Math 20500, Autumn 2007 "Integration in Several Variables"
Instructor: Yon-Seo Kim, Class homepage: http://www.math.uchicago.edu/
Class Hours: Sec 45] MWF 11:30am - 12:20pm at Eckhart 308, Sec 55] MWF 12:30pm - 1:30pm at Eckhart 308
email: email@example.com (please start the message title with Math205-)
Office: E306, Office Hours MF 10:00am - 11:30am or by appointments.
College Fellow: Francis Chung, email: firstname.lastname@example.org OH W 1:30pm - 3:30pm
Office: Basement of 5720 Woodlawn, Discussion section: Thursdays 6pm-7pm at E203
Textbook: Advanced Calculus by Patrick M. Fitzpatrick, 2nd ed.
This course will cover Chapters 6-8 and 18-20. Course materials include:
Integration of several variables, Taylor polynomial approximation, Change of variables, Line/Surface Integrals.
Homework will be assigned every week and will be collected one-week after the homework is assigned.
There will be one midterm and the final exam. Midterm is scheduled at the end of 4th week.
Grading basis: Homework -30%, Midterm -30%, Final Exam -40%
Notes and Comments on Past/Future Lectures
Sep 24th: 6.1-6.2; Darboux sums, Archimedes-Riemann Theorem, Integrability of a function. Note
Sep 26th: 6.3-6.4; Properties and Applications of Integral. Note
Sep 28th: 6.5-6.6; The first and second Fundamental Theorems. Note
Oct 1st: Fundamental Theorems, Application to differential equations. Note
Oct 3rd: Integration by parts. Convergence theorems. Note
Oct 5th: Approximation formulas. Note
Oct 8th: Taylor polynomials. Note
Oct 10th: Taylor series expansion. Note
Oct 12th: Cauchy Integral Remainder Theorem. Note
Oct 15th: Binomial Expansion. Weierstrass Approximation Theorem. Note
Oct 17th: Weierstrass Approximation Theorem cont'd.
Oct 19th: Midterm Exam. Solutions: section 45. section 55
Oct 22nd: Integration in Several variables. Note
Oct 24th: Further properties of Integration in R^n. Note
Oct 26th: Set of Jordan content zero. Note
Oct 29th: Volume of a subset in R^n. Note
Oct 31st: Fubini's Theorem. Note
Nov 2nd: Fubini's Theorem cont'd. Note
Nov 5th: Change of Variables. Note
Nov 7th: Change of Variables cont'd. Note
Nov 9th: Line integral. Note
Nov 12th: Surface integral. Note
Nov 14th: Surface integral contd. Note
Nov 16th: Green's Formula. Note
Nov 19th: Green's Formula and Application. Note
Nov 21st: Stokes Formula. Note
Nov 23rd: Stokes Formula contd.
Nov 26th: Review Session.
Nov 28th: Review Session.
Final Exam Solutions: Sec 45   Sec 55
HW#1 (due Oct 5th): 6.1- 2,3,5,7. 6.2- 3,6,8,9,12. 6.3- 1,2,4,6. 6.4- 3,4,7. 6.5- 2,5,6. 6.6- 3,4,6,8,11.
HW#2 (due Oct 12th): 7.1- 2,3,5. 7.2- 2,3,5,8,9. 7.3- 1,3,6,10,13. 7.4- 3,7,10.
HW#3 (due Oct 22nd): 8.1- 2,4,5. 8.2- 2,5,7,8,11,12. 8.3- 2,5. 8.4- 3,5. 8.5- 4,6.
HW#4 (due Oct 29th): 8.6- 3,4. 8.7- 4,5,6. 18.1- 1,2,5,6,10,12.
HW#5 (due Nov 5th): 18.2- 1,2,4,5,6,8,10. 18.3- 1,2,5,6,9,12.
HW#6 (due Nov 12th): 19.1- 2,3,4,5,9. 19.3- 1,2,4,5,7,10,11.
HW#7 (due Nov 19th): 20.1- 1,3,6,7,8,9,10,11.
HW#8 (due Nov 28th): 20.2- 5,6,8,12,16,18. 20.3- 2,3,4,6,8,10,11,12,13,14,15,16.
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REM: Logarithms were useful because highly involved problems like
Could be worked using only add, subtract, multiply, and divide
THE SAME APPLIES TO FUNCTIONS - The easiest to evaluate are polynomials since they only involve add, subtract, multiply and divide.
