Paper 3, Section II, A

Vector Calculus | Part IA, 2015

The vector field F(x)\mathbf{F}(\mathbf{x}) is given in terms of cylindrical polar coordinates (ρ,ϕ,z)(\rho, \phi, z) by

F(x)=f(ρ)eρ\mathbf{F}(\mathbf{x})=f(\rho) \mathbf{e}_{\rho}

where ff is a differentiable function of ρ\rho, and eρ=cosϕex+sinϕey\mathbf{e}_{\rho}=\cos \phi \mathbf{e}_{x}+\sin \phi \mathbf{e}_{y} is the unit basis vector with respect to the coordinate ρ\rho. Compute the partial derivatives F1/x,F2/y\partial F_{1} / \partial x, \partial F_{2} / \partial y, F3/z\partial F_{3} / \partial z and hence find the divergence F\nabla \cdot \mathbf{F} in terms of ρ\rho and ϕ\phi.

The domain VV is bounded by the surface z=(x2+y2)1z=\left(x^{2}+y^{2}\right)^{-1}, by the cylinder x2+y2=1x^{2}+y^{2}=1, and by the planes z=14z=\frac{1}{4} and z=1z=1. Sketch VV and compute its volume.

Find the most general function f(ρ)f(\rho) such that F=0\nabla \cdot \mathbf{F}=0, and verify the divergence theorem for the corresponding vector field F(x)\mathbf{F}(\mathbf{x}) in VV.

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