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 у and ф = G(x, у), ψ = G(x', y), with Ѳ_y G(z, у) = -4πδ(у - 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 G_D(x, x' must be symmetric in x and x'.

(b) For Neumann boundary conditions, use the boundary condition (1.45) for G_N(x, x') to show that G_N(x, x') is not symmetric in general, but that G_N(x, x') – F(x) is symmetric in x and x', where

F(x) = 1/S∫_SG_N(x, y) da_y

(c) Show that the addition of F(x) to the Green function does not affect the potential Ф(х). See problem 3.26 for an example of the Neumann Green function.