Legendre transform: Difference between revisions

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[[http://en.wikipedia.org/wiki/Legendre_transform Legendre transform]]
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).
 
If one has the function <math>f(x,y);</math> one can write
 
:<math>df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy</math>
 
Let <math>p= \partial f/ \partial x</math>, and <math>q= \partial f/ \partial y</math>, thus
 
:<math>df = p~dx + q~dy</math>
 
If one subtracts <math>d(qy)</math> from <math>df</math>, one has
 
:<math>df- d(qy) = p~dx + q~dy -q~dy - y~dq</math>
or
:<math>d(f-qy)=p~dx - y~dq </math>
 
Defining the function <math>g=f-qy</math>
then
 
:<math>dg =  p~dx - y~dq</math>
 
The partial derivatives of <math>g</math> are
 
:<math>\frac{\partial g}{\partial x}= p, ~~~ \frac{\partial g}{\partial q}= -y</math>.
 
==See also==
*[[Thermodynamic relations]]
==References==
<references/>
;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]]

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