Abstract :
The magnetostatic integral operator in R3 is of considerable interest in both
scattering and potential theory. As well as for its scalar pendant, the electrostatic
integral operator, its eigenvalues, and its eigenfunctions play a fundamental role in
the theory of permanent magnetization. While the eigenvalues of both operators
are closely related, explicit expressions for the eigenfunctions of the magnetostatic
operator are known only in the case when the underlying surface is a sphere. For
the cases of ellipsoids and spheroids, we present explicit expressions for the
eigenvalues in terms of Lam´e functions and associated Legendre functions and
expressions for eigenfunctions in terms of surface ellipsoidal and surface spheroidal
harmonics, respectively. As an application the permanent magnetization of ellipsoids
is discussed.
