Legendre transform

The Legendre transform (Adrien-Marie Legendre) is used to perform a change change of variables (see, for example, Ref. 1, Chapter 4 section 11 Eq. 11.20 - 11.25):

If one has the function $f(x,y);$ one can write

$df=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy$

Let $p=\partial f/\partial x$, and $q=\partial f/\partial y$, thus

$df=pdx+qdy$

If one subtracts $d(qy)$ from $df$, one has

$df-d(qy)=pdx+qdy-qdy-ydq$

or

$d(f-qy)=pdx-ydq$

Defining the function $g=f-qy$ then

$dg=pdx+qdy$

The partial derivatives of $g$ are

$\frac{\partial g}{\partial x}=p,\frac{\partial g}{\partial q}=-y$.

See also

• Thermodynamic relations

References

1. Mary L. Boas "Mathematical methods in the Physical Sciences" John Wiley & Sons, Second Edition.
2. Robert A. Alberty "Use of Legendre transforms in chemical thermodynamics", Pure and Applied Chemistry 73 pp. 1349-1380 (2001)