The microcanonical ensemble involved the thermodynamic variables N, V and E as its variables. However, it is often convenient and desirable to work with other thermodynamic variables as the control variables. Legendre transforms provide a means by which one can determine how the energy functions for different sets of thermodynamic variables are related. The general theory is given below for functions of a single variable.
Consider a function f(x) and its derivative
The equation y=g(x) defines a variable transformation from x to y.
Is there a unique description of the function f(x) in terms of the
variable y? That is, does there exist a function 
that is
equivalent to f(x)?
Given a point 
, can one determine the value of the function 
given
only 
? No, for the reason that the function 
for any
constant c will have the same value of as shown in the figure below.
However, the value
can be determined uniquely if we specify the slope of the line tangent to
f at , i.e., and the y-intercept, 
of this line.
Then, using the equation for the line, we have
This relation must hold for any general x:
Note that f'(x) is the variable y, and 
, where 
is
the functional inverse of g, i.e., 
. Solving for 
gives
where is known as the Legendre transform of f(x). In shorthand
notation, one writes
however, it must be kept in mind that x is a function of y.
