On Two Useful Identities in the Theory of Ellipsoidal Harmonics

Two identities on ellipsoidal harmonics, which appear naturally in the theory of boundary value problems, are stated and proved. The first involves the ellipsoidal analogue of the Beltrami operator in spherical coordinates (also known as surface Laplacian). The second identity includes the tangential surface gradient operator defined as the cross product of the unit normal with the gradient operator on an ellipsoidal surface. In both cases, the basic spectral properties of these two operators, as they act on the surface ellipsoidal harmonics, are provided.