Editing Differential form (section)

=== Integration over chains ===
Let {{math|''M''}} be an {{math|''n''}}-manifold and {{math|''ω''}} an {{math|''n''}}-form on {{math|''M''}}.  First, assume that there is a parametrization of {{math|''M''}} by an open subset of Euclidean space.  That is, assume that there exists a diffeomorphism
<math display="block">\varphi \colon D \to M</math>
where {{math|''D'' ⊆ '''R'''<sup>''n''</sup>}}.  Give {{math|''M''}} the orientation induced by {{math|''φ''}}.  Then {{Harv|Rudin|1976}} defines the integral of {{math|''ω''}} over {{math|''M''}} to be the integral of {{math|''φ''<sup>∗</sup>''ω''}} over {{math|''D''}}.  In coordinates, this has the following expression.  Fix an embedding of {{math|''M''}} in {{math|'''R'''<sup>''I''</sup>}} with coordinates {{math|''x''<sup>1</sup>, ..., ''x''<sup>''I''</sup>}}.  Then
<math display="block">\omega = \sum_{i_1 < \cdots < i_n} a_{i_1,\ldots,i_n}({\mathbf x})\,dx^{i_1} \wedge \cdots \wedge dx^{i_n}.</math>
Suppose that {{math|''φ''}} is defined by
<math display="block">\varphi({\mathbf u}) = (x^1({\mathbf u}),\ldots,x^I({\mathbf u})).</math>
Then the integral may be written in coordinates as
<math display="block">\int_M \omega = \int_D \sum_{i_1 < \cdots < i_n} a_{i_1,\ldots,i_n}(\varphi({\mathbf u})) \frac{\partial(x^{i_1},\ldots,x^{i_n})}{\partial(u^{1},\dots,u^{n})}\,du^1 \cdots du^n,</math>
where
<math display="block">\frac{\partial(x^{i_1},\ldots,x^{i_n})}{\partial(u^{1},\ldots,u^{n})}</math>
is the determinant of the [[Jacobian matrix and determinant|Jacobian]].  The Jacobian exists because {{math|''φ''}} is differentiable.

In general, an {{math|''n''}}-manifold cannot be parametrized by an open subset of {{math|'''R'''<sup>''n''</sup>}}.  But such a parametrization is always possible locally, so it is possible to define integrals over arbitrary manifolds by defining them as sums of integrals over collections of local parametrizations.  Moreover, it is also possible to define parametrizations of {{math|''k''}}-dimensional subsets for {{math|''k'' &lt; ''n''}}, and this makes it possible to define integrals of {{math|''k''}}-forms.  To make this precise, it is convenient to fix a standard domain {{math|''D''}} in {{math|'''R'''<sup>''k''</sup>}}, usually a cube or a simplex.  A {{math|''k''}}-'''chain''' is a formal sum of smooth embeddings {{math|''D'' → ''M''}}.  That is, it is a collection of smooth embeddings, each of which is assigned an integer multiplicity.  Each smooth embedding determines a {{math|''k''}}-dimensional submanifold of {{math|''M''}}.  If the chain is
<math display="block">c = \sum_{i=1}^r m_i \varphi_i,</math>
then the integral of a {{math|''k''}}-form {{math|''ω''}} over {{math|''c''}} is defined to be the sum of the integrals over the terms of {{math|''c''}}:
<math display="block">\int_c \omega = \sum_{i=1}^r m_i \int_D \varphi_i^*\omega.</math>

This approach to defining integration does not assign a direct meaning to integration over the whole manifold {{math|''M''}}.  However, it is still possible to assign such a meaning indirectly because every smooth manifold may be smoothly [[Triangulation (topology)|triangulated]] in an essentially unique way, and the integral over {{math|''M''}} may be defined to be the integral over the chain determined by a triangulation.