Answer
There does not exist such a vector field $G$.
Work Step by Step
Consider a vector field $G$ such that $div [curl (G)]=0$
$F=A i+B j+C k $ and $div F=\dfrac{\partial A}{\partial x}+\dfrac{\partial B}{\partial y}+\dfrac{\partial C}{\partial z}$
Given: $curl G=\lt x \sin y, \cos y, z-xy \gt$
$div[curl(G)]=\dfrac{\partial A}{\partial x}+\dfrac{\partial B}{\partial y}+\dfrac{\partial C}{\partial z}=\sin y-\sin y+1 $
$\implies div [curl (G)]=1 \ne 0$
Thus, we get there does not exist such a vector field $G$.
