(a) Let $\mathcal{M}$ be a manifold with coordinates $x^{\mu}$. The commutator of two vector fields $\boldsymbol{V}$ and $\boldsymbol{W}$ is defined as

$$[\boldsymbol{V}, \boldsymbol{W}]^{\alpha}=V^{\nu} \partial_{\nu} W^{\alpha}-W^{\nu} \partial_{\nu} V^{\alpha}$$

(i) Show that $[\boldsymbol{V}, \boldsymbol{W}]$ transforms like a vector field under a change of coordinates from $x^{\mu}$ to $\tilde{x}^{\mu}$.

(ii) Show that the commutator of any two basis vectors vanishes, i.e.

$$\left[\frac{\partial}{\partial x^{\alpha}}, \frac{\partial}{\partial x^{\beta}}\right]=0$$

(iii) Show that if $\boldsymbol{V}$ and $\boldsymbol{W}$ are linear combinations (not necessarily with constant coefficients) of $n$ vector fields $\boldsymbol{Z}_{(a)}, a=1, \ldots, n$ that all commute with one another, then the commutator $[\boldsymbol{V}, \boldsymbol{W}]$ is a linear combination of the same $n$ fields $Z_{(a)}$.

[You may use without proof the following relations which hold for any vector fields $\boldsymbol{V}_{1}, \boldsymbol{V}_{2}, \boldsymbol{V}_{3}$ and any function $f$ :

$$\begin{aligned} {\left[\boldsymbol{V}_{1}, \boldsymbol{V}_{2}\right] } &=-\left[\boldsymbol{V}_{2}, \boldsymbol{V}_{1}\right] \\ {\left[\boldsymbol{V}_{1}, \boldsymbol{V}_{2}+\boldsymbol{V}_{3}\right] } &=\left[\boldsymbol{V}_{1}, \boldsymbol{V}_{2}\right]+\left[\boldsymbol{V}_{1}, \boldsymbol{V}_{3}\right] \\ {\left[\boldsymbol{V}_{1}, f \boldsymbol{V}_{2}\right] } &=f\left[\boldsymbol{V}_{1}, \boldsymbol{V}_{2}\right]+\boldsymbol{V}_{1}(f) \boldsymbol{V}_{2} \end{aligned}$$

but you should clearly indicate each time relation $(1),(2)$, or (3) is used.]

(b) Consider the 2-dimensional manifold $\mathbb{R}^{2}$ with Cartesian coordinates $\left(x^{1}, x^{2}\right)=$ $(x, y)$ carrying the Euclidean metric $g_{\alpha \beta}=\delta_{\alpha \beta}$.

(i) Express the coordinate basis vectors $\partial_{r}$ and $\partial_{\theta}$, where $r$ and $\theta$ denote the usual polar coordinates, in terms of their Cartesian counterparts.

(ii) Define the unit vectors

$$\hat{\boldsymbol{r}}=\frac{\partial_{r}}{\left\|\partial_{r}\right\|}, \quad \hat{\boldsymbol{\theta}}=\frac{\partial_{\theta}}{\left\|\partial_{\theta}\right\|}$$

and show that $(\hat{\boldsymbol{r}}, \hat{\boldsymbol{\theta}})$ are not a coordinate basis, i.e. there exist no coordinates $z^{\alpha}$ such that $\hat{\boldsymbol{r}}=\partial / \partial z^{1}$ and $\hat{\boldsymbol{\theta}}=\partial / \partial z^{2}$.