Anonymous
Posted July 9, 2002
This excellent and concise treatment of the essential theorems of differential and integral calculus of several variables, is not the book described by the publisher's blurb displayed on this page. That blurb refers instead to another lovely book, Plateau's Problem, by Frederick Almgren.
1 out of 1 people found this review helpful.
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Posted July 8, 2002
This is one of the quickest and clearest introductions to the three essential results of advanced calculus: the inverse function theorem, the change of variables theorem for multiple integration, and 'Stokes' theorem' (a several variable version of the fundamental theorem of calculus due actually to Lord Kelvin) for differential forms on manifolds. There is a self contained introduction to the algebraic concept of alternating forms preceding the treatment of integrals of forms. There are a few gaps in some of the arguments, most of which are filled in a section of addenda at the back, but this is a really useful, efficient place to learn the results you simply must know to do most advanced mathematics. The proof of the inverse function theorem here is for finite dimensions, but you can learn the less frequently used infinite dimensional version when you need it.
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Posted January 26, 2010
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Overview
This little book is especially concerned with those portions of ”advanced calculus” in which the subtlety of the concepts and methods makes rigor difficult to attain at an elementary level. The approach taken here uses elementary versions of modern methods found in sophisticated mathematics. The formal prerequisites include only a term of linear algebra, a nodding acquaintance with the notation of set theory, and a respectable first-year calculus course (one which at least mentions the least upper bound (sup) and greatest lower bound (inf) of a set of real numbers). Beyond this a certain (perhaps latent) rapport with abstract mathematics will be found almost essential.
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