Problem
Problem 6.110 Let $M$ be a $C^{\infty}$ manifold equipped with a linear connection $\nabla$. Let $\left\{X_{1}, \ldots, X_{n}\right\}$ be a basis of $T_{p} M$, and $\left\{\theta^{1}, \ldots, \theta^{n}\right\}$ its dual basis. The divergence of $Z \in \mathfrak{X}(M)$ is defined by
$$
(\operatorname{div} Z)(p)=\sum_{i} \theta^{i}\left(\nabla_{X_{i}} Z\right)
$$
(i) Prove that $(\operatorname{div} Z)(p)$ does not depend on the chosen basis.
(ii) Show that the divergence of a $C^{\infty}$ field on $\mathbb{R}^{3}$ is the same as the definition given in Advanced Calculus.