# Paper 2, Section II, B

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

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

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

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

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

and

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

(c) Consider the equation

$\nabla^{2} u=f$

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

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

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

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