Editing Curl (mathematics)

{{Short description|Circulation density in a vector field}}
{{redirect|Rotor (operator)|the geometric algebra concept|Rotor (mathematics)|other uses|Rotation operator (disambiguation)}}
[[File:Uniform curl.svg|thumb|Depiction of a two-dimensional vector field with a uniform curl.]]
{{Calculus |Vector}}

In [[vector calculus]], the '''curl''', also known as '''rotor''', is a [[vector operator]] that describes the [[Differential (infinitesimal)|infinitesimal]] [[Circulation (physics)|circulation]] of a [[vector field]] in three-dimensional [[Euclidean space]]. The curl at a point in the field is represented by a [[vector (geometry)|vector]] whose length and direction denote the [[Magnitude (mathematics)|magnitude]] and axis of the maximum circulation.<ref name="Mathworld" /> 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 [[derivative|differentiation]] for vector fields. The corresponding form of the [[fundamental theorem of calculus]] is [[Kelvin–Stokes theorem|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 {{math|curl '''F'''}} is more common in North America. In the rest of the world, particularly in 20th century scientific literature, the alternative notation {{math|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 operator|del]] (nabla) operator, as in {{nowrap|<math>\nabla \times \mathbf{F}</math>,}}<ref>[[ISO/IEC 80000|ISO/IEC 80000-2 standard]] Norm ISO/IEC 80000-2, item 2-17.16</ref> 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|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 <math>\nabla \times</math> for the curl. 

The name "curl" was first suggested by [[James Clerk Maxwell]] in 1871<ref>[http://www.clerkmaxwellfoundation.org/MathematicalClassificationofPhysicalQuantities_Maxwell.pdf Proceedings of the London Mathematical Society, March 9th, 1871]</ref> but the concept was apparently first used in the construction of an optical field theory by [[James MacCullagh]] in 1839.<ref>{{cite book|url = https://archive.org/details/collectedworks00maccuoft |title = Collected works of James MacCullagh|date = 1880 | location = Dublin|publisher = Hodges}}</ref><ref>[http://jeff560.tripod.com/c.html Earliest Known Uses of Some of the Words of Mathematics] [[tripod.com]]</ref>

==Definition==
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{{Multiple image
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| 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''}}
| align             = 
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| header            = Right-hand rule
}}

<!---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.

== Usage ==
In practice, the two coordinate-free definitions described above are rarely used because in virtually all cases, the curl [[operator (mathematics)|operator]] can be applied using some set of [[curvilinear coordinates]], for which simpler representations have been derived.

The notation <math>\nabla\times\mathbf{F}</math> has its origins in the similarities to the 3-dimensional [[cross product]], and it is useful as a [[mnemonic]] in [[Cartesian coordinate system|Cartesian coordinates]] if <math>\nabla</math> is taken as a vector [[differential operator]] [[del]]. Such notation involving [[operator (physics)|operators]] is common in [[physics]] and [[algebra]].

Expanded in 3-dimensional [[Cartesian coordinate system|Cartesian coordinates]] (see ''[[Del in cylindrical and spherical coordinates]]'' for [[Spherical coordinate system|spherical]] and [[Cylindrical coordinate system|cylindrical]] coordinate representations), <math>\nabla\times\mathbf{F}</math> is, for <math>\mathbf{F}</math> composed of <math>[F_x,F_y,F_z]</math> (where the subscripts indicate the components of the vector, not partial derivatives):
<math display="block">
\nabla \times \mathbf{F} =
\begin{vmatrix} \boldsymbol{\hat\imath} & \boldsymbol{\hat\jmath} & \boldsymbol{\hat k} \\[5mu]
{\dfrac{\partial}{\partial x}} & {\dfrac{\partial}{\partial y}} & {\dfrac{\partial}{\partial z}} \\[5mu]
F_x & F_y & F_z \end{vmatrix}
</math>
where {{math|'''i'''}}, {{math|'''j'''}}, and {{math|'''k'''}} are the [[unit vector]]s for the {{math|''x''}}-, {{math|''y''}}-, and {{math|''z''}}-axes, respectively. This expands as follows:<ref>{{Cite book|last=Arfken|first=George Brown |title=Mathematical methods for physicists | date=2005|publisher=Elsevier| others=Weber, Hans-Jurgen | isbn=978-0-08-047069-6 | edition=6th | location=Boston | oclc=127114279 | page = 43}}</ref>
<math display="block">
\nabla \times \mathbf{F} =
\left(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}\right) \boldsymbol{\hat\imath} +
\left(\frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} \right) \boldsymbol{\hat\jmath} +
\left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right) \boldsymbol{\hat k}
</math>

