Stokes theorem on manifolds

Stokes 定理

Geometry & Topology · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Other names / variants: generalized Stokes theorem

Translation evidence: Literature usage

Meaning in context

For an oriented smooth n-manifold with boundary, n at least one, and a smooth compactly supported (n−1)-form ω, the integral of dω over the manifold equals the integral of the pullback of ω to its boundary, with outward-normal-first boundary orientation.

对定向光滑 $n$ 维带边流形（$n\geq1$）及光滑紧支集 $(n-1)$ 形式 $\omega$，有 $\int_M d\omega=\int_{\partial M}j^*\omega$，边界取外法向在前的诱导定向。

Example

On the unit disk oriented by $dx\wedge dy$, $\omega=(x\,dy-y\,dx)/2$ satisfies $\int_D d\omega=\int_{\partial D}\omega=\pi$ for counterclockwise boundary orientation.

Usage and distinctions

The form must be smooth on the whole manifold, including its boundary. Compactness of the manifold makes the support condition automatic; noncompact manifolds need support or other justified conditions at infinity.

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Definition sources

1. Marco Gualtieri — Geometry and Topology I§6.4, Proposition 6.17 and Theorem 6.18, printed pp. 60–61 · Checked 2026-10-15

Translation sources

1. Peking University — 2022 mathematics teaching plans课程号00130190，内容提要三、外微分形式：Stokes 定理；PDF p. 77, printed p. 74 · Checked 2026-10-15

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