parametrize a disc

Question

$S$ is the disc of radius 1 centered at the origin located on the $xy$ axis, oriented downward. First parametrize the given surface using $(x,y,z)= G(u,v)$ with $(u,v)$ in $W$ and then calculate $\frac{\partial G}{\partial u} \times \frac{\partial G}{\partial v}$ and calculate the unit normal $\hat{n}$ to the surface at any generic point.

I was thinking let $(x,y,z)=(\cos(v), \sin(v), 0)$, is it correct? But then $\frac{\partial G}{\partial u}$=0 and the cross product is just 0 and that's weird....

Answer 1

It should be $(x,y,z) = G(u,v) = (u\cos v, u\sin v, 0)$, with $0 < u < 1$, and $0 < v < 2*\pi$.

Then $U = dG/du = (\cos v, \sin v, 0)$, and $V = dG/dv = (-u\sin v, u\cos v, 0)$. So $dG/du\times dG/dv = (0, 0, u)$, and $U\times V = u$. So $n = (0,0,u)/||u|| = (0, 0, 1)$ and $(0, 0, -1)$

Answer 2

In your parametrization, you are missing the radius of the disc. What you described is a parametrization of a circle. You want your parametrization to look like $(x,y,z)=(u\cos v, u \sin v, 0)$. It is easy to see what the intervals for u,v should be, saying they are in W does not cut it.