A general Gauss theorem for evaluating singular integrals over polyhedral domains

A general Gauss divergence theorem with applications to convolution integrals of the form ∫ f(x̄)h(|x̄− ā|)dVn, where the integration extends over an n-dimensional polyhedral domain, is presented. The kernel h(|x̄ − ā|) may be singular, but the given integral must remain integrable. As a result of the Gauss theorem, the given integral is reduced to an integral over the boundary of the n-dimensional polyhedral domain, which can be expressed as a sum of similar integrals over (n − 1)−dimensional polyhedral domains. The technique is illustrated with the evaluation of potential integrals for uniform and linear source distributions on polygonal domains, which is known to be of particular importance in the numerical treatment of electromagnetic problems.