Problem
Problem 6.109 Let $(M, g)$ be a Riemannian manifold, $T_{p} M$ the tangent space at $p \in M$ and $T_{p}^{*} M$ its dual space. The musical isomorphisms $b$ and $\sharp$ are defined (see Problem 6.40) by
$$
\text { b: } T_{p} M \rightarrow T_{p}^{*} M, \quad X \mapsto X^{\downarrow}, \quad X^{\natural}(Y)=g(X, Y)
$$
and its inverse $\omega \mapsto \omega^{\sharp}$, respectively. The gradient of a function $f \in C^{\infty} M$ is defined as
$$
\operatorname{grad} f=(\mathrm{d} f)^{\#} \text {. }
$$
(i) Prove that $g(\operatorname{grad} f, X)=X f, X \in \mathfrak{X}(M)$.
Given local coordinates $\left\{x^{i}\right\}$ :
(ii) Compute $\left(\partial / \partial x^{i}\right)^{b}$.
(iii) Calculate $\left(\mathrm{d} x^{i}\right)^{\sharp}$.
(iv) Write grad $f$ in local coordinates.
(v) Verify that in the particular case of $\mathbb{R}^{3}$ equipped with the Euclidean metric, we recover the classical expression of $\operatorname{grad} f$.