Paper 2, Section II, B

Vector Calculus | Part IA, 2020

(a) State the value of xi/xj\partial x_{i} / \partial x_{j} and find r/xj\partial r / \partial x_{j} where r=xr=|\boldsymbol{x}|.

(b) A vector field u\boldsymbol{u} is given by

u=ar+(ax)xr3\boldsymbol{u}=\frac{\boldsymbol{a}}{r}+\frac{(\boldsymbol{a} \cdot \boldsymbol{x}) \boldsymbol{x}}{r^{3}}

where a\boldsymbol{a} is a constant vector. Calculate the second-rank tensor dij=ui/xjd_{i j}=\partial u_{i} / \partial x_{j} using suffix notation and show how dijd_{i j} splits naturally into symmetric and antisymmetric parts. Show that

u=0\nabla \cdot \boldsymbol{u}=0


×u=2a×xr3\nabla \times u=\frac{2 a \times x}{r^{3}}

(c) Consider the equation

2u=f\nabla^{2} u=f

on a bounded domain VR3V \subset \mathbb{R}^{3} subject to the mixed boundary condition

(1λ)u+λdudn=0(1-\lambda) u+\lambda \frac{d u}{d n}=0

on the smooth boundary S=VS=\partial V, where λ[0,1)\lambda \in[0,1) is a constant. Show that if a solution exists, it will be unique.

Find the spherically symmetric solution u(r)u(r) for the choice f=6f=6 in the region r=xbr=|\boldsymbol{x}| \leqslant b for b>0b>0, as a function of the constant λ[0,1)\lambda \in[0,1). Explain why a solution does not exist for λ=1\lambda=1

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