Although expressed in terms of coordinates, the result is invariant under proper rotations of the coordinate axes but the result inverts under reflection.

In a general coordinate system, the curl is given by<ref name="Mathworld">{{MathWorld |title=Curl |urlname=Curl}}</ref>
<math display="block">(\nabla \times \mathbf{F} )^k = \frac{1}{\sqrt{g}} \varepsilon^{k\ell m} \nabla_\ell F_m</math>
where {{mvar|ε}} denotes the [[Levi-Civita symbol#Levi-Civita tensors|Levi-Civita tensor]], {{math|∇}} the [[covariant derivative]], <math> g</math> is the [[determinant]] of the [[metric tensor]] and the [[Einstein summation convention]] implies that repeated indices are summed over. Due to the symmetry of the [[Christoffel symbols]] participating in the covariant derivative, this expression reduces to the [[partial derivative]]:
<math display="block">(\nabla \times \mathbf{F} ) = \frac{1}{\sqrt{g}} \mathbf{R}_k\varepsilon^{k\ell m} \partial_\ell F_m</math>
where {{math|'''R'''<sub>''k''</sub>}} are the local basis vectors. Equivalently, using the [[exterior derivative]], the curl can be expressed as:
<math display="block"> \nabla \times \mathbf{F} = \left( \star \big( {\mathrm d} \mathbf{F}^\flat \big) \right)^\sharp </math>

Here {{music|flat}} and {{music|sharp}} are the [[musical isomorphism]]s, and {{math|<small>★</small>}} is the [[Hodge star operator]]. This formula shows how to calculate the curl of {{math|'''F'''}} in any coordinate system, and how to extend the curl to any [[orientation (space)|oriented]] three-dimensional [[Riemannian metric|Riemannian]] manifold. Since this depends on a choice of orientation, curl is a [[Chirality (mathematics)|chiral]] operation. In other words, if the orientation is reversed, then the direction of the curl is also reversed.

==Examples==

=== Example 1 ===
Suppose the vector field describes the [[velocity field]] of a [[fluid flow]] (such as a large tank of [[liquid]] or [[gas]]) and a small ball is located within the fluid or gas (the center of the ball being fixed at a certain point). If the ball has a rough surface, the fluid flowing past it will make it rotate. The rotation axis (oriented according to the right hand rule) points in the direction of the curl of the field at the center of the ball, and the angular speed of the rotation is half the magnitude of the curl at this point.<ref>{{citation|first1=Josiah Willard | last1=Gibbs|author-link1=Josiah Willard Gibbs| first2=Edwin Bidwell|last2=Wilson| author-link2=Edwin Bidwell Wilson | title=Vector analysis |series=Yale bicentennial publications| url = http://hdl.handle.net/2027/mdp.39015000962285?urlappend=%3Bseq=179 | year=1901 | publisher=C. Scribner's Sons | hdl=2027/mdp.39015000962285?urlappend=%3Bseq=179}}</ref>
The curl of the vector field at any point is given by the rotation of an infinitesimal area in the ''xy''-plane (for ''z''-axis component of the curl), ''zx''-plane (for ''y''-axis component of the curl) and ''yz''-plane (for ''x''-axis component of the curl vector). This can be seen in the examples below.

