Analysis on Manifolds

Rating:
Summary: Fun
Review: I ploughed through this book years ago. I just noticed that a couple of reviews were only posted this year.
I thought I would do the same.

This was a great read by the way.

I suspect that everyone who picked up this book at some point was looking for a way to circumvent Spivak's terse exposition. I don't blame them.

..and then Browder came out with his analysis text. So with advanced calculus in view, these (more or less) recent publications make the subject even more accessible to undergraduates.

..and now Spivak doesn't look so hard, all of a sudden.

Munkres presentation is certainly original. Motivating examples are bountiful, and the figures are excellent.

The perfect prequel to Boothby.

Enjoy.

Rating:
Summary: A good introduction but not the best
Review: One thing i like about this book is the way Munkres presents the counterexamples : why theorem 5.11 wont work if we ease one statement from the hypothesis. Also, the material is accessible and the exercises hard -- both of which, IMHO, are important benchmark for a good math text.

However, compared to his classic textbook of topology, Munkres did not perform as well in connecting with the readers. The text is very hard to read, and is not suitable for self study. This is useful only as a class text, or as a reference for those who already knew (or passed) the subject.

Rating:
Summary: A masterpiece yet accessible on this topic
Review: This book covers a natural extention to my course on analysis in R^n--only content similar to first one sixth of the book got treated at the end of the course. Having read first half (just before manifold) in a continuous fashion (span of nearly a week for 4 hours-ish p.d.), I find this one exceptionally clearly-written, (unlike some point in Spivak's Calculus on Manifold), and in content it serves as a detailed amplification on Spivak's (Sp seems to try to keep the proofs elegant and concise more than possible, making a couple of important theorems render indigestible).

Other noticeable features are:

1) Mistake-free.

2) Proofs are truncated into stages with explicit objectives in each, making them well-structured on paper and easy to recall in future, and in this way techniques in proofs become highlighted into some elementary theorems (to get most job done) so that the scope of applications are much widened.

3) Motivations scattered throughout the book for integrity.

4) Examples given illustrate as counterexample of how theorem fails with some condition changed or missing.

5) The level of presentation is uniform throughout the book: strictly speaking, only a good single-variable analysis course (Rudin will do, and also helpful to refer to the overlapping topics) and some motivation are needed, all essential concepts of linear algebra, topology are introduced afresh and uniquely and in the favorable context: either indispensible in later proofs (can act as a practice of it) or results proven motivate its introduction and properties, though some knowledge beforehand can help you to appreciate more, and focus on mainbody.

6) Each proof is not necessarily the shortest in methods, you may say, but looks most natural and appropriate at this level. Actually, most time it's quite concise whilst, in main theorems, all details are laid out without undue omission. (In contrast, some authors waffle lavishly between substance, but say bare minimum (sometimes unjustified) when it comes to proofs.) Length is also due to partition of proof into stages, which is way clearer in mind than a gluster of dense but appearingly short arguments. And richness and details of proofs themselves are good for getting hang of techniques.

All in all, Munkres is clearly a master, while reading it, you just feel it cannot get any better. Clarity, style, and organisation put the book far above its peers, and an undeniably outstanding first course in multivariable analysis and manifold alike.

Although exam-irrelevant, I will surely continue the journey of reading it, in a belief that it'll serve as a solid step-stone to embark on diff geometry or GR with ease, which is my original purpose. hope you can share my enjoyment.

Rating:
Summary: Extreme clarity, elucidative proofs
Review: This book is one of the most well written mathematics books I've ever read. I found it at a booksale at a local college bookstore, and I sat reading it for 15 minutes in the store. I found the subject matter enthrauling, and I would recommend it to anyone as an introduction to higher analysis. Perhaps the only objection that I might have to this book is the way that he segments his proofs. It seems to discretize what should be a well-flowing argument. Still, the proofs are excellent in spite of this barrier (though some might consider it an asset).