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# Form Integration

A differential k-form can be integrated on an -dimensional manifold. The basic example is an -form in the open unit ball in . Since is a top-dimensional form, it can be written and so

 (1)

where the integral is the Lebesgue integral.

On a manifold covered by coordinate charts , there is a partition of unity such that

1. has support in and

2. .

Then

 (2)

where the right-hand side is well-defined because each integration takes place in a coordinate chart. The integral of the -form is well-defined because, under a change of coordinates , the integral transforms according to the determinant of the Jacobian, while an -form pulls back by the determinant of the Jacobian. Hence,

 (3)

is the same integral in either coordinate chart.

For example, it is possible to integrate the 2-form

 (4)

on the sphere . Because a point has measure zero, it is enough to integrate on , which can be covered by stereographic projection . Since

 (5)

the pullback map of is

 (6)

the integral of on is

 (7)

Note that this computation is done more easily by Stokes' theorem, because .

de Rham Cohomology, Stokes' Theorem, Submanifold, Top-Dimensional Form, Volume Form

This entry contributed by Todd Rowland

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## Cite this as:

Rowland, Todd. "Form Integration." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/FormIntegration.html