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 is defined as . The diagram exemplifies the concept for a function of single variable. Given a slope , one determines the maximum value of shown by the double arrows. This maximum value occurs at the point marked by the arrow, and gives the Legendre transform . If is differentiable at the point where this maximum occurs, then is given by the intersection point between the tangent to the graph and the vertical axis, and . Eschrig used the Legendre transform to derive new properties of the density-functional theory for finite temperature applications.