![]() ![]() 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. The text by Flanders on differential forms is a bit terse, but once you understand the calculations it's quite deep. It has a few hundred more problems for you to chew on. That text is very centered around the concept of differential forms, worth a look. I recently got Cartan's advanced calculus text which is available as a Dover. On the other hand, the text by Hans Sagan is really something, very complete lots of details. You might like Kaplan's text, it's more on the math for scientist and engineers side of the advanced calculus spectrum. I've used Edwards as a primary text for an advanced calculus course and I will agree it is a bit short on problems. That should be more than enough to get you started - good luck! It's a book anyone can learn something new from. Even if you're using a "purer" treatment like Spivak, it's a book you simply must have. 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. I think this is the book that'll serve your needs best of the ones on this list. Vector Calculus, Linear Algebra, and Differential Forms. Similar in content, but easier and much more modern, is J.H. 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. It even ends with an abstract treatment of classical mechanics. ![]() 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. Then again, these were honor students at Harvard University in the late 1960s - arguably the best undergraduates the world has ever seen. 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. 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. 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. ![]() Munkres is more of a standard textbook and covers the same material with much more detail. It's quite rough going, but it's worth the effort if you've got the patience. Spivak's book is basically a problem course with quite a few pictures. The standard books for learning this material are Calculus On Manifolds by the legendary Michael Spivak and Analysis on Manifolds by James Munkres. Indeed, I think eventually separate books on both subjects will be obsolete and there'll be unified presentations of both. Of necessity, there's going to be a lot of overlap between such textbooks and differential topology books. ![]() What you're really asking for is a textbook giving a modern presentation of vector calculus/calculus of functions of several variables. 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. ![]()
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