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

$df=p~dx+q~dy$

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

$df-d(qy)=p~dx+q~dy-q~dy-y~dq$

or

$d(f-qy)=p~dx-y~dq$

Defining the function $g=f-qy$ then

$dg=p~dx+q~dy$

The partial derivatives of $g$ are

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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. R. A. Alberty "Use of Legendre transforms in chemical thermodynamics", Pure and Applied Chemistry 73 pp. 1349-1380 (2001)