So I have the following vector field: $$\vec{F}(x,y,z)=(x,y,z)$$ I have to calculte the Surface integral: $\iint_{\partial G}\vec{F}d\vec{P}$ using both the direct method with parametrization and with Divergence (Gauss's) theorem.
$$G: x^2+y^2 \leq1, -1\leq z \leq x^2+y^2$$
So with Divergence theorem I first calculate the divergence: $$div(\vec{F})=3$$ I used Polar coordiantes: $$x=rcos\phi , y=rsin\phi, z=z, J=r$$ $$\phi\in[0,2\pi] r\in[0,1] z\in[-1,r^2]$$
And the using the divergence theorem:
$$\iint_{\partial G}\vec{F}d\vec{P}= \iiint_{\Delta}div(\vec{F})dP= \iiint_{\Delta}div(\vec{F}) 3r dP=\frac{9\pi}{2}$$
If I made any mistakes here, i would appreciate them found out.
So for the parametrization i had problems, I used Polar corodinates and put $r=1$ since $r\in[0,1]$: $$\vec{r}(\phi,z)=(cos\phi, sin\phi, z)$$ But I think here is a mistake, i get $z\in[-1,1]$ and using the parametrization I get a different result from the one I got at using Divergence theorem:
$$\iint_{\partial G}\vec{F}d\vec{P}=\iint_{\delta}\vec{F}(\vec{r}(\phi,z))*(\vec{r_{\phi}}(\phi,z)x\vec{r_z}(\phi,z))d\vec{P}=\int_{0}^{2\pi}d\phi\int_{-1}^{1}1dz=4\pi$$
Any help fidning my mistakes would be appreciated. Thank you in advance.