The Helmholtz theorem and scalar potential expansion

The classical Helmholtz theorem which decomposes a given vector field to cu rl-free and divergence-free components and presents the field in terms of a scalar and a vector potential is reformulated so that the divergence-free part is further decomposed in two parts with respect to either one or two g iven unit vectors. It is shown that these decompositions follow in a straig htforward way from certain operator identities, The field is represented in terms of three scalar potential functions, two of which can be related to Hertzian potentials and TE/TM decomposition when decomposing time-harmonic electromagnetic field vectors outside the source region. Applying the decom position to time-harmonic sources as well as the fields, equations between scalar source and field potentials can be formulated which gives an alterna tive method of solving electromagnetic problems.