Seeking Advanced Vector Analysis Text

Question

I would like to get reccomendations for a text on "advanced" vector analysis. By "advanced", I mean that the discussion should take place in the context of Riemannian manifolds and should provide coordinate-free definitions of divergence, curl, etc. I would like something that has rigorous theory but also plenty of concrete examples and a mixture of theoretic/concrete exercises.

The text that I have seen that comes closest to what I'm looking for is Janich's Vector Analysis. The Hatcheresque style of writing in this particular text though isn't really suitable for me.

Looking forward to your reccomendations, thanks.

Answer 1

I have actually found something that comes pretty close to what I was looking for: Morita's Geometry of Differential Forms. While not a full-blown Riemannian geometry text, it seems to strike a nice balance between theory and computation and discusses many of the same topics discussed in the Janich book referenced in my question. In addition to concrete examples, it also has detailed solutions to the exercises.

Answer 2

See Willard Gibbs, archive text for an old text on Vector analysis, also referenced in Wikipedia, available free, and downloadable...At the very least, it should be of historical importance?

Most of my search returned Janich's text as a reference, though there were more "introductory" level texts to choose from. Perhaps a search on AMS website will return some more timely texts at the caliber you're looking for.

You might want to check out this text in Vector Calculus by Paul Matthews; its TOC seemed more comparable to what you are looking for than an (out-of-print) text I found entitled "Advanced Vector Analysis". At any rate, you can "look inside" to peruse the table of contents, etc., of Matthews text, rated slightly higher than Janich's.

Added:

In light of the text you mention in your answer to your question, you might find this pdf handout for a differential geometry class at Stanford interesting: Stokes' Theorem on Riemannian manifolds (or Div, Grad, Curl, and all that). Or, for that handout (and a whole bunch! of such handouts), see this page. (My apologies if this material is too "basic" for your needs!)

Answer 3

You might want to check out Tensor Analysis on Manifolds by Bishop and Goldberg.

Answer 4

Physicists often have the problem that their theories, like general relativity, are very elegant in a coordinate free formulation; but they still need coordinates all the time because they have to compute concrete solutions to concrete problems. So books about mathematically well defined physical theories that make heavy use of differential geometry are actually a good source for what you are looking for, switching between the coordinate free form and concrete coordinates, with a lot of concrete problems.

Try, for example, the classic:

• Misner, Thorne, Wheeler: Gravitation.

This is a scary 1500 pages tome, but it is that long because it takes a lot of space and time to explain basic mathematical concepts in differential geometry. You don't have to read all the later chapters about special applications to general relativity. Although I'd like to recommend that you do: It is less work than it looks at first sight, because the text is easy to read.