When I first learnt this 9 years ago, I hated it. It was a “here’s the formula, use it” kind of course. Second time around, I learn the subject in the context of differential forms and boy what a world of difference.
[tex] \int_{\partial\Omega} \omega = \int_{\Omega} d\omega[/tex]
and
[tex] \Lambda^0 \; \stackrel{ grad } {\longrightarrow} \; \Lambda^1 \; \stackrel{ curl } {\longrightarrow} \; \Lambda^2 \; \stackrel{ div } {\longrightarrow} \; \Lambda^3 [/tex]
is indeed a most beautiful theory.
The funny thing was that I had to teach two tutorial sessions on the same subject while I was relearning the material, and so I had a very clear idea of what was happening. But how much did that benefit my students, I would never know.
We’re learning differential forms with an eye toward Stoke’s Theorem right now in my undergrad Analysis class, based on baby Rudin.
It’s horrible: exactly as you put it, a “here’s the formula, use it” kind of approach. Other books aren’t terribly helpful— e.g. “differential forms are tensor fields of type (0, k) that are completely antisymmetric”.
Yes, Rudin is completely rigourous but not very intuitive. A better viewpoint is from linear algebra, looking at Exterior Powers. Try Janich’s book Vector Analysis.
Oh, wow! I remember picking up that book once. It looked innocent enough… I got a little worried when it picked up immediately with charts and manifolds… then I put it back down quickly when it mentioned that it presupposed knowledge of the Rank Theorem. That was only a handful of pages into the book.
One day I’ll read it; it looks like a good book. But now that I have words ‘exterior powers’ maybe I can find a more approachable work.
Thanks.
Yes, Janich use the most general context. p-forms lives in the tangent space of manifolds. I just skimmed parts which I didn’t understand. For purposes of Rudin, just take everything as [tex] R^n [/tex].
p-forms are linear algebra stuff so Rudin does not discuss them. They look like:
[tex] \omega = v_1 \wedge \ldots \wedge v_p [/tex]
It’s alternating, so when you permute the indices, there is a sign change corresponding to the sign of the permutation.
After that, we consider differential forms following Rudin. It’s quite fun, really.
Dear,Sir
I am intersted with yours book and also picking it.
Regards,
good education provider, sir