Curl and Divergence

Divergence and curl are simply two ways of asking "what's the flow doing right here?" Picture a tiny box floating in a river. Divergence asks: is more water leaving the box than entering? If so, there's a source inside, and the divergence is positive. Curl asks: would a tiny paddle wheel at that spot start to spin? The formulas look like a wall of partial derivatives, but each piece is just "how fast does this component change in that direction." Divergence adds up three of them into a single number; curl arranges six of them into a vector. Bear with the bookkeeping; the meaning is what matters.

Divergence is the outflow per unit volume. Think about why the formula is $P_x + Q_y + R_z$: if $P$, the $x$-component, grows as you move in the $x$-direction, then water leaves the box's right face faster than it enters the left face, so there's a net outflow in $x$. Add the same for $y$ and $z$. Air heating up and expanding has positive divergence; water draining has negative divergence. Water barely compresses, so its flows have $\nabla\cdot\vec F = 0$: whatever enters a box must leave it.

Curl is the rotation a tiny paddle wheel would feel. You've already met its $z$-component: $Q_x - P_y$, the "spin" from Green's theorem and the conservative-field test. The full curl just does that check about all three axes. For example, $\langle -y, x, 0\rangle$ has curl $\langle 0, 0, 2\rangle$: spinning counterclockwise about the $z$-axis.

A common mistake. It's tempting to think curl means the flow lines are curved. It doesn't. The first animation further down shows a river with perfectly straight flow lines that still spins a paddle wheel, and the challenge problem shows water circling a drain with zero curl. Curl is about the difference in push on opposite sides of a tiny wheel, not about the shape of the path.

Two identities:

• $\operatorname{curl}(\nabla f) = \vec 0$: gradient fields never swirl. Each component of the curl is a difference of mixed partials, like $f_{yx} - f_{xy}$, and Clairaut makes every one of them zero. That's the 3D version of the conservative-field test.
• $\operatorname{div}(\operatorname{curl}\vec F) = 0$: swirl fields have no sources. Again, the mixed partials cancel in pairs.

Maxwell's equations are statements about the divergence and curl of the electric and magnetic fields; fluid dynamics uses $\nabla\cdot\vec v = 0$ for incompressible flow and the curl for vorticity (tornadoes, eddies).