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

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.

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. For example, the one example Spivak gives on how to take a derivative, he identifies that derivative of the projection mapping with the i-th standard basis element. That is, he is identifying the dual space of R^n with R^n itself, all the while not telling us. While this is a nice trick that can help us take derivatives faster, this should have been mentioned in the text. Chapter 4 is rough as well. Many times Spivak will say that a theorem is obvious in an easy case, then give you a sentence on how to generalize it to the more general case. There have been many times where I have had to write my own proofs because his were lacking in detail. My margins are full of notes and missing steps because of this. For me, this wasn't too bad because I learned the material really well by being so involved, but I can easily imagine many readers being left in the dust. However, this is a good book, and these flaws only detract one star in my opinion.

The next thing we need to ask is what do you need to read this. You will need a very solid understanding of single variable analysis. If you haven't read Rudin's Principles of Mathematical Analysis, Third Edition yet, now is a great time to get yourself a copy. Also, you will need a strong linear algebra background. My personal favorite is Axler's Linear Algebra Done Right, but many people like the book by Hoffman and Kunze. One you have this background, this is the best place to go if you're looking for a quick lesson in multivariable analysis and your first introduction to manifolds.

After you're done with this book, you're going to have to buy a more serious book on manifolds. This book if good for getting your feet wet, but there are so many essential things left out. I prefer the books by John Lee over Spivak's Vol. I, and so I recommend you look into those.

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)!