(a) For $\mathbf{x} \in \mathbb{R}^{n}$ and $r=|\mathbf{x}|$, show that

$$\frac{\partial r}{\partial x_{i}}=\frac{x_{i}}{r}$$

(b) Use index notation and your result in (a), or otherwise, to compute

(i) $\nabla \cdot(f(r) \mathbf{x})$, and

(ii) $\nabla \times(f(r) \mathbf{x})$ for $n=3$.

(c) Show that for each $n \in \mathbb{N}$ there is, up to an arbitrary constant, just one vector field $\mathbf{F}(\mathbf{x})$ of the form $f(r) \mathbf{x}$ such that $\nabla \cdot \mathbf{F}(\mathbf{x})=0$ everywhere on $\mathbb{R}^{n} \backslash\{\mathbf{0}\}$, and determine $\mathbf{F}$.