Maxwell's Relations

Using Schwarz' Theorem:

d u = T d s - P d v ⟶ \frac{\partial T}{\partial v} |_{s} = - \frac{\partial P}{\partial s} |_{v}

d h = T d s + v d P ⟶ \frac{\partial T}{\partial P} |_{s} = \frac{\partial v}{\partial s} |_{P}

d f = - s d T - P d v ⟶ \frac{\partial s}{\partial v} |_{T} = \frac{\partial P}{\partial T} |_{v}

d g = - s d T + v d P ⟶ \frac{\partial s}{\partial P} |_{T} = - \frac{\partial v}{\partial T} |_{P}