=== Example 2 ===
{{Multiple image
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| image1            = Uniform curl.svg
| image2            = Curl of uniform curl.png
| alt1              = 
| caption1          = 
| caption2          = 
| footer            = Vector field {{math|1='''F'''(''x'',''y'')=[''y'',−''x'']}} (left) and its curl (right).
}}

The [[vector field]]
<math display="block">\mathbf{F}(x,y,z)=y\boldsymbol{\hat{\imath}}-x\boldsymbol{\hat{\jmath}}</math>
can be decomposed as
<math display="block">F_x =y, F_y = -x, F_z =0.</math>

Upon visual inspection, the field can be described as "rotating". If the vectors of the field were to represent a linear [[force]] acting on objects present at that point, and an object were to be placed inside the field, the object would start to rotate clockwise around itself. This is true regardless of where the object is placed.

Calculating the curl:
<math display="block">\nabla \times \mathbf{F} =0\boldsymbol{\hat{\imath}}+0\boldsymbol{\hat{\jmath}}+ \left({\frac{\partial}{\partial x}}(-x) -{\frac{\partial}{\partial y}} y\right)\boldsymbol{\hat{k}}=-2\boldsymbol{\hat{k}}
</math>

The resulting vector field describing the curl would at all points be pointing in the negative {{Math|''z''}} direction. The results of this equation align with what could have been predicted using the [[Right-hand rule#A rotating body|right-hand rule]] using a [[Cartesian coordinate system#In three dimensions|right-handed coordinate system]]. Being a uniform vector field, the object described before would have the same rotational intensity regardless of where it was placed.

{{clear}}
=== Example 3 ===
{{Multiple image
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| image1            = Nonuniform curl.svg
| image2            = Curl of nonuniform curl.png
| alt1              = 
| caption1          = 
| caption2          = 
| footer            = Vector field {{math|1='''F'''(''x'', ''y'') = [0, −''x''<sup>2</sup>]}} (left) and its curl (right).
}}

For the vector field
<math display="block">\mathbf{F}(x,y,z) = -x^2\boldsymbol{\hat{\jmath}}</math>

the curl is not as obvious from the graph. However, taking the object in the previous example, and placing it anywhere on the line {{math|1=''x'' = 3}}, the force exerted on the right side would be slightly greater than the force exerted on the left, causing it to rotate clockwise. Using the right-hand rule, it can be predicted that the resulting curl would be straight in the negative {{math|''z''}} direction. Inversely, if placed on {{math|1=''x'' = −3}}, the object would rotate counterclockwise and the right-hand rule would result in a positive {{math|''z''}} direction.

Calculating the curl:
<math display="block">{\nabla} \times \mathbf{F} = 0 \boldsymbol{\hat{\imath}} + 0\boldsymbol{\hat{\jmath}} + {\frac{\partial}{\partial x}}\left(-x^2\right) \boldsymbol{\hat{k}} = -2x\boldsymbol{\hat{k}}.</math>

The curl points in the negative {{math|''z''}} direction when {{math|''x''}} is positive and vice versa. In this field, the intensity of rotation would be greater as the object moves away from the plane {{math|1=''x'' = 0}}.

{{clear}}

===Further examples===
* In a vector field describing the linear velocities of each part of a rotating disk in [[uniform circular motion]], the curl has the same value at all points, and this value turns out to be exactly two times the vectorial [[angular velocity]] of the disk (oriented as usual by the [[right-hand rule]]). More generally, for any flowing mass, the linear velocity vector field at each point of the mass flow has a curl (the [[vorticity]] of the flow at that point) equal to exactly two times the ''local'' vectorial angular velocity of the mass about the point.
* For any solid object subject to an external physical force (such as gravity or the electromagnetic force), one may consider the vector field representing the infinitesimal force-per-unit-volume contributions acting at each of the points of the object. This force field may create a net ''[[torque]]'' on the object about its center of mass, and this torque turns out to be directly proportional and vectorially parallel to the (vector-valued) integral of the ''curl'' of the force field over the whole volume.
* Of the four [[Maxwell's equations]], two—[[Faraday's law of induction|Faraday's law]] and [[Ampère's circuital law|Ampère's law]]—can be compactly expressed using curl. Faraday's law states that the curl of an [[electric field]] is equal to the opposite of the time rate of change of the magnetic field, while Ampère's law relates the curl of the magnetic field to the current and the time rate of change of the electric field.