To approximate near x = 0:
Requires a Polynomial with:
a) the same y – intercept:
b) the same slope:
c) the same concavity:
the same rate of change of
Objective. The purpose of this demo is to use a graph of the function y = f(x) and its n th Taylor Polynomial, pn (x) to illustrate the approximation of y = f(x) by a Taylor Polynomial centered at x = a. We include an option for the visualization of the error function Rn (x) = f(x) - pn (x).
Level: This demo is appropriate for any course in which Taylor Polynomials are discussed.
Prerequisites: Students should be familiar with computing derivatives of a function at x = a and with the definition of n th Taylor Polynomial centered at x = a.
(1) Browser Based:
(2) Mathematica Notebook
(3) MATLAB M-files
(4) Maple 6
Instructor's Notes: An important area in mathematics is the computation of approximate values for functions at particular points. One of the first encounters students have with such approximations is using the slope of a secant line to a graph to estimate the slope of a tangent line. Then the equation of a tangent line at a point is used for a linear approximation to the function in a neighborhood of the point. As they study Taylor Polynomials the more general problem of approximating a function by a polynomial is encountered.
Suppose we are interested in approximating a function y = f(x) near x = a by a polynomial of degree n:
The strategy we use to find the coefficients is to require a high degree of "match" at x = a. Specifically, if we require that the polynomial and its first n derivatives at x = a match the function and its first n derivatives at x = a, the result of these requirements is that we construct a formula for the n th Taylor Polynomial for f, centered at x = a:
When a = 0, the polynomial is called an n th Maclaurin polynomial for f.
Students spend so much time learning this rather complicated formula, they tend to lose sight of why they would want to use a polynomial to approximate a function. They also never seem to really grasp exactly what it is they have found or how good the approximation might be.
Other issues involved in approximation are "How GOOD is the approximation?" and "Over what interval can I expect the approximation to be good?"
Rn (x) is sometimes called the n th remainder of f.
By plotting the error function, we can visualize the "goodness" of the approximation for various values of n as well as the interval over which the approximation could be considered "good." These ideas lead to a discussion of the interval of convergence for a Taylor series. In Examples 1-6 above, it is not hard to convince students that the interval of convergence is , while in Example 7 the interval of convergence is (-1,1). More details about error analysis may be appropriate in a numerical analysis class, however, the pictures supply a visual foundation and hence an intuitive idea about what we mean when we say that the Taylor Series converges to f(x) (in an appropriate interval) if and only if the n th remainder tends to 0 as n increases without bound.
Approximation to f(x) = cos(x) by Maclaurin Polynomials and the error.
Approximation to f(x) = log(x+1) by Maclaurin Polynomials and the error.
My experience has been that this demo plants a visual image of the idea of Taylor approximation that students readily recall. As a result they seem to remember that the n th Taylor Polynomial at x = a agrees with the function and its first n derivatives at x = a. I surveyed my Calculus 3 students before we began a discussion of Taylor Series. The students who had seen the demo in the previous year all remembered much more clearly and could put into words what a Taylor Polynomial is used for, and why we might want to use one.
Mathematica Notebook: Preview and download an interactive Mathematica notebook here.
MATLAB M-Files: Two MATLAB M-files, sinmovie.m and logmovie.m, illustrate the approximation of f(x) = sin(x) and f(x) = log(x),respectively, by Taylor Polynomials. These files were developed by David R. Hill. Preview the animations and download from here.
Maple 6 Worksheet: Preview and download an interactive Maple 6 worksheet from here.
Mathcad Worksheet: Preview and download an interactive Mathcad worksheet from here.
TI-89 Calculator Program: TI-89 animation of Taylor Polynomial approximations. View and download from here. The TI-89 program was developed by Lila F. Roberts.
Credits. This demo was submitted by
Associate Professor of Mathematics
158 Harmon Drive
Northfield, VT 05663
and is included in Demos with Positive Impact with her permission.
Each segment of this module consists of sample homework assignments created around one or another of the MIT Mathlets. Each segment is followed by questions, my remarks about the features of the Mathlets, and examples that illuminate their use. At the end there are a couple of exercises.1. In this module…
In this module, you will watch video segments where you will be instructed to pause and engage in a variety of activities, as well as think about the questions posed.
