curl

Drop a tiny paddle wheel into a flowing fluid and watch it. If the flow is faster on one side than the other, the wheel spins. Curl is the vector that measures this local spinning at every point of a vector field — the tendency of the field to circulate around the point. Its direction is the axis the little wheel turns about (by the right-hand rule) and its magnitude is twice the angular speed of the spin.

For F = (P, Q, R) the curl is the vector field curl F = nabla cross F, computed as the cross product of del with F: its components are (partial R/partial y - partial Q/partial z, partial P/partial z - partial R/partial x, partial Q/partial x - partial P/partial y). Unlike divergence, curl produces a vector, not a scalar. Its honest definition is again a limit: the component of curl F along a direction n is the circulation of F around a tiny loop perpendicular to n, divided by the loop's area, as the loop shrinks to a point. So curl is circulation-per-unit-area. A field with curl F = 0 everywhere is called irrotational; on a simply connected region such a field is exactly a gradient, that is, conservative.

Curl is the other half of the backbone of electromagnetism. Faraday's law, curl E = -partial B/partial t, says a changing magnetic field circulates the electric field, and Ampere's law with Maxwell's correction, curl B = mu_0 J + mu_0 epsilon_0 partial E/partial t, says currents and changing electric fields circulate the magnetic field. In fluid mechanics curl v is the vorticity, the local spin of the flow. Stokes' theorem then lifts the pointwise curl up to total circulation around a whole closed loop, pairing curl with circulation just as the divergence theorem pairs divergence with flux.