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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.

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Maybe you want to have a look at some books on Riemannian geometry instead of looking at texts on "advanced vector analysis" (whatever that is)? If I remember correctly, Gallot-Hulin-Lafontaine do everything as coordinate-free as possible. –  t.b. Jun 13 '11 at 21:37
    
@Theo Buehler: +1. Have been studying from Gallot-Hulin-Lafontaine for a bit now and its fantastic. Find it much better than Petersen. –  Dactyl Jun 14 '11 at 9:22
    
@Theo Thanks for the reccomendation; I'll take a look –  ItsNotObvious Jun 14 '11 at 11:34
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4 Answers 4

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!)

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Thanks for the reccomendations. The Matthews text isn't really a match though. I've done a fair amount of searching for this myself and the Janich text (which I do own) is a perfect example of the content that I'm looking for. The "Vector Analysis" section of Chapter 11 of Amman and Escher's Analysis III also provides an (abbreviated) example –  ItsNotObvious Jun 13 '11 at 19:32
    
Got it...There's not a lot that "jumps out" on this topic; I tried to search for the AMS text guide/topic-references; I'll keep poking around...Perhaps you'll need to make to with Janich, and supplement with material suiting your style a bit better. As soon as you mentioned "Hatcheresque" - I cringed, so I'm rooting for you on this! –  amWhy Jun 13 '11 at 19:59
    
Hmmm...I could only find Amman and Escher's Analysis III in German (sad to say English is my native language, though I can make my way through spanish...German, unfortunately, remains -- well -- incomprehensible to me!) I'll try taking a closer look at the content of Janich's text; I did come across lecture notes, etc. that addressed Vector Analysis, some of which, I believe, accompanied/complimented Janich (as course text). –  amWhy Jun 13 '11 at 20:10
    
<a href="books.google.com/…; is an English reference to A&E Analysis III a (very) good book. –  ItsNotObvious Jun 13 '11 at 20:25
    
That handout and the others that you reference are definitely on-target. Thanks! –  ItsNotObvious Jun 20 '11 at 13:22
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up vote 2 down vote accepted

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.

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Feel free to accept your own answer! –  amWhy Jun 15 '11 at 21:56
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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.

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That's an interesting suggestion; I've have tried similar explorations in the past though and it didn't really work out because there was not enough correspondence between the physics-view and the math-view, but I'll take a closer look at MTW. I find that physics texts contain very interesting math but the presentation is so often lacking in rigor that it doesn't satisfy. –  ItsNotObvious Jun 14 '11 at 18:54
    
@3Sphere: There are also texts about general relativity for mathematicians, but MTW has chapters about classical electromagnetism, too, with an explicit explanation of the correspondence of vector calculus and the formulation with differential forms. –  Tim van Beek Jun 15 '11 at 13:07
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You might want to check out Tensor Analysis on Manifolds by Bishop and Goldberg.

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