Solution (source code)

= Solution

Take the <curl> component by component:
$$
\mathbf F=\left(\partial_y\frac{y^3}{3}-\partial_z\frac{x^3}{3},\partial_z\frac{z^3}{3}-\partial_x\frac{y^3}{3},\partial_x\frac{x^3}{3}-\partial_y\frac{z^3}{3}\right)
=\boxed{(y^2,z^2,x^2).}
$$
Its <divergence> is zero, as expected because the <divergence of a curl is zero>.