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Legendre Transformation

Given a function of two variables

df = {\partial f\over\partial x}\,dx+{\partial f\over\partial y}\, dy\equiv u\,dx+v\,dy,
\end{displaymath} (1)

change the differentials from $dx$ and $dy$ to $du$ and $dy$ with the transformation
g\equiv f-ux
\end{displaymath} (2)

$\displaystyle dg$ $\textstyle =$ $\displaystyle df-u\,dx-x\,du = u\,dx+v\,dy-u\,dx-x\,du$  
  $\textstyle =$ $\displaystyle v\,dy-x\,du.$ (3)

$\displaystyle x$ $\textstyle \equiv$ $\displaystyle -{\partial g\over\partial u}$ (4)
$\displaystyle v$ $\textstyle \equiv$ $\displaystyle {\partial g\over\partial y}.$ (5)

© 1996-9 Eric W. Weisstein