How to compute $\iint_S \mathrm{curl} F\cdot n dS$ where $S$ is the portion of $z=x^2+y^2$ below $z=16$, $n$ is the normal vector pointing down, and $F=(z,x,y^2)$?
I thought of the stokes theorem, but I'm not sure what to compute. So I was trying directly computing $$\mathrm{curl} \ F=(2y,1,1)$$ but what is a normal vector I can use? the surface parametrization could be difficult.
To compute $\iint_S curl \hspace{1mm}\mathbf{F}\cdot{\mathbf{N}}dS$, you can use Stokes' theorem, noting that a parametrisation of the boundary, C, is: $\mathbf{r}(t) = \ (4\cos{t}, 4\sin{t}, 16) $:
$$\iint_S curl \hspace{1mm}\mathbf{F}\cdot{\mathbf{N}}dS = \int_C \mathbf{F}(\mathbf{r}(t))\cdot{d\mathbf{r}}\\ = \int_C (16, 4\cos{t}, 16\sin^2{t}) \cdot{(-4\sin{t}, 4\cos{t},0)}\hspace{1mm}dt\\ = \int_0^{2\pi}-64\sin{t}\hspace{0.5mm} +16\cos^2{t}\hspace{1mm}dt = 16\pi. $$
Hope this helps.