certain simple conditions on the materials of the medium described below, that for an arbitrary surface enclosing a volume : Equivalently, in differential form (by the divergence theorem): This general form is commonly simplified for a number of special cases. In particular, one usually assumes that and are localized (i.e. have compact support), and that there are no incoming waves from infinitely far away. In this case, if one integrates throughout space then the surface-integral terms cancel (see below) and one obtains: This result (along with the following simplifications) is sometimes called the Rayleigh-Carson reciprocity theorem, after Lord Rayleigh's work on sound waves and an extension by Carson (1924; 1930) to applications for radio frequency antennas. Often, one further simplifies this relation by considering point-like dipole sources, in which case the integrals disappear and one simply has the product of the electric field with the corresponding dipole moments of the currents. Or, for wires of negligible thickness, one obtains the applied current in one wire multiplied by the resulting voltage across another and vice versa; see also below. Another special case of the Lorentz reciprocity theorem applies when the volume entirely contains both of the localized sources (or alternatively if intersects neither of the sources). In this case: Reciprocity for electrical networks Above, Lorentz reciprocity was phrased in terms of an externally applied current source and the resulting field. Often, especially for electrical networks, one instead prefers to think of an externally applied voltage and the resulting currents. The Lorentz reciprocity theorem describes this case as well, assuming ohmic materials (i.e. currents that respond linearly to the applied field) with a 3×3 conductivity matrix that is required to be symmetric, which is implied by the other conditions below. In order to properly describe this situation, one must carefully distinguish between the externally applied fields (from the driving voltages) and the total fields that result (King, 1963). More specifically, the above only consisted of external "source" terms introduced into Maxwell's equations. We now denote this by to distinguish it from the total current produced by both the external source and by the resulting electric fields in the materials. If this external current is in a material with a conductivity , then it corresponds to an externally applied electric field where, by definition of : Moreover, the electric field above only consisted of the response to this current, and did not include