the reciprocity of Green's functions

Here is a surprising and useful fact. Put a point source at A and measure the response at B; now swap — put the same source at B and measure at A. Reciprocity says you get the same number. Shout from the doorway and someone at the window hears you exactly as loudly as you would hear them shouting back from the window. Source and listener can trade places without changing what is heard.

Mathematically this is the symmetry G(x, y) = G(y, x): the response felt at x to a unit source at y equals the response felt at y to a unit source at x. It is not an accident — it follows from the self-adjointness (symmetry) of the operator together with the boundary condition. The proof is Green's second identity applied to G(., y) and G(., z): plug both into integral of (u L v - v L u) over the region, which equals a boundary integral that vanishes because of the boundary condition each Green's function satisfies, and the leftover delta-terms give G(z, y) = G(y, z). So reciprocity is exactly the statement that the integral operator with kernel G is symmetric — a fingerprint of a self-adjoint problem.

This matters both as a check and as a tool. As a check, any candidate Green's function you build had better be symmetric. As a tool, reciprocity is Maxwell-Betti reciprocity in elasticity, Rayleigh's reciprocity in acoustics, and the reciprocity of antennas in electromagnetism: it lets you compute a hard configuration by computing an easier swapped one. Honest caveat: reciprocity requires a self-adjoint operator and symmetric boundary conditions. For non-self-adjoint problems — for example an operator with a first-order drift (convection) term, or for the time-dependent causal Green's function — the symmetry is broken, replaced by a relation between G and the Green's function of the adjoint operator.