Solution
= Solution
A differentiable <gradient field> must have equal mixed derivatives. Here $\partial F_y/\partial z=3z^2$ but $\partial F_z/\partial y=-x$, which are not identically equal on any open region. Equivalently,
$$
\nabla\times\mathbf F=(-x-3z^2,\ x+y-3z^2,\ 0).
$$
Thus \b[no scalar potential exists for this field on an open region]. Vanishing of the <curl> on a special lower-dimensional set would not supply a three-dimensional potential.