The problem required me to find a parametrization of the surface of the upper portion cut from the sphere $x^2+y^2+z^2=8$ by the plane $z=-2$.
So I used the cylindrical coordinates here. With $r=\sqrt{x^2+y^2}$, I have that
$$z=\sqrt{8-r^2}$$
for the sphere, and by putting $z=-2$ into the equation, I have got $r=2$.
Then it is possible for me to say that the parametrization is
$$\vec{r}(r,\theta)=r\cos\theta\hat{i}+r\sin\theta\hat{j}+\sqrt{8-r^2}\hat{k}$$
with $2\le r\le \sqrt{8}$ and $0\le\theta <2\pi$.
Edit: With the help of @Ted Shifrin, I am trying to use spherical coordinates:
I proposed that the parametrization be
$$(\sqrt{8}\sin\phi\cos\theta,\sqrt{8}\sin\phi\sin\theta,\sqrt{8}\cos\phi)$$
with $0\le\theta <2\pi$ and $0\le\phi\le 3\pi/4$.
Is it correct?