== Identities ==
{{Main|Vector calculus identities}}

In general [[curvilinear coordinates]] (not only in Cartesian coordinates), the curl of a cross product of vector fields {{math|'''v'''}} and {{math|'''F'''}} can be shown to be
<math display="block">\nabla \times \left( \mathbf{v \times F} \right) = \Big( \left( \mathbf{ \nabla \cdot F } \right) + \mathbf{F \cdot \nabla} \Big) \mathbf{v}- \Big( \left( \mathbf{ \nabla \cdot v } \right) + \mathbf{v \cdot \nabla} \Big) \mathbf{F} \ . </math>

Interchanging the vector field {{math|'''v'''}} and {{math|∇}} operator, we arrive at the cross product of a vector field with curl of a vector field:
<math display="block"> \mathbf{v \ \times } \left( \mathbf{ \nabla \times F} \right) =\nabla_\mathbf{F} \left( \mathbf{v \cdot F } \right) - \left( \mathbf{v \cdot \nabla } \right) \mathbf{F} \ , </math>
where {{math|∇<sub>'''F'''</sub>}} is the Feynman subscript notation, which considers only the variation due to the vector field {{math|'''F'''}} (i.e., in this case, {{math|'''v'''}} is treated as being constant in space).

Another example is the curl of a curl of a vector field. It can be shown that in general coordinates
<math display="block"> \nabla \times \left( \mathbf{\nabla \times F} \right) = \mathbf{\nabla}(\mathbf{\nabla \cdot  F}) - \nabla^2 \mathbf{F} \ , </math>
and this identity defines the [[vector Laplacian]] of {{math|'''F'''}}, symbolized as {{math|∇<sup>2</sup>'''F'''}}.

The curl of the [[gradient]] of ''any'' [[scalar field]] {{mvar|φ}} is always the [[zero vector]] field
<math display="block">\nabla \times ( \nabla \varphi ) = \boldsymbol{0}</math>
which follows from the [[antisymmetric tensor|antisymmetry]] in the definition of the curl, and the [[symmetry of second derivatives]].

The [[divergence]] of the curl of any vector field is equal to zero: 
<math display="block">\nabla\cdot(\nabla\times\mathbf{F}) = 0.</math>

If {{mvar|φ}} is a scalar valued function and {{math|'''F'''}} is a vector field, then
<math display="block">\nabla \times ( \varphi \mathbf{F}) = \nabla \varphi \times \mathbf{F} + \varphi \nabla \times \mathbf{F} </math>

== Generalizations ==
The vector calculus operations of [[gradient|grad]], curl, and [[divergence|div]] are most easily generalized in the context of differential forms, which involves a number of steps. In short, they correspond to the derivatives of 0-forms, 1-forms, and 2-forms, respectively. The geometric interpretation of curl as rotation corresponds to identifying [[bivector]]s (2-vectors) in 3 dimensions with the [[special orthogonal Lie algebra]] <math>\mathfrak{so}(3)</math> of infinitesimal rotations (in coordinates, skew-symmetric 3&nbsp;×&nbsp;3 matrices), while representing rotations by vectors corresponds to identifying 1-vectors (equivalently, 2-vectors) and {{nowrap|<math>\mathfrak{so}(3)</math>,}} these all being 3-dimensional spaces.

=== Differential forms ===
{{main|Differential form}}

In 3 dimensions, a differential 0-form is a real-valued function <math>f(x,y,z)</math>; a differential 1-form is the following expression, where the coefficients are functions:
<math display="block">a_1\,dx + a_2\,dy + a_3\,dz;</math>
a differential 2-form is the formal sum, again with function coefficients:
<math display="block">a_{12}\,dx\wedge dy + a_{13}\,dx\wedge dz + a_{23}\,dy\wedge dz;</math>
and a differential 3-form is defined by a single term with one function as coefficient:
<math display="block">a_{123}\,dx\wedge dy\wedge dz.</math>
(Here the {{mvar|a}}-coefficients are real functions of three variables; the [[wedge product]]s, e.g. <math>\text{d}x\wedge\text{d}y</math>, can be interpreted as [[Bivector|oriented plane segments]], <math>\text{d}x\wedge\text{d}y=-\text{d}y\wedge\text{d}x</math>, etc.)

