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Comment by bmitc | Hacker News Reader

For anyone wanting to learn more about this, I highly recommend Advanced Calculus: A Differential Forms Approach by Harold Edwards and An Introduction to Manifolds by Loring Tu. The former reads almost like a novel and is a real treat of mathematical exposition. It's also a little quirky which is always nice. Tu's book is simply the gold standard of an introduction to the mathematics of manifolds and differential forms. It is the most concise and straightforward introduction to the full theory. It is also a wonderful book.

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I love Hubbard & Hubbard, which is also great as it's an introductory text. It's been used often at Harvard Math 55 and some much simpler courses:

http://matrixeditions.com/#vec

That is indeed a very good book. Although I would say that some of the notation in it is non-standard, for better or for worse. Tu is the most consistent author I have ever seen with notation, and that matters a lot in smooth manifold theory and differential geometry.

Another good book is Advanced Calculus: A Geometric View by James Callahan.

> Another good book is Advanced Calculus: A Geometric View by James Callahan.

Thank you for this! For those of us who had real difficulty with Advanced Calculus, Callahan's methodical, visual, generous approach is deeply felt and appreciated. I did not know of this book until now and immediately found myself absorbed. It's embarrassing to admit, but as one who loves mathematics yet seems to struggle and stagnate more often than everyone around me, I often want to ask for, and indeed need, a bit of hand-holding. Callahan is a wonderful guide in that sense. Thanks again.

No problem. It is a great book.

Definitely check out Edwards' book I mentioned above as well. It is a gem of a book. Although it doesn't use matrices and instead uses linear expansions, it is still brilliant. The first three chapters give an exposition of the theory, and then the next three go back and prove things. So if anything, take a look at the first three chapters and then the later ones on applications and extensions. It also has a geometrical viewpoint.

Spivak's Calculus on Manifolds (aka little Spivak) is another really good treatment.

Reads like Baby Rudin. (Good for some, bad for others.)

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