So I have the spherical Cap formed by the interception of the sphere $x^2+y^2+z^2=5$ and the plane $z=1$. Given the field $\vec{F}=(xy^2z,yz^3,y^4)$, I want to calculate the flux through the cap:
$$ \iint_{S}\vec{F}\cdot\vec{n}dS $$
I already know that $\vec{n}=\frac{1}{\sqrt{5}}(x,y,z)$, so, evaluated at the surface in question:
$$ \vec{F}\cdot\vec{n}=\frac{5}{\sqrt{5}}zy^2 $$
By deffinition we have $\vec{n}=\frac{\vec{r}_u \times \vec{r}_v}{||\vec{r}_u \times \vec{r}_v||}$. So as far as I understand, I can write $||\vec{r}_u \times \vec{r}_v||=\sqrt{5}$.
This last bit allows me to write $dS=||\vec{r}_u \times \vec{r}_v||dA=\sqrt{5}dA$.
Now I have 2 options: Either I work in sperical coordinates, or cilyndrical. As it turns out I have a problem with both.
For spherical coordinates I end up with an answer that is $20\sqrt{5}\pi$ where the answer is actuall $20 \pi$. This indicates to me that probably my reasoning about the part "$||\vec{r}_u \times \vec{r}_v||=\sqrt{5}$" is wrong.
For cilyndrical coordinates I used
$$ x=r\cos(\theta) \qquad y=r\sin(\theta) \qquad z=\sqrt{5-r^2} $$
with $r\in [0,2]$ because of the "domain" of the region. And $\theta \in [0,2\pi)$. Unfortunately the asnwer doesn't match either.
I would appreciate any kind of indication of where things went wrong and/or sugestions of how to make things right!