The [[exterior derivative]] of a {{math|''k''}}-form in {{math|1='''R'''<sup>3</sup>}} is defined as the {{math|(''k'' + 1)}}-form from above—and in {{math|'''R'''<sup>''n''</sup>}} if, e.g.,
<math display="block">\omega^{(k)}=\sum_{1\leq i_1<i_2<\cdots<i_k\leq n} a_{i_1,\ldots,i_k} \,dx_{i_1}\wedge \cdots\wedge dx_{i_k},</math>
then the exterior derivative {{math|''d''}} leads to
<math display="block"> d\omega^{(k)}=\sum_{\scriptstyle{j=1} \atop \scriptstyle{i_1<\cdots<i_k}}^n\frac{\partial a_{i_1,\ldots,i_k}}{\partial x_j}\,dx_j \wedge dx_{i_1}\wedge \cdots \wedge dx_{i_k}.</math>

The exterior derivative of a 1-form is therefore a 2-form, and that of a 2-form is a 3-form. On the other hand, because of the interchangeability of mixed derivatives, 
<math display="block">\frac{\partial^2}{\partial x_i\,\partial x_j} = \frac{\partial^2}{\partial x_j\,\partial x_i} , </math>
and antisymmetry,
<math display="block">d x_i \wedge d x_j = -d x_j \wedge d x_i</math>

the twofold application of the exterior derivative yields <math>0</math> (the zero <math>k+2</math>-form).

Thus, denoting the space of {{math|''k''}}-forms by <math>\Omega^k(\mathbb{R}^3)</math> and the exterior derivative by {{math|''d''}} one gets a sequence:
<math display="block">0 \, \overset{d}{\longrightarrow} \;
\Omega^0\left(\mathbb{R}^3\right) \, \overset{d}{\longrightarrow} \;
\Omega^1\left(\mathbb{R}^3\right) \, \overset{d}{\longrightarrow} \;
\Omega^2\left(\mathbb{R}^3\right) \, \overset{d}{\longrightarrow} \;
\Omega^3\left(\mathbb{R}^3\right) \, \overset{d}{\longrightarrow} \, 0.</math>

Here <math>\Omega^k(\mathbb{R}^n)</math> is the space of sections of the [[exterior algebra]] <math>\Lambda^k(\mathbb{R}^n)</math> [[vector bundle]] over '''R'''<sup>''n''</sup>, whose dimension is the [[binomial coefficient]] <math>\binom{n}{k}</math>; note that <math>\Omega^k(\mathbb{R}^3)=0</math> for <math>k>3</math> or <math>k<0</math>. Writing only dimensions, one obtains a row of [[Pascal's triangle]]:

<math display="block">0\rightarrow 1\rightarrow 3\rightarrow 3\rightarrow 1\rightarrow 0;</math>

the 1-dimensional fibers correspond to scalar fields, and the 3-dimensional fibers to vector fields, as described below. Modulo suitable identifications, the three nontrivial occurrences of the exterior derivative correspond to grad, curl, and div.

