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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(x,y);} one can write

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle df} , one has

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

or

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

Defining the function 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 g=f-qy} then

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 dg = p~dx + q~dy}

The partial derivatives of 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 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)