Generalized Helmholtz Decomposition and Static Electromagnetics

Abstract

The well-known decomposition of vector fields to solenoidal and irrotational parts, known as the Helmholtz decomposition, is generalized in terms of more general linear operators involving two arbitrary symmetric, positive-definite and complete (non-singular) dyadics. It is seen that, in terms of the generalized decomposition, potential expansions for the static electric and magnetic fields in anisotropic media can be formed in a straightforward manner. The decomposition theorem is further generalized in a form applicable to the static electromagnetic fields in the most general linear medium (bi-anisotropic medium) characterized by four medium dyadics. In this case, the decomposition theorem and potential expressions are presented in a compact form in terms of six-vectors.