Evaluate the integral for the given data. Describe the kind of surface. Show the details of your work.
$$F = [0,x,0]$$ $$S: x^{2} + y^{2} + z^{2} = 1, x \geq 0, y \geq 0 , z \geq 0$$
I'm trying to find a parameterized version of S.
$x = cos \theta sin \theta, y = sin \theta sin \theta$ and z = $cos \theta$ so I get $$r(u,v) = [cos \theta sin \theta, sin \theta sin \theta, cos \theta]$$ Is that right? Obv not since the terms are wrong, but what should the term be?
When I use the r as an input of F I get:
$$F = [0, cos \theta sin \theta, 0]$$
But then I can't differentiate r with respect to u because I wrote everyhting in $\theta$
hint
The parametrisation of $ S $ should be
$$x=\cos(\theta)\sin(\phi)$$ $$y=\sin(\theta)\sin(\phi)$$ $$z=\cos(\phi)$$ with $$0\le \theta \le \frac{\pi}{2}$$ and $$0\le \phi \le \frac{\pi}{2}.$$