Form Integration

A differential k-form can be integrated on an n-dimensional manifold. The basic example is an n-form alpha in the open unit ball in R^n. Since alpha is a top-dimensional form, it can be written alpha=fdx_1 ^ ... ^ dx_n and so


where the integral is the Lebesgue integral.

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

1. rho_i has support in U_i and

2. sumrho_i=1.



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


is the same integral in either coordinate chart.

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

 alpha=zdx ^ dy-ydx ^ dz+xdy ^ dz

on the sphere S^2. Because a point has measure zero, it is enough to integrate alpha on S^2-(0,0,1), which can be covered by stereographic projection phi:R^2->S^2-(0,0,1). Since


the pullback map of alpha is

 phi^*(alpha)=4/((1+r^2)^2)dx ^ dy,

the integral of alpha on S^2 is


Note that this computation is done more easily by Stokes' theorem, because dalpha=3dx ^ dy ^ dz.

See also

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.

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