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Legendre transform
2011-12-15T07:29:11Z
<p>TermPaperServices: </p>
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<div>The '''Legendre transform''' is used to perform a change [http://www.ghostpapers.com Term Paper] 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).<br />
<br />
If one has the function <math>f(x,y);</math> one can write<br />
<br />
:<math>df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy</math><br />
<br />
Let <math>p= \partial f/ \partial x</math>, and <math>q= \partial f/ \partial y</math>, thus<br />
<br />
:<math>df = p~dx + q~dy</math><br />
<br />
If one subtracts <math>d(qy)</math> from <math>df</math>, one has<br />
<br />
:<math>df- d(qy) = p~dx + q~dy -q~dy - y~dq</math><br />
or<br />
:<math>d(f-qy)=p~dx - y~dq </math><br />
<br />
Defining the function <math>g=f-qy</math><br />
then<br />
<br />
:<math>dg = p~dx - y~dq</math><br />
<br />
The partial derivatives of <math>g</math> are<br />
<br />
:<math>\frac{\partial g}{\partial x}= p, ~~~ \frac{\partial g}{\partial q}= -y</math>.<br />
<br />
==See also==<br />
*[[Thermodynamic relations]]<br />
==References==<br />
<references/><br />
;Related reading<br />
*[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)]<br />
[[category: mathematics]]</div>
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