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dE = F dot dx [1]
This to me is, or I want it to be (grin), the great unifying concept,
but I'm not sure how to word the concept succinctly or also determine
whether I can always sleuth it in a physics problem.
We are all
familiar with F dot dx from mechanics, and using it to explicitly
write down the mechanical potential or kinetic energy. But it seems
also to be highly general, a way to tie thermo and mechanics
together, dE being a sum of as many terms "as you can think of that
apply to the problem." The dot notation reinforces the gradient
notion JD brought up. Eschewing signs, we have terms such as T dot dS
(entropy), P dot dV (mechanical, just another form of mechanics' F
dot dx?), mu dot dN (particle exchange), and the aforementioned V dot
dQ (charge transfer). To be conservative, the general term "F" must
itself be the gradient of a potential, no? So we have 2 gradient in
what I have called a sum of generalized F dot dx terms?