Editing Differential form (section)

== Integration ==
A differential {{math|''k''}}-form can be integrated over an oriented {{math|''k''}}-dimensional manifold.  When the {{math|''k''}}-form is defined on an {{math|''n''}}-dimensional manifold with {{math|''n'' &gt; ''k''}}, then the {{math|''k''}}-form can be integrated over oriented {{math|''k''}}-dimensional submanifolds.  If {{math|1=''k'' = 0}}, integration over oriented 0-dimensional submanifolds is just the summation of the integrand evaluated at points, according to the orientation of those points. Other values of {{math|1=''k'' = 1, 2, 3, ...}} correspond to line integrals, surface integrals, volume integrals, and so on.  There are several equivalent ways to formally define the integral of a differential form, all of which depend on reducing to the case of Euclidean space.

=== Integration on Euclidean space ===
Let {{math|''U''}} be an open subset of {{math|'''R'''<sup>''n''</sup>}}.  Give {{math|'''R'''<sup>''n''</sup>}} its standard orientation and {{math|''U''}} the restriction of that orientation.  Every smooth {{math|''n''}}-form {{math|''ω''}} on {{math|''U''}} has the form
<math display="block">\omega = f(x)\,dx^1 \wedge \cdots \wedge dx^n</math>
for some smooth function {{math|''f'' : '''R'''<sup>''n''</sup> → '''R'''}}.  Such a function has an integral in the usual Riemann or Lebesgue sense.  This allows us to define the integral of {{math|''ω''}} to be the integral of {{math|''f''}}:
<math display="block">\int_U \omega\ \stackrel{\text{def}}{=} \int_U f(x)\,dx^1 \cdots dx^n.</math>
Fixing an orientation is necessary for this to be well-defined.  The skew-symmetry of differential forms means that the integral of, say, {{math|''dx''<sup>1</sup> ∧ ''dx''<sup>2</sup>}} must be the negative of the integral of {{math|''dx''<sup>2</sup> ∧ ''dx''<sup>1</sup>}}.  Riemann and Lebesgue integrals cannot see this dependence on the ordering of the coordinates, so they leave the sign of the integral undetermined.  The orientation resolves this ambiguity.

=== 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.

=== Integration using partitions of unity ===
There is another approach, expounded in {{Harv|Dieudonné|1972}}, which does directly assign a meaning to integration over {{math|''M''}}, but this approach requires fixing an orientation of {{math|''M''}}.  The integral of an {{math|''n''}}-form {{math|''ω''}} on an {{math|''n''}}-dimensional manifold is defined by working in charts.  Suppose first that {{math|''ω''}} is supported on a single positively oriented chart.  On this chart, it may be pulled back to an {{math|''n''}}-form on an open subset of {{math|'''R'''<sup>''n''</sup>}}.  Here, the form has a well-defined Riemann or Lebesgue integral as before.  The change of variables formula and the assumption that the chart is positively oriented together ensure that the integral of {{math|''ω''}} is independent of the chosen chart.  In the general case, use a partition of unity to write {{math|''ω''}} as a sum of {{math|''n''}}-forms, each of which is supported in a single positively oriented chart, and define the integral of {{math|''ω''}} to be the sum of the integrals of each term in the partition of unity.

It is also possible to integrate {{math|''k''}}-forms on oriented {{math|''k''}}-dimensional submanifolds using this more intrinsic approach.  The form is pulled back to the submanifold, where the integral is defined using charts as before.  For example, given a path {{math|''γ''(''t'') : [0, 1] → '''R'''<sup>2</sup>}}, integrating a {{math|1}}-form on the path is simply pulling back the form to a form {{math|''f''(''t''){{thin space}}''dt''}} on {{math|[0, 1]}}, and this integral is the integral of the function {{math|''f''(''t'')}} on the interval.

=== Integration along fibers ===
{{main|Integration along fibers}}

[[Fubini's theorem]] states that the integral over a set that is a product may be computed as an iterated integral over the two factors in the product.  This suggests that the integral of a differential form over a product ought to be computable as an iterated integral as well.  The geometric flexibility of differential forms ensures that this is possible not just for products, but in more general situations as well.  Under some hypotheses, it is possible to integrate along the fibers of a smooth map, and the analog of Fubini's theorem is the case where this map is the projection from a product to one of its factors.

