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In taking a course from this book, I found that it is not really suitable for study. Its treatment of calculus on manifolds is too quick and not rigorous enough to really be helpful to a student who is seeing this material for the first time (the supposed target audience). His proofs often contain errors or omissions (not major, but annoying when learning), and his definitions are hidden in the text, marked only by boldface font. This makes the book infuriating to read, since once the defined term has been found, it can require some effort to track down the actual definition, which can even be on a separate page! Finally, his exercises are generally quite awful. They focus on computations which rarely shed much light on the book's material or on the exercise. One such problem is titled "A First Course in Complex Analysis," and uses differential forms to derive the Cauchy integral formula. While cute, this did not reinforce strength in forms or complex analysis--it was simply a tedious set of computations.
There are many other books on differential geometry out there, and a reader can do much better than this one. Many of his definitions are very imprecise, particularly when he comes to differential forms, which are probably one of the most unfamiliar parts of his books to the average undergraduate. Also, he does very little to indicate how further study might proceed. While he mentions closed and exact forms, he never mentions how these notions can be used to study the geometry of a space (de Rham cohomology). He defines chains and integration of forms on chains, and even proves Stokes's theorem, but he does not mention the important dual relation of chains and forms (singular homology). Finally, his speed and quick prose forces the reader to work much harder than he or she should, thereby "inflating" the required prerequisites he tries to keep so low.
To understand this book you'll need to know some basic analysis and linear algebra, and understand the concept of a function to a reasonable degree...
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.
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)!
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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