Download Module slides [ PDF. 1MB ]
Download Complete Module 3 video [ ZIP. 17MB, 540p ]
After completing this module, the participant will be able to use Mathlets to:
The final use we will look at during this short course is the use of Mathlets as part of homework. Let me introduce you to the idea and benefits of using Mathlets in homework during this next video segment.3.1. First Assignment: Vector Addition
This exercise would be appropriate in a class discussing parametrized curves.Launch
You see a wheel with a light at the end of a spoke. Animate the rolling wheel using the [>>] key. You can return the wheel to the start position using the [<<] key. You can also control directly using the slider.
(a) Just as in a lecture, you have to lead students through the elements of a Mathlet. It is a good idea to encourage students to express the meaning of elements of the Mathlet themselves:
(b)-(c) Do you think that having the parametric expression for the light present on the screen lessens the value of these questions?
(d)-(f) Now we start experimenting. Homework in Mathematics classes often involves somewhat random computations, designed to give the student practice at carrying out some manipulation or algorithm. The Mathlet provides a graphical reason for wanting to do such calculations.Think About
What are your comments about the Wheel project? Explain how some part of this may be useful in one of the classes you teach.3.2 Second Assignment: Taylor Polynomials Launch
Mathlet: Taylor Polynomials
There is a lot to play with here! When you have explored the Mathlet a little, settle down on the menu item .
(a) I thought it was important to focus on the Taylor series at a single point, maybe beginning at the point , before moving to the animation offered by this Mathlet.
(b) Perhaps I should have interposed an animation with , to let students see the more familiar tangent line before they see the best approximating parabola. The parabola flattens out to a straight line at the points of inflection. This is what we envision in teaching about points of inflection, but without a tool like this it is hard to convey to the student, and my guess is that identifying the transition from concave to convex with the points of inflection will come as something of a revelation.
(c) Luckily the zeros of come out nicely, but this is still a substantial computation for beginners, one which might be resented without the payoff of verifying an observation.
(d) This was a surprise to me! You can substitute into , but the result, , does not expand in any very obvious way. With enough insight you can realize that , so
agreeing with the coefficients on the Mathlet. It is actually easier to start from the Taylor expansion guessable from the displayed coefficients and work back; so the Mathlet provides a useful hint.
This choice of menu item was rather random; similar exercises could be constructed around any of the others. The last one is the standard example of a function which is not real-analytic. It is very interesting to see how the Taylor coefficients all become zero as decreases to 0. Quite often I find myself surprised and puzzled by behaviors displayed by a Mathlet; this is a good example.Think About
What are your thoughts about the Taylor Polynomials project? What parts of this project may be useful in one of the classes you teach?4. Mathlets in Homework, Segment 2
Here are some examples of homework problems involving various Mathlets.4.1. First Assignment: Secant Approximation
This assignment might be given in a calculus course. The early sections would be appropriate when the derivative is being introduced.Launch
Accept the default menu choice . Play with the two sliders and the two check boxes.
Set . (Notice that you can set a slider at a value marked by a hashmark buy clicking on the hashmark.)
Explain how parts of this may be useful in one of the classes you teach.4.2 Second Assignment: Frequency Response
This assignment would be appropriate in an engineering-oriented ordinary differential equations class.Launch
This Mathlet deals with a spring-mass-dashpot system. The input signal is a force acting directly on the mass, given by ; the system response is the displacement of the mass from its neutral position. We are interested only in the periodic system response.
Observe the various sliders and their functions.
Mathlets in Homework(1) It is very important to be explicit about what you want students to do: is it enough for them to make observations from the Mathlet, or do you expect them to prove (or calculate) things?
The rest of the parts of this problem are only reasonable as homework after you have modeled this kind of think in lecture. There are various approaches.
Here is how I like to teach this: Write for the characteristic polynomial, so the equation is where is the differentiation operator. The input signal is . The equation has real coefficients, so if we have a solution of the complex equation then is a solution of the original equation. Now, , from which it follows that , so we can take . The amplitude of is then , and this is what we are asked to maximize. Maximizing this is minimizing the magnitude of , or of its square . You can do this by setting the derivative with respect to , , equal to zero. This gives and where
This solves (4), and with our values for , , and , it gives the answer to (2), .
