Curl (mathematics)

In vector calculus, the curl is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional Euclidean space. The curl at a point in the field is represented by a vector whose length and direction denote the magnitude and axis of the maximum circulation.[1] The curl of a field is formally defined as the circulation density at each point of the field.

Depiction of a two-dimensional vector field with a uniform curl.

A vector field whose curl is zero is called irrotational. The curl is a form of differentiation for vector fields. The corresponding form of the fundamental theorem of calculus is Stokes' theorem, which relates the surface integral of the curl of a vector field to the line integral of the vector field around the boundary curve.

The alternative terminology rotation or rotational and alternative notations rot F or the cross product with the del (nabla) operator ∇ × F are sometimes used for curl F. The ISO/IEC 80000-2 standard recommends the use of the rot notation in boldface as opposed to the curl notation.[2]

Unlike the gradient and divergence, curl as formulated in vector calculus does not generalize simply to other dimensions; some generalizations are possible, but only in three dimensions is the geometrically defined curl of a vector field again a vector field. This deficiency is a direct consequence of the limitations of vector calculus; when expressed via the wedge operator of geometric calculus, the curl generalizes to all dimensions. The unfortunate circumstance is similar to that attending the 3-dimensional cross product, and indeed the connection is reflected in the notation ∇× for the curl.

The name "curl" was first suggested by James Clerk Maxwell in 1871[3] but the concept was apparently first used in the construction of an optical field theory by James MacCullagh in 1839.[4][5]


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