Customer Reviews

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

5 star:		(15)
4 star:		(10)
3 star:		(4)
2 star:		(6)
1 star:		(0)

 

 

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.

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)!

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.

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.

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.

This book may only be 100 pages, but you'll get your money's worth. It is a clear and concise introduction to multivariable analysis and differential geometry. Use either this or Chapters 9 and 10 from Rudin (or in my case both) and you're golden in R^n for n>=1.

This book is just the best monograph on the subject available. The author states his goal clearly since the beginning and includes the exact amount of fundamental material needed to get to the main result: Stoke's theorem. The approach is not as general or abstract as it could be, but obviously Dr. Spivak decided to sacrifice that to make the reading more comprehensible, because his goal was not an encycopaedic treaty of analysis and differential geometry, but a detailed explanation of how this important subject is understood by modern mathematicians.

Just a word of warning: this book is not for beginners. If you are a newcomer to multivariate calculus or if you are not in love with abstract mathematics then this book could give you a headache.

A key thing about this book: it's basically typed up lecture notes. Especially as it gets further along, it displays a notable lack of rigour. Some of the problems are not necessarily provable using information from the book. Furthermore, theorems don't clearly state the assumptions under which they operate. In chapter 5, the author resorts to basically presenting a laundry list of facts about differential forms on manifolds, so it's hard to get much beyond a basic idea of what's going on.

I believe this book may be good for a course where the instructor can answer student's questions about ideas that aren't addressed rigorously. For self-study, it could be, at best, a supplement to another book. I'm actually a little baffled about why it seems to be so well-reviewed.

This book is really a great introduction to the concepts of multilinear algebra, and also works out the main theorems on advanced calculus.I believe that it will be a very helpfull introduction to those that want to study differential topology and geometry. A good place to find more material on the subject, and also a treatment of related topics on differential geometry, is do Carmo's book Differential Forms and Applications.

Many years ago, when I was a freshman in a Physics class, my Calculus teacher gave me this small book. It changed the way I viewed mathematics. Spellbound, I turned page after page enjoying the beauty of the theorems and the logic of the whole construction. This books explains the reason behind Stokes and Gauss theorems and introduces many useful concepts. It is a must read book for anybody seriously interested in the modern Calculus. It does not require exotic mathematical background, and any reader having some classes in classical Calculus can read it.

I recently reread this book and was happy to recall the magic of this great introduction to the real mathematics.