I'm learning about surface integral, and one problem uses the surface of a hemisphere with radius $2$. $(x^2+y^2+z^2=4, z \geq 0)$ So I now want to find the parametrization $r(t, s)$. I thought I could do the following; First, look at it from a topdown perspective, then the $x$ and $y$ coordinates will be $2\cos s$ and $2\sin s$ respectively, with $s$ ranging from $0$ to $2\pi$. Then, $z$ will simply be $t$ ranging from $0$ to $2$.
Looking at it again, I think I know what's wrong -- I'm basically constructing a cylinder right now, not a sphere, since my $x-y$ length is always $2$ although it should get smaller when getting higher up. However, how should I best bring $t$ into this?
In general, how can I come up with a parametrization "fast"? During an exam I wouldn't have the time to think about it for too long.
The simplest parametrization of parts of a sphere uses spherical coordinates:
$$\begin{align}x&=r\sin\theta\cos\varphi,\\ y&=r\sin\theta\sin\varphi,\\ z&=r\cos\theta, \end{align}$$
where $r$ is the radius, $\varphi\in[0,2\pi)$ and $\theta\in[0,\pi]$ for an entire sphere. Here, $\varphi$ describes the "horizontal" rotation (the axis of rotation being the $z$-axis), while $\theta$ is the angle between the $z$-axis and the line connecting the parametrized point to the origin. A hemisphere is obtained by restricting $\theta$ to $[0,\frac{\pi}{2}]$. Here is a vizualization from Wikipedia:
Coming up with this quickly is not something you're supposed to do in an exam. In an exam, you're supposed to already know the most important coordinate systems: polar, spherical and cylindrical coordinates should be at your disposal without having to derive them manually.