Eigenfunctions of the Curl Operator, Rotationally Invariant Helmholtz Theorem, and Applications to Electromagnetic Theory and Fluid Mechanics

Abstract

In the present paper we introduce eigenfunctions of the curl operator. The expansion of vector fields in terms of these eigenfunctions leads to a decomposition of such fields into three modes, one of which corresponds to an irrotational vector field and two of which correspond to rotational circularly polarized vector fields of opposite signs of polarization. Under a rotation of coordinates, the three modes which are introduced in this fashion remain invariant. Hence we have introduced the Helmholtz decomposition of vector fields in an irreducible, rotationally invariant form.These expansions enable one to handle the curl and divergence operators simply. As illustrations of the use of the curl eigenfunctions, we solve four problems. The first problem that is solved is the initial value problem of electromagnetic theory with given time- and space-dependent sources and currents and we show that the radiation and longitudinal modes uncouple in a very simple way. In the second problem we show how fluid motion...