Analysis On Manifolds (Advanced Books Classics)

Spivaks book Calculus on Manifolds became famous because of the rather ingenious proof of Stoke's Theorem in his original course notes. The book was published, it seems, simply to make that proof and the associated machinery, which at the time were somewhat novel, widely accessible. That it is a book intended for differential geometers and topologists considering teaching this materal, rather than a book intended for self-study, is evinced by the staggering number of errors. Rudin likewise has significant pedagogical drawbacks. Munkres is the only book of the three with a mathematically sound and complete treatment of Stokes’ Theorem.Many reviewers have claimed that the errors in Spivak are predominantly typographical, and in any event do not affect the the development of the mathematics; hence, they claim that this book, being significantly shorter, is “more advanced” than Munkres. This is untrue. The mathematical errors in Spivak were so glaring that the book is now published with an addenda fixing some, but not all, of the incorrect mathematics. For instance, there is a circularity in the development of the integration apparatus which is not addressed in the addenda: in an attempt to bound the image of a set under a map, it is assumed that a diffeomorphism sends a set of measure zero to another set of measure zero; at this point in the text, nothing of the sort has been proved, and the text actually uses this special case in proving the general case that all diffeomorphisms send all sets of measure zero to other sets of measure zero.Now, this is just one example. And anyone who has seen the material more carefully developed can spot this problem quickly, and prove the special case on their own. But certainly this makes the book unsuitable to the uninitiated.I should say that Rudin's undergraduate text is no better in this regard. In Rudin, the change of variables theorem is proved in a very limited case. He then attempts to build the integration of differential forms over chains machinery as in Spivak, but in doing so he implicitly uses a more general version of the change of variables theorem than was actually proved.Many students complain that Munkres is pedantic--I tend to agree. In particular, Munkres tends to prove things in a style familiar from Dummit and Foote, where all intermediate structures are explicitly given names. This makes each individual proof step easy to follow, but it makes the proof *strategy* difficult to divine. But it is the only one where Stoke's Theorem is rigorously and completely proved, without relying on circular or incomplete lemmas. To those students who claim Munkres is "too easy," and instead celebrate Spivak or Rudin, I would simply point out that those books were obviously sufficiently obtuse to them that they missed the mathematical holes therein.I scored the book 4/5 because the proofs are so long that they are entirely unintuitive. In order to truly take ownership of this material, it is necessary to either work all the exercises on a second reading, or work all the proofs yourself on a second reading, or switch to a more advanced but more terse book on a second reading (like Loomis’ or Tu’s books on the same subject).

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.

Yes, that headline is somewhat contradictory. Allow me to explain.As an undergraduate math major decades ago, I took the standard multivariable calculus courses, which were somewhat less rigorous treatments culminating in the fundamental theorem for line integrals, Gauss' divergence theorem, and Green's/Stokes' theorems. Then I took the typical real analysis courses, which were a rigorous treatment of analysis on R (with some complex analysis). But, somehow, I ended up never taking any rigorous coursework on multivariable analysis.Since it's been so long, I was wary of trying to wade into Spivak or Rudin, especially without the benefit of a formal course. I tried Vector Calculus by Baxandall and Liebeck, but had some difficulties with some of the notation (possibly Britishisms? not sure) in differential calculus, and was also hoping for a "novice introduction" (if there is such a thing) to differential forms.So, for the reviews that complained that Munkres is too pedantic, that's exactly what I was looking for, and so far, I've been quite pleased. Being a much shorter book, it obviously goes much faster than Baxandall and Liebeck, but I've also appreciated the detail that he puts in to a small space. (I also slightly prefer his writing style over Baxandall/Liebeck.)I'm still working my way through, so I suspect I'll have to slow down significantly once I hit manifolds (something Munkres touches on in the Preface; he says the latter half of the book is more sophisticated). But I am hoping that the detail combined with somewhat more familiar notation will help ease understanding.

User-friendly, crystal clear, highly recommended.

This is my favorite book of all time. I'm a big fan of Dr. Munkres and I really like his step-by-step proof style (some people may consider it pedantic). I would recommend this book to everyone who is looking for a rigorous modern treatmen of multivariable calculus.The only flag on this book is the paper quality. It is not bad at all, but I really was expecting something better. I don't know why the paper quality on math textbooks (at least) has been getting worse through the time. I hope this gets better on the future. Although, the book has a pretty well printing. Very readable.

wenn es um die Analysis auf Mannigfaltigkeiten geht. Da das Buch für den amerikanischen Markt konzipiert wurde, besteht die erste Hälfte des Buches aus dem Stoff der Analysis, den man bei uns eigentlich in den beiden ersten Semestern Analysis verdaut haben sollte. Ab Kapitel 5 geht es dann los. Munkres hat seine Beweise sauber aufgeschrieben und bei etwas komplexeren Herleitungen diese in Teilschritte zerlegt, denen man sehr gut folgen kann. Es wird ein klarer Weg zu den Integralsätzen beschritten, der zum Erlernen des Stoffes gerade für Anfänger hilfreich ist.

Classic book but it's an Indian edition book.