Vector Calculus | Part IA, 2008

Let F=ω×(ω×x)\mathbf{F}=\boldsymbol{\omega} \times(\boldsymbol{\omega} \times \mathbf{x}), where x\mathbf{x} is the position vector and ω\boldsymbol{\omega} is a uniform vector field.

(i) Use the divergence theorem to evaluate the surface integral SFdS\int_{S} \mathbf{F} \cdot d \mathbf{S}, where SS is the closed surface of the cube with vertices (±1,±1,±1)(\pm 1, \pm 1, \pm 1).

(ii) Show that ×F=0\boldsymbol{\nabla} \times \mathbf{F}=0. Show further that the scalar field ϕ\phi given by

ϕ=12(ωx)212(ωω)(xx)\phi=\frac{1}{2}(\boldsymbol{\omega} \cdot \mathbf{x})^{2}-\frac{1}{2}(\boldsymbol{\omega} \cdot \boldsymbol{\omega})(\mathbf{x} \cdot \mathbf{x})

satisfies F=ϕ\mathbf{F}=\boldsymbol{\nabla} \phi. Describe geometrically the surfaces of constant ϕ\phi.

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