Determining Whether a Line Integral Is Independent of the Path

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.