Divergence as an exterior derivative

Divergence relative to a volume form: Let $\mu$ be a positive volume form and $X$ a smooth vector field on a smooth oriented manifold, with boundary allowed. The divergence relative to $\mu$ is the smooth scalar function determined by ${\mathcal{L}}_X\mu =({\mathrm{div}}_{\mu }X)\mu \mathrm{.}$ 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.

Form calculus extends locally across a manifold boundary: On smooth manifolds with boundary, the coordinate exterior derivative, pullback naturality, graded Leibniz rule, support containment, and Cartan identity hold for smooth forms: $d(F^{\ast }\alpha )=F^{\ast }(d\alpha ),d(\alpha \wedge \beta )=d\alpha \wedge \beta +(-1{)}^{\mathrm{deg}\alpha }\alpha \wedge d\beta ,$ $\mathrm{supp}d\alpha \subseteq \mathrm{supp}\alpha ,{\mathcal{L}}_X\alpha =d({\iota }_X\alpha )+{\iota }_{X}d\alpha \mathrm{.}$ For arbitrary smooth vector fields at boundary points, ${\mathcal{L}}_X$ is defined by local Euclidean extensions; a two-sided flow inside the manifold is not required.