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92 of 95 people found the following review helpful:
5.0 out of 5 stars
The Mathematician's Calculus, October 30, 2001
This review is from: Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus (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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32 of 33 people found the following review helpful:
5.0 out of 5 stars
One reviewer said :"by carefully developing only what is essential." which is best thing to say about this book, January 19, 2006
This review is from: Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus (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!
Simply, if you love studying Math, (some say torture urself with Math), then that's the right book for you.
I can not but give 5 stars for this book. Overpriced, not many examples, WHATEVER, The name of the book is calculus on Manifolds (not advanced calc 2 or real analysis 2), and thats what you will absolutely find in the book.
*** Update ***
now that I'm done with the book. It has been a great experience, especially it's my first exposure to manifolds (also differentials). However, I think this book really lacks examples. If I was not studying this book as independent study with a professor, I would have learned some wrong concepts on my own (especially in the section about n-cubes, examples by the author were REALLY needed there to clear any confusion). The way I studied this book is that I read it, try to rewrite all the proofs on my own rigorously including all the left-out details, then go to my professor, he will give more intuition, and I try to come up with examples in his office. It's been great, I learned a lot. I still think lack of examples is a problem. Though wud not want to change my 5 stars.
Now I think studying this book as second (at least not first) exposure to the material would be a lot better, That's if you are studying it on your own! However, IF you have extra time and IF you can discuss the material with a professor everytime you read a section, and He can direct you to develop the right examples, then this book is GREAT (and I think can be covered in one semester)!
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27 of 29 people found the following review helpful:
5.0 out of 5 stars
Great book for a first course in higher mathematics., July 12, 2005
This review is from: Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus (Paperback)
Students will need a flare for the abstract in tackling the contents of the book.
It is not suggested for applied sciences such as engineering or the faint of heart.
However, for those wishing an introduction to differential manifolds at a basic level the book is recommended.
Spivak recasts notions of differentiability and integration in a more general setting still in Rn ( limits continuity higher dimensional derivatives measure theory partitions of unity etc) and then introduces the concepts of forms and tensors and associated properties and operations. He now gets to the good stuff by introducing manifolds ( structures with patches that look locally like Rn and are sewed together the right way ) and applying the more abstract theory developed earlier to these structures. One learns how to integrate forms on manifolds.
Stokes theorem is the result. Some other useful ideas like orientation are introduced as well.
A very nice compact book useful for time immemorial.
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