...Prove Their Worth...

"Problems worthy of attack
prove their worth
by hitting back." - Piet Hein

A kind of running diary and rambling pieces on my struggles with assorted books, classes, and other things, as they happen. You must be pretty bored to be reading this...

Wednesday, August 21, 2002

Holy donkey poo. I've just finished working out what an integral means (a particular integral, not 'integral' in general). Damn, this thing was hairy. Hmm. I guess a brief bit of catch-up again is advisable. To my surprise, I've actually Accomplished A Goal, and I now know how to write down Maxwell's equations on a (sort of) arbitrary semi-Riemannian oriented manifold. It's really very simple: dF = 0, and *d*F = J. Explaining what that means is a different question altogether, though.

Anyway. I'm now reading the next chapter, with the rather terrifying title "DeRham Cohomology Theory in Electromagnetism". So far, it's all about getting down and naughty with 'potentials', both scalar and vector. A scalar potential is a function Phi such that E = dPhi, where E is the electric one-form (you can get away with calling it a field if you squint at it). It's called 'potential' because it has a lot to do with such things as 'potential energy' and whatnot. Moving on. If all you've got is Phi, you'd need to integrate to get E. And it turns out you have to integrate E along a path, say from point p to point q, to get Phi. That's all fine and dandy. But can we actually do that? What can stop us? (This is all the stuff the book asks.) Well, we'll be in deep shit if there isn't a path between p and q, for starters! That is, our manifold (or space, or whatever) damn well better not be scattered around the room in chunks - we'll need to be 'connected'. Next, it's fairly obvious that 'in general', we'll get different Phis if we go and integrate along different paths. This is Bad Mojo - we want phi to be defined reasonably uniquely (it'd be perverse to have an infinite number of Phis for every E). So the question is, what conditions need to be in place for our integral to be 'path independent', that is, giving the same answer for any reasonable path? The way to attack this turns out to find a crafty way of writing down an integral that integrates along all paths, and then differentiate it with respect to the 'change in paths', kind of, and then see what will make it zero. That way, we'll get the conditions on E that will make it path-independent.

Actually doing that is a total bitch, requiring strong calculus-fu. There's all the differentiations, and chain rules, and integration by parts, and products, and watching after notation, and hair-pulling, and other indignities, all to do one damn integral. By the time I got done with it (which was ten minutes ago, and I started last night), I felt like poor Polly Nomial. A quote from the linked Saga:

...She was being watched, however: that smooth operator, Curly Pi, was lurking inner product. As his eyes devoured her curvilinear coordinates, a singular expression crossed his face. Was she convergent? He wondered. He decided to integrate improperly at once. Hearing an improper fraction behind her, Polly rotated and saw Curly approaching with his power series extrapolated. She could tell at once from his degenerate conic and his dissipative terms that he was bent to no good.

It was like that, but scarier, different in the details (for one, I'm not a female nor a polynomial) and it lasted a long time. Ahem. Anyway, in the end it turns out if dE = 0, it's going to be path-independent. This is great, because dE has to be 0 in the first place, because that's a consequence of Maxwell's equations.

I'm a lazy bum, so I still don't have anything ready to post on the stuff I listed earlier.


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