The formulas for divergence and curl are easy to compute and easy to misread. Here is what they measure, in the language of a flowing fluid with velocity field $\mathbf{F} = \langle P, Q \rangle$. Divergence is net outflow per unit area. Curl is circulation per unit area: how strongly the flow turns a tiny paddle wheel.
Key idea
Near a point, for a tiny region of area $\Delta A$:
$$\text{net outflow} \approx (\nabla \cdot \mathbf{F})\,\Delta A, \qquad \text{circulation counterclockwise} \approx (Q_x - P_y)\,\Delta A.$$
$\nabla \cdot \mathbf{F} > 0$: a source (more flows out than in). $\nabla \cdot \mathbf{F} < 0$: a sink. $Q_x - P_y > 0$: a paddle wheel turns counterclockwise; $< 0$: clockwise. In space, the curl vector points along the axis of the spin (right-hand rule), and its length is twice the paddle wheel's angular speed.
Why it works
Outflow from a small box $[x, x + \Delta x] \times [y, y + \Delta y]$. Out through the right side flows about $P(x + \Delta x, y)\,\Delta y$; in through the left side about $P(x, y)\,\Delta y$. Net: $$[P(x + \Delta x, y) - P(x, y)]\,\Delta y \approx P_x\,\Delta x\,\Delta y.$$ Top and bottom give $Q_y\,\Delta x\,\Delta y$ the same way. Total: $(P_x + Q_y)\,\Delta A$.
Circulation around the same box, counterclockwise: bottom $P(x, y)\,\Delta x$, top $-P(x, y + \Delta y)\,\Delta x$, right $Q(x + \Delta x, y)\,\Delta y$, left $-Q(x, y)\,\Delta y$. These add to about $(Q_x - P_y)\,\Delta A$.
A rigid rotation $\mathbf{v} = \omega\langle -y, x \rangle$ has $Q_x - P_y = \omega + \omega = 2\omega$: curl is twice the angular velocity.
Circulation around the same box, counterclockwise: bottom $P(x, y)\,\Delta x$, top $-P(x, y + \Delta y)\,\Delta x$, right $Q(x + \Delta x, y)\,\Delta y$, left $-Q(x, y)\,\Delta y$. These add to about $(Q_x - P_y)\,\Delta A$.
A rigid rotation $\mathbf{v} = \omega\langle -y, x \rangle$ has $Q_x - P_y = \omega + \omega = 2\omega$: curl is twice the angular velocity.
Worked examples
Worked example: Shear flow
Find the divergence and scalar curl of $\langle y, 0 \rangle$ and describe the flow.
- $P_x + Q_y = 0 + 0 = 0$: no spreading, no bunching.
- $Q_x - P_y = 0 - 1 = -1$: a paddle wheel turns clockwise, at every point.
- All arrows are horizontal, but they get longer as $y$ grows. The difference in speed across the wheel is what turns it.
Answer: divergence ${}0$, curl $-1$
Worked example: Radial flow
Describe $\langle x, y \rangle$ with divergence and curl.
- $P_x + Q_y = 1 + 1 = 2$: every point is a source; any small region loses fluid at rate ${}2$ per unit area.
- $Q_x - P_y = 0 - 0 = 0$: no spin anywhere.
Answer: divergence ${}2$, curl ${}0$
Worked example: A whirlpool with no curl
Find the scalar curl of $\mathbf{F} = \left\langle \frac{-y}{x^2 + y^2}, \frac{x}{x^2 + y^2} \right\rangle$ away from the origin.
- $Q_x = \frac{(x^2 + y^2) - x \cdot 2x}{(x^2 + y^2)^2} = \frac{y^2 - x^2}{(x^2 + y^2)^2}$.
- $P_y = -\frac{(x^2 + y^2) - y \cdot 2y}{(x^2 + y^2)^2} = \frac{y^2 - x^2}{(x^2 + y^2)^2}$.
- $Q_x - P_y = 0$. The arrows circle the origin, but their length ${}1/r$ drops off just fast enough that a paddle wheel away from the origin does not turn. (All the spin is concentrated at the origin; see Green's theorem for regions with holes.)
Answer: ${}0$ for $(x, y) \ne (0, 0)$
Common mistake
Judging curl by the shape of the arrows. Curved arrows do not mean curl, and straight arrows do not mean zero curl. The whirlpool field above circles the origin with zero curl; the shear field $\langle y, 0 \rangle$ has straight arrows and curl $-1$. Curl compares how the field changes across a point, so compute $Q_x - P_y$ rather than looking at the overall pattern.