EIGENFUNCTIONS OF THE CURL OPERATOR, ROTATIONALLY INVARIANT HELMHOLTZ THEOREM, AND APPLICATIONS TO ELECTROMAGNETIC THEORY AND FLUID DYNAMICS

Air Force Requirements, such as the knowledge of the upper atmosphere environment of vehicles and the knowledge of the propagation characteristics of radio and radar signals, require the solutions of the equations of motion of fluid dynamics and of electromagnetic theory which are often very complicated. This report presents a new mathematical approach to the obtaining of such solutions. The vector field is represented in such a form that new techniques may be used to find the appropriate solutions. Some problems of fluid dynamics and electromagnetic theory are solved as an illustration of the new approach. In this report, eigenfunctions of the curl operator are introduced. 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 the Helmholtz decomposition of vector fields has been introduced 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, four problems are solved.