Solution
= Solution
The <vector triple product> removes the nested <cross products>:
$$
F_i=\omega_i\omega_jx_j-|\boldsymbol\omega|^2x_i.
$$
Since $\boldsymbol\omega$ is constant, the <divergence> is
$$
\partial_iF_i=\omega_i\omega_i-3|\boldsymbol\omega|^2=-2|\boldsymbol\omega|^2.
$$
The cube has <volume> $2^3=8$. With the outward <unit normal>, the <divergence theorem> therefore gives
$$
\boxed{\int_S\mathbf F\cdot d\mathbf S=-16|\boldsymbol\omega|^2.}
$$