — e'°°*]. Construction of Differential Models 47 N .·S(é) If —-·-···········"'""""" *'"' """"'/?(¢) IO5) 0 it Fig. This means that I (t) = I (0) e'°°* and.--. 9 of tz S(t)==S(O)=N——I(0). Figure 9 provides diagrams that illustrate the changes with the passage of time in the
.
(41) is given by the formula I (t) = Ce"°°* —l— cxS (0) te'°°'. we get C = I (0) and. I (0) > I*. the set of all solutions to Eq. hence. This implies that for all t's belonging to the interval [O. (41) by em.48 Differential Equations in Applications number of individuals in each of the three groups. (41) If we now multiply both sides of Eq.C and. therefore. Equation (39). since by its very meaning the function I = I (t) is continuous. T] the disease spreads to the individuals susceptible to it. we get the equation é. Hence. Case 2. (lem) = as (0). (42) Assuming that t = O. implies that S (t) = S (0)e'°°' for OQ t < T. Eq. Ie°°' = otS (0) t —i. (42) takes the form I (t) = [I (0) + 0tS (O) tl e'°". (43) with Og t < T. we arrive at the differential equation ij. therefore. Substituting this into Eq. In this case there must exist a time interval OQ t < T in which I (t) > I* for all values of t.--}-0tI =otS (O) e·°°'.
. (40).
(43). I* = [I (O) -1. we arrive at the equation * = S (0) S (°°) I [I (O)—|-S(O) ln Sho) il Sw) . (44). for such individuals the following chain of equalities holds true: S (T) = S (oo) = S (0) e'°°T. Hence. the aforesaid implies that at t= T its right-hand side assumes the value 1*. (44) But S (T) = 1imS (t) =S (oo) f->00 is the number of individuals that are susceptible to the disease and yet avoid falling ill.Ch. __ 1 S (0) Thus. (45) to predict the time when the epidemic will stop. Substituting T given by (45) into Eq. if we can point to a definite value of S (oo). that is. The answer to the first question is important because starting from T the individuals susceptible to the disease cease to become infected. If we turn to Eq.ocS (O) Tle'°°T. we can use Eq. 4-0770
. 1. Construction of Differential Models 49 We devote our further investigations to the problem of finding the specific value of T and the moment of time tmax at which the number of infected individuals proves maximal.
x= S(0)¤X1>{—li -—-' (0)/S (0>l} = S (tmax)• This relationship shows. that at time tmax the number of individuals susceptible to the disease coincides with the number of infected individuals..i'L.50 Differential Equations in Applications or . this equation yields ig-=(as (0) . (43).. (46) Since I * and all the terms on the right-hand side of (46) are known. we get Im. we turn to Eq..a1 (0) ». we can use this equation to determine S (oo). The moment in time at which I attains its maximal value is given by the following formula: . To answer the second question.azs (0) q e—<¤ :0.. (43). _i_ _ I (0) tmu "' ¤= li S0) If we now substitute this value into Eq. In accordance with the posed question.
. = i& ifi S(0¤) $(0) +1H S(00) ' which can be rewritten in the form 1* __ I (0) ——S(cO) —§—1nS(0o) --—————S(0) -}-1nS (0). for one.
1. the individuals susceptible to the disease cease to be infected and I (t) =I*e*°¤(*·T).Ch. 1. Figure 10 gives a rough sketch of the diagrams that reflect the changes with the passage of time of the number of individuals in each of the three groups considered. Qi
.Z' T Z Fig.10 The Pursuit Curve Let us examine an example in which differential equations are used to choose a cor· rect strategy in pursuit problems. Construction of Differential Models 51 N ·······**""""*'***"""' **** """""""" ""'"''' " ''`"""'"`""`'' —R(t} [ __-ml _ mv. 10 But if t > T.— sm I 0 Ht) tH7U.
To solve the problem we first introduce polar coordinates.52 Differential Equat. r and 9. in such a manner that the pole O is located at the point at which the submarine was located when discovered and that the point at which the destroyer was located when the submarine was discovered lies on the polar axis r (Figure 11). We wish to {ind the trajectory (the pursuit curve) that the destroyer must follow to pass exactly over the submarine if the latter immediately dives after being detected and proceeds at top speed and in a straight line in an unknown direction.ions in Applications d8 »/I x fi \\ ¤ \\ \ x dr r if 0 Fig. Further reasoning is based on
. The destroyer's speed is twice the submarine's speed. At a certain moment in time the fog lifts and the submarine is spotted floating on the surface three miles from the destroyer. 11 We assume that a destroyer is in pursuit of a submarine in a dense fog.
Then it must move about O in such a way that both moving objects remain at the same distance from point O all the time.
. if the two still have not met.Ch. Let us decompose the destroyer's velocity vector (of length 2v) into two components. the radial component U. we find that this distance is either one mile or three miles.. Obviously.2v or from the equation x _ 3-]-:: T `"` -5. the destroyer must take up a position in which it will be at the same distance from pole O as the submarine. distance x can be found either from the equation 27 _ 3--:1: 72-. Construction of Differential Models 53 the following considerations. Only in this case will the destroyer eventually pass over the submarine when circling pole O. 1. The aforesaid implies that the destroyer must first go straight to point O until it finds itself at the same distance x from O as the submarine. Solving these equations. v. First. the destroyer must circle pole O (clockwise or counterclockwise) and head away from the pole at a speed equal to that of the submarine. Now. and the tangential component vt (Figure 11).' where v is the submarine's speed a11d 20 the destr0yer's speed.
we have vt:]/(2v)2-02:]/-3:0. that is. The latter component.54 Diflerential Equations in Applications The radial component is the speed at which the destroyer moves away from pole O. I. is equal to the product of the angular ve— locity dt)/dt and radius r. in turn. . r W .. the solution of the initial problem is reduced to the solution of a system of two differential equations. which. d0 "t=*a. dr/r = d9/ V3.. Solving the last differential equation.V 3 v. by excluding the variable t. Thus. we find that r=Ce°/Vg.. can be reduced to a single equation. as is known.· But since v. that is .v. dr d9 ·· -52. o
. _· ' while the tangential component is the linear speed at which the destroyer circles the pole. must be equal to v. with C an arbitrary constant.2.
It is the number of personnel that will play a decisive part in the construction of these models.. Thus.r = 1 at 0 = 0 and r = 3 at 9 =—:n. the destroyer must move two or six miles along a straight line toward the point where the submarine was discovered and then move in the spiral r = e9/VT or the spiral r = 3e(9+")/V? 1. to specify the criteria that would allow taking into account. Suppose that two opposing forces. Later these models were generalized so as to describe battles involving regular troops or guerilla forces or the two simultaneously. not only the number of
. Below we consider these three models. to fulfill its mission.11 Combat Models During the First World War the British engineer and mathematician Frederick W.Ch. nv and y. respectively. when comparing the opposing sides. Lanchester (1868-1946) constructed several mathematical models of air battles. practically speaking. 1. we conclude that in the first case ii C = 1 and in the second C = 3e"/V. Construction of Differential Models 55 If we now allow for the fact that destroyer beginsits motion about pole O starting from the polar axis r at a distance of sz: miles away fromO. thatis. are in combat. since it is difficult. We denote the number of personnel at time t measured in days starting from the first day of combat operations by sz: (t) and y (t).
but we can already point to a number of factors that enable describing the rate of change in the number of personnel on the two sides. more than that. Precisely. Of course. We also assume that x(t) and y(t) change continuously and.z: (t) and y (t) as functions of t. It is then clear that the total rate of change of x (t) is given by
. these assumptions simplify the real situation because as (t) and y (t) are integers. level of military equipment. But at the same time it is clear that if the number of personnel on each side is great.56 Differential Equations in Applications personnel but also combat readiness. we denote by OLR the rate at which side x suffers losses from disease and other factors not related directly to combat operations. we may assume that during small time intervals the change in the number of personnel is also small (and does not constitute an integral number). an increase by one or two persons will have an infinitesimal effect from the practical standpoint if compared to the effect produced by the entire force. preparedness and experience of the officers. insufficient to specify concrete formulas for . of course. and a great many other factors. morale. Therefore. and by CLR the rate at which it suffers losses directly in combat operations involving side y. are differentiable as functions of time. The above reasoning is. Finally by RR we denote the rate at which reinforcements are supplied to side x.
A.S.
. CLR. Drew. The article shows how the examples agree with the models considered.(OLR +01. and xo and yo are the personnel in :1: and y prior to combat. We use the following notation: a. We are now ready to set up the three combat models suggested by Lanchester. Coleman. P (t) and O (t) are terms that allow for the possibility of reinforcements being supplied to :2: and y in the course of one day.<l=é1£l = . g. 1.* The iirst refers to combat operations involving regular troops: rw: —a$ (t)""b!/ U) +P (t)» 9%-gl:-cx(:)-dy(¢)+Q(z). Braun. * Examples of combat operations are given in the article by C. C. 109-131). and RR and then examine the resulting differential equations.Ch. Coleman.S. (47) A similar equation holds true for y (t). c. The results should indicate the probable winner.12) + RR. and D. Construction of Differential Models 57 the equation . pp. and h are nonnegative constants characterizing the effect of various factors on personnel losses on both sides (x and y). The problem now is to {ind the appropriate formulas for the quantities OLR. eds. 1983. d. New York: Springer. M. "Combat models" (see Dijjercntial Equation Models. b.
specified by the equations d<r(¢> . the third model.··——··¢$(¢)—€¢¤(¢)y(¢)+P(¢)· di !jg%—=—dy(*)—h¢(¢)y(¢)+Q(¢) describes combat operations involving only guerilla forces.58 Differential Equations in Applications In what follows we call this system the Atype differential system (or simply the Atype system).. which we call the C-type system. The second model. has the form d ··%%Q= ··¤¤*¤ (t)-—. Each of these differential equations describes the rate of change in the number of personnel on the opposing sides as a function of various factors and has the form (47). q.€'¤¤(¢)y(¢)+P(¢)» d —%€L= -· cw (¢)—dy(¢> +Q (¢) and describes combat operations involving both regular troops and guerilla forces. We will call this system the B-type system. Finally. Losses in personnel that are not directly related to combat operations and are determined by the terms —a.2: (t) and —dy (t) make it possible to describe the constant fractional loss rates (in the absence of combat operations and reinforcements)
.
The factor b characterizes the effectiveness of side y in combat. C == Txpx. second. In considering the A-type system. 1. while the presence of the terms —by (t). the equation 1 dx 2w··b shows that constant b measures the average effectiveness of each man on side y. first. One way is to write these coefficients in the form b == I'. we assume. Construction of Differential Models 59 via the equations 1 dx ___ — 1 Q_ _ YHT" a' Y dz "' d' If the Lanchester models contain only terms corresponding to reinforcements and losses not associated with combat operations. there is no simple way in which we can calculate the effectiveness coefficients b and c. that only the personnel directly involved in combat come under fire. this means that there are no combat operations.Ch. Of course. Under such assumptions Lanchester introduced the term -——by (t) for the regular troops of side zz: to reflect combat losses. that each side is within the range of the fire weapons of the other and./py. —c. Thus. -gx (t) y (t).z: (t). and —-hx (t) y (t) means that combat operations take place. The same interpretation can be given to the term ——cx (t). (48)
.
to the number y (t) of personnel of the opposing side. and py and px are the probabilities that each shot fired by sides y and x. Suppose that the guerilla forces amounting to x (t) personnel occupy a certain territory R and remain undetected by the opposing side. And although the latter controls this territory by firepower. whereas in the B-type system they are nonlinear.-y
. on the one hand. on the other. it cannot know the effectiveness of its actions. proves accurate. according to which the probability of an accurate shot fired by side y is directly proportional to the so-called territorial effectiveness A. (t) y (t). respectively. lt is also highly probable that the losses suffered by the guerilla forces x are. Nevertheless.60 Differential Equations in Applications where ry and rx are the coefficients of firepower of sides y and x. more difficult to estimate than the coeffiient b in the first relationship in (48). Hence. We note further that the terms corresponding to combat losses in the A-type system are linear. the term corresponding to the losses suffered by the guerilla forces x has the form —ga. proportional to the number of personnel sc (t) on R and. in general. to find g we can use the firepower coefficient ry and also allow for the ideas expounded by Lanchester. where the coefficient reflecting the effectiveness of combat operations of side y is. The explanation lies in the following. respectively.
the mathematical model is reduced to the following differential system: dd éz-. Construction of Differential Models 61 of a single shot Bred by y and inversely proportional to the area Ax of the territory R occupied by side x. we arrive at the following relationship: b [y2 (t) -— y:] = c [x2 (t) — xg]. Here by ATU we denote the area occupied by a single guerilla frghter. neither side receives any reinforcement. é: -cx. (49) Below we discuss in greater detail each of the three differential models. h=rx—x-{Ji-. (50) Dividing the second equation by the first. (51) Integrating. we find that dy __ isi -5 -. (52)
. with a high probability we can assume that the formulas for finding g and h are AA g=ry-f. Thus.Ch. Case A (differential systems of the A-type and the quadratic law). -by.by . If. 1. Let us assume that the regular troops of the two opposing sides are in combat in the simple situation in which the losses not associated directly with combat operations are nil. in addition.
the equation byz —. y > 0).cxz = K (53) obtained from Eq. (52) specifies a hyperbola (or a pair of straight lines if K = 0) and we can classify system (50) with great precision. 12 This relationship explains why system (50) corresponds to a model with a quadratic law.) A'>0. The arrows on the curves point in
. Figure 12 depicts the hyperbolas for different values of K.62 Differential Equations in Applications ya. y wins K = 0) even Wl//b K < Q . If by K we denote the constant quantity by§ —— c:c§. for obvious reasons we consider only the first quadrant (:1: > 0.z* wins 0r {/-—/(/c $(5) Fig. We can call it a differential system with a hyperbolic law.
we proceed as follows. (53) the variable y can never vanish. we agree to say that side y (or ac) wins if it is the first to wipe out the other side x (or y).. while the variable sc vanishes at y (t) = l/K/b. 1. To answer the question of who wins in the constructed model (50). in accordance with the quadratic law. To derive formulas that would provide us with a time dependence.> > ry sa The left-hand side of (55) demonstrates that changes in the personnel ratio yo/xo give an advantage to one side. For example. we can rewrite (54) in the form yo 2 fx Px <. Note also that Eq.Ch. As a result we
. it must strive for a situation in which K is positive. Construction of Differential Models 63 the direction in which the number of personnel changes with passage of time. (54) Using (48). byg > c:1:§. in our case the winner is side y if K > 0 since in accordance with Eq. for y to win a victory. For example. Thus.2:0 from one to two gives y a four—fold advantage. (53) denotes the relation between the personnel numbers of the opposing sides but does not depend explicitly on time. a change in the yo/. that is. We differentiate the first equation in (50) with respect to time and then use the second equation in this system.
(57) and (58) in
.64 Differential Equations in Applications ·" . yo cos Bt — gg.9 % (U i. If for the initial conditions we take dx $(0)=%» ·g. (56) can be obtained in the form sc (t) = :1:0 cos [St — yyo sin Bt. 13 arrive at the following differential equation: dh: EF '·" bC. T Fig. M2 •-•-—·-1--·-•-•—-Zv-•· *¤¤I1¥l¤Q 0(. (57) where B = VE and Y = In a similar manner we can show that y (t) L-.sin Bt.'IZ T-"' O. (58) Figure 13 depicts the graphs of the functions specihed by Eqs.L=0= —b1/0. then the solution to Eq.
we get the equation i!!. by§ > cxg.£ dx -_ g ' which after being integrated yields g ly (¢) —. Construction of Differential Models 65 the case where K >O (i.ytl = h lx (¢) —. Under these limitations the B—type differential system assumes the form dd f-: ——g¤¤y. Case B (difierential systems of t.e. The only requirement is that vyo be greater than xo.. (61) 5—0770
.(60) This linear relationship explains why the nonlinear system (59) corresponds to a model with a linear law for conducting combat operations. if we exclude the possibility of losses not associated with combat and if neither side receives reinforcements.. -5%: —h¤¤y.(59) Dividing the second equation in (59) by the first. as in the previous case.he B-type and the linear law). The dynamical equations that model combat operations between two opposing sides can easily be solved._ _ We note in conclusion that for side y to win a victory it is not necessary that yu be greater than xo. 1.Ch. Equation (60) can be rewritten in the form gy — hx = L. or YU 0 > xo).wo].
We assume that this is side y.z* wirzs 0. Figure 14 provides a geometrical interpretation of the linear law (61) for different values of L. side y wins as a result of combat operations.yAy ' ( )
. 14 with L = gy. EUC/Z 4/y L·<Q.. that if L is positive. we see that side y wins if _§/L "xArxAx xo > ryA. gyo — hxo must be positive. This implies.//7 mw) Fig.90} DQ _ywz'ns l=0. or Ll $0 > 8 ° If we now turn to (49). ·. . as we already know. while if L is negative.) -— hay.66 Differential Equations in Applications . side sc wins. Let us now study in greater detail the situation where one of the sides wins. Then.. for one.
In the C-model the guerilla forces face regular troops. we arrive at the differential equation 12 -. the strategy of y is to make the ratio yo/xo as large as possible and the ratio Ax/A y as small as possible. We note. 1.Ch. In this case we have the following system of differential equations: where :2: (t) is the number of personnel in the guerilla forces. finally. From the practical standpoint it is more convenient to write inequality (62) in the form Ay!/0 '°xArx Axxo > "yAry • We see that in a certain sense the products A yyo and Axxo constitute critical values. and y (t) the number of personnel in the regular troops. Dividing the second equation in (63) by the first.. that by combining (61) and (59) we can easily derive formulas that give the time dependence of changes in the personnel numbers of both sides. Construction of Differential Models 67 Thus. Case C (differential systems of the C—type and the parabolic law). We introduce the simplifying assumptions that neither is supplied with reinforcements and that the losses associated directly with combat operations are nil. dx " gy ' 5*
.
z· wins ·M/20 . Basing our reasoning on the parabolic law for conducting combat operations and assuming M positive. we conclude that the victory of regular
.2c. they are defeated if M is positive. the system of differential equations (63) corresponds to a model with a parabolic law for conducting combat operations. 15 Integrating with the appropriate limits yields the following relationship: gyz (t) =. (64) where M : gy§ -—. Thus.= 20:1: (t) -§—M.z'(t') Fig. The guerilla forces win if M is negative. /*/<Q . Experience shows that regular troops can defeat guerilla forces only if the ratio yo/:::0 considerably exceeds unity.68 Differential Equations in Applications _y(£) /V/>Q ywi/ZS /'/=Q eve/z 1//Qi.zv0. (64) for different values of lll. Figure 15 depicts the parabolas defined via Eq.
Iilthe load is. Construction of Differential Models 69 troops is guaranteed if (yo/x0)2 is greater than (2c/g) . 16
. let us consder an idealized model of a pendulum clock consisting of a rod of length Z and a load of mass m attached to the lower end of the rod (the rod's mass is assumed so small that it can be ignored in comparison to m) (Figure 16).. If we allow for (48) and (49)._ 1.1:. 1.*.12 Why Are Pendulum Clocks Inaccurate? To answer this question..def1eoted by an angle ot 7 { `X CX\\E v \\ I 8 \\\ l x> |/ t r' nz L·=—-rz Fig.Ch. we can rewrite this condition in the form y 2 T A p_" 1 > 2 i ·y't .
we have _o J. LS .. and g is the acceleration of gravity. (65) where v is the speed at which the load moves. we have v = ds/dt == l(d9/dt) and (65) leads us to tl1e following differential equation: -g. This is the
.. (66) in the form ` dt: —· I ... in accordance with energy conservation we obtain il§°i=mg(lcos9—lcosu)..(-'§)2=g(e0Se—¤0Sa). 1/ 23 ]/cos6—-—cosot If we denote the period of the pendulum by T. Since s == l9.70 Differential Equations in Applications and then released.19. we can rewrite Eq_. (ee) If we now allow for the fact that -6 decreases with the passage of time t (for small fs). 4 1/ 29 ]/cost)-cosoc ' (I or T Q d6 I/28 0 ]/cosB·—cosoz ( ) The last formula shows that the period of the pendulum depends on ot...
-d6=kcosq>dcp. practically speaking. This implies that when 6 increases from O to ct.. since cos9=l—2sin2g— . Construction of Differential Models 71 main reason why pendulum clocks are inaccurate since. 1.. 8 _ on _ 6 0 I/-sin? -2——s1n2 -5"' °° d9 0 I/k2—si112 -5 with k = sin (cx/2)... Indeed. we have ___ G T... the variable cp grows from O to rt/2. every time that the load is deflected to the extreme position the deflection angle differs from cx.. with goes-.. Now instead of variable 9 we introduce a new variable cp via the formula sin (6/2) = lc sin cp. We note that formula (67) can be written in a simpler form.@.Ch..... or 2 I/k'——sin' 2k cos cp dcp 2 COS? }/1-k' s1n°q>
...._..2 I/L S . cosoc:-1-—-2 sin2E°2— .
k= which can be obtained from Eq.. differing from the elliptic integral of the second kind wb E'(k. where the function w dw F k..... all further discussion of the pendulum prob— lem will be related to an approach considered when conservative systems are studied in mechanics. T"4l' g § ]/1—k2sin"q> :. (60) by differentiating with respect to t..4 1/.é.. 0 Elliptic integrals cannot be expressed in terms of elementary functions. ip)-: S l/1——k2sin2q> dqn._'L.F(k.72 Differential Equations in Applications The last relationship enables us to rewrite formula (68) in the form [Ta/2 d . Here we only note that the starting point in our studies will be the differential equation -gag--f-kSiH9=O. ___ .
. it/2)... = S ————————-——— ( 0 1/1—k2 sinzcp is known as the elliptic integral of the first kind. Therefore.
CP = r.
.13 The Cycloidal Clock We have established that clocks with ordinary (circular) pendulums are inaccurate. 1. Suppose that the x axis is the straight line along which the generating circle rolls and that the radius of this circle is r (Figure 17). y-=MS=CP——CN. Then. which Galileo Galilei called the cycloid (from the Greek word for "circular"). But OP =MP ==r9. SP =MN=rsin9. Below we give its solution. the point occupies position M. Is there any pendulum whose period is independent of the swing? This problem was first formulated and solved as early as the 17th century.Ch. Construction of Differential Models 73 1. CN = rcostl. but first let us turn to the derivation of the equation of a remarkable curve. Let us also assume that initially the point that traces the cycloid is at the origin and that after the circle turns through an angle 6. This is a plane curve generated by a point on the circumference of a circle (called the generating circle) as it rolls along a straight line without slippage. on the basis of geometrical reasoning we obtain x = OS ·-= OP -— SP.
The very method of constructing a cycloid implies that the cycloid consists of congruent arcs each of which corresponds to a full revolution of the generating circle.-i)— l/2ry-—y2.* * The reader may find many interesting facts about the cycloid and related curves in G.z· _ 3 P ZJz'r· ` _ Fig.74 Differential Equations in Applications . parametrically the cycloid is spec_ified by the following equations: x=r(9-—sin9). 1980) (in Russian).N.9 ML1 0 A . (69) If we exclude parameter 9 from these equations. we arrive at the following equation of a cycloid in a rectangular Cartesian coordinate system Oxy: 33 =. y=r(1—cos6).
.—rcos·1(-K. 17 Hence. Berman's book The Cycloid (Moscow: Nauka.
epara_te arcs are linked at points where they have a common vertical tangent. which have proved to be extremely important for physics and engineering. The cycloid has many other interesting properties. correspond to the lowest possible positions occupied by the point on the generating circle that describes the cycloid. These points. and the segment of that straight line between successive cusps is known as the base of an arc of the cycloid. 1. For example. Construction of Differential Models 75 The 's. The cycloid possesses the following properties: (a) the area bounded by an are of a cycloid and the respective base is thrice the area of the generating circle (Galileo's theorem). the proiile of pinion teeth
. The highest possible positions occupied by the same point lie exactly in the middle of each are and are known as the vertices. The last result is quite unexpected. The distance along the straight line between successive cusps is 2:nr. whose calculation is not very simple. while the length of an arc of a cycloid is expressed as an integral multiple of the radius (or diameter) of the generating circle. since to calculate the length of such a simple curve as the circumference of a circle it is necessary to introduce the irrational number at. known as cusps.Ch. (b) the length of one cycloid arc is four times the length of the diameter of the generating circle (Wren's theorem).
astronomer. and mathematician Christian Huygens (1629-1695) to build an accurate clock in 1673. We take a small metal ball and send it rolling down the slope. Ignoring friction. Let us assume that a trough in the form of a cycloid is cut out of a piece of wood as shown in Figure 18. From the practical standpoint the problem can be solved in the following manner. 18 and of many typos of eccentrics. the particle is at rest). Let us now turn to the problem that enabled the Dutch physical scientist. at time t = to.76 Differential Equations in Applications :4 /1 ¥" m léei V Fig. Huy— gens found that the cycloid possesses such an isochronous (from the Greek words for "equal" and "time") or tautochronous property (from tautos for "identical"). and other mecha11ical parts of devices have the shape of a cycloid. let us try to determine
. cams. The problem consists of building in the vertical plane a curve for which the time of descent of a heavy particle sliding without friction to a fixed horizontal line is the same wherever the particle starts on the curve (it is assumed that initially.
Ch. When the ball reaches a pointN (9).e. In our case the last formula assumes the form v. which in view of Eq. (69) can be found in the following manner: h=y—y0=r(1——cos6) —— r (1 — cos 60) = r (cos 90 —— cos 9).-: l/2gr (cos 90. we can rewrite this formula (which is actually a differential equation) in the form dT :2 2r sin (9/2) df} ]/2gr (cos 90 — cos 6) ·
. K. we arrive at the formula Tg-: V 2gr (cos 90 ——cos 6) Since for a cycloid ds = 2r sin (9/2) dB. i. point M. starting from point M. Let :1:0 and y0 be the coordinates of the initial position of the ball. and 90 the corresponding value of parameter 9. the distance of descent along the vertical will be h. 1. We know that the speed of a falling object is given by the formula U= l/EEE. Construction of Differential Models 77 the time it takes the ball to reach the lowest possible point. say. where g is the acceleration of gravity. since speed is the derivative of distance s with respect to time T.cos 9) On the other hand.
This constitutes the motion of a cycloidal pendulum
. Obviously.. the time interval T in the course of which the ball rolls down from point M to point K is given by the formula T = TL which shows that period T is independent of 9. the period does not depend on the initial position of the ball. we obtain JT T__ S 2r sin (B/2) dB U `|/2gr(cos00-—cos0) ___ TIS ___2l/LS dc0s(9/2) g 0 '. two balls that begin their motion simultaneously from points M and N will roll down and find themselves at point K at the same moment of time. Proceeding then in the opposite direction and traversing its path.. in its motion down the slope the ball passes point K and. the ball completes a full cycle.78 Diiierential Equations in Applications Integrating this equation within appropriate limits. Since we agreed to ignore friction. continues its motion up the slope to point M1 lying at the same level as point M.·/ cos? -%°——-cos2g— == JT Thus. M. that is. by inertia.
ruction of Differential Models 79 \ \\ \ \ ~` \\ \\ ` \\\` ///)" Fig. which sets it apart from the simple (circular) pendulum. 1. Const. If the ball on the string is deflected to an arbitrary point M. 19 with a period of oscillations T0 = lm Vr/g. (70) A peculiar feature of the cycloidal pendulum. The length of the string must be twice the diameter of the generating circle of the cycloid. is that its period does not depend on the amplitude (or swing). it begins to swing back g and forth with a period that is independ-
. To this end it is sufficient to make a template (say. Let us now show how an ordinary circular pendulum can be made to move in a tautochronous manner without resorting to trough and similar devices with considerable friction. out of wood) consisting of two semiarcs of a cycloid witha common cusp (Figure 19). The template is fixed in the vertical plane and a string with a ball is suspended from the cusp O.Cli.
Hence. Even if due to friction a11d air drag the amplitude of the \§_ oscillations diminu ishes.
. the period of small oscillations of a circular pendulum with a string length Z: 4r is practically the same as that of a cycloidal pendulum of the same length.which moves along an arc of a cirFig· 20 cumference. lf these oscillations are very small.80 Differential Equations in Applications y ent of the position of point M. C if. By way of an example let us study the small oscillations that a pendulum executes along the arc AB of a cycloid (Figure 20). The path AB of a cycloidal pendulum will differ little from the path CE of a circular pendulum with a string length equal to [ir. the period '° remains unchanA ged. when the arc differs little from the arc of a cycloid. the effect of the guiding template is practically nil and the pendulum oscillates almost like an ordinary circular pendulum with a string (the "rod") whose length is 4r. For a circuB lar pendulum.the isochronous property is satisfied approximately only for small amplitudes.
