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

A vector field $\mathbf{u}$ is given by

$$\mathbf{u}=\frac{\mathbf{k}}{r}+\frac{(\mathbf{k} \cdot \mathbf{x}) \mathbf{x}}{r^{3}}$$

where $\mathbf{k}$ is a constant vector. Calculate the second-rank tensor $d_{i j}=\partial u_{i} / \partial x_{j}$ using suffix notation, and show that $d_{i j}$ splits naturally into symmetric and antisymmetric parts. Deduce that $\boldsymbol{\nabla} \cdot \mathbf{u}=0$ and that

$$\boldsymbol{\nabla} \times \mathbf{u}=\frac{2 \mathbf{k} \times \mathbf{x}}{r^{3}}$$