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

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Change of variables on oriented manifolds

Statement

Let $F : M \to N$ be a diffeomorphism of oriented smooth $n$ -manifolds and $ω \in Ω_{c}^{n} (N)$ . If $F$ preserves orientation everywhere, $\int_{M} F^{*} ω = \int_{N} ω$ ; if it reverses orientation everywhere, $\int_{M} F^{*} ω = - \int_{N} ω$ . If the sign varies between components, apply the appropriate signed equality on each component and add.

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]

Orientation reversal changes the integral sign: Let $- M$ have the opposite orientation on every component of an oriented smooth manifold $M$ . For every compactly supported top form, $\int_{- M} ω = - \int_{M} ω$ , in all dimensions.

[F3]

Pullback of forms is smooth functorial and preserves wedges: For a smooth map $F : M \to N$ , pullback sends smooth differential forms on $N$ to smooth differential forms on $M$ , is functorial, and satisfies $F^{*} (α \land β) = F^{*} α \land F^{*} β .$

Proof

Given: The objects and hypotheses in the statement above.

1.1

The support of $F^{*} ω$ is $F^{- 1} (supp ω)$ because the tangent maps are isomorphisms. It is compact, being the continuous image of the compact support under the inverse homeomorphism. Pull back the target partition and use the charts $ϕ_{i} \circ F$ ; these remain subordinate and locally finite.

F3given

2.1

If orientation is preserved, the corresponding source and target charts have the same sign and the same localized coefficient: $(ϕ_{i} F)^{- 1} = F^{- 1} ϕ_{i}^{- 1}$ and functoriality cancels the pullbacks. Thus their finite chart sums agree. Choice independence makes this the asserted intrinsic equality.

F1F3step 1.1

3.1

For global reversal, reverse the source orientation to apply the preceding equality and then negate its integral. More generally the sign is locally constant because a continuous nonzero determinant has constant sign on each connected component; apply that argument componentwise. In dimension zero the diffeomorphism is a bijection of finite supports with the corresponding point signs; empty support and zero forms give zero.

F2step 2.1∎

Depends on

Used by

Change of variables on an oriented circle Example
Reflection reverses the signed form integral Example
Integration on an oriented embedded submanifold Proposition
The general Stokes theorem Theorem

Dependency tree · two levels

10 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 Proposition 16.6(d), pp.407–408 (standard reference, not scraped)