DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-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.

Divergence relative to a volume form

Definition

Let $μ$ be a positive volume form and $X$ a smooth vector field on a smooth oriented manifold, with boundary allowed. The divergence relative to $μ$ is the smooth scalar function determined by $L_{X} μ = ({div}_{μ} X) μ .$ At a boundary point use the local-extension Lie derivative of Form calculus extends locally across a manifold boundary. The nonzero top form spans each top exterior-power fiber, so the scalar is unique. Smooth existence and its coordinate formula are discharged by Coordinate formula and well-definedness of divergence ↗.

Depends on

Used by

Divergence as an exterior derivative Lemma
Coordinate formula and well-definedness of divergence Proposition

Dependency tree · two levels

11 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

Encyclopedia of Mathematics, Divergence, Comments paragraph beginning Let M be an n-dimensional manifold (standard reference, not scraped)