Vector Calculus | Part IA, 2002

(a) Show, using Cartesian coordinates, that ψ=1/r\psi=1 / r satisfies Laplace's equation, 2ψ=0\nabla^{2} \psi=0, on R3\{0}.\mathbb{R}^{3} \backslash\{0\} .

(b) ϕ\phi and ψ\psi are smooth functions defined in a 3-dimensional domain VV bounded by a smooth surface SS. Show that

V(ϕ2ψψ2ϕ)dV=S(ϕψψϕ)dS\int_{V}\left(\phi \nabla^{2} \psi-\psi \nabla^{2} \phi\right) d V=\int_{S}(\phi \nabla \psi-\psi \nabla \phi) \cdot d \mathbf{S}

(c) Let ψ=1/rr0\psi=1 /\left|\mathbf{r}-\mathbf{r}_{0}\right|, and let VεV_{\varepsilon} be a domain bounded by a smooth outer surface SS and an inner surface SεS_{\varepsilon}, where SεS_{\varepsilon} is a sphere of radius ε\varepsilon, centre r0\mathbf{r}_{0}. The function ϕ\phi satisfies

2ϕ=ρ(r).\nabla^{2} \phi=-\rho(\mathbf{r}) .

Use parts (a) and (b) to show, taking the limit ε0\varepsilon \rightarrow 0, that ϕ\phi at r0\mathbf{r}_{0} is given by

4πϕ(r0)=Vρ(r)rr0dV+S(1rr0ϕnϕ(r)n1rr0)dS,4 \pi \phi\left(\mathbf{r}_{0}\right)=\int_{V} \frac{\rho(\mathbf{r})}{\left|\mathbf{r}-\mathbf{r}_{0}\right|} d V+\int_{S}\left(\frac{1}{\left|\mathbf{r}-\mathbf{r}_{0}\right|} \frac{\partial \phi}{\partial n}-\phi(\mathbf{r}) \frac{\partial}{\partial n} \frac{1}{\left|\mathbf{r}-\mathbf{r}_{0}\right|}\right) d S,

where VV is the domain bounded by SS.

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