Legendre Transformation

Given a function of two variables

$\begin{displaymath} df = {\partial f\over\partial x}\,dx+{\partial f\over\partial y}\, dy\equiv u\,dx+v\,dy, \end{displaymath}$

(1)

change the differentials from

and

with the transformation

$\begin{displaymath} g\equiv f-ux \end{displaymath}$

(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)

Then

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