Johnsarelli

1

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Derivation of the heat equation

I have seen differential approaches like yours in the past, with a variety of levels of rigor. Some explicitly invoke Taylor's theorem, for instance, while others rely roughly on the intuitive idea of differentials and linearization such as yours. However, none of these approaches is really a rigorous derivation by modern standards. The two derivations are in some sense equivalent, except that infinitesimals have been rejected in favor of integrals over finite (not infinitesimal) spatial regions. The latter is completely rigorous.

I don't think there is an error in your understanding of the problem, in light of the fact that the derivation is "correct", but there were some stylistic problems with it (besides the lack of rigor). The symbol d should not be used for both the spatial variation in T and its temporal variation. It is better to call one d_{x and the other dt, or some other suitable notation. Of course, you implicitly kept track of this in the derivation, but it should be made explicit. I also have a problem with the way the derivation of Q2 was handled, but it is difficult to pin down. It might be better to use Δx here rather than dx, at least until the limit is taken at the very end when you write down the formula for Q2. Then the identity is justified by Taylor's theorem (or finite differences if you prefer). Of course, you may as well just invoke Fourier's law instead since there is already an article treating that topic. silly rabbit (talk) 12:24, 15 May 2008 (UTC)}