Alternative vector potential formulations for magnetostatics

A generalized Laplace-Beltrami boundary value problem (BVP) is proposed and from which a vector BVP is derived. The new BVP is different from that derived from Maxwell's equations in magnetostatics. Nevertheless, when the Coulomb (1981) gauge is applied we obtain a BVP of physical meanings. Theorems of uniqueness of solution are presented. It is shown that the two BVPs may have the same solutions and thus both of them can be taken as alternative and generalized formulations for the magnetostatics. Variational formulations for these two BVPs are also presented.