In a Green's function \(G(x,x')\), the primed variable \(x'\) marks the source point — the location of a unit impulse — while \(x\) is the field point where the response is measured. This can be seen as a matrix element \(\langle x | L^{-1} | x' \rangle\) of the inverse operator. For the 1D Poisson equation \(-\frac{d^2 u}{dx^2}=f\), we obtain \(G(x,x')=-\frac12|x-x'|\) using \(\frac{d^2}{dx^2}|x-x'|=2\delta(x-x')\). Additional examples illustrate the same concept.