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I am currently trying to teach myself multivariable calculus using C.H. Edwards' "Advanced Calculus of Several Variables", but the text unfortunately doesn't have very many problems with solutions. I've attempted a number of the problems, but I'm not sure if my solutions are correct.

Could someone suggest any other books that I could use that have lots of problems that are of the same caliber as those in this book?

Thanks! :-)

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5 Answers 5

up vote 13 down vote accepted

There are a number of rigorous textbooks on multivariable calculus for honors students/"weak" advanced students at the same level or higher then Edwards, Nargles.

What you're really asking for a textbook giving a modern presentation of vector calculus/calculus of functions of several variables.Of necessity,there's going to be a lot of overlap between such textbooks and differential topology books. Indeed, I think eventually separate books on both subjects will be obsolete and there'll be unified presentations of both. The standard books for learning this material are Calculus On Manifolds by the legendary Micheal Spivak and Analysis on Manifolds by James Munkres. Spivak's book is basically a problem course with quite a few pictures. It's quite rough going,but it's worth the effort if you've got the patience. Munkres is more of a standard textbook and covers the same material with much more detail.The main problem is that given your question,you really want something with applications as well and not merely rigorous theory, in which case neither is really going to completely fill your needs.

Notorious for it's level of difficulty is Advanced Calculus by Lynn Loomis and Schlomo Sternberg, now available for free at Sternberg's website,which is a huge gift to all mathematics students of all levels. This book was written for an honors course in advanced calculus at Harvard in the late 1960's and it's unimaginable that they actually taught UNDERGRADUATES this material at this level. Then again,these were honor students at Harvard University in the late 1960's-argueably the best undergraduates the world has ever seen. In any event,for mere mortals,this is a wonderful first year graduate text and probably the most complete treatment of the material that's ever been written. It even ends with an abstract treatment of classical mechanics. It's well worth the effort,but boy,you better make sure you got a firm grasp of undergraduate analysis of one variable and linear algebra first.

Similar in content,but easier and much more modern is J.H. Hubbard and B.B. Hubbard. Vector Calculus, Linear Algebra, and Differential Forms.I think this is the book that'll serve your needs best of the ones on this list. Beautifully written,wonderfully illustrated with many,many applications,philosophical digressions and unusual sidebars, like Kantorovich's Theorum and historical notes on Bourbaki,this is the book we all wish our teachers had handed us when we first got serious about mathematics.Even if you're using a "purer"treatment like Spivak,it's a book you simply must have. It's a book anyone can learn something new from.

That should be more then enough to get you started-good luck!

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Thank you! Hubbard & Hubbard sounds excellent. –  nargles324 Dec 25 '12 at 20:24
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why in world did somebody downvote this? Strange. –  James S. Cook Dec 25 '12 at 20:54
    
@James Beats ne-but seems to happen a lot when I post and whoever it is is too cowardly to admit it. I think it's not a single person. –  Mathemagician1234 Dec 26 '12 at 0:31
    
@Mathemagician1234: Pavel did not ask the question; nargles324 did. –  Jonas Meyer Dec 27 '12 at 17:42
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www.math.harvard.edu/~shlomo/docs/Advanced_Calculus.pdf‎ the link to the book. –  yiyi Sep 9 '13 at 1:00

C.H. Edwards (coauthored with Penney) also has a text: Multivariable Calculus (And just Calculus: Edwards and Penney). But I can't say it's at an advanced level. Perhaps you can find it in a library, or inter-library load, to see if it has exercises that suit your needs.

This book combines traditional mainstream calculus with the most flexible approach to new ideas and calculator/computer technology. It contains superb problem sets and a fresh conceptual emphasis flavored by new technological possibilities. Chapter topics cover polar coordinates and parametric curves, infinite series; vectors and matrices, curves and surfaces in space, partial differentiation, multiple integrals, and vector calculus. For individuals interested in the study of calculus. - Editor review

I know I first learned calculus from Edwards & Penney (earlier edition), and had no complaints.

I'm not familiar with the text you mention, a Dover Book. But one suggestion would be to check and see if Edwards cites any references (e.g., is there a bibliography? suggested reading?, etc.)

Actually, there is a suggested reading section at the end book by Edwards and that may point you to some sources of more exercises; it also discusses and suggests different texts for the various topics he covers in your text.

One text listed as a reference in your book is Spivak's Calculus, and Spivak's Calculus on Manifolds:A Modern Approach To Classical Theorems Of Advanced Calculus. The Calculus text has a problem/solution text which can be purchased separately, and might be helpful for self-study purposes.

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Thanks! I'll definitely look at Spivak's two texts. –  nargles324 Dec 25 '12 at 20:23

Interestingly, no one has suggested the excellent (and I had thought well known) book

Harold M. Edwards, Advanced Calculus: A Differential Forms Approach (1994).

Besides its engaging writing style, the solutions (or at least good hints) to every exercise in the book are given at the end of the book (about 40 pages of carefully written solutions).

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+1 for a classic that sadly,most of us didn't put down. Edwards is wonderful,but it seriously needs to be updated in its treatment of linear algebra; he uses linear equations instead of matricies to present it. Why Edwards didn't take the opportunity to do that when the book was reissued and he corrected it is beyond me. –  Mathemagician1234 Mar 9 '13 at 22:32

There are many excellent texts. I've used Edwards as a primary text for an advanced calculus course and I will agree it is a bit short on problems.

You might like Kaplan's text, it's more on the math for scientist and engineers side of the advanced calculus spectrum. On the other hand, the text by Hans Sagan is really something, very complete lots of details. I recently got Cartan's advanced calculus text which is available as a Dover. That text is very centered around the concept of differential forms, worth a look. It has a few hundred more problems for you to chew on. The text by Flanders on differential forms is a bit terse, but once you understand the calculations it's quite deep.

I suppose the question is what are you after? Analysis? Basics of Differential Forms? Multivariate integration? If I have any trouble with Edwards, it is that the analysis is a bit scattered in that text. As an example, the proof of the implicit or inverse mapping theorems ultimately rests on an iterative sequence converging to the desired map. However, ideas about convergence of series of functions are relegated to the appendix. That said, I learned many things from Edwards and I do think it is a great place to start.

The Dover text by Rosenlicht is a good analysis supplement to Edwards. Easy to read and it'll help you level-up analytically without much distraction.

If you want something lower level, take a look at Susan Colley's Vector Calculus, the first or 2nd ed. are pretty cheap.

I totally agree that Munkreses and $Hubbard^2$ are worth a look. The text by Sternberg has tons of problems also.

James Calahan wrote a beautiful text a few years ago, it's missing some generality, but the study of the interplay between linear algebra and implicit function theory is very pretty and I hope it finds a way into all the next generation advanced texts. It also makes some effort to explain the rudiments of Morse theory which is a bit unusual in a good way.

Hope this helps.

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The Calahan text is unique and very interesting indeed-basically it tries to explain classical vector analysis in R^3 ALONE from a modern point of view. –  Mathemagician1234 Dec 26 '12 at 5:44

A recent book titled "Functions of several real variables" has many examples and solved problems that illustrate all notions and theorems. Its treatment of the subject is rigorous and its style modern and inviting.

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