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

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Closed forms have zero boundary integral

Statement

Assume ${A C}_{ω}$ . For oriented $M^{n}$ with boundary, $n \geq 1$ , if $η \in Ω_{c}^{n - 1} (M)$ is closed, then $\int_{\partial M} j^{*} η = 0$ . When $M$ is compact, no separate support assumption on the smooth closed form is needed.

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.

1.1F1

Closedness says $d η = 0$ . General Stokes identifies the boundary integral with $\int_{M} d η$ , which is the integral of the zero top form and hence zero. An empty boundary is included.

2.1step 1.1algebra∎

For compact $M$ every closed support is compact. Thus the same argument applies to every smooth closed $(n - 1)$ -form. For $n = 1$ it gives the signed sum of boundary values of a locally constant function; for the zero form all terms vanish.

Depends on

The general Stokes theorem

Used by

A nonzero period obstructs exactness and bounding Corollary

Dependency tree · two levels

7 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 16.14, p.414 (standard reference, not scraped)