Multidimensional Real Analysis I: Differentiation [Repost]

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Multidimensional Real Analysis I: Differentiation [Repost]

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Think of this two volume series as the Mother of All Multivariable Calculus books. It's NOT an intro to multivariable calculus for someone who has finished a couple semesters of calculus; you'll need a good stiff course (see my review of Derivatives and Integrals of Multivariable Functions by Guzman) in m.v. calculus, a dose of linear algebra, and mathematical maturity at the junior/senior undergrad level to tackle it. But if you want to go deeper - much deeper - than a first course in multivariable calculus, this is a great book.

[…]The book is refreshingly free of errors (a few trivial typos are about the extent of them, at least as far as I've gotten) and well translated from the Dutch. There are hundreds of problems after the main text; although solutions aren't given in general, the material is well enough explained so that the reader should be able to solve them. (one of the authors has a web site giving corrections to the text plus some solutions - see […] ).

All theorems are proved in full detail, but be aware of two things: first, the proofs are "mathematicians proofs"; short and slick methods are favored over pedagogically softer ones. (example: one of the main theorems of m.v. calculus is the chain rule. Most undergrad texts would simply prove it head on from the definitions of derivative and composition of functions, but here the authors rely on a slick piece of machinery in the form of something called Hadamard's theorem.) Second, the reader WILL have to take out her pencil and paper and fill in some details, but the good news is that the text gives enough information to make this possible in all cases.