Interpretations from Helmholtz’ theorem in classical electromagnetism Available to Purchase

Helmholtz’ theorem is a profound mathematical theorem which provides the definitive relationship between a vector field and mathematically defined source functions. The usual presentations of electromagnetic theory establish the principle sources of the electromagnetic field vectors, but leave the question of completeness unanswered, that is, whether all sources are included. Helmholtz’ theorem provides the basis for investigating the existence of other possible sources, and in at least one case shows a possible source not normally considered. Helmholtz’ theorem also provides a significant result pertaining to the meaning of the inverse square radial field, of which the Newtonian gravitational and Coulombic electrostatic fields are the classical examples. It can be established that the inverse square relation is determined by more elementary properties of the field and of the source function to which it relates.