Analysis on Manifolds

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.


Fun   June 10, 2004
Alex (MTL)
21 out of 28 found this review helpful

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.


Excellent.   December 26, 2006
math student (nyc)
2 out of 2 found this review helpful

I've just finished all but the last half of the last section, which deals with abstract manifolds, and I've done most of the problems in the book. It is important to note that the book only deals with manifolds that are subsets of euclidean n-space.

Anyway, the book is well-written. It demands some maturity and basic linear algebra, analysis and topology. I found only two misprints which are basically of no consequence. Figures abound and are excellent. I've got only two complaints:

(1) The author never mentions that the set of all C^r scalar maps on an open set in R^n is closed under sums, products and quotients. This is used constantly in the latter parts of the book but is never proven. The proof can be found in Spivak's book. The first time this fact is needed is in the proof of the inverse function theorem (det(Df(x)) is a continuous function of x if f is C^r), and also during the construction of a partition of unity. There are more subtle points than this that are left to the reader, but I feel that it should have been proven or given as an exercise if only for the sake of completeness.

(2) The book isn't hard (though it isn't totally easy), but the very last section on abstract manifolds seems harder to read than all the rest of the book, because the author does less to elucidate things here of all places, where more elucidation is needed. He's trying in several pages to generalize results on euclidean submanifolds obtained throughout the whole book to abstract manifolds. I feel that the exposition ought to have been much more thorough here, or much more informal, or that this section should have just been completely omitted.

Nonetheless I feel I'm now ready to take a course in abstract differentiable manifolds. The problems in the book are good, and there are only at most ten or twelve problems in every section, so the reader isn't overburdened as reading the text well and carefully is a task in its own right.

I've profited considerably by completing this book and I highly recommend it.


A good introduction but not the best   April 16, 2004
bal gombak (Cambridge, MA USA)
18 out of 22 found this review helpful

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.


A Readable Introduction   June 30, 2000
10 out of 13 found this review helpful

This is an extremely readable introduction to the subject of calculus on arbitrary surfaces or manifolds. The author develops the subject from the beginning assuming only basic calculus and linear algebra - and then introduces concepts of integration and tensor analysis as the book progresses. Each segment is accompanied by a series of problems that does well to reinforce concepts. All in all, a good introduction.