Calculating range of parameters for parametrization of hemisphere.

Question

I have a hemisphere $x^2+y^2+z^2=1\ ;z\geqslant 0.$ I want to represent it by the vector representation $$\vec r(u,v)=\sin u \cos v\ \hat i+\sin u\sin v \ \hat j+\cos u \ \hat k$$

I am having a very silly problem of figuring out the ranges of $u\ \&\ v$.

For $u$ , I can guess that since $\cos u\geqslant 0\ ;\ u\in [-\pi/2,\pi/2]$ . but I am not getting how to calculate the range of $v$ . Could someone help?

Answer 1

Correction: Take $u\in[0, \pi/2]$ then $v$ is simply $0$ to $2\pi$