Legendre transformation

In mathematics, two differentiable functions f and g are said to be Legendre transforms of each other if their first derivatives are inverse functions of each other:

Related Topics:
Mathematics - Differentiable - Derivative - Inverse function

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:Df = left( Dg ight)^{-1}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

f and g are then said to be related by a Legendre transformation. Legendre transformations are named after Adrien-Marie Legendre. They are unique up to an additive constant which is usually fixed by the additional requirement that

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:f(x) + g(y) = leftlangle x,y ight angle.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The Legendre transformation is its own inverse, and is related to integration by parts.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~