Again, define  and  to be conjugate pairs, and the  to be the natural variables of the internal energy.
Since all of the natural variables of the internal energy  are extensive quantities
it follows from Euler's homogeneous function theorem that the internal energy can be written as:
From the equations of state, we then have:
This formula is known as an Euler relation, because Euler's theorem on homogeneous functions leads to it. (It was not discovered by Euler in an investigation of thermodynamics, which did not exist in his day.).
Substituting into the expressions for the other main potentials we have:
As in the above sections, this process can be carried out on all of the other thermodynamic potentials. Thus, there is another Euler relation, based on the expression of entropy as a function of internal energy and other extensive variables. Yet other Euler relations hold for other fundamental equations for energy or entropy, as respective functions of other state variables including some intensive state variables.