Static and dynamic potential integrals for linearly varying source distributions in two- and three-dimensional problems

A technique for the evaluation of static and dynamic potentials due to source distributions defined on domains with simple shape is presented. The domains considered are polyhedral regions and, in two-dimensional problems, plane polygons, on which uniform or linearly varying source distributions are defined. It is shown how three-dimensional (two-dimensional) potential integrals are always reducible to surface (line) integrals with nonsingular kernel, by use of a nonlinear transformation for the integration variables that permits analytic integration. In the static case the integration on the boundary is performed analytically and closed form results are given. In the dynamic case the expressions of the boundary integrals are given in a form suitable for numerical integration. The use of matrix notation allows for very compact expressions readily translatable into computer programs.