nLab
integration of differential forms

Idea

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).

Description

See differential form for basic definitions.

Integration of top-dimension pseudoforms (pseudoforms to measures)

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

  $$\hat{\omega }={\omega }_U\wedge \mathrm{d}{x}^1\wedge \cdots \wedge \mathrm{d}{x}^n.$$\hat{\omega} = \omega_U \wedge \mathrm{d}x^1 \wedge \cdots \wedge \mathrm{d}x^n .

  In this situation, it is convenient also to write

  $$\omega ={\omega }_U\mathrm{d}{x}^1\cdots \mathrm{d}{x}^n;$$\omega = \omega_U \mathrm{d}x^1 \cdots \mathrm{d}x^n ;

  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 $R^n$ that we'll also call $U$; so use the latter formula to interpret

  (1)$${\int }_U\omega ={\int }_U{\omega }_U(x^1,\dots ,x^n)\mathrm{d}{x}^1\cdots \mathrm{d}{x}^n,$$\int_U \omega = \int_U \omega_U(x^1,\ldots,x^n)\, \mathrm{d}x^1 \cdots \mathrm{d}x^n ,

  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

  $$\omega =\sum _U{w}_U{\omega }_U,$$\omega = \sum_U w_U \omega_U ,

  where $w_U$ is a weight function defined on $U$ and ${\omega }_U$ is the restriction of $\omega$ to $U$. Then we have

  $${\int }_X\omega =\sum _U{\int }_U{w}_U(x^1,\dots ,x^n){\omega }_U(x^1,\dots ,x^n)\mathrm{d}{x}^1\cdots \mathrm{d}{x}^n,$$\int_X \omega = \sum_U \int_U w_U(x^1,\ldots,x^n) \omega_U(x^1,\ldots,x^n)\, \mathrm{d}x^1 \cdots \mathrm{d}x^n ,

  or more generally,

  $${\int }_E\omega =\sum _U{\int }_U{\chi }_E(x^1,\dots ,x^n)w_U(x^1,\dots ,x^n){\omega }_U(x^1,\dots ,x^n)\mathrm{d}{x}^1\cdots \mathrm{d}{x}^n$$\int_E \omega = \sum_U \int_U \chi_E(x^1,\ldots,x^n) w_U(x^1,\ldots,x^n) \omega_U(x^1,\ldots,x^n)\, \mathrm{d}x^1 \cdots \mathrm{d}x^n

  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:

Theorem

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 $\mid {\omega }_U\mid$ in place of ${\omega }_U$ in (1) and the following equations, but then the map from forms to measures would not be linear.

Measures to pseudoforms

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:

• Given a point $a$, choose one of the two local orientations at $a$.
• Given $n$ linearly independent vectors $(v_1,\dots ,v_n)$ at $a$, develop them into a coordinate system on a neighbourhood $U$ of $a$.
• For sufficiently large natural number $k$, the coordinate cube $C_k$ of points with coordinates in $[0,1/k{]}^n$ exists (lies within $U$).
• Let $L$ be ${\lim }_{k\to \mathrm{\infty }}{k}^n\omega (C_k)$.
• If the coordinate system on $U$ is positively oriented at $a$, then let $\omega (v_1,\dots ,v_n)$ be $L$; if the coordinate system on $U$ is negatively oriented at $a$, then let $\omega (v_1,\dots ,v_n)$ be $-L$.
• Extend the definition to $n$ arbitrary vectors by continuity (which necessarily maps a linearly dependent tuple of vectors to zero).

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 $\omega (v_1,\dots ,v_n)$ 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).

Integration of more general forms

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 $R:U\to X$), 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.

Integration on more general domains

One often sees the definition of integration given for parametrised submanifolds, that is submanifolds where $U$ is an open subset of $R^p$. 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:

$${\int }_{\sum _i{\alpha }_i{U}_i}\omega ≔\sum _i{\alpha }_i{\int }_{U_i}\omega .$$\int_{\sum_i \alpha_i U_i} \omega \coloneqq \sum_i \alpha_i \int_{U_i} \omega .

Properties

• Stokes theorem

Related concepts

• Lie integration

• cohomological integration

References

An exposition with an eye towards applications in physics is section 3 of

• Theodore Frankel, The Geometry of Physics - An Introduction