Maxwell relations between thermodynamic quantitiesFor thermoelastic materials, show that the following relations hold: where is the Helmholtz free energy and is the Gibbs free energy. Also show that 
Proof:
Recall that
and
(Note that we can choose any functional dependence that we like, because the quantities e, η, are the actual quantities and not any particular functional relations).
The derivatives are
and
Hence,
From the above, we have
and
Hence,
More Maxwell relationsFor thermoelastic materials, show that the following relations hold: and 
Proof:
Recall,
and
Therefore,
and
Also, recall that
and
Hence,
and
