Prove that \(\int_C (x^3 - y^3)\,dx + (x^3 + y^3)\,dy \ge 0\) for every smooth, simple closed curve \(C \subset \mathbb{R}^2\), oriented counterclockwise.
Solution
Set \[ P(x,y)=x^3-y^3,\qquad Q(x,y)=x^3+y^3. \] If \(D\) is the bounded region enclosed by the positively oriented smooth simple closed curve \(C\), Green’s theorem gives \[ \int_C P\,dx+Q\,dy =\iint_D\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dA. \] Now \[ \frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y} =3x^2-(-3y^2)=3(x^2+y^2)\ge0. \] Therefore \[ \int_C (x^3-y^3)\,dx+(x^3+y^3)\,dy =3\iint_D(x^2+y^2)\,dA\ge0. \]