Cosmic Variance had three interesting &ldquot;greatest&rdquot; discussion threads:

In the comments to the Greatest Physics Textbook, Clifford (the original poster) joked that no self-respecting mathematician ever read Schey’s *Div, Grad, Curl and All That.* I don’t know about anyone else, but that’s the book I learned the subject from. The book gives incredibly hand-wavy proofs, and if I remember right it trumpets its lack of rigor, but it does a good job of giving the intuition behind the Green, Gauss, and Stokes theorems. Reading it made reading something like Spivak’s *Calculus on Manifolds* much easier.

Is it just me, or was the first time you saw this stuff slightly mystifying, but when you see the material in the Manifolds context, the results seem almost obvious?

(I mean aside from the being new to the material vs not being new to the material)

Actually, I think I was the opposite. I tried to learn it from the manifolds point of view, and didn’t get any feel for what it was about. What finally allowed to learn it was hearing the sentence “the determinant of a matrix is a volume”, hearing the sentence “the derivative of volume is surface area”, and then reading Div, Grad, Curl and All That which explains all of the vector calculus theorems from the “derivative of volume is surface area” point of view. Then I was able to make sense of it from the manifolds point of view.

(Unless by “manifolds point of view”, you just mean that differential forms are much easier to understand than curl. Then I agree. In fact, I’ve completely forgotten everything I’ve ever known about curl.”

I guess I mean the latter. I too have forgotten everything I ever knew about curl except for how to spell it.

For good intuition about calculus on manifolds I recommend Misner, Thorne and Wheeler’s ‘Gravitation’ which has nice pictures of forms. When I think about differential forms I actually use a ‘dual’ form of MWT’s diagrams that I’ve described (

veryhandwavingly) here.My link vanished. ‘here’ should point to: http://homepage.mac.com/sigfpe/Mathematics/forms.pdf