Consider the electrostatic Green functions of Section $1\mathrm{.}10$ for Dirichlet and Neumann boundary conditions on the surface $S$ bounding the volume $V.$ Apply Green's theorem $(1\mathrm{.}35)$ with integration variable $y$ and $=G(x,y),=G(x^{\text{'}},y),$ with $gra{d}_y^{2}G(z,y)=-4(y-z).$ Find an expression for the difference $\{G(x,x^{\text{'}})][$:$G(x^{\text{'}},x)\}$ 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 must be symmetric in $x$ and $x^{\text{'}}.$

$($b$)$ For Neumann boundary conditions, use the boundary condition $(1\mathrm{.}45)$ for $G_N(x,x^{\text{'}})$ to show that $G_N(x,x^{\text{'}})$ is not symmetric in general, but that $,($:$x^{\text{'}}\}$ is symmetric in $x$ and $x^{\text{'}},$ where

$F(x)=\frac{1}{S}o{\displaystyle \underset{S}{\overset{\phantom{}}{}}}{G}_N(x,y)d{a}_y$

$($c$)$ Show that the addition of $F(x)$ to the Green function does not affect the potential $(x).$ See problem $3\mathrm{.}26$ for an example of the Neumann Green function.