Comments on: Div, Grad, Curl and All That

By: sigfpe

sigfpe — Fri, 09 Sep 2005 18:52:42 +0000

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

By: sigfpe

sigfpe — Fri, 09 Sep 2005 18:52:10 +0000

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 (very handwavingly) here.

By: michael

michael — Fri, 09 Sep 2005 15:32:07 +0000

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

By: Walt

Walt — Fri, 09 Sep 2005 02:49:42 +0000

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.”

By: michael

michael — Thu, 08 Sep 2005 16:27:33 +0000

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)