Ch. the period of small oscillations of a circular pendulum can be expressed in terms of the length of the string: T = 2% l/{E This formula is derived (in a different way) in high school physics.14 The Brachistochrone Problem The problem concerning the brachistochrone (from the Greek words for "shortest" and "time"). if in (70) we put r == Z/4. In conclusion we note that the cycloid is the only curve for which a particle moving along it performs isochronous oscillations. a curve of fastest descent.
. 1. 1. Among the various curves passing through these two points we must {ind A Fig. Construction of Differential Models 81 Now. Take two points A and B lying in a vertical plane but not on a single vertical line (Figure 21). 21 6-0170 . was proposed by the Swiss mathematician John Bernoulli (1667-1748) in 1696 as a challenge to mathematicians and consists of the following.
82 Differential Equations in Applications A U. von Leibniz (1646-1716). Let us turn to Figure 22. The problem is famous not only from the general scientific viewpoint but also for being the source of ideas in a completely new field of mathematics. and Jakob Bernoulli (1654-1705).z' I 'O | I 1 ar 1 i1Z I%: {n C1 5 Fig. in which a ray of light is depicted as propagating from point A to point P with
. It was solved by John Bernoulli himself and also by Gottfried W. the calculus of variations. Sir Isaac Newton (1642-1727).2* n c-. The problem was tackled by the best mathematicians. 22 the one for which the time required for a particle to fall from point A to point B along the curve under the force of gravity is minimal. al OQ | . Solution of the brachistochrone problem can be linked with that of another problem originating in optics. Guillaume L'Hospital (16611704).
where generally 60
. with a constant.c-x)¤ l 12 If we assume that the ray of light propagates from pointA to point B along this path in the shortest possible time T.Ch. obviously. But then I __ C--. 1.r]/z»¤+. which initially was discovered in experiments in the form sin oil/sin oiz = a. The importance of this principle lies not only in the fact that it can be taken as a rational basis for deriving Snell's law but also. or the principle of least time. in that it can be applied to finding the path of a ray of light in a medium of variable density. Construction of Differential Models 83 a velocity U1 and then from point P to point B in a denser medium with a lower velocity v2.23 vi VPKF? vi 7 or sincil __ sinoiz —Z""—?5I" The last formula expresses Snell's wellknown law of refraction. the derivative dT/dx must vanish. be found from the following formula: T: 1/a¤v+x¤ _. for one. The total time Trequired for the ray to propagate from point A to point B can. The above assumption that light chooses a path from A to B that would take the shortest possible time to travel is known as Fermat's principle.
but it decreases when we pass from an up— per layer to a lower layer. OE"` `````' " .05 _____ v4 I 1% Fig. .84 Differential Equations in Applications Z4 lil ``—``—"``` " vg O.2. ...1. The incident ray is refracted more and more strongly as it passes from layer to layer and moves closer and closer to the vertical line.. the velocity of light changes (decreases) continuously and we conclude
.... .2.. . In each layer the speed of light is constant... For the sake of an example let us consider Figure 23.1. Applying Snell's law to the interfaces between the layers. we get sin cz.. 23 the light travels not along straight—line segments.... which depicts a ray of light propagating through a layered medium..1 _ sinotz __ sin ons _ sin oa. Then.__ ____ v GT . U1 _ V2 _ va _ V4 · Now let us assume that the layer thicknesses decrease without limit while the number of layers increases without limit... in the limit.. .....
as fol-
.. -. As the ray travels through Eartl1's atmosphere of increasing density.. Let us go back to the brachistochrono problem... We imagine that a ball (like a ray of light propagating in media) is capable of choosing the path of descent from point/1 to pointB with the shortest possible time of descent..-.. 24 (soe Figure 24) that @2:% (7. its velocity decreases and the ray bends. Then. i\ |x \ Fig.) U with a = const. A similar situation is observed (with certain reservations) when a ray of Sun light falls on Earth. We introduce a system of coordinates in the vertical plane in the way shown in Figure 25. Construction of Differential Models 85 \I \ OC ~/1 ".-. 1.Ch.
On the other hand. 25 lows from the above reasoning. This means that v= l/2gy...COSB Z SGGB 1/1+ta¤¤|$ . geometric construction enables us to show that SlHG..__L___ 1/4+<y*>2 ° Combining the last two relationships with (71) yields E/[1 +(y')21== C· (72)
. The energy conservation principle implies that the speed gained by the ball at a given level depends only on the loss of potential energy as the ball reaches the level and not on the shape of the trajectory followed. formula (71) is valid.86 Differential Equations in Applications A Iw n 1 I {H xI \n n I Ot \\\ I\B Fig.
0). An elegant solution of this problem amounting to setting up a second-order linear differential equation is given below. > 77. as can easily be demonstrated. and nl. We wish to know the difference of the limits of these two sequences. Out of ml. Vm0n0• Treating ml and nl in the same manner as ml. It be-
. Let ml. and nl. and nl. we construct two new numbers ml and nl that are. a remarkable curve: it is not only isochronous——it is brachistochronous. are convergent. which. and nl. we arrive at two sequences of real numbers. 1. we put mz :2%. Continuing this process indefinitely. indeed. ln other words.15 The Arithmetic Mean. . and the Associated Differential Equation Let us consider the following curious problem first suggested by the famous German mathematician Carl F. {mh} and {nh} (lc = O. 1..88 Differential Equations in Applications of the cycloid (69). Gauss (1777-1855). respectively. be two arbitrary positive numbers (ml. the Geometric Mean. ni.). nz: l/mlnl. the arithmetic mean and the geometric mean of ml. 2. The cycloid is.. we put mii .
and nl. It obviously depends on ml. 721. Let a be the sought difference. Eq. a fact expressed analytically as follows: cz =f (ml. introduced above.1-}-. Now. including a. xl = ni/my ·» U Z 1/lc (L no/m0)» yl = 1/f (1. by the same number k. nl. nl. each of the numbers ml. This means that a is a first—order homogeneous function in ml. (73) leads to the following relationship: dy __ ___ 2 2 dyl dxl §:-_` (1—l—:c)2 yl-l. will be multiplied by k. nl). and nl.Ch.2: dxl dw '
. The definition of a also implies that a =f(ml. Construction of Differential Models 89 longs to the German mathematician Carl W.. if we multiply ml. and..r——x3) ' On the other hand. and nl.y1"`"£"""-`—-`—. ml.1_l_x · (73) Since xl is related to sc via the equation __ 2 i/5 $1* "W? i we find that dm _ 1—¤= __<¤¤r—=¤¥>(1+¢¤)" dn: _` (1+x)2l/Q _ 2(. a 2 mo}: (is no/mo) : m1]((1¤ nl/m1)' Introducing tlie notation x = nl.. hence.).where f is a function. Borchardt (1817-1880). we find that __ m __ 2yl y-. 1. nl/ml)./ml.
.752) —%'Q.... etc. hence.2:3) -3.. mi)-=y —-—---' '".Ch.. we find that only y E O is such a solution. since y must be the constant solution to Eq..1: 1——x2 <1+¤=>1/5Z<<¤+¤·=i> V?{(i+¢2> Vi? 1-:1:.·· "··' .§. g. Construction of Differential Models 91 we find that :::2 is transformed into x3.—xy=O.. The differential equation (74) is remarkable not only because it enabled reducing
. assuming that d dy . the difference of the limits of the sequences {mn} and {nk} is zero. Thus. 0 we can easily find the value of this number.]—<¤y-a* (y>» we arrive at the following formula: as. (74) If we note that 2 ¤=r<m. (y): 1--as 1--..x. * >< >< —————-—. a y ). ¢* (y) = 0This means that y satisfies the differential equation (:1: —-— . we find that 1 —. tends to zero and. 1. (74). Hence. <1+xn>1/xi (" If we now send rz to infinity. Indeed.—[(¤¤—-wa) .
..92 Differential Equations in Applications the initial problem to an obvious one but also because it is linked directly to the solution of the problem of calculating the period of small oscillations of a circular pendulum.><(2f1)" k2n)' 1].><(2n—1)2 n y-1+21 22><4"><62>< X(2")2 Z2 7].:*.:*. then sr/2 S ·········—————""`° 0 1/1-kzsinzqn __:m ·' OO i2><32><5"><. where _ OO 12><32><52><.. is a solution to the differential equation (74). at/2). the period of small oscillations of a circular pendulum can be found from the formula T = 4]/UQF (lc. ( n/) 0 l/1—k2sin2cp It has been found that if 0 { lc < 1. As demonstrated earlier. with arr/2 d F ky 2 —-T S ...><(2n—1)2 ——?l1—l-E1 2°X42><6°X .
..
in accordance with Newton's second law.*5* (75) y __. Hence.7..-:0. We wish to derive the equation of the object's motion that ignores forces of friction (air drag). Let us select the coordinate axes as shown in Figure 26.16 On the Flight of an Object Thrown at an Angle to the Horizon Suppose that an object is thrown at an angle cx to the horizon with an initial velocity vo. 26
. Fig. Factoring out m yields daz ___ day _ w—O· W.Ch. m%:——mg.·:=. Construction of Differential Models 93 1. At an arbitrary point of the trajectory only the force of gravity P equal to mg. acts on the object.? 0 A . we can write the differential equations of the motion of the object as projected on the :4: and y axes as follows: d2 d2 mq. 1. with m the mass of the object and g the acceleration of gravity."Et=" m v va tm..
we find that the equations of the object's motion are sc = (vo cos ot) 23. and the shape of the trajectory. The second problem can be solved by calculating the value of x at a value of t equal to the time of flight. For example. the maximum height that the object reaches in its flight. we can find the time of the object's flight up to the moment when the object hits Earth.CC=O. The first equa-
. y=—-0.94 Differential Equations in Applications The initial conditions imposed on the object's motion are . that is. (77). (75) and allowing for the initial conditions (76). The second equation in (77) implies that this happens when t[v0 since--€§]=O. The second value provides the solution.:·-UOCOSOC. at t-—=O. The first problem can be solved by finding the value of time t at which y = 0. the range of the flight. Integrating Eqs. t Tgzvosinoz. (77) A number of conclusions concerning the character of the object's motion can be drawn from Eqs. y = (vo sin cx) t — gtz/2. ·—?. either t = O or t = (2120 sin oc)/g.
for one thing. which in rectangular Cartesian coordinates can be written as follows: y= x tan oc. Substituting this value of t into the second equation in (77). we find that the maximum height reached by the object is v§ sinz on/2g. we arrive at the equation -—gt + vo sin oc =O.gi? seczct. which yields t = (vo sin ot)/y.
. Noting that fg-: —·gt—|—v0 sin oc. The solution to the fourth problem has already been found. 1. The solution to the third problem can be obtained immediately by formulating the maximum condition for y. 8* ___ E ° This implies.. Construction of Differential Models 95 tion in (77) implies that the range of the flight is given by the formula (vo cos ct) (2v. the trajectory is represented by a parabola since Eqs. Namely. But this means that at the point where y is maximal the derivative dy/dt vanishes. or ot = !i5° In this case the range is vg/g. that the range is greatest when 2ot == 90°. (77) represent parametrically a parabola. sin oc) _ 1:% sin 20:.Ch.
2* Fig. with O<ot<1 and g the acceleration of gravity. although it is associated (consciously or subconsciously) with the "floating" of astronauts in the cabin of a spacecraft. Let us determine the pressure that the person exerts on the cabin's floor and the acceleration that the elevator must undergo so that this pressure will vanish.17 Weightlessness The state of weightlessness (zero g) can be achieved in various ways.96 Differential Equations in Applications 1. Suppose that a person of weight P is standing in an elevator that is moving downward with an acceleration co = osg. Let us consider the following problem. Two forces act on the person in the elevator (Figure 27): the force of gravity P and the force Q that the floor exerts on the perQ P . 27
.
we conclude that Q < P. The differential equation of the person's motion can be written in the form dzx mq?--P—Q. (78) Since d20:/dtz =-— to = ug and m = P/g. For this it is sufficient to put Q = O in (79). O < oc < 1. Thus. the pressure that the person exerts on the floor of the cabin is determined by the force Q = P (1 -{— ot). On the other hand.Ch. the pressure that the person exerts on the floor is nil. for Q to vanish the acceleration of the elevator must be equal to the acceleration of gravity. that is. when the cabin is falling freely with an acceleration equal to g. the pressure that the person exerts on the floor of the cabin of an elevator moving downward is determined by the force Q = P (l — OL). we can rewrite Eq. (78) as follows: d2 Q:. lt is this state that is called weightless7-0770
. Thus.-P—mTi£-:P(1—ot). We conclude that in this case ot = 1. when the elevator is moving upward with an acceleration to = osg. Let us now establish at what acceleration the pressure vanishes. 1. (79) Since 0 < ot < 1. Construction of Differential Models 97 son (equal numerically to the force of pressure of the person on the floor).
Of course. The previous problem implies that in the state of weightlessness the pressure on the walls of the spacecraft is zero. weightlessness is experienced not only during a free fall in an elevator. h=l
. and h is the altitude at which the spacecraft travels (reckoned from Earth's surface). Let us now turn our attention to Figure 28. What must be the speed of a spacecraft moving around the Earth as an artificial satellite for a person inside it to be in the state of weightlessness? One assumption in this problem is that the spacecraft follows a circular orbit of radius r + h. The as axis is directed along the principal normal rz to the circular trajectory of the spacecraft. Q == O. where r is Earth's radius. In the state of weightlessness all points of an object experience the same acceleration. We use the differential equation of the motion of a particle as projected on the principal normal: TL mvz T: E Frm. too.98 Differential Equations in Applications ness. For illustration let us consider the following problem. so that the force Q acting on an object inside the spacecraft is zero. Hence. In it the various parts of a person's body exert no pressure on each other. so that the person experiences extraordinary sensations.
any two objects separated by a distance r and having masses rn and M are at-
. we find that the required speed is given by the formula -. km F " (r+h)2 ' where m is the mass of the spacecraft. in accordance with Newton's law of gravitation. The above formula then yields lc = gr? Hence. that is. p Z F: Jeri.· 8 U "' " l/ r+h 1. where h = 0. and constant lc can be determined from the following considerations. which. At Earth's surface.100 Differential Equations in Applications Here force P is equal to the force F of attraction to Earth. is inversely proportional to the square of the distance r -\~h from the center of Earth. the force of gravity F is equal to mg. lf we now substitute the obtained value of P into (80) and note that Q = 0.18 Kepler's Laws of Planetary Motion In accordance with Newton's law of gravitation. (r+h>" ' where g is the acceleration of gravity at Earth's surface.
The effect of other planets on this motion will be ignored. 1. let us describe the motion ofthe planets in the solar system assuming that m is the mass of a planet orbiting the Sun and M is the Sun's mass.Ch. 29 tracted with a force GmM F :71 (81) where G is the gravitational constant. Let the Sun be at the origin of the coordinate system depicted in Figure 29 and the planet be at time t at a point with running coordinates x and y. Basing ourselves on this law.2* P ar Fig. Construction of Differential Models 101 y \\V /*005 gv m I\_ F ___ FSL/7§0 // I w' / ry /I M/I 7 0 . The attractive force F acting on the planet can be decomposed into two components: one parallel to the :1: axis and equal to F cos cp and the other parallel to the y axis and equal to
.
(84) under the initial conditions (85).g)=v0a'tt=O.102 Differential Equations in Applications F sin rp. (82) and (83) as follows: •• k •• k x Z — "°.•• i x-_ — (x2_|_y2)3/2 9 y __' l-— (x2_*_y2)3/2 ° (84) Without loss of generality we can assume that x:a.2:/r. we arrive at the following equations: most: —-Fcos cp = —£?¥cos cp. Finally. we can rewrite Eqs. we arrive at the differential equations •• T. (85) We see that the problem has been reduced to the study of Eq.y=—-O. (82) my: —Fsinq>:—§I}%sin cp. The special features of Eqs. allowing for the fact that r = l/x2 -I— y2. (84) suggest that the most convenient coordinate system here is the one using polar coordinates: x = r cos cp and
.x•=O. (83) Bearing in mind that sin cp = y/r and cos cp = . Using formula (81) and Newton's second law.` ¤ U Z _ jig/" » where constant It is equal to GM.E..
we have reduced the problem of studying Eqs. We also note that Eq. (87) by sin cp. in polar coordinates they assume the form (91) Thus. <93> where C1 is a constant possessing an interesting geometric interpretation. we arrive at the equation 2rqi + riii Z 0. Precisely. and subtracting the second product from the first. (89) Multiplying both sides of Eq. (90) can be rewritten in the form d• -H-— (rzqn) = 0. (89) and (90) under the initial conditions (91). (88) by cos cp. (84) under the initial conditions (85) to that of studying Eqs. (92) yields rah . we find that ro. (92) But Eq. Let S be the area of the sector limited by the segments OP and OQ and the
.. (eo) As for the initial conditions (85). both sides of Eq.rqlz = -k/rz.C. suppose that an object moves from point P to point Q along the arc PQ (Figure 30).104 Differential Equations in Applications adding the products.
Hence.2 Fig. we conclude that the areal velocity is a constant. dS _1 2 dcp _ 1 2' 7F* 2 " at —?r "'• (94) The derivative dS/dt constitutes what is known as the areal velocity. 0 or dS = (1/2) rz dcp. 30 arc From the calculus course we know tl1at cp S--—i— rzd . too. This law of areas constitutes one of the three Kepler laws. and since in view of (93) rzqn is a constant. 1. in turn. In full it can be formulated as follows: each planet moves along a plane curve around the Sun in such a manner that the radius vector connecting
.Ch.. means that the object moves in such a manner that the radius vector describes equal areas in equal time intervals. But this.. 2 (P. Construction of Differential Models 105 3 Cl { 0 P .
or rp = avg/rz. (89) and (90) with the initial conditions (91) imposed on them.evt _ k dt_dr dt _p dr—_ r3 —F' or dp __ a2v§ _ k Pw"—_r W Separating the variables in the last differential equation and integrating. we can rewrite this equation in the form £-§rE.. (95) This transforms Eq. (89) into °' a2v2 lc ':'%'"Tr· Assuming that : p. rzcp = avo.Q. that r = a and cp : vo/a at t = O. we return to Eqs. Hence. we {ind that v2 lc C-'¤··ij'"T·
. But then condition (93) implies that C1 = avo. which deals with the shape of the planetary trajectories. we arrive at the following relationship: TM r w+C2· Since p = : O at r : a. To derive the next Kepler law. for one.106 Differential Equations in Applications the S un with the planet describes equal areas in equal time intervals. The initial conditions imply.
Construction of Differential Models 107 Thus. The result is T: a. 1. (95).(XJ°2···l·2Br···1. The following cases are possible
.T r 22%+*: ' tv or. we finally have _ a2v§/lc rv 1+ecos<p (97) From analytic geometry we know that this is the equation of a conic in terms of polar coordinates. if we consider only the positive value of the square root. we hud that —§%·-—"L`7° -I/.2v§/lc 1+¤<>¤S(<¤>+6'3> ' where e = il/cx + [32/B = avg/lc —-— 1. The last equation can be integrated by substituting 1/u for r.("<·"T)+T"··. dr ___ l/ 2k 2]:: a2v2 ar.§" (99 Dividing Eq. (96) by Eq. Thus. The constant C3 can be determined from the condition that r = a at cp = O. with e the eccentricity of the conic. where __ 1 ___ 2k B_ lc Q--? a3v§ ' — a2v§.Ch. It is easy to verify that C3 = O. we arrive at the equation Lz ____ k azvz v§ lc T.
We. say. (2) a hyperbola if e > 1. As for recurrent comets. their orbits resemble "prolate" ellipses whose eccentricity is smaller than unity but very close to it. Say. or vg = 2k/a. Astronomical observations have shown that for all the planets belonging to the solar system the value of vg is smaller han 2l. but in the majority of cases these ellipses are close to circles.c/a. Its latest apparition was in the period between the end of 1985 and the beginning of 1986. or vg = lc/o. or vg<2k/cz._respec-
. e differs little from zero. Note that the orbits of the Moon and the artificial satellites of Earth are also ellipses. or vg > 2k/a. (3) a parabola if e : 1. since they never return to the same place. Let us now establish the physical meaning of eccentricity e. Celestial bodies that move along parabolic and hyperbolic orbits may be observed only once. First we note that the components x and y of a planet's velocity vector Valong the as and y axes. that is. arrive at another of Kepler's laws: the planets describe ellipses with the Sun at one focus. (4) a circle if e = O. Halley's comet appears in Earth's neighborhood approximately every 76 years. Halley's comet. therefore. like.108 Differential Equations in Applications here: (1) an ellipse if e<1.
m (rzcpz + rz).Ch. and the size v of vector V satisfy the relationship U2 I WL Q2.2 °2 km Assuming that cp = 0 and combining (97) with (100). Construction of Differential Models 109 tively. 1. then formulas (98) and (99) yield 1 2. OO km km <>¤ km T lf by E we denote the total energy of the system. which in view of energy conservation must be constant. we get __ azvz/lc mr2a2v2 _ km __ '·· ¢'··E·
. (98) The potential energy of a system is minus one times the amount of work needed to move the planet to infinity (where the potential energy is zero by convention). which when combined with (86) yields w:#&+h From this it follows that the kinetic energy of a planet of mass m is given by the formula % mvz = é. Hence.
if we could impart such a "blow" to Earth that it would increase Earth's total energy to a positive value. E < O.2v§. E >O. This law deals with the period of revolution of planets around the Sun. or E = —mk2/2a. respectively. is :1:2 12 @+%:1*
. Thus. Earth would go over to a hyperbolic orbit and leave the solar system forever. we naturally restrict our discussion to the case of elliptic orbits.110 Differential Equations in Applications Excluding r from the last two relationships. E = O. the shape of a planet's orbit is completely determined by the value of E. parabola. we {ind that P 2a2v§ " Z l/1 +L nm Equation (97) for the shape of the orbit finally assumes the form T: azvg/lc 1+ ]/1—{—E (2a2v§/mkz) cos cp This formula implies that the orbit is an ellipse. Let us now turn to Kepler's third law. Say. or circle if. as is known. whose equation in terms of Cartesian coordinates. hyperboba. Taking into account the results obtained in deriving the previous Kepler law.
These forces may be the weight of the beam itself or an external force or the two forces acting simultaneously. we conclude. Then. Clearly. Usually a bent symmetry axis is called the elastic line.:g'—:T€"· This constitutes the analytical description of Kepler's third law: the squares of the periods of revolution of the planets are pro— portional to the cubes of the major axes of the planets' orbits. that at§n == av0T/2. Finally. on the basis of (94) and (95). T is the time it takes the planet to complete one full orbit about the Sun. Suppose that forces acting on the beam in the vertical plane containing the symmetry axis bend the beam as shown in Figure 33. taking into account (102).19 Beam Deflection Let us consider a horizontal beam AB (Figure 32) of a constant cross section and made of a homogeneous material.a.442 Differential Equations in Applications Let us denote the period of revolution of a planet by T.-{. By definition.2§2»n2 43-[2 T2: —. The problem of deter-
. since the area limited by an ellipse is Jtgn. 1. we arrive at the following result: 4. the symmetry axis will also bend due to the action of these forces. The symmetry axis of the beam is indicated in Figure 32 by a dashed line.
Figure 36 illustrates the case of a concentrated load. For example. if we know the equation of the elastic line. the load can vary along the entire length of the beam or only a part of this length (Figure 35). There are also various ways in which loads can be applied to beams. Let us consider a horizontal beam OA (Figure 37). Figure 35 depicts a beam lying freely on supports A and B. Note that there can be various types of beam depending on the way in which beams are fixed or supported. Thus. which becomes the elastic line (also depicted in Figure 38 by a curved dashed line).Ch. F2. External forces F1. The positive direction of the y axis is downward from point O. The deflection y of the elastic line from the III axis is known as the sag of the beam at point x. 1. the origin. Such a beam is said to be a cantilever. we can always find the sag 8-0770
. (and the weight of the beam if this is great) bend the symmetry axis. Figure 34 depicts a beam whose end A is rigidly fixed and end B is free. Suppose that its symmetry axis (the dashed line in Figure 37) lies on the :1: axis. Another type of beam with supports is shown in Figure 36. For example. a uniformly distributed load is shown in Figure 34. Of course. Construction of Differential Models 113 mining the shape of this line plays an important role in elasticity theory. with the positive direction being to the right of point O.
.2:. 34 of the beam. The bending moment is defined as the algebraic sum of the moments of the forces that act from one side of the beam at point x. 32 AB %"""Ill§!. 33 Al B §§ Fig.114 Differential Equations in Applications A £m/# m#//1///0/////100// B Fig. Below we show how this can be done in practical terms. In calculating the moments we assume that the forces acting on the beam upward result in negative moments while those acting downward result in positive moments."'"/M Fig. Let us denote by M (x) the bending moment in the cross section of the beam at coordinate ..
the slope y' of the elastic line is extremely small and. J is the moment of inertia of the cross section of the beam at point as about the horizonta-1 straight line passing through the center of mass of the cross section. A horizontal homogeneous steel beam of length l lies freely on two supports and
. Now. (103) we can take the approximate equation EJ y" = M (:1:). therefore. which is usually the case in practice. (104) To illustrate how Eq. if we suppose that the sag of the beam is small. The product EJ is commonly known as the flexural rigidity. wl F5 W `4 y (D Fig. 38 pends on the type of material of the beam. in what follows we assume this product constant.116 Differential Equations in Applications 5 F 0 3 fl. (104) is used in practice. we consider the following problem. instead of Eq.
. Thus.1/2 from point Q and generates a positive moment. Since we are dealing with a two-support beam. The bending moment M (sc) is the algebraic sum of the moments of these forces acting on one side from point Q (Figure 39). 1.. each support acts on the beam with an upward reaction force equal to half the weight of the beam (or pl/2). In Figure 39 the elastic line is depicted by the dashed curve. At a distance x from point Q a force equal to pl/2 acts on the beam upward and generates a negative moment. a force equal to px acts on the beam downward at a distance .--··' .. On the other hand. We wish to determine the equation of the elastic line and the maximal sag. which is p kgf per unit length.. Let us first consider the action of forces to the left of point Q. the net bending
.Ch.¤<&·~v} Fig.2: 1QI 9 *°°` . 39 sags under its own weight. Construction of Differential Models 117 pi/2 pz/2 L-a: 0XA —` __________ .
we find that a force equal to p (l — :1. we can easily write the basic equation (104). (107) so as to {ind y: y=Oatx=Oand.
.2=i-E2 . which in our case assumes the form Ely": Jl? -—%E (107) Since the beam does not bend at points O and A. The net bending moment in this case is calculated by the formula M(x>=·p (i——¤¤>i?—§L<i—x> . while a force equal to pl/2 acts on the beam upward at a distance l — az from point Q and generates a negative moment.r.118 Differential Equations in Applications moment at point Q is given by the formula z21 Mw: ···§Lx+Px(%)=f%·—%£ (105) If we consider the forces acting to the right of point Q.moments prove to be equal.sz:)/2 from point Q and generates a positive moment. 2 . knowing how to find a bending moment. Now.2 (106) Formulas (105) and (106) show that the bending.) acts on the beam downward at a distance (l -.atx:l.. we write the boundary conditions for Eq. ...py=O..
basing our reasoning on symmetry considerations (the same can be done via direct calculations). logging trucks move along forest roads some of the time. in this example./`.20 Transportation of Logs In transporting logs to saw mills.Ch. For instance. It is used to calculate the maximal sag. 1. Ignoring the question of how traffic should be arranged that loaded and empty traffic trucks meet only at such sections. thirty-meter logs can be transported. 1. we conclude that the maximal sag will occur at :2: = Z/2 and is equal to 5pl'*/384E. let us establish how wide the turns in the road must be and what trajectory the driver must try to follow so that. say. (108) This constitutes the equation of the elastic line.Sections of the road are made wider so that trucks can pass each other. (107) then yields y = @1% (xi -— 2Zx3—|— Pcs). It is assumed that the truck is sufficiently maneuverable to cope with a limited section of the road. The width of the forest road is usually such that only one truck can travel along the road. Construction of Differential Models 119 Integration of Eq. with E =-— 21 >< 105 kgf/cm2 and J = 3 >< 104 cm4. Usually a logging truck consists of a tractor unit and a trailer connected freely
.
but also has a platform with two posts. By XY we denote a log for which AX : kh. Thus. The points A and B correspond to the axes of the platforms a distance h apart. Point C corresponds to a small axis that connects
. The other ends of the logs are put on this platform. 40 to each other. The tractor unit has a front (driving) axle and two back axles above which a round platform carrying two posts and a rotating beam are fastened. The chassis connects the back platform with the axis that links the trailer with the tractor unit. One end of each log is placed on this platform.9 C Q P XBAY s R Q P A. F Fig.120 Differential Equations in Applications S . Schematically the logging truck is depicted in Figure 40. the length of the chassis can change during motion. The trailer's chassis consists of two metal cylinders one of which can partly move inside the other. which enables the tractor unit and the trailer to move independently to a certain extent. The trailer has only two back axles. This platform can rotate in the horizontal plane about a symmetrically positioned vertical axis.
