= Solution
Assume the components of $\mathbf F$ have continuous second <partial derivatives>. Expanding the <divergence> of its <curl> gives
$$
\begin{aligned}
\nabla\cdot(\nabla\times\mathbf F)
={}&\partial_x\partial_yF_z-\partial_x\partial_zF_y
+\partial_y\partial_zF_x-\partial_y\partial_xF_z\\
&+\partial_z\partial_xF_y-\partial_z\partial_yF_x.
\end{aligned}
$$
Each positive term cancels a negative term because the corresponding mixed <derivatives> commute. Consequently
$$
\boxed{\nabla\cdot\mathbf G=\nabla\cdot(\nabla\times\mathbf F)=0}.
$$
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