Alphabeta Math
ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedverified 2026-09-26 (gpt-6-sol)
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Reflection reverses the signed form integral

Example

Assume countable choice ACω. For f∈Cc∞(R) and reflection r(x)=−x, with the increasing orientation, ∫Rr∗(f dx)=−∫Rf dx,∫Rr∗(f ∣dx∣)=∫Rf ∣dx∣. Densities retain the sign of f; the absolute value here belongs to the coordinate density.

Facts & Assumptions

[A1]

Countable choice is assumed for the published manifold integration suppliers (The Axiom of Countable Choice (ACω)).

[F1]

Change of variables on oriented manifolds: Let F:M→N be a diffeomorphism of oriented smooth n-manifolds and ω∈Ωcn(N). If F preserves orientation everywhere, ∫MF∗ω=∫Nω; if it reverses orientation everywhere, ∫MF∗ω=−∫Nω. If the sign varies between components, apply the appropriate signed equality on each component and add.

[F2]

Orientation-free density integration and its properties: Compactly supported smooth density integration is independent of charts and partition, linear, local, nonnegative on nonnegative densities and strictly positive for a nonzero nonnegative density. It is invariant under every diffeomorphism, without choosing an orientation. The finite-parametrization formula holds under the hypotheses of prop-integration-of-top-forms-by-finite-parametrizations, with orientation preservation omitted and absolute Jacobians used.

Verification

Given: The objects and hypotheses in the statement above.

1.1A1F1

The derivative of reflection is −1, so r∗(f dx)=−f(−x)dx. The map is a globally orientation-reversing diffeomorphism, and its compact pullback support is the reflected support. Under [A1], oriented change of variables gives the first identity.

2.1A1F2step 1.1∎

For the density the Jacobian factor is ∣−1∣=1, giving f(−x)∣dx∣. Under [A1], diffeomorphism invariance of density integration gives the second identity. Empty support, zero f, and signed f all satisfy the same formulas.

Depends on

Used by

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Dependency tree · two levels

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Sources