Coordinate formula for riemannian divergence

Given: A smooth vector field $X={\sum }_i{X}^i{\mathrm{\partial }}_i$.

Riemannian divergence: The Riemannian divergence is defined in a local orientation by ${\mathcal{L}}_X{\mathrm{vol}}_g=({\mathrm{div}}_{g}X){\mathrm{vol}}_g$. Use def-divergence-relative-to-a-volume-form with the local form of def-riemannian-volume-form-on-an-oriented-manifold. On overlaps, changing orientation multiplies the nonvanishing form by a locally constant sign. The Lie derivative multiplies by that same sign, so its scalar quotient is unchanged and glues even on nonorientable manifolds. Equivalently this differentiates the positive density of lem-the-riemannian-volume-density-is-coordinate-independent and divides by it; the equivalence is local in a density frame. At a boundary use local smooth extensions. In dimension zero every vector field, and hence divergence, is zero.

Coordinate formula and well-definedness of divergence: If $\mu =\rho d{x}^1\wedge \cdots \wedge d{x}^n$ with $\rho$ nowhere zero and $X={\sum }_i{X}^i{\mathrm{\partial }}_i$, then ${\mathrm{div}}_{\mu }X={\rho }^{-1}{\sum }_{i=1}^n{\mathrm{\partial }}_i(\rho X^i)\mathrm{.}$ This defines a smooth global function, also at boundary points. In dimension zero $X=0$ and divergence is zero.