Construction of Differential Models 121 the tractor unit with the trailer. while RR and SS are the trailer axles. All axles have the same length 2L. but in the simplest case of log transportation a = O. we put it equal to 2W. FF is the front axle of the tractor unit. which is commonly understood to be the maximum deflection of the rear part of the logging truck (for the sake of simplicity we assume this part to be point X) from the trajectory along which the logging truck moves. As for the width of the load in its rear. is exactly above the road's center line. Usually a-:0. so that for the sake of simplicity we assume that the width of the logging truck is 2L. and PP and QQ are the back axles of the tractor unit. Here it is convenient to fix a coordinate system
. Point A in Figure 41 is determined by the angle X that the truck AC makes with the initial direction. 1. Let us suppose that the road has a width of 2[iih and that usually a turn in the road is simply an arc of a circle of radius h/cz centered at point O (Figure 41). with AC=ah. Next. the driver operates the truck in such a way that point A. For the sake of simplicity we assume that a logging truck enters a turn in the road in such a way that the tractor unit and the trailer are positioned along a single straight line. In what follows we will need the concept of the sweep of the logging truck.3. corresponding to the axis of the front platform.Ch.
// \ x /' ' 0 . The required halfwidth h of the road. = X —— 9. As for angle BAC in Figure 41. Usually this angle is the logging truck's angle of lag.122 Differential Equations in Applications X \\\\ Zrzifial __ g_8 \_\____ dirccfian '/' \\ // \ /0 x x y // // A /// //// \ // /`/// \ / /. which determines the sweep of the logging truck at a turn and is known as the halfwidth of the road at the outer curb of a
. denoting it by u we {ind that u.z· Fig. 41 Oxy in such a way that the horizontal axis points in the initial direction and the vertical axis is perpendicular to it. In a general situation the load carried by the truck will make an angle 6 with the initial direction.
further..0 we have N ° :=¢ 142°. means that the center line AC of the tractor unit constitutes a tangent to the arc of a circle at point A.—:T—T. is defined as the algebraic sum OX — OA —l— W.1 we have N° z 19°.
. The log length 7th will be greater than h. For practical considerations we must consider only such cx's that lie between 0 and 1. so that OA is perpendicular to AC. This requirement. for one thing. and angle X is determined by the motion of point A along the arc of the circle. while the halfwidth of the road at the inner curb of a turn is defined as the algebraic sum OA + L — OP. Thus. and the greater the value of oc.—. the greater the maneuverability of the logging truck. reasoning on practical grounds. (109) where h is measured in meters. Construction of Differential Models 123 turn. that in building a road the curvature of a turn in the road is determined by an angle N° that corresponds to an arc length of a turn of approximately 30 m. at h = 9 m and ot -:0. while at h = 12 m and cx = 1. Note. 1. We stipulate that in its motion a logging truck's wheel either experiences no lateral skidding at all or the skidding is small. In our notation.Ch. but again. O i80° 30u N · . where OP is the perpendicular dropped from point O onto AB.
it is assumed that 0 { a < 0. employing the fact that X = u. Thus.:—tanxp. Y: —£—cosX +hsin0. point B moves in the direction BC and dY -ai-—. 0. varies between 1 and 3.—sin X—hcos 9. '—'r"*———. (111) where mp is the angle which BC makes with the initial direction.»T*-Th · (**2) where 0<b<1. (110) Since the trailer's wheels do not skid either.. the coordinates of point A in Figure 41 are x= —E·sinX. Since the wheels of the tractor unit do not skid laterally. As for constant a.£cosX. + 0 and studying triangle ABC.124 Differential Equations in Applications it must not be greater than 3h. Finally. and xp. and u are
.5. we note that in each case the value of h is chosen differently. but it varies between 9 and 12 m. we arrive at the following chain of equalities: sin (X—1|>) ___ sin (0—1|>) __ sin u. a ex The coordinates of point B are X=—. the value of 2. y:-. Next.
1. X dependence proves to be extremely complicated. Spectrum 7. where the angle of lag plays the role of the sought function.. Nevertheless. —{-(-Ex-cosX+hT1T(-s1n9)s1n1p-O. No. 1: 19-26 (1974/75). . If we combine (111) with (110). To this end we can * See A. hinders an effective study. However. %3—cos9)cos1p It dB .Ch.-. Math. which.B. the resulting u vs. Eq.. 114 dx _1 0t(1——ac0Su) ( ) with the initial condition u (0) :.22. we find that (—-Q-S1nx+z. "The sweep of a logging truck". Tayler.
. Carrying out the necessary calculations..· O.* _ By substituting v for tan (u/2) in the differential equation (114).. W9 arrive at sin(X——1p)== on —g§—cos(9—1p). of course. Construction of Differential Models 125 functions of X.. (113) If we exclude variable 142 from this relationship fora fixed value of a by employing (112) and the fact that X =-· u —l— B. (114) can easily be integrated and studied numerically.. we can mtegrate the equation in closed form. we arrive (since 9 : 0 at X = 0) at the differential equation 2.
1437181 _4 .3146894 1 .2 .128 Differential Equations in Applications approximately given by the relationship /\J YN @(1-——a) But if ot :1. This requirement is especially important for small oa's and in the neighborhood of X = O.8 . = u (X).6 .2299778 .0 a = 0. we have ¤(1+¤2> Y: 1—a" ° For the step AX in the variation of the independent variable X we must take a number that does not exceed C.2824244 .3 Step of independent variable = 0. Below we give the protocol for calculating the values of the function u.3347111
. Solution of differential equation by Runge-Kutta second-order method Parameters: alpha = 1.5 Initial values: of independent variable = 0 of function = 0 Xu 00 .
For Qi
. Now let us determine the sweep of the logging truck by using the concept of halfwidth of the road at the outer curb of a turn (this quantity was defined earlier as the algebraic sum OX — OA —}— W (Figure 41)). while the dashed curves show what must be the value of B -— L/h to guarantee the necessary "margin" at the inner curb of the turn.W/h and oc for different values of Z.—-sin u) . the maximal halfwidth Bh of the road can be determined from the following formula for B: 11W B = I/'"2+w"tr+v7· In Figure 43 the solid curves reflect the relationship that exists between B-. 1. -·· This shows that the sweep decreases as X grows since the angle of lag. u..2--2-g. Construction of Differential Models 131 influences the angle of lag of the logging truck. For clarity of exposition the scales on the u and X axes are chosen to be different. increases. Thus. First we note that OX2 = (-g—sin X—9»h cos 9)2 —}—(£—c0sX-|—7tn sin6)2 =h2 (2%-1-}.Ch.
Thus.2 m. and L/h = 0.Ch. then for logs of approximately 24 m in length (reckoning from the axis of the front platform) we have X = 2. in accordance with (109) N° is approximately 28°.. W/h = 0. If the width of the logging truck is 2.—(1-—cos C)|a=0-..4 m and width of the load in the rear is 1. If a logging truck in which the distance between the front and back platforms is 12 m follows a turn along an arc of a circle of radius 60 m. Then (115) yields _. . from Figure 43 it follows that for all values of cz we have B = 0..2 for the inner curb. then oc = 0. = 0. since the value of C decreases with a.%-(1—·cosu)-|—%'— 1L <—.(115) Next.(1—c0s C) +-5.:2——K6%2—. 2 [1--2-<—.05. 1. Construction of Differential Models 133 every position of the logging truck on the road we must have [5:.. Theoretically the necessary halfwidth of the road at the
. which corresponds to the simplest case of logging.1.2 and.45 for the outer curb of the turn and B = 0. Here is a typical example that illustrates these results. Now let us dwell on the results that follow from the above line of reasoning. the angle of lag u proves to be the greatest when a.
since in this case the angle of lag grows.2.4 m. If the logging truck transports logs whose length is 14.134 Differential Equations in Applications outer curb of the turn is equal to 5. as can easily be seen.4 m.76 m so that the driver can drive the truck along a curve whose length at the turn is approximately equal to the length of the road's center line. we see that an increase in the length of the logs by 9.4 m (reckoned from the axis of the front platform) and whose bunch width in the rear is 1. if we compare the two cases considered here. For one.4 m and that at the inner curb to 2. while the same quantity at the inner curb. The theory developed above shows that the sweep of the logging truck is the greatest when the truck enters the turn. The value of [5 then proves to be the same for the inner and outer curbs of the turn and equal to 0. is equal to 2. as in the previous case. This conclusion also holds true in the situation
. Practice has shown that an inexperienced driver is not able to drive his truck along such a curve and needs a road whose total width at a turn is at least 10. then in the case at hand Pt = 1.64 m. This reasoning shows that the longer the logs transported the wider the road at a turn must be. Hence.4 mi.8 m (if the load transported is 24 m long) for a truck width of 2.22.8 m. the necessary halfwidth of the road at the outer curb of the turn is 2.6 m requires widening the road at a turn by 2.
+ Q". for fairly large values of a the values on >1 become possible. We note that in the case of simple log transportation.Ch. that is. they must obey the following relationship: ot (1 —— a cos C) = sin C. is not much greater than unity. the maximal value of on is (1 — a2)**/2. We note in conclusion that for ot > 0.5 considerable economy in the width of a road is achieved by increasing a (see Figure 43).-+2 gmc.25 at a == 0.5. But if there exists a nonzero initial angle of lag C0 caused by the zigzag nature of the turn. with the practical extremal value of ot being 1. --27. However. Then the required width of the road is determined from the following formula for B: rs: 1/ xw. it is impossible to pass a turn with on greater than unity. Construction of Differential Models 135 when the truck enters one section of a zigzag turn after completing the previous one at the point of inflection. The results represented graphically in Figure 43 correspond to the case where prior to entering a turn the tractor unit and the trailer are positioned on a single straight line.
. Thus. If the load is such that 2. 1. a-:-0. we must select the initial condition in the differential equation (114) in the form u (0) == ——C0. $. the value of oc is chosen such that the required halfwidth of the road at the inner curb of any turn is always smaller than at the outer curb.
Below with concrete examples we show how in solving practical problems one can use the simplest approaches and methods of the qualitative theory of ordinary differential equations. for
. that is.1 Curves of Constant Direction of Magnetic Needle Let us see how in qualitative integration. one can use a general property of such equations whose analogue. Hence. to study differential models of real phenomena and processes we need methods that will enable us to extract the necessary information from the properties of the differential equation proper. 2. the overwhelming majority of differential equations are not integrable in closed (analytical) form. the process of establishing the general nature of solutions to ordinary differential equations. However.Chapter 2 Qualitative Methods of Studying Differential Models In solving the problems discussed in Chapter 1 we constructed differential models and then sought answers by integrating the resulting differential equations. as noted in the Preface.
the slope K of the tangent to the integral curve at point M (:2:. The collection of all line elements in D is said to be the field of directions or the line element field.Ch. we say that at each point M (:c. The reader will recall that curves at Earth's surface can be specified along which the direction of a magnetic needle is constant. y). y) is assumed singlevalued and continuous over the set of variables sc and y within a certain domain D of the (ec. To each point M (x. y) is an interior point and which makes an angle 9 with the positive direction of the x axis such that K = tan 9 = f (sc. y). Thus. 11). (U6) where the function f (1:. let us consider the first-order ordinary differential equation -3%-=f(¤¤. y)-plane. is the property of the magnetic field existing at Earth's surface. y) belonging to the domain D of function f (x. From this it follows that geometrically the differential equation (116) expresses the fact that the direction ofa tangent at every point of an integral curve coincides with the direction of the field at this point.
. Qualitative Methods 137 example. Graphically a linear element is depicted by a segment for which point M (az. 2. y) the differential equation assigns a value of dy/dx. Bearing this in mind. y) of D the differential equation (116) defines a direction or a line element.
the set of points in the (. y)-plne. The form of this equation suggests that the family of isoclinals is given by the equation x2+y2=v. which cannot be integrated in closed (analytical) form. its isoclinals are given by the equation f (J:. that is. that is. As for the differential equation (116). Let us consider. y)-plane at which the direction of the field specified by the differential equation (116) is the same. y) = vi where v is a varying real parameter. the differential equation {11%-=w2—l-y·'*. v>O. The isoclinals of the magnetic field at Earth's surface are curves at each point of which a magnetic needle points in the same direction. we can approximately establish the behavior of the integral curves of a given differential equation. At each point of
. Knowing the isoclinals.2:. the isoclinals are concentric circles of radius I/v centered at the origin and lying in the (as.138 Differential Equations in Applications To construct the field of directions it has proved expedient to use the concept of isoclinals (derived from the Greek words for "equal" and "sloping"). for example.
However. This information alone is sufficient to convey an idea of the behavior of the integral curves of the given differential equation (Figure 44).Ch. even with more complicated equations knowing the isoclinals may prove to be expedient in solvinga specific problem. 44 such an isoclinal the slope of the tangent to the integral curve that passes through this point is equal to the squared radius of the corresponding circle. 2. We arrived at the final result quickly because the example was fairly simple. Qualitative Methods 139 9 V gg? it Fig. Let us consider a geometric method of integration of differential equations of the
.
6* .·—*f(9?» ll). points of inflection should be sought among the points at which y" vanishes. (116).. This means that the points of these curves may prove to be points of maxima or minima for the integral curves of the initial differential equation.%¥+Ji§. :..
. specifies curves at whose points dy/dx = 0. that is.—·+*—5'.0. y) 6f(==.'+—w*y 0 .!%.. we find that .·(_.(L) Equation (I 0).. y) 6f(¢» y) » y ·· '—5. the "zero" isoclinal equation. This oxplains why out of the entire set of isoclinals we isolate the "zero" isoclinal. for * It is assumed here that integral curves that {ill a certain domain possess the property that only (que integral curve passes through each point of the omain. For greater precision in constructing integral curves it is common to find the set of inflection points of these curves (provided that such points exist). _ 6f(¤=. 11) _ '—3.140 Differential Equations in Applications type (116). Employing Eq. As is known. The method is based on using the geometric properties of the curves given by the equations f(¤¤· y)=0· (I0) 0f(=v. (L) are the possible point—of-inflection curves. y)_ The curves specified by Eq.* Note..
The straight lines (I 0) and (L) break down the (sc.Ch. The equation of curve (I 0) in this case has the form as + y = 0. Sm in which the first and second derivatives of the solution to the differential equation have definite signs. that a point of inflection of an integral curve is a point at which the integral curve touches an isoclinal. and S3 (y' < 0. Thezcurves consisting of extremum points (maxima and minima points) and points of inflection of integral curves break down the domain of f into such subdomains S1. The points ofminima of the integral curves lie on the straight line (I 0). is not a point-0f—inflection curve. A direct check verifies that curve (I 0) is not an integral curve. S 2 (y' < 0. or y = -:1:. As for curve (L). to the left ofthe straight line (L). we find that it is an integral curve and. to the right of the straight line (I 0). . There are
. In each specific case these subdomains should be found. Qualitative Methods 141 one. hence. As an example let us consider the differential equation y' = x + y. between the straight lines (I0) and (L). y" >O). y)·plane into three subdomains (Figure 45): S1 (y' >O. while to the left they point downward (left to right in Figure 45).. 2. T 0 the right of (I 0) the integral curves point upward. This enables giving a rough picture of the behavior of integral curves. y" > 0). S2. y" < 0). whose equation in this case is y -l— x —}— 1 = 0.
since it separates one family of integral curves from another.
. Note that in the given case the integral curve (L) is a kind of "dividing" line. The behavior of the integral curves on the whole is shown in Figure 45. To the right of the straight line (L) the curves are convex downward and to the left convex upward. Such a curve is commonly known as a separatrix.142 Differential Equations in Applications ' SI 0 Y —¤ S5 \ K -1 Fig. 45 no points of inflection.
we tacitly assumed that the differential equation in question had a solution. These theorems are important for both theory and practice. Many methods of numerical solution of differential equations have been developed.Ch. that is. The problem of when a solution exists and of when it is unique is solved by the so·called existence and uniqueness theorems. which narrows their practical potential.1. and although they have the common drawback that each provides only a concrete solution. Thus. Often their proof is constructive.2 Why Must an Engineer Know Existence and Uniqueness Theorems? When speaking of isoclinals and point—of— inflection curves in Section 2. It must be
. Qualitative Methods 143 2. they are widely used. existence and uniqueness theorems lie at the base of not only the above-noted qualitative theory of differential equations but also the methods of numerical integration. the methods by which the theorems are proved suggest methods of finding approximate solutions with any degree of accuracy. 2. They serve as a basis for creating new methods and theories. Existence and uniqueness theorems are highly important because they guarantee the legitimacy of using the qualitative methods of the theory of differential equations to solve problems that emerge in science and engineering.
J r.) —— yo. then for every point (xo. let us take two simple examples. No. ( > that is defined on a certain interval containing point xo. (116) function f is defined and continuous on a bounded domain D in the (x. 1: 41-44 (1976). Sci. y)* See C. yo) E D there exists a solution y (x) to the initial-value problem ** dy — — 117 E. y)—plane. To illustrate what has been said. that before numerically integrating a differential equation one must always turn to existence and uniqueness theorems.144 Differential Equations in Applications noted. This is essential to avoid misunderstandings and incorrect conclusions. Existence and uniqueness theorem If in Eq. J. such a problem issaid to be an initial-value problem. y).* but hrst let us formulate one variant of existence and uniqueness theorems.— f (ac. however. Technol. ** If we wish to find a solution of a differential equation satisfying a certain initial condition (in our case the initial condition is y (xo) = yo). (116) function f is defined and continuous on a bounded domain D in the (x. y (wo..E. 7. "Why teach existence and uniqueness theorems in the first course of ordinary differential equations?". Roberts. Educ. Int. Existence theorem If in Eq.
. Math.
Let us consider the following problem. y0) E D can be extended to a point that lies as close to the boundary of D as desired. yi) and step h = 0. then every solution to Eq. then for every point (xo. y1)|<L|y2-—y1l. from the problem considered on p. Extension theorem If the hypotheses of the existence theorem or the existence and uniqueness theorem are satisfied. In the first case the extension is not necessarily unique while in the second it is. 2. with L a positive constant. 162 concerned with a conservative system consisting of an object oscillating horizontally in 10-0770
. y (—— 1) = 0.21 (118) on the interval l-.+1 = yi -}— hf (xi. that is.1. 3].Ch.1. Using the numerical Euler integration method with the iteration scheme y. yi)-—f(x. solve the initial-value problem y' == -—x/y. yo) E D there exists a unique solution y (x) to the initial-value problem (117) defined on a certain interval containing point xo. for example. liisv. (116) with the initial data (xo. Note that the problem involving the equation y' = —x/y emerges. Qualitative Methods 145 plane and satisfies in D the Lipschitz condition in variable y.
The fact is that as the solution y = y (as) approaches the sv axis.0441 —— 1:2. the angle
. allowing for the concrete form of the differential equation.21 -1 Integrating.Ch. we arrive at an erroneous result. the initial-value problem (118) has a solution y (x) defined on a certain interval containing point xo = -1. As a result of numerical integration we have arrived at a solution of (118) defined on an interval (a. we have yI S mm.*:1 1.0441 .4. Indeed. b).0218. If we employ only numerical integration. according to the extension theorem. Thus. can be extended to a value of y (cc) close to the value y (x) =-· O. since in the initial equation the variables can be separated. However. by resorting to the existence theorem (and to the extension theorem) we were able to "cut ofi" the segment on which there is certain to be no solution of the initialvalue problem. in accordance with the existence theorem. This solution.—§ we 0. we get y = I/1. with a< -1 and 1. Thus. we can specify the true interval in which the solution to the initial-value problem (118) exists. Qualitative Methods 149 ing the :1: axis.3< b<1. Hence. 2. a solution to the initial—value problem (118) exists only for | :2: | < I/1.
in the time that the independent variable x changes by 0. and we find ourselves on an integral curve that differs from the original.150 Differential Equations in Applications of slope of the curve tends to 90°. The following example is even more instructive. 1]. whose beginning in the case at hand has the following form: 10 REM Euler's method 20 DEF FNF(X. the value of y is able to "jump over" the as axis.1. Y) 110 X ¤= X —{— DX The results are presented graphically in Figure 47.1 and an iteration scheme E/z+1 = Hz + hl (xi+1/21 yi+1/2)? with yi+1/2 = yi + hf ( ¤?z» yi)/2We solve the initial-value problem (119) using the above program. Therefore. The approach here consists in first employing the Euler method and then an improved Euler method with a step h = 0. This happens because the Euler method allows for the angle of slope only at the running point.
. Y) =3·•=X·•=SGN(Y)·•·ABS(Y) A (1/3) 30 GOSUB 1100: BEM Coordinate axes 40 REM Next value of function 100 Y = Y —l— DX*FNF(X. We wish to solve the initialvalue problem y' = 3x?/yi y (-1) == --1 (M9) on the segment [— 1.
Separating the variables.Ch.2:3 if :1:<O. This result already suggests that the solution via the Euler method gives the function yl (sc) == 1:3 while the solution via the improved Euler method gives . according to the extension theorem. y = . which means that the solution of the initial—value problem considered on the segment [-1. this solution can be extended to any segment. the existence theorem implies that the initial—value problem (119) has a solution defined on a segment containing point xl.i. Let us now turn to the existence and uniqueness theorem in connection with this problem. the functionf (sv.1:3 if . y) = 3. -1 -1 or. Further. y2(x) Zl -. 2. we note that since the function f (. since 6f (ac. and. we get Q! x S 1]**/3 dn:-3S §d§.2:.1:>O Both yl and yl are solutions to the initialvalue problem (119). y)/0y = :cy'2/3. = --1. y)=
.::%/y` is continuous in the entire (1:. 1] is not unique. Qualitative Methods 153 initial-value problem. First. y)-plane.x". finally.
O]. a domain does contain points belonging to the x axis. Hence. by resorting to the existence and uniqueness theorem (and the extension theorem) we were able to understand the results of numerical integration. if we are speaking of the uniqueness of the solution of the initial—value problem (119) on the [-1. from the existence and uniqueness theorem (and the extension theorem) it follows that in this case the solution to the initial—value problem can be extended in a unique manner at least to the x axis. we already know that as soon as y vanishes. there can be several such solutions. it is easy to show that the function does not satisfy the Lipschitz condition. 0) in a unique manner. But since the straight line y = O constitutes a singular integral curve of the differential equation y' = Bx:}/y. the solution exists and is defined only on the segment [-1. however. that is. Generally. Thus.
. however. If. there is no way in which we can extend the solution to the initial-value problem (119) beyond point O(O.154 Differential Equations in Applications 3x:}/y satisfies the Lipschitz condition in variable y in any domain not containing the 1: axis. 1] segment.
dx/dt)-plane (Figure 50). 72 when we considered pendulum clocks. dx/dt)—plane. 2.
. Thus. Each new state of the system corresponds to a new point in the phase plane. This point is called a representative point. the changes in the state of the system can be represented by the motion of a certain point in the phase plane.3 A Dynamical Interpretation of Second-Order Differential Equations Let us consider the nonlinear differential equation dzx dx 1at:'('"»m') (*20) whose particular case is the second-order differential equation obtained on p. the trajectory of the representative point is known as the phase trajectory. Qualitative Methods 155 2. The values of zz: and dx/dt at each moment in time characterize the state of the system and correspond to a point in the (:1:.Ch. The phase plane depicts the set of all possible states of the dynamical system considered. Then the differential equation (120) is the equation of motion of the particle. We take a simple dynamical system consisting of a particle of unit mass that moves along the :1: axis (Figure 49) and on which a force f (sz. dx/dt) acts. which is known as the plane of states or the phase (az.
156 Differential Equations in Applications da: f("'°>Ez/T) O J; Fig. 49 a' 1 B-? (J;) gl?)-plane 0 as Fig. 50 and the rate of motion of this point as the phase velocity. If we introduce the variable y = dx/dt, Eq. (120) can be reduced to a system of two differential equations: d gé- =y, =.f(¤>» y)· (121) If we take t as a parameter, then the solution to system (121) consists of two functions, :1: (t) and y (t), that in the phase (:1:, y)-plane define a curve (a phase trajectory).
Ch. 2. Qualitative Methods 157 It can be shown that system (121) and even a more general system %f—=X(r¤» y)» {1]%-·=Y(¤¤» y)· (122) where the functions X and Y and their partial derivatives are continuous in a domain D, posseses the property that if .1: (t) and y (t) constitute a'solution to the differential system, we can write w=w(¢+C),y=y(¢+C). (123) where C is an arbitrary real constant, and (123) also constitute a solution to the same differential system. All solutions (123) with different values of C correspond to a single phase trajectory in the phase (.1:, y)plane. Further, if two phase trajectories have at least one common point, they coincide. Here the increase or decrease in parameter t corresponds to a certain direction of motion of the representative point along the trajectory. In other words, a phase trajectory is a directed, or oriented, curve. When we are interested in the direction of the curve, we depict the direction of the representative point along the trajectory by placing a small arrow on the curve. Systems of the (122) type belong to the class of autonomous systems of differential equations, that is, systems of ordinary differential equations whose right-hand
158 Differential Equations in Applications sides do not explicitly depend on time t. But if at least in one of the equations of the system the right-hand side depends explicitly on time t, then such a system is said to be nonautonomous. In connection with this classification of differential equations the following remark is in order. If a solution .2: (t) to Eq. (120) is a nonconstant periodic solution, then the phase trajectory of the representative point in the phase (x, y)—plane is a simple closed curve, that is, a closed curve without self—intersections. The converse is also true. If differential systems of the (122) type are specified in the entire (x, y)—plane, then, generally speaking, phase trajectories will completely cover the phase plane without intersections. And if it so happens that X (xm yo) ;'— Y ($01 y0) Z 0 at a point Mo (xo, yo), the trajectory degenerates into a point. Such points are called singular. In what follows we consider primarily only isolated singular points. A singular point Mo (xo, yo) is said to be isolated if there exists a neighborhood of this point which contains no other singular points €X0€Pl> M0 (xm yo)From the viewpoint of a physical interpretation of Eq. (120), the point Mo (xo, 0) is a singular point. At this point y =0 and f (xo, 0) = 0. Thus, in this case the isolated singular point corresponds to the state of
Ch. 2. Qualitative Methods 159 a particle of unit mass in which both the speed dx/dt and the acceleration dy/di = dzsc/dig of the particle are simultaneously zero, which simply means that the particle is in the state of rest or in equilibrium. In view of this, singular points are also called points of rest or points of equilibrium. The equilibrium states of a physical system constitute very special states of the system. Hence, a study of the types of singular points occupies an important place in the theory of differential equations. The first to consider in detail the classification of singular points of differential systems of the (122) type was the distinguished Russian scientist Nikolai E. Zhukovsky (1847-1921). In his master's thesis "The kinematics of a liquid body", presented in 1876, this problem emerged in connection with the theory of velocities and accelerations of fluids. The modern names of various types of singular points were suggested by the great French mathematician Jules H. Poincaré (1854-1912). Now let us try to answer the question of the physical meaning that can be attached to phase trajectories and singular points of differential systems of the (122) type. For the sake of clarity we introduce a twodimensional vector field (Figure 51) defined by the function V (rv. y) = X (wl y)i + Y (rv, y)i.
160 Differential Equations in Applications P -1 X Y Fig. 51 where i and j are the unit vectors directed along the x and y axes, respectively, in a Cartesian system of coordinates. At every point P (x, y) the field has two components, the horizontal X (x, y) and the vertical Y (sc, y). Since dx/dt -—= X (x, y) and dy/dt: Y (as, y), the vector associated with each nonsingular point P (x, y) is tangent at this point to a phase trajectory. lf variable t is interpreted as time, vector V can be thought of as the vector of velocity of a representative point moving along a trajectory. Thus, we can assume that the entire phase plane is filled with representative points and that each phase trajectory constitutes the trace of a moving representative point. As a result we arrive at an analogy with the two-dimensional motion of an incompressible fluid. Here, since system (122) is autonomous, vector V
case are simply the trajectories of the moving particles of fluid and the singular points O. the aim of the qualitative theory of ordinary differential equations of the (122) type is to build a phase portrait as complete as possible directly from the functions X (x. y). the motion of the fluid is steady-state. These features constitute the main part of the phase portrait. y) is time-independent and. The phase trajectories in this. (2) the different patterns of phase trajectories near singular points. Qualitative Methods 1. y) and Y (sc. which in the given case correspondmto periodic motion.Ch. therefore. two possibilities may realize themselves: the particles that are in the vicinity of singular points remain there with the passage of time or they leave the vicinity for other parts of the plane). (3) the stability or instability at singular points (i. and (4) the presence of closed trajectories. and O" (see Figure 51) are those where the fluid is at rest. as noted earlier.61 at each fixed point P (sc. Since. The most characteristic features of the fluid motion shown in Figure 51 are (1) the presence of singular points. 11-0770
. O'. differential equations cannot generally be solved analytically. 2.e. or the complete qualitative behavior pattern of Qthe phase trajectories of a general—type system (122).
