theorem 10.69 Mean-value property

open in the book · parts/02-mathematical-methods/08-pdes.tex:1847 · p. 359

Rests on

depends_on definition 10.66 Harmonic function ¶
- depends_on definition 7.98 Functions of class $C^{1}$ ¶
  - depends_on definition 7.20 Continuity at a point ¶
    - depends_on definition 7.16 Limit ¶
      - depends_on definition 7.2 Absolute value ¶
      - depends_on definition 7.9 Real function ¶
    - depends_on equation 7.7 eq:ana-limit-left ¶
    - depends_on equation 7.5 eq:ana-limit-right ¶
  - depends_on definition 7.97 Partial derivative; gradient ¶
    - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 7.16 Limit ¶ ↺
      - depends_on definition 7.9 Real function ¶ ↺
    - depends_on definition 5.15 Basis ¶
      - depends_on definition 5.12 Subspace generated by a set of vectors ¶
      - depends_on definition 5.14 Linear independence ¶
- depends_on equation 10.24 eq:pde-poisson-equation ¶
depends_on lemma 10.58 Darboux's equation for spherical means ¶
- depends_on definition 7.98 Functions of class $C^{1}$ ¶ ↺
- depends_on theorem 7.133 Gauss ¶
  - depends_on definition 7.127 Simple regions ¶
    - depends_on definition 7.98 Functions of class $C^{1}$ ¶ ↺
    - depends_on definition 6.9 Compact set ¶
      - depends_on definition 6.5 Open cover ¶
  - depends_on remark 7.128 What the derivations below take as given ¶
    - depends_on definition 7.125 Multiple integral ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶
      - depends_on definition 7.39 Darboux sums and the definite integral ¶
    - depends_on theorem 7.40 Continuous functions are integrable ¶
      - depends_on definition 7.39 Darboux sums and the definite integral ¶ ↺
      - depends_on theorem 7.24 Extreme value theorem ¶
      - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
      - proves proof ch:05-real-analysis@proof-23 ¶
    - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶ ↺
  - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
    - depends_on definition 7.41 Antiderivative ¶
      - depends_on corollary 7.36 cor:ana-mvt-consequences ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
    - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
      - depends_on theorem 7.40 Continuous functions are integrable ¶ ↺
      - depends_on theorem 7.24 Extreme value theorem ¶ ↺
      - proves proof ch:05-real-analysis@proof-24 ¶
    - proves proof ch:05-real-analysis@proof-25 ¶
  - proves proof ch:05-real-analysis@proof-79 ¶
- proves proof ch:08-pdes@proof-22 ¶
proves proof ch:08-pdes@proof-28 ¶
proves proof ch:08-pdes@prooflink-4 ¶

Supports

depends_on corollary A.145 Harmonic functions are smooth ¶
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 ¶ ↺
depends_on theorem 10.77 Strong maximum principle for harmonic functions ¶

Neighborhood

Every logical edge within two steps of this node.

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partly declared
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not graded
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Edges

type	direction	node	provenance	where
`depends_on`	→	Harmonic function	declared	parts/02-mathematical-methods/08-pdes.tex:1857
`depends_on`	→	Darboux's equation for spherical means	declared	parts/02-mathematical-methods/08-pdes.tex:1857
`depends_on`	←	Harmonic functions are smooth	declared	appendices/A-long-proofs.tex:7959
`depends_on`	←	Converse of the mean-value property	declared	appendices/A-long-proofs.tex:7552
`depends_on`	←	Strong maximum principle for harmonic functions	declared	parts/02-mathematical-methods/08-pdes.tex:2101
`proves`	←	ch:08-pdes@proof-28	declared	parts/02-mathematical-methods/08-pdes.tex:1860
`proves`	←	ch:08-pdes@prooflink-4	declared	parts/02-mathematical-methods/08-pdes.tex:1898