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Here's my attempt to formulate thermodynamics without the usual
I was reading this, but missed a link to your partial derivatives page (maybe it's there) you also cited. So obviously, my example is doubly wrong:
E(S,T) -> dE = (dE/dS)(const T) dS + (dE/dT)(const S) dT 
= (!T)dS + (!Cv)dT 
(! means "not")
So it might seem that the most general way of approaching thermo problems is to assume that all variables are always included, for example E = E(S,V,N,P,T), and an equation of state (eg PV=NkT). I haven't looked at this yet, but I don't recall thermo texts with this approach. At first glance it would appear that this approach would not lead to elegant relationships, but this may be a premature conclusion.
The reason I was reviewing this was to reconsider the simple heating/cooling of water (for example) without resorting to the dE = d'Q + d'W formulation, as per previous discussions here. To use J Denker's terminology, uncramped vs cramped (if I followed it correctly). But I'm finding the going tough. The standard treatment is simply "you heat the water, d'Q = CvdT, and if you want the entropy change, solve TdS = CvdT."
"you heat the water, d'Q = CvdT"becomes "you heat the water, dE = Cv dT".