The vector field is given by the spherical coordinates $\vec A = kr^3\,\vec {e}_r$, where $k$ is a constant and $r=\sqrt {x^2+y^2+z^2}$.
I thought about using the Gauss-Ostrogradski theorem, where the flow is given by: $$ \iiint_V \operatorname{div}{\!\vec A} \; dV $$
But i got lost in the math and couldn't find a proper answer. Do you guys have any tips or ideas on how to solve this problem?
A little insight on my math: I got confused on how to calculate the divergent of $\vec A$ because of the base vector $\vec {e}_r$, since I am not used to it. But, my idea was to use the following intervals on the triple integral: ($\sqrt {R^2-x^2-y^2}$,$-\sqrt {R^2-x^2-y^2}$) and integrate in relation to dz;($\sqrt {R^2-x^2}$,$-\sqrt {R^2-x^2}$) and integrate in relation to dy; ($R$,$-R$) and integrate in relation to dx. I have no idea if it's correct or not. If anyone can please help or solve this for me, it would be really helpfull.
It is easier if you do the whole thing without resorting to cartesian coordinates. Let $\phi$ denote the flux. Then \begin{align} \phi&=\oint_{S}\vec E\cdot \vec{dS}\\ &=\oint_{S}kr^3\vec{e_r}\cdot dS\vec{e_r}\\ &=kr^3\oint_{S}dS\\ &=kr^3\times 4\pi r^2\\ &=4\pi k r^5 \end{align} `