(a) The function satisfies in the volume and on , the surface bounding .
Show that everywhere in .
The function satisfies in and is specified on . Show that for all functions such that on
Hence show that
(b) The function satisfies in the spherical region , with on . The function is spherically symmetric, i.e. .
Suppose that the equation and boundary conditions are satisfied by a spherically symmetric function . Show that
Hence find the function when is given by , with constant.
Explain how the results obtained in part (a) of the question imply that is the only solution of which satisfies the specified boundary condition on .
Use your solution and the results obtained in part (a) of the question to show that, for any function such that on and on ,
where is the region .