Curl and Divergence — Question 3

Question 3 – Solution

Strategy. Use divergence for local volume change and curl for twice the infinitesimal angular-velocity vector.

Step 1: Expansion rate $$\nabla\cdot\mathbf F=0+0+2z=2z.$$ At $P$, this is $$\boxed{(\nabla\cdot\mathbf F)(P)=4},$$ so the field has positive local expansion there.

Step 2: Curl $$\nabla\times\mathbf F =\langle 0-0,0-0,2-(-2)\rangle =\boxed{\langle 0,0,4\rangle}.$$ This curl is constant, so its value at $P$ is the same vector.

Step 3: Interpret The curl points along the positive $z$-axis, indicating counterclockwise rotation in horizontal planes when viewed from $+z$. The associated infinitesimal angular-velocity vector is half the curl: $$\boxed{\vec\omega=\langle 0,0,2\rangle}.$$

Verification The horizontal field $\langle-2y,2x\rangle$ is rigid rotation with angular speed $2$, which matches $|\nabla\times\mathbf F|/2=2$.