= Solution
Direct differentiation gives $\boxed{\Phi=x^3y^2z+C}$ with $\nabla\Phi=\mathbf G$, so $\mathbf G$ is a <conservative vector field> and its integral depends only on the endpoints. In contrast,
$$
\nabla\times\mathbf F=\left(-2x^3y,\ 3x^2z(2y-z),\ 3x^2z(2y-z)\right)
$$
is nonzero, for example at $(1,1,1)$. Thus \b[$\mathbf F$ is not conservative]. Both earlier paths lie in the surface $y=z$, where the two fields happen to coincide. This illustrates why <coinciding path integrals need not imply a conservative field>.
Back to article page
