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! 🙂
Best Answer
There are a number of rigorous textbooks on multivariable calculus for honors students/"weak" advanced students at the same level or higher than Edwards, Nargles.
What you're really asking for is 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 Michael 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 its level of difficulty is Advanced Calculus by Lynn Loomis and Shlomo 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 1960s 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 1960s - arguably 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 Theorem 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 than enough to get you started - good luck!