Problem
Problem 6.111 Let $(M, g)$ be a Riemannian manifold, and let:
(a) $\nabla$ be the Levi-Civita connection of $g$.
(b) $\operatorname{grad} f$ be the gradient of the function $f \in C^{\infty} M$.
(c) $\operatorname{div} X$ be the divergence (see the previous problem) of the vector field $X \in$ $\mathfrak{X}(M)$. For a local field of orthonormal frames $\left(e_{i}\right), i=1, \ldots, n$, we have $\operatorname{div} X=\sum_{i} g\left(\nabla_{e_{i}} X, e_{i}\right)$
(d) $H^{f}$ be the Hessian of $f \in C^{\infty} M$, defined as the second covariant differential $\nabla(\nabla f)$, that is,
$$
H^{f}(X, Y)=X Y f-\left(\nabla_{X} Y\right) f, \quad X, Y \in \mathfrak{X}(M)
$$
(e) $\Delta f$ be the Laplacian of $f \in C^{\infty} M$, defined by
$$
\Delta f=\operatorname{div} \operatorname{grad} f
$$
Moreover, suppose $\operatorname{dim} M=3$.
Prove the following formulas for $f, h \in C^{\infty} M, X, Y \in \mathfrak{X}(M)$ :
1. $\operatorname{grad}(f h)=f \operatorname{grad} h+h \operatorname{grad} f$.
2. $\operatorname{div}(f X)=X f+f \operatorname{div} X$.
3. $H^{f h}=f H^{h}+h H^{f}+\mathrm{d} f \otimes \mathrm{d} h+\mathrm{d} h \otimes \mathrm{d} f$.
4. $\Delta(f h)=f \Delta h+h \Delta f+2 g(\operatorname{grad} f, \operatorname{grad} h)$.