Because integrating a differential form over a submanifold requires fixing an orientation, a prerequisite to integration along fibers is the existence of a well-defined orientation on those fibers.  Let {{math|''M''}} and {{math|''N''}} be two orientable manifolds of pure dimensions {{math|''m''}} and {{math|''n''}}, respectively.  Suppose that {{math|''f'' : ''M'' → ''N''}} is a surjective submersion.  This implies that each fiber {{math|''f''{{i sup|&minus;1}}(''y'')}} is {{math|(''m'' &minus; ''n'')}}-dimensional and that, around each point of {{math|''M''}}, there is a chart on which {{math|''f''}} looks like the projection from a product onto one of its factors.  Fix {{math|''x'' ∈ ''M''}} and set {{math|1=''y'' = ''f''(''x'')}}.  Suppose that
<math display="block">\begin{align}
\omega_x &\in {\textstyle\bigwedge}^m T_x^*M, \\[2pt]
\eta_y &\in {\textstyle\bigwedge}^n T_y^*N,
\end{align}</math>
and that {{math|''η''<sub>''y''</sub>}} does not vanish.  Following {{Harv|Dieudonné|1972}}, there is a unique
<math display="block">\sigma_x \in {\textstyle\bigwedge}^{m-n} T_x^*(f^{-1}(y))</math>
which may be thought of as the fibral part of {{math|''ω''<sub>''x''</sub>}} with respect to {{math|''η''<sub>''y''</sub>}}.  More precisely, define {{math|''j'' : ''f''{{i sup|&minus;1}}(''y'') → ''M''}} to be the inclusion.  Then {{math|''σ''<sub>''x''</sub>}} is defined by the property that
<math display="block">\omega_x = (f^*\eta_y)_x \wedge \sigma'_x \in {\textstyle\bigwedge}^m T_x^*M,</math>
where
<math display="block">\sigma'_x \in {\textstyle\bigwedge}^{m-n} T_x^*M</math>
is any {{math|(''m'' &minus; ''n'')}}-covector for which
<math display="block">\sigma_x = j^*\sigma'_x.</math>
The form {{math|''σ''<sub>''x''</sub>}} may also be notated {{math|''ω''<sub>''x''</sub> / ''η''<sub>''y''</sub>}}.

Moreover, for fixed {{math|''y''}}, {{math|''σ''<sub>''x''</sub>}} varies smoothly with respect to {{math|''x''}}.  That is, suppose that
<math display="block">\omega \colon f^{-1}(y) \to T^*M</math>
is a smooth section of the projection map; we say that {{math|''ω''}} is a smooth differential {{math|''m''}}-form on {{math|''M''}} along {{math|''f''{{i sup|&minus;1}}(''y'')}}.  Then there is a smooth differential {{math|(''m'' &minus; ''n'')}}-form {{math|''σ''}} on {{math|''f''{{i sup|&minus;1}}(''y'')}} such that, at each {{math|''x'' ∈ ''f''{{i sup|&minus;1}}(''y'')}},
<math display="block">\sigma_x = \omega_x / \eta_y.</math>
This form is denoted {{math|''ω'' / ''η''<sub>''y''</sub>}}.  The same construction works if {{math|''ω''}} is an {{math|''m''}}-form in a neighborhood of the fiber, and the same notation is used.  A consequence is that each fiber {{math|''f''{{i sup|&minus;1}}(''y'')}} is orientable.  In particular, a choice of orientation forms on {{math|''M''}} and {{math|''N''}} defines an orientation of every fiber of {{math|''f''}}.

The analog of Fubini's theorem is as follows.  As before, {{math|''M''}} and {{math|''N''}} are two orientable manifolds of pure dimensions {{math|''m''}} and {{math|''n''}}, and {{math|''f'' : ''M'' → ''N''}} is a surjective submersion.  Fix orientations of {{math|''M''}} and {{math|''N''}}, and give each fiber of {{math|''f''}} the induced orientation.  Let {{math|''ω''}} be an {{math|''m''}}-form on {{math|''M''}}, and let {{math|''η''}} be an {{math|''n''}}-form on {{math|''N''}} that is almost everywhere positive with respect to the orientation of {{math|''N''}}.  Then, for almost every {{math|''y'' ∈ ''N''}}, the form {{math|''ω'' / ''η''<sub>''y''</sub>}} is a well-defined integrable {{math|''m'' &minus; ''n''}} form on {{math|''f''{{i sup|&minus;1}}(''y'')}}.  Moreover, there is an integrable {{math|''n''}}-form on {{math|''N''}} defined by
<math display="block">y \mapsto \bigg(\int_{f^{-1}(y)} \omega / \eta_y\bigg)\,\eta_y.</math>
Denote this form by
<math display="block">\bigg(\int_{f^{-1}(y)} \omega / \eta\bigg)\,\eta.</math>
Then {{Harv|Dieudonné|1972}} proves the generalized Fubini formula
<math display="block">\int_M \omega = \int_N \bigg(\int_{f^{-1}(y)} \omega / \eta\bigg)\,\eta.</math>

