I am working on this problem:
Use Stoke's Theorem to evaluate $\int_CF\bullet dr$. $C$ is the boundary of the portion of the paraboloid $x=y^2+z^2$ with $x\geq 4$, n to the back, $F= \langle yz,y-4,2xy \rangle$.
I know that Stokes' Theorem states: $$\int_{\partial S}F(x,y,z)\bullet dr=\iint_S(\nabla \times F)\bullet ndS$$
I found $\nabla\times F$. According to my calculations it is $\langle 2x,-y,-z \rangle$. The only thing is that I am completely forgetting how to find n. I feel like this should be pretty simple - but I cannot figure out how to do it. Can anybody help me figure that out?
Also, once I find n and finish figuring out the integrand, how do I figure out the limits of integration? Do I change it to cylindrical coordinates and do a double integral or a triple integral?
Also, I don't understand what this is finding - is it the flux?
I'm sorry if that was too many questions for one question. Thank you for your help!
$\boldsymbol{n}$ is the unit normal vector to the surface. To find it, you need to parametrise the paraboloid with two parameters, say $u$ and $v$, as $\boldsymbol{r}(u,v)$, and then the normal will be \begin{equation*}\frac{\partial\boldsymbol{r}}{\partial u}\times\frac{\partial\boldsymbol{r}}{\partial v}\end{equation*} Your limits of integration will be determined by the ranges for the parameters $u$ and $v$ to cover the whole paraboloid.
Finally, yes, Stokes' theorem is used to relate the flux integral over a surface to the line integral around its boundary. So you can see it as finding the flux.