In thermodynamic identity, we say dU=TdS-pdV, and only Maxwell's relation is required for U to be a proper behaved thermodynamics variable.
But from multivariable calculus, that result requires dS and dV to be orthogonal to each other. And I never see people prove that or trying to justify the Maxwell's relation otherwise. My guess is that the vector field (T,P) is defined in a conjugated space, so that dS, dV is no longer required to be orthogonal. But I don't have reliable reference to justify if my guess is canonical or not. Anyone knows it?
