Paper 3, Section II, C

Vector Calculus | Part IA, 2013

(a) Prove that

×(F×G)=F(G)G(F)+(G)F(F)G\nabla \times(\mathbf{F} \times \mathbf{G})=\mathbf{F}(\nabla \cdot \mathbf{G})-\mathbf{G}(\nabla \cdot \mathbf{F})+(\mathbf{G} \cdot \nabla) \mathbf{F}-(\mathbf{F} \cdot \nabla) \mathbf{G}

(b) State the divergence theorem for a vector field F\mathbf{F} in a closed region ΩR3\Omega \subset \mathbb{R}^{3} bounded by Ω\partial \Omega.

For a smooth vector field F\mathbf{F} and a smooth scalar function gg prove that

ΩFg+gFdV=ΩgFndS,\int_{\Omega} \mathbf{F} \cdot \nabla g+g \nabla \cdot \mathbf{F} d V=\int_{\partial \Omega} g \mathbf{F} \cdot \mathbf{n} d S,

where n\mathbf{n} is the outward unit normal on the surface Ω\partial \Omega.

Use this identity to prove that the solution uu to the Laplace equation 2u=0\nabla^{2} u=0 in Ω\Omega with u=fu=f on Ω\partial \Omega is unique, provided it exists.

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