Compute:
$\int_S x^2dy \wedge dz + y^2 dz \wedge dx + z^2 dx \wedge dy$
Where $S$ is the sphere $x^2 + y^2 + z^2 = r^2$ oriented by the outward pointing unit normal.
So my attempt at a solution:
Let $\omega = x^2dy \wedge dz + y^2 dz \wedge dx + z^2 dx \wedge dy$
Then $d\omega = (2x + 2y + 2z) dx \wedge dy \wedge dz$
If we define $S^1 = [(x,y,z) \in \mathbb{R}^3 : x^2 + y^2 +z^2\leq r^2]$
The $\partial S^1 = S$
So by Stoke's Theroem (the general version)
$\int_{\partial S^1} \omega$ = $\int_S d\omega$
$= 2\int_S (x + y + z) dx \wedge dy \wedge dz$
Now I'm not really sure where to go from here. Should I use a coordinate change? Also, did I apply the theorem properly? Any help would be appreciated.
Thank you.