The curl of a vector field measures its local tendency to circulate around a point. Its formula can be understood by asking how much a field pushes along the boundary of a tiny loop.

Let

$$\mathbf F=(P,Q,R).$$

First look only at a small rectangle of side lengths $\Delta x$ and $\Delta y$ in the $xy$-plane, traversed counterclockwise as viewed from the positive $z$-axis.

Circulation around a tiny rectangle

Along the bottom edge, the field contributes approximately

$$P(x,y),\Delta x.$$

Along the right edge, it contributes

$$Q(x+\Delta x,y),\Delta y.$$

The top and left edges are traversed in the negative coordinate directions, so their contributions are approximately

$$-P(x,y+\Delta y),\Delta x$$

and

$$-Q(x,y),\Delta y.$$

Adding all four gives

$$\begin{aligned} \text{circulation} &\approx \bigl(Q(x+\Delta x,y)-Q(x,y)\bigr)\Delta y\ &\quad- \bigl(P(x,y+\Delta y)-P(x,y)\bigr)\Delta x. \end{aligned}$$

For a tiny rectangle,

$$Q(x+\Delta x,y)-Q(x,y) \approx \frac{\partial Q}{\partial x}\Delta x,$$

and

$$P(x,y+\Delta y)-P(x,y) \approx \frac{\partial P}{\partial y}\Delta y.$$

Therefore

$$\text{circulation} \approx \left( \frac{\partial Q}{\partial x}

\frac{\partial P}{\partial y} \right)\Delta x\Delta y.$$

Dividing by the rectangle's area shows that

$$\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}$$

measures local circulation per unit area in the $xy$-plane. This is the $z$-component of curl.

Repeating the same argument in the other coordinate planes produces

$$\boxed{ \nabla\times\mathbf F

\left( \frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z}-\frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y} \right)}. $$

Direction and local rotation

The curl vector points along the axis about which the local circulation is measured, with its sign determined by the right-hand rule. A useful physical picture is a tiny paddle wheel placed in a velocity field: if the surrounding flow tends to turn the wheel, the field has nonzero curl there.

Curl is local. A field can have complicated global circulation even when its curl vanishes at every point of its domain; the shape and topology of the domain can matter.

Example

For

$$\mathbf F(x,y,z)=(-y,x,0),$$

we have $P=-y$, $Q=x$, and $R=0$. Hence

$$\nabla\times\mathbf F

(0,0,1-(-1)) =(0,0,2).$$

The arrows circulate counterclockwise in the $xy$-plane, and the positive $z$-component records that orientation. The value $2$ is the infinitesimal circulation per unit area for loops perpendicular to the $z$-axis.

Stokes' theorem later turns this local circulation density into a global statement relating curl across a surface to circulation around its boundary.