Rating: 
Summary: Somewhat Terse
Review: A useful companion to Spivak's big introduction to geometry book. I think the midpoint between that book and this one would be an excellent source for learning geometry. This book uses only equations, whereas in Spivak's bigger book, it's hard to figure out what rigorous mathematical concept he's trying to introduce, as everything is done with words and pictures.
Rating: 
Summary: It's short, but is it sweet?
Review: All you have to do is read 130 pages. Then you'll know Stokes' Theorem. Tempting, isn't it?This book is perfectly rigorous, except for a few annoying gaps. It's clear, except that Spivak does not always go out of his way to convey intuition. I think this book would be easier to read if it were longer, and written more like Spivak's Calculus. But there is something strangely satisfying about having so much knowledge in such a small space. You should read this book along with Mathematical Analysis, by Browder.
Rating: 
Summary: Not suitable as textbook or for self study
Review: I do not recommend this book. It's not complete and the subject is too abstract and difficult for such a skimpy treatment to get through to the beginning students. At best it might be used as supplemental reading, then it's too expensive (a 100 page paperback.) I suggest instead the books by Buck (easiest to read), Munkres (up-to-date & user friendly) and the magnificent (but difficult) Loomis & Sternberg.
Rating:
Summary: A classical book about a classical subject
Review: I doubt that any person who works in this field (advanced calculus) has never heard about this book. It is a very good presentation of that part of analysis that treats Stokes' Theorem from a mathematically rigorous point of view. It pressuposes very little on the part of the reader (only calculus of a single variable and a bit of linear algebra, that can be learnt quickly). It is incredible that it does not pressupose, for instance, topology. That's part of the reason for its success. The other is Spivak's good fluency with his conversational style, and a conciseness and elegance that made a book with only 130 pages so readable and at the same time containing so much information. For a course, this book is better used as collateral reading; but it does so well in explaining the concepts that perhaps it is better to read the ideas first in this book than in the adopted bibliography. A word about the topics treated: its first chapter covers the basic topology, the second treats vector differential calculus, the third multiple integration, and the last two manifold calculus, obtaining Stokes' theorem as the greatest jewel of the development. Because the objective of the book is really in the last two chapters, the treatment of vector differential calculus is pretty incomplete, so cannot be used as the only source - but Spivak makes this clear from the beginning. Of course, the theory is a bit abstract, and I believe that the better profit of this book is made only on Senior level or in Graduate school. Finally, about the author: I know all mathematical books of Spivak (not many) and he has an unique way of writing that I have not met yet in anyone else's book; it makes mathematics a very dramatic, emotional and poetic subject (although most people think differently about it), so the reading is interesting even if you don't like his way of doing mathematics.
Rating:
Summary: A beautiful introduction to multivariate calculus
Review: I read Michael Spivak's book Calculus on Manifolds afterhaving studied Walter Rudin's Principles of MathematicalAnalysis. In a few short chapters, Spivak takes you on a tour of a very beautiful piece of mathematics that culminates in the proof of the foundational Stokes' Theorem. I would highly recommend this wonderful book to anyone interested in studying mathematical analysis. It is an especially useful resource to people interested in differential geometry and in partial differential equations. Spivak's coverage of multivariate calculus is more geometric and more intuitive than Rudin's. For this reason, I think that these two books provide complementary coverage of calculus of several variables. These volumes open the door to the serious study of mathematical analysis.
Rating:
Summary: The Master Piece
Review: This book is designed only for those who want to learn about differential manifolds deeply. This little tiny book is a like a holy bible for differential manifolds, Spivak knows what to put inside the book. One thing that I can say about all spivak's book are proofs. He never run away from proof, he likes proof. So for those who are hate mathematics, don't buy spivak's books because it won't meet your standard criteria.
Rating:
Summary: Simply the best
Review: This book is just the best monograph on the subject available. The author states his goal clearly since the beginning and includes the exact amount of fundamental material needed to get to the main result: Stoke's theorem. The approach is not as general or abstract as it could be, but obviously Dr. Spivak decided to sacrifice that to make the reading more comprehensible, because his goal was not an encycopaedic treaty of analysis and differential geometry, but a detailed explanation of how this important subject is understood by modern mathematicians. Just a word of warning: this book is not for beginners. If you are a newcomer to multivariate calculus or if you are not in love with abstract mathematics then this book could give you a headache.
Rating:
Summary: Not fit for an introduction
Review: This book is not fit for an introduction to tensors, manifolds, or integration on chains. Spivak is scarce with textual explanations, and his proofs are built for brevity, not pedagogical insight.
I first used this text as an undergraduate introductory course to Stoke's Theorem on manifolds, and I found the book to be frustrating at best. Minimal preparation for approaching Spivak would be at least a year of Graduate real analysis (lebesgue integration and differential forms). Also, a mastery of undergraduate linear algebra is crucial; and some topology is beneficial.
The one thing I CAN praise Spivak for is the problems. 75% of the material to be learned in Spivak is contained in the problems that conclude each section. The problems contain numerous definitions and theorems which are essential in the reading of the book. There are none/few concrete examples anywhere (problems or text) -- Munkres's Analysis on Manifolds is superb in this area, however.
Spivak is raved as a classic text in this field. Just don't make it the first one you read.
Rating:
Summary: A great little book
Review: This book is really a great introduction to the concepts of multilinear algebra, and also works out the main theorems on advanced calculus.I believe that it will be a very helpfull introduction to those that want to study differential topology and geometry. A good place to find more material on the subject, and also a treatment of related topics on differential geometry, is do Carmo's book Differential Forms and Applications.
Rating:
Summary: This is not an "introduction" to calculus on manifolds
Review: This is not an introduction to calculus on manifolds. It is most useful only when you have been introduced to the material elsewhere and want to get the author's perspective on it. The author certainly has a very subtle perspective on the material. I never thought I'd hear myself say this, but this material is covered much more clearly in "Principles of Mathematical Analysis" by Rudin. I think this is true for two reasons. First, Rudin breaks down the key results into smaller pieces; second, his choice of notation is more consistent and clearer. Another good book on the same topic is "Advanced Calculus" by Loomis.
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