(3) and (6) The complex gain is , so we are asked to find the angular frequency for which the real part of is zero and the imaginary part is positive. This occurs when , or (independent of !). With our values of and this gives .
(5) For to be real, the contents of the square root have to be non-negative. So .
This is a lot of computation and somewhat abstract. It is very reassuring to students to see the phenomena displayed on the Mathlet.
The complex gain is (in this problem) given by . The trajectory of this complex-valued function of is what is displayed in the [Nyquist plot]. It contains both the gain and the phase lag, and the Nyquist plot shows both and the relationship between them.
Technically, we have not drawn Bode plots: engineers would draw the log of the frequency horizontally, and the log of the gain vertically. My experience is that using log plots is a step too far for students at the stage at which I see them in my classes.
Also, a Nyquist plot, properly speaking, shows the trajectory of the complex gain for negative as well as positive.
For more Mathlets addressing these more advanced issues, see the Mathlets Bode and Nyquist Plots and Nyquist Plot .5. Mathlets in Homework, Conclusion
If you decide to assign Mathlets as part of homework there are a few things to consider. Watch this video segment where I discuss some of the items you need to consider.
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This optional assignment is worth some extra points, the number of which will be determined by the quality of what you turn in.
There is no penalty for not doing this assignment.
All write ups for this assignment must be typed or written very neatly by hand. It might help you to use a graphing calculator or computer algebra system such as Mathematica, Wolfram Alpha. Sage. or Maple.
Recall is the vector space of continuous real valued functions with domain . Define an inner product on by for all .
Here are some possible topics to explore (you may find other interesting things to include in your write up):
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"The book of the universe is written in the language of mathematics."Although the language of mathematics has evolved over time, the statement has as much validity today as it did when it was written. In Mathematics 1b you will become more well-versed in the language of modern mathematics and learn about its applications to other disciplines. Math 1b is a second semester calculus course for students who have previously been introduced to the basic ideas of differential and integral calculus. Over the semester we will study three (related) topics, topics that form a central part of the language of modern science:
We will start the semester by studying infinite sums. You already are aware that a rational number such as can be represented by an infinite sum, ( , for the case at hand). Actually, irrational numbers such as e and have representations as infinite sums as well. In fact, we will find that many functions, such as and can be represented by infinite polynomials known as power series. Polynomial approximations based on these power series representations are widely used by engineers, physicists, and many other scientists.
In your previous math courses you may have seen functions represented by integrals. For example, can be represented by . Integrals can be used in many contexts. The definite integral enables us to tackle many problems, including determining the net change in amount given a varying density. In the second unit of the course we will revisit integration. First we'll study the integration analogues of both the Product Rule and Chain Rule for differentiation and briefly touch on some alternative transformations of integrals that enable us to tackle them more efficiently. The goal is not to transform you into an integration automaton (we live in the computer age), but to have you acquire familiarity with the techniques and the ability to apply them to some standard situations. More important is the ability to apply the integration as appropriate in problem solving; we will devote time to developing your skill in doing this.
We will end with differential equations, equations modeling rates of change. Differential equations permeate quantitative analysis throughout the sciences (in physics, chemistry, biology, enviromental science, astronomy) and social sciences. In a beautiful and succinct way they provide a wealth of information. By the end of the course you will appreciate the power and usefulness differential equations and you will see how the work we have done with both series and integration comes into play in analyzing their solutions.
Single Variable Calculus: Concepts and Contexts by James Stewart. Second edition, Brooks/Cole 2001. This text is available at the Harvard Coop. There will be supplementary material available as well.
Each section of Math 1b has a Course Assistant who will be in class, collect and correct homework assignments, and hold weekly problem sessions. These problem sessions are part of the course and will be generally be devoted to working problems and amplifying the lecture material. The pace of the course is rather fast, so these sessions should be particularly valuable to you in learning the material. A schedule of all problem sessions will be posted outside the Calculus Office (SC 308) and posted on the course web site; feel free to go to any Math 1b Course Assistant's Problem Session. Periodically there may will be group exercises scheduled during problem sessions - `homework' exercises meant to be worked on as a group and facilitated by a Course Assistant. You will be notified by e-mail when problem sessions will be utilized in this way.
