There is the Inverse Function Theorem, the Implicit Function Theorem, and various kinds of multiple integrals and change-of-variable formulas. A limitation of the book is that it deals only with submanifolds of Euclidean spaces except for an appendix that sketches the general case in metric spaces. I think this is a reasonable approach for this kind of course. The author makes the exposition easy to follow by gradually building up the types of manifolds, first dealing with parallelepipeds, then open sets, then parameterized manifolds, then general manifolds.
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Accessible to readers with knowledge of basic calculus and linear algebra. Sections include series of problems to reinforce concepts. Having read first half just before manifold in a continuous fashion span of nearly a week for 4 hours-ish p. Other noticeable features are: 1 Mistake-free. 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.
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. 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.
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. 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. The result: we have that chapters 2, 3, and 4 take up just under pages, more pages than Spivak entirely.
This is before manifolds even make their appearance, i. 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.
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.
While Munkres nearly bored me to death, Spivak developed the integral swiftly and clearly, and so I was captivated. 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. I have yet to find a proof in Munkres that I think is better than the analogue in Spivak. The problem sets are not very good here either, but his exposition works well.
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. If, however, you have an insatiable desire to use this book, buy Spivak and get this one at your university library. A good introduction but not the best By Suhaimi Ramly on Apr 15, One thing i like about this book is the way Munkres presents the counterexamples : why theorem 5.
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.
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. This is used constantly in the latter parts of the book but is never proven. 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.
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. A Readable Introduction A Customer on Jun 30, 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. An amazing book! By Bryan Urizar on May 30, I would definitely recommend this book to anyone! However, he covered a lot more material and it was basically all that is found in do Carmo.
Hope you guys enjoy it just as much as I did! 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.
I enjoyed reading the book and I liked Munkres teaching style, especially the many examples in the book are helpful for a good understanding. Excellent textbook on a difficult subject By Abheek Saha on Apr 06, 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. The one shortcoming of this book is that none of the exercises have any solutions. Fun By Alex on Jun 10, 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. So with advanced calculus in view, these more or less recent publications make the subject even more accessible to undergraduates. Munkres presentation is certainly original.
Motivating examples are bountiful, and the figures are excellent. The perfect prequel to Boothby. Perfect for any undergrad By Abelian on Apr 24, I used this book for my second course in analysis as an undergraduate and loved it. Very clear exposition of the material with some examples and counterexamples. Great book for anyone wanting to move on to study topology or differential geometry.
Wang on Oct 09, Munkres is well-regarded as the author of the advanced undergraduate topology text "Topology: A First Course". This book on rigorous calculus on several variables is somehow not particularly well-known. This is unfortunate, because there is really a dearth of textbooks on this topic. The book every professor seems to favor is a thin, but challenging-to-digest volume by Spivak, "Calculus on Manifolds. I wish I had this book as a reference years ago when I was learning this subject for the first time.
However, the book tends towards the elementary which is not necessarily a bad thing! To sum up, the main virtues of this text are its clarity and elementary approach, but sometimes it is too slow and it spoonfeeds students a bit too much. Petrillo on Apr 21, this book is vastly better then browder or spivak.
Satisfied customer By Justin C. Goding on Sep 23, Very satisfied with both the service and book. It was used but in very good condition. Good textbook, terrible kindle textbook By Lowell T on Aug 07, The book is a great introduction to multivariable calculus although a bit slow paced for my taste. However, the kindle version is atrocious.
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Analysis Manifolds by Munkres James R
Analysis on Manifolds