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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Product rule for volume-form divergence

Statement

For a smooth scalar function $f$ and smooth vector field $X$ , ${div}_{μ} (f X) = d f (X) + f {div}_{μ} X .$ The formula holds also on manifolds with boundary.

Facts & Assumptions

[F1]

Coordinate formula and well-definedness of divergence: If $μ = ρ d x^{1} \land \dots \land d x^{n}$ with $ρ$ nowhere zero and $X = \sum_{i} X^{i} \partial_{i}$ , then ${div}_{μ} X = ρ^{- 1} \sum_{i = 1}^{n} \partial_{i} (ρ X^{i}) .$ This defines a smooth global function, also at boundary points. In dimension zero $X = 0$ and divergence is zero.

Proof

Given: The objects and hypotheses in the statement above.

1.1F1algebra

In any chart the coordinate divergence formula gives ${div}_{μ} (f X) = ρ^{- 1} \sum_{i} \partial_{i} (ρ f X^{i}) = \sum_{i} X^{i} \partial_{i} f + f ρ^{- 1} \sum_{i} \partial_{i} (ρ X^{i})$ . This is the ordinary finite product rule.

2.1F1step 1.1∎

The first term is $d f (X)$ and the second is $f {div}_{μ} X$ , so the coordinate-invariant equality follows. The formula is valid for $f = 0$ , constant $f$ , $X = 0$ , and in dimension zero, where both sums and $d f (X)$ are zero; the cited coordinate formula includes boundaries.

Depends on

Coordinate formula and well-definedness of divergence

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

EoM Divergence Comments definition; direct algebraic consequence of the coordinate formula (standard reference, not scraped)