Curl: Rotation and Circulation
Divergence asks whether a field has sources or sinks. Curl asks a different question: does the field rotate?
Imagine placing a tiny paddle wheel in a vector field. If the arrows push one side of the wheel harder than the other, it spins. The curl measures how fast — and in which direction.
The intuition
- Non-zero curl = the field swirls. A vortex, a whirlpool, a rotating wind pattern.
- Zero curl = no rotation. The field might be strong, might vary, but there’s no net spinning tendency.
Key surprise: a field can look like it curves without having curl. A circular flow where faster water is on the outside and slower on the inside can balance perfectly — no net rotation of a tiny paddle wheel, even though the large-scale flow goes in circles.
Hover over the field. The paddle wheel shows the local curl — spinning anticlockwise for positive, clockwise for negative, still for zero.
The formula
In two dimensions, the curl of $\mathbf{F} = (F_x, F_y)$ is a scalar:
$$(\nabla \times \mathbf{F})_z = \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}$$
It measures the tendency to rotate in the $xy$-plane. In three dimensions, the curl is a full vector (with $x$, $y$, and $z$ components), showing rotation about each axis.
For the vortex $\mathbf{F} = (-y, x)$: curl $= 1 - (-1) = 2$. Positive — anticlockwise rotation everywhere.
For the source $\mathbf{F} = (x, y)$: curl $= 0 - 0 = 0$. The field spreads out but doesn’t rotate.
Divergence and curl together
These two operations give complementary views of a vector field:
| Divergence | Curl | |
|---|---|---|
| Question | Sources/sinks? | Rotation? |
| Measures | Net outflow | Net circulation |
| Analogy | Balloon inflating | Paddle wheel spinning |
| Source field | Positive | Zero |
| Vortex field | Zero | Positive |
A field can have divergence without curl (sources), curl without divergence (whirlpools), both, or neither.
Why curl matters
Curl appears in the other two of Maxwell’s four equations:
- Faraday’s law: $\nabla \times \mathbf{E} = -\partial \mathbf{B}/\partial t$ — a changing magnetic field makes the electric field curl
- Ampère–Maxwell law: $\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0\, \partial \mathbf{E}/\partial t$ — electric current and changing electric fields make the magnetic field curl
The curl tells you where fields circulate. Electric fields circulate when magnets change. Magnetic fields circulate around currents — and around changing electric fields.