Paper 3, Section II, C

Vector Calculus | Part IA, 2011

The vector fields A(x,t)\mathbf{A}(\mathbf{x}, t) and B(x,t)\mathbf{B}(\mathbf{x}, t) obey the evolution equations

At=u×(×A)+ψBt=(B)u(u)B\begin{aligned} \frac{\partial \mathbf{A}}{\partial t} &=\mathbf{u} \times(\boldsymbol{\nabla} \times \mathbf{A})+\nabla \psi \\ \frac{\partial \mathbf{B}}{\partial t} &=(\mathbf{B} \cdot \nabla) \mathbf{u}-(\mathbf{u} \cdot \boldsymbol{\nabla}) \mathbf{B} \end{aligned}

where u\mathbf{u} is a given vector field and ψ\psi is a given scalar field. Use suffix notation to show that the scalar field h=ABh=\mathbf{A} \cdot \mathbf{B} obeys an evolution equation of the form

ht=Bfuh\frac{\partial h}{\partial t}=\mathbf{B} \cdot \nabla f-\mathbf{u} \cdot \nabla h

where the scalar field ff should be identified.

Suppose that B=0\boldsymbol{\nabla} \cdot \mathbf{B}=0 and u=0\boldsymbol{\nabla} \cdot \mathbf{u}=0. Show that, if un=Bn=0\mathbf{u} \cdot \mathbf{n}=\mathbf{B} \cdot \mathbf{n}=0 on the surface SS of a fixed volume VV with outward normal n\mathbf{n}, then

dH dt=0, where H=Vh dV.\frac{\mathrm{d} H}{\mathrm{~d} t}=0, \quad \text { where } H=\int_{V} h \mathrm{~d} V .

Suppose that A=ar2sinθeθ+r(a2r2)sinθeϕ\mathbf{A}=a r^{2} \sin \theta \mathbf{e}_{\theta}+r\left(a^{2}-r^{2}\right) \sin \theta \mathbf{e}_{\phi} with respect to spherical polar coordinates, and that B=×A\mathbf{B}=\boldsymbol{\nabla} \times \mathbf{A}. Show that

h=ar2(a2+r2)sin2θh=a r^{2}\left(a^{2}+r^{2}\right) \sin ^{2} \theta

and calculate the value of HH when VV is the sphere rar \leqslant a.

[ In spherical polar coordinates ×F=1r2sinθerreθrsinθeϕ/r/θ/ϕFrrFθrsinθFϕ\left[\text { In spherical polar coordinates } \nabla \times \mathbf{F}=\frac{1}{r^{2} \sin \theta}\left|\begin{array}{ccc} \mathbf{e}_{r} & r \mathbf{e}_{\theta} & r \sin \theta \mathbf{e}_{\phi} \\ \partial / \partial r & \partial / \partial \theta & \partial / \partial \phi \\ F_{r} & r F_{\theta} & r \sin \theta F_{\phi} \end{array}\right|\right.

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