Solution
= Solution
Taking the <curl> component by component gives
$$
\nabla\times\mathbf F=\bigl(\partial_yx-\partial_z(x^2+2y),\ \partial_z(2xy+z)-\partial_xx,\ \partial_x(x^2+2y)-\partial_y(2xy+z)\bigr)=\boxed{\mathbf0}.
$$
= Solution
Taking the <curl> component by component gives
$$
\nabla\times\mathbf F=\bigl(\partial_yx-\partial_z(x^2+2y),\ \partial_z(2xy+z)-\partial_xx,\ \partial_x(x^2+2y)-\partial_y(2xy+z)\bigr)=\boxed{\mathbf0}.
$$