Differential forms and the differential can be defined on any Euclidean space, or indeed any manifold, without any notion of a Riemannian metric. On a [[Riemannian manifold]], or more generally [[pseudo-Riemannian manifold]], {{math|''k''}}-forms can be identified with [[p-vector|{{math|''k''}}-vector]] fields ({{math|''k''}}-forms are {{math|''k''}}-covector fields, and a pseudo-Riemannian metric gives an isomorphism between vectors and covectors), and on an ''oriented'' vector space with a [[nondegenerate form]] (an isomorphism between vectors and covectors), there is an isomorphism between {{math|''k''}}-vectors and {{math|(''n'' − ''k'')}}-vectors; in particular on (the tangent space of) an oriented pseudo-Riemannian manifold. Thus on an oriented pseudo-Riemannian manifold, one can interchange {{math|''k''}}-forms, {{math|''k''}}-vector fields, {{math|(''n'' − ''k'')}}-forms, and {{math|(''n'' − ''k'')}}-vector fields; this is known as [[Hodge duality]]. Concretely, on {{math|'''R'''<sup>3</sup>}} this is given by:
* 1-forms and 1-vector fields: the 1-form {{math|''a<sub>x</sub> dx'' + ''a<sub>y</sub> dy'' + ''a<sub>z</sub> dz''}} corresponds to the vector field {{math|(''a<sub>x</sub>'', ''a<sub>y</sub>'', ''a<sub>z</sub>'')}}.
* 1-forms and 2-forms: one replaces {{math|''dx''}} by the dual quantity {{math|''dy'' ∧ ''dz''}} (i.e., omit {{math|''dx''}}), and likewise, taking care of orientation: {{math|''dy''}} corresponds to {{math|1=''dz'' ∧ ''dx'' = −''dx'' ∧ ''dz''}}, and {{math|''dz''}} corresponds to {{math|''dx'' ∧ ''dy''}}. Thus the form {{math|''a<sub>x</sub> dx'' + ''a<sub>y</sub> dy'' + ''a<sub>z</sub> dz''}} corresponds to the "dual form" {{math|''a<sub>z</sub> dx'' ∧ ''dy'' + ''a<sub>y</sub> dz'' ∧ ''dx'' + ''a<sub>x</sub> dy'' ∧ ''dz''}}.

Thus, identifying 0-forms and 3-forms with scalar fields, and 1-forms and 2-forms with vector fields:
* grad takes a scalar field (0-form) to a vector field (1-form);
* curl takes a vector field (1-form) to a pseudovector field (2-form);
* div takes a pseudovector field (2-form) to a pseudoscalar field (3-form)

On the other hand, the fact that {{math|1=''d''{{isup|2}} = 0}} corresponds to the identities
<math display="block">\nabla\times(\nabla f) = \mathbf 0</math>
for any scalar field {{mvar|f}}, and
<math display="block">\nabla \cdot (\nabla \times\mathbf v)=0</math>
for any vector field {{math|'''v'''}}.

Grad and div generalize to all oriented pseudo-Riemannian manifolds, with the same geometric interpretation, because the spaces of 0-forms and {{math|''n''}}-forms at each point are always 1-dimensional and can be identified with scalar fields, while the spaces of 1-forms and {{math|(''n'' − 1)}}-forms are always fiberwise {{math|''n''}}-dimensional and can be identified with vector fields.

Curl does not generalize in this way to 4 or more dimensions (or down to 2 or fewer dimensions); in 4 dimensions the dimensions are
{{block indent |text = 0 → 1 → 4 → 6 → 4 → 1 → 0;}}
so the curl of a 1-vector field (fiberwise 4-dimensional) is a ''2-vector field'', which at each point belongs to 6-dimensional vector space, and so one has
<math display="block">\omega^{(2)}=\sum_{i<k=1,2,3,4}a_{i,k}\,dx_i\wedge dx_k,</math>
which yields a sum of six independent terms, and cannot be identified with a 1-vector field. Nor can one meaningfully go from a 1-vector field to a 2-vector field to a 3-vector field (4 → 6 → 4), as taking the differential twice yields zero ({{math|1=''d''{{isup|2}} = 0}}). Thus there is no curl function from vector fields to vector fields in other dimensions arising in this way.

However, one can define a curl of a vector field as a ''2-vector field'' in general, as described below.

