Past exam of the mathematical methods II course of the University of Oxford / 2015 / 1 / Solution

For a continuously differentiable vector field on a volume $V$ with piecewise smooth boundary $S$, the divergence theorem stateswhere $\hat{\mathbf{n}}$ is the outward unit normal. For the polynomial field here,Therefore the outward flux through the unit cube isAs a direct boundary check, the $x=1$ face contributes $2$, the $y=1$ face contributes $-1$, and the $z=1$ face contributes $1/2$. All three opposite faces contribute zero, reproducing the result and fixing the orientation sign.