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.

- Message me to let me know (smaurer1@swarthmore.edu)
- 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.

- (A' => B') => (A => B)
- Change of Variable in Limits
- Completing the Square, Gaussian Elimination, and Quadratic Forms
- Derivatives and Gradients
- Dimensions of Surfaces and Tangents
- First Multivariate Improper Integrals
- Gravity Problems
- Iterating a Double Improper Integral that Isn't Absolutely Convergent
- Lagrange Multipliers and Economics
- Last Assignment on Differential Forms
- Limits of Sets
- Matrix Norms and Quadratic Forms
- More Contour Integration and Exact Differentials
- More Improper Integrals
- More Integration of Forms
- More on Quadratic Forms and Symmetric Matrices
- Neighborhood and Open-Set Continuity
- Orders of Growth
- Pictures of Functions
- Some Starter Inverse/Implicit Function Problems
- Vacuously True

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.

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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

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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

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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.

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Solutions for this problem set are available.

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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.

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