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In vector calculus, the curl is a vector operator that describes the infinitesimal rotation of a 3-dimensional vector field. At every point in the field, the curl of that point is represented by a vector. The attributes of this vector (length and direction) characterize the rotation at that point. The direction of the curl is the axis of rotation, as determined by the right-hand rule, and the magnitude of the curl is the magnitude of rotation. If the vector field represents the flow velocity of a moving fluid, then the curl is the circulation density of the fluid. 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 ''rotor'' or ''rotational'' and alternative notations rot F and ∇ × F are often used (the former especially in many European countries, the latter, using the del operator and the cross product, is more used in other countries) for ''curl'' and curl F. 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 similar phenomenon as in 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〔(Proceedings of the London Mathematical Society, March 9th, 1871 )〕 but the concept was apparently first used in the construction of an optical field theory by James MacCullagh in 1839.〔(Collected works of James MacCullagh )〕 ==Definition== The curl of a vector field F, denoted by curl F, or ∇ × F, or rot F, at a point is defined in terms of its projection onto various lines through the point. If is defined to be the limiting value of a closed line integral in a plane orthogonal to where is a line integral along the boundary of the area in question, and |''A''| is the magnitude of the area. If is the unit vector perpendicular to the plane (see caption at right), then the orientation of C is chosen so that a tangent vector forms a positively oriented basis for R3 (right-hand rule). The above formula means that the curl of a vector field is defined as the infinitesimal area density of the ''circulation'' of that field. To this definition fit naturally * the Kelvin-Stokes theorem, as a global formula corresponding to the definition, and * the following "easy to memorize" definition of the curl in curvilinear orthogonal coordinates, e.g. in Cartesian coordinates, spherical, cylindrical, or even elliptical or parabolical coordinates: ::, ::, ::. Note that the equation for each component, can be obtained by exchanging each occurrence of a subscript 1, 2, 3 in cyclic permutation: 1→2, 2→3, and 3→1 (where the subscripts represent the relevant indices). If (''x''1, ''x''2, ''x''3) are the Cartesian coordinates and (''u''1,''u''2,''u''3) are the orthogonal coordinates, then : is the length of the coordinate vector corresponding to ''ui''. The remaining two components of curl result from cyclic permutation of indices: 3,1,2 → 1,2,3 → 2,3,1. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Curl (mathematics)」の詳細全文を読む スポンサード リンク
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