Editing Differential form (section)

== Applications in physics ==
<!-- This section is linked from [[Maxwell's equations]] -->

Differential forms arise in some important physical contexts. For example, in Maxwell's theory of [[electromagnetism]], the '''Faraday 2-form''', or [[electromagnetic field strength]], is
<math display="block">\mathbf{F} = \frac 1 2 f_{ab}\, dx^a \wedge dx^b\,,</math>
where the {{math|''f''<sub>''ab''</sub>}} are formed from the electromagnetic fields <math>\vec E</math> and <math>\vec B</math>; e.g., {{math|1=''f''<sub>12</sub> = ''E''<sub>''z''</sub>/''c''}}, {{math|1=''f''<sub>23</sub> = −''B''<sub>''z''</sub>}}, or equivalent definitions.

This form is a special case of the [[curvature form]] on the {{math|[[U(1)]]}} [[principal bundle]] on which both electromagnetism and general [[gauge theories]] may be described.  The [[connection form]] for the principal bundle is the vector potential, typically denoted by {{math|'''A'''}}, when represented in some gauge.  One then has
<math display="block">\mathbf{F} = d\mathbf{A}.</math>

The '''current {{math|3}}-form''' is
<math display="block"> \mathbf{J} = \frac 1 6 j^a\, \varepsilon_{abcd}\, dx^b \wedge dx^c \wedge dx^d\,,</math>
where {{math|''j''<sup>''a''</sup>}} are the four components of the current density. (Here it is a matter of convention to write {{math|''F''<sub>''ab''</sub>}} instead of {{math|''f''<sub>''ab''</sub>}}, i.e. to use capital letters, and to write {{math|''J''<sup>''a''</sup>}} instead of {{math|''j''<sup>''a''</sup>}}. However, the vector rsp. tensor components and the above-mentioned forms have different physical dimensions. Moreover, by decision of an international commission of the [[International Union of Pure and Applied Physics]], the magnetic polarization vector has been called <math>\vec J</math> for several decades, and by some publishers {{math|'''J'''}}; i.e., the same name is used for different quantities.)

Using the above-mentioned definitions, [[Maxwell's equations]] can be written very compactly in [[geometrized units]] as
<math display="block">\begin{align}
        d {\mathbf{F}} &= \mathbf{0} \\
  d {\star \mathbf{F}} &= \mathbf{J},
\end{align}</math>
where <math>\star</math> denotes the [[Hodge star]] operator.  Similar considerations describe the geometry of gauge theories in general.

The {{math|2}}-form <math>{\star} \mathbf{F}</math>, which is [[duality (mathematics)|dual]] to the Faraday form, is also called '''Maxwell 2-form'''.

Electromagnetism is an example of a {{math|[[U(1)]]}} [[gauge theory]].  Here the [[Lie group]] is {{math|U(1)}}, the one-dimensional [[unitary group]], which is in particular [[abelian group|abelian]].  There are gauge theories, such as [[Yang–Mills theory]], in which the Lie group is not abelian.  In that case, one gets relations which are similar to those described here. The analog of the field {{math|'''F'''}} in such theories is the curvature form of the connection, which is represented in a gauge by a [[Lie algebra]]-valued one-form {{math|'''A'''}}.  The Yang–Mills field {{math|'''F'''}} is then defined by
<math display="block">\mathbf{F} = d\mathbf{A} + \mathbf{A}\wedge\mathbf{A}.</math>

In the abelian case, such as electromagnetism, {{math|1='''A''' ∧ '''A''' = 0}}, but this does not hold in general.  Likewise the field equations are modified by additional terms involving exterior products of {{math|'''A'''}} and {{math|'''F'''}}, owing to the [[Maurer–Cartan form|structure equations]] of the gauge group.