Calculate and find the flux integral

Question

Find the flux integral $$\iint_S \operatorname{rot} \vec{F} {N}\,dS$$ where $S$ is the half sphere $x^2+y^2+z^2=4$ and $z \ge 0$ with an aligned unit standard $\vec{N}$ (normal) and $\vec{F} =(3x-y, yz, xy)$.
I know that I should use Stokes theorem and know that rot is the same as curl and $curl \vec{F} = \nabla \times \vec{F}$ = $x-y-y+1$ but I don't know how to continue on from here. Does working with a half sphere mean that I have 2 normals? How do I calculate the normal?
Edit: I've calculated curl: $\nabla \times F = \det \begin{bmatrix}\mathbf i& \mathbf j& \mathbf k\\\frac {\partial F}{\partial x}&\frac {\partial F}{\partial y})&\frac {\partial F}{\partial z}\\3x-y&yz&xy \end{bmatrix} = (x-y)\mathbf i -(y\mathbf j + (1)\mathbf k$

Answer 1

As per Stokes' theorem, $\displaystyle \iint_S (\nabla \times \vec{F}) \cdot \hat{n} \, dS = \int_C \vec {F} \cdot dr \ $, where $C$ is the boundary curve of the surface $S$.

$S$ is $x^2+y^2+z^2 = 4, z \geq 0$. Its boundary curve is circle $x^2+y^2 = 4$ at $z = 0$. Parametrize the boundary curve in polar coordinates as $r(t) = (2 \cos t, 2 \sin t, 0), 0 \leq t \leq 2\pi$.

$\vec{F} = (3x-y, yz, xy)$ so $\vec F(r(t)) = (6 \cos t - 2 \sin t, 0, 4 \sin t \cos t)$

Now find $r'(t)$, do the dot product and complete the line integral. That is same as the flux of $curl \vec F$ over the given hemisphere.