Solution
= Solution
The <divergence> is $\nabla\cdot F=2xyz+2xyz+2xyz=6xyz$. With outward orientation, the <divergence theorem> converts the <surface flux> into a <volume integral>:
$$
\boxed{\int_SF\cdot dS=6\int_0^1\int_0^1\int_0^1xyz\,dx\,dy\,dz=6\left(\frac12\right)^3=\frac34.}
$$
Directly, the three faces lying in coordinate <planes> contribute zero, and each opposite face contributes $1/4$, confirming the sign and normalization.