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Riemannian divergence
Definition
The Riemannian divergence is defined in a local orientation by .
Use Divergence relative to a volume form with the local form of 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 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.
Source locator
Lee, pp.423–424, definition and Exercise 16.31.
Depends on
Used by
- Coordinate formula for riemannian divergence Proposition
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
- John M. Lee, Introduction to Smooth Manifolds, second edition (standard reference, not scraped)