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I have already taken up to Multivariable Calculus, Linear Algebra and Diff Eq. I want to learn Calculus on Manifolds by myself, could you recommend a good introductory book on this subject?

Should I look at other things first, like topology, to get a better background?

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Looking at topology may seem like a good idea, but I think sticking with manifolds is the right track for you to take. –  picakhu Dec 16 '10 at 4:52
    
The first three chapters of Shlomo Sternberg's Lectures on differential geometry present a good overview of the subject. –  Baudrillard Dec 16 '10 at 19:11

7 Answers 7

up vote 9 down vote accepted

A good place to start learning about manifolds is to read a book by Spivak called "Calculus on Manifolds." This book is very small, and the first three chapters are a short review on multivariate calculus, but I should say that the focus is much more on mathematical rigor than a normal calculus course so it would be worth your time to look through that I imagine. The last chapter introduces manifolds, how to integrate on them, and eventually culminates in the modern version of Stoke's Theorem.

If you want however to get a much more in depth view on manifolds, you will have to learn some topology. A good free online book to learn from, that I myself originally used, is called "Topology Without Tears." It can be downloaded for free at: http://uob-community.ballarat.edu.au/~smorris/topology.htm

It covers about everything a first course in topology covers, while being fairly straightforward and easy to understand. It is especially good for people who have not dealt with mathematical proof very much yet, because the author is very clear and detailed in his explanations and often explains about common proof techniques.

If you prefer a different book, or a more commonly used book than "Topology Without Tears," Munkre's Topology book is fine.

I can recommend two books on introductory manifolds which will be a lot more in depth than Spivak, but are still at the introductory level:

John Lee's "Introduction to Smooth Manifolds" and Boothby's "Introduction to Differentiable manifolds"

These books are about Smooth Manifolds, which are a type of manifold in which calculus can be done.

Lastly, if you learn topology and think you want to learn about manifolds from a topological point of view

William Fulton's "Algebraic Topology"

is a good place to look.

I hope this helps! I've read a lot of books, so if you are looking for anything in particular that is slightly different from what I've said please feel free to email me.

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As an alternative to Fulton, also on topological approach of manifolds, you could consider John Lee's "Introduction to Topological Manifolds" for a nice intro (presupposing little to no topology and introducing it). I love this book! –  5space May 20 at 3:08

Munkres (the author of the very clear text on Topology) wrote a book called Analysis on Manifolds. This is basically like Spivak, but twice as long and with more pictures. Everyone has fond memories of Spivak after they've used it, because of how small and "cool" it is. But at the end of the day (especially for self-learning) I think Munkres is probably more straightforward and less frustrating.

http://www.amazon.com/Analysis-Manifolds-Advanced-Books-Classics/dp/0201315963

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I agree completely.But once you've learned it,working your way through Spivak is well worth the effort to master it. –  Mathemagician1234 Sep 25 '11 at 21:02

What you're really asking for a textbook giving a modern presentation of vector calculus/calculus of functions of several variables.Of necessity,there's going to be a lot of overlap between such textbooks and differential topology books. Indeed, I think eventually separate books on both subjects will be obsolete and there'll be unified presentations of both. The standard books for learning this material are Calculus On Manifolds by the legendary Micheal Spivak and Analysis on Manifolds by James Munkres. Spivak's book is basically a problem course with quite a few pictures. It's quite rough going,but it's worth the effort if you've got the patience. Munkres is more of a standard textbook and covers the same material with much more detail.

Notorious for it's level of difficulty is Advanced Calculus by Lynn Loomis and Schlomo Sternberg, now available for free at Sternberg's website,which is a huge gift to all mathematics students of all levels. This book was written for an honors course in advanced calculus at Harvard in the late 1960's and it's unimaginable that they actually taught UNDERGRADUATES this material at this level. Then again,these were honor students at Harvard University in the late 1960's-argueably the best undergraduates the world has ever seen. In any event,for mere mortals,this is a wonderful first year graduate text and probably the most complete treatment of the material that's ever been written. It even ends with an abstract treatment of classical mechanics. But you better make sure you got a firm grasp of undergraduate analysis of one variable and linear algebra first.

