calculus on manifolds by Michael Spivak

books:

	• Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus Michael Spivak Westview Press, 1971 - 160 pages average customer review:based on 28 reviews view larger image
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highly recommended

Not very satisfying or rigorous

A key thing about this book: it's basically typed up lecture notes. Especially as it gets further along, it displays a notable lack of rigour. Some of the problems are not necessarily provable using information from the book. Furthermore, theorems don't clearly state the assumptions under which they operate. In chapter 5, the author resorts to basically presenting a laundry list of facts about differential forms on manifolds, so it's hard to get much beyond a basic idea of what's going on.

I believe this book may be good for a course where the instructor can answer student's questions about ideas that aren't addressed rigorously. For self-study, it could be, at best, a supplement to another book. I'm actually a little baffled about why it seems to be so well-reviewed.

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calculus on manifolds by Michael Spivak

Excellent classic covering differential and integral calculus first on subsets of R^{n}, then on manifolds in R^{n}. The paperback version still retains a few typos, but other than that, it's a very clear exposition.

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Good, but not great.

Spivak's book came with high praise, and for the first 3/4 of the book I certainly agreed. Spivak has a style that is at once conversational and precise. He doesn't skimp on the details, but strives to make the content as understandable as possible. His proofs aren't always as slick and polished as those you may find in Rudin, but they are well thought out, and the exercises complement them very well.

There are, however, two problems with Spivak's book. The first being that for the most part, it is rather easy. The level of most of the exercises is rather low compared to the material he is presenting, and as the book goes on, they help you adjust to the material less well. Similarly, Spivak's readability falls apart eventually. He introduces complicated notation and concepts without sufficient explanation which serves to make the last (crucial) part of the book nearly incomprehensible compared to the earlier chapters. It reads as if the end was put together in a rush, or at least by a different person. This puts a large dent in the enjoyment to be had from the book, and relegates it from the place of essential reference, to merely a good read.

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Tip of the iceberg...

This book would serve well as a self-study introduction to smooth manifolds for a student just finishing a high school calculus course. For those serious about the subject, pick up John M. Lee's "Introduction to smooth manifolds."

Intersting indeed. Indeed.

Hi, so I was looking for a book on auto manifolds and here I am. This all seems pretty out there to me, but the cover looks pretty nice, so 5 stars it is. Great work Mr. Spivak

reviews: page 1, 2, 3, 4, 5, 6

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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