Solution
= Solution
Compute all three components of the Cartesian <curl> explicitly:
$$
\begin{aligned}
(\nabla\times\mathbf F)_x&=\partial_y(4xyz)-\partial_z(2xz^2)=4xz-4xz=0,\\
(\nabla\times\mathbf F)_y&=\partial_z(2yz^2)-\partial_x(4xyz)=4yz-4yz=0,\\
(\nabla\times\mathbf F)_z&=\partial_x(2xz^2)-\partial_y(2yz^2)=2z^2-2z^2=0.
\end{aligned}
$$
Thus $\boxed{\nabla\times\mathbf F=\mathbf0}$ throughout $\mathbb R^3$. There are no excluded singular points in this polynomial field.