Alphabeta Math
ExampleConstruction: AI-adaptedVerification: 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.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Reflection reverses the signed form integral

Example

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

[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.

1.1

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. Oriented change of variables gives the first identity.

F1
2.1

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

F2step 1.1∎

Depends on

Used by

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Dependency tree · two levels

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Sources