Vector Calculus | Part IA, 2003

Suppose VV is a region in R3\mathbb{R}^{3}, bounded by a piecewise smooth closed surface SS, and ϕ(x)\phi(\mathbf{x}) is a scalar field satisfying

2ϕ=0 in V, and ϕ=f(x) on S.\begin{aligned} \nabla^{2} \phi &=0 \text { in } V, \\ \text { and } \phi &=f(\mathbf{x}) \text { on } S . \end{aligned}

Prove that ϕ\phi is determined uniquely in VV.

How does the situation change if the normal derivative of ϕ\phi rather than ϕ\phi itself is specified on SS ?

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