Analytic Integrals with $\nabla\partial R^{-1}/\partial n$ Kernels for Linear Triangular Mesh Elements

Analytic expressions are presented for surface integrals whose kernels are the gradient of the normal derivative of the static Green's function. The derived integrals are valid over linear triangular mesh elements for constant and linear basis functions. Numerical results are validated in double precision using an arbitrary-precision math library. Near-machine precision accuracy is achieved for the cases presented.