Generalized helmholtz decomposition and static electromagnetics

The well-known decomposition of vector fields to solenoidal and irrotationa l parts, known as the Helmholtz decomposition, is generalized in terms of m ore general linear operators involving two arbitrary symmetric, positive-de finite and complete (non-singular) dyadics. It is seen that, in terms of th e generalized decomposition, potential expansions for the static electric a nd magnetic fields in anisotropic media can be formed in a straightforward manner. The decomposition theorem is further generalized in a form applicab le 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 comp act form in terms of six-vectors.