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Integral of a compactly supported smooth density
Definition
Assume . Let be a compactly supported smooth density on , with boundary allowed. Choose a chart partition and write . For define The zero extensions are Riemann integrable, including at genuine faces, by Riemann-integrable half-space extensions of chart coefficients. The compact-support/local-finiteness argument of Local finiteness near compact support applies to density supports as closed sets, so the sum is finite. For sum the scalar density values over the finite support, without orientation signs. Empty support gives zero. Choice independence is discharged by Orientation-free density integration and its properties ↗.
Depends on
Used by
- E-valued Dolbeault operators and their L2 structure Definition
- Hermitian metric and L² pairing on a compact Riemann surface Definition
- Riemannian volume of a compactly supported smooth density Definition
- Second-order Laplace-type operators on metric vector bundles Definition
- Formal adjoints of differential operators on metric vector bundles Lemma
- Harmonic representatives for compact de Rham and Dolbeault complexes Theorem
- Measurable integration extends smooth density integration Theorem
- Orientation-free density integration and its properties Theorem
Dependency tree · two levels
28 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
- Lee density-integral construction pp.431–432; Nicolaescu Proposition 3.4.3 (standard reference, not scraped)