Paper 3, Section II, C

Vector Calculus | Part IA, 2016

(a) For smooth scalar fields uu and vv, derive the identity

(uvvu)=u2vv2u\nabla \cdot(u \nabla v-v \nabla u)=u \nabla^{2} v-v \nabla^{2} u

and deduce that

ρxr(v2uu2v)dV=x=r(vunuvn)dSx=ρ(vunuvn)dS\begin{aligned} \int_{\rho \leqslant|\mathbf{x}| \leqslant r}\left(v \nabla^{2} u-u \nabla^{2} v\right) d V=\int_{|\mathbf{x}|=r}\left(v \frac{\partial u}{\partial n}-u \frac{\partial v}{\partial n}\right) d S \\ &-\int_{|\mathbf{x}|=\rho}\left(v \frac{\partial u}{\partial n}-u \frac{\partial v}{\partial n}\right) d S \end{aligned}

Here 2\nabla^{2} is the Laplacian, n=n\frac{\partial}{\partial n}=\mathbf{n} \cdot \nabla where n\mathbf{n} is the unit outward normal, and dSd S is the scalar area element.

(b) Give the expression for (×V)i(\nabla \times \mathbf{V})_{i} in terms of ϵijk\epsilon_{i j k}. Hence show that

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

(c) Assume that if 2φ=ρ\nabla^{2} \varphi=-\rho, where φ(x)=O(x1)\varphi(\mathbf{x})=O\left(|\mathbf{x}|^{-1}\right) and φ(x)=O(x2)\nabla \varphi(\mathbf{x})=O\left(|\mathbf{x}|^{-2}\right) as x|\mathbf{x}| \rightarrow \infty, then

φ(x)=R3ρ(y)4πxydV.\varphi(\mathbf{x})=\int_{\mathbb{R}^{3}} \frac{\rho(\mathbf{y})}{4 \pi|\mathbf{x}-\mathbf{y}|} d V .

The vector fields B\mathbf{B} and J\mathbf{J} satisfy

×B=J\nabla \times \mathbf{B}=\mathbf{J}

Show that J=0\nabla \cdot \mathbf{J}=0. In the case that B=×A\mathbf{B}=\nabla \times \mathbf{A}, with A=0\nabla \cdot \mathbf{A}=0, show that

A(x)=R3J(y)4πxydV\mathbf{A}(\mathbf{x})=\int_{\mathbb{R}^{3}} \frac{\mathbf{J}(\mathbf{y})}{4 \pi|\mathbf{x}-\mathbf{y}|} d V

and hence that

B(x)=R3J(y)×(xy)4πxy3dV\mathbf{B}(\mathbf{x})=\int_{\mathbb{R}^{3}} \frac{\mathbf{J}(\mathbf{y}) \times(\mathbf{x}-\mathbf{y})}{4 \pi|\mathbf{x}-\mathbf{y}|^{3}} d V

Verify that A\mathbf{A} given by ()(*) does indeed satisfy A=0\nabla \cdot \mathbf{A}=0. [It may be useful to make a change of variables in the right hand side of ()(*).]

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