TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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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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Differentiation of a pulled-back form along a time-dependent flow

Statement

If $Φ_{t, s}$ is the local evolution of $X_{t}$ and $ω_{t}$ is a smooth time-dependent form, then $\frac{d}{d t} Φ_{t, s}^{*} ω_{t} = Φ_{t, s}^{*} ({\dot{ω}}_{t} + L_{X_{t}} ω_{t}) .$

Facts & Assumptions

Given: The manifolds, forms, vector fields, maps, and coordinates explicitly named in the statement.

[F1]

The preceding result states that For $ω \in Ω^{k} (M)$ , $L_{X} ω$ is the Lie derivative of $ω$ regarded as an alternating covariant tensor. (The Lie derivative of a differential form).

Proof

technique · direct

1.1F1given

The cocycle writes $Φ_{t + h, s}^{*} ω_{t + h} = Φ_{t, s}^{*} (Φ_{t + h, t}^{*} ω_{t + h})$ . Divide the increment by $h$ .

2.1step 1.1∎

As $h \to 0$ , variation of $ω_{t}$ gives ${\dot{ω}}_{t}$ and the short evolution gives $L_{X_{t}} ω_{t}$ , proving the formula.

Depends on

Used by

Time-dependent pullback differentiation for a translation Example
Moser pullback differentiation equation Lemma
Time-dependent Hamiltonian evolution is symplectic Proposition

Dependency tree · two levels

9 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

John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)