Editing Differential form (section)

== Intrinsic definitions ==
{{see also|Exterior algebra}}

Let {{math|''M''}} be a [[smooth manifold]]. A smooth differential form of degree {{math|''k''}} is a [[section (fiber bundle)|smooth section]] of the {{math|''k''}}th [[exterior algebra|exterior power]] of the [[cotangent bundle]] of {{math|''M''}}.  The set of all differential {{math|''k''}}-forms on a manifold {{math|''M''}} is a [[vector space]], often denoted <math>\Omega^k(M)</math>.

The definition of a differential form may be restated as follows.  At any point <math>p\in M</math>, a {{math|''k''}}-form <math>\beta</math> defines an element
<math display="block"> \beta_p \in {\textstyle\bigwedge}^k T_p^* M,</math>
where <math>T_pM</math> is the [[tangent space]] to {{math|''M''}} at {{math|''p''}} and <math>T^*_p(M)</math> is its [[dual space]].  This space is {{Clarify|date=December 2021|reason=Please specify the isomorphism. The linked MO answer does not give the isomorphism explicitly and is unclear for non-specialists. The definition of the isomorphism needs to be clear to be able to chose between two commonly used definitions of the wedge product.|text=[[natural isomorphism|naturally isomorphic]]<ref>{{Cite web|url=https://mathoverflow.net/q/68033|title = Linear algebra – "Natural" pairings between exterior powers of a vector space and its dual}}</ref>}} to the fiber at {{math|''p''}} of the dual bundle of the {{math|''k''}}th exterior power of the [[tangent bundle]] of {{math|''M''}}. That is, <math>\beta</math> is also a linear functional <math display="inline">\beta_p \colon {\textstyle\bigwedge}^k T_pM \to \mathbf{R}</math>, i.e. the dual of the {{math|''k''}}th exterior power is isomorphic to the {{math|''k''}}th exterior power of the dual:
<math display="block">{\textstyle\bigwedge}^k T^*_p M \cong \Big({\textstyle\bigwedge}^k T_p M\Big)^*</math>

By the universal property of exterior powers, this is equivalently an [[alternating form|alternating]] [[multilinear map]]:
<math display="block">\beta_p\colon \bigoplus_{n=1}^k T_p M \to \mathbf{R}.</math>
Consequently, a differential {{math|''k''}}-form may be evaluated against any {{math|''k''}}-tuple of tangent vectors to the same point {{math|''p''}} of {{math|''M''}}.  For example, a differential {{math|1}}-form {{math|''α''}} assigns to each point <math>p\in M</math> a [[linear functional]] {{math|''α''<sub>''p''</sub>}} on <math>T_pM</math>. In the presence of an [[inner product]] on <math>T_pM</math> (induced by a [[Riemannian metric]] on {{math|''M''}}), {{math|''α''<sub>''p''</sub>}} may be [[Riesz representation theorem|represented]] as the inner product with a [[tangent vector]] <math>X_p</math>. Differential {{math|1}}-forms are sometimes called [[covariance and contravariance of vectors|covariant vector fields]], covector fields, or "dual vector fields", particularly within physics.

The exterior algebra may be embedded in the tensor algebra by means of the alternation map. The alternation map is defined as a mapping  
<math display="block">\operatorname{Alt} \colon {\bigotimes}^k T^*M \to {\bigotimes}^k T^*M.</math>
For a tensor <math>\tau</math> at a point {{math|''p''}},
<math display="block">\operatorname{Alt}(\tau_p)(x_1, \dots, x_k) = \frac{1}{k!}\sum_{\sigma \in S_k} \sgn(\sigma) \tau_p(x_{\sigma(1)}, \dots, x_{\sigma(k)}),</math>
where {{math|''S''<sub>''k''</sub>}} is the [[symmetric group]] on {{math|''k''}} elements.  The alternation map is constant on the cosets of the ideal in the tensor algebra generated by the symmetric 2-forms, and therefore descends to an embedding
<math display="block">\operatorname{Alt} \colon {\textstyle\bigwedge}^k T^*M \to {\bigotimes}^k T^*M.</math>

This map exhibits <math>\beta</math> as a [[antisymmetric tensor|totally antisymmetric]] [[covariance and contravariance of vectors|covariant]] [[tensor field]] of rank {{math|''k''}}. The differential forms on {{math|''M''}} are in one-to-one correspondence with such tensor fields.