Curl

The curl of a vector field, denoted or (the notation used in this work), is defined as the vector field having magnitude equal to the maximum "circulation" at each point and to be oriented perpendicularly to this plane of circulation for each point. More precisely, the magnitude of is the limiting value of circulation per unit area. Written explicitly,

(1)

where the right side is a line integral around an infinitesimal region of area that is allowed to shrink to zero via a limiting process and is the unit normal vector to this region. If , then the field is said to be an irrotational field. The symbol is variously known as "nabla" or "del."

The physical significance of the curl of a vector field is the amount of "rotation" or angular momentum of the contents of given region of space. It arises in fluid mechanics and elasticity theory. It is also fundamental in the theory of electromagnetism, where it arises in two of the four Maxwell equations,

			(2)
			(3)

where MKS units have been used here, denotes the electric field, is the magnetic field, is a constant of proportionality known as the permeability of free space, is the current density, and is another constant of proportionality known as the permittivity of free space. Together with the two other of the Maxwell equations, these formulas describe virtually all classical and relativistic properties of electromagnetism.

In Cartesian coordinates, the curl is defined by

(4)

This provides the motivation behind the adoption of the symbol for the curl, since interpreting as the gradient operator , the "cross product" of the gradient operator with is given by

(5)

which is precisely equation (4). A somewhat more elegant formulation of the curl is given by the matrix operator equation

(6)

(Abbott 2002).

The curl can be similarly defined in arbitrary orthogonal curvilinear coordinates using

(7)

and defining

(8)

as

			(9)
			(10)

The curl can be generalized from a vector field to a tensor field as

(11)

where is the permutation tensor and ";" denotes a covariant derivative.