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