=== Curl geometrically ===
2-vectors correspond to the exterior power {{math|Λ<sup>2</sup>''V''}}; in the presence of an inner product, in coordinates these are the skew-symmetric matrices, which are geometrically considered as the [[special orthogonal Lie algebra]] {{math|<math>\mathfrak{so}</math>(''V'')}} of infinitesimal rotations. This has {{math|1=<big><big>(</big></big>{{su|p=''n''|b=2}}<big><big>)</big></big> = {{sfrac|1|2}}''n''(''n'' − 1)}} dimensions, and allows one to interpret the differential of a 1-vector field as its infinitesimal rotations. Only in 3 dimensions (or trivially in 0 dimensions) we have {{math|1=''n'' = {{sfrac|1|2}}''n''(''n'' − 1)}}, which is the most elegant and common case. In 2 dimensions the curl of a vector field is not a vector field but a function, as 2-dimensional rotations are given by an angle (a scalar&nbsp;– an orientation is required to choose whether one counts clockwise or counterclockwise rotations as positive); this is not the div, but is rather perpendicular to it. In 3 dimensions the curl of a vector field is a vector field as is familiar (in 1 and 0 dimensions the curl of a vector field is 0, because there are no non-trivial 2-vectors), while in 4 dimensions the curl of a vector field is, geometrically, at each point an element of the 6-dimensional Lie algebra {{nowrap|<math>\mathfrak{so}(4)</math>.}}

The curl of a 3-dimensional vector field which only depends on 2 coordinates (say {{math|''x''}} and {{math|''y''}}) is simply a vertical vector field (in the {{math|''z''}} direction) whose magnitude is the curl of the 2-dimensional vector field, as in the examples on this page.

Considering curl as a 2-vector field (an antisymmetric 2-tensor) has been used to generalize vector calculus and associated physics to higher dimensions.<ref>{{cite arXiv| last1=McDavid|first1=A. W.| last2=McMullen|first2=C. D.| date=2006-10-30 | title=Generalizing Cross Products and Maxwell's Equations to Universal Extra Dimensions| eprint=hep-ph/0609260}}</ref>

==Inverse==

{{Main|Helmholtz decomposition}}

In the case where the divergence of a vector field {{math|'''V'''}} is zero, a vector field {{math|'''W'''}} exists such that {{math|1='''V''' = curl('''W''')}}.{{citation needed|date=April 2020}} This is why the [[magnetic field]], characterized by zero divergence, can be expressed as the curl of a [[magnetic vector potential]].

If {{math|'''W'''}} is a vector field with {{math|1=curl('''W''') = '''V'''}}, then adding any gradient vector field {{math|grad(''f'')}} to {{math|'''W'''}} will result in another vector field {{math|'''W''' + grad(''f'')}} such that {{math|1=curl('''W''' + grad(''f'')) = '''V'''}} as well. This can be summarized by saying that the inverse curl of a three-dimensional vector field can be obtained up to an unknown [[irrotational field]] with the [[Biot–Savart law]].

== See also ==
*[[Helmholtz decomposition]]
*[[Hiptmair–Xu preconditioner]]
*[[Del in cylindrical and spherical coordinates]]
*[[Vorticity]]
{{Clear}}

==References==
{{Reflist}}

==Further reading==
* {{cite book |author=Korn, Granino Arthur and [[Theresa M. Korn]] |title=Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review |url=https://archive.org/details/mathematicalhand00korn_849/page/n173/mode/2up |url-access=limited |date=January 2000 |publisher=Dover Publications |location=New York |pages=157–160 |isbn=0-486-41147-8 }}
* {{cite book |title=Div, Grad, Curl, and All That: An Informal Text on Vector Calculus |first=H. M. |last=Schey |year=1997 |location=New York |publisher=Norton |isbn=0-393-96997-5 }}

== External links ==
* {{springer|title=Curl|id=p/c027310}}
* {{cite web | title = Multivariable calculus | url = https://mathinsight.org/thread/multivar | website = mathinsight.org | access-date = February 12, 2022}}
* {{cite web |title=Divergence and Curl: The Language of Maxwell's Equations, Fluid Flow, and More |date=June 21, 2018 |url=https://www.youtube.com/watch?v=rB83DpBJQsE | archive-url=https://ghostarchive.org/varchive/youtube/20211124/rB83DpBJQsE| archive-date=2021-11-24 | url-status=live|via=[[YouTube]] }}{{cbignore}}

{{Calculus topics}}

[[Category:Differential operators]]
[[Category:Linear operators in calculus]]
[[Category:Vector calculus]]
[[Category:Analytic geometry]]