There are Leibniz rules for $\nabla(fg),\nabla\cdot(f\textbf{G}),\nabla\times(f\textbf{G})$ and all can be proved by coordinate expansion (not so tedious).
I wonder whether there are Leibniz rules for $\nabla(\textbf{F}\cdot\textbf{G}),\nabla\cdot(\textbf{F}\times\textbf{G}),\nabla\times(\textbf{F}\times\textbf{G})$. It seems that none of them holds when I test by Mathematica. If so, are there formulas to expand these ones?
Any help will be appreciated.