2007-07-02

i finally got around to reading the proof of the poincaré lemma. moosehead_beer, do you want a proof post? how familiar are you with differential forms ( $dx \wedge dy$ )?

like does this make sense:
$\begin{multline*}d(f\,dx + g\,dy + h\,dz) =\\ \bigg({\partial h \over \partial y} - {\partial g \over \partial z}\bigg)dy\wedge dz + \bigg({\partial f \over \partial z} - {\partial h \over \partial x}\bigg)dz\wedge dx + \bigg({\partial g \over \partial x} - {\partial f \over \partial y}\bigg)dx\wedge dy\end{multline*}$

i think i can get away with no more than that.

poincaré lemma in three dimensions states that (all functions are C^∞)

1. If grad f = 0, then f is constant. (obvious)
2. If curl (f, g, h) = 0 then (f, g, h) = grad F. (semi-obvious)
3. If div (f, g, h) = 0 then (f, g, h) = curl (F, G, H).
4. All f = div(F, G, H). (obvious)

it does some other stuff in higher dimensions.

2007-07-02

(no subject)