Similar in content,but easier and much more modern is J.H. Hubbard and B.B. Hubbard. Vector Calculus, Linear Algebra, and Differential Forms.Beautifully written,wonderfully illustrated with many,many applications,philosophical digressions and unusual sidebars, like Kantorovich's Theorum and historical notes on Bourbaki,this is the book we all wish our teachers had handed us when we first got serious about mathematics.Even if you're using a "purer"treatment like Spivak,it's a book you simply must have. It's a book anyone can learn something new from.

Lastly,I want to mention something. The very first course trying to present to undergraduates the essentials of calculus on manifolds was given at Princeton University in the early 1960's and was taught by Nornan Steenrod,along with Nickerson and Spence.Spivak was an undergraduate in that course and famously,it inspired his textbook. Although those notes circulated for many years, they've never been published in book form.

Until now. The venerable Dover Books has acquired the rights to the notes and they will finally be officially published by them this summer. I've never seen the notes in detail-but given thier heritage,they're well worth checking out.Also,since Dover is putting them out in a very cheap edition,there's no good reason not to own a copy. So be on the lookout for that.

That should be more then enough to get you started-good luck!

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Why did someone downvote this?!? Ok, now someone's just being personally spiteful on here. –  Mathemagician1234 Sep 17 '11 at 22:13
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Honestly-I want to know who down-voted this post and why. If they can't give me a plausible reason based on content why they did it-they should remove it immediately. –  Mathemagician1234 Sep 25 '11 at 21:01

May be this : Spivak,Calculus-Manifolds-Approach

It is an reprint from year 1995, although maybe an more new could also you serve.

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For the record, I really disliked this book. I thought it had way to many equations, and didn't give me any sense of what was going on conceptually. I know that many people like it, though. –  David Speyer Dec 16 '10 at 13:48
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I agree with David. I am a fan of most of Spivak's books, but this one struck me as being essentially all technicalities. (My negative impression of it as an undergraduate has been recorded for posterity in the Chicago Undergraduate Mathematics Bibliography, the existence of which would be rather embarrassing to me now except for the fact that most the opinions I expressed therein happen to be the same as the ones I currently hold!) –  Pete L. Clark Jul 26 '11 at 7:55
    
@Pete That list is where I first heard of you,friend. I still recommend the list to anyone who asks me for a knowledgable list of math book reviews(in addition to my own opinions,of course!) So don't be embarrassed by it. In fact,if you ever find the time,I'd consider updating and expanding it! –  Mathemagician1234 Oct 1 '11 at 18:25

Spivak and Munkres are both not bad, but I think Advanced Calculus by Loomis and Sternberg is even better. It rigorously covers linear algebra, calculus in severable variables, metric spaces, and multilinear algebra. Then it moves on to manifolds and integration, and ends with applications to classical mechanics.

Moreover, it's free! See http://www.math.harvard.edu/~shlomo/

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A classic text,but really a graduate text for mere mortals.It's an eternal monument to how incredibly good the honors undergraduates at Harvard in the 1960's were. –  Mathemagician1234 Oct 1 '11 at 20:09

If you are looking for a book with tons of proofs, I would strongly recommend J.H. Hubbard and B.B. Hubbard. Vector Calculus, Linear Algebra, and Differential Forms.

Super good as an introduction book if you can find someone to go to when you are stuck. http://www.amazon.com/Vector-Calculus-Linear-Algebra-Differential/dp/0130414085

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Differential forms and connections by Darling is not too bad. I can't remember how basic it is though. He does spend a lot of time going through what differential forms are. My approach to learning is to grab a bunch of books and read each one in turn until i get confused or stuck and then to go to the other to see if it makes more sense. There is also topology from the differentiable viewpoint by milnor. but this is more on smooth manifolds than calculus.

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