The core theory presented is the solution to the steady-state diffusion equation in spherical coordinates using separation of variables and superposition principles. It defines spherical harmonics ($Y_{nm}$) through discrete eigenvalues derived from periodicity constraints in $\phi$ and orthogonality conditions in $\theta$, establishing their role as basis functions for representing scalar fields with specific symmetries. The domain is transport phenomena within mass/heat transfer, specifically addressing the theoretical relationship between point source solutions (Green's functions), multipole expansions (dipoles, sources/sinks superposition), and boundary value problems via the method of images in infinite or finite domains.
The core theory presented is the solution to the steady-state diffusion equation in spherical coordinates using separation of variables and superposition principles. It defines spherical harmonics ($…