To get the (standard) Legendre transform U ∗ ∗ of the internal energy U with respect to volume V, the function u ( p , S , V , { N i } ) = p V − − U is defined first, then it shall be maximized or bounded by V. To do this, the condition ∂ ∂ u ∂ ∂ V = p − − ∂ ∂ U ∂ ∂ V = 0 → → p = ∂ ∂ U ∂ ∂ V needs to be satisfied, so U ∗ ∗ = ∂ ∂ U ∂ ∂ V V − − U is obtained. This approach is justified because U is a linear function with respect to V (so a convex function on V) by the definition of extensive variables. The non-standard Legendre transform here is obtained by negating the standard version, so − − U ∗ ∗ = H = U − − ∂ ∂ U ∂ ∂ V V = U + P V .