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Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus

39 customer reviews
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: 5.5 x 0.4 x 8.3 inches
  • Shipping Weight: 9.1 ounces (View shipping rates and policies)
  • Average Customer Review: 4.1 out of 5 stars  See all reviews (39 customer reviews)
  • Amazon Best Sellers Rank: #192,620 in Books (See Top 100 in Books)

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139 of 145 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!
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22 of 22 people found the following review helpful By U of M Math Student on June 18, 2011
Format: Paperback Verified Purchase
Let me start by saying that I think this book is the best for an advanced undergraduate or graduate student who wants to learn multivariable analysis and get an introduction to manifolds. There are several reasons for this. The first thing I think this book does well is that it has interesting problems. Unlike other competitors (i.e. Munkres), who offer no interesting problems in many sections, this book is absolutely loaded with great problems. One of the problems is even called "A first course in complex variables." Let that tell you about the quality of exercises.

Another thing I like about this book is that it swiftly builds up the multivariable analysis theory without too many pit stops. One thing I hated about Munkres is that he too way too long to develop the multivariable riemann integral. Munkres takes three steps to developing it (rectangles, Jordan-measurable sets, and then open sets), and on each stage he reproves all of the facts that we know the integral should have. Spivak, on the other hand, develops the integral over rectangles, tells you in a sentence how to generalize it to Jordan-measurable sets (that's all that was needed), and then uses partitions of unity to define the more general integral. Spivak's method is faster, gives us a good look at how partitions of unity can be used, and uses the fact that the reader should be able to prove and predict the properties that the integral should have based on the assumption that we've dealt with the single variable case before. The makes Spivak a much quicker and interesting read than any other book on the subject.

While I do like this book, it is not without flaws. The general opinion is that this book is a little too terse on explanations sometimes.
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Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus
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