Integration on Manifolds

Core Idea

Integration of differential forms on oriented manifolds generalizes multiple integrals from ℝⁿ to curved spaces. An n-form on an oriented n-manifold is integrated by pulling back to coordinate charts via a partition of unity — the transformation law for forms automatically handles the Jacobian determinant from the change of variables formula. This framework unifies line integrals, surface integrals, and volume integrals into a single coordinate-free theory.

Explainer

In multivariable calculus on ℝⁿ, you integrate functions f by computing ∫ f dx¹...dxⁿ. Under a change of variables x = φ(u), this becomes ∫ f(φ(u)) |det Dφ| du¹...duⁿ — the Jacobian determinant appears. On a manifold, there are no preferred coordinates, so you need an object whose transformation law automatically includes the Jacobian. Differential n-forms are exactly that object: under a coordinate change, an n-form transforms by det(Jacobian) — without the absolute value, which is why you need an orientation to fix the sign.

The construction of ∫_M ω proceeds in three steps. First, choose an atlas {(Uα, φα)} of positively oriented charts and a subordinate partition of unity {ρα}. Second, write ω = Σα ρα ω, where each term is supported in a single chart. Third, compute each ∫_M ρα ω by pulling back to ℝⁿ: in coordinates, (φα⁻¹)*(ρα ω) = fα dx¹ ∧ ... ∧ dxⁿ, and ∫_M ρα ω = ∫ fα dx¹...dxⁿ as an ordinary integral. The sum of these integrals is ∫_M ω. A crucial verification: this is independent of the choice of atlas and partition of unity (because the transformation law for forms handles the change-of-variables automatically).

When a manifold has a Riemannian metric g, there is a canonical volume form dVg: in positively oriented coordinates, dVg = √det(gij) dx¹ ∧ ... ∧ dxⁿ. To integrate a function f : M → ℝ, you integrate the n-form f · dVg. The factor √det(gij) accounts for the "stretching" of the coordinate grid relative to the intrinsic geometry — on a sphere in spherical coordinates, this gives the familiar sin θ factor. But forms of degree less than n (like 1-forms on surfaces) can be integrated over appropriate submanifolds without any metric at all.

Integration on manifolds is the essential link between local differential data and global geometric/topological invariants. The total curvature of a surface (the Gauss-Bonnet integral), the de Rham cohomology pairing, characteristic classes of vector bundles, and the action functional in physics are all integrals of differential forms over manifolds. Stokes' theorem — the next topic — provides the fundamental relationship between integrals over a manifold and integrals over its boundary, completing the framework that unifies Green's theorem, the divergence theorem, and the classical Stokes theorem into a single statement.