Editing Differential form (section)

== Operations ==
As well as the addition and multiplication by scalar operations which arise from the vector space structure, there are several other standard operations defined on differential forms. The most important operations are the [[exterior product]] of two differential forms, the [[exterior derivative]] of a single differential form, the [[interior product]] of a differential form and a vector field, the [[Lie derivative]] of a differential form with respect to a vector field and the [[covariant derivative]] of a differential form with respect to a vector field on a manifold with a defined connection.

=== Exterior product ===
The exterior product of a {{math|''k''}}-form {{math|''α''}} and an {{math|''ℓ''}}-form {{math|''β''}}, denoted {{math|''α'' ∧ ''β''}}, is a ({{math|''k'' + ''ℓ''}})-form. At each point {{math|''p''}} of the manifold {{math|''M''}}, the forms {{math|''α''}} and {{math|''β''}} are elements of an exterior power of the cotangent space at {{math|''p''}}. When the exterior algebra is viewed as a quotient of the tensor algebra, the exterior product corresponds to the tensor product (modulo the equivalence relation defining the exterior algebra).

The antisymmetry inherent in the exterior algebra means that when {{math|''α'' ∧ ''β''}} is viewed as a multilinear functional, it is alternating. However, when the exterior algebra is embedded as a subspace of the tensor algebra by means of the alternation map, the tensor product {{math|''α'' ⊗ ''β''}} is not alternating. There is an explicit formula which describes the exterior product in this situation.  The exterior product is
<math display="block">\alpha \wedge \beta = \operatorname{Alt}(\alpha \otimes \beta).</math>
If the embedding of <math>{\textstyle\bigwedge}^n T^*M</math> into <math>{\bigotimes}^n T^*M</math> is done via the map <math>n!\operatorname{Alt}</math> instead of <math>\operatorname{Alt}</math>, the exterior product is
<math display="block">\alpha \wedge \beta = \frac{(k + \ell)!}{k!\ell!}\operatorname{Alt}(\alpha \otimes \beta).</math>
This description is useful for explicit computations. For example, if {{math|1=''k'' = ''ℓ'' = 1}}, then {{math|''α'' ∧ ''β''}} is the {{math|2}}-form whose value at a point {{math|''p''}} is the [[alternating bilinear form]] defined by
<math display="block"> (\alpha\wedge\beta)_p(v,w)=\alpha_p(v)\beta_p(w) - \alpha_p(w)\beta_p(v)</math>
for {{math|''v'', ''w'' ∈ T<sub>''p''</sub>''M''}}.

The exterior product is bilinear: If {{math|''α''}}, {{math|''β''}}, and {{math|''γ''}} are any differential forms, and if {{math|''f''}} is any smooth function, then
<math display="block">\alpha \wedge (\beta + \gamma) = \alpha \wedge \beta + \alpha \wedge \gamma,</math>
<math display="block">\alpha \wedge (f \cdot \beta) = f \cdot (\alpha \wedge \beta).</math>

It is ''skew commutative'' (also known as ''graded commutative''), meaning that it satisfies a variant of [[anticommutativity]] that depends on the degrees of the forms: if {{math|''α''}} is a {{math|''k''}}-form and {{math|''β''}} is an {{math|''ℓ''}}-form, then
<math display="block">\alpha \wedge \beta = (-1)^{k\ell} \beta \wedge \alpha .</math>
One also has the [[Differential graded algebra|graded Leibniz rule]]:<blockquote><math>d(\alpha\wedge\beta)=d\alpha\wedge\beta + (-1)^{k}\alpha\wedge d\beta.</math></blockquote>

=== Riemannian manifold ===
On a [[Riemannian manifold]], or more generally a [[pseudo-Riemannian manifold]], the metric defines a fibre-wise isomorphism of the tangent and cotangent bundles.  This makes it possible to convert vector fields to covector fields and vice versa.  It also enables the definition of additional operations such as the [[Hodge star operator]] <math>\star \colon \Omega^k(M)\ \stackrel{\sim}{\to}\ \Omega^{n-k}(M)</math> and the [[Hodge star#Codifferential|codifferential]] <math>\delta\colon \Omega^k(M)\rightarrow \Omega^{k-1}(M)</math>, which has degree {{math|−1}} and is [[Differential operator#Adjoint of an operator|adjoint]] to the exterior differential {{math|''d''}}.

==== Vector field structures ====
On a pseudo-Riemannian manifold, {{math|1}}-forms can be identified with vector fields; vector fields have additional distinct algebraic structures, which are listed here for context and to avoid confusion.

Firstly, each (co)tangent space generates a [[Clifford algebra]], where the product of a (co)vector with itself is given by the value of a quadratic form – in this case, the natural one induced by the [[metric tensor|metric]]. This algebra is ''distinct'' from the [[exterior algebra]] of differential forms, which can be viewed as a Clifford algebra where the quadratic form vanishes (since the exterior product of any vector with itself is zero). Clifford algebras are thus non-anticommutative ("quantum") deformations of the exterior algebra. They are studied in [[geometric algebra]].

Another alternative is to consider vector fields as derivations. The (noncommutative) algebra of [[differential operator]]s they generate is the [[Weyl algebra]] and is a noncommutative ("quantum") deformation of the ''symmetric'' algebra in the vector fields.

=== Exterior differential complex ===
One important property of the exterior derivative is that {{math|1=''d''{{i sup|2}} = 0}}. This means that the exterior derivative defines a [[cochain complex]]:
<math display="block">0\ \to\ \Omega^0(M)\ \stackrel{d}{\to}\ \Omega^1(M)\ \stackrel{d}{\to}\ \Omega^2(M)\ \stackrel{d}{\to}\ \Omega^3(M)\ \to\ \cdots \ \to\ \Omega^n(M)\ \to \ 0.</math>

This complex is called the de Rham complex, and its [[cohomology]] is by definition the [[de Rham cohomology]] of {{math|''M''}}.  By the [[Poincaré lemma]], the de Rham complex is locally [[exact sequence|exact]] except at {{math|Ω<sup>0</sup>(''M'')}}.  The kernel at {{math|Ω<sup>0</sup>(''M'')}} is the space of [[locally constant function]]s on {{math|''M''}}.  Therefore, the complex is a resolution of the constant [[sheaf (mathematics)|sheaf]] {{math|{{underline|'''R'''}}}}, which in turn implies a form of de Rham's theorem: de Rham cohomology computes the [[sheaf cohomology]] of {{math|{{underline|'''R'''}}}}.