In this paper, we have tied three fields together with a common mathematics based on the fact that the underlying extensive function must be convex. From such a common basis, gradient constructions are derived which have useful geometrical interpretations and allow results from one field to be applied through analogy to the others.
Apparent differences in the way common tangents are applied to compositions and to interfaces are resolved by consideration of the particular metric in use.
The analogies lead to the notion of the chemical Wulff shape which is constructed on chemical free energy normalized by the same euclidian metric which is used to normalize surface tension. This construction suggests a promising means to determine phase boundaries without resorting to numerical differentiation.
Finally, the common mathematical structure presents a unified way of studying, teaching, and understanding three important topics in materials science.