But if we study Earth's motion over several million years. For example. lf as is the displacement of the object (mass m) from the state of equilibrium and the force with which the two springs act on the object (the restoring force) is proportional to :1:. Systems of this kind are called conservative. can be assumed to hold true for such systems. A simple example of a conservative system is one consisting of an object moving horizontally in a vacuum under forces exerted by two springs (Figure 52). Differential Equations in Applications 2. that the sum of kinetic and potential energies remains constant. The law of energy conservation. namely. we must allow for energy dissipation related to tidal ilows of water in seas and oceans. rotating Earth may be seen as a conservative system if we take a time interval of several centuries. 1i>o.162.4 Conservative Systems in Mechanics Practice gives us ample examples of the fact that any real dynamical system dissipates energy. The dissipation usually occurs as a result of some form of friction. the equation of motion has the form m§-Q-1+kx=o. But in some cases it is so slow that it can be neglected if the system is studied over a fairly small time interval.
.
since the restoring force exerted by them is a linear function of x.Ch. the equation of motion for such a nonconservative system is m!£+¢i'€+kx=—. since the damping force is a linear function of velocity dx/dt. If f and g are such arbitrary functions that f(0) = O and g (O) = 0. Qualitative Methods 163 rn \ Fig. 2. If an object of mass m moves in a medium that exerts a drag on it (the damping force) proportional to the object's velocity. daz dx m'·(W·+§(·&T)+f(x)~0» (125) can be interpreted as the equation of motion of an object of mass m under a restoring force ——j (x) and a damping force —g (dx/dt).0 c>o (124) d¢2 dz ' ' Here we are dealing with linear damping. 52 Springs of this type are known as linear. the more general equation. Generally. these forces 11*
.
(129) Then. The result is 33 i—nzz/3---1.m 2: ·-S f(E) dE 2 • 2 'yf) . assuming that as = xo at t = to and y = yo. hence Eq. From Eq. we can integrate Eq. (128) This equation can be written as my dy = -j(x) dx. (126) we can pass to the autonomous system 3 __ gg _ _ f(¤¤) in "y' at — m · (127) If we now exclude time t from Eqs. (127)...7 xo
. (125) can be con— sidered the basic equation of nonlinear mechanics. Let us briefly examine the special case of a nonlinear conservative system described by the equation d2 mg? +1* (w) =0» (126) where the damping force is zero and. hence.164 Differential Equations in Applications are nonlinear. (129) from to to t. we arrive at a differential equation for the trajectory of the system in the phase plane: dy __ __ f (¤¤) —(E— my . energy is not dissipated..
Qualitative Methods 165 which may be rewritten as 11 —§—my2+S f(§) d§=*g·"'»!/3+S 1'(¤¤) dw00 (130) Note that my'!/2 := m(d:1:/dt)2/2 is the kinetic energy of a dynamical system and x V(w) = S f(E)d§ (131) 0 is the system's potential energy. while the straight lines sc = 0:. Eq. Thus... 0). (130) expresses the law of energy conservation: émz/2+V(¤¤) =E» (132) where E = my?/2 -}.2:0) is the total energy of the system. Clearly.Ch. The singular points of system (127) are the points M`. are the roots of the equation f (as) = 0. (::1. As noted earlier. different values of E correspond to diiierent curves of constant energy in the phase plane.V (. Thus. (126). where :1:. 2. are parallel
. (128).. (132) is the equation of the phase trajectories of system (127). Equation (128) implies that the phase trajectories of the system intersect the ac axis at right angles.. the singular points are points of equilibrium of the dynamical system described by Eq. Eq. since it is obtained by integrating Eq.
In addition. let us introduce the (. Note that since dx/dt = y. 72) —(E£+ksinx==0 (134) dia 3 '°
. with the z axis lying on the same vertical line as they axis of tl1e phase plane. (132) in the form y = 1]/% [E—V(¤v)l. The above reasoning is fairly general and makes it possible to investigate the equation of motion of a pendulum in a medium without drag. z)plane (one such straight line is depicted in Figure 53).2: axis. if we write Eq. We then plot the graph of the function z = V (1:) and several straight lines z = E in the (x.2:. This enables us to mark the respective values of y in the phase plane. which has the form (see p. Then for a definite sc we multiply E — V (x) by 2/m and allow for formula (133). We mark a value of E — V (:1:) on the graph. In this case. (133) we can easily plot the phase trajectories. Eq. Indeed. z)-plane. the plane of energy balance (Figure 53). (132) shows that the phase trajectories are symmetric with respect to the . the positive direction along any trajectory is determined by the motion of the representative point from left to right above the az axis and from right to left below the x axis.166 Differential Equations in Applications to the as axis.
. Zn. 0). (j rt. O). (134) is dd. —£—=—-ksinx.168 Differential Equations in Applications -—k sin :1:. 0 In the (x. In this case the autonomous system corresponding to Eq. . After determining a value of E —-— V (cv) we can draw a sketch
. O). z = E = 2k. (135) The singular points are (O. we arrive at an equation for the phase trajectories. is shown). This equation is a particular case of Eq. -—%. (132) with m = 1. .yz-1+k (1 ·—-cos ac) = E. z)—plane we plot the function z = V (x) as well as several straight lines z = E (in Figure 54 only one such line. where the potential energy determined by (131) is specified by the relationship 3 V(x) = S f(§)d§==k(1——cos:z:). —a—?=y. and the differential equation of the phase trajectories of system (135) assumes the form dy __ _ k sin x YE" `—§_' Separating the variables and integrating. (.|.
0). .. where rz = 1. belong to the vorteaypoint type. the corresponding phase trajectories prove to be closed and Eq. . == 1. As for the above example. On the other hand. while the singular points (jam.. if E >2lc. (134) has no periodic solutions. Finally. . m = 0. 2. is a separatrix. .170 Difierential Equations in Applications The resulting phase portrait shows (see Figure 54) that if the energy E varies from O to 2k. . 2:1tm. . (134) acquires periodic solutions. the value E = 2lc corresponds to a phase trajectory in the phase plane that separates two types of motion. 2. .. O). 1. rz. belong to the saddle-point type. .. while the closed trajectories lying inside the regions bou_nded by separatrices correspond to oscillations of the pendulum. 1. The wavy lines lying outside the separatrices correspond to rotations of a pendulum. With some we will get acquainted shortly. . 2. O). There are different types of singular points. m = O. the respective phase trajectories are not closed and Eq. that is. 0). A singular point of an autonomous differential system of the (122) type is said to be a vortex point if there exists a neighborhood of this point completely filled with nonintersecting
. the singular points (5. 2. Figure 54 shows that in the vicinity of the singular points (j2rcm. the behavior of the phase trajectories —differs from that of the phase trajectories in the vicinity of the singular points (—_l_—:rm.
while in the neighborhood of the singular points (j2¤tm. Now let us establish the effect of linear friction on the behavior of the phase trajectories of a conservative system.Ch.. A saddle point is a singular point adjoined by a finite number of phase trajectories ("whiskers") separating a neighborhood of the singular point into regions where the trajectories behave like a family of hyperbolas defined by the equation xy = const. 2. it can be shown that the phase trajectories are such as shown in Figure 55. 2. c>O. Qualitative Methods 171 phase trajectories surrounding the point. m = O. . the pendulum is able to oscillate about the position of equilibrium. the pattern of phase trajectories resembles the one depicted in Figure 56. that is. If we now compare the phase portrait of a conservative system with the last two portraits of nonconservative systems. we see that saddle points have not changed (we consider only small neighborhoods of singular points). 1. the closed phase trajectories have transformed into spirals (for low friction) or into trajectories that "enter" the singular points in certain directions (for high fric-
. . O). But if friction is so high that oscillations become impossible. The equation is %%—+c·—°gl§i—+ksinx=O. lf friction is low.
h gy 5
.
l
174 Differential Equations in Applications tion). In the first case (spirals) we have singular points of the focal-point type and in the second, of the nodal-point type. A singular point of a two-dimensional autonomous differential system of the general type (122) (if such a point exists) is said to be a focal point if there exists a neighborhood of this point that is completely filled with nonintersecting phase trajectories resembling spirals that "wind" onto the singular point either as t-» +<x> or as t-> -0o. A nodal point is a singular point in whose neighborhood each phase trajectory behaves like a branch of a parabola or a half-line adjoining the point along a certain direction. Note that if a conservative system has a periodic solution, the solution cannot be isolated. More than that, if l` is a closed phase trajectory corresponding to a periodic solution of the conservative system, there exists a certain neighborhood of l` that is completely filled with closed phase trajectories. Note, in addition, that the above definitions of types of singular points have a purely qualitative, descriptive nature. As for the analytical features of these types, there are no criteria, unfortunately, in the general case of systems of the (122) types, but for some classes of differential equations such criteria can be formulated.
Ch.· 2. Qualitative Methods 175 A simple example is the linear system i§%':a1$+b1!/» 'g'%='a2$+bzy• where al, bl, az, and bz are real constants. If the coefficient matrix of this sytem is nonsingular, that is, the determinant at b1` 07 Iaz bz lf the origin O (O, O) of the phase plane is the only singular point of the differential system. Assuming the last inequality valid, we denote the eigenvalues of the coefficient matrix by X1 and X2. It can then be demonstrated that (1) if K1 and K2 are real and of the same sign, the singular point is a nodal point, (2) if X1 and K2 are real and of opposite sign, the singular point is a saddle point, (3) if X], and A2 are not real and are not pure imaginary, the singular point is a focal point, and (4) if kl and K2 are pure imaginary, the singular point is a vortex point. Note that the first three types of singular points belong to the so—called coarse singular points, that is, singular points whose nature is not affected by small perturbations of the right—hand sides of the initial differential system. On the other hand,
176 Differential Equations in Applications a vortex point is a fine singular point; its nature changes even under small perturbations of the right-hand sides. 2.5 Stability of Equilibrium Points and of Periodic Motion As we already know, singular points of different types are characterized by different patterns of the phase trajectories in sufficiently small neighborhoods of these points. There is also another characteristic, the stability of an equilibrium point, which provides additional information on the behavior of phase trajectories in the neighborhood of singular points. Consider the pendulum depicted in Figure 57. Two states of equilibrium are shown: (a) an object of mass m is in a state of equilibrium at the uppermost point, and (b) the same object is in a state of equilibrium at its lowest point. The first state is unstable and the second, stable. And this is why. If the object is in its uppermost state of equilibrium, a slight push is enough to start it moving with an ever increasing speed away from the equilibrium position and, hence, away from the initial position. But if the object is in the lowest possible state, a push makes it move away from the position of equilibrium with a decreasing speed, and the weaker the push the smaller the distance by which the
Ch. 2. Qualitative Methods {77 I I I I I I I Q; Fig. 57 object is displaced from the initial position. The state of equilibrium of a physical system corresponds to a singular point in the phase plane. Small perturbations at an unstable point of equilibrium lead to large displacements from this point, while at a stable point of equilibrium small perturbations lead to small displacements. Starting from these pictorial ideas, let us consider an isolated singular point of system (122), assuming for the sake of simplicity that the point is at the origin O (O, O) of the phase plane. We will say that this singular point is stable if for every positive R there exists a positive rg}? such that 12-0770
8-9 (in Russian).G. and all will prove that real movements require selecting out of the possible solutions of the equations of motion only those that correspond to stable states. Numerous examples can be added to this list. Chetaev (1902-1959) wrote: * .180 Differential Equations in Applications Nikolai G. The crankshaft must be so designed that it does not break from the vibrations that appear in real conditions of motor operation.
.. Chetaev. it is advisable to change the design of the corresponding device in such a way that the state of motion corresponding to this solution becomes unstable. the gun and the projectiles must be constructed in such a manner that the trajectories of projectile flight are stable and the projectiles fly correctly. Stability of Motion (Moscow: Nauka. if we wish to avoid a certain solution..If a passenger plane is being designed. Returning to the pendulum depicted in Figure 57. 1965: pp. we note the following curious and somewhat unexpected fact. To ensure that an artillery gun has the highest possible accuracy of aim and the smallest possible spread. Moreover. a certain degree of stability must be provided for in the future movements of the plane so that it will be stable in {light and accidentfree during take-off and landing. Research has shown that the upper (unstable) position of equilibrium can be made stable by * See N.
1981) (in Russian). * Now let us turn to a concept no less important than the stability of an equilibrium point. Qualitative Methods 181 introducing vertical oscillations of the point of suspension. 2.G. More than that. Strizhak. Thus. the concept of stability of periodic movements (solutions). In other words. Methods of Investigating Dynamical Systems of the "Pendulum" Type (Alma-Ata: Nauka.Ch. In the phase plane these solutions are represented by closed trajectories that completely fill a certain region. at the x axis) will move apart to a certain finite distance * The reader can find many cxancplts cf stabilization of different 15 pcs of pcndulums in the book by T. Let us assume that we are studying a conservative system that has periodic solutions. the period of oscillations in a conservative system depends on the initial data. a horizontal position) by properly vibrating the point of suspension. Generally. the period of traversal of different trajectories by representative points is different. to each periodic motion of a conservative system there corresponds a motion of the representative point along a fixed closed trajectory in the phase plane. Geometrically this means that two closely spaced representative points that begin moving at a certain moment t = to (say. not only the upper (vertical) position of equilibrium can be made stable but also any other of the pendulum's positions (for one.
.
y2 = r2 will lie inside the circle :::2 + y2 = R2 for all t > to (Figure 58). if a singular point is not stable. it is said to be unstable. a singular point is said to be asymptotically stable if it is stable and if there exists a circle x2 —I— y2 = rf.178 Differential Equations in Applications . Without adhering to rigorous reasoning we can say that a singular point is stable if all phase trajectories that are near the point initially remain there with the passage of time.
. Also. Finally.9 V WI W `J Fig. such that each trajectory that at time t = to lies inside the circle converges to the origin as t-> —{—oo. 58 every phase trajectory originating at the initial moment t = to at a point P lying inside the circle :1:2 —i.
O). no conditions are imposed on how the phase trajectory must approach point O (O. the well-known Soviet specialist in the field of mathematics and mechanics 12•
. the singular points. which illustrates the behavior of the phase trajectories for pendulum oscillations in a medium with low drag. 2. The fact is that if a device is designed without due regard for stability considerations. The introduced concept of stability of an equilibrium point is purely qualitative. too. Qualitative Methods 179 A singular point of the vortex type is always stable (but not asymptotically). in Figure 56 the singular points. it is additionally necessary that every phase trajectory tend to the origin with the passage of time.Ch. focal points. As for the concept of asymptotic stability. which are nodal points. in this case. The concepts of stability and asymptotic stability play an important role in applications. Emphasizing the importance of the concept of stability. if compared with the notion of simple stability. A saddle point is always unstable. since no mention of properties referring to the behavior of phase trajectories has been made. are also asymptotically stable. However. are asymptotically stable. which in the iinal analysis may lead to extremely unpleasant consequences. In Figure 55. when built it will be sensitive to the very smallest external perturbations.
59 with the passage of time. If knowing an a-neighborhood (with a as small as desired) of a point M moving along a closed trajectory I` (Figure 59) * ensures that we: know a moving 6 (s)-neighborhood of the same point M such that every representative point that initially lies in the 6 (s)-neighborhood will never leave the e-neighborhood with the passage of time. But it also may happen that these points do not separate.
. The essence of this concept lies in the following. To distinguish between these two possibilities. the concept of stability in the sense of Lyapunov is introduced for periodic solutions.182 Differential Equations in Applications y8 (:*2 P5 8 Y} g l·`ig. then the periodic solution * An s—neighborhood of a point M is understood to be a disk of radius 6 centered at point M.
for instance. which means that under small variations of initial data the representative point transfers from one phase trajectory to another lying as close as desired to the initially considered trajectory. orbital stability. in terms of an elliptic integral of the fnrst kind taken from 0 to at/2. Qualitative Methods 183 corresponding to l` is said to be stable in the sense of Lyapunov. In the second the oscillation period depends on the initial data and is expressed. 2. when we consider the differential equation that describes the horizontal movements of an object of mass ni in a vacuum with two linear springs acting on the object (Figure 52). An example of periodic solutions that are unstable in the sense of Lyapunov but are orbitally stable is the solutions of the differential equation (134).Ch. Examples of periodic solutions that are stable in the sense of Lyapunov are those that emerge. we must bear in mind that they still possess some sort of stability.
. it is said to be unstable in the sense of Lyapunov. as we know. which describes the motion of a circular pendulum in a medium without drag. In the first case the oscillation period does not depend on the initial data and is found by using the formula T = 2nl/m/lc. If a periodic solution is not stable in the sense of Lyapunov. When it comes to periodic solutions that are unstable in the sense of Lyapunov.
Suppose that l` is a phase trajectory of system (122). Rouche. V. we note that the question of whether periodic movements are stable in the sense of Lyapunov is directly linked to the question of isochronous vibrations. Lukashevich. Press. Amel'kin. This method is known as Lyapunov's direct. for example. and M. Stability Theory by Lyapunov`s Direct Method (New York: Springer. and A. A. * 2. This idea lies at the base of one of two methods used in studying stability problems.P.
. P. Nonlinear Vibrations in Second-Order Systems (Minsk: Belorussian Univ. N. Lyapunov (1857-1918).6 Lyapunov Functions Intuitively it is clear that if the total energy of a physical system is at its minimum at a point of equilibrium. 1982) (in Russian). 1977). method for stability investigations. ** The reader will End many interesting examples of stability investigations involving differential models in the book by N. the point is one of stable equilibrium. ** We illustrate Lyapunov's direct method using the (122) type of system when the origin is a singular point. the book by V. y) that is continuous together with its first partial derivatives 0V/6x and 6V/6y * See. Habets. Laloy. We consider a function V = V (x. Sadovskii. both suggested by the famous Russian mathematician Aleksandr M.184 Differential Equations in Applications Finally. or second.
2:. 2. The following concepts are important for the practical application of the method. y) are the righthand sides of system (122). For example. But if at points of G we have V (sc. y) is continuous together with its first partial derivatives 6V/Ox and 0V/dy in a domain G containing the origin in the phase plane. then on this curve the function V (x. Formula (136) is essential in the realization of Lyapunov's direct method. y) is said to be nonnegative (nonpositive). Suppose that V = V(x. y) varies along l` is given by the formula dV GV dx OV dy _ OV ·g·. y)-plane is positive definite.='g jg'l"'5. y)/> 0 (Q O). y) and Y (sc.
. y (1:)) moves along curve I`. If the representative point (sc (t).Ch. O) = O. with V (0. with the result that the rate at which V (:1:. Qualitative Methods 185 in a domain containing I` in the phase plane.. while the function V (as. y) = $2 is nonnegative since it vanishes on the entire y axis. y) may be considered a function of t. This function is said to be positive (negative) definite if at all points of G except the origin V (sc.· §—·gX($» H) where X (. the function V defined by the formula V (sc. y) = x2 + y2 and considered in the (az. the function V = V (1:. y) it is positive (negative).
One such surface is shown in Figure 61. where the section of the surface with the plane z = C results not in a curve but in a ring.%x<x. we can require that at all points of the domain G\O the following inequality hold true: OV 2 0V 2 (H) HE) * °· This means that the equation z = V (x. y) does not increase along the trajectory I` in the neighborhood of the origin. y) is such that Wo. the function V = V (zz.186 Differential Equations in Applications If V (x. which is
. then the origin. We note here that in view of (136) the requirement that W be nonpositive means that dV/dtg 0 and. hence. lf a positive definite function V (0:. y) is positive definite. y>=2—'-§. y) with a positive definite V may specify a surface of a more complex structure. y). V is said to be the Lyapunov function of system (122).0) (Figure 60). Generally.y)-plane at point O (0. the equation z = V (ay. Here is a result arrived at by Lyapunov: if for system (122) there exists a Lyapunov function V (x. y> +i'i§-ii Y as y> we is nonpositive. y) can be interpreted as the equation of a surface resembling a paraboloid that touches the (x.
60 z. y) is such that function Wdefined via (137) is negative definite. then the origin is asymptotically stable. If the positive definite function V = V (sc.Ch. 61 a singular point. is stable. W lI\\& y Fig. Qualitative Methods {87 z cv 0 5/ Fig. We will shew with an example how to
. 2.
The kinetic energy of the object (of unit mass) is y2/2 and the potential energy (i.188 DiHerential Equations in Applications apply the above result. which in view of (124) can be written in the form -(3.+clg-?+kx=0. y)-plane is the only singular point. (138) The reader will recall that in this equation c > 0 characterizes the drag of the medium in which the object moves and k > 0 characterizes the properties of the spring (the spring constant). the energy stored by the spring) is S kg dg Z TQ. 0 This implies that the total energy of the system is VMS.e.azz. (138) has the form -3-?=y. c>0. $4:-—kx-cy.(140) It is easy to See that V specified by (140)
. The autonomous system corresponding to Eq. y)=% y2+é— M2.. Let us consider the equation of motion of an object of unit mass under a force exerted by a spring.. (139) For this system the origin of the phase (x.
2. Lyapunov Functions (Moscow: Nauka. Many studies have been devoted to this problem and a number of interesting results have been obtained in recent yea1·s.g-Y<¤¤» y) =/~¤¤¤y+y(—k¤>-vy>= —cy2<0. This is not always the case. Qualitative Methods 189 is a positive definite function.
. 1970) (in Russian). And since in the given case 0 -g-§X(¢» y>+. The formulated Lyapunov criterion is purely qualitative and this does not provide a procedure for Ending the Lyapunov function even if we know that such a function exists.A. Barbashin. O) is stable. however.7 Simple States of Equilibrium The dynamical interpretation of secondorder differential equations already implies " The interested reader can refer to the book by E.Ch. This makes it much more difficult to determine whether a concrete system is stable or not. this function is the Lyapunov function of system (139).* 2. In the above example the result was obtained fairly quickly. which means that the singular point O (O.. The reader must bear in mind that the above criterion of Lyapunov must be seen as a device for finding effective indications of equilibrium.
then the type of the singular point. It is also clear that. which in the given
. We need criteria that will enable us to determine the type of a singular point from the form of the initial differential equation. differential equations cannot be integrated in closed form. It so happens that the simplest case in establishing the type of a singular point is when the Jacobian or functional determinant 6X 6X T9? 75] J ("'* y): ay ar "`6T T9? is nonzero at the point. using the example of an object of unit mass subjected to the action of linear springs and moving in a medium with linear drag.190 Differential Equations in Applications that investigation of the nature of equilibrium states or. generally speaking. But first let us discuss a system of the (122) type. the singular points provides a key for establishing the behavior of integral curves. y*) ak 0. how some results of the qualitative theory of differential equations can be used to this end. as a rule it is extremely difficult to {ind such criteria. Unfortunately. but it is possible to isolate certain classes of differential equations for which this can be done fairly easily. which is the same. y*) is a singular point of system (122) and if J* = J (x*. Below we show. If (x* .
If the phase tra-
. or a focal point. But if functions X and Y are linear in variables as and y. y*) is a saddle point. And.Ch.2:*. y*) = O. only if D* = D (. generally. Note. the singular point (x*. a sufficiently small neighborhood of this point is filled with trajectories that either spiral into this point or converge to it in certain directions. there can be an infinite number of such conditions. the condition D* = 0 becomes sufficient for the singular point (. If J * > 0 but the singular point (42:*. For instance. that the condition D* = 0 is generally insufficient for the singular point (x*. the singular point may be a vortex point.z:*. Qualitative Methods 191 case is called a simple singular point. y*) is not a vortex point. For a vortex point to be present certain additional conditions must be met. if J * is negative. 2. and if . depends largely on the sign of constant J *. y*) to be a vortex point. y) =q.-7 vanishes at the singular point. the singular point is reached as i—> --o¤ and is unstable while if D* < 0. The singular point may be a vortex point only if the divergence 6X GY D(w. the singular point is reached as i-> +¤o and proves to be stable. however.+—g. conditions that include higher partial derivatives. Here. y*) to be a vortex point. if D* > 0.I`* is positive. a nodal point. that is.
6X (=¤*» y*) ~ 6X (=¤*» y*) ~ ° ( 0:1: w+ 6y y where . y*) ~ 6Y(¤>*.7=y—y*.· of what is known as exceptional directions. y*) can be found from what is known as the characteristic equation 0Y(=¤*.}* == D*2 — 4. .192 Differential Equations in Applications jectories that reach the singular point are spirals. =. the partial derivatives in Eq. but if the integral curves converge to a singular point along a certain tangent. (142) If X and Y contain linear terms. the tangents to the trajectories of the differential system (122) at a singular point (x*. the point is a nodal point (Figure 62).10*. if we introduce the slope 2.) A.}*.}*. Equation (141) is homogeneous. = O. —l— Y..)2 — 4.-:. Irrespective of the sign of Jacobian . (143) The discriminant of this equation is A = (X. we have the following quadratic equation for finding K: XQK2 + (X. y*)~ tg.::-. (141) act as coefficients of x and y in the system obtained from system (122) after introducing the substitution (142). Bx x + Oy y —— 141) Q. — Y. we are dealing with a focal point.
.. —— Y. Hence.
For one thing. (143) is not an identity. we have only one exceptional straight line. while the neighborhood of the singular point is divided into two sectors.194 Differential lriquations in Applications Hence. if there are two real exceptional directions. The existence of real exceptional directions means (provided that J * is positive) that there is a singular point of the nodal type. one of which is completely filled with trajectories that "enter" the singular point and touch ll and the other is completely filled with trajectories that "enter" the singular point and touch Z2. In the iirst case the singular point is either a vortex point or a focal point. or there is one 2—fold direction. (143) always gives two real exceptional directions. The boundary between the sectors
. The singular point divides the exceptional straight line into two half—lines. or there are two. ll and Z2. there are no real exceptional directions. Eq. if J* <O. it can be proved that there are exactly two trajectories (one on each side) whose tangent at the singular point is one of the exceptional straight lines (directions) while all the other trajectories "enter" the singular point touching the other exceptional straight line (Figure 62a). If A = O and Eq. which corresponds to a saddle point. But if J * > 0. It can be obtained from the previous case when the two exceptional directions coincide. The pattern of the trajectories for this case is illustrated by Figure 62b.
one of which touches I1 at the singular point and the other touches L2 at this point. 2. and then all the straight lines passing through the singular point are exceptional and there are exactly two trajectories (one on each side) that touch each of these straight lines at the singular point. the differential equation describing the motion of a unitmass object under the action of linear springs in a medium with linear drag has the form 2 . Qualitative Methods 195 consists of two trajectories. with Eq.8 Motion of a Unit—Mass Object Under the Action of Linear Springs in a Medium with Linear Drag As demonstrated earlier. (145) 13*
. (144) we can associate an autonomous system of the form -£§—=y.'}E.i+c%§+kx=o. lf in Eq. we arrive at an identity. -%=·—kx—cy. (144) So as not to restrict the differential model (144) to particular cases we will not tix the directions in which the forces ——c (dx/dt) and —-kx act.Ch. (143) all coeflicients vanish. As shown earlier. 2. This point (Figure 62c) is similar to the point with one 2-fold real exceptional direction.
196 Differential Equations in Applications If we now exclude the trivial case with lc = O, which we assume is true for the meantime, the differential system (145) has an isolated singular point at the origin. System (145) is a particular case of the general system (122). In our concrete example the Jacobian J (:1:, y) = lc, and the divergence D (xr, y) : ——c. The characteristic equation assumes the form X2-{-cli.-|-k=0, where A =· cg — 4k is the discriminant of this equation. In accordance with the results obtained in Section 2.7 we arrive at the following cases. (1) lf lc is negative, the singular point is a saddle point with one positive and one negative exceptional direction. The phase trajectory pattern is illustrated in Figure 63, where we can distinguish between three different types of motion. When the initial conditions correspond to point a, at which the velocity vector is directed to the origin and the velocity is sufficiently great, the representative point moves along a trajectory toward the singular point at a decreasing speed; after passing the origin the representative point moves away from it at an increasing speed. If the initial velocity decreases to a critical value, which
Ch. 2. Qualitative Methods 197 y=.i: 0 E` ·¤ `C a Fig. 63 corresponds to point b, the representative point approaches the singular point at a decreasing speed and "reaches" the origin in an infinitely long time interval. Finally, if the initial velocity is lower than the critical value and corresponds, say, to point C, the representative point approaches the origin at a decreasing speed, which vanishes at a certain distance xl from the origin. At point (xl, O) the velocity vector reverses its direction and the representative point moves away from the origin. If the phase point corresponding to the initial state of the dynamical system lies in either one of the other three quadrants, the interpretation of the motion is obvious.
198 Differential Equations in Applications yar: x Fig. 64 fj Fig. 65 (2) If lc > O, then J * is positive and the type of singular point depends on the value of c. This leaves us with the following possibilities: (2a) lf c = O, that is, drag is nil, the singular point is a vortex point (Figure 64). The movements are periodic and their amplitude depends on the initial conditions.
