Answer
curl $(F\times G)=F (div G)-G (div F)+(G \cdot \nabla)F-(F \cdot \nabla)G$
Work Step by Step
Definition of curl: $(F\times G)=\nabla \times (F\times G)$
Apply the product rule as follows:
curl $(F\times G)=\dot{\nabla}\times (\dot{F}\times G)+\dot{\nabla}\times (F\times \dot{G})$
we know that $a \times (b \times c)=b(a \cdot c)-c(a \cdot b)$
Now, $curl (F\times G)=[F(\dot{\nabla} \cdot G)-G(\dot{\nabla} \cdot \dot{F})]+[F(\dot{\nabla} \cdot \dot{G})-\dot {G}(\dot{\nabla} \cdot F)]$
or,
$curl (F\times G)=F (div G)-G (div F)+\dot {F}(\dot{\nabla} \cdot G)-\dot {G}(\dot{\nabla} \cdot F)$
Therefore,
curl $(F\times G)=F (div G)-G (div F)+(G \cdot \nabla)F-(F \cdot \nabla)G$
Hence, the result has been verified.
