At the back of Spivak's wonderful Calculus text, there is a chapter that simply fascinated me when I encountered it as an undergraduate. It is an annotated bibliography listing books that a reader who had learned calculus might want to read next. Among many intriguing suggestions, it contains a mention of Calculus on Manifolds, which is described as an attempt to introduce readers to a broad swath of mathematics.
How could I resist? I found the little book for sale, bought it, and set to. The book is fairly short, after all!
First lesson: when it comes to mathematics books, "short" does not necessarily mean "a quick read". This one isn't. Already on the first page or two, I realized that I'd need to slow down and read carefully.
But golly, it was neat! It treated Fubini's theorem for Riemann integrals, for one thing, something that had been hand-waved through in my classes. It gave a full proof of the change-of-variables theorem for multivariable integration. And then it really got going, defining differentiable manifolds, differential forms, manifolds with border, integration on chains, and getting all the way to the general Stokes Theorem. Quite a ride!
Of course, I didn't get through it all. I went back to it as a graduate student, and learned a little more and a little better. I went back to it again when I had to teach vector calculus. I have never been able to overcome a certain disdain for the vector versions of the theorems of Gauss, Green, and Stokes, which are just special cases of the main theorem in this book. I even tried to teach future engineers about differential forms, with predictably terrible results.
Calculus on Manifolds is incredibly dense, makes no concessions to the reader, contains very little physical motivation. It is also elegant, beautiful, and full of serious mathematics, the sort of book that repays the reader's efforts. Don't give it to just anyone. But when the right students come along, those who will put in the work and who will appreciate the elegance, set them up with this little book, and just watch the sparks fly.
Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.