Solution
= Solution
Add the four outward <fluxes>:
$$
\boxed{\Phi_{\mathrm{closed}}=-\frac16-\frac14-\frac13+1=\frac14}.
$$
For a check, $\nabla\cdot\mathbf F=2(x+y+z)$. The <divergence theorem> gives the same result from
$$
\iiint_{x,y,z\geq0,\ x+y+z\leq1}2(x+y+z)\,dx\,dy\,dz=\frac14.
$$