Let $\Gamma_{\mu u}^{\alpha}$ be the Levi-Civita connection (implying the symmetry $\Gamma_{\mu u}^{\alpha}=\Gamma_{v \mu}^{\alpha}$ ).

(i) Show that $abla_{\mu} abla_{v} f=abla_{v} abla_{\mu} f$.

(ii) Consider a one-form field $v$ with coefficients of the form

$$\begin{equation*}
v_{\mu}=h abla_{\mu} f=h \partial_{\mu} f, \tag{5.396}
\end{equation*}$$

where $f=f(x), h=h(x)$ are some smooth functions. Show that

$$\begin{equation*}
v_{[\mu} abla_{\sigma} v_{v]}=0 \tag{5.397}
\end{equation*}$$