Ch. 2. Qualitative Methods 199 (2b) If c > O, that is, damping is positive, the divergence D : OX/0.2; + GY/fiy is negative and, hence, the representative point moves along a trajectory toward the origi11 and reaches it in an infinitely long time interval. More precisely: (2b,) If A <O, that is, cg < 4k, the sin-gular point proves to be a focal point (Figure 65) and, hence, the dynamical system performs damped oscillations about the state of equilibrium with a decaying amplitude. (2b2) If A :0, that is, c2 :4k, the singular point is a nodal point with a single negative exceptional direction (Figure 66). The motion in this case is aperiodic and corresponds to the so—called critical damping. (2b,,) If A > O, that is, cg > 4k, the singular point is a nodal point with two negative exceptional directions (Figure 67). Qualitatively the motion of the dynamical system is the same as in the previous case and corresponds to damped oscillations. From the above results it follows that when c >O and k>O, that is, drag is positive and the restoring force is attractive, the dynamical system tends to a state of equilibrium and its motion is stable. (2c) lf c < O, that is, damping is negative, the qualitative pattern of the phase trajectories is the same as in the case (2b), the only difference being that here the
These cases belong to those of complex singular points. (144) assumes the form dx dy W z ya `(T? = ""cy• This implies that the straight line y = 0 is densely populated by singular points. (144) assumes the form dzx az? *-0The respective autonomous system is da: dy
. the fact that c = O does not generally mean that system (122) possesses a vortex point. if k = c = 0. and the fact that k = 0 does not mean that the system of a general type has no singular point. the autonomous system (145) corresponding to Eq. we note that if k = 0 (c qé O). Eq. Finally. which we consider below. Returning to the dynamical system considered in this section. with the phase trajectory pattern shown in Figure 69. However. Note that the diagrams can also be interpreted as a summary of the results of studies of the types of singular points of system (122) when J* #0 at c = —D* and lc = J*.202 Diflerential Equations in Applications point on the values of parameters c and k.
it also influences the operation of steam and gas turbines. For one thing. as in the previous case. the specific heat capacity of the gas is cp and the nozzle's— varying cross-sectional area is denoted by A.Ch. that is. The respective phase trajectory pattern is shown in Figure 70.1: axis is densely populated by singular points. such flow emerges in the vicinity of a wing and fuselage of an airplane.9 Adiabatic Flow of a Perfect Gas Through a Noazle of Varying Cross Section A study of the flow of compressible viscous media is highly important from the practical viewpoint. 2. all its properties are assumed to be uniform in a single cross section of the nozzle. the . Below we discuss the flow of. (146) where q is the friction coefficient depend_ing basically on the Reynolds number but assumed constant along the nozzle. Qualitative Methods 203 Here. Friction in the boundary layer is caused by the tangential stress 1: given by the formula t = qpvz/2. a perfect gas through a nozzle with a varying cross section (Figure 7-1). the nuclear reactors. 2. jet engines. The flow is interpreted as one-dimensional. p the
.
drag. and v the flow velocity. One of the basic equations describing this type of flow is the well-known conti-
. combustion.204 Differential Equations in Applications _. and condensation are excluded. chemical transformations. we assume that adiabaticity conditions are met. 71 flux density. evaporation. that is.z· Fig.5* . Finally. 70 as / Fig./-.
we can write the equation for the flow energy as c. (149) where h is the enthalpy of the flow (the thermodynamic potential) at absolute temperature T. From this equation it follows that dp dA dv _ 7+7-}--7-0. 2. or dh —i— d (02/2) : O. In our case the flow is adiabatic. (149) dh = c. We note that generally such an equation links the external work done on the system and the action of external heat sources with the increase in enthalpy (heat content) in the flow and the kinetic and potential energies. therefore. (150)
..dT and. which in the given cse is written in the form w = pA v.wvz/2. the energybalance equation can be written in the form 0=w(h-l—dh)—-—wh —{— w [v2/2 + d (122/2)] ——. (148) Let us now turn to the equation for the energy of steady—state flow.Ch.dT + d (v2/2) = O. Qualitative Methods 205 nuity equation. But in Eq.. (147) where the flux variation rate w is assumed constant. hence.
1: dA = w dv. we can always assume that the momentum equation for the flow has the form -—A dp -—. The term dA dp in Eq.206 Differential Equations in Applications Now let us derive the momentum equation for the flow. we can write the momentum equation in the form pA+pdA—(p+dp)(A-l—dA)—1:dA = w dv. (153) Noting that pvz/2 = ·ypM2. (152) If we denote by D the hydraulic diameter. (151) where p is the static pressure. where x is the coordinate along the nozzle's axis. we note that its variation along the nozzle's axis is determined by a function F such that D = F (:1:). where Y is the specific heat ratio of the medium. and M
. From the definition of the hydraulic diameter it follows that dA 4da: -1-I-zi . Assuming that the divergence angle of the nozzle wall is small. Note that here the common approach to problems involving a steadystate flow is to use Newton's second law. (151) is of a higher order than the other terms and. therefore. or —A dp —-— dA dp — 1 dA = w dv.
(155) Denoting the square of the Mach number by y. when the Mach number becomes equal to unity.. (148). including compressibility and viscosity elfects. (150).+ YM-'*(4q—j—-|——. and (155). Kestin and S.-. Qualitative Methods 207 is the Mach number. This means that the integral curves of the last equation intersect what is known as the sonic line and have vertical tangents at the intersection points. 292: 172-175. Flow patterns derived for the flow of gases through nozzles.* The denominator of the right-hand side of Eq. 25. we arrive at the following differential equation: .. (154) Combining this with Eqs.
. we arrive at the following represen— tation for the momentum equation (152): dp da: dv? __ -1. 179 (1953)." Aircraft Engm.-.. Since the right-hand * See J. that is. employing Eqs..Ch. we can write formula (146) thus: 1 := qypM2. (156) vanishes at y = 1.)-O.(156) dx (i-·y) F (¤¤) ' where the prime stands for derivative of the respective quantity. (147) and (153).K. "One-dimensional high—speed flows. No. 2. and performing the necessary algebraic transformations. Zaremba.1 I @(1+-L——y) (wry-F (=•¤>> -'k'-:-—-£-.
Hence. that is. on the other hand. (156) are specified by the equations y* :1. appears only if F" (x*) is negative. the integral curves "ilip over" and the possibility of inflection points is excluded. provided that the proiile contains an inflection point. at a certain distance from the throat of the nozzle. that is. F"(:1:*) > O. Thus. Since q is a sufficiently small constant. through a saddle point or a nodal point. the transition from subsonic flow to supersonic (and back) can occur inside the nozzle only through a singular point with real exceptional directions. (156) reverses its sign in the process of intersection. a nodal point emerges in the part of the nozzle that lies behind an inflection point of the nozzle's profile or. a saddle point appears nea1· the throat of the nozzle. The coordinates of the singular points of Eq. the section on which the integral curves intersect the sonic line with vertical tangents must be the exit section of the nozzle.208 Differential Equations in Applications side of Eq.
. A nodal point. in practical terms. Thus. The physical meaning of this phenomenon implies that along integral curves the value of a: must increase continuously. A saddle point appears if J* is negative. F'(¤¤*> = v<1» which imply that these points are situated in the divergiug part of the nozzle.
(156) cannot be integrated in closed form. The case of a nodal point (Figure 73) allows for a continuous transition only from supersonic to subsonic flow. we must employ numerical methods of integration in any further discussion. Since Eq. which lie on different sides of a separatrix. 2. l). Such a construction is indeed possible since the characteristic equation provides us with the direction of the two tangents at the singular point S (:1:*. hence. It is advisable in this connection to begin the construction of the four separatrices of a saddle point as integral curves by allowing for the fact that the singular point itself is a point. Qualitative Methods 209 From the characteristic equation F(¤¢*) K2 + 2qv (V +1)% —-2(Y +i)F"(¢*) =0 we can see that the slopes of the two exceptional directions are opposite in sign in the case of a saddle point and have the same sign (are negative) in the case of a nodal point. provide no in14-0770
. so to say. This means that only a saddle point allows for a transition from supersonic to subsonic velocities and from subsonic to supersonic velocities (Figure 72).Ch. from which these integral curves emerge. the corresponding points move along curves on and [5 that strongly diverge and. lf this is ignored and the motion monitored as beginning at points a and b in Figure 72.
(156) vanishes.Ch. if we exclude the possibility of equilibrium points of the vortex and focal types emerging in this picture. which points to the presence of extrema. the error may be minimized if we allow for the convergence of integral curves in the direction in which the values of sc diminish. if we move along an integral curve that "emerges" from point S and we assume that the initial segment of the curve coincides with a segment of an exceptional straight line. Qualitative Methods 211 other hand. Figure 72 illustrates the pattern of integral curves in the vicinity of a singular point. then it appears that the neighborhood of a singular point can be broken down into 14*
.10 Higher-Order Points of Equilibrium In previous sections we studied the types of singular points that emerge when the Jacobian J * is nonzero. 2. Then in the vicinity of a singular point there may be an iniinitude of phase—trajectory patterns. But suppose that all the partial derivatives of the functions X and Y in the right—hand sides of system (122) vanish up to the nth order inclusive. The straight line passing through point xt (the throat of the nozzle) corresponds to values at which the numerator in the right-hand side of Eq. 2. However.
but first let us make several assumptions to simplify matters.212 Differential Equations in Applications a finite number of sectors belonging to three standard types. and Y. We assume that the origin is shifted to the singular point. :1:* = y* = O. that is. In addition. y) 'I'x(¤¤. y) (x2__'_y2)(7l+1)/2 ' (x2. Below we describe these sectors in detail.. y) Wy (rv. the right-hand sides of system (122) can be written in the form X (¤v» y) = X n (rv. we assume that the functions <D(=·=» y) (Dx (¢» y) (Dy (=¤. y) + 'I' (rn y>» where X . Under these assumptions the following assertions hold true. are polynomials of degree n homogeneous in variables as and y (one of these polynomials may be identically zero). and the functions (D and W have in the neighborhood of the origin continuous iirst partial derivatives. (1) Every trajectory of the system of equations (122) with right-hand sides of the (157) form that "enters" the origin along a certain tangent touches one of the ex-
. These are the hyperbolic. y) (x2_)_y2)('Tl+1)/2 ' (x2_·_y2)71/P ' are bounded in the neighborhood of the origin. parabolic._|_y2)7l·/2 ' 1 'I'<=·=. y) + (D (re y)» (157) Y (rm y) = Ya (w.. and elliptic.
Ch. 2. Qualitative Methods 213 ceptional straight lines specified by the equation wYn (an y) —— !/Xu (fe y) = 0- (158) Since the functions X ,, and Y,, are homogeneous, we can rewrite Eq. (158) as an equation for the slope A = y/x. Then the exceptional straight line is said to be singular if Xa (vs y) = Yum y) =0 on this straight line. Some examples of such straight lines are shown in Figure 62. The straight lines defined by Eq. (158) but not singular are said to be regular. (2) The pattern of the phase trajectories of (122) in the vicinity of one of the two rays that "emerge" from the origin and together form an exceptional straight line can be studied by considering a small disk (centered at the origin) from which we select a sector limited by two radii lying sufficiently close to the ray on both sides of it. Such a sector is commonly known as a standard domain. More than that, in the case of a regular exceptional straight line, which corresponds to a linear factor in Eq. (158), the standard domain considered in a disk of a sufficiently small radius belongs t.o either one of two types: attractive or repulsive. (2a) The attractive standard domain is characterized by the fact that each tra-
214 Diflerential Equations in Applications Fig. 74 Fig. 75 jectory passing through it reaches the origin along the tangent that coincides with the exceptional straight line (Figure 74). (2b) The repulsive standard domain is characterized by the fact that only one phase trajectory passing through it reaches the singular point along the tangent that coincides with the exceptional straight line. All ot.her phase trajectories of (122) that enter the standard domain through the boundary of the disk leave the disk by crossing one of the radii that limit the domain (Figure 75).
Ch. 2. Qualitative Methods 215 Let us turn our attention to the following fact. If the disk centered at the origin is small enough, the two types of standard domains can be classified according to the behavior of the vector (X, Y) on the boundary of the domain. '[`he behavior of vector (X, Y) can be identified here with the behavior of vector (X ny Yn). More than that, it can be demonstrated that if, as assumed, a fixed exceptional direction does not correspond to a multiple root of the characteristic equation, the vector considered on one of the radii that limit the domain is directed either inward or outward. Then if in the first case the vector considered on the part of the boundary of the standard domain that is the arc of the circle is also directed inward, and in the second case outward, the standard domain is attractive. But if the opposite situation is true, the standard domain is repulsive. It must be noted that in any case the vector considered on the part of the boundary that is the arc of the circle is always directed either inward or outward since it is almost parallel to the radius. Standard domains corresponding to singular exceptional directions or multiple roots of the characteristic equation have a more complicated nature, but since to some extent they constitute a highly rare phenomenon, we will not describe them here. lf we now turn, for example, to a saddle
216 Differential Equations in Applications point, we note that it allows for four repulsive standard domains. In the neighborhood of a singular point of the nodal type there are two attractive standard domains and two repulsive. (3) If there exist real-valued exceptionaldirection straight lines, the neighborhood of a singular point can be divided into a finite number of sectors each of which is bounded by the two phase trajectories of (122) that "enter" the origin along definite tangents. Each of such sectors belongs to one of the following three types. (3a) The elliptic sector (Figure 76) contains an infinitude of phase trajectories in the form of loops passing through the origin and touching on each side of the boundary of the sector. (3b) The parabolic sector (Figure 77) is filled with phase trajectories that connect the singular point with the boundary of the neighborhood. (3c) The hyperbolic sector (Figure 78) is filled with phase trajectories that approach the boundary of the neighborhood in both directions. More precisely: (Zia) Elliptic sectors are formed between two phase trajectories belonging to two successive standard domains, both of which are attractive. (4b) Parabolic sectors are formed between two phase trajectories belonging to" two
Ch. 2. Qualitative Methods gn Fig. 76 Fig. 77 Fig. 78 successive standard domains one of which is attractive and the other repulsive. All phase trajectories that pass through the latter domain touch at the singular point of the exceptional straight line that defines the attractive domain.
where the Jacobian J * is nonzero.11 Inversion with Respect to a Circle and Homogeneous Coordinates Above we described methods for establishing the local behavior of phase trajectories of differential systems of the (122) type in the neighborhood of singular points. in the book by V.V Ame1'kin.P.218 Differential Equations in Applications (4c) Hyperbolic sectors are formed between two phase trajectories belonging to two successive repulsive standard domains. For example-. Lukashevich.* 2. Press. and A. 1982) (in Russian . Nonlinear Vibrations in Second—0rder Systems (Minsk: Belorussian Univ. for examp e. it is easy to distinguish the four hyperbolic sectors at a saddle point and the four parabolic sectors at a nodal point. lf a singular point does not allow for the existence of real exceptional directions.
.A. N. And although in many cases all required information can be extracted by following these methods. Sadovskii. there may be a need to study the " Methods that make it possible to distinguish between a vortex {Joint and a focal point are discussed. the phase trajectories in its neighborhood always possess the vortex or focal structure. Elliptic sectors do not appear in the case of simple singular points.
Qualitative Methods 219 trajectory behavior in infinitely distant parts of the phase plane. __ n {L.§··_F§i·. Note that in the majority of
. y) of the phase plane into point M ' (§. the slopes of asymptotic directions are the slopes of tangents at the new origin § = 1] = O. Hence.{32+*12 ' y. which is defined by the following formulas: . say by inversion. It is well known that such a transformation maps circles into circles (straight lines are considered circles passing through the point at infinity). Transformation (159) maps every finite point M (az. with points M and M' lying on a single ray that emerges from the origin and obeying the condition OM >< OM' =-· rz (Figure 79). Geometrically this transformation constitutes what has become known as inversion with respect to ci circle and maps the origin into the point at infinity and vice versa.§"+n" $y (g2··`_. as x2 —1~ y2 —> oo. straight lines passing through the origin are invariant under transformation (159). For one thing. 2.__ E. A simple way of studying the asymptotic behavior of the phase trajectories of the differential system (122) is to introduce the point at infinity by transforming the initial differential system in an appropriate manner. 1]) of the same plane.Ch.
80 cases the new origin serves as a singular point. f Fig. The reasons for this are discussed below.
.220 Differential Equations in Applications ml ·¤ wv. 79 N Fig.
We also note that completion of the (:1:. we can always establish the point where the curve intersects the respective unit circle in the (1:. 1])-plane. y)-plane. by considering this curve in. 2. y)-plane with the point at infinity is topologically equivalent to inversion of the stereographic projection (Figure 80). if we map the plane onto the sphere. more precisely. a vector field on the plane transforms into a vector field on the sphere and the point at infinity may prove to be a singular point on the sphere. Although inversion with respect to a circle is useful. it proves cumbersome and inconvenient when the point at infinity has a complicated structure. ij)-plane. that is. The fact is that since a unit circle is mapped. In such cases
. 1])-plane. Conversely. The projection center N is the antipodal point of S. into itself. via transformation (159). any further investigations can be carried out in the usual manner. y)-plane. in which the points on a sphere are mapped onto a plane that is tangent to the sphere at point S. has a definite tangent at the origin of the new (Q. a unit circle in the (Q. this can be started by considering the (§. Qualitative Methods 221 As for the question of how to construct in the old (sc. It is clear that the projection center N corresponds to the point at infinity in the (41:.Ch. say. y)-plane a curve that has a definite asymptotic direction.
But if z = 0. y)-plane is associated with a triple of real numbers (§. y = 1]/z. Such a straight line may carry several singular points. y)-plane completed with the straight line at infinity is called the projective plane.·'= 0. eq. This implies that every point of the projective plane is mapped continuously and in a one-to-one manner onto a pair of antipodal points on the unit sphere. The (x. Under this transformation. If we now consider a pencil of lines and describe its center with a sphere of. more convenient. If a point (x. say. y) is not at infinity. kz) for every real k q'-= O. z) and (k§. transformation of the (:1:.222 Differential Equations in Applications another. the projective plane may be interpreted as the set of all pairs of antipodal points on a unit sphere. and the nature of these points is usually simpler than that of a singular point introduced by inversion with respect to a circle. lm. we have a straight line at infinity. To visualize the projective plane. unit radius. then each line of the pencil intersects the sphere at antipodal points. each point of the (x. 1]. we need only consider one-half of the sphere and assume its points to be the points of the
. y)—plane is used by introducing homogeneous coordinates: as = §/z. Thus. z) that are not simultaneously zero and no difference is made between the triples (§. z =.
which touches it at pole S (Figure 81). Qualitative Methods 233 é'·=z=·—0 **0 AM. the lower one) onto the ot-plane. the projective plane.9 Fig. Each pair of the antipodal points of the boundary corresponds to a line at infinity. the gaseous fuel—air mixture is forced through rotating
. If we orthcgonally project this hemisphere (say. 81 projective plane. 2.Ch. and the completion of the Euclidean plane with this line transforms the plane into a closed surface.12 Flow of ra Perfect Gas Through a Rotating Tube of Uniform Cross Section In some types of turboprop helicopters and airplanes and in jet turbines. 2. we ¢=z=0 lL i iA' Ew . the projective plane is mapped onto a unit disk whose antipodal points on the boundary are assumed identical.
where v is the speed of a gas particle with respect to the tube." Aeronaut. Zaremba.K. "Adiabatic one-dimensional flow of a perfect gas through a rotating tube of uniform cross section. 82 tubes of uniform cross section installed in the compressor blades and linked by a hollow vertical axle.* It is * See I . Figure 82 depicts schematically a single rotating tube of a compressor blade. Kestin and S.
. 4: 373-399 (1954). In a blade the gaseous mixture participates in a rotational motion with respect to the axis with constant angular velocity co and moves relative to the tube with an acceleration v (dv/dr). Quart. and r is the coordinate measured along the rotating compressor blades.224 Differential Equations in Applications d CK 5 C ' V QH ` e Fig. To establish the optimal conditions for rotation we must analyze the flow of the mixture through a rotating tube and link the solution with boundary conditions determined by the tube design.
We ignore here the effect produced by pressure variations in the tube (if they exist at all) and by variations in pressure drop acting on the cross section plane. the gas is accelerated thanks to the combined action of the pressure drop and dynamical acceleration in the rotation compressor blade. 2. The boundary of the cavity occupied by the gas is denoted by a. generally speaking. to the cross section b by rl.Ch. both of which are the result of the Coriolis force. requires experimental verification. the gas is assumed perfect with a speciiic heat ratio y. Qualitative Methods 225 assumed that the fuel-air mixture. and all the processes that the gas mixture undergoes are assumed reversibly adiabatic (exceptions are noted below). It is assumed that the gas expands through a nozzle with an outlet cross section b that at the same time is the inlet of a tube of uniform cross section. whose initial state is known. is supplied along the hollow axle to a cavity on the axle in which the flow velocity may be assumed insignificant. since the existence of a lateral pressure drop may is-0770
. The expansion of the gas mixture from state a. we denote the velocity of the gas after expansion by vi and the distance from the rotation axis O. to state b is assumed to proceed isentropically. When passing through the tube. The last assumption. whose uniform cross—sectional area will be denoted by A and whose hydraulic diameter by D.
But if the diameter of the tube is small compared to the tube's length.226 Differential Equations in Applications serve as a cause of secondary flows. It is now clear that the equations of momentum and energy balance for a compressible mixture traveling along a tube of uniform cross section must be modified so as to allow for forces of inertia that appear in a rotating reference frame. we consider two modes of passage of the mixture through the nozzle. This produces a thrust force caused by the presence of a torque. such an assumption is justified. As for the continuity equation. it remains the same. Denoting the external (atmospheric) pressure by Pa. Below for the sake of simplicity we assume that the exit nozzle is a converging one and has an exit (throat) of area A*.
. Now let us suppose that starting from cross section c at the right end of the tube the gas is compressed isentropically and passes through a diverging nozzle. In the process it transfers into a state of rest with respect to the compressor blade in the second cavity at a distance r3 from the rotation axis and reaches a state with pressure Pd and temperature Td. From the second cavity the gaseous mixture expands isentropically into a converging or converging—diverging nozzle in such a manner that it leaves the cavity at right angles to the tube's axis.
that is. to a situation in which the P3-to-Pd ratio exceeds the critical value. Qualitative Methods 227 The first applies. 2./Pa < (2/(v + i)"'"'*'The pressure P3 at the nozzle's throat has a fixed value that depends on Pd but not on P3.Ch. P3 —-= P3.. Hence. In further analysis the flow is assumed adiabatic everywhere and isentropic everywhere except in the tube between cross sections b and c. let us now consider the continuity equation. or P. the equation of 15*
./Pd > (2/(v + i))""""'· In this case the flow at the nozzl0°s throat is subsonic and. pressure P3 at the throat is equal to the atmospheric pressure. The second case applies to a situation in which the Pa-to-Pd ratio is below the critical value.. As in the case where we derive the differential equation that describes the adiabatic flow of a perfect gas through a nozzle of varying cross section. P3 -= (2/(V + i))""""' PaIn the latter case the flow at the nozzle's throat has the speed of sound Us = (2/(v + i))"2 aa where ad depends only on temperature Td. hence. or P.
4.wzf dr
. _ 1p— V . V3 A .. the equation of continuity in this case has the form . (160) with V the speciiic volume and xp the flux density. 83 momentum. . To derive the equation of motion..4 r' + dr Fig. or '\|> = mr/A.228 Differential Equations in Applications x U 7 U+d0 . For instance.. and the equation of energy balance.2.1. We note that the dynamical effect of the rotational motion of the compressor blade can be used to describe the flow with respect to the moving tube. or the momentum equation.const...0 P+6Z'/D v/ [. we turn to Figure 83. . where m' is the flux mass. where in accordance with the D'A1embe1·t principle the force of inertia dI = {3.
Here ·t : Z. Hence. If we define the speed of sound a via the equation h:`QlT> we find that ada-l——·Y—%vdv—— -3%-}—w2rdr=O. remains constant along the entire tube. where h is the enthalpy. (02/2V). As a first approximation we can assume that 2. it is simple to derive if we use the first law of thermodynamics for open systems and bear in mind that the amount of work performed by the system is wzr dr.Ch. we can write the momentum equation in the form A dv 2}»Av2 A ·I7dT°U·H·. V dP —|—v dv-}. depends on the Reynolds number R. Thus dh—|—vdv—-w2rdr=O. Qualitative Methods 229 is assumed to act in the positive direction of r.
.·-—-· dT°·l······I7ZU2T°dr. or after certain manipulations. With this in mind. and the friction force dF = (4A/D) dr. the force of pressure A dP. the element of mass dm 2 (A/V) dr moves with an acceleration v(dv/dr) caused by the combined action of the force of inertia dI.ED?-L v2dr—w2r dr= 0. where 7. (161) As for the energy-balance equation. 2.
while the domains containing the straight lines (17ia) and (171b) must be attractive. 2.Ch. hence. is negative below the straight line (171c) and positive above. In view of the symmetry of the field specified by vector (X4. As for the radii lying between the straight lines (171c) and (171a). Thus. the above facts remain valid for standard domains obtained from those mentioned earlier by a rotation about the singular point through an angle of 180°. Y4).
. Geometrically this means that the righthand side of the differential equation El. Y4 (gv ll) dg X4 (gs determining the behavior of integral curves in a disk of sufficiently small radius and centered at the origin fixes an angle greater than the angle of inclination of the radii that lie in the first quadrant between the straight lines (171b) and (171c). Qualitative Methods 233 has the same sign in both the first quadrant and the neighborhood of the straight lines as the left-hand side of (171c) and. the standard domain containing the straight line (171c) must be repulsive. on them the right—hand side of this differential equation fixes an angle that is smaller than the angle of inclination of these radii (Figure 84). The explanation lies in the fact that (172) may change sign only when an exceptional straight line is crossed.
there are exactly two integral curves that "enter" the singular point along the tangent (171c) and an infinitude of integral curves that touch the coordinate axes (171a) and (171b) at the point of rest.234 Differential Equations in Applications / \ em 7 X —¤~ f `\ TY ' //4/ /K // x \ jr +\ Ai `é\ /7 Ev Fig. These sectors are parabolic because they lie between two successive standard domains. 84 Thus. one of which is attractive and the other repulsive. More than that. Each of the first and third quadrants is divided into two sectors by the integral curves that touch the exceptional straight line (171c) at the origin. We see then that the second and fourth quadrants contain elliptic sectors since they lie between two successive attractive standard domains (Figure 85). all the integral curves except those that touch the straight line (171c) touch the
.
2: -—>oo or y ->cx>.Ch. 2. we see that there exists exactly one integral curve having an asymptotic direction with a slope (q —{. y)—plane. Turning to this plane. we can analyze the differential equation (168) solely in the first quadrant of the (sc. Indeed.2) G2/2m. that is. All other integral curves allow for the asymptotic direction of one of the coordinate axes. along each of them in the movement toward infinity not only does y/x —>0 or :1:/y -—>0 but so does y -+0 or cc —->0 (with . 85 coordinate axes (171a) and (17lb) at the singular point. Qualitative Methods 235 7 is Fig. Basing our reasoning on physical considerations. respec-
. it is easy to prove that all these integral curves asymptotically approach one of the coordinate axes.
and ·a special investigation is required to clarify it.2: Fig.236 Diflerential Equations in Applications 0 . where some integral curves have been plotted at great values of :1: and y.
.13 Isolated Closed Trajectories We already know that in the case of a singular point of the vortex type a certain region of the phase plane is completely filled with closed trajectories. However. This is illustrated in Figure 86. Note that the pattern of integral curves at a finite distance from the origin depends primarily on the position of the singular points. It can also be proved that the integral curves that approach the coordinate axes asymptotically (Figure 86) leave the first quadrant at a finite distance from the origin. 86 tively). 2.
we introduce polar coordinates r and 9. 2. and in other fields. in various aspects of automatic control. To solve this system. Then. Qualitative Methods 237 a more complicated situation may occur when there is an isolated closed trajectory. Interestingly. where x = rcos 6 and y = rsin 9. only nonlinear differential equations and systems can have isolated closed trajectories. radiophysics. a trajectory in a certain neighborhood of which there are no other closed trajectories. astronomy. Below we study the possibility of isolated periodic solutions emerging in processes that occur in electric circuits. Isolated periodic solutions correspond to a broad spectrum of phenomena and processes occurring in biology. differentiating the relationships x2 —|— y2 == rz and 9 =
. Such solutions emerge in differential models in economics. we also consider as a model the nonlinear differential system {j—§°—=—y+=v(i ——·w"—y2). dy (173) Ei-:-:z:—|—y(l —-0:2-y2).Ch. This case is directly linked with the existence of isolated periodic solutions. that is. and the theory of device design. in airplane design. medicine. oscillation theory.
adding the products. we arrive at dx dy __ dr dy _ dx _ _ dB` "w+ya7··rw· xr. and allowing for the first relationship in (174). ·aT—1. (175) and (176) imply that system (173) can be reduced to the form dr df) _ 717--T(1··—T°2). Since at the moment we are only interested in constructing trajectories. (176) System (173) has only one singular point. Then Eqs.238 Differential Equations in Applications tan·1(y/x) with respect to t.
