definition A.134 Mean-value property

open in the book · appendices/A-long-proofs.tex:7531 · p. 2860

ground object -- no derivation owed

Rests on

depends_on definition 6.2 Open set ¶
- depends_on definition 6.1 Topological space ¶
  - depends_on definition 3.31 Empty set ¶
  - depends_on definition 3.35 Union, intersection, difference ¶
    - depends_on definition 3.8 Conjunction ¶
    - depends_on definition 3.9 Disjunction ¶
    - depends_on definition 3.7 Negation ¶
  - depends_on definition 3.30 Subset ¶
  - depends_on equation 3.51 eq:set-indexed ¶
depends_on equation 10.54 eq:pde-spherical-mean ¶

Supports

depends_on corollary A.146 Locally uniform limits of harmonic functions ¶
depends_on proposition A.143 Reproduction identity ¶
depends_on theorem A.135 Converse of the mean-value property ¶
- depends_on corollary A.146 Locally uniform limits of harmonic functions ¶ ↺
- depends_on corollary A.145 Harmonic functions are smooth ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Open set	declared	appendices/A-long-proofs.tex:7541
`depends_on`	→	eq:pde-spherical-mean	declared	appendices/A-long-proofs.tex:7541
`depends_on`	←	Locally uniform limits of harmonic functions	declared	appendices/A-long-proofs.tex:7973
`depends_on`	←	Reproduction identity	declared	appendices/A-long-proofs.tex:7850
`depends_on`	←	Converse of the mean-value property	declared	appendices/A-long-proofs.tex:7552