Chapter 6.10, Problem 13P

Draw a figure similar to Figure 10.6 but with $q$ outside the surface. A vector (liker in the figure) from $q$ to the surface now intersects it twice, and for each solid angle $d\Omega$ there are two $d{\sigma }^{\prime }s,$ one where $r$ enters and one where it leaves the surface. Show that $D\cdot nd\sigma$ is given by ( 10.21) for the $d\sigma$ where $r$ leaves the surface and the negative of ( 10.21) for the $d\sigma$ where $r$ enters the surface. Hence show that the total ${\displaystyle \oint D}\cdot nd\sigma$ over the closed surface is zero.