Knowlpedia

Definition

Integration of differential forms

Integration assigns a real number to a compactly supported top-degree form on an oriented manifold.

Definition

Let MM be an nn-dimensional . The integral of a ω\omega is the number obtained by choosing an orientation-preserving atlas and a (ρi)(\rho_i), writing

(xi−1)∗(ρiω)=fi dx1∧⋯∧dxn,(x_i^{-1})^*(\rho_i\omega)=f_i\,dx^1\wedge\cdots\wedge dx^n,

and setting

∫Mω=∑i∫xi(Ui)fi(x) dx.\int_M\omega=\sum_i\int_{x_i(U_i)}f_i(x)\,dx.

Compact support makes only finitely many terms relevant after a suitable locally finite choice, and the change-of-variables theorem makes the result independent of all choices.

Basic properties

Integration is linear. Reversing the orientation of MM changes the sign of every integral. If F:M→NF:M\to N is an orientation-preserving of oriented nn-manifolds and ω\omega is compactly supported on NN, then

∫MF∗ω=∫Nω.\int_M F^*\omega=\int_N\omega.

This change-of-variables identity is the coordinate independence built into the definition Lee, chapter on integration.

Forms of lower degree

A kk-form is not integrated over all of an nn-manifold when k≠nk\ne n. Instead, if SS is an oriented kk-manifold and f:S→Mf:S\to M is a smooth parametrization for which the integral is defined, one sets

∫fω:=∫Sf∗ω.\int_f\omega:=\int_S f^*\omega.

This formulation includes integration over oriented submanifolds and smooth singular simplices.

Stokes' theorem and cohomology

The compatibility between integration and the is :

∫Mdη=∫∂Mη\int_M d\eta=\int_{\partial M}\eta

with the induced and appropriate compact-support hypotheses. Consequently, integrals of closed forms over cycles depend only on cohomology and homology classes; this pairing underlies the .

References
  1. John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. DOI record. Relevant: the chapters on orientations, integration on manifolds, and Stokes' theorem.
  2. Loring W. Tu, An Introduction to Manifolds, 2nd ed., Springer, 2011. DOI record. Relevant: the chapter on integration.