Comments on "Meshfree solutions of volume integral equations for electromagnetic scattering by anisotropic objects"

In [1], the author encloses a scatterer of volume by a tight-fitting circular cylinder of volume V_e. A small cylindrical volume V_o is defined around, and centered on, the observation point for the total field quantity, to be an exclusion region for the integration singularity. Several formulations for the integral equations are given, depending on the material properties' nature. The main numerical problem is integration over V_e - V_o of the various scalar Green's function derivatives. The other problem addressed in the paper is treatment of the singularities in V_o. The author tries to use the Green-Gauss theorem, stated in [1, Eq. (32)], to convert the integral over V_e - V_o to a surface integral. Use of the theorem is not straightforward because of the exclusion region, and the first problem shows up in [1, Eq. (34)]. The domain of the surface integral is shown as Γ_e + Γ_o. Taking these domains separately implies an integration over V_e plus an integration over , which isn't what's wanted. Taking the conventional approach to a 3D integration in a simple volume leads to a result that also shows how the Green-Gauss Theorem should be applied in pieces.