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

Michael Spivak
4.1 out of 5 stars  See all reviews (36 customer reviews)

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Book Description

January 22, 1971 0805390219 978-0805390216
This little book is especially concerned with those portions of ”advanced calculus” in which the subtlety of the concepts and methods makes rigor difficult to attain at an elementary level. The approach taken here uses elementary versions of modern methods found in sophisticated mathematics. The formal prerequisites include only a term of linear algebra, a nodding acquaintance with the notation of set theory, and a respectable first-year calculus course (one which at least mentions the least upper bound (sup) and greatest lower bound (inf) of a set of real numbers). Beyond this a certain (perhaps latent) rapport with abstract mathematics will be found almost essential.

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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: #94,136 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews
129 of 134 people found the following review helpful
5.0 out of 5 stars The Mathematician's Calculus October 30, 2001
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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44 of 47 people found the following review helpful
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
5.0 out of 5 stars A beautiful introduction to multivariate calculus November 29, 1996
By A Customer
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

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
4.0 out of 5 stars Excellent little book, but... April 29, 2000
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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Most Recent Customer Reviews
4.0 out of 5 stars A good book for a graduate student
I used this book in my first semester of differential geometry. I appreciate that Spivak takes a cohomological approach to integration on manifolds. Read more
Published 9 months ago by louis.math
5.0 out of 5 stars Great book
Used as a reference for my real analysis of several variables course. Exercises are tough and require a fair amount of thought, which is exactly how math books should be. Read more
Published 10 months ago by dclark
5.0 out of 5 stars Great Text
This isn't the greatest book ever, but when used in the right course it can be wonderful. It is also a great book when you go back for a refresher, also.
Published on January 18, 2012 by Rob S
2.0 out of 5 stars Not a great book
I used it for my undergrad 2nd year analysis course and felt it's not suitable for being a textbook at all. Read more
Published on August 7, 2011 by Xuan Zheng
4.0 out of 5 stars Awesome content, horrible production.
Everything good to say about the content of this book is referred to those with positive reviews. I, however, would like to comment on the production quality of this book. Read more
Published on July 13, 2011 by Black Milk
4.0 out of 5 stars The best we have right now, but don't blink
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... Read more
Published on June 18, 2011 by U of M Math Student
4.0 out of 5 stars Good, over-rated, but good
This is a good book if you want to learn the basics of Stokes' theorem on manifolds (in euclidean R^d) space. Read more
Published on February 21, 2011 by LB
3.0 out of 5 stars newspaper quality
I'm giving this review not so much to discuss the contents of the book, which I think is very good and would give *****, but the production quality of this edition which is truly... Read more
Published on July 17, 2010 by The Rizzler
5.0 out of 5 stars Great book. If you're prepared that is.
In the preface Spivak says that this book is accessible to anyone that has had good courses in calculus and linear algebra. Read more
Published on July 2, 2010 by Fluffy, Destroyer of Men
5.0 out of 5 stars More Advanced Than It Claims
Be warned: Spivak is speaking a different language from any I was taught. I have spent hundreds of hours on this book, and have filled three binders with notes on the first three... Read more
Published on May 4, 2010 by Robert S. Cruikshank Jr.
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