In the microcanonical ensemble, the entropy S is a natural function of N,V and E, i.e., S=S(N,V,E). This can be inverted to give the energy as a function of N,V, and S, i.e., E=E(N,V,S). Consider using Legendre transformation to change from S to T using the fact that
The Legendre transform 
of E(N,V,S) is
The quantity 
is called the Hemlholtz free energy
and is given the symbol A(N,V,T). It is the fundamental energy in
the canonical ensemble.
The differential of A is
However, from A=E-TS, we have
From the first law, dE is given by
Thus,
Comparing the two expressions, we see that the thermodynamic relations are
