Paper 3, Section II, A

Vector Calculus | Part IA, 2015

(i) Starting with the divergence theorem, derive Green's first theorem

V(ψ2ϕ+ψϕ)dV=VψϕndS\int_{V}\left(\psi \nabla^{2} \phi+\nabla \psi \cdot \nabla \phi\right) d V=\int_{\partial V} \psi \frac{\partial \phi}{\partial n} d S

(ii) The function ϕ(x)\phi(\mathbf{x}) satisfies Laplace's equation 2ϕ=0\nabla^{2} \phi=0 in the volume VV with given boundary conditions ϕ(x)=g(x)\phi(\mathbf{x})=g(\mathbf{x}) for all xV\mathbf{x} \in \partial V. Show that ϕ(x)\phi(\mathbf{x}) is the only such function. Deduce that if ϕ(x)\phi(\mathbf{x}) is constant on V\partial V then it is constant in the whole volume VV.

(iii) Suppose that ϕ(x)\phi(\mathbf{x}) satisfies Laplace's equation in the volume VV. Let VrV_{r} be the sphere of radius rr centred at the origin and contained in VV. The function f(r)f(r) is defined by

f(r)=14πr2Vrϕ(x)dSf(r)=\frac{1}{4 \pi r^{2}} \int_{\partial V_{r}} \phi(\mathbf{x}) d S

By considering the derivative df/drd f / d r, and by introducing the Jacobian in spherical polar coordinates and using the divergence theorem, or otherwise, show that f(r)f(r) is constant and that f(r)=ϕ(0)f(r)=\phi(\mathbf{0}).

(iv) Let MM denote the maximum of ϕ\phi on Vr\partial V_{r} and mm the minimum of ϕ\phi on Vr\partial V_{r}. By using the result from (iii), or otherwise, show that mϕ(0)Mm \leqslant \phi(\mathbf{0}) \leqslant M.

Typos? Please submit corrections to this page on GitHub.