I have a vague notion that thermodynamics is best captured in some language like differential geometry or something of the sort, but I am unfamiliar with said language. That being said, it seems to come up over and over again and, unfrortunately, my lack of knowledge hurts me here. In particular, I am studying Callen's derivation (Chapter 2) of the so-called "entropic intensive parameters" (simply the partial derivatives of the entropy with respect to its natural variables $U,V,X_k$ -- we restrict to a simple system for simplicity) and how these relate to the "energetic intensive parameters" (simply the partial derivatives of the entropy with respect to its natural variables $S,V,X_k$). One finds, says Callen, that \begin{equation} \left(\frac{\partial S}{\partial U} \right)_{X_k} \stackrel{(1)}{=} \frac{1}{\left(\frac{\partial U}{\partial S} \right)_{X_k}} = \frac{1}{T} \end{equation} and \begin{equation} \left(\frac{\partial S}{\partial X_i} \right)_{U,X_k; \, k\neq i} \stackrel{(1)}{=} \frac{-\left(\frac{\partial U}{\partial X_i} \right)_{S,X_k; \, k\neq i}}{\left(\frac{\partial U}{\partial S} \right)_{X_i,X_k; \, k\neq i}} =: \frac{-P_i}{T} \end{equation} where in both cases the (1) has used that all of the thermodynamic variables are related as some $\psi(S,U,N,X_k) = const$. That is, I believe the development above uses/assumes that system is described on such a $\psi(S,U,N,X_k) = const$ surface (so that the so-called reciprocal and reciprocity relations can be used). Is this correct? It also seems tacit in Callen's development that this $\psi$ should be such that we can always get one as a function (and not some relation?) of all the others...is this understanding correct too?