the "external" field Therefore, we now denote the field from before as where the total field is given by Now, the equation on the left-hand side of the Lorentz reciprocity theorem can be rewritten by moving the from the external current term to the response field terms and also adding and subtracting a term, to obtain the external field multiplied by the total current For the limit of thin wires, this gives the product of the externally applied voltage (1) multiplied by the resulting total current (2) and vice versa. In particular, the Rayleigh-Carson reciprocity theorem becomes a simple summation: where and denote the complex amplitudes of the AC applied voltages and the resulting currents, respectively, in a set of circuit elements (indexed by ) for two possible sets of voltages and Most commonly, this is simplified further to the case where each system has a single voltage source at and Then the theorem becomes simply or in words: The current at position (1) from a voltage at (2) is identical to the current at (2) from the same voltage at (1). Conditions and proof of Lorentz reciprocity The Lorentz reciprocity theorem is simply a reflection of the fact that the linear operator relating and at a fixed frequency (in linear media): where is usually a symmetric operator under the "inner product" for vector fields and (Technically, this unconjugated form is not a true inner product because it is not real-valued for complex-valued fields, but that is not a problem here. In this sense, the operator is not truly Hermitian but is rather complex-symmetric.) This is true whenever the permittivity and the magnetic permeability , at the given , are symmetric 3×3 matrices (symmetric rank-2 tensors) – this includes the common case where they are scalars (for isotropic media), of course. They need not be real – complex values correspond to materials with losses, such as conductors with finite conductivity (which is included in via ) – and because of this, the reciprocity theorem does not require time reversal invariance. The condition of symmetric and matrices is almost always satisfied; see below for an exception. For any Hermitian operator under an inner product , we have by definition, and the Rayleigh-Carson reciprocity theorem is merely the vectorial version of this statement for this particular operator that is, The Hermitian property of the operator here can be derived by integration by