Editing Curl (mathematics) (section)

== 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>