Problems are an integral part of the course; it is virtually impossible to learn the material and to do well in the course without working through the homework problems in a thoughtful manner. Don't just crank through computations and write down answers; think about the problems posed, the strategy you employ, the meaning of the computations you perform, and the answers you get. It is often in this reflection that the greatest learning takes place.
An assignment will be given at each class meeting. Unless otherwise specified, the assignment is due at the following class meeting and will be returned, graded, at the subsequent class. If you miss a class, then you are responsible for obtaining the assignment and handing it in on time. Solutions put together by the course assistants will be available on the course website. When your homework assignments are returned to you, you can consult the solutions for help with any mistakes you might have made. Problem sets must be turned in on time. When computing your final homework grade, your lowest two homework scores will be dropped if you are in a TTh section and your lowest three homework scores will be dropped if you are in a MWF section.
Note that homework problems will sometimes look a bit different from problems specifically explicitly discussed in class. To do mathematics you need to think about the material, not simply follow recipes. (Following preset recipes is something computers are great at. We want you to be able to do more than this.) Giving you problems different from those done in class is consistent with our goal of teaching you the art of applying ideas of integration and differentiation to different contexts. Feel free to use a calculator or computer to check or investigate problems for homework. However, an answer with the explanation `` because my calculator says so" will not receive credit. Use the calculator as a learning tool, not as a crutch. Calculators will not be allowed on examinations due in part to equity issues.
You are welcome to collaborate with other students on solving homework problems; in fact, you are encouraged to do so, and we will provided you with contact information for your classmates in order to faciliate that. However, write-ups you hand in must be your own work, you must be comfortable explaining what you have written, and there must be a written acknowledgement of collaboration with the names of you coworkers.
Odd-numbered problems are solved in the Student Solution Manual ; some coies will be put on reserve in the Cabot Science Library. After working on the problems on your own, you are free to consult this manual provided you acknowledge the use of this manual in your submitted work. (This is a standard rule of ethics.)
Exams are common and given in the evenings. Please keep these exam dates free from conflicts:
October 16 (Tuesday)
There will be an optional Technique re-Test available on Tues. Nov. 20th: 7:30 - 8:30 in SC C. The higher of your two scores counts in the computation of your course grade. The first test is not optional.
Calculators will not be allowed on examinations, due in part to equity issues. We will make sure that problems on the exams require minimal calculation to allow you to spend your time demostrating your mathematical knowledge as opposed to your calculating ability. We expect you to express your ideas, line of reasoning, and answers clearly.
Your course grade will be determined as follows:Take the higher of
Sources of Academic Support:
In addition to your section leader's office hours (hours that you are free to come talk with him or her without appointment) and your Course Assistant's problem session, there is a Math Question Center in Loker Common. The Math Question Center is open from 8:00 to 10:00 PM every evening except for Fridays and Saturdays. The Math Question Center is staffed by both section leaders and course assistants. You can go there for help or simply to find other students with whom to discuss your work.
A schedule of all Math 1b problem sessions will be posted on the course website. You are welcome to go to any and as many problem sessions as you like.
Tentative week-by-week syllabus
Infinite series in general. N'th term test for convergence. Harmonic series.
Determining convergence by comparison to another series or to an improper integral.
Alternating Series Test and accompanying error estimate. Absolute convergence. The Ratio Test.
Power Series. Getting new power series from old ones by substitution, differentiation and integration.
Representations of Functions as Power Series.
Taylor polynomials and approximating functions by polynomials.
The Taylor remainder and Taylor's Inequality.
Taylor series and MacLaurin Series.
Applications of Taylor Polynomials.
Series Review and Recap
Functions as integrals. The Fundamental Theorem of Calculus.
Integration by substitution.
Integration by Parts.
Partial fractions and additional techniques of integration involving more sophisticated substitutions.
Using series to integrate.
Approximating definite integrals.
Applications of integration. Slicing problems: Total mass from density, total population from
population density, etc.
Areas and Volumes
Applications of integration:
volumes, arc length, average value, work, hydrostatic pressure and fluid force.
Modeling with differential equations.
What does it mean to solve a differential equation?
Getting information without solving
Directions fields and Euler's method.
Autonomous first order differential equations: Qualitative analysis of solutions.
Solving separable differential equations.
Exponential growth versus logistic growth.
Using series to solve differential equations.
Systems of differential equations: for example, predator-prey systems.
Vibrating springs: second order linear homogeneous differential equations