Integration on Manifolds — Mathematics Knowledge System

Overview

Integration on manifolds generalises the Riemann and Lebesgue integrals to smooth manifolds. The correct objects to integrate are not functions but differential forms: an n-form on an oriented n-dimensional manifold can be integrated coordinate-independently. The fundamental theorem of calculus generalises to Stokes' theorem, which unifies the classical theorems of Green, Gauss, and Stokes. For Riemannian manifolds, the metric also gives a canonical volume form enabling integration of functions.

Intuition

On a curved surface in R³ you cannot simply 'lay a coordinate grid' and integrate, because coordinates distort area. A differential n-form carries its own notion of signed volume element that transforms correctly under coordinate changes — it is exactly what you need to integrate intrinsically. The wedge product ∧ encodes orientation: swapping two covectors changes the sign, mirroring the signed-area interpretation of the determinant.

Formal Definition

Definition

Let M be a smooth oriented n-manifold (possibly with boundary ∂M). A differential n-form ω ∈ Ω^n(M) on M is integrated as follows.

\int_M \omega = \sum_\alpha \int_{\phi_\alpha(U_\alpha)} (\phi_\alpha^{-1})^* (\rho_\alpha \omega)

Integration via partition of unity {ρ_α} subordinate to atlas {(U_α, φ_α)}

INT-FORM

\int_M d\omega = \int_{\partial M} \omega

Stokes' theorem (ω ∈ Ω^{n-1}(M), M compact oriented n-manifold with boundary)

STOKES

d\text{vol}_g = \sqrt{\det(g_{ij})} \, dx^1 \wedge \cdots \wedge dx^n

Riemannian volume form in local coordinates

VOL-FORM

Notation

Notation	Meaning
$\Omega^k(M)$	Space of smooth k-forms on manifold M
$d : \Omega^k \to \Omega^{k+1}$	Exterior derivative
$\int_M \omega$	Integral of an n-form over an oriented n-manifold M
$d\text{vol}_g$	Riemannian volume form associated to metric g
$\varphi^* \omega$	Pullback of form ω under smooth map φ

Proofs

Well-definedness of ∫_M ω (independent of partition of unity)

$\text{Let } \{\rho_\alpha\}, \{\sigma_\beta\} \text{ be two partitions of unity}$ (Both subordinate to respective atlases)
$\sum_\alpha \int (\phi_\alpha^{-1})^*(\rho_\alpha \omega) = \sum_{\alpha,\beta} \int (\phi_\alpha^{-1})^*(\rho_\alpha \sigma_\beta \omega)$ (Since Σ_β σ_β = 1)
$= \sum_{\beta,\alpha} \int (\psi_\beta^{-1})^*(\sigma_\beta \rho_\alpha \omega)$ (By the change-of-variables formula for forms: pulling back by ψ_β⁻¹ introduces the Jacobian, which equals the transformation rule for n-forms, so the sum is symmetric)
$= \sum_\beta \int (\psi_\beta^{-1})^*(\sigma_\beta \omega)$ (Using Σ_α ρ_α = 1)

Theorems

Theorem 1: Stokes' Theorem

\text{Let } M \text{ be a compact oriented } n\text{-manifold with boundary and } \omega \in \Omega^{n-1}(M). \text{ Then: } \int_M d\omega = \int_{\partial M} \omega

Theorem 2: De Rham's Theorem

\text{Integration of forms induces an isomorphism } H^k_{\text{dR}}(M) \xrightarrow{\sim} H^k(M; \mathbb{R}) \text{ between de Rham and singular cohomology}

Theorem 3: Change of Variables on Manifolds

\text{If } \varphi : M \to N \text{ is an orientation-preserving diffeomorphism and } \omega \in \Omega^n(N), \text{ then } \int_M \varphi^* \omega = \int_N \omega

Theorem 4: Divergence Theorem (Gauss)

\int_M \text{div}(X) \, d\text{vol}_g = \int_{\partial M} \langle X, \nu \rangle \, d\text{vol}_{g|_{\partial M}}

Applications

Maxwell's equations in their covariant form dF = 0 and d*F = J are integrated via Stokes' theorem over 2-manifolds and 3-manifolds in spacetime.

Worked Examples

Write F·dr as a 1-form ω = y dx - x dy.
Compute dω:
$d\omega = dy \wedge dx - dx \wedge dy = -dx\wedge dy - dx\wedge dy = -2\,dx\wedge dy$
Apply Stokes' theorem:
$\oint_C \omega = \int_D d\omega = \int_D (-2)\, dx\wedge dy = -2 \cdot \text{Area}(D) = -2\pi$

Answer: ∮_C F·dr = -2π.

Practice Problems

Difficulty 8/10

Prove that for a closed form ω (dω = 0), ∫_M dω = 0, and deduce that the integral ∫_C ω over a cycle C depends only on the cohomology class [ω].

Difficulty 9/10

Let M be a compact orientable Riemannian n-manifold without boundary. Show that every harmonic function f (Δf = 0) is constant.

Difficulty 9/10

Prove that every exact form is closed (d² = 0), and give an example showing the converse fails globally but holds locally (Poincaré lemma).

Common Mistakes

Common Mistake

Integration of a function on a manifold is the same as integration of a 0-form

A function f is a 0-form, but to integrate it we need a volume form: ∫_M f = ∫_M f · dvol_g. The volume form provides the 'measure element' and depends on the Riemannian metric or the choice of orientation. Without it, ∫_M f is not defined on a general manifold.

Common Mistake

Stokes' theorem requires the manifold to be embedded in Euclidean space

Stokes' theorem is an intrinsic theorem — it holds for any compact oriented manifold with boundary, with no embedding required. The proof uses only the structure of differential forms and the definition of the integral.

Historical Background

Green (1828) and Gauss (1813) stated special cases of the divergence theorem. Stokes attributed the general theorem to Kelvin (1850); it appeared as an examination question at Cambridge set by Stokes. Cartan (1899–1945) developed the calculus of exterior differential forms that provides the modern framework. De Rham (1931) proved that integration of forms provides the isomorphism between de Rham cohomology and singular cohomology.

1813
Gauss states the divergence theorem in special cases
Carl Gauss
1828
Green's theorem in the plane
George Green
1850
Stokes' theorem (attributed to Kelvin) appears in Cambridge exams
George Stokes, Lord Kelvin
1899
Cartan begins developing exterior differential forms
Élie Cartan
1931
De Rham proves his cohomology theorem via integration of forms
Georges de Rham

Summary

Differential n-forms are the correct objects to integrate on oriented n-manifolds, transforming correctly under coordinate changes.
The Riemannian volume form dvol_g = √det(g_{ij}) dx¹∧...∧dxⁿ enables integration of functions on a Riemannian manifold.
Stokes' theorem ∫_M dω = ∫_{∂M} ω unifies the fundamental theorem of calculus, Green's theorem, and the divergence theorem.
De Rham's theorem: integration gives an isomorphism H^k_{dR}(M) ≅ H^k(M; R).
Closed forms are locally exact (Poincaré lemma); global obstructions are measured by de Rham cohomology.

References

BookLee, J.M. (2013). Introduction to Smooth Manifolds (2nd ed.). Springer.
BookTu, L.W. (2011). An Introduction to Manifolds (2nd ed.). Springer.