Curl measures how much a flowing field spins at each point
Drop a tiny paddle wheel into moving water and it may start to turn. The curl of a vector field captures that spin as an arrow: its length gives the strength of the swirl, its direction the axis. James Clerk Maxwell named it in 1871, though the idea had surfaced decades earlier.
In vector calculus the curl, also called the rotor, is an operator that takes a three-dimensional vector field and returns another one describing circulation. At every point it produces an arrow aligned with the axis of greatest rotation, with a length showing how strong that rotation is. Formally it is the density of circulation at each point. Fields with zero curl everywhere are described as irrotational.
One way to picture the definition is to shrink a small flat loop around a point, add up how much the field pushes along that loop, and divide by the enclosed area. The limit, taken with the loop's plane set perpendicular to a chosen direction, gives the curl's component along that direction. An equivalent picture uses a tiny enclosing shell instead of a loop, and a third shows the curl as twice the rotation rate of an infinitesimal ball of the field.
Curl is a kind of derivative for vector fields, and it has its own version of the fundamental theorem of calculus. Stokes' theorem states that adding up the curl across a surface gives the same result as measuring circulation around that surface's edge. English speakers usually write curl F, while much of the world, especially in 20th-century scientific writing, used rot F, short for rate of rotation. Modern authors often sidestep the split with a cross product involving the nabla symbol, which also shows how curl relates to divergence and gradient.
The concept seems to have appeared first in 1839, when James MacCullagh used it while building a theory of optical fields. Unlike divergence and gradient, curl in its familiar form does not carry over neatly to other dimensions; only in three is the curl of a vector field itself a vector field. Recasting it through the wedge product of geometric calculus lets it generalise to any number of dimensions.
Source: Curl (mathematics)
