Extended helmholtz theorem and its application in electromagnetic theory

Abstract

It is proved that in a regular boundary system of rectangular, cylindrical or spherical coordinate, an arbitrary vector function can be separated into three orthogonal parameters: TE model field, TM model field and irrotational field. Each of these components can be determined fully by a scalar function. On the basis of this theorem, the completeness of vector wave function system {L, M andN} is proved also. Thus it is explained that a vector function space can be projected into three uncrossed subspaces not only in Euclidean space but also in the subspace of vector wave function space.