Integration of Differential Forms

Orientability

Theorem

A smooth -manifold is orientable if and only if there exists an differential -form which is nowhere vanishing on .

Integral of a Differential -Form on

Integral of a Differential -Form on

As in , every differential -form can be represented as where is a smooth function on (See example). The integral of over a region is defined as where the right-hand side is the Lebesgue integral of over .

Definition

Let be an oriented smooth -manifold, and be a differential -form on . The integral of over is defined as where is a smooth chart of , and the right-hand side is the integral of over the region .

Change of Variables Formula

Suppose is a region in a smooth manifold , and is an orientation-preserving diffeomorphism onto its image. Then where is a differential form on .

Stokes' Theorem

Suppose is a smooth manifold with boundary, and is a differential form on . Then

Proposition

For a compact smooth -manifold with boundary , a vector field tangent to the boundary , and an -form . Then

Proof By Cartan’s formula, we have Therefore, where the second equality holds by Stokes’ theorem. Note that is an -form on , so it vanishes, . For any vector fields , fix some , Note that since is tangent to the boundary , is a vector in , which is of dimension . Therefore, must be linearly dependent, and thus on .

Remark

We can also achieve this directly by definition of Lie derivative. Since is compact and the vector field is tangent to the boundary, the associated flow of is global (i.e. exists for all and defines a one-parameter group of diffeomorphisms of onto itself). Without loss of generality, assume each is orientation-preserving. Then by definition of Lie derivative, Since is compact and all objects are smooth, we can interchange the differentiation with respect to and the integration over : Use the change of variables theorem for integration of differential forms, we obtain: Thus, where the last equality holds because is independent of .