 # Curl (mathematics)

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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.

Curl F is a notation common today to the United States and Americas. In many European countries, particularly in classic scientific literature, the alternative notation rot F is traditionally used, which is spelled as "rotor", and comes from the "rate of rotation", which it represents. To avoid confusion, modern authors tend to use the cross product notation with the del (nabla) operator $\nabla \times \mathbf {F}$ 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 $\nabla \times$ for the curl.