Editing Curl (mathematics) (section)

=== 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>