Legendre transformations on the entropy

Question

Response to the following problem:

Another set of characteristic functions for a single-substance system can be defined by performing the Legendre transformations on the entropy S(U, V) rather than on the internal energy U ( V, S). The thermodynamic potentials turn out to be particularly useful in statistical mechanics and the theory of irreversible thermodynamics, in contrast to equilibrium thermodynamics.

(a) Show that Legendre transformation of S( U, V) that produces the characteristic function J(1/T, V), known as the Massieu function, is given by the transform

J = U/T +S = -(A/T), and  dJ = (U/T²)dT + (P/T)dV.

(b) Show that Legendre transformation of J(1/T, V) that produces the thermodynamic potential Y(1/T, P/T), known as the Planck junction, is defined by the transform.

Y = H/T + S = -(G/T), and  dY = (H/T²)dT - (V/T)dP.