Problem Sets for Honors Multivariate Calculus

It's hard to find satisfactory books for an honors multivariate calculus course. In particular, the problem sets may not cover just what you want. The problems may only ask for proof of more theorems, without first exercising the students in understanding the concepts and definitions. An honors course could have challenging computational problems as well as theory problems. Or there might not be enough theory problems. For whatever reasons, you may feel, as I did, that you need to provide extra problems. If any of the problem sets listed below fill you needs, you are welcome to use them.

I used C. H. Edwards, Jr, Advanced Calculus of Several Variables, a Dover paperback. Some of my additional problems I sent out by email, but others I made into handouts using Plain TeX. Below are brief descriptions of each handout. (Click on the title immediately below to go directly to the description). After each description you can click on a file name to download the handout.

Most problems on these handouts are my creations, though occasionally they knowingly come from elsewhere (usually Buck's Advanced Calculus) and surely others are at least influenced by materials I saw elsewhere long ago. I have not put any copyright on these problems. If they fit in with your idea of an honors multivariate calculus course I am delighted if you can use them. If you do use them, I have just two requests.

  1. Message me to let me know (smaurer1@swarthmore.edu)
  2. If you hand out a whole sheet, let your students know where you got it.

Each problem set comes in some subset of three forms, a dvi (TeX) file, a pdf (Acrobat) file and a ps (Postscript) file. If you have TeX installed, you are better off downloading the dvi file: it is the smallest of the three files. If you have the free Acrobat Reader, you can view and print the pdf file. If you have a version of Ghostscript, your can view and print the ps file.

Some of the pdf and ps files don't look good on the screen at normal sizes. This is because the dvi files from which they were created were produced with nonscalable fonts before I realized this made a difference. However, all the files have printed clearly in my experience.

Here is the list of problem sets. Click on a title to reach the anotation below.



Handout Titles

(A' => B') => (A => B)

Many claims in mathematics are of this form, e.g., if the Inverse Function Thm is true (which is a statement of the form A'=>B') then the Implicit Function Thm is true. This problem set discusses the importance of this sort of statement and suggests how best to go about proving it.

dvi version,
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Change of Variable in Limits

Many places in analysis, and especially when proving inverse function theorems, one must change the independent variable in a limit calculation. E.g., if f is invertible, b=f(a), and b+k = f(a+h), then we might want to look at

lim_{k->0} f(a+h) instead of lim_{h->0} f(a+h)
and claim the limits are equal. This problem set takes the students through a set of exercises that clarifies what is going on and shows them how to justify such variable changes.

dvi version,
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Completing the Square, Gaussian Elimination, and Quadratic Forms

In single variable calculus, to determine if a critical point of a function is a max or a min, you determine the sign of the second derivative. In several variables, you determine what sort of quadratic is associated with the matrix of 2nd-order partials. Analyzing this quadratic can be done several ways; one generalizes the method of completing the square from high school algebra, and at the same time is intimately related to Gaussian elimination. This problem set develops these relationships through examples. The other main way to analyze the quadratic form is explored in Matrix Norms and Quadratic Forms

ps version,
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Derivatives and Gradients

This is a fairly standard set of computational problems to familiarize the student with derivative and gradient concepts and notation for functions from R^n. Our honors text was weak on such standard material, so I provided some. There are some problems in this set on Oh and oh notation, and on justifying the fact that water runs downhill in the direction of steepest descent.

dvi version,
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Dimensions of Surfaces and Tangents

Through a series of definitions and examples, this problem set connects the intuitive idea of dimension of a surface to the formal definition of dimension of a subspace that my students had seen in their previous linear algebra course. The intermediate step is to define affine space and the dimension of an affine space.

dvi version,
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First Multivariate Improper Integralss

For improper multivariate integrls (unbounded domain or unbounded functions) Fubini's Theorem can be false (the integral might not equal the associated iterated integral, or the two iterated integrals might not be equal). This problem set illustrates some of the things that can happen.

ps version,
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Gravity Problems

For gravitational attraction, balls act as if all the mass is at the center. Proving this is a excellent way to illustrate several themes in multivariate calculus. Students tend to think this result is true because the center is the centroid. But most volumes do not act as if all the gravitational attraction came from the centroid. This problem set explores these ideas.

