Legendre transform: Difference between revisions

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The '''Legendre transform''' (Adrien-Marie Legendre)
The '''Legendre transform''' is used to perform a change of variables (see, for example, Ref. <ref>Mary L. Boas "Mathematical methods in the Physical Sciences" John Wiley & Sons, Second Edition ISBN 0471044091</ref> Chapter 4 section 11 Eq. 11.20 - 11.25).
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 <math>f(x,y);</math> one can write
If one has the function <math>f(x,y);</math> one can write
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:<math>\frac{\partial g}{\partial x}= p, ~~~ \frac{\partial g}{\partial q}= -y</math>.
:<math>\frac{\partial g}{\partial x}= p, ~~~ \frac{\partial g}{\partial q}= -y</math>.


==Example==
==See also==
==See also==
*[[Thermodynamic relations]]
*[[Thermodynamic relations]]
==References==
==References==
#Mary L. Boas "Mathematical methods in the Physical Sciences" John Wiley & Sons, Second Edition.
<references/>
#[http://www.iupac.org/publications/pac/2001/7308/7308x1349.html Robert A. Alberty "Use of Legendre transforms in chemical thermodynamics", Pure and Applied Chemistry '''73''' pp. 1349-1380 (2001)]
;Related reading
*[http://www.iupac.org/publications/pac/2001/7308/7308x1349.html Robert A. Alberty "Use of Legendre transforms in chemical thermodynamics", Pure and Applied Chemistry '''73''' pp. 1349-1380 (2001)]
[[category: mathematics]]
[[category: mathematics]]

Latest revision as of 10:33, 15 December 2011

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

If one has the function one can write

Let , and , thus

If one subtracts from , one has

or

Defining the function then

The partial derivatives of are

.

See also[edit]

References[edit]

  1. Mary L. Boas "Mathematical methods in the Physical Sciences" John Wiley & Sons, Second Edition ISBN 0471044091
Related reading