Journal of Engineering Mathematics 23: 53-79, 1989 © 1989 Kluwer Academic Publishers. Printed in the Netherlands 53 Limiting forms for surface singularity distributions when the field point is on the surface T.E. BROCKETT, M.-H. KIM and J.-H. PARK University of Michigan, Department of Naval Architecture and Marine Engineering, Ann Arbor, Michigan 48109, U.S.A. Received 20 April 1988; accepted 31 August 1988 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. 1. Introduction Two identities are frequently employed to construct solutions for problems in mathematical physics: * Green's [10] Third Identity for a scalar (or vector if the gradient of the scalar be taken) field and * an equation for a vector field that has its antecedents in Stokes' [27] definition of a scalar and vector potential as well as the Maxwell Equations [19] (see also [23], [24] and [5]). A value of the dependent variable of interest at a point within the field may be constructed in terms of integrals over the surfaces bounding the field as well as over the volume. Often (e.g., for potential flow of an incompressible fluid) the field value is obtained from the integrals over only the bounding surface since equations governing the field give zero values for the weight functions within the volume integrals. For other problems, e.g., the no-slip boundary condition for viscous flow, there is a zero value to the weight function in the surface integral. Considerable literature exists for formulating such solutions for scalar fields when the field point is away from the bounding surfaces (see e.g., [10], [14], [17], [23] and [28]). In general the resulting integrals are well behaved for field points in the volume away from the bounding surface. To apply boundary conditions or compute certain values of interest on the boundary, limiting forms of the integrals must be derived. The integrand becomes explicitly singular if the field point were set to a point on the boundary. Mangler [18] defines two formalisms that