Chapter 16.5, Problem 29E

Prove the identity, assuming that the appropriate partial derivatives exist and are continuous. If f is a scalar field and F, G are vector fields, then f $F,F\cdot G,$ and $F\times G$ are defined by

$\begin{array}{l}\left(fF\right)\left(x,y,z\right)=f\left(x,y,z\right)F\left(x,y,z\right)\\ \left(F\cdot G\right)\left(x,y,z\right)=F\left(x,y,z\right)\cdot G\left(x,y,z\right)\\ \left(F\times G\right)\left(x,y,z\right)=F\left(x,y,z\right)\times G\left(x,y,z\right)\end{array}$

$\text{curl}\left(\text{curl }F\right)=\text{grad}\left(\text{div }F\right)-{\nabla }^{2}F$