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 with vorticity can be described as a superposition of the circularly polarized modes. We then obtain exact solutions of the incompressible Navier–Stokes equations for particular superpositions of the circularly polarized modes. In the third problem we obtain particular exact vorticity solutions of the equations describing a viscous incompressible fluid on the rotating earth. The fourth problem that is solved is the propagation of small disturbances in all generality in a compressible, viscous fluid in which the pressure is a function of the density only. We also give the Helmholtz decomposition in terms of spherical coordinates, the vector components themselves and their arguments being given in terms of these coordinates. Such a decomposition should prove useful in problems with spherical symmetry.