Editing Curl (mathematics) (section)

==Definition==
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{{Multiple image
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| image1            = Curl.svg
| footer            = The components of {{math|'''F'''}} at position {{math|'''r'''}}, normal and tangent to a closed curve {{math|''C''}} in a plane, enclosing a planar [[vector area]] {{nowrap|<math>\mathbf{A} = A\mathbf{\hat{n}}</math>.}}
}}

{{Multiple image
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| image1            = Curlorient.svg
| caption1          = Convention for vector orientation of the line integral
| image2            = Right hand rule simple.png
| caption2          = The thumb points in the direction of <math>\mathbf{\hat{n}}</math> and the fingers curl along the orientation of {{math|''C''}}
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| alt1              = 
| header            = Right-hand rule
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<!---DEL CROSS F IS NOT A DEFINITION, IT'S AN ALTERNATIVE NOTATION. IT DOESN'T MEAN ANYTHING SO YOU CAN'T BASE A DEFINITION ON IT----->
The curl of a vector field {{math|'''F'''}}, denoted by {{math|curl '''F'''}}, or <math>\nabla \times \mathbf{F}</math>, or {{math|rot '''F'''}}, is an operator that maps {{math|[[Smooth function|''C<sup>k</sup>'']]}} functions in {{math|'''R'''<sup>3</sup>}} to {{math|''C''<sup>''k''−1</sup>}} functions in {{math|'''R'''<sup>3</sup>}}, and in particular, it maps continuously differentiable functions {{math|'''R'''<sup>3</sup> → '''R'''<sup>3</sup>}} to continuous functions {{math|'''R'''<sup>3</sup> → '''R'''<sup>3</sup>}}. 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 <math>\mathbf{\hat{u}}</math> is any unit vector, the component of the curl of {{math|'''F'''}} along the direction <math>\mathbf{\hat{u}}</math> may be defined to be the limiting value of a closed [[line integral]] in a plane perpendicular to <math>\mathbf{\hat{u}}</math> 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 {{math|''p''}} as<ref>Mathematical methods for physics and engineering, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press, 2010, {{ISBN|978-0-521-86153-3}}</ref><ref>Vector Analysis (2nd Edition), M.R. Spiegel, S. Lipschutz, D. Spellman, Schaum's Outlines, McGraw Hill (USA), 2009, {{ISBN|978-0-07-161545-7}}</ref>
<math display="block">(\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}</math>
where the [[line integral]] is calculated along the [[Boundary (topology)|boundary]] {{math|''C''}} of the [[area]] {{math|''A''}} containing point p, {{math|{{abs|''A''}}}} being the magnitude of the area. This equation defines the component of the curl of {{math|'''F'''}} along the direction <math>\mathbf{\hat{u}}</math>. The infinitesimal surfaces bounded by {{math|''C''}} have <math>\mathbf{\hat{u}}</math> as their [[Normal vector|normal]]. {{math|''C''}} is oriented via the [[right-hand rule]].

The above formula means that the component of the curl of a vector field along a certain axis is the ''infinitesimal [[area density]]'' of the circulation of the field in a plane perpendicular to that axis. This formula does not ''a priori'' define a legitimate vector field, for the individual circulation densities with respect to various axes ''a priori'' need not relate to each other in the same way as the components of a vector do; that they ''do'' indeed relate to each other in this precise manner must be proven separately.

To this definition fits naturally the [[Kelvin–Stokes theorem]], as a global formula corresponding to the definition. It equates the [[surface integral]] of the curl of a vector field to the above line integral taken around the boundary of the surface.

Another way one can define the curl vector of a function {{math|'''F'''}} at a point is explicitly as the limiting value of a vector-valued surface integral around a shell enclosing {{math|''p''}} divided by the volume enclosed, as the shell is contracted indefinitely around {{math|''p''}}.

More specifically, the curl may be defined by the vector formula
<math display="block">(\nabla \times \mathbf{F})(p) \overset{\underset{\mathrm{def}}{}}{{}={}} \lim_{V \to 0}\frac{1}{|V|}\oint_S \mathbf{\hat{n}} \times \mathbf{F} \ \mathrm{d}S</math>
where the surface integral is calculated along the boundary {{math|''S''}} of the volume {{math|''V''}}, {{math|{{abs|''V''}}}} being the magnitude of the volume, and <math>\mathbf{\hat{n}}</math> pointing outward from the surface {{math|''S''}} perpendicularly at every point in {{math|''S''}}.

In this formula, the cross product in the integrand measures the tangential component of {{math|'''F'''}} at each point on the surface {{math|''S''}}, and points along the surface at right angles to the ''tangential projection'' of {{math|'''F'''}}. Integrating this cross product over the whole surface results in a vector whose magnitude measures the overall circulation of {{math|'''F'''}} around {{math|''S''}}, and whose direction is at right angles to this circulation. The above formula says that the ''curl'' of a vector field at a point is the ''infinitesimal volume density'' of this "circulation vector" around the point.

To this definition fits naturally another global formula (similar to the Kelvin-Stokes theorem) which equates the [[volume integral]] of the curl of a vector field to the above surface integral taken over the boundary of the volume. 

Whereas the above two definitions of the curl are coordinate free, there is another "easy to memorize" definition of the curl in curvilinear [[orthogonal coordinates]], e.g. in [[Cartesian coordinates]], [[spherical coordinates|spherical]], [[cylindrical coordinates|cylindrical]], or even [[Elliptic coordinate system|elliptical]] or [[parabolic coordinates]]: <math display="block">\begin{align}
& (\operatorname{curl}\mathbf F)_1=\frac{1}{h_2h_3}\left (\frac{\partial (h_3F_3)}{\partial u_2}-\frac{\partial (h_2F_2)}{\partial u_3}\right ), \\[5pt]
& (\operatorname{curl}\mathbf F)_2=\frac{1}{h_3h_1}\left (\frac{\partial (h_1F_1)}{\partial u_3}-\frac{\partial (h_3F_3)}{\partial u_1}\right ), \\[5pt]
& (\operatorname{curl}\mathbf F)_3=\frac{1}{h_1h_2}\left (\frac{\partial (h_2F_2)}{\partial u_1}-\frac{\partial (h_1F_1)}{\partial u_2}\right ).
\end{align}</math>

The equation for each component {{math|(curl '''F''')<sub>''k''</sub>}} can be obtained by exchanging each occurrence of a subscript 1, 2, 3 in cyclic permutation: 1&nbsp;→&nbsp;2, 2&nbsp;→&nbsp;3, and&nbsp;3&nbsp;→&nbsp;1 (where the subscripts represent the relevant indices).

If {{math|(''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>)}} are the [[Cartesian coordinate system|Cartesian coordinates]] and {{math|(''u''<sub>1</sub>, ''u''<sub>2</sub>, ''u''<sub>3</sub>)}} are the orthogonal coordinates, then 
<math display="block">h_i = \sqrt{\left (\frac{\partial x_1}{\partial u_i} \right )^2 + \left (\frac{\partial x_2}{\partial u_i} \right )^2 + \left (\frac{\partial x_3}{\partial u_i} \right )^2}</math> 
is the length of the coordinate vector corresponding to {{math|''u<sub>i</sub>''}}. The remaining two components of curl result from [[cyclic permutation]] of [[index notation|indices]]: 3,1,2&nbsp;→&nbsp;1,2,3&nbsp;→&nbsp;2,3,1.