(i) For $r=|\mathbf{x}|$ with $\mathbf{x} \in \mathbb{R}^{3} \backslash\{\mathbf{0}\}$, show that

$$\frac{\partial r}{\partial x_{i}}=\frac{x_{i}}{r} \quad(i=1,2,3) .$$

(ii) Consider the vector fields $\mathbf{F}(\mathbf{x})=r^{2} \mathbf{x}, \mathbf{G}(\mathbf{x})=(\mathbf{a} \cdot \mathbf{x}) \mathbf{x}$ and $\mathbf{H}(\mathbf{x})=\mathbf{a} \times \hat{\mathbf{x}}$, where $\mathbf{a}$ is a constant vector in $\mathbb{R}^{3}$ and $\hat{\mathbf{x}}$ is the unit vector in the direction of $\mathbf{x}$. Using suffix notation, or otherwise, find the divergence and the curl of each of $\mathbf{F}, \mathbf{G}$ and $\mathbf{H}$.