On the correct way of taking the surface limit in the integral equation formulations of electromagnetics

Surface integral equation formulations are widely used when considering problems of electromagnetic scattering by homogeneous or partially homogeneous material bodies. We demonstrate that two common ways of extracting principal parts from the surface integral equations of electromagnetics, viz., the flat disk approach and the hemispherical indentation approach, give different results. By examining a third, conical indentation method, which may act as either of the former two approaches, we have concluded that the hemispherical indentation technique is flawed. However, if the principal parts I/sub q/ and I/sub M/ are treated collectively, a correct result is obtained. We believe that the problem with the hemisphere method is that the flat surface limit is never properly recovered in this approach: irrespective of how small the hemisphere radius becomes, the field point Q faces a dent in S, not a flat surface. However, the hemisphere method may be saved with the proper use of the mean value theorem of integral calculus: when the average value of the whole vector integrand, rather than the field only, is taken outside the integral sign, the resulting principal parts agree with those obtained by the flat disk method.