It is also possible to integrate forms of other degrees along the fibers of a submersion.  Assume the same hypotheses as before, and let {{math|''α''}} be a compactly supported {{math|(''m'' &minus; ''n'' + ''k'')}}-form on {{math|''M''}}.  Then there is a {{math|''k''}}-form {{math|''γ''}} on {{math|''N''}} which is the result of integrating {{math|''α''}} along the fibers of {{math|''f''}}.  The form {{math|''α''}} is defined by specifying, at each {{math|''y'' ∈ ''N''}}, how {{math|''γ''}} pairs with each {{math|''k''}}-vector {{math|'''v'''}} at {{math|''y''}}, and the value of that pairing is an integral over {{math|''f''{{i sup|&minus;1}}(''y'')}} that depends only on {{math|''α''}}, {{math|'''v'''}}, and the orientations of {{math|''M''}} and {{math|''N''}}.  More precisely, at each {{math|''y'' ∈ ''N''}}, there is an isomorphism
<math display="block">{\textstyle\bigwedge}^k T_yN \to {\textstyle\bigwedge}^{n-k} T_y^*N</math>
defined by the interior product
<math display="block">\mathbf{v} \mapsto \mathbf{v}\,\lrcorner\,\zeta_y,</math>
for any choice of volume form {{math|''ζ''}} in the orientation of {{math|''N''}}. If {{math|''x'' ∈ ''f''{{i sup|&minus;1}}(''y'')}}, then a {{math|''k''}}-vector {{math|'''v'''}} at {{math|''y''}} determines an {{math|(''n'' &minus; ''k'')}}-covector at {{math|''x''}} by pullback:
<math display="block">f^*(\mathbf{v}\,\lrcorner\,\zeta_y) \in {\textstyle\bigwedge}^{n-k} T_x^*M.</math>
Each of these covectors has an exterior product against {{math|''α''}}, so there is an {{math|(''m'' &minus; ''n'')}}-form {{math|''β''<sub>'''v'''</sub>}} on {{math|''M''}} along {{math|''f''{{i sup|&minus;1}}(''y'')}} defined by
<math display="block">(\beta_{\mathbf{v}})_x = \left(\alpha_x \wedge f^*(\mathbf{v}\,\lrcorner\,\zeta_y)\right) \big/ \zeta_y \in {\textstyle\bigwedge}^{m-n} T_x^*M.</math>
This form depends on the orientation of {{math|''N''}} but not the choice of {{math|''ζ''}}.  Then the {{math|''k''}}-form {{math|''γ''}} is uniquely defined by the property
<math display="block">\langle\gamma_y, \mathbf{v}\rangle = \int_{f^{-1}(y)} \beta_{\mathbf{v}},</math>
and {{math|''γ''}} is smooth {{Harv|Dieudonné|1972}}.  This form also denoted {{math|''α''<sup>♭</sup>}} and called the '''integral of {{math|''α''}} along the fibers of {{math|''f''}}'''.  Integration along fibers is important for the construction of Gysin maps in de Rham cohomology.

Integration along fibers satisfies the '''projection formula''' {{Harv|Dieudonné|1972}}.  If {{math|''λ''}} is any {{math|''ℓ''}}-form on {{math|''N''}}, then
<math display="block">\alpha^\flat \wedge \lambda = (\alpha \wedge f^*\lambda)^\flat.</math>

===Stokes's theorem===
{{main|Stokes's theorem}}
The fundamental relationship between the exterior derivative and integration is given by the [[Stokes' theorem]]: If {{math|''ω''}} is an ({{math|''n'' − 1}})-form with compact support on {{math|''M''}} and {{math|''∂M''}} denotes the [[manifold#Manifold with boundary|boundary]] of {{math|''M''}} with its induced [[Orientation (mathematics)|orientation]], then
<math display="block">\int_M d\omega = \int_{\partial M} \omega.</math>

A key consequence of this is that "the integral of a closed form over homologous chains is equal": If {{math|''ω''}} is a closed {{math|''k''}}-form and {{math|''M''}} and {{math|''N''}} are {{math|''k''}}-chains that are homologous (such that {{math|''M'' − ''N''}} is the boundary of a {{math|(''k'' + 1)}}-chain {{math|''W''}}), then <math>\textstyle{\int_M \omega = \int_N \omega}</math>, since the difference is the integral <math>\textstyle\int_W d\omega = \int_W 0 = 0</math>.

For example, if {{math|1=''ω'' = ''df''}} is the derivative of a potential function on the plane or {{math|'''R'''<sup>''n''</sup>}}, then the integral of {{math|''ω''}} over a path from {{math|''a''}} to {{math|''b''}} does not depend on the choice of path (the integral is {{math|''f''(''b'') − ''f''(''a'')}}), since different paths with given endpoints are [[homotopic]], hence homologous (a weaker condition). This case is called the [[gradient theorem]], and generalizes the [[fundamental theorem of calculus]]. This path independence is very useful in [[contour integration]].

