flux of a vector field on the surface of a sphere

Question

I attached a picture of the question, but basically have to find the flux of a field on the surface of a sphere. Ive tried the divergence theorem but it doesnt seem to be working.

Answer 1

The divergence theorem works perfectly. $$ \iint_S(\mathbf{F}\cdot\mathbf{n})dS=\iiint_D(\nabla\cdot\mathbf{F})dV. $$ $$ \nabla\cdot\mathbf{F}=y^2+z^2+x^2. $$ Now we need to compute $\iiint_D(\nabla\cdot\mathbf{F})dV$. We just note, that $$ \iiint_D(\nabla\cdot\mathbf{F})dV=\frac{1}{8}\iiint_\Omega(\nabla\cdot\mathbf{F})dV,\tag 1 $$ where $\Omega$ is the whole sphere $x^2+y^2+z^2\leq1$. $$ \iiint_\Omega(\nabla\cdot\mathbf{F})dV=\iiint_\Omega(x^2+y^2+z^2)dV=\int_0^1r^24\pi r^2dr=4\pi\int_0^1r^4dr=4\pi\frac{r^5}{5}\bigg|_0^1=\frac{4\pi}{5}. $$ Taking $(1)$ into account we get the final result $\frac{\pi}{10}$.