Topological considerations in coupling magnetic scalar potentials to stream functions describing surface currents

The problem discussed is that of coupling cuts for magnetic scalar potentials to cuts for stream functions describing currents on the orientable boundary of a good conductor or on thin, possibly nonorientable, conducting sheets. The solution of this problem is of interest both for its own sake and for the insight it gives into the general case of volume distributions of current. It is shown how the topological formalism clearly articulates the lumped parameter aspects. It is also shown that cuts for stream functions can be made on a nonorientable surface and that the discontinuities in the magnetic scalar potential can be systematically related to discontinuities in stream functions by a suitable choice of cuts.>