PropositionStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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Linearity and additivity of the form integral

Statement

For compactly supported smooth top forms $ω, η$ on an oriented $M^{n}$ and $a, b \in R$ , $\int_{M} (a ω + b η) = a \int_{M} ω + b \int_{M} η .$ Also $\int_{M} ω = \sum_{C} \int_{C} ω ∣_{C}$ , where $C$ ranges over connected components with their restricted orientations; only finitely many meet $supp ω$ .

Facts & Assumptions

[F1]

Independence of atlas, partition and refinement: The compact-support integral on an oriented manifold is independent of the chart cover, coordinate maps, subordinate partition, and refinement. If $U \subset M$ is open and contains $supp ω$ , with its restricted orientation, then $\int_{U} ω ∣_{U} = \int_{M} ω$ .

[F2]

Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in $R^{m}$ : Let $Q = \prod_{j < m} [a_{j}, b_{j}]$ be nondegenerate. For integrable $f, g : Q \to R$ and scalars $α, β$ , the function $α f + β g$ is integrable and its integral is $α \int_{Q} f + β \int_{Q} g$ . If $f \leq g$ , then $\int_{Q} f \leq \int_{Q} g$ . Also $∣ f ∣$ is integrable and $∣ \int_{Q} f ∣ \leq \int_{Q} ∣ f ∣$ . If $a_{r} < c < b_{r}$ , cutting $Q$ at the coordinate hyperplane $x_{r} = c$ gives two nondegenerate subrectangles; integrability on $Q$ is equivalent to integrability on both restrictions, and their integral values add to the integral over $Q$ .

Proof

Given: The objects and hypotheses in the statement above.

1.1

Choose a common chart partition for the compact union of the two supports. In positive dimension, each chart coefficient for $a ω + b η$ is the corresponding linear combination. Riemann linearity, followed by summation of finitely many terms, gives the first formula. Empty supports and zero scalars cause no exception.

F1F2

2.1

Manifolds have connected small ball or half-ball neighborhoods. Every connected component is therefore open. Its components form an open cover, so a compact support meets only finitely many of them. Choose the chart cover inside the components, group the finite sum accordingly, and use locality.

F1step 1.1

3.1

In dimension zero use the finite sums $\sum_{p} ε (p) ω (p)$ ; both distributivity and grouping are identities of finite sums. A singleton contributes its one signed value.

step 1.1algebra∎

Depends on

Used by

Localization of Stokes by a partition of unity Lemma
Computing form integrals by finite parametrizations Proposition
Positivity of the oriented integral Proposition

Dependency tree · two levels

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Sources

Lee Proposition 16.6(a), pp.407–408 (standard reference, not scraped)