In thermodynamics, we often ask questions about the energy of the
system. Here we will discuss the energy that we've already introduced
as well as alternate formulations of the energy of a system.

###
The Thermodynamic Identity

Suppose that we seek the energy of a system *U* in terms of its usual
variables, *σ*, *V*, and *N*. Unfortunately, we can't write a
closed solution for *U* in terms of those three variables. But not all
is lost. We can utilize the mathematical tool known as the differential.
Then we get:

So far, this may not look helpful. But if you glance back at our
previous definitions of temperature,
pressure, and chemical potential, we can rewrite the above:

*dU*(*σ*, *V*, *N*) = *τ* *dσ* - *p* *dV* + *μ* *dN*

The result is known as the Thermodynamic Identity, and is the most
basic equation in our study of thermodynamics. Notice that there is
great parallel structure to the equation. All of the extensive
variables appear as differentials, while the intensive variables
appear alone. Note that *U* is still a function of just the three
extensive variables, since we can think of the other three "variables"
as derivable from the three extensive.

###
Legendre Transform

We can use another mathematical tool here to make the Thermodynamic Identity even
more useful. The Legendre Transform allows us to make a variable change in our
definition of *U*. After all, suppose we don't want the energy as a function of the
three variables above, *σ*, *V*, and *N*.

We will utilize the Legendre Transform minimally, and not delve into the underlying
mathematics. The basic idea is that you can define a new function that is related to
the original by an added product of two correlated terms. Let us make this explicit
by using it.