I have been teaching Advanced Calculus at the University of Pittsburgh for many years. The course is intended both for advanced undergraduate students and the first year graduate students who have to pass the Preliminary Exam. This is a difficult course as you can see from the problems that we have on our exam:


What bothers me quite a lot is the lack of a good collection of problems for functions of several variables. There are plenty of excellent collections for functions of one variable and for metric spaces, but there is almost nothing regarding good problems for functions of several variables. The only exception that I know is:

P. N. de Souza, J.-N. Silva, Berkeley problems in mathematics. Third edition. Problem Books in Mathematics. Springer-Verlag, New York, 2004.

This is an amazing collection of problems covering many areas of mathematics and what is important the problems have complete solutions.

Question. Do you know a good collection of problems for functions of several variables?

By this I mean a collection of problems that require deep understanding of the problem rather than a standard application of formulas and theorems. I believe that most of the problems in our Preliminary Exam in Analysis fell into this category. Ideally, I would prefer to have a collection with solutions or hints, as it would be very helpful for students (and for me as well).

There are many non published collections of problems available online. I am also interested in links to such collections.

While this might seem as a question that is not research level, I think otherwise. We teach Advanced Calculus to students and if we want them to be ready to do research in Analysis, we need to teach them with such problems.

Edit. I actually knew all the references listed in the answers. Clearly, the answers show that there is no good source of "ready to use" problems in Advanced Calculus of several variables.

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    $\begingroup$ I see a perfect fit with textbook-recommendation tag; why the close votes then? $\endgroup$ – Mateusz Kwaśnicki Feb 28 '20 at 19:47
  • $\begingroup$ @MateuszKwaśnicki Do you know any such collection of problems? I do not mind if it is written in Polish. $\endgroup$ – Piotr Hajlasz Feb 28 '20 at 22:21
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    $\begingroup$ Fichtenholz wrote a three-volume textbook Differential and Integral Calculus, available in a number of languages, but not in English. This used to be "the textbook" for real analysis here in Wrocław, and quite likely not only here. It contains a lot of examples and problems, covering some more advanced topics and including a lot of really difficult exercises, but it is certainly more of a textbook than a collection of problems. And I am pretty much sure you know it already. $\endgroup$ – Mateusz Kwaśnicki Feb 28 '20 at 22:58
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    $\begingroup$ For what it's worth, while Volume I of Introduction to Calculus and Analysis by Richard Courant and Fritz John is an excellent source of "lower level" single variable calculus problems, the problems in Volume II (for which 119 pages are devoted to their solutions) are almost entirely straightforward and are primarily designed to reinforce mastery of the text material and techniques, and thus they are probably NOT the type of problems you're looking for. $\endgroup$ – Dave L Renfro Feb 29 '20 at 9:32

Section 3.3 of Putnam and Beyond by Răzvan Gelca and Titu Andreescu (Springer, 2007) is entitled "Multivariable Differential and Integral Calculus" and has a number of interesting, non-routine problems.


You might want to have a look at "Multidimensional Real Analysis" parts I (differentiation) and II (integration). They have plenty of problems, and the books are aimed at mid/advanced undergraduates: the books were/are used at Utrecht University for a second-year math course, when students had already taken real (1D) analysis and vector calculus (without all the rigorous proofs).


By this assumptions, I think you can find many interesting problems in the real analysis books that have chapters about several variable calculus. For example, I suggest the book:

Problems and Solutions in Real Analysis, 2nd Edition, by Masayoshi Hata.

There are a lot of good problems with detailed solutions about several variable calculus in this book.

Also, you can see the section of problem and solution of the Mathematical Monthly journals and select the relevant problems.

  • $\begingroup$ Nice book, but there are only 5 problems for several variables :( $\endgroup$ – Piotr Hajlasz Mar 17 '20 at 0:40
  • $\begingroup$ Yes! But there are many similar books. Also, these gives idea for designing problem by yourself. $\endgroup$ – Shah Rooz Mar 17 '20 at 7:31

Take a look at Schaum's Outline of Advanced Calculus. It has many solved problems, and many unsolved problems

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    $\begingroup$ "By this I mean a collection of problems that require deep understanding of the problem rather than a standard application of formulas and theorems." That's precisely what Schaum's Outline problems are not. $\endgroup$ – M. Vinay Feb 29 '20 at 5:40

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