elements of ), and Rayleigh-Carson reciprocity is equivalent to the statement that Unlike it is not generally possible to give an explicit formula for the Green's function (except in special cases such as homogeneous media), but it is routinely computed by numerical methods. Lossless magneto-optic materials One case in which is not a symmetric matrix is for magneto-optic materials, in which case the usual statement of Lorentz reciprocity does not hold (see below for a generalization, however). If we allow magneto-optic materials, but restrict ourselves to the situation where material absorption is negligible, then and are in general 3×3 complex Hermitian matrices. In this case, the operator is Hermitian under the conjugated inner product and a variant of the reciprocity theorem still holds: where the sign changes come from the in the equation above, which makes the operator anti-Hermitian (neglecting surface terms). For the special case of this gives a re-statement of conservation of energy or Poynting's theorem (since here we have assumed lossless materials, unlike above): The time-average rate of work done by the current (given by the real part of ) is equal to the time-average outward flux of power (the integral of the Poynting vector). By the same token, however, the surface terms do not in general vanish if one integrates over all space for this reciprocity variant, so a Rayleigh-Carson form does not hold without additional assumptions. The fact that magneto-optic materials break Rayleigh-Carson reciprocity is the key to devices such as Faraday isolators and circulators. A current on one side of a Faraday isolator produces a field on the other side but not vice versa. Generalization to non-symmetric materials For a combination of lossy and magneto-optic materials, and in general when the ε and μ tensors are neither symmetric nor Hermitian matrices, one can still obtain a generalized version of Lorentz reciprocity by considering and to exist in different systems. In particular, if satisfy Maxwell's equations at ω for a system with materials and satisfy Maxwell's equations at for a system with materials where denotes the transpose, then the equation of Lorentz reciprocity holds. This can be further generalized to bi-anisotropic materials by transposing the full 6×6 susceptibility tensor. Exceptions to reciprocity For nonlinear media, no reciprocity theorem generally holds. Reciprocity also does not generally apply for time-varying ("active") media; for example, when is modulated in time by some external process. (In both of these