These thermodynamic relations are also known as coefficient relations.
T = ( ∂ U ∂ S ) V {\displaystyle T=\left({\frac {\partial U}{\partial S}}\right)_{V}}
P = − ( ∂ U ∂ V ) S {\displaystyle P=-\left({\frac {\partial U}{\partial V}}\right)_{S}}
T = ( ∂ H ∂ S ) p {\displaystyle T=\left({\frac {\partial H}{\partial S}}\right)_{p}}
V = ( ∂ H ∂ p ) S {\displaystyle V=\left({\frac {\partial H}{\partial p}}\right)_{S}}
S = − ( ∂ F ∂ T ) V {\displaystyle S=-\left({\frac {\partial F}{\partial T}}\right)_{V}}
p = − ( ∂ A ∂ V ) T {\displaystyle p=-\left({\frac {\partial A}{\partial V}}\right)_{T}}
S = − ( ∂ G ∂ T ) p {\displaystyle S=-\left({\frac {\partial G}{\partial T}}\right)_{p}}
V = ( ∂ G ∂ p ) T {\displaystyle V=\left({\frac {\partial G}{\partial p}}\right)_{T}}
