Customer Reviews

Analysis On Manifolds (Advanced Books Classics)

5 star:		(6)
4 star:		(4)
3 star:		(0)
2 star:		(1)
1 star:		(0)

 

 

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.

I hate to ruin all the fun, but I have to disagree with everyone who likes this book. There are a few things that Munkres does that saves this book from being a complete failure, but overall the sheer lack of interesting problems, the heavy emphasis given to only computation in the beginning of the book, and Munkres's bloated expository style put this far behind its older brother, Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus.

Let's talk about the problems first. Spivak heavily integrated his problems into the text, so much so that it is almost impossible to read the book without doing his problem sets. This might have been a problem if the problem sets were boring or impossible. But Spivak crams exciting problems into almost every set, and they are all doable. In Analysis on Manifolds, you're lucky to get even one interesting problem in a set. Let us take the problem sets from both books after the subsections introducing the derivative. In Munkres, there are seven questions, each of them being a computational problem. In Spivak, there are the computational problems, but there is also a problem exploring properties of functions being equal up to n-th order, and we have to prove ourselves that a function f:R to R^2 is differentiable if and only if both its component functions are differentiable. Whereas Spivak's problems are insightful and give the reader a look at what's to come, Munkres's problems feel like a afterthought. The fact is that this same problem set in Munkres could have easily been pulled out of a standard Calc 3 book. This is a problem throughout the entire book. The first truly interesting problem in this book comes in the beginning of chapter 4 ( section 16, problem 3 (b) ). You can't learn math without good problems, and the sheer lack of them in this book is reason enough to switch to Spivak.

Another thing wrong with this book is his bloated exposition on multivariable analysis. Multivariable analysis borrows heavily from single variable analysis in terms of how proofs are constructed and the motivation behind them. It follows that if you've studied single variable analysis out of a book like Rudin (a standard assumption), then you should easily be able to pick up on how multivariable analysis works. Spivak understands this; Munkres doesn't. The result: we have that chapters 2, 3, and 4 take up just under 140 pages, more pages than Spivak entirely. This is before manifolds even make their appearance, i.e. the good part of the book. This excess length in Munkres is due to his bloated proofs and painfully slow route he uses to develop the integral over open sets. First he defines the integral over a rectangle, then over compact sets, then considers the limits of those integrals, all the while proving all the standard properties that we know the integral has every step of the way. Spivak on the other hand, defines the integral of a rectangle, then tells you in a sentence how to define it for compact sets (that's all that is needed). Then, it is up to you to prove what properties you know this new integral should have. Finally, Spivak uses partitions of unity to define the integral over a open set, and it is obvious from there that this new integral still has all the desired properties. Not only that, but we get some extra use with partitions of unity that foreshadows how we'll use it them to define the integral on a manifold. While Munkres nearly bored me to death, Spivak developed the integral swiftly and clearly, and so I was captivated. This is only the definition of integrability I'm talking about, but Munkres does this many more times such as the statement and proof of Fubini's theorem.

The proofs in Munkres are also bloated. Many people say that they are more natural or expository. A good expositor like Spivak or Rudin will know how to convey the essential idea of a proof in a short amount of space, while simultaneously providing a concise and complete proof. Munkres tries to give ideas, but he includes every single detail to every proof. This is not only unnecessary, but it kills the flow of the book. There are so many times when Munkres will actually cloud the essential idea of a proof by a bunch of unnecessary and technical steps. Let us consider the proof of the Riemann condition for integrability. Munkre's proof is over a page, while Spivak's proof takes up just about an inch. His proof doesn't leave any part to the reader, and it conveys the key idea much better as well. I have yet to find a proof in Munkres that I think is better than the analogue in Spivak.

Let's talk about manifolds now. This is where I think Munkres's bloated style comes in handy. Everyone's first exposure to manifolds is painful, and Munkres's slow-but-clear style really eases the reader into manifolds and how they work. The problem sets are not very good here either, but his exposition works well. Spivak's problems remain solid throughout, but chapters 4 and 5 are known to take no prisoners. So this is what I think turn many people looking for other sources, and Munkres is often the one they come to.

Another thing that I think Munkres does well is that he includes examples in the text, something that Spivak clearly lacks. But unfortunately these are the only positive things I have to say about the book. This book commits too many expository crimes in my eyes, and I cannot recommend anyone buying it. A student is much better off battling through Spivak and talking with other students who already know the material. They'll learn more, and Spivak will take them farther in their understanding. If, however, you have an insatiable desire to use this book, buy Spivak and get this one at your university library.

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.

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.

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.

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.

The book of Munkres is a really good introduction to Manifolds in R^n.

It starts almost from the very beginning and introduces the reader to all kind of topics in multivariate calculus, integration theory and theories needed for manifolds, in particular integration on manifolds. Some of the main topics are algebra and topologyin R^n, the chain rule, the inverse function theorem, the implicit function theorem, the n-dimensional Riemann integral, the change of variables theorem, manifolds, integration on manifolds and Stokes' theorem. I enjoyed reading the book and I liked Munkres teaching style, especially the many examples in the book are helpful for a good understanding.

I took a course in advanced analysis, in which we covered the first few chapters of this book (upto the implicit function theorem). Since then I have been going it alone and have finished integration; on my way to manifolds. An excellent book for a reading course, very lucidly written. Since there are many things which are "obvious" in R, but can really cause difficulties in R^n (path dependence of continuity, for example), the author does a great job of identifying these particular issues (note the entire chapter on change of variables) and pointing out how the difficulties arise.

The one shortcoming of this book is that none of the exercises have any solutions. If the author provided solutions for even 5% of the problems (see, for example, Oksendal's Stochastic Differential Equations), it would have been enormously beneficial for somebody going it alone, like me.

I would definitely recommend this book to anyone! One thing I love about the book is how Munkres divides the longer proofs into parts, so its easier to keep track of what you're proving. However, starting from differential forms I felt that Munkres wasn't really doing as a good job as in the first half of the book. And so I would recommend using do Carmo's Differential Forms and Applications along with it. The course I took was a second year course in Analysis and the professor's area of interest is Differential Geometry so maybe he expected a lot more from us and hence deviated a lot from the book and thus, I didn't find Munkres as useful in these later sections. This same professor used Spivak's Calculus on Manifolds and Differential Forms and Applications from do Carmo the first time he taught it, but the second time only used Munkres. However, he covered a lot more material and it was basically all that is found in do Carmo. It's also very beneficial if you go through do Carmo's book as its definitely a classic and it will lead you into do Carmo's books on differential geometry which are one of the best out there. So I must admit, I'm probably biased in my review and hence the 4 stars, but its definitely a book I'll be keeping. Hope you guys enjoy it just as much as I did! :)

this book is vastly better then browder or spivak. it is also more thorough in its discussion of elementary results, (though less thorough in generalities). very helpful for undergraduates.