. (175) Multiplying the second equation in (173) into sc and the first into y. and allowing for the second relationship in (174). we find that rz gil. we find that dr r -8-2-: rz (1 -1*2). ff aT··'2'a2'· (174) Multiplying the first equation into sc and the second into y. O (0. Each of these equations can easily be integrated and the entire family of solutions. we can assume that r is positive. 0). subtracting one product from another. rz.
These two relationships define a closed trajectory. a circle x2 —l. there can be two additional types of limit cycles. COS + tg) sin to) x = -·T ———— » I! = +*——_————-——' V 1+C¢'"' I/1+ 06** If now in the first equation in (178) we put C-:0. A limit cycle is said to be (orbitally) unstable if all neighboring trajectories spiral away from it as t—»—|—o<>. from the inner side) spiral on to it and all neighboring trajectories
. in terms of the old variables x and y. it is clear that r is greater than unity and tends to unity as t —> +oo. it is clear that r is less than unity and tends to unity as t -» +oo. is given by the formulas 1 ' 9:-1t+t°' (178) or. more precisely. But if C is positive. And a limit cycle is said to be (orbitally) half-stable if all neighboring trajectories on one side (say. This means that there exists only one closed trajectory r == _1 which all other trajectories approach along spirals with the passage of time (Figure 87). Qualitative Methods 239 as can easily be seen. 2. we get r=1 and 6:-—t-{—t0. In fact.Ch. Closed phase trajectories possessing such properties are known as limit cycles or. (orbitally) stable limit cycles.y2 L-= 1. If C is negative.
this cannot be done. In the above example we were able to find in explicit form the equation of a closed phase trajectory. Note that a closed trajectory of (122).240 Differential Equations in Applications . on the outer side) spiral away from it as t->-|—¤¤. there are no closed trajectories in the region either.7 7 . implies that if there are no singular points of a differential system within a region of the phase plane. if such a trajectory exists.z* Fig. but generally. 87 on the other side (say. of course. contains within its interior at least one singular point of the system. This. for one thing. Hence the importance in the theory of ordinary differential equations of criteria that enable at least specifying the regions where a limit cycle may occur.
.
·••"""°`_—"-°`7<\\ X0`F 'x (i I1 l1 \/ \\ // Fig. then F is either closed or approaches a closed trajectory along a spiral with the passage of time. With each boundary point of D we associate a vector V(¢¤» y) = X (rv. Then the PoincaréBendixson criterion holds true. We illustrate this in Figure 88. 88 Let D be a bounded domain that lies together with its boundary in the phase plane and does not contain any singular points of system (122). Here D consists of two closed curves I`. y)i+Y(¤¢. Qualitative Methods 241 T1 f/. and remains there at all 16-0770
. 2. if a trajectory T` that emerges at the initial moment t= to from a boundary point. ij I` is a trajectory of (122) that at the initial moment t:-· to emerges from a point that lies in D and remains in D for all t) to. y)iThen. namely.Ch. enters D. and I`2 and the circular domain between them.
the domain D lying between the circles with radii r = 1/2 and r = 2 contains no singular points. success depends both on the type of system and on the experience of the researcher. according to the above criterion. which is associated with the points on the boundary of D. is always directed into D. since no general methods exist for building the appropriate domains and. it is a circle of radius r = 1. O). it will along a spiral approach a closed trajectory F0 that lies entirely i11 D. O (O. however. At the same time we must bear in mi11d that
. then. The differential system (173) provides a simple example illustrating the application of the above criterion in finding limit cycles. that great difficulties are generally encountered in a system of the (122) type when we wish to realize practically the Poincaré-Bendixson criterion. system (173) has only one singular point. and. The curve I`0 must surround a singular point of the differential system. therefore. Such a closed trajectory does indeed exist. The first equation in (177) implies that dr/dt is positive on the inner circle and negative on the oute1·. therefore. Note. Indeed. not lying in D.242 Differential Equations in Applications moments tgto. This means that the circular domain lying between circles with 1·adii r = 1/2 and r : 2 must contain a closed trajectory of the differential system (173). Vector V. point P.
. y)=(i——y)F(¤¤). which constitutes a differential model for describing an adiabatic OIl9"(llI1"l|3l1Sl(l113l flow of a perfect gas of constant specific heat ratio througli a nozzle with drag.* then no limit cycles of the dijjerential system (122) can exist in D. 16*
.. y) that is continuous together with its first partial derivatives and is such that in a simply connected domain D of the phase plane the sum 0 (BX) 0 (BY) 0. then for this equation we have X(¤v. ln this respect the most widespread condition is the Dulac criterion: if there exists a function B (x. At B (x.Ch.1 I Y(¤v. y) Ei 1 the criterion transforms into the Bendixson criterion. Qualitative Methods 243 finding the conditions in which no limit cycles exist is no less important than establishing criteria for their existence. y)=4y (1l+ y) <vqy—F (·¤¤))· If we put v—1 ·* Btw.—— 0]} * That is.. lf we turn to the differential equation (156). y)= {4yF(¤v) [1+ —q. 2. positive definite or negative definite.2: + Oy is a function of fixed sign.
and does not pass through the singular points of this system. say. Eq. is characterized by a certain direction speciiied by an angle 9 (Figure 89).244 Difierential Equations in Applications we find that 6(BX) 6(BY) _ Yq dx + dy = F(x) >O and. Let us discuss one more concept that can be employed to establish the existence of limit cycles. hence.e. hence.y)=X(¤v» 1/)i+Y(¤>. if P (x. with i and j the unit vectors directed along the Cartesian axes. Let I` be a simple closed curve (i. This number n is said to be the index of the closed curve I` (or the index of cycle I`). a curve without self~intersections) that is not necessarily a phase trajectory of system (122). angle 9 acquires an increment A9 = 2nn. lies in the phase plane. that is. vector V performs an integral number of cycles in the process. If point P (x. (156) has no closed integral curves. the vecto1· V(¤v. or zero.y)i. a negative integer. y) moves along I`. with n a positive integer. y) is a point of I`. counterclockwise a11d completes a full cycle. Then. The concept is that of the index of a singular point. lf we begin to contract I` in such a manner that under this deformation I` does not
. is a nonzero vector and.
89 pass through singular points of the given vector Held. remain an integer. (2) the index of a closed curve surrounding several singular points is equal to the sum of the indices of these points. and (3) the index of a closed curve encompassing only ordinary points is zero. The index has the following properties: (1) the index of a closed trajectory of the diyjerential system (122) is equal to +1. vary continuously and. This means that under continuous deformation of the curve the index of the cycle does not change. the index of the cycle must. on the one hand. Qualitative Methods 245 V y0 F 0 eZ` Fig.
.Ch. on the other. 2. This property leads to the notion of the index of a singular point as the index of a simple closed curve surrounding the singular point.
are not (see Figure 90). Figure 91 depicts singular points with
. and h is the number of hyperbolic sectors. The trajectories may intersect L or touch it. We can still use formula (179) to calculate the index of a singular point. while the C-type points. For practical purposes the following simple method may be suggested. a closed trajectory must enclose either one singular point with an index -}-1 or several singular points with a net index equal to -{-1. that since the index of a closed trajectory of system (122) is always +1. In the latter case only exterior points of contact (type A) or interior points of contact (type B) are taken into account.246 Differential Equat·ions in Applications This implies. The index of a singular point is calculated by the formula n:1+§#. for one thing. but now e is the number of interior points of contact and h the number of exterior points of contact of trajectories of (122) with cycle L. points of inflection. This fact is often used to prove the absence of limit cycles. Suppose that L is a cycle that does not pass through singular points of (122) and is such that any trajectory of (122) has no more than a finite number of points common to L. uw) where e is the number of elliptic sectors.
Earlier it was noted that constructing a complete picture of the behavior of the phase trajectories of the differential system (122) is facilitated by introducing the point at infinity via transformations (159). the point at infinity must be a singular point with a nonzero index. But if instead of inversion we employ homogeneous coordinates. Qualitative Methods 247 O am QM B'<\>L Fig. the not index of the points is +2. Thus. That
. 2. if the net index of all the singular points of a differential system (possessing a finite number of such po. 90 indices 0. +3. Topological considerations provide a very general theorem which states that when a continuous vector field with a finite number of singular points is specified on a sphere. respectively. the net index of all the singular points is already +1.ints) that lie in a finite phase-plane domain is distinct _from +2.Ch. +1. and -2. +2.
.
or less than G0. If we turn to Eq. It can also be shown that. 2. (168). Qualitative Methods 249 this is so can be seen from the fact that if the plane is projected onto a sphere with the center of projection placed at the center of the sphere. equal to G0. or not a single point of intersection with the parabola fixed by the equation 1 -—py + qG2x2 = O (which. in turn. then. This implies. one double point. that the net index of finite singular points is zero. for one thing. as shown in Figure 85. A saddle point and a nodal point. which describes adiabatic one-dimensional flow of a perfect gas through a rotating tube of uniform cross section. for the singular point at infinity we have e = 2 and h == O. the following combinations of finite singular points occur. (a) G > G0. two points on the sphere correspond to a single point on the projective plane. and the circumference of the great circle parallel to the plane corresponds to a straight line at infinity. is equivalent to G being greater than G0.
. (b) G = G0.Ganz == 0 has two points. A higher-order singular point with two hyperbolic (h —. with G0 = 2mq*/2/p).Ch. It follows from this that the index of this point is +2 and does not depend on the values of the constants in Eq. depending on whether the straight line specified by the equation my . (168).= 2) sectors and two parabolic (e = O).
92
.. Figure 92 depicts schematically a dynatron oscillator. We see that in three cases the net index is zero. with the ia-va characteristic shown by a solid curve in Figure 93. as it should be.14 Periodic Modes in Electric Circuits We will show how limit cycles emerge in a dynatron oscillator (Figure 92).L__ Fig. Singular points are absent.250 Differential Equations in Applications (c) G <G0. and economics} we will show how such phenomena emerge in the study of electric circuits. Here ia is the current and va the voltage in the t é'rC e '|' + r .. Although phenomena linked with generation of limit cycles can be illustrated by examples from mechanics. An analysis of the operation of such an oscillator leads to what is known as the Van der Pol equation. biology. 2.
As follows from Figure 93. i·l·ir+iL·l—io=O. 2. and ic = Cv. The circuit consists of resistance r.. inductance L._(.. with ir = v/r. The characteristic of the tube may be approximated with a thirddegree polynomial i = ow + YU3. which is connected in series with the screen—grid tube. and capacitance C connected in parallel and known as the tank circuit.. the dynatron. The real circuit can be replaced in this case with an equivalent circuit shown schematically in Figure 94.
.z —l.Ch.bvz) + o>§v = O. Y > O. 1 _ 3y __ _1___ 2 'E·'+'?F""¤ 2·`"b¤ rc —""<·· we can write the p1·evious equation in the form all.. In accordance with one of Kirchhoff's laws. Qualitative Methods 251 screen-grid tube. on > O. As a result of simple manipulations we arrive at the following differential equation: " <¤ 1 3v 2 1-. Here i and v stand for the coordinates in a system whose origin is shifted to the point of inflection O. " +l·zr+'.z'+'@"" lvi LC -*0If we now put cz. L (<liL/dt) == v. which is shown by a dashed curve in Figure 93.
Since b>0. (180) can have no closed integral curves. Thus. the representative point moves along a trajectory toward the singular point. and J* (0. y) (180) The only finite singular point of this equation is the origin. ot< —-1/r. y) == —-(a -4. hence. a trajectory that emerges from the point at infinity cannot reach the singular point at the origin no matter what the value of t including the case where t-= —l-oo. This implies that if we can prove that a trajectory originating at the point at infinity resembles a spiral
. that is. will consider only the case where a is negative.bvz). that is. 0) = m§>0. We. hence. dy __ dv __ `EZ""Y(Uv y)v `(`E"·'I/(vv y)v we see that as t grows. D (v.Ch. lf we now consider the differential system corresponding to the differential equation (180). Qualitative Methods 253 equation the following first-order differential equation: dy ___ __ (¢+bv2)y+<¤3v __ Y(v. 0) == —a > 0 and. This implies that D (0. therefore. the singular point is either a nodal point or a focal point. Eq. 2. we assume that a>0 and conclude that divergence D does not reverse its sign and. y) TY" y "` V (v.
Here. (168) was studied./z. Such a procedure yields an app1·oximation to the limit cycle." Appl. A numerical method suggested by the Dutch physicist and mathematician Van der Pol (1889-1959) consists in building a trajectory that originates at a point positioned at a great distance from the origin and in checking whether this trajectory possesses the above-mentioned property. we employ the more convenient (in the present problem) homogeneous-coordinate transformations v = E. Kestin and S.254 Differential Equations in Applications winding onto the origin as t-><x>. Below we give a proof procedure* based on analytical considerations and the investigation of the properties of singular points at infinity. this will guarantee that at least one limit cycle exists. "Geometrical methods in the analysis of ordinary differential equations. Res. in contrast to the method by which Eq. (181) The straight line at infinity is fixed by the equation z = O. Sei. but it can be employed only in the case where concrete numerical values are known. Zaremba. y = 1]/z.
. B3: 144 189 (1953). To reduce the number of * See the paper by J.K. The existence proof for such a cycle can be obtained either numerically or analytically.
Note. 2.. in the following manner: dt == zz dt)." "l·" · af: **2It is convenient to introduce a new parameter. as 9 grows) the representative point moves along this trajectory toward the only singular point z = iq = O. d (183) HT)-. 9. speciijes t]1e regular exceptional direction z = O. (180) is transformed to dz __ dn (az2—|—b)·q—l-0J?z2 717*. that the straight line at infinity. y : 1]/z.Ch. which can be establishedby studying the appropriate
. This can be done if we assume that § = 1. we first exclude the point Q = z = 0.T|Z —— . Then v == 1/z. Qualitative Methods 255 variables to two. constitutes a trajectory of the differential system (183) and as t grows (hence. In this case the differential system associated with Eq. The characteristic equation. with two repulsive regions corresponding to this direction. which in this case has the form —b1jz = O. z -= 0. (182) Then this system of equations (3311 be written as gg: —-(azz-}—b)1]—c0§z`·*—1]2z2=i'!_. {irst.
I. The region lying between the curve fixed by the equation 9 = 0 and the axis fr] = O is topologically equivalent to two repulsive regions.256 Diflerential Equations in Applications z QT I E 3:0 \ 112/ x \ \ '? / \/ j/ 4)// "Z' Jr E Fig. additional reasoning is required. which makes it possible to fix three directions. on different sides of the symmetry axis z = O. Thus. on each side of the straight line mq = 0 there is at least
. The second exceptional direction 1] = 0 is singular and. therefore. II. 95 vector field (Figure 95). and III. The locus of points 9 == U is a curve that touches the straight line iq == 0 at the origin and passes through the second and third quadrants (Figure 95).
+7.Ch. Hence. since for small values of z and positive tj we have la/2 I2 wi/luz I. t.—7+. 2.+·. This means that on each side of the straight line 1] = O there can be only one phase trajectory that touches this straight line at the origin and belongs to the region considered.here can be no curves that touch the axis 1] = 0 at the origin and pass through the first or fourth quadrant. If we now consider the differential equation dn __ a b wt ii . lt is also clear that the qualitative picture is symmetric about the axis z == O. it is easy to see that the representative points moving along these trajectories will also move apart as z decreases.._ jy. Qualitative Methods 257 one t1·ajectory that touches the straight line at the origin. Such curves are also absent from the region that lies to the left of the curve fixed by the equation iy = O since in this case 9 is positive and l7—U770
.—f(¤» Z).. we note that for small values of | TII 0j __ 1 m?. if we take two phase trajectories with the same value of 1]. More than that. ia-? (ir?) <" in the second quadrant between curve ay = O and axis fr] = O.
By studying the signs of P and Q we can establish the pattern of the vector iield (Figure 96) and the phase-tra-
. ?. and the characteristic equation here is z3 = 0..+ §+w. while the curve fixed by the equation Q (§. that link the previous point with the point 7. In ou1· reasoning we did not discuss the point lf. arrows IV).. == z = U proves to be singula1·. z) = 0 has a multiple point at the origin. The point E.Zz_{_ré(aZz·bz 2 2. where the variable 9 is defined in (182)... the exceptional direction z = O is singular. let us put 1] = 1 in (181). do. = z = O. Then the differential system associated with the differential equation (180) can be written as follows: .·=Z (¤Z2+b§2+w§§Z2)=Q. This reasoning suggests that the singular point rj = z == O is a saddle point. The other two phase trajectories reach the saddle point as t-—>-—· oo. The curve fixed by the equation P (E.258 Differential Equations in Applications TZ has the same sign as z (Figure 95. too.9§_.§Z)—-1. To conclude our investigation. z = O. Hence. = z = 0. z) = Ol (Figure 96) touches the axis z = O at the origin and has a cuspidal point there. The two phase trajectories that reach this point as t —> 5}-oo are segments of the straight line at infinity.
2. no general methods exist for doing this. if the latter exist. the phasetrajectory pattern of the corresponding differential system is determined by the type of singular points and closed integral curves (phase trajectories). in addition to establishing the types of singular points. The reader will recall that a curve or an arc of a curve with a continuous tangent iS said to be a curve (arc) without contact if it touches the vector (X.15 Curves Without Contact In fairly simple cases the complete pattern of the integral curves of a given differential equation or. This proves the existence of at least one limit cycle for the Van der Pol differential equation. which is the same. Hence. Y) specified
.260 Differential Equations in Applications This follows from the fact that in the first and fourth quadrants IZ/E I> IQ/P IThe above reasoning shows that the Van der Pol equation has no phase trajectories that tend to infinity as t grows but has an infinitude of phase curves that leave infinity as t grows. separatrices. Sometimes the qualitative picture can be constructed if. that link the singular points. we can find the curves. Unfortunately. it is expedient in qualitative integration to employ the so—called curves without contact.
Various auxiliary inequalities can be used in qualitative integration of differential equations. But if in D we have the strict inequality f(¤v.2:. The definition implies that vector (X. denoting by yl (x) a solution of the first equation such that 2/1(·T0) = I/os with (xm yo) €D» and by y2(x) a solution of the second equation with the same initial data. Thus. g'g".1:0 in D and the curve y = y2 (0:) is a curve without contact. then. say. a curve without contact may be intersected by the phase trajectories of (122) only in one direction when t grows and in the opposite direction when t diminishes.z:. y) < s (¤v» y). if two differential equations are known. knowing the respective curve may provide information about the pattern of a particular part of a phase trajectory. We have already
.2:) for x Q xo in D. then y1(¤¢)< y2 (rv) fer x>. As an example let us consider the differential equation (168). 2. y) { g (. Qualitative Methods 261 by the differential system (122) nowhere. y) in a domain D. Y) must point in one direction on the entire curve. Therefore. Fo1· example. we can prove that yl (sr) { y2 (.f(·Tv y)v 'g%`T—"g(xv y) and it is known that f(.Ch.
if the representative point emerges from any interior point of A and moves along an integral curve with t decreasing. which are points of intersection of the straight line my — Gzx = 0 and the parabola 1 . y) = 2y (my . a saddle point and a nodal point. y) < 0. the equation allows for two finite singular points. we can see that one of the exceptional straight lines passes through A.6%).
.py + 062332. it is easy to see that vector (X. it cannot leave A without passing through one of the singular points. But since inside A we have X (as. Y) is directed on the boundary of A outward except at the singular points. This implies that an integral curve that is tangent to this straight line at the saddle point must enter A and then proceed to the nodal point. Y (rv. If we put X (fm y) = 1 —. Hence. The segments of the straight line and parabola that link these two finite singular points are curves without contact and they specify a region of the plane which we denote by A. Finding the slopes of the exceptional directions for the saddle point.py —{— qG2:c2 = O. the singular point that is attractive is the nodal point.262 Differential Equations in Applications shown that if G > G0.
The Contact Curve Earlier we noted that in the qualitative solution of differential equations it is expedient to use curves without contact. then.* li the topographical system is selected in such a manner that each value of para~ meter C is associated with a unique curve. enveloping each other. Hence. One approach is linked to the selection of an appropriate topographical system of curves. y) = C. the importance of various particular methods and approaches in solving this problem. however.
. assuming for the sake of deiiniteness that the curve corresponding to a deiinite value of C envelopes all the curves with smaller values of C (which means that the "size" oi the curves grows with C) and that no singular points of the corresponding * Other definitions of a topographical system also exist in the mathematical literature. with C a real-valued parameter. 2. Qualitative Methods 263 2. Here a topographical system of curves dehned by an equation cb (x. but essentially they differ little from the one given here. and continuously dijjerentiable simple closed curves that completely jill a doubly connected domain G in the phase plane. is understood to be a family of nonintersecting.Ch. It must be noted.16 The Topographical System of Curves. that there is no general method of building such curves that would be applicable in every case.
say.264 Differential Equations in Applications differential system lie on the curves belonging to the topographical system. d€D/dt is positive on a certain curve belonging to the topographical system. This implies.¤(¢>» t\>(¢)) 0'D(¤¤.w*X ('"· yl then the curves (cycles in our case) without contact are the curves of the topographical system on which the derivative d€D/dt is a function of fixed sign. And if d€D/dt is negative. lf. with the passage of time i. being a cycle without contact. the region cannot contain any closed trajectories. in addition. then such a curve. Limit cycles may exist only in the annular regions where the
. all the phase trajectories enter this region. for one thing. the limit cycles of the differential system. that if in an annular region completely filled with curves belonging to the topographical system the derivative d<D/dt is a function of fixed sign. the finite region bounded by it. that is. d<1>(<. If we consider the function (D (cp (t). we arrive at the following conclusion. y) '—i?":—". possesses the property that all phase trajectories intersecting it leave. ip (t)) where x = cp (t) and y = ip (t) are the parametric equations of the trajectories of the differential system of the (122) type and calculate the derivative with respect to t.
we consider the differential system dd gif-=y—·:v+w3. a stable focal point. To illustrate this reasoning. O).:.Ch. —H%·=—rv—y+y3. (184) which has only one singular point in the finite part of the plane. respectively. 2. O (0. we arrive a the_ following conclusion. Qualitative Methods 265 derivative d€D/dt is a function with alternating signs... For the topographical system of curves we select the family of concentric circles centered at point O (O. Noting that the greatest and smallest values of the expression in parentheses are 1 and 1/2. For r greater than I/2 the value of the derivative dCD/dt is positive Wh11G for r less than unity it is negative.which in polar coordinates x = r cos 6 and y = r sin 6 assumes the form -S4?)-—. ——2r2-{— 2r'* (cost 9 —[—sin*= 9) = —2r2 ·\~2r'* ( cos 49). with C a positive parameter. that is. O). On the basis of the Poincaré-Bendixson criterion 18-0770
. In our case the derivative d€D/dt is given by the relationship d<D -5 =—· — 2 (wz + y2) —l— 2 (¤¢" + 1%*). the family of curves specified by the equation x2 + y2 == C.
To return to the problem oi building curves without contact in the general case. Allowing now for the type of the singular point O (O.266 Differential Equations in Applications we conclude (if we replace t with —~t in system (184)) that the annular region bounded by the circles x2 + y2 -:1 and x2 + y2 == 2 contains a limit cycle of system (184). as can easily be seen. in the annular region with boundaries x2 —|— y2 :1 and x2 —|— y2 = 2 the expression 3 (x2 + y2) -— 2 always retains its sign. then in G there can be no more than one simple closed curve consisting of trajectories of (122) and containing within it the interior boundary of G. we Hind that OX/0x + 6Y/dy = 3 (x2 —i— y2) —— 2 and. ln our case. let us examine a somewhat different way
. and this limit cycle is unique. O). we conclude that the cycle is an unstable limit cycle. selecting B (x. y) E 1 for system (184). y) that is continuous together with its first partial derivatives and is such that in adoubly connected domain G belonging to the domain of definition of system (122) the function 6(BX) 0(BY) 01: + Oy is of fixed sign. To prove the uniqueness it is sufficient to employ the Dulac criterion for a doubly connected domain: if there exists a function B (x.
-@/23 X I.G1! . or L . y) =-·· C) is called a contact curve. the locus of points at which the trajectories of the differential system of the (122) type touch the curves belonging to the topographical system (defined by the equation (D (1. Introduction of this concept can be explained by the fact that if the derivative dd)/dt vanishes on a set of points of the phase plane. This approach is based on the notion of a contact curve. when 0 IJ 6iD the slopes coincide. The equation of a contact curve has the (185) form. Of particular interest here is the case where the topographical system can be selected in such a manner that either the contact curve itself or a real branch of this curve proves to be a simple closed curve. Thus. Qualitative Methods 267 of using the topographical system of curves. Gy ° Hence. this set constitutes the locus of points at which the trajectories of the differential system touch the curves belonging to the topographical system. Then the topographical system con-
.Ch. 2. Indeed. the slope of the tangent to a trajectory of the differential system is Y/X and the slope of the tangent to a curve belongirg to the topographical system is —-(Gil 0:1:)/(GCD/Gy).
If the derivative d€D/dt is nonpositive (nonnegative) on the "greatest" curve specified by the equation CD (sc. if
.y2 =·. Specifically. y) = C2. then the annular region bounded by these curves contains at least one limit cycle of the differential system studied.y4 is a closed curve (Figure 99). in the last example (see (184)% the contact curve specified by the equation :1:2 -1. Indeed.268 Differential Equations in Applications /\ x \/ @" \/ \/ Fig. y) = C1 and nonnegative (nonpositive) on the "smallest" curve specified by the equation (D (az. 98 tains the "greatest" and "smallest" curves that are touched either by a contact curve or by a real branch of this curve (in Figure 98 the contact curve is depicted by a dashed curve). which enables us to find the °°gI'eatest" and "smallest" curves in the selected topographical system that are touched by the contact curve..154 -{.
we conclude that the "greatest" and "smal1est" curves of the topographical
.Ch. 99 we note that in polar coordinates the equation of the contact curve has the form rz = (c0s' 9 + sin' 6)*. Qualitative Methods 269 H Fig. respectively. 2. we can easily {ind the parametric representation of this curve: x __ cos 9 y __ sin 9 l/c0s* 9—|—sin*9 ' _ }/cos49—{—sin·*9 Allowing now for the fact that the greatest and smallest values of r2 are. 2 and l.
These circles. For instance. we note that in building the topographical system of curves we can sometimes employ the right-hand side of system (122).17 The Divergence of a Vector Field and Limit Cycles Returning once more to the general case. Then. the contact curve or a real branch of this curve coincides with one of the curves belonging to the topographical system.270 Differential Equations in Applications system :::2 —{— y2 == C are the circles :1:2 —}— y2 == 1 and :::2 + y2 == 2. If the "greatest" and "smallest" curves merge. —|—oo). with A 6 (-2. (186) assumes the form 3 (x2 —|— y2) — 2 = it. Eq. respectively. as we already know. It has been found that the differential system may be such that the equation 0X GY where A is a real parameter. specifies the topographical system of curves. specify an annular region containing the limit cycle of system (184). that is. we arrive at the topographical system of curves x2 + y2 == A used above. if we turn to the differential system (184). such a curve is a trajectory of the differential system. assuming that A = (7t -[— 2)/3. A remark is in order.
. 2.
%: —y+x (i——¤>2—y2>. proves to be a limit cycle of system (173). its real branch :v" + y2 == 1 coincides with one of the curves belonging to the topographical system. which assumes the form azz —|. We will not dwell any further on these results. Qualitative Methods 271 Let us consider. 237239. This branch. defines the topographical system of curves. with it E (-—<x>. The contact curve in this case is given by the equation (:1:** -{— y2) X (. for example.$2-*92)For this system GX -5. 2. as we see..K)/4. and Eq.C11. (186) assumes the form 2 4 (:1:2 + y2) :: X. we introduce a new parameter A = (2 —. then the latter equation. If instead of parameter R.+%. but only note that the following fact may serve as justification for what
.2:2 -|— y2 —— 1) == 0 and. 2**4 ($2+!!2). The last two examples. gé/·= $+9 (1. of course. the differential system (173): 3. a particular situation. At the same time the very idea of building a topographical system of curves by employing the concept of divergence proves to be fruitful and leads to results of a general nature.y2 = A. 2). illustrate. as shown on pp.
