Page "Reciprocity (electromagnetism)" Paragraph 36

The inverse of the operator, i. e. in ( which requires a specification of the boundary conditions at infinity in a lossless system ), has the same symmetry as and is essentially a Green's function convolution.

So, another perspective on Lorentz reciprocity is that it reflects the fact that convolution with the electromagnetic Green's function is a complex-symmetric ( or anti-Hermitian, below ) linear operation under the appropriate conditions on ε and μ.

More specifically, the Green's function can be written as giving the n-th component of at from a point dipole current in the m-th direction at ( essentially, gives the matrix 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.