proposition 10.67 The Newtonian potential is the fundamental solution

open in the book · parts/02-mathematical-methods/08-pdes.tex:1790 · p. 359

Rests on

depends_on definition 17.38 Schwartz space and tempered distributions ¶
depends_on lemma 10.44 Green's identities ¶
- 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 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-14 ¶
proves proof ch:08-pdes@proof-27 ¶

Supports

depends_on example 10.49 The method of images ¶
depends_on lemma A.94 Fundamental solution of $\pp_{\bar z}$ ¶
depends_on proposition A.522 The Newtonian potential inverts the Laplacian ¶
- depends_on theorem A.523 Helmholtz decomposition: existence ¶
depends_on proposition 10.71 Green's representation formula ¶
depends_on remark 10.73 What compact support is doing, and the rate that replaces it ¶
depends_on theorem 7.137 Helmholtz decomposition ¶
- depends_on remark A.527 The hypotheses, and what happens without them ¶
- depends_on remark 30.46 Why the plane-wave route, and not the potentials ¶
- depends_on remark 30.27 Two results this chapter borrows from Part II ¶
- depends_on remark 10.73 What compact support is doing, and the rate that replaces it ¶ ↺
- depends_on remark 10.74 Where the decomposition is used, and what it costs ¶

Neighborhood

Every logical edge within two steps of this node.

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partly declared
a check failed
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declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Schwartz space and tempered distributions	declared	parts/02-mathematical-methods/08-pdes.tex:1798
`depends_on`	→	Green's identities	declared	parts/02-mathematical-methods/08-pdes.tex:1798
`depends_on`	←	The method of images	declared	parts/02-mathematical-methods/08-pdes.tex:1320
`depends_on`	←	Fundamental solution of $\pp_{\bar z}$	declared	appendices/A-long-proofs.tex:5733
`depends_on`	←	The Newtonian potential inverts the Laplacian	declared	appendices/A-long-proofs.tex:25502
`depends_on`	←	Green's representation formula	declared	parts/02-mathematical-methods/08-pdes.tex:1913
`depends_on`	←	What compact support is doing, and the rate that replaces it	declared	parts/02-mathematical-methods/08-pdes.tex:2003
`depends_on`	←	Helmholtz decomposition	declared	parts/02-mathematical-methods/05-real-analysis.tex:4673
`proves`	←	ch:08-pdes@proof-27	declared	parts/02-mathematical-methods/08-pdes.tex:1801