State the value of $\partial x_{i} / \partial x_{j}$ and find $\partial r / \partial x_{j}$, where $r=|\mathbf{x}|$.

Vector fields $\mathbf{u}$ and $\mathbf{v}$ in $\mathbb{R}^{3}$ are given by $\mathbf{u}=r^{\alpha} \mathbf{x}$ and $\mathbf{v}=\mathbf{k} \times \mathbf{u}$, where $\alpha$ is a constant and $\mathbf{k}$ is a constant vector. Calculate the second-rank tensor $d_{i j}=\partial u_{i} / \partial x_{j}$, and deduce that $\boldsymbol{\nabla} \times \mathbf{u}=\mathbf{0}$ and $\boldsymbol{\nabla} \cdot \mathbf{v}=0$. When $\alpha=-3$, show that $\boldsymbol{\nabla} \cdot \mathbf{u}=0$ and

$$\nabla \times \mathbf{v}=\frac{3(\mathbf{k} \cdot \mathbf{x}) \mathbf{x}-\mathbf{k} r^{2}}{r^{5}}$$