Verify Stokes's Formula for... Verify Stokes's Formula for $\textbf{F}(x,y,z)=(3y,-xz,yz^2)$, where $S$ is the surface of the paraboloid $2z=x^2+y^2$ bounded by the plane $z=2$. 
So I need to compute the integral using the formula $\iint_S \text{curl}~\textbf{F} \dot ~\eta ~d\sigma$ and the formula $\int_\Gamma \textbf{F}~\dot~\textbf{T}~ds$ and get the same answer. 
I have a formula that says $$\int_\Gamma \textbf{F} \cdot \textbf{T}~ds = \int_a^b \textbf{F}(\gamma(t)) \cdot \gamma'(t)dt,$$ so I thought that the line that I am integrating across is $4=x^2+y^2$, so I said $\gamma(t)=(2\text{cos}(t),2\text{sin}(t),2)$, with $0 \leq t \leq 2\pi$. From here it's a pretty straight forward computation, but I'm not sure I set it up correctly. Does this look right?
Now for the first integral, I have $$\iint\limits_S \text{curl}~\textbf{F} \cdot \eta ~d\sigma = \iint\limits_R \text{curl}~\textbf{F}(\textbf{r}(u,v)) \cdot \frac{\partial \textbf{r}}{\partial u}\times\frac{\partial \textbf{r}}{\partial v} du \, dv.$$ However, I can't figure out what $\textbf{r}(u,v)$ should be. I tried using polar coordinates but it was pretty ugly. Any tips?

Your setup for the line integral is correct. For the second, there are some ways to parametrize. I'm going to show two. $$\text{First: } \begin{align} x & = u, \\ y & = v, \\ z & = \frac{u^2+v^2}{2}, \end{align} \quad -2 \leq u,v \leq 2, \; u^2+v^2 \leq 4.$$ Notice this is the parametrization of any graph. $$\text{Second: } \begin{align} x & = r \cos \theta, \\ y & = r \sin \theta, \\ z & = \frac{r^2}{2}, \end{align} \quad 0 \leq r \leq 2, \; 0 \leq \theta \leq 2 \pi.$$ This is in terms of polar coordinates. I recommend the second.
