= Solution
The first field has vanishing <curl> and is the <gradient> of a <monomial>:
$$
\boxed{\nabla\times\mathbf F_1=\mathbf0,\qquad
\phi_1=x^3y^2z+C,\qquad\mathbf F_1=\nabla\phi_1}.
$$
For the second field, direct component <differentiation> gives
$$
\boxed{\nabla\times\mathbf F_2=
(-2x^3y,\,3x^2z(2y-z),\,3x^2z(2y-z))}.
$$
It is not identically zero. Thus \b[$\mathbf F_1$ is conservative and $\mathbf F_2$ is not] on $\mathbb R^3$. The conditional path-integral requests therefore apply to $\mathbf F_1$.
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