The use of tensor densities in equivalent circuits for field problems

Equivalent electrical networks to represent fields described by certain partial differential equations have for many years been used as a standard analogue technique. In these networks the dependent and independent variables are usually represented by voltage drops and currents in the branches, called `across¿ and `through¿ variables, and these analogue quantities on the network give the distribution of the field. In setting up equivalent networks for this purpose Kron added two steps to the existing concepts, namely (i) he derived circuits in terms of general co-ordinates, the topology of each network being independent of the co-ordinate system chosen, and (ii) by applying Stokes´s theorem to the system of vectors he showed that the networks thus derived could be interpreted as containing the field quantities flowing in the filled space. The quantities measured on the network then correspond to line, surface or volume integrals of the vector field quantites. With this interpretation the networks become more realistic models of the fields since the whole space is filled and the choice of co-ordinate system does not affect the physics of the phenomena. The present paper describes the mathematical basis of this fuller interpretation of analogue networks.