proposition A.522 The Newtonian potential inverts the Laplacian

open in the book · appendices/A-long-proofs.tex:25493 · p. 3047

Rests on

depends_on definition 17.38 Schwartz space and tempered distributions ¶
depends_on proposition A.521 Every derivative passes onto $g$