Figure 1: The Legendre transform is a valuable tool in classical mechanics and thermodynamics, and involves mapping a function of a coordinate to a function of a “derivative” of a coordinate. In some cases, the transform can be useful in converting a poorly behaved function into a well-behaved one. The Legendre transform of a functional $f[x]$ is defined as $f^{*}[q]={sup}_x\{\langle x,q\rangle -f[x]\}$. The diagram exemplifies the concept for a function of single variable. Given a slope $q$, one determines the maximum value of $qx-f(x)$ shown by the double arrows. This maximum value occurs at the point marked by the arrow, and gives the Legendre transform $f^{*}(q)$. If $f(x)$ is differentiable at the point where this maximum occurs, then $f^{*}(q)$ is given by the intersection point between the tangent to the graph and the vertical axis, and $q=df/dx$. Eschrig used the Legendre transform to derive new properties of the density-functional theory for finite temperature applications.