Curl operator Definition and 13 Threads

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. The curl of a field is formally defined as the circulation density at each point of the field.
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.Unlike the gradient and divergence, curl does not generalize as 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 is a phenomenon similar to the 3-dimensional cross product, and the connection is reflected in the notation ∇× for the curl.
The name "curl" was first suggested by James Clerk Maxwell in 1871 but the concept was apparently first used in the construction of an optical field theory by James MacCullagh in 1839.

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