Derive the characteristic functions and Maxwell relations. Starting point: First law of thermodynamics for a reversible process with only volume work. $\backslash$delta Q $=$ TdS; $\backslash$delta W $=-$pdV and dU $=$ TdS$-$pdV$.$ Systematics: One begins with the defining equations of the thermodynamic state equations for H$,$ A$,$ and G and proceeds by coefficient comparison with the total differential of these equations, then applying the Schwarz theorem.