I'm looking for a better discussion than I've found regarding
thermodynamic functions. In particular, in reading various
treatments, it would appear that E = E(S,V) and E = E(V,T) are valid,
but E = E(S,T) is not. I was thinking originally that there was some
rule about mixing intensive and extensive variables, but the 2nd of
the above 3 functional forms seems to kill that idea.
The first of the 3 leads to the most recognizable differential form :
dE = (dE/dS)dS + (dE/dV)dV = TdS - PdV
The second is less obvious:
dE = (dE/dT)dT + (dE/dV)dV = CvdT + (?)dV
Through several manipulations, the dE/dV can be cast as T*(dP/dT) - P, or
dE = CvdT + T*(dP/dT) dV - PdV
which leads to (using the top equation):
TdS = CvdT + T*(dP/dT) dV
But why can't I use the 3rd functional form, E(S,T)? It would seem no
more or less valid than E(V,T) from an intensive/extensive
standpoint, but I'd get
dE = (dE/dS)dS + (dE/dT)dT = TdS + CvdT
which is clearly at odds with the above.
 Interpret the parenthetical quantities as garden variety partial
derivatives with the appropriate variable held constant.