 Solution

The divergence theorem states that, for a sufficiently smooth vector field $\mathbf{F}$ on a volume $\Omega$ with piecewise smooth closed boundary $\partial \Omega$,Here $\hat{\mathbf{n}}$ is the outward unit normal. The left-hand side is the total outward flux; the right-hand side adds the local divergence throughout the volume. The orientation and the requirement to include the complete closed boundary are essential.