aah, work!

Work until Wednesday! It'll make me pay for actually relaxing/concentrating on Noteables during Tech Week (SUCH a good concert)...sorry, HRSFA. I have two papers to write, and am really having trouble doing anything about this fact. Oh well, at least I seem to be really helping my tutee in Math 25, more than I think I managed to help people in 23.

(mathy) And I thought of a much better intuitive way of thinking of smooth functions and the Implicit Function theorem: smooth functions are locally (almost) linear. So if you have a linear function from n+m dimensions to m dimensions, then it kills n dimensions, so the preimage of a point will be an n-dimensional plane; the Implicit Function theorem just says that if you have a /smooth/ (well, continuously differentiable) function from n+m dimensions to m dimensions, then it kills n dimensions, so the preimage of a point will be /locally/ like an n-dimensional hyperplane, i.e. will be an n-manifold, and can locally be parametrized. silkspinner, has anyone ever told us this in this way, and have I just missed it? It's so nice and intuitive.