Determine if $\int_{C}(x^{2}y\,dx+xy^{2}\,dy)$ is independent of the path anywhere in the plane.

Solution Let $P=x^{2}y$ and $Q=xy^{2}$. Then $\dfrac{\partial P}{\partial y}=x^{2}$ and $\dfrac{\partial Q}{\partial x}=y^{2}$. Since these two functions are not equal (except on the graph of the equation $x^{2}=y^{2}$, which is not an open set), the line integral is not independent of the path anywhere in the $xy$-plane.