Limiting forms for surface singularity distributions when the field point is on the surface

Abstract

Scalar and vector mathematical identities involving an integral of singularities distributed over a surface and sometimes over a field can be employed to define field values of a quantity of interest. As the volume excluding the singular point from the field tends to zero, the field value is derived. The expressions that result become singular as the point of interest in the field approaches the boundary. Derivation of limiting integral expressions as the field point tends to the surface having a distribution of first and second degree singularities is the main task reported. The limiting expressions for vector values require evaluation as generalized Cauchy Principal-Value Integrals for which some aspect of symmetry in a local region excluding the singularity is required. A contribution from the integral over the local region doubles the value of the identities at a point on the boundary. For a doublet distribution, a singular term arises from the local-region integration that cancels a similar singularity in the integral over the remaining surface. This local contribution for doublets depends explicitly upon the shape of the local region as well as non-orthogonality of the surface coordinate axes. The resulting expressions for surface integrals reproduce known relations for line integrals in two-dimensional fields.