Vector Calculus · Challenge problem

Let $u(x, y) = x^2 - y^2$ and let $C$ be the ellipse $\dfrac{x^2}{4} + y^2 = 1$ with outward unit normal $\mathbf n$. Evaluate $\displaystyle\oint_C u\,\dfrac{\partial u}{\partial n}\,ds$, where $\dfrac{\partial u}{\partial n} = \nabla u\cdot\mathbf n$. Give your answer in terms of $\pi$.