To calculate the flux of the vector field $$\vec F=\left(2\pi x+\frac{2x^2y^2}{\pi} \right) \vec{\imath}+\left(2\pi xy-\frac{4y}{\pi}\right)\vec{\jmath}$$ along the outward normal across the ellipse $x^2+16y^2=4$
I know that to get the normal vector, I have to calculate the gradient of the surface. $$\hat{n} = \dfrac{\langle 2x, 32y\rangle}{||\langle 2x, 32y\rangle||}$$ Then, $$\int_S {F} \cdot \mathbf{n} \ dS $$ can be obtained but its a very tedious calculation.How can I simplify this?Are there any other approaches?