# Stoke’s Theorem and Vector Analysis

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.

$\int_{\partial\Omega} \omega = \int_{\Omega} d\omega$

and

$\Lambda^0 \; \stackrel{ grad } {\longrightarrow} \; \Lambda^1 \; \stackrel{ curl } {\longrightarrow} \; \Lambda^2 \; \stackrel{ div } {\longrightarrow} \; \Lambda^3$

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.

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### 6 Responses to Stoke’s Theorem and Vector Analysis

1. Alex says:

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”.

2. tpc says:

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.

3. Alex says:

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.

4. tpc says:

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 $R^n$.

p-forms are linear algebra stuff so Rudin does not discuss them. They look like:
$\omega = v_1 \wedge \ldots \wedge v_p$
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.

5. mohamed says:

Dear,Sir
I am intersted with yours book and also picking it.
Regards,