LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedaudited 2026-09-07

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Form calculus extends locally across a manifold boundary

Statement

Let $M$ and $N$ be smooth manifolds with boundary (possibly empty boundary), let $F : N \to M$ be any smooth map, and let $X$ be any smooth vector field on $M$ . The coordinate exterior derivative, pullback naturality, support containment, and Cartan identity hold for every smooth form $α \in Ω^{*} (M)$ . For homogeneous $α \in Ω^{p} (M)$ and $β \in Ω^{q} (M)$ , the graded Leibniz rule also holds: $d (F^{*} α) = F^{*} (d α), d (α \land β) = d α \land β + (- 1)^{p} α \land d β,$ $supp d α \subseteq supp α, L_{X} α = d (ι_{X} α) + ι_{X} d α .$ For arbitrary smooth vector fields at boundary points, $L_{X}$ is defined by local Euclidean extensions; a two-sided flow inside the manifold is not required.

Facts & Assumptions

[F1]

Smooth functions and tensor fields extend locally across the boundary: Every smooth function or tensor field on a manifold with boundary extends smoothly across each boundary point to some neighbourhood in its double; the extension is not canonical.

[F2]

Half-space extensions agreeing on a relatively open set have the same derivatives there: If two smooth Euclidean extensions agree on a relatively open subset of $H^{n}$ , then all of their derivatives agree at every point of that subset.

[F3]

The local coordinate formula for the exterior derivative: Let $(U, x^{1}, \dots, x^{n})$ be a smooth chart on a smooth manifold and $ω$ a smooth $k$ -form on $U$ , with $k \geq 0$ . Summing over increasing $k$ -tuples $I$ , and writing $d x^{I} = d x^{i_{1}} \land \dots \land d x^{i_{k}}$ , if $ω = \sum_{I} ω_{I} d x^{I}$ , then $d ω = \sum_{I} d ω_{I} \land d x^{I} .$

[F4]

The exterior derivative commutes with pullback: For every smooth map $F : M \to N$ and every form $ω$ on $N$ , $d (F^{*} ω) = F^{*} (d ω) .$

[F5]

The exterior derivative is a graded derivation: Let $M$ be a smooth manifold. The exterior derivative is an $R$ -linear map $d : Ω^{*} (M) \to Ω^{*} (M)$ of degree one. For homogeneous smooth forms $α \in Ω^{p} (M)$ and $β \in Ω^{q} (M)$ , $d (α \land β) = d α \land β + (- 1)^{\deg α} α \land d β .$

[F6]

Cartan's magic formula: For every vector field $X$ and differential form $ω$ , $L_{X} ω = d (ι_{X} ω) + ι_{X} (d ω) .$

[F7]

The exterior derivative does not enlarge support: For every form $ω$ , $supp (d ω) \subseteq supp (ω)$ .

Proof

Given: The objects and hypotheses in the statement above.

1.1

Extend the finitely many coordinate coefficients of forms and vector fields across a boundary point. Two extensions agreeing on the half-space have all derivatives equal there. Hence the coordinate formula for $d$ , which uses only first derivatives, restricts independently of the extension. The same holds for the coordinate Lie derivative, whose coefficients involve first derivatives of the field and form.

F1F2F3

2.1

For a smooth map between boundary charts, extend its coordinate components locally and extend the target form near the image point. Shrink the source neighborhood so the extended map lands in that target extension domain. The boundaryless pullback identity restricts to $d F^{*} α = F^{*} d α$ , with independence assured by equality of derivatives.

F2F4step 1.1

2.2

The graded Leibniz identity for extensions restricts to the asserted identity. At a point outside the support the form vanishes on a relative neighborhood; all its derivatives, including their one-sided limits at the face, vanish. Thus its derivative vanishes there and support cannot increase.

F2F5F7step 1.1

3.1

Apply Cartan’s formula on each extension neighborhood and restrict; equality of first derivatives gives the same result for any extensions. These local equalities agree on overlaps by the coordinate tensor laws. Degree zero, the zero form, and empty manifolds cause no exception, and the construction in dimension one uses exactly the same one-sided derivatives.

F2F6step 1.1∎

Depends on

Used by

A nonzero period obstructs exactness and bounding Corollary
Divergence relative to a volume form Definition
Compact-support Stokes on Euclidean space Lemma
Divergence as an exterior derivative Lemma
Localization of Stokes by a partition of unity Lemma

Dependency tree · two levels

25 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 Stokes proof pp.412–414 and published extension/calculus dependencies (standard reference, not scraped)