The mathematical study of potentials is known as potential theory- The study of harmonic functions on manifolds.
This mathematical formulation arises
from the fact that, in physics, the scalar potential is irrotational, and thus has a vanishing Laplacian — the
very definition of a harmonic function.
In physics, a potential may refer to the scalar potential or to the vector potential. In either case, it is a field defined in space, from which many important physical properties may be derived.