Editing Differential form (section)

=== 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>