I have the vector field $\mathbf{F}(x, y) = y \mathbf{i} - x \mathbf{j}$ and the surface $S$ bounded by $x^2 + y^2 + z^2 = 9$ with $z \geq 0,$ and need to verify the Stokes theorem.
The line integral is easy to calculate and is $- 2 \pi.$ But I am confused on how to evaluate the double integral $$ \iint_{S} \text{curl} \hspace{.1cm}\mathbf{F} \cdot d\mathbf{s}$$ because I don't know how express $S$ (a half sphere) in two dimensions. How can I do that?
There are a handful of ways you might parametrize this surface, the three most common being by Cartesian coordinates, cylindrical coordinates, or spherical coordinates. The implicit equation for the surface is $x^2+y^2+z^2=9$ with $z\geq 0$. Solving for $z$ as a function of $x$ and $y$ then, the Cartesian parametrization of the surface is:
$$\begin{cases}\mathbf{r}(x,y)=x\mathbf{i}+y\mathbf{j}+z(x,y)\mathbf{k}=x\mathbf{i}+y\mathbf{j}+\sqrt{9-x^2-y^2}\mathbf{k},\\ -\sqrt{9-x^2}\leq y\leq \sqrt{9-x^2},\\ -3\leq x\leq 3.\end{cases}$$
This is probably the least elegant way to do the problem in my opinion but it's worth going through the calculations at least once (think of it as eating your vegetables).
Now, since the surface is a hemisphere, the obvious first guess of what coordinates best simplify the problem would be spherical coordinates of course. We have surface of constant radial distance from from the origin, so we parametrize the surface by polar and azimuthal angles much like we parametrize the surface of the Earth by latitude and longitude.
$$\begin{cases}\mathbf{r}(\theta,\phi)=3\sin{\theta}\cos{\phi}\mathbf{i}+3\sin{\theta}\sin{\phi}\mathbf{j}+3\cos{\theta}\mathbf{k},\\ 0\leq\theta\leq\frac{\pi}{2},\\ 0\leq\phi\leq\pi.\end{cases}$$
As it turns out though, cylindrical coordinates are the most efficient of the three choices considered here for carrying the calculations because of the cylidrical symmetry of the vector field $\mathbf{F}$. I leave setting up this parametrization as an exercise to the questioner with the incentive that if you make the effort to set it up, you'll be saving effort in the long run.
