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

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Coordinate formula and well-definedness of divergence

Statement

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.

Facts & Assumptions

[F1]

Divergence relative to a volume form: 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 lem-exterior-and-cartan-calculus-extend-to-manifolds-with-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 prop-divergence-is-well-defined-and-has-the-coordinate-formula.

[F2]

The local coordinate formula for the exterior derivative: Let $(U, x^{1}, \dots, x^{n})$ be a smooth chart on a smooth manifold and $ω$ a smooth $k$ -form on $U$ , with $k \geq 0$ . Summing over increasing $k$ -tuples $I$ , and writing $d x^{I} = d x^{i_{1}} \land \dots \land d x^{i_{k}}$ , if $ω = \sum_{I} ω_{I} d x^{I}$ , then $d ω = \sum_{I} d ω_{I} \land d x^{I} .$

Proof

Given: The objects and hypotheses in the statement above.

1.1

For $n \geq 1$ , $d μ = 0$ by degree. Cartan’s boundary-compatible identity, used in the defining Lie derivative, gives $L_{X} μ = d (ι_{X} μ)$ . Here $ι_{X} μ = \sum_{i} (- 1)^{i - 1} ρ X^{i} d x^{1} \land \dots \land \hat{d x^{i}} \land \dots \land d x^{n}$ .

F1algebra

2.1

The exterior coordinate formula differentiates this to $(\sum_{i} \partial_{i} (ρ X^{i})) d x^{1} \land \dots \land d x^{n}$ . Divide by the nowhere-zero smooth $ρ$ . The quotient is smooth; on overlaps two such quotients multiply the same nonvanishing $μ$ to give the same $L_{X} μ$ , so they agree. Boundary extensions give the same first derivatives, as in the definition.

F1F2step 1.1

3.1

For $n = 0$ the tangent fibers are zero, so $X = 0$ , the Lie derivative is zero, and its quotient by the nonzero scalar $μ$ is zero. The coordinate sum is empty. For any dimension the zero vector field and the empty manifold introduce no exception.

F1step 2.1∎

Depends on

Used by

Product rule for volume-form divergence Proposition

Cited to discharge well-definedness by Divergence relative to a volume form.

Dependency tree · two levels

8 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; Lee defining divergence equation p.423; coordinate derivation from published Cartan formula (standard reference, not scraped)