dvi version,
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Iterating a Double Improper Integral that Isn't Absolutely Convergent

This handout steps students through a proof that the double integral of e^{-xy}*sin(x) over the first quadrant can be evaluated by iterated integration in either order, thus allowing one to prove that the integral of sin(x)/x from 0 to infinity is pi/2. The double integral is not absolutely convergent, so the equality of the iterated integrals is not automatic. This problem provides a good example of the sort of careful analysis a working mathematician might be called on to do for one special problem or another.

dvi version,
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Lagrange Multipliers and Economics

One of the nicest areas of application of lagrange multipliers is to economics, but this area is often not treated at all in multivariate texts, at least honors ones. This problem set fills the gap.

dvi version,
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Last Assignment on Differential Forms

Some problems to tie up the loose ends on integration of forms in R^2 and R^3. An application of the inverse function theorem to show that there really is a 1-form dtheta everywhere except 0; a bit on the heat equation; and more on reducing gravitational attraction to point masses (partial overlap with the gravity handout).

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Limits of Sets

Coming into this course, students are familiar with limits of sequences and limits of functions as x->a. This problem set explores the idea of the set of limit points of a set.

Solutions for this problem set are available.

dvi version,
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Matrix Norms and Quadratic Forms

Our text introduced matrix norms in the context of proving the second order conditions for max and min. But the concept is more generally useful, even for nonsymmetric matrices, so I wrote a set of problems to get at the more general situation as well.

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More Contour Integration and Exact Differentials

By Green's Theorem, it is possible to show that some 1-forms have the same integral around many different closed curves, even though the direct computation of the different path integrals may vary widely in difficulty. This problem set gives specific examples of this phenomenon, and also extends student understanding of the related theory in some other ways.

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More Improper Integrals

This problem set gives concrete examples. Students check if improper integrals in several variables are convergent, and they see what can go wrong otherwise (i.e., things you expect to be equal are not).

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More Integration of Forms

Steps students through the proofs of Green's Theorem in the rectangular case and the proof that, if w is a 1-form in R^2, then dw=0 everywhere is a sufficient condition for w=df everywhere. The handout then proceeds, first in R^2 and then R^3, to the important idea that if a flux is caused by a one-point gravity-like flow (inverse square from a single point), then it doesn't matter where inside the curve or surface that singular point is. (I was leading up to a conceptually simple proof in class that gravity of spherically symmetric solids acts as if all the mass is at the center. See also the handout "Final Problems").

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More on Quadratic Forms and Symmetric Matrices

The Hessian matrix of 2nd-order partials is almost always symmetric. Symmetric matrices are diagonalizable with real, orthogonal eigenvalues. These facts are relatively easy to prove using dot product arguments, but my students had not seen these facts in their previous honors linear algebra course -- not enough time. In this problem set they work out the proofs.

ps version,
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Neighborhood and Open-Set Continuity

A comparison of the epsilon-delta definition with the "neighborhood" definition and the "open set" definition.

ps version,
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Orders of Growth

The concepts and notation of Big O and little o provide a wonderful way to clear your paper of convergence details, which details can be especially nasty in several variables. If you work through this long, sequenced problem set, you should come to understand this notation very well, and become accomplished at using it.

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Pictures of Functions

This problem set starts with definitions of the implicit, explicit, and parametric geometric representation of functions, and goes on to problems that also bring in the tangents spaces to these representations.

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Some Starter Inverse/Implicit Function Problems

Most texts state the inverse and implicit function theorems, and honors texts prove them. But most texts are short on concrete examples. This problem set gives some concrete examples of the theorems (and of situations in which these theorems don't quite apply. There is also a concrete example of the method for proving the implicit theorem by reducing it to the inverse theorem. There is also a problem showing how the implicit function theorem underlies a lot of marginal analysis in economic models.

dvi version,
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Vacuously True

When A => B and A is always false, the implication A => B is said to be vacuously true. This problem set gives a more careful definition and several examples to illustrate why this concept is useful in mathematics.

Solutions for this problem set are also available from the author.

dvi version..... pdf version..... return to list

  
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Written June 1998. Minor correction 11/06/08.