The sphere $x^2 + y^2 + z^2 = a^2$ intersects the plane $x + 2y + z = 0$ in a curve $C$. Calculate $\int_C \vec{v} \cdot d\vec{r}$, where $\vec{v} = 2yi -zj +2xk$
So I solved this question by taking the curl of $\vec{v}$ and dot-producting that with the normal vector getting an answer of $\pm 5a^2\pi$ but in the solution they keep it in terms of $dS$ and get $\pm \frac{5}{\sqrt 6}a^2\pi$ I don't understand where the $ \sqrt 6$ comes from if anyone could shed some light on that thank you!
I think you have made a simple mistake in using unit normals! As it seems you have just used the vector $(1,2,1)$ instead of $\frac{1}{\sqrt{6}}(1,2,1)$ for the unit normal of the plane $x + 2y + z = 0$. :)