Verify Stokes’ Theorem for the vector field $\boldsymbol{F}(x, y, z) =\langle 3y, 4z, −6x \rangle$, if the surface $S$ is the part of the paraboloid $z = 9 − x^2 − y^2$ that lies above the $xy$-plane, oriented upward.
I'm a bit confused on how to parametrise this in terms of $t$. I want to do $ \int F dr$ first. I know how to do everything but I'm a bit confused with parametrising.
I was thinking that $r(t) = \langle 3\cos(t), 3\sin(t), 0\rangle$
Is that correct? When we parametrise, we want to define a curve or surface in terms of either one or two variables. In this case, the boundary curve of the paraboloid is on the $xy$-plane, so $z=0$.
Yes, that is correct. Recall that the circulation you want to do first is a scalar quantity. Then you get $d\boldsymbol{r}$ by differentiating the parameterization vector $\boldsymbol{r}$, and the integral is $$\oint_C \boldsymbol{F}\cdot d\boldsymbol{r}. $$
Here $C$ is the mentioned curve enclosing a surface $S$ that can be parameterized by $\boldsymbol{r}(u,v) = \langle u, v, 9 - u^2 - v^2\rangle$.