Functional Differential Geometry (2012) [pdf]

Enough of you care about this to vote it to the front page? Who are you people? In that case, I have two favorite books on this topic.

Bamberg and Sternberg, A Course in Mathematics for Physics Students. It's a redo of calculus using differential geometry from the start. A very pretty way to do E&M, or calculations on the surface of the Earth, or vector flows.

Grady and Polimeni, Discrete Calculus. This is how you do calculus on graphs, which is how you do scoring algorithms on graphs. You know, for big data.

Discrete exterior calculus in Python! (almost forgot) http://arxiv.org/abs/1103.3076 They have demos that show how extraordinarily powerful it is.

That ought to keep you busy for a few months, you odd ducks.

The Sussman way, by Gerald Sussman and Jack Wisdom with Will Farr.

I know differential geometry is, as they say, needed for general relativity, they say also for quantum field theory.

I think a functional (or CS) perspective is an interesting approach, but I think you can lose some nice results in introductory differential geometry by following just this course. For example, the Gauss-Bonet theorem doesn't appear to be covered, which is an incredibly beautiful result linking the geometry and topology of manifolds.

For a more classical introduction to differential geometry requiring only multivariate calculus and some real analysis/point set topology, Do Carmo's "Differential Geometry of Curves and Surfaces" is a great textbook.

I put together a summary (key definitions/theorems) from an undergraduate course following Do Carmo at [2].

[1]: http://www.amazon.com/Differential-Geometry-Curves-Surfaces-...

[2]: http://ajtulloch.github.io/PDFs/MATH3968LectureNotes.pdf

Wait, Bamberg wrote a book on differential forms and didn't tell me when I asked him for self-study references (he recommended Hubbard&Hubbard IIRC)? Crazy. (Context: I took a Hilbert Space class from him and only then realized what I had missed by skipping his vector calculus class for the standard intro to analysis.) Maybe he made the generous assumption that I'd already thoroughly reviewed the available books and had found his wanting?

Oh well. Either way, I'll have to check it out. I never did get far in Flanders (which left too much implicit, making it difficult to read) and H&H didn't seem to emphasize the E&M applications I was looking for (admittedly I only skimmed it). It appears that said applications are the focus of Bamberg's book, so it would likely be a better fit.

I second the Bamberg and Sternberg recommendation. Those books are great introductions to differential geometry and E&M. You can go directly to part 2 if you already have a strong calculus foundation, but part 1 will give that to you and make a gentle introduction to part 2 where differential geometry is explored in more detail.