(d) If … at all points then … at all points

True or false, with a good reason.

(a) If $\iint \mathbf{F}\cdot ndS=0$ for every closed surface, $\mathbf{F}$ is constant.

(b) If $\mathbf{F}=\mathrm{grad}f$ then $÷\mathbf{F}=0$.

(c) If $|\mathbf{F}|\le 1$ at all points then $\iiint ÷\mathbf{F}dV\le$ area of the surface $S$.

(d) If $|\mathbf{F}|\le 1$ at all points then $|÷\mathbf{F}|\le 1$ at all points.

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §15.5 The Divergence Theorem, #27