Consider the electrostatic Green functions of Section 1.10 for Dirichlet and Neumann boundary conditions on the surface S bounding the volume V. Apply Green's theorem (1.35) with integration variable y and ϕ=G(x,y), ψ=G(x′,y), with ∇y2G(z,y)=−4πδ(y−z). Find an expression for the difference [G(x,x′)−G(x′,x)] in terms of an integral over the boundary surface S.
((a)) For Dirichlet boundary conditions on the potential and the associated boundary condition on the Green function, show that GD(x,x′) must be symmetric in x and x′.
((b)) For Neumann boundary conditions, use the boundary condition (1.45) for GN(x,x′) to show that GN(x,x′) is not symmetric in general, but that GN(x,x′)−F(x) is symmetric in x and x′, where
F(x)=S1∮SGN(x,y)day
((c)) Show that the addition of F(x) to the Green function does not affect the potential Φ(x). See problem 3.26 for an example of the Neumann Green function.