[BUG] multivariable chain rule

[Smith, Alexander J.]

Say p is a function of variables u, v...p(u,v).

Then dp makes sense even if there is no inner product on the (u,v)
space:

dp=p_u du+p_v dv. I suggest to students in a Calc III class that
they accept dp  as a "place-holder" for what happens if u and v should happen to be
functions of t...

dp/dt=p_u du/dt+p_v dv/dt.

One problem with insisting  dp = (grad p) dot (d rvec) is that this
assumes there is an inner product present. Mathias reminded us last
year of the physical application where in thermodynamics you have
equations of state like f(p,V,T) = 0 and you look at things like the
partial of p with respect to V assuming T=constant, etc. There is no
inner product and gradient around, but the chain rule still makes sense. 

I am not precisely sure what Tevian means when he says there are
excellent reasons for NOT restricting the notion of directional
derivative to unit vectors, but I think I agree for my own
reasons. When I read his statement I am reminded that one-forms like
dp make sense on a general manifold, but gradients only make sense
if there is extra structure present (Riemannian manifold, for
example). So he must be right.
-------------- next part --------------
An HTML attachment was scrubbed...
URL: http://frontend.science.oregonstate.edu/pipermail/bug/attachments/20051115/c4071fc3/attachment.html