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 $df$, one has

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle df-d(qy)=p~dx+q~dy-q~dy-y~dq}

or

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle d(f-qy)=p~dx-y~dq}

Defining the function Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle g=f-qy} then

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

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)