(i) Starting with the divergence theorem, derive Green's first theorem
(ii) The function satisfies Laplace's equation in the volume with given boundary conditions for all . Show that is the only such function. Deduce that if is constant on then it is constant in the whole volume .
(iii) Suppose that satisfies Laplace's equation in the volume . Let be the sphere of radius centred at the origin and contained in . The function is defined by
By considering the derivative , and by introducing the Jacobian in spherical polar coordinates and using the divergence theorem, or otherwise, show that is constant and that .
(iv) Let denote the maximum of on and the minimum of on . By using the result from (iii), or otherwise, show that .