parts. For a finite integration volume, the surface terms from this integration by parts yield the more-general surface-integral theorem above. In particular, the key fact is that, for vector fields and integration by parts (or the divergence theorem) over a volume enclosed by a surface gives the identity: This identity is then applied twice to to yield plus the surface term, giving the Lorentz reciprocity relation. Conditions and proof of Lorenz reciprocity using Maxwell's equations and vector operations We shall prove a general form of the electromagnetic reciprocity theorem due to Lorenz which states that fields and generated by two different sinusoidal current densities respectively and of the same frequency, satisfy the condition Let us take a region in which dielectric constant and permeability may be functions of position but not of time. Maxwell's equations, written in terms of the total fields, currents and charges of the region describe the electromagnetic behavior of the region. The two curl equations are: Under steady constant frequency conditions we get from the two curl equations the Maxwell's equations for the Time-Periodic case: It must be recognized that the symbols in the equations of this article represent the complex multipliers of , giving the in-phase and out-of-phase parts with respect to the chosen reference. The complex vector multipliers of may be called vector phasors by analogy to the complex scalar quantities which are commonly referred to as phasors. An equivalence of vector operations shows that for every vectors and If we apply this equivalence to and we get: If products in the Time-Periodic equations are taken as indicated by this last equivalence, and added, This now may be integrated over the volume of concern, From the divergence theorem the volume integral of equals the surface integral of over the boundary. This form is valid for general media, but in the common case of linear, isotropic, time-invariant materials, is a scalar independent of time. Then generally as physical magnitudes and Last equation then becomes In an exactly analogous way we get for vectors and the following expression: Subtracting the two last equations by members we get and equivalently in differential form Q.E.D. Surface-term cancellation The cancellation of the surface terms on the right-hand side of the Lorentz reciprocity theorem, for an integration over all space, is not entirely obvious but can be derived in a number of ways. A rigorous treatment of the surface integral