Riemann integration, Lebesgue integration
line integral/contour integration
integration of differential forms
integration over supermanifolds, Berezin integral, fermionic path integral
Kontsevich integral, Selberg integral, elliptic Selberg integral
integration in ordinary differential cohomology
integration in differential K-theory
Here we discuss the integration of a differential form (possibly twisted in some way) on a topological manifold (possibly with additional structure) over an appropriately structured submanifold (or formal linear combination thereof).
See at differential form for basic definitions.
Let $X$ be an $n$-dimensional topological manifold, and let $\omega$ be a continuous $n$-pseudoform on $X$. Suppose that $X$ is paracompact and Hausdorff, so that we may find a locally finite cover of $X$ with a subordinate partition of unity and a continuous coordinate chart on each patch. (When $X$ is differentiable, or even smooth, then these may also be chosen to be differentiable or smooth, which may be convenient but is not necessary.) Then $\omega$ defines a measure on $X$ as follows:
On each coordinate patch $U$, fix the orientation given by the coordinates to turn $\omega$ into an untwisted $n$-form $\hat{\omega}$; then write $\hat{\omega}$ in coordinates as
In this situation, it is convenient also to write
in other words, we interpret $\mathrm{d}x^1 \cdots \mathrm{d}x^n$ as the absolute value of $\mathrm{d}x^1 \wedge \cdots \wedge \mathrm{d}x^n$.
The coordinates on $U$ define a diffeomorphism between $U$ and an open subset of $\mathbf{R}^n$ that we'll also call $U$; so use the latter formula to interpret
where the right-hand side is now interpreted in the usual way as as integral with respect to Lebesgue measure.
Using the partition of unity, write
where $w_U$ is a weight function defined on $U$ and $\omega_U$ is the restriction of $\omega$ to $U$. Then we have
or more generally,
for $E$ a measurable subset of $X$ and $\chi_E$ the characteristic function of $E$.
A priori, this definition depends not only on the particular coordinate patches chosen but also on the partition of unity chosen to go with them. Furthermore, the defintion could be done just as easily (perhaps even more easily) for something other than an $n$-pseudoform. But the (perhaps surprising) fact that justifies it all is this:
When $\omega$ is an $n$-pseudoform, the definition of $\int_E \omega$ is independent of the coordinates and partition chosen. Furthermore, the map from $n$-pseudoforms to measures is linear.
Note that, if $\omega$ were an $n$-form instead of a pseudoform, then the definition would depend on the orientation of the coordinates chosen. We could fix that by using the absolute value ${|\omega_U|}$ in place of $\omega_U$ in (1) and the following equations, but then the map from forms to measures would not be linear.
It may also be enlightening to consider how to go back from a measure to an $n$-pseudoform. If $\omega$ is an absolutely continuous Radon measure on $X$, then it defines an $n$-pseudoform (which we may also call $\omega$) as follows:
Again, this definition is independent of the coordinate system chosen (as long as it extends the given vectors); or if that's not true, then we messed up and need to add further restrictions to the absolutely continuous Radon measure $\omega$. The definition is not independent of the orientation chosen, of course; thus we get a pseudoform rather than an untwisted form. You might try to ignore the orientation and take $\omega(v_1,\ldots,v_n)$ to be $L$ always, but that does not define an exterior form, as is most easily seen if two vectors are switched (which does not change $L$). Instead, this would define an absolute differential form (which is equivalent to a pseudoform when, as here, the degree equals the dimension).
One can integrate forms other than $n$-pseudoforms, of course, but only over certain structures within the manifold $X$. Specifically, if $R$ is a $p$-dimensional submanifold of $X$ (that is a $p$-dimensional manifold $U$ equipped with a map $R\colon U \to X$), then we would like to integrate $p$-forms or $p$-pseudoforms (defined on $X$) over $R$. Here is how we do this:
We may integrate a $p$-form $\eta$ over $R$ if $R$ is oriented, that is if $U$ is oriented. We pull back $\eta$ from $X$ to $U$, then use the orientation on $U$ to turn $\eta$ into a $p$-pseudoform, which we can then integrate on the $p$-dimensional manifold $U$.
We may integrate a $p$-pseudoform $\eta$ over $R$ if $R$ is pseudooriented, that is if it is equipped with a map that, for each point $a$ on $U$, takes a local orientation of $X$ at $R(a)$ to a local orientation of $U$ at $a$, continuously in $a$ and taking opposite orientations to opposite orientations. Then locally, we turn $\eta$ into a $p$-form on $X$ using a local orientation on $X$, pull that back to $U$, and use the corresponding local orientation on $U$ to turn that back into a $p$-pseudoform, which we can then integrate on $U$.
Thus, while integration of $n$-pseudoforms is the most basic, integration of general $p$-forms is actually a bit simpler than integration of general $p$-pseudoforms. Integration of other twisted or vector-valued forms can also be done, again given appropriate structure on $R$.
