Vector Calculus | Part IA, 2004

State (without proof) the divergence theorem for a vector field F\mathbf{F} with continuous first-order partial derivatives throughout a volume VV enclosed by a bounded oriented piecewise-smooth non-self-intersecting surface SS.

By calculating the relevant volume and surface integrals explicitly, verify the divergence theorem for the vector field

F=(x3+2xy2,y3+2yz2,z3+2zx2)\mathbf{F}=\left(x^{3}+2 x y^{2}, y^{3}+2 y z^{2}, z^{3}+2 z x^{2}\right)

defined within a sphere of radius RR centred at the origin.

Suppose that functions ϕ,ψ\phi, \psi are continuous and that their first and second partial derivatives are all also continuous in a region VV bounded by a smooth surface SS.

Show that

V(ϕ2ψ+ϕψ)dτ=SϕψdSV(ϕ2ψψ2ϕ)dτ=SϕψdSSψϕdS\begin{aligned} \int_{V}\left(\phi \nabla^{2} \psi+\nabla \phi \cdot \nabla \psi\right) d \tau &=\int_{S} \phi \boldsymbol{\nabla} \psi \cdot \mathbf{d} \mathbf{S} \\ \int_{V}\left(\phi \nabla^{2} \psi-\psi \nabla^{2} \phi\right) d \tau &=\int_{S} \phi \boldsymbol{\nabla} \psi \cdot \mathbf{d} \mathbf{S}-\int_{S} \psi \boldsymbol{\nabla} \phi \cdot \mathbf{d} \mathbf{S} \end{aligned}

Hence show that if ρ(x)\rho(\mathbf{x}) is a continuous function on VV and g(x)g(\mathbf{x}) a continuous function on SS and ϕ1\phi_{1} and ϕ2\phi_{2} are two continuous functions such that

2ϕ1(x)=2ϕ2(x)=ρ(x) for all x in V, and ϕ1(x)=ϕ2(x)=g(x) for all x on S\begin{aligned} \nabla^{2} \phi_{1}(\mathbf{x}) &=\nabla^{2} \phi_{2}(\mathbf{x})=\rho(\mathbf{x}) \quad \text { for all } \mathbf{x} \text { in } V, \text { and } \\ \phi_{1}(\mathbf{x}) &=\phi_{2}(\mathbf{x})=g(\mathbf{x}) \quad \text { for all } \mathbf{x} \text { on } S \end{aligned}

then ϕ1(x)=ϕ2(x)\phi_{1}(\mathbf{x})=\phi_{2}(\mathbf{x}) for all x\mathbf{x} in VV.

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