. y) such that the equation 0(BX) 0(BY) _ —*a?·+*a. *"a.-.272 Differential Equations in Applications has been said: if a diyferential system of the (122) type has a limit cycle L.:. We can easily verify that there is not a single real lt for which the equation 2 — 5. For this system the divergence of the vector field is 0X GY ··5Tz:·—_"'··5!·.4xy + 7y2 + 3.y3.2:1.'—("2+y"—') >< (-— 23x2-1-16xy-25y2-20)-14.Bxyz —{·.1:2 ·— 3y2 = lt specifies a trajectory of the initial differential system. Let us consider. which has a limit cycle specified by the equation x2 —I— y2 = 1. %—= ——¢v°+=v2y+¤¤y2.·` ·· X specifies a curve that has a finite real branch coinciding with cycle L. y) = 3:::2 —. the differential system -%. then 0(BX) 0 (BY) ___ . there exists a real constant lt and a positive and continuously diyfferentiable in the phase plane function B (sc. for example.·T— 2*** 5$2—·· 3yz.—+_a. But if we take a function B (x.-· 2:1:3 —{— cozy.
.
.
. In conclusion we note that in studying concrete differential models it often proves expedient to employ methods not discussed in this book.Ch. 2. on the erudition and experience of the researcher. on how deep the appropriate mathematical tools are developed.iYL Z . of course.)Q+ . and. Qualitative Methods 273 and the equation _E°. Everything depends on the complexity of the differential model.Q..21. 14 Ox Oy specifies a curve whose finite real branch $2 + y2 -1 =·-· 0 proves to be the limit cycle.
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Summary
Based on the best-selling series by the Aufmann team, this hardcover text for the combined introductory and intermediate algebra course adheres to the formula that has made the Aufmann developmental texts so reliable for both students and instructors. The text's clear writing style, emphasis on problem-solving strategies, and proven Aufmann Interactive Method--in an objective-based framework--offer guided learning for both lecture and self-paced courses. The completely integrated learning system is organized by objectives. Each chapter begins with a list of learning objectives, which are woven throughout the text, in Exercises, Chapter Tests, and Cumulative Reviews, as well as through the print and multimedia ancillaries. The result is a seamless, easy-to-follow learning system. This special MEDIA ENHANCED EDITION now comes with Enhanced WebAssign and flash videos for every end of chapter test question available online through the student website.
Table of Contents
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Real Numbers and Variable Expressions
Introduction to Integers
Operations with Integers
Rational Numbers
Exponents and the Order of Operations Agreement
Variable Expressions
Translating Verbal Expressions into Variable Expressions
Solving Equations and Inequalities
Introduction to Equations
General Equations
Application Problems
Geometry Problems
Markup and Discount Problems
Applications: Problems Involving Percent
Inequalities in One Variable
Absolute Value Equations and Inequalities
Linear Functions and Inequalities in Two Variables
The Rectangular Coordinate System
Introduction to Functions
Linear Functions
Slope of a Straight Line
Finding Equations of Lines
Parallel and Perpendicular Lines
Inequalities in Two Variables
Systems of Equations and Inequalities
Solving Systems of Linear Equations by Graphing and by the Substitution Method
Solving Systems of Linear Equations by the Addition Method
Solving Systems of Equations by Using Matrices and by Using Determinants
Application Problems
Solving Systems of Linear Inequalities
Polynomials
Introduction to Polynomials
Multiplication of Monomials
Multiplication of Polynomials
Integer Exponents and Scientific Notation
Division of Polynomials
Factoring
Common Factors
Factoring Polynomials of the Form x2 + bx + c
Factoring Polynomials of the Form ax2 + bx + c
Special Factoring
Solving Equations by factoring
Rational Expressions
Introduction to Rational Expressions
Operations on Rational Expressions
Complex Fractions
Rational Equations
Proportions and Variation
Literal Equations
Rational Exponents and Radicals
Rational Exponents and Radical Expressions
Operations on Radical Expressions
Radical Functions
Solving Equations Containing Radical Expressions
Complex Numbers
Quadratic Equations and Inequalities
Solving Quadratic Equations by Factoring or by Taking Square Roots
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Algebra 1
By Spencer Perry
Release Date: 2014-04-26
Genre: Study Aids
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Description
This book is a general reference guide for Algebra 1. Each of the six chapters cover a specific area of interest in Algebra 1: Foundations of Functions, Properties & Attributes of Functions, Symbols & Unknown Variables, Manipulating & Solving Equations, Properties of Linear Functions, and Quadratic Functions & Their Functions. At the end of each chapter, there are quizzes to help review the ideas presented in the chapter. The book also contains many interactive graphs to help reinforce the topics covered in each chapter.
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Pre-Calculus: Trigonometry
Pre-Calculus: Trigonometry
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University of California, Irvine
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Reading: Unit Circle and Solving Right Triangles: Reading and Exercises
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Sunday, 7 August 2016
Iteration and the new GCSE
So my blog frequency has become significantly lower recently - believe it or not I have been even busier than normal writing and sourcing resources for our new Year 7 mixed ability course, putting together topic tests for Year 7 and Year 11 (thanks AQA for all of your work putting your own topic tests together - I have stolen most of them!) and then writing the homework booklets for all three of my Year 11 schemes for term 1. All in all today is actually the first day since we broke up (bear in mind that Leicestershire broke up on 15/07/16) that I haven't been doing school work of some description - as a reward for finishing the homework booklets a day early I gave myself the weekend off!
One of the things that I have had to sort out as part of writing the tests and homework booklets is finding sources of questions on iteration and numerical methods for solving equations, so I thought I would share some of the better ones here, and also offer some tips on designing your own.
1) Check out A-Level worksheets - I dug through some of my old Core 3 resources (unfortunately I haven't taught A-Level for the last two years since moving to my new school) and found an ample supply of iterative formulae that were used. Some of them weren't suitable (too many natural logarithms and exponential functions) but many were with just some small adaptations. In particular a lot of A-Level questions ask pupils to show there is a root in a given interval using a change of sign approach and also ask pupils to justify why a given formula will converge to a solution. As far as I have seen the GCSE will not ask pupils to use a change of sign to show there is a root in a given interval,although to be fair it wouldn't be a bad thing to do with pupils as a way of tying roots of equations, graphs and iteration together. In addition it will definitely not require pupils to justify why a given iterative formula will converge, as this requires knowledge of calculus - although again it might be nice for the best mathematicians to look at this as a way of linking rates of change to iterative formulae. For some examples questions made from A-Level worksheets check out my Year 11 Higher or Higher+ term 1 and 2 homework booklets - there are a few pages on Iterative methods with a few exam style questions all taken from A-Level worksheets or similar.
2) Exam board website - we are using the AQA exam board and they have a multitude of resources available for use with iteration. If you don't know AQA's site it is well worth getting yourself signed up for it. Browse to the New GCSE (8300) and select the Numerical methods section under Higher GCSE Algebra resources and you will find worksheets with some decent enough questions, as well as their topic test with some more. The one I really like though is their 'bridging' material, which can again be found under the New GCSE (8300) page. They have a lovely document in there called Pocket 4, which is all about iterative formulae. Although billed as a KS3 bridging material I would definitely save some of the later activities and use them during the actual GCSE teaching.
3) Linked Pair Pilot - Although trial and improvement is not mentioned specifically in the new GCSE specifications, it is still being used under the guise of a numerical method. The Linked Pair Pilot papers, in particular the Applications 2 paper, has some nice examples of trial and improvement used to solve practical problems in geometry and other areas, which is nicely in keeping with the aims of the new GCSE. Often they have the tables printed on a separate page as well, which means you can feel free to not use them for the more confident mathematicians, just giving them the page with the question setup on instead.
5) Design your own - It isn't actually that tricky to design iteration questions, although there are a couple of things to beware of to ensure the question will work. Start with a polynomial set equal to 0; cubics are good as they can't be solved using other GCSE techniques (except if it has an obvious factorisation) and are guaranteed to have at least one root. From here you can do one of two things:
(a) Use the Newton-Raphson formula:
The examples of exam questions I have seen using this formula have had the subtraction simplified to give a single fraction as the iterative formula, however I cannot see any reason why pupils couldn't be given the formula with the basic substitution already done and told to do a 'show that', i.e.
(b) Rearrange - the classic method for generating iterative formula is to rearrange the equation
f(x) = 0 into the form x = g(x). This is being used a lot in the new GCSE practice and sample materials which include asking pupils to show how a given rearrangement can be arrived at:
If you use this approach to design your own question then a word of caution - not all possible rearrangements will find all of the roots. The best things do here is to check the graph of the rearranged function for the gradient in the locale of the root. The rule goes that if the gradient of the rearranged function around the root you are looking for is in the range (-1,1) then the formula will converge to the root there - if not then it wont. For example for the problem above the graphs of the original function and the rearranged function look like this:
where the red graph is the original cubic and the blue graph is the square root function. You can see that there are actually three roots to the cubic, corresponding to the three points that the root function intercepts the line y = x. However the given rearrangement wont find the root that is slightly bigger than 2, as the gradient of the root curve is greater than 1 around that point. The rearrangement will quite happily find the other roots in the intervals (0,1) and (-1,0) as the gradients are close to 0 around these points. It is definitely worth just checking this if you are going to design your own rearrangement questions as you wouldn't want to give your pupils rearrangement that doesn't work!
1 comment:
One of the most popular posts on my own blog is that on Iterative Techniques (updated July 2016); I think there is a real need for resources on this as it is completely new at this level. Teaching Year 10 last year, I found AQA's Bridging the Gap you mentioned excellent. | 677.169 | 1 |
Algebra 2 Lesson Plan In this unit, heterogeneous
groupings of students will
create a PowerPoint
presentation on the properties
of conic sections.
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CGN 3421 - Computer Methods GurleyNumerical Methods - Lecture 1page 48 of 53Matrix methods - solving simultaneous equations•We're familiar with the equation relating force and displacement for a spring as . •We've also learned how to analyze a simple truss structure in statics•All a truss is permitted to do is stretch or compress. So one way of analyzing truss type structures is tovisualize the structure as a group of attached springs. For example, the simple truss below can be repre-sented as springs. •Usually engineers know what the constant is for each spring, and the forces acting on the structure.
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Ch 1: High School Algebra: Solving Math Word Problems
About This Chapter
Watch online math video lessons to learn strategies for solving word problems. Take the self-assessment quizzes that follow each lesson to verify your understanding of the key concepts presented in this chapter.
If the dreaded word problem is the only thing standing between you and homework freedom, check out the lessons included in this chapter. Over the course of a few short videos, experienced instructors outline techniques you can use to improve your understanding of what's being asked and visualize the type of equation needed to find the solution. There are even multiple-choice quizzes attached to the end of each lesson, so you can put these strategies to the test and confirm your understanding of each lesson's main ideas. By the time you reach the end of this chapter, you should be familiar with the following:
Processes involved in solving single- and multi-step word problems
Methods for gathering and visualizing your thoughts
Tips for rephrasing word problems
Video
Objectives
Solving Word Problems: Steps & Examples
Master the step-by-step approach used to solve word problems. Get practice applying these procedures to a sample problem.
Solving Word Problems with Multiple Steps
Find out how to break down multi-step word problems and solve them one step at a time.
Restating Word Problems Using Words or Images
Learn why creating drawings, notes and diagrams can help you tackle word problems.
Personalizing a Word Problem to Increase Understanding
Explore the benefits of relating word problems to situations you might encounter in your own life | 677.169 | 1 |
No need to wait for office hours or assignments to be graded to find out where you took a wrong turn.Introductory Combinatorics, 4th Edition. Richard A. Brualdi.Solutions Manuals are available for thousands of the most popular college and high school textbooks in subjects such as Math, Science ( Physics, Chemistry, Biology ), Engineering ( Mechanical, Electrical, Civil ), Business and more.Introductory Combinatorics 5th Edition by Brualdi, Richard,.
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Calculus Concentrate by Russell A. Gordon
Description: The text represents one person's attempt to put the essential ideas of calculus into a short and concise format. It may not appeal to a wide range of mathematicians, but it should provide most students with a good foundation in calculus.
Similar booksYet Another Calculus Text by Dan Sloughter Introduction to calculus based on the hyperreal number system for readers who are already familiar with calculus basics. It covers hyperreals, continuous functions, derivatives, geometric interpretation, optimization, integrals, applications, etc. (11700 views)
Sequences and Power Series: Guidelines for Solutions of Problems by Leif Mejlbro - BookBoon Guidelines for solutions of problems concerning sequences and power series. It is not an alternative textbook, but it can be a useful supplement to the ordinary textbooks. The text presupposes some knowledge of calculus of functions in one variable. (8514 views)
The Calculus for Beginners by John William Mercer - Cambridge University Press The author has been guided by the conviction that it is much more important for the beginner to understand clearly what the processes of the Calculus mean, and what it can do for him, than to acquire facility in performing its operations. (7484 views) | 677.169 | 1 |
MAT 260: Problem Solving in Mathematics
About the course
Summary.
This course is intended for students preparing for the Putnam exam, a
competitive national mathematics exam for undergraduates which is held each
December. Students actively solve challenging problems in plane geometry, basic
number theory, and calculus, and write precise arguments. Relevant preparation
for problem-solving is provided in the course.
Grading.
Grades will be based primarily on class participation and effort in
solving problems. There is no final exam and no fixed homework, but you
are expected to try your best to solve some of the problems offered for
next week's discussion. Familiarity with basic concepts of analysis, algebra,
and geometry is assumed — please consult with me if you are not sure
about your background.
Books.
For the most recent collection of Putnam problems, check out The William Lowell Putnam Mathematical Competition 1985-2000:
Problems, Solutions, and Commentary, by Kiran Kedlaya, Bjorn Poonen, and
Ravi Vakil, available online for example from this link. Another very useful book is Putnam and Beyond, by Razvan Gelca and Titu Andreescu, available online from this link.
Time and location
Putnam Competition 2017
This year's William Lowell Putnam Competition will take place on Saturday, December 2, 2017, from 10am to 1pm and from 3pm to 6pm. The deadline for registration is October 15, 2017. If you are interested in participating, please talk to me.
Policy Statements
Disability Support Services.
If you have a physical, psychological, medical or learning disability that
may impact your course work, please contact Disability Support Services,
Educational Communications Center Building, room 128, at (631) 632-6748. They
will determine with you what accommodations, if any, are necessary and
appropriate. All information and documentation is confidential. Students who
require assistance during emergency evacuation are encouraged to discuss their
needs with their professors and Disability Support Services. For procedures and
information go to this website.
Academic Integrity.
Each student must pursue his or her academic goals honestly and be
personally accountable for all submitted work. Representing another person's
work as your own is always wrong. Faculty are required to report any suspected
instances of academic dishonesty to the Academic Judiciary. For more
comprehensive information on academic integrity, including categories of
academic dishonesty, please refer to the academic
judiciary website.
Critical Incident Management.
Stony Brook University expects students to respect the rights, privileges,
and property of other people. Faculty are required to report to the Office of
Judicial Affairs any disruptive behavior that interrupts their ability to
teach, compromises the safety of the learning environment, or inhibits
students' ability to learn. | 677.169 | 1 |
Communication in mathematics
Tag Archives: Labs
If you read my introduction, you probably already knew which lab I would pick. This lab is talking about Integration. It explains different way we can integrate, such as: Riemann Sum (Rectangle), Trapezoid Sum (Trapezoid, and Simpson's Rule (parabola (dotted)). Integrations help us to calculate the approximate area underneath a given curve.
A question that will come up while working on this lab may be about comparing the few integration techniques listed previously. We will also be working with the difference of left hand, right hand and midpoint integration.
The reason I would like to work on this lab is because I have taken two and half calculus classes working with integrals and I would like to see if this opens my mind more to what exactly we are doing when we integrate. | 677.169 | 1 |
This guide book to mathematics contains in handbook form the fundamental working knowledge of mathematics which is needed as an everyday guide for working scientists and engineers, as well as for students. Easy to understand, and convenient to use, this guide book gives concisely the information necessary to evaluate most problems which occur in concrete applications. For the 4th edition, the concept of the book has been completely re-arranged. The new emphasis is on those fields of mathematics that became more important for the formulation and modeling of technical and natural processes, namely Numerical Mathematics, Probability Theory and Statistics, as well as Information Processing.
Ingram "Handbook of Mathematics" contains the fundamental working knowledge of mathematics which is needed as an everyday guide for both students and professionals in the areas of physics, mathematics, and engineering. Covering material from the introductory level through to more advanced applications, this classic handbook also presents tables of many important functions. 95 illus Pub: 5/97.
From the reviews of the fourth edition:
"The handbook by I. N. Bronshtein et al. will be a great resource book for statisticians. This book has 745 figures and 142 tables. … This book is well composed, and the items are very precise with excellent references for readers to follow up for more details. … I recommend them strongly to the statistical community. I enjoyed reading … . I am sure both applied and theoretical statisticians will benefit significantly … and hence they should consider having them in their collection." (Ramalingam Shanmugam, Journal of Statistical Computation and Simulation, Vol. 76 (2), 2006) | 677.169 | 1 |
The Khan Academy is a non-profit that produces a lot of learning videos on youtube on quite a few different subjects. They seem to be popular for math in particular and take you from basic math to Linear Algebra. | 677.169 | 1 |
Geometry: Euclid and Beyond
Offers an opportunity to understand one of the great thinkers of western civilization, Euclid. This book includes such topics as the introduction of coordinates, the theory of area, geometrical constructions and finite field extensions, history of the parallel postulate, various non-Euclidean geometries, and the regular and semi-regular polyhedra. | 677.169 | 1 |
The module will assume prior knowledge equivalent to the following module. If you have not taken these modules you should consult the module descriptor
Level HE1 (FHEQ Level 4) Essential Mathematics
Module overview
This module builds on the Essential Mathematics module to develop further mathematical and computational skills as an aid to understanding and exploring physics concepts. The mathematics Units of Assessment are taught in lecture-based classes with associated workshop sessions, and cover multi-variable calculus, Fourier Series
The computational part of the course consists of a series of assessed exercises, with classroom support, which develop computational problem solving skills, and link in with the mathematics covered elsewhere in the module and in the prerequisite module.
Module aims
enable students to classify and solve simple first- and second-order ordinary differential equations, including the concepts and appreciation of convergence tests of numerical series.
Enable students to compute the coefficients of Fourier series.
provide an understanding of functions of more than one variable, their derivatives, and the location stationary points of functions of two variables, and to be able to classify them as maxima, minima or saddle points.
enable the use multiple integrals to calculate surface and volume properties
develop skills in and experience of developing computational solutions to problems in mathematics and physics.
produce well-structured and well-though-out program solutions to problems, drawing on examples from the mathematical physics part of the module,
present results from the programs in appropriate graphical format.
Learning outcomes
Attributes Developed
Test numerical and functional series for their convergence properties
KC
Be able to solve simple first- and second-order ordinary differential equations.
KC
Be able to compute and manipulate partial derivatives
KC
Be able to compute Fourier series coefficients
KC
Be able to use matrices to represent and solve sets of linear equations
K
Be able to evaluate derivatives and integrals of two- and multi-variable functions and be able to apply these to find maxima and minima and to the calculation of physical quantities such as volume, mass, moments of inertia and centre of gravity of various geometric shapes with both homogeneous and inhomogeneous densities.
KC
Be able to use computational techniques to solve unseen problems in mathematics and physics, confidently using appropriate syntax and algorithm design,
equip students with subject knowledge
develop skills in applying subject knowledge to unseen problems in mathematics, including problems with a direct physical application
ensure that students are able to take problems in
ability to design and implement computational solutions to given mathematical problems
ability to tackle unseen mathematical problems using known methods
Thus, the summative assessment for this module consists of:
5 computational exercises covering different programming skills, and surveying different areas of the mathematics syllabus with deadlines spread equally throughout semester
a mid-semester mathematics test (1 hr)
a final examination in mathematics (1.5 hrs)
Formative assessment
Verbal feedback is given in tutorial sessions. Continuous feedback given in supervised computational classes. Mid-semester maths test provides feedback as well as contributing to | 677.169 | 1 |
Representation of real numbers on a line. Complex numbers—basic properties, modulus, argument, cube roots of unity. Binary system of numbers. Conversion of a number in decimal system to binary system and vice-versa. Arithmetic, Geometric and Harmonic progressions. Quadratic equations with real coefficients. Solution of linear inequations of two variables by graphs. Permutation and Combination. Binomial theorem and its applications. Logarithms and their applications.
2. MATRICES AND DETERMINANTS :
Types of matrices, operations on matrices. Determinant of a matrix, basic properties of determinants. Adjoint and inverse of a square matrix, Applications-Solution of a system of linear equations in two or three unknowns by Cramer's rule and by Matrix Method.
3. TRIGONOMETRY :
Angles and their measures in degrees and in radians. Trigonometrical ratios. Trigonometric identities Sum and difference formulae. Multiple and Sub-multiple angles. Inverse trigonometric functions. Applications-Height and distance, properties of triangles.
4. ANALYTICAL GEOMETRY OF TWO AND THREE DIMENSIONS: Point in a three dimensional space, distance between two points. Direction Cosines and direction ratios. Equation two points. Direction Cosines and direction ratios. Equation of a plane and a line in various forms. Angle between two lines and angle between two planes. Equation of a sphere.
5. DIFFERENTIAL CALCULUS :
Concept of a real valued function–domain, range and graph of a function. Composite functions, one to one, onto and inverse functions. Notion of limit, Standard limits—examples. Continuity of functions—examples, algebraic operations on continuous functions. Derivative of function at a point, geometrical and physical interpretation of a derivative—applications. Derivatives of sum, product and quotient of functions, derivative of a function with respect to another function, derivative of a composite function. Second order derivatives. Increasing and decreasing functions. Application of derivatives in problems of maxima and minima.
Definition of order and degree of a differential equation, formation of a differential equation by examples. General and particular solution of a differential equations, solution of first order and first degree differential
equations of various types—examples. Application in problems of growth and decay.
7. VECTOR ALGEBRA :
Vectors in two and three dimensions, magnitude and direction of a vector. Unit and null vectors, addition of vectors, scalar multiplication of a vector, scalar product or dot product of two vectors. Vector product or cross product of two vectors. Applications—work done by a force and moment of a force and in geometrical problems. 'B'—GENERAL KNOWLEDGE (Maximum Marks—400)
The question paper on General Knowledge will broadly cover the subjects : Physics, Chemistry, General Science, Social Studies, Geography and Current Events.
- The syllabus given below is designed to indicate the scope of these subjects included in this paper. The topics mentioned are not to be regarded as exhaustive and questions on topics of similar nature not specifically mentioned in the syllabus may also be asked. Candidate's answers are expected to show their Temperature and Heat, change of State and Latent Heat, Modes of transference of Heat.
Sound waves and their properties, Simple musical instruments. Rectilinear propagation of Light, Reflection and refraction.
Spherical mirrors and Lenses, Human Eye..
Material used in the preparation of substances like Soap, Glass, Ink, Paper, Cement, Paints, Safety Matches and Gun-Powder.
Elementary ideas about the structure of Atom, Atomic Equivalent and Molecular Weights, Valency.
Section 'C' (General Science)
Difference between the living and non-living. Basis of Life—Cells, Protoplasms and Tissues. Growth and Reproduction in Plants and Animals.
Elementary knowledge of Human Body and its important organs.
Common Epidemics, their causes and prevention.
Food—Source of Energy for man. Constituents of food, Balanced Diet. The Solar System—Meteors and Comets, Eclipses.
Achievements of Eminent Scientists.
Section 'D' (History, Freedom Movement etc.)
A broad survey of Indian History, with emphasis on Culture and Civilisation. Freedom Movement in India.
Elementary study of Indian Constitution and Administration.
Elementary knowledge of Five Year Plans of India. Panchayati Raj, Co-operatives and Community Development.
Bhoodan, Sarvodaya, National Integration and Welfare State, Basic Teachings of Mahatma Gandhi, Socialism and Communism. Role of India in the present world.
Section 'E' (Geography)
The Earth, its shape and size. Lattitudes and Longitudes, Concept of time. International Date Line. Movements of Earth and their effects.
Origin of Earth. Rocks and their classification; Weathering— Mechanical and Chemical, Earthquakes and Volcanoes.
Ocean Currents and Tides Atmosphere and its composition; Temperature and Atmospheric Pressure, Planetary Winds, Cyclones and Anti-cyclones; Humidity; Condensation and Precipitation; Types of Climate, Major Natural regions of the World.
Regional Geography of India—Climate, Natural vegetation. Mineral and Power resources; location and distribution of agricultural and Industrial activities.
Important Sea ports and main sea, land and air routes of India. Main items of Imports and Exports of India.
Section 'F' (Current Events)
Knowledge of Important events that have happened in India in the recent years.
Current important world events.
Prominent personalities—both Indian and International including those connected with cultural activities and sports. | 677.169 | 1 |
How you speak and write can say a lot about you. Grammar makes a lasting impression, but learning it can seem like a never-ending parade of complicated and contradictory rules! That's why we at The Princeton Review created Grammar Smart—instead of boring you with countless rules and confusing grammatical terms, this book takes a fun approach to showing the logic behind each correct sentence.
Even in a world where every cell phone is also a calculator, basic math competency is a must! In this book, you'll learn how to efficiently solve common problems and effortlessly perform foundational math operations like addition, subtraction, multiplication, and division. Once you've got that down, we'll go over how to handle the scary stuff—like exponents, square roots, geometry, and algebra. | 677.169 | 1 |
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Equation graph plotter - EqPlot 1.3.10 review
Equation graph plotter - EqPlot, with this expert application it is possible to graphically review equations, by putting a large number of equations.
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Publisher's description
EqPlot plots 2D graphs from complex equations. The application comprises algebraic, trigonometric, hyperbolic and transcendental functions. EqPlot can be used to verify the results of nonlinear regression analysis program.
Graphically Review Equations:
EqPlot gives engineers and researchers the power to graphically review equations, by putting a large number of equations at their fingertips. Up to ten equations could be plotted at the same time, so that intersections and domains could be studied visually.
Understandable and convenient interface:
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JEE Main Syllabus 2015
Joint Entrance Exam is scheduled by CBSE in the month of April, every year, and candidates will have not even a month to prepare for it after their board exams. However, this should not be troubling, as most of JEE Main syllabus remains same as that of class XI and XII; CBSE Board. From below, you can see the entire syllabus of JEE Main.
JEE Main Syllabus - Importance
There is no need to state how important syllabus is, however, optimum utilization, of it, should be ensured. Candidates can do a lot more with JEE Main syllabus, rather than just knowing the topics. With the help of it, candidates can also know which topics possess more weightage and which can be ignored for a time being. As per this information, candidates can prepare a well planed routine to complete the courses to crack JEE Main.
The exam will cover topics from Mathematics, Physics and Chemistry. Detailed topics are listed below:
JEE Main Syllabus (Mathematics):-
Sets, Relations And Functions: Sets and their representation; Union, intersection and complement of sets and their algebraic properties; Power set; Relation, Types of relations, equivalence relations,functions; One-one, into and onto functions, composition of functions
Complex Numbers and Quadratic Equations: Complex coefficients, nature with given roots.
Matrices And Determinants: MatPermutations And Combinations: Fundamental principle of counting, permutation as an arrangement and combination as selection, Meaning of P (n,r) and C (n,r), simple applications
Mathematical Induction: Principle of Mathematical Induction and its simple applications
Binomial Theorem And Its Simple Applications: Binomial theorem for a positive integral index, general term and middle term,properties of Binomial coefficients and simple applications
Integral Calculus: Integral as an anti derivative. Fundamental integrals involving algebraic, trigonometric, exponential and logarithmic functions. Integration by substitution, by parts and by partial fractions. Integration using trigonometric identities. Integral as limit of a sum. Fundamental Theorem of Calculus. Properties of definite integrals. Evaluation of definite integrals, determining areas of the regions bounded by simple curves in standard form
Differential Equations: Ordinary differential equations, their order and degree. Formation of differential equations. Solution of differential equations by the method of separation of variables, solution of homogeneous and linear differential equations
Coordinate Geometry: Cartesian system of rectangular coordinates 10 in a plane, distance formula, section formula, locus and its equation, translation of axes, slope of a line, parallel and perpendicular lines, intercepts of a line on the coordinate axes.
Three Dimensional Geometry: Coord
Vector Algebra: Vectors and scalars, addition of vectors, components of a vector in two dimensions and three dimensional space, scalar and vector products, scalar and vector triple product.
