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

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A compactly supported primitive has zero total derivative integral

Statement

Assume ${A C}_{ω}$ . If $M^{n}$ is oriented and boundaryless, $n \geq 1$ , and $η \in Ω_{c}^{n - 1} (M)$ , then $\int_{M} d η = 0$ . In particular, on a compact such manifold every exact smooth top form has zero integral. The compact-support assumption is on the primitive $η$ , not merely on $d η$ .

Facts & Assumptions

[F1]

The general Stokes theorem: Assume ${A C}_{ω}$ . Let $M$ be an oriented smooth $n$ -manifold with boundary, $n \geq 1$ , and let $η \in Ω_{c}^{n - 1} (M)$ . With $j : \partial M ↪ M$ and the outward-normal-first orientation, $\int_{M} d η = \int_{\partial M} j^{*} η .$ An empty boundary contributes zero; in dimension one its integral is a finite signed sum of point values.

Proof

Given: The objects and hypotheses in the statement above.

1.1

View $M$ as a manifold with empty boundary. General Stokes applies to the compactly supported primitive and gives $\int_{M} d η = \int_{\emptyset} η = 0$ , including the zero primitive.

2.1

If $M$ is compact, the closed support of any smooth primitive is a compact subset of $M$ , so the first conclusion applies to every exact top form. This holds for n=1 as well; no negative-degree form or dimension-zero Stokes assertion is used.

step 1.1algebra∎

Depends on

The general Stokes theorem

Used by

A nonzero period obstructs exactness and bounding Corollary
Nonzero total integral obstructs exactness on a closed manifold Corollary

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Lee Corollary 16.13, p.414 (compact-support version from Theorem 16.11) (standard reference, not scraped)