The differentials of four state functions are often used in thermodynamics. E (often using the symbol U), H, A, and G. Consider E first:

where the shorthand notation and is used. By comparing the above two expressions, we see that

The so-called "Maxwell's Relations" are derived from the fact that second order derivatives do not depend on the order of differentiation:

Similarly,
If we do the same thing with H, A, and G, we obtain

Now let's try an example of using the above equations to determine a thermodynamic relationship. Notice our expression for H has S, p, and N as the natural variables. Suppose we wish to determine . We cannot obtain this by reading from any of the above equations. To get rid of the S-dependence in our expression for dH, we simply write dS in terms of dT, dp, and dN: