Paper 3, Section II, B

Vector Calculus | Part IA, 2019

Show that for a vector field A\mathbf{A}

×(×A)=(A)2A\nabla \times(\boldsymbol{\nabla} \times \mathbf{A})=\boldsymbol{\nabla}(\boldsymbol{\nabla} \cdot \mathbf{A})-\nabla^{2} \mathbf{A}

Hence find an A(x)\mathbf{A}(\mathbf{x}), with A=0\boldsymbol{\nabla} \cdot \mathbf{A}=0, such that u=(y2,z2,x2)=×A\mathbf{u}=\left(y^{2}, z^{2}, x^{2}\right)=\nabla \times \mathbf{A}. [Hint: Note that A(x)\mathbf{A}(\mathbf{x}) is not defined uniquely. Choose your expression for A(x)\mathbf{A}(\mathbf{x}) to be as simple as possible.

Now consider the cone x2+y2z2tan2α,0zhx^{2}+y^{2} \leqslant z^{2} \tan ^{2} \alpha, 0 \leqslant z \leqslant h. Let S1S_{1} be the curved part of the surface of the cone (x2+y2=z2tan2α,0zh)\left(x^{2}+y^{2}=z^{2} \tan ^{2} \alpha, 0 \leqslant z \leqslant h\right) and S2S_{2} be the flat part of the surface of the cone (x2+y2h2tan2α,z=h)\left(x^{2}+y^{2} \leqslant h^{2} \tan ^{2} \alpha, z=h\right).

Using the variables zz and ϕ\phi as used in cylindrical polars (r,ϕ,z)(r, \phi, z) to describe points on S1S_{1}, give an expression for the surface element dSd \mathbf{S} in terms of dzd z and dϕd \phi.

Evaluate S1udS\int_{S_{1}} \mathbf{u} \cdot d \mathbf{S}.

What does the divergence theorem predict about the two surface integrals S1udS\int_{S_{1}} \mathbf{u} \cdot d \mathbf{S} and S2udS\int_{S_{2}} \mathbf{u} \cdot d \mathbf{S} where in each case the vector dSd \mathbf{S} is taken outwards from the cone?

What does Stokes theorem predict about the integrals S1udS\int_{S_{1}} \mathbf{u} \cdot d \mathbf{S} and S2udS\int_{S_{2}} \mathbf{u} \cdot d \mathbf{S} (defined as in the previous paragraph) and the line integral CAdl\int_{C} \mathbf{A} \cdot d \mathbf{l} where CC is the circle x2+y2=h2tan2α,z=hx^{2}+y^{2}=h^{2} \tan ^{2} \alpha, z=h and the integral is taken in the anticlockwise sense, looking from the positive zz direction?

Evaluate S2udS\int_{S_{2}} \mathbf{u} \cdot d \mathbf{S} and CAdl\int_{C} \mathbf{A} \cdot d \mathbf{l}, making your method clear and verify that each of these predictions holds.

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