Assume that the function f is convex in x for all y, so that one may perform the Legendre transform on f in x, with p the variable conjugate to x (for information, there is a relation ∂ ∂ f ∂ ∂ x | x ¯ ¯ = p where x ¯ ¯ is a point in x maximizing or making p x − − f ( x , y ) bounded for given p and y). Since the new independent variable of the transform with respect to f is p, the differentials dx and dy in df devolve to dp and dy in the differential of the transform, i.e., we build another function with its differential expressed in terms of the new basis dp and dy.