Note that, if $X$ is thought of a submanifold of itself, then it has a natural pseudoorientation that takes each local orientation to itself, and so we recover the original definition of integration of $n$-pseudoforms on $X$. Also, if $X$ is an oriented manifold, then it may be viewed as an oriented submanifold of itself, giving a definition of integration of $n$-forms on $X$.
To integrate on unoriented submanifolds of arbitrary dimension, use absolute differential forms. To integrate nonlinear differential forms, use cogerm differential forms.
One often sees the definition of integration given for parametrised submanifolds, that is submanifolds where $U$ is an open subset of $\mathbf{R}^p$. This amounts to a combination of the concepts above, with the two uses of $U$ (as a coordinate patch in $X$ or as the source of a submanifold of $X$) identified. The theorem that the integral of an $n$-pseudoform on $X$ is independent of the coordinates chosen now becomes a theorem that the integral of a parametrised submanifold is independent of the parametrisation (up to some details about orientation), which in the end returns the result that one can integrate forms over arbitrary submanifolds (given an orientation or pseudoorientation as above).
We can also integrate on a formal linear combination of submanifolds, which is handy since these are the chains of de Rham homology?. This is straightforward:
We discuss here a general abstract formulation of differential forms, their integration and the Stokes theorem in cohesive (∞,1)-toposes, hence the axiomatics of cohesive homotopy type theory (following Bunke-Nikolaus-Völkl 13, theorem 3.2).
Let $\mathbf{H}$ be a cohesive (∞,1)-topos. We write $(\Pi \dashv \flat \dashv \sharp)$ for the defining triple of adjoint modalities (shape modality $\dashv$ flat modality $\dashv$ sharp modality).
As usual (see at structures in a cohesive ∞-topos – de Rham cohomology) we deduce from this the secondary cohesive modalities $\Pi_{dR}$ and $\flat_{dR}$: for a pointed object with loop space object $G$ (for instance the delooping $\mathbf{B}G$) $\flat_{dR}G$ is defined by the long homotopy fiber sequence
where the morphism on the right is the $\flat$-counit. Here $\theta_G$ is the Maurer-Cartan form on $G$.
(Beware that elsewhere we write $\flat_{dR}\mathbf{B}G$ for what here we write $\flat_{dR}G$.)
Dually $\Pi_{dR}$ is defined.
Write now
for tangent cohesive (∞,1)-topos of $\mathbf{H}$, extending it by its stabilization $Stab(\mathbf{H})$ given by spectrum objects in $\mathbf{H}$.
We assume that there is an interval object
“exhibiting the cohesion” (see at continuum) in that there is a (chosen) equivalence between the shape modality $\Pi$ and the localization $L_{\Delta^1}$ at the the projection maps out of Cartesian products with this line $\Delta^1\times (-) \to (-)$
This is the case for instance for the “standard continuum”, the real line in $\mathbf{H} =$ Smooth∞Grpd.
It follows in particular that there is a chosen equivalence of (∞,1)-categories
between the flat modal homotopy-types and the $\Delta^1$-homotopy invariant homotopy-types.
Given a stable homotopy type $\hat E \in Stab(\mathbf{H})\hookrightarrow T \mathbf{H}$ cohesion provides two objects
which may be interpreted as de Rham complexes with coefficients in $\Pi(\flat_{dR} \Sigma \hat E)$, the first one restricted to negative degree, the second to non-negative degree. Moreover, there is a canonical map
which interprets as the de Rham differential $\mathbf{d}$. See at differential cohomology diagram for details.
Throughout in the following we leave the “inclusion” $\iota$ of “differential forms regarded as $\hat E$-connections on trivial $E$-bundles” implicit.
Integration of differential forms is the morphism
which is induced via the homotopy cofiber property of $\flat_{dR}\Omega \hat E$ from the counit naturality square of the flat modality on $[(\ast \coprod \ast \stackrel{(i_0, i_1)}{\to} \Delta ^1 ), -]$, using that this square exhibits a null homotopy due to the $\Delta^1$-homotopy invariance of $\flat \hat E$.
(Bunke-Nikolaus-Völkl 13, (9), (10)
(Bunke-Nikolaus-Völkl 13, theorem 3.2
In $\mathbf{H} =$ Smooth∞Grpd let $\hat E \in Stab(\mathbf{H})$ be given, under the stable Dold-Kan correspondence, by the traditional truncated de Rham complex $\hat E \coloneqq \Omega^{\bullet \geq n }$.
Then on this object the general cohesive integration map of def. 1 reduces on the $-n$th homotopy group to the tradition fiber integration of differential forms as above:
(Bunke-Nikolaus-Völkl 13, lemma 4.2 (7)
An exposition with an eye towards applications in physics is in section 3 of
Discussion in the abstract context of cohesion and differential cohomology is in