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**lllll** Calculate the flux of F across S.

Let $\displaystyle F = \frac{r}{|r|}$ where $\displaystyle r=x\vec{i}+y\vec{j}+z\vec{k}$, S consists of the hemisphere $\displaystyle z=\sqrt{1-x^2-y^2}$ and the disk $\displaystyle x^2+y^2 \leq 1$ in the xy plane

so for F I got:

$\displaystyle \frac{x\vec{i}+y\vec{j}+z\vec{k}}{\sqrt{x^2+y^2+z^ 2}}$

therefore my divergence will be:

$\displaystyle \frac{\partial}{\partial x} = \frac{y^2+z^2}{(x^2+y^2+z^2)^{3/2}}$

$\displaystyle \frac{\partial}{\partial y} = \frac{x^2+z^2}{(x^2+y^2+z^2)^{3/2}}$

$\displaystyle \frac{\partial}{\partial z} = \frac{x^2+y^2}{(x^2+y^2+z^2)^{3/2}}$

now putting everything together I get:

$\displaystyle \frac{2(x^2+y^2+z^2)}{(x^2+y^2+z^2)^{3/2}}= \frac{2}{\sqrt{x^2+y^2+z^2}}$

now converting into cylindrical coordinates I get:

$\displaystyle \int_{\theta=0}^{\theta=2\pi} \int_{\phi=0}^{\phi=\pi/2} \int_{\rho=0}^{\rho=1} = \frac{2\rho^2 \sin(\phi)}{\rho} \ d\rho \ d\phi \ d\theta = 2\pi$

now I don't know if this is correct since $\displaystyle x^2+y^2 \leq 1$ is throwing me off.