This theorem also underlies the duality between [[de Rham cohomology]] and the [[homology (mathematics)|homology]] of chains.

=== Relation with measures ===
{{details|Density on a manifold}}

On a ''general'' differentiable manifold (without additional structure), differential forms ''cannot'' be integrated over subsets of the manifold; this distinction is key to the distinction between differential forms, which are integrated over chains or oriented submanifolds, and measures, which are integrated over subsets. The simplest example is attempting to integrate the {{math|1}}-form {{math|''dx''}} over the interval {{math|[0, 1]}}. Assuming the usual distance (and thus measure) on the real line, this integral is either {{math|1}} or {{math|&minus;1}}, depending on ''orientation'': {{nowrap|<math display="inline"> \int_0^1 dx = 1</math>,}} while {{nowrap|<math display="inline"> \int_1^0 dx = - \int_0^1 dx = -1</math>.}} By contrast, the integral of the ''measure'' {{math|{{abs|''dx''}}}} on the interval is unambiguously {{math|1}} (i.e. the integral of the constant function {{math|1}} with respect to this measure is {{math|1}}). Similarly, under a change of coordinates a differential {{math|''n''}}-form changes by the [[Jacobian determinant]] {{math|''J''}}, while a measure changes by the ''absolute value'' of the Jacobian determinant, {{math|{{abs|''J''}}}}, which further reflects the issue of orientation. For example, under the map {{math|''x'' ↦ −''x''}} on the line, the differential form {{math|''dx''}} pulls back to {{math|−''dx''}}; orientation has reversed; while the [[Lebesgue measure]], which here we denote {{math|{{abs|''dx''}}}}, pulls back to {{math|{{abs|''dx''}}}}; it does not change.

In the presence of the additional data of an ''orientation'', it is possible to integrate {{math|''n''}}-forms (top-dimensional forms) over the entire manifold or over compact subsets; integration over the entire manifold corresponds to integrating the form over the [[fundamental class]] of the manifold, {{math|[''M'']}}. Formally, in the presence of an orientation, one may identify {{math|''n''}}-forms with [[densities on a manifold]]; densities in turn define a measure, and thus can be integrated {{Harv |Folland |1999 |loc = Section 11.4, pp.&nbsp;361&ndash;362}}.

On an orientable but not oriented manifold, there are two choices of orientation; either choice allows one to integrate {{math|''n''}}-forms over compact subsets, with the two choices differing by a sign. On a non-orientable manifold, {{math|''n''}}-forms and densities cannot be identified —notably, any top-dimensional form must vanish somewhere (there are no [[volume form]]s on non-orientable manifolds), but there are nowhere-vanishing densities— thus while one can integrate densities over compact subsets, one cannot integrate {{math|''n''}}-forms. One can instead identify densities with top-dimensional [[Volume form#Relation to measures|pseudoform]]s.

Even in the presence of an orientation, there is in general no meaningful way to integrate {{math|''k''}}-forms over subsets for {{math|''k'' < ''n''}} because there is no consistent way to use the ambient orientation to orient {{math|''k''}}-dimensional subsets.  Geometrically, a {{math|''k''}}-dimensional subset can be turned around in place, yielding the same subset with the opposite orientation; for example, the horizontal axis in a plane can be rotated by 180 degrees.  Compare the [[Gram determinant]] of a set of {{math|''k''}} vectors in an {{math|''n''}}-dimensional space, which, unlike the determinant of {{math|''n''}} vectors, is always positive, corresponding to a squared number.  An orientation of a {{math|''k''}}-submanifold is therefore extra data not derivable from the ambient manifold.

On a Riemannian manifold, one may define a {{math|''k''}}-dimensional [[Hausdorff measure]] for any {{math|''k''}} (integer or real), which may be integrated over {{math|''k''}}-dimensional subsets of the manifold. A function times this Hausdorff measure can then be integrated over {{math|''k''}}-dimensional subsets, providing a measure-theoretic analog to integration of {{math|''k''}}-forms. The {{math|''n''}}-dimensional Hausdorff measure yields a density, as above.

=== Currents ===
The differential form analog of a [[distribution (mathematics)|distribution]] or generalized function is called a '''[[Current (mathematics)|current]]'''.  The space of {{math|''k''}}-currents on {{math|''M''}} is the dual space to an appropriate space of differential {{math|''k''}}-forms.  Currents play the role of generalized domains of integration, similar to but even more flexible than chains.