Statistics and Probability: Measures of Dispersion: Calculation of mean, median, mode of grouped and ungrouped data calculationLaws Of Motion: ForceWork, Energy And Power: Work done by a constant force and a variable force; kinetic and potential energies, work energy theorem, power. Potential energy of a spring, conservation of mechanical energy, conservative and non-conservative forces; Elastic and inelastic collisions in one and two dimensions
Rotational Motion Center of mass of a two-particle system, Center
Gravitation The universal law of gravitation. Acceleration due to gravity and its variation with altitude and depth. Kepler's laws of planetary motion. Gravitational potential energy; gravitational potential. Escape velocity. Orbital velocity of a satellite. Geo-stationary satellites
Thermodynamics: Thermal equilibrium, zeroth law of thermodynamics, concept of temperature. Heat, work and internal energy. First law of thermodynamics. Second law of thermodynamics: reversible and irreversible processes. Carnot engine and its efficiency
Kinetic Theory Of Gases: Equation of state of a perfect gas, work doneon compressing a gas.Kinetic theory of gases -assumptions
Current Electricity: Electric current, Drift velocity, Ohm's law, Electrical resistance, Resistances of different materials, V-I characteristics of Ohmic and nonohmic conductors, Electrical energy and power, Electrical resistivity, Colour code for resistors; Series and parallel combinations of resistors; Temperature dependence of resistance. ElectricElectrostatics Electric charges: Conservation of charge, Coulomb's law-forces between two point charges, forces Electric Conductors and insulators, Dielectrics and electric polarization, capacitor, combination of capacitors in series and in parallel, capacitance of a parallel plate capacitor with and without dielectric medium between the plates, Energy stored in a capacitor
Magnetic Effects Of Current And Magnetism: B Force on a current carrying conductor in a uniform magnetic field. Force between two parallel current carrying conductors-definition of ampere. Torque experienced by a current loop in uniform magnetic field; Moving coil galvanometer, its current sensitivity and conversion to ammeter and voltmeter. Current loop as a magnetic dipole and its magnetic dipole moment. Bar magnet as an equivalent solenoid, magnetic field lines; Earth's magnetic field and magnetic elements. Para-, dia- and ferro- magnetic substances. Magnetic susceptibility and permeability, Hysteresis, Electromagnets and permanent magnets.
Optics: Refifving powers.
Chemical Thermodynamics: Fund
Enthalpies of bond dissociation, combustion, formation, atomization, sublimation, phase transition, hydration,ionization and solution. Second law of thermodynamics; Spontaneity of processes; DS of the universe and DG of the system as criteria for spontaneity, DG0 (Standard Gibbs energy change) and equilibrium constant.
Electrochemical cells:.
Chemical Kinetics Rate – Arrhenius theory, activation energy and its calculation, collision theory of bimolecular gaseous reactions (no derivation).
Section B: Inorganic Chemistry
Classification Of Elements And Periodicity In Properties: Modem periodic law and present form of the periodic table, s, p, d and f block elements, periodic trends in properties of elements atomic and ionic radii, ionization enthalpy, electron gain enthalpy, valence, oxidation states and chemical reactivity
General Principles And Processes Of Isolation Of Metals: Modes of occurrence of elements in nature, minerals, ores; Steps involved in the extraction of metals -concentration, reduction (chemical and electrolytic methods) and refining with special reference to the extraction of Al, Cu, Zn and Fe; Thermodynamic and electrochemical principles involved in the extraction of metals.
Hydrogen: Position of hydrogen in periodic table, isotopes, preparation, properties and uses of hydrogen; Physical and chemical properties of water and heavy water; Structure, preparation, reactions and uses of hydrogen peroxide; Hydrogen as a fuel.
S - Block Elements (Alkali And Alkaline Earth Metals): Group - 1 and 2 Elements General introduction, electronic configuration and general trends in physical and chemical properties of elements, anomalous properties of the first element of each group, diagonal relationships. Preparation and properties of some important compounds - sodium carbonate and sodium hydroxide; Industrial uses of lime, limestone, Plaster of Paris and cement; Biological significance of Na, K, Mg and Ca.
P - Block Elements: Group - 13 to Group 18 Elements
General Introduction: Electronic configuration and general trends in physical and chemical properties of elements across the periods and down the groups; unique behaviour of the first element in each group. Groupwise study of the p – block elements | 677.169 | 1 |
Algebra in 15 Minutes a Day by LearningExpress LLC Editors
By LearningExpress LLC Editors
You do not have to be a genius to turn into an algebra ace-you can do it in precisely quarter-hour an afternoon jam-packed with brief and snappy classes, Junior ability developers: Algebra in quarter-hour an afternoon makes studying algebra effortless. it truly is real: making experience of algebra does not need to take many years . . . and it does not need to be tricky! in precisely one month, scholars can achieve services and straightforwardness in the entire algebra suggestions that regularly stump scholars. How? every one lesson supplies one small a part of the larger algebra challenge, in order that each day scholars construct upon what was once discovered the day prior to. enjoyable factoids, catchy reminiscence hooks, and beneficial shortcuts ensure that each one algebra idea turns into ingrained. With Junior ability developers: Algebra in quarter-hour an afternoon, sooner than you recognize it, a suffering pupil turns into an algebra pro-one step at a time. in precisely quarter-hour an afternoon, scholars grasp either pre-algebra and algebra, together with: Fractions, multiplication, department, and different simple math Translating phrases into variable expressions Linear equations actual numbers Numerical coefficients Inequalities and absolute values platforms of linear equations Powers, exponents, and polynomials Quadratic equations and factoring Rational numbers and proportions and masses extra! in exactly quarter-hour an afternoon, scholars grasp either pre-algebra and algebra, together with: Fractions, multiplication, department, and different simple math Translating phrases into variable expressions Linear equations genuine numbers Numerical coefficients Inequalities and absolute values structures of linear equations Powers, exponents, and polynomials Quadratic equations and factoring Rational numbers and proportions and lots more and plenty extra! as well as the entire crucial perform that children have to ace school room assessments, pop quizzes, category participation, and standardized checks, Junior ability developers: Algebra in quarter-hour an afternoon presents mom and dad with a simple and available technique to support their youngsters exce
Finally, subtract: 20 – 16 = 4. The expression 2(6 + 4) – 42 is equal to 4. TIP: If there is more than one operation inside a set of parentheses, use the order of operations to tell you which operation to perform first. In the expression (5 + 4(3)) – 2, addition and multiplication are both inside parentheses. Because multiplication comes before addition in the order of operations, we begin by multiplying 4 and 3. qxd:JSB 12/18/08 11:45 AM Page 49 single-variable expressions 49 Practice 1 Evaluate each expression. | 677.169 | 1 |
Introduction
Tip:
"GrafEq" has the same pronounciation as "graphic".
GrafEq is designed for two kinds of mathematicians: the teacher and the student.
Teachers will appreciate features such as the
variable font size and the ability to cut and paste expressions and graphs to text documents;
students will appreciate the standard mathematics notation and
ease of use of the program.
Everyone will appreciate GrafEq's mathematical
prowess with its broad range of relations that can be displayed with confidence.
GrafEq is an exceptional tool for exploring mathematics.
As an introduction, here are some characteristics underlying the design of GrafEq:
Mathematical
Algebraic specifications are accepted in a broad variety of intuitive formats.
For instance, a linear equation can be entered in the form
y = mx + b, (y-k)/(x-l) = m,
or ax + by + c = 0.
Honest
Any inaccuracy is explicitly revealed.
For instance, although a pixel is very small, it is still an area rather than a point;
therefore, the 'coordinates' of a pixel are reported using a range of values,
such as (5.210±0.002, 17.001±0.002).
Valid
Graphs plotted are correctly. For instance,
the graph of y≤sqrt(16-x2) differs from the graph of
y≯sqrt(16-x2), because '≤' is not equivalent
to '≯'.
Reliable
No solutions will be missed: A pixel is off only if it contains no solutions at all;
it stays on if it may possibly contain one or more solutions for the graph's
active relations.
Extraneous solutions are eliminated when graphing is complete.
The program can even use multiple colours to distinguish the pixels that have been
proven to contain solutions, the pixels that have been proven to not contain solutions,
and the pixels that are currently undecided.
Hint:
A closed and open version of a relation will often
have identical graphs on the computer's monitor.
For example: the graph of x<5 will be identical to that of x≤5
unless the edge x=5 falls right between two adjacent pixels.
Multi-faceted
Each GrafEq graph can be presented in four forms:
Symbolic - an algebraic definition, typeset using mathematics notation, and
presented in one or more algebraic
relation windows. (Each relation can contain one or more constraints,
and each constraint can be an equation, an inequality, a set description, or
a conditional definition.)
Structural - a flow chart, or tree interpretation, presented in one or more
structural relation windows. (Each relation is associated with an algebraic
window and a structural window.)
Graphical - a cartesian or polar representation, presented in one or more
view windows.
Printed - physical copies can be printed directly from GrafEq by means of
its page editing window, or indirectly from other programs such as
word processors or drawing applications by first copying the relations and/or
graph views from GrafEq, and then pasting into the other program.
This manual provides a convenient and complete reference to the various features
and some anticipated application of such features.
Various tutorials, from beginner to expert levels, are also useful tools and a starting
point for a rewarding journey of mathematics exploration.
Hint:
This manual presumes that the reader will have access to general information
about the computer and its operating system, as well as mathematical terminology and concepts.
If the illustrations in this manual do not match your display,
check the settings in the File / Preference manual.
The screen shots are from a computer running Windows 98.
GrafEq will run on a black and white display, in which case some references
to colours should instead be references to patterns. | 677.169 | 1 |
Algebra 1 homework 35.1
Pre-algebra Algebra Integrated math Geometry Algebra 2 Trigonometry Precalculus Calculus Statistics Probability College algebra Discrete math Linear algebra Differential equations Business math Advanced mathematics Science. On review days you will complete one review worksheet, and one review quiz. The worksheet and quiz are a little longer. But, you are only going to get one chance on the quiz. Do your best to get as many questions correct as possible. We want to see how much of the material you remember correctly so far.
How well you do on the review quizzes are a strong indicator of how well you will algebra 1 homework 35.1 on the actual Algebra 1 ECA test. Algebra 1 homework 35.1 Luck. This is due the first Tuesday after Mardi Algebra 1 homework 35.1. I look forward to reading and, perhaps, learning some interesting topology. Contributors, all leading experts in their fields, provide theoretical discussions, practical insights and recommendations, historical perspectives and.
The Asian American Educational Experience. Placement exam solutionsQuiz 1 is on Monday, September 12. Quiz 1 with solutionsQuiz 2 is algehra Friday, September 30. Quiz 2 with solutionsQuiz 3 is on Monday, October 31. Quiz 3 with solutionsQuiz 4 is on Monday, November 21. Quiz 4 with solutionsExam 1 (Chapter 1) is on Monday, September 19. Get help and answers to writing and introduction for a report math problem including algebra, trigonometry, geometry, calculus, trigonometry, fractions, solving expression, simplifying expressions and more.Get answers to math questions.
Math 110 Homework Assignments College AlgebraMath 110Homework AssignmentsWeek 1Read sections algebra 1 homework 35.1 and 1.3byThursday, January 27. For those of you who have forgotten someof the details about factoring and working with rational expressions,sections P5 and P will be very helpful. When reading a section,it helps if you first read an example, then copy it to a piece of paperand try to work the example with the book closed. If you getstuck, then you can easily find out where you went wrong. SOLUTION: 1 3 5 are the first three terms of the first differences of a quadratic sequence.
The 7th term of the quadratic sequence is 35.1.determine the th and 5th terms of the quadrati. The 7th term of the hlmework sequence is 35.1.determine the th and 5th terms of the quadratic sequence (4)2.Determine the nth term of the quadratic sequence(5)Answer by KMST(4487) ( Show Source). For a few examplesPre-algebra Algebra Integrated math Geometry Algebra hoomework Trigonometry Precalculus Calculus Statistics Probability College algebra Discrete math Linear algebbra Differential equations Business math Advanced mathematics Science.
Students should be working to complete the Review Packet for the final exam ( ans key). The written portion ofthe final exam will be given on Friday, June 1st. The multiple choiceportion will take place during your scheduled exam time during examweek. HW: 5th Period (8th Period did not meet today due to Lunch on the Lawn) Finish the Using the Quadratic Formula WS for Tuesday.You can also begin working on the Review Homeworm for the final exam ( ans key). | 677.169 | 1 |
Course organization: This can be found on the assignment sheet (available
from Prof. Gottlieb, or MathSci 205 or 242).
Amendments to this are as follows: Instead of three hour exams at 100
points per exam, there will be only two hour exams at 100 points per
exam. About 150 additional points will be given on quizzes. 50 points
will be given for homework. With the 150 points on the final, this
adds up to the standard 550 points possible over the semester.
Each quiz will be worth 10 points, 5 points of which will be given for
attending the class when the quiz is given.
GOAL OF THE COURSE: Prof. Gottlieb will design the quizzes and parts of the hour
exams to encourage students to learn to read carefully and
accurately. The ability to read and speak and write with
mathematical precision is the most important attribute
necessary for success in mathematics.
EXAM 1 : February 24, Tuesday. Bring books and calculators and handouts to the exam. It covers the first chapter and section 1 of chapter 2.
EXAM 2 : April 16, Tuesday. Bring books and calculators and handouts to the
exam. It covers Chapter 2 and Chapter 3, except for 3.5 and 3.6.
ASSIGNMENTS
Assignment 1 : Using the algebra rule sheet, write out the meanings of equations 1 -10 and 15, 16, and 17 in
words. Do not use symbols or abbreviations. Also give an example of the
rule by using the numbers on the number sheet. If you cannot express a rule
in words, then give 26 examples of it instead.
Assignment 2 : Lessons 1 and 2 . Due 1/27/98
Assignment 3 : Lessons 3 - 4 and Lessons 5 - 6 . Due 2/3/98
Assignment 4 : Lessons 7 - 8 . Due 2/10/98
Assignment 5 : Lessons 9 - 10 and 11 - 12. Due 2/17/98
Assignment 6 : Lessons 13 -14 and 15 - 16. Due 2/24/98
Assignment 7 : Lesson 17. Due 3/4/98
Assignment 8 : Lessons 18, 19 - 20, 21 . Due 3/17/98
Assignment 9 : Lessons 22, 25 - 26 . Due 3/24/98
Assignment 10 : Lessons 27 - 28, 29 . Due 3/28/98
Assignment 11 : Lessons 30 - 31, 32 . Due 4/7/98
Assignment 12 : Lessons 38 - 39, 40 - 41. Due 4/14/98
Assignment 13 : Lessons 33 - 34, 35, 43. Due 4/28/98
QUIZZES
1. Which numbers on the number sheet are equal to each other?
2. Is f(x) = x^2 / x the same function as g(x) = x ?
3. How many solutions can there be for the simultaineous set of equations
y = f(x) and y = g(x) where f is a linear function and g is a quadratic
function?
4. How many entries in the chapter summary on page 82 have not been covered
in the class? (I. e., how many are defined in section 5?)
5. Different in the two classes. Essentially the difference between 3 as a
number and 3 as a function.
6. Add up the score of your test.
7. Given values for f(0), g(0), f'(0), g'(0), use quotient and product rules
to find the derivatives at 0 for quotients and products.
8. Given u^2 + v^2 = 1 + u^3 . Find du/dv when u = v = 1.
9. Least amount of information to give h'(3) when h(x) = f(g(x)).
10. Given g(x) and f'(x) , what is h'(1) equal to if h(x) = g(x)f(x) ?
11. A rocket ship is going at 1000 miles per hour after one hour.
How fast is it going after two hours if its fuel consumptiom is proportional to
time and its position is invesely proportional to it fuel consumption?
12. Intemediate question based on trying to solve 11.
13. Two function's graphs intersect at (2,2). Both functions are increasing
and concave upward at x = 2 . What is the sign of the derivative of the
product of the two functions at x = 2 ?
14. Add up test score
15. Two graphs intersect at x = 2 with the line y = x . What is f(2), f'(2),
f''(2) and similarly for g. Now y = f(x) is concave upward and y = g(x) is
concave downward. | 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 |
Description
For upper-level undergraduate and graduate two-semester sequence courses, or for any of several different one-term courses, depending on course emphasis.This text presents the fundamental numerical techniques used in engineering, applied mathematics, computer science, and the physical and life sciences in a manner that is both interesting and understandable. Each topic is illustrated by examples that range in complexity from very simple to moderate and are supported by geometrical or graphical illustrations where appropriate. A detailed algorithm is given for each method.show more | 677.169 | 1 |
Bob Miller's humor-laced, step-by-step learning tips make even the most difficult math problems routine. Based on more than 28 years of teaching and student feedback, his easy-to-grasp strategies give students much-needed confidence.
Third semester calculus is easier than Calc II--that's only part of the bonus this guide to mathematical fulfillment brings to today's attention--challenged student. Even vectors and integrals present no problem Bob Miller's Calc for the Clueless: Calc III by Bob Miller today - and if you are for any reason not happy, you have 30 days to return it. Please contact us at 1-877-205-6402 if you have any questions.
More About Bob Miller
Bob Miller was a lecturer in mathematics at City College of New York for more than 30 years. He has also taught at Westfield State College and Rutgers. His principal goal is to make the study of mathematics both easier and more enjoyable for students.
Bob Miller takes very difficult concepts and makes them extremely easy to grasp. It is the one math book that will not make you tear your hair out of your head! The author uses a bit of humor and emotion to make math understandable to the average layman. The books are meant to be a supplement to whatever math text you are using. After Bob explains the concepts you will breeze through your textbook's examples. A great book at a great price!
ARgH! Sep 7, 2005
In Calc III, the very first example has a mathematical typo! I spent ten minutes going crazy, not understanding how he reached his answer. The midpoint between 3 and -4 on the z axis is not and can never be -3.5!
My preview of next semester calc3 Dec 19, 2002
very easy to understand - very informative discriptions -
Useless, mostly Apr 26, 2002
Got this book to help me study for my Math103 (calc 3 in duke notation) final, and it really serves no purpose. In the author's attempts to make calculus "easy" he ignores many many things that are important if one is to do well in calc 3 (or at least calc 3 as it is taught here). I've just started going through the book, and have only covered vectors and quadric surfaces, but the book has already completely skipped on covering the "universal" formula for distance between 2 things, as well as not covering curvature, the components of acceleration... I can only imagine how much more the author skips later on. I realize that the author could not satisfy the requirements for every calc3 class at every college, but come on: this is stuff that comes straight from what is probably the most used calculus text, Edwards + Penney. If you have a decent textbook for your calc3 class, you're better of just taking the time to study from it : you'll end up in much better shape.
Multivariable simplicity Jan 21, 2002
After the stress of Taylor and Sequential Calculus in Calculus II, I assumed that the Calculus of several variables would be trivial. In essence, it was. However, the prohibition of calculator usage, coupled with the oddities of 3-D drawing made Calculus III a little more difficult. However, Bob Miller came to the rescue! With his relaxed prose style and user-friendly interface, Stokes, Gauss, and the like became quite simple. I was also concurrently in a Matrix Theory course, and his review of the idea of the R^n field and Abelian rules helped clarify other issues. All he needs now is to write a Linear Algebra | 677.169 | 1 |
books.google.com - This book provides a concise introduction to numerical concepts in engineering analysis, using FORTRAN, QuickBASIC, MATLAB, and Mathematica to illustrate the examples. Discussions include:matrix algebra and analysissolution of matrix equationsmethods of curve fitmethods for finding the roots of polynomials... Analysis
teaches readers to become proficient in FORTRAN or QuickBASIC programming to solve engineering problems
provides an introduction to MATLAB and Mathematica, enabling readers to write supplementary m-files for MATLAB and toolkits for Mathematica using C-like commands The book emphasizes interactive operation in developing computer programs throughout, enabling the values of the parameters involved in the problem to be entered by the user of the program via a keyboard. In introducing each numerical method, Engineering Analysis gives minimum mathematical derivations but provides a thorough explanation of computational procedures to solve a specific problem. It serves as an exceptional text for self-study as well as resource for general reference. | 677.169 | 1 |
Homework textbook answers
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Ms mission is to empower. Guided textbook solutions created by Chegg experts Learn from step by step solutions for over 22,000 ISBNs in Math, Science, Engineering, Business and moreSolutions in Algebra 1 (9780133706185). Nnect to a Tutor Now for Math help, Algebra help, English, ScienceMy Dear Aunt Sally. Er 2? Don't run up against deadlines with your homework. Pearson Prentice Hall and our other respected imprints provide educational materials, technologies, assessments and related services across the secondary curriculum! Ig Ideas Learning, LLC. Swers to ALL your math homework. Ch answer shows how to solve a textbook. Is the place to go to get the answers you need and to ask the questions you wantCPM Educational Program is a California nonprofit 501(c)(3) corporation dedicated to improving grades 6 12 mathematics instruction. Answers. The best multimedia instruction on the web to help you with your homework and study! WAMAP is a web based mathematics assessment and course management platform? UdyDaddy is the easiest way to complete your homework at any time and score A grades. 1: Variables and Expressions: Exercises: p. Rrect results and step by step solutions for all your math textbook problems. Teacher Login Registration : Teachers: If your school or district has purchased print student editions, register now to access the full online version of the book. S use is provided free to Washington State public educational institution students and. Engage students with immersive content, tools, and experiences. Click your Geometry textbook below for homework help. L Rights ReservedTomorrow's answers today. Learn Spanish with our free online tutorials with audio, cultural notes, grammar, vocabulary, verbs drills, and links to helpful sites. Get answers to your college and high school textbooks. R answers explain actual Geometry textbook homework problems. Rt of the world's leading collection of online homework, tutorial, and assessment products, Pearson. Illion step by step textbook solutions for math, science, engineering and business courses. 1 2: Order of Operations and Evaluating ExpressionsTutorvista provides Online Tutoring, Homework Help, Test Prep for K 12 and College students.
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WAMAP is a web based mathematics assessment and course management platform. Get answers to your college and high school textbooks. 1: Variables and Expressions: Exercises: p. 1 2: Order of Operations and Evaluating ExpressionsLearn Spanish with our free online tutorials with audio, cultural notes, grammar, vocabulary, verbs drills, and links to helpful sites. R answers explain actual Geometry textbook homework problems. S use is provided free to Washington State public educational institution students and. Click your Geometry textbook below for homework help! Er 2. Solutions in Algebra 1 (9780133706185). Ch answer shows how to solve a textbook! The best multimedia instruction on the web to help you with your homework and study. Illion step by step textbook solutions for math, science, engineering and business courses. | 677.169 | 1 |
Cultural Approaches to Negotiation. This section, various ways of analyzing cultural differences will be discussed as they relate to negotiation. ..
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MTHED 428 builds upon experiences in early undergraduate courses to enhance prospective and/or practicing teachers' mathematical knowledge by supporting them to build deep and connected understandings of rational number, ratio, proportion, variable, expressions, and equations and be able to call upon those understandings in order to interpret grades 4-8 students' mathematical understandings. In particular, students in this course will learn that rational number arise as an extension of whole numbers and can be represented in many forms and interpreted as ratios, measures, quotients, operators, and part-whole relationships. Students will also build understandings of equivalence and the mathematical concepts and relationships that underlie previously learned computational algorithms. Students will understand that ratios involve coordinating two quantities and multiplicative relationships, and that a proportion is a statement of equality between two ratios. Students will learn how number concepts in prekindergarten–grade 4 connect to algebra topics in grades 4-8. Topics in this area include different views and uses of variable, the nature of and use of algebraic expressions and how expressions and equations differ, multiple strategies for manipulating and representing algebraic expressions and equations, and how expressions and equations can be used to represent real-world situations.
Students will also learn what research has documented about how the concepts of rational number, ratio, proportion, variable, expressions, and equations develop in grades 4-8; the challenges that grades 4-8 learners face in learning this content; connections to previously-learned mathematical content from grades PreK-3; and how grades 4-8 students' understandings of the targeted concepts form essential foundational understandings for mathematical learning in grades 9-12. Students will engage in mathematical reasoning and justification and utilize technological tools appropriate for use in grades 4-8 mathematics.
General Education: None
Diversity: None
Bachelor of Arts: None
Effective: Summer 2015
Prerequisite:
formal admission to CEAED major or permission of | 677.169 | 1 |
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TExEs Mathematics 7-12 235Read more...
Domain I. Number concepts. Competency 1. The teacher understands the real number system and its structure, operations, algorithms, and representations --
Competency 2. The teacher understands the complex number system and its structure, operations, algorithms, and representations --
Competency 3. The teacher understands number theory concepts and principles and uses numbers to model and solve problems in a variety of situations --
Domain II. Patterns and algebra. Competency 4. The teacher uses patterns to model and solve problems and formulate conjectures --
Competency 5. The teacher understands attributes of functions, relations, and their graphs --
Competency 6. The teacher understands linear and quadratic functions, analyzes their algebraic and graphical properties, and uses them to model and solve problems --
Competency 7. The teacher understands polynomial, rational, radical, absolute value, and piecewise functions, analyzes their algebraic and graphical properties and uses them to model and solve problems --
Competency 8. The teacher understands exponential and logarithmic functions, analyzes their algebraic and graphical properties, and uses them to model and solve problems --
Competency 9. The teacher understands trigonometric and circular functions, analyzes their algebraic and graphical properties, and uses them to model and solve problems --
Competency 10. The teacher understands and solves problems using differential integral calculus. Domain III. Geometry and measurement. Competency 11. The teacher understands measurement as a process --
Competency 12. The teacher understands geometries, in particular Euclidean geometry, as axiomatic systems --
Competency 13. The teacher understands the results, uses and applications of Euclidean geometry --
Competency 14. The teacher understands coordinate, transformational, and vector geometry and their connections --
Domain IV. Probability and statistics. Competency 15. The teacher understands how to use appropriate graphical and numerical techniques to explore data, characterize patterns, and describe departures from patterns --
Competency 16. The teacher understands concepts and applications of probability --
Competency 17. The teacher understands the relationships among probability theory, and statistical inference and how statistical inference is used in making and evaluating predictions --
Domain V. Mathematical processes and perspectives. Competency 18. The teacher understands mathematical reasoning and problem solving --
Competency 19. The teacher understanding mathematical connections both within and outside of mathematics and how to communicate mathematical ideas and concepts. --
Domain VI. Mathematical learning, instruction, and assessment. Competency 20. The teacher understands how children learn mathematics and plans, organizes, and implements instruction using knowledge of students, subject matter, and statewide curriculum( Texas essential knowledge and skills [TEKS]) --
Competency 21. The teacher understands assessment and uses a variety of formal and informal assessment techniques to monitor and guide mathematics instruction and to evaluate student progress --
Sample tests.
Other Titles:
TExES :
Responsibility:
by Sharon Wynne.
Abstract: | 677.169 | 1 |
In some cases, the more of something there is, the more rapidly it may
change (as the number of births is proportional to the size of the population).
In other cases, the rate of change of something depends on how much there
is of something else (as the rate of change of speed is proportional to
the amount of force acting). (1 of 6)
Symbolic statements can be manipulated by rules of mathematical logic to
produce other statements of the same relationship, which may show some
interesting aspect more clearly. Symbolic statements can be combined to
look for values of variables that will satisfy all of them at the same
time. (2 of 6)
Any mathematical model, graphic or algebraic, is limited in how well it
can represent how the world works. The usefulness of a mathematical model
for predicting may be limited by uncertainties in measurements, by neglect
of some important influences , or by requiring too much computation. (3
of 6)
Standard 6-4 page 154, Grades 9-12
Recognize that a variety of problem situations can be modeled by the
same type of function
When a relationship is represented in symbols, numbers can be substituted
for all but one of the symbols and the possible value of the remaining
symbol computed. Sometimes the relationship may be satisfied by one value,
sometimes more than one, and sometimes maybe not at all. (5 of 6) | 677.169 | 1 |
Revision Courses
Long Maths
Course Purpose
The 11Plus Specialist Mathematics Full Day course provides students with a solid understanding in the specialist 11Plus Numerical Reasoning (Long and Short Mathematics). Over 10,000 students have completed 11PlusDIY Mock Exam and we continually see students failing to time manage Short Maths questions and failing to score well in Longer Mathematics questions that include cascading, where getting the first question incorrect usually leads to the following questions being incorrectly answered.
This full day course has been designed from our experience to refine and refresh all Short Maths core basics for the students, showing them how to score high marks in the exam in an effective and timely manner (Short Maths should be time managed). We believe its doubly important to instill Short Maths skills because students who score highly in Short Maths, can be tutored to translate to these skills to the longer multi-part maths questions in a logical and efficient format. Long Mathematics (This is completed in the afternoon of the full day course)
The second part of the course will help students to decipher the Long Maths syllabus necessary to complete an 11 Plus exam. Experience shows that many students cannot complete Long Maths in a timely manner and it differentiates students.
Capturing Long Maths skills early will allow students to understand and answer complex questions quickly which should underpin successful exam completion.
Course Outcome
Students should be able to understand and answer Long Maths questions in a time efficient and effective manner.
Course Content
Students are provided with a booklet of Long Maths course materials. The course will guide students through 11 Plus Long Maths subject areas, with a series of learning and testing to show students 'how to answer questions effectively and gain time' in the 11 Plus exam with improved accuracy.
Students will complete a multiple choice exam paper to prove learning and highlight subjects that require more attention.
Additional work will be provided for after the course to improve understanding.
Course Length
The full course includes 6 hours of in class tuition (plus full course pack for homework). Long Mathematics will be 3 hours of this.
Course Cost
The Full One Day Course Standard Price is £150.
A discounted Price £120 (applies to the first 10 students booking only).
Course Example
Below are examples of the Long Mathematics questions that will be taught and understood by students on this course. | 677.169 | 1 |
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