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Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus Paperback – January 22, 1971

ISBN-13: 978-0805390216 ISBN-10: 0805390219

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Product Details

  • Paperback: 160 pages
  • Publisher: Westview Press (January 22, 1971)
  • Language: English
  • ISBN-10: 0805390219
  • ISBN-13: 978-0805390216
  • Product Dimensions: 8.3 x 5.5 x 0.4 inches
  • Shipping Weight: 7.8 ounces (View shipping rates and policies)
  • Average Customer Review: 4.1 out of 5 stars  See all reviews (36 customer reviews)
  • Amazon Best Sellers Rank: #56,502 in Books (See Top 100 in Books)

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4.1 out of 5 stars
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And that's why I gave it 2 stars instead of 1.
Xuan Zheng
If however you can get the earlier one, it is worth paying extra to buy a book and not something that looks like a newspaper flyer.
The Rizzler
It is a clear and concise introduction to multivariable analysis and differential geometry.
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Most Helpful Customer Reviews

129 of 134 people found the following review helpful By "dpapaioa" on October 30, 2001
Format: Paperback
When you are in college, the standard calculus 1,2, (maybe 3) courses will teach you the material useful to engineers. If you want to become a mathematician (pure or applied), you must pretty much forget the material in these courses and start over. That's where you need Spivak's "Calculus on Manifolds". Spivak knows you learned calculus the wrong way and devotes the first three chapters in setting things right. Along the way he clears all the confusion arising from inconsistent notation between partial derivatives, total derivatives, Laplacians, and the like.
Chapter four contains the main objective of the book: Stokes Theorem. I think Spivak does a great job in minimizing the pain students feel when faced with tensor algebra for the first time, by carefully developing only what is essential. By first developing the notions of vector fields and forms on Euclidean spaces rather than manifolds, he eases the assimilation of these concepts. There is a slight price to pay by not developing the notion of tangent spaces in terms of germs and derivations (the modern approach), but this is quite justified for the level of the book. The student who completes chapter four (including the exercises) is well-equipped to study differential geometry.
Chapter five is a brief introduction to differential geometry, a teaser if you will, for the amazing ramifications of the tools developed in the book.
As Spivak remarks in the introduction, the exercises are the most important part of the book. Spivak rewards the students in the exercises by leaving many interesting developments to them like the indefinite integral of a Gaussian and Cauchy's integral formula.
This book is a gem for the student of mathematics.
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Format: Paperback
So far Im at chapter 2 (just finished it). So Im going to update this once im done with the book.

Let me say first this is not a book to read while you are lying on bed, You absolutely need a pen, a paper, and write down the theorems, and then rewrite all the proofs, and write on your own the skipped steps. Note the author says more than one time "clearly", and those "clearly" are kinda clear, however proving them will take space, and I think they need to be proven anyway, to get a better grasp on material.! (sometimes if you think the clearly is not near clear, then maybe your thinking wrong, rethink about the problem).

Anyway, whats BEST about this book, is that it "is carefully developing only what is essential" to get to manifolds (which I never studied b4). But comparing this book to other books, Other books introduce LOTS and LOTS of material, that you really might not need to know ALL of it to get to manifolds. I am not saying all those extra material are not important, but to simply study the subject of manifolds, you really do not "need" them.

this book is five chapters:
1)Functions on Euclidean Spaces
2) differentiation
3) Integration
4) Integration on chains
5) Integration on Manifolds

IT might sound trivial for grad math books, but this book does NOT have solution to the exercices at end of book, however, some of the excerices have hints just right after the statement of the problem, and I think they are kinda solvable.

True, not so many examples provided in the book, however, if you sit and write and prove theorems, then you should be able to create your own example, and more like discover things!
Read more ›
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22 of 22 people found the following review helpful By A Customer on November 29, 1996
Format: Paperback
I read Michael Spivak's book Calculus on Manifolds after
having studied Walter Rudin's Principles of Mathematical
Analysis. 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.
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23 of 24 people found the following review helpful By "seanpool" on April 29, 2000
Format: Paperback
This is a very thin book, especially with paper cover. The content, though, is not thin at all. As creamy as one could wish for. Don't let the size fool you.
Before buying this book, I suggest you try reading one or two pages (excluding Chapter 1) on the stuff that you think you are best familiar with. If you can understand every paragraph within 30 minutes without having to go back and forth, you must have been a grad student in math for 3 years and about to get a Ph.D. in analysis. I'm not kidding!
Having said the above, I think this is a wonderful little book. Its notations are the best I have seen. No confusions at all, at least not for me. People also do refer to this book a lot.
One thing I find quite bothersome is the treatment of measure zero. I think Spivak spent too few pages on it. Well, speaking about spending too few pages, if you see a proof going for more than two pages in this book, be prepared. Take a bath, eat a good dinner and sit tight before going through it. :) Almost forgot: you ABSOLUTELY need to do the exercises.
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