See Koenig Born Diagram for a short background and explanation of the symbols used
Maxwell's relations arise from the fact that dU,dH,dA and dG are exact differentials This leads to the Maxwell's relations which are
(dV/dT)P = (dS/dP)T
(dT/dV)S = (dP/dS)V
(dV/dS)P = -(dT/dP)S
(dS/dV)T = -(dP/dT)V
All d's refer to partial derivatives and the quantity outside the brackets is kept constant.