In vector calculus, the curl, also known as rotor, 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 respectively. 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 notation curl F is more common in English-speaking countries. In the rest of the world, particularly in 20th century scientific literature, the alternative notation rot F is traditionally used, which comes from the "rate of rotation" that it represents. To avoid confusion, modern authors tend to use the cross product notation with the del (nabla) operator, as in ∇ × F {\displaystyle \nabla \times \mathbf {F} } , which also reveals the relation between curl (rotor), divergence, and gradient operators. 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; on the other hand, when expressed as an antisymmetric tensor field via the wedge operator of geometric calculus, the curl generalizes to all dimensions. The circumstance is similar to that attending the 3-dimensional cross product, and indeed the connection is reflected in the notation ∇ × {\displaystyle \nabla \times } 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.
Definition
The curl of a vector field F, denoted by curl F, or ∇ × F {\displaystyle \nabla \times \mathbf {F} } , or rot F, is an operator that maps Ck functions in R3 to Ck−1 functions in R3, and in particular, it maps continuously differentiable functions R3 → R3 to continuous functions R3 → R3. It can be defined in several ways, to be mentioned below: One way to define the curl of a vector field at a point is implicitly through its components along various axes passing through the point: if u ^ {\displaystyle \mathbf {\hat {u}} } is any unit vector, the component of the curl of F along the direction u ^ {\displaystyle \mathbf {\hat {u}} } may be defined to be the limiting value of a closed line integral in a plane perpendicular to u ^ {\displaystyle \mathbf {\hat {u}} } divided by the area enclosed, as the path of integration is contracted indefinitely around the point. More specifically, the curl is defined at a point p as
( ∇ × F ) ( p ) ⋅ u ^
=
d e f lim A → 0 1 | A | ∮ C ( p ) F ⋅ d r {\displaystyle (\nabla \times \mathbf {F} )(p)\cdot \mathbf {\hat {u}} \ {\overset {\underset {\mathrm {def} }{}}{{}={}}}\lim _{A\to 0}{\frac {1}{|A|}}\oint _{C(p)}\mathbf {F} \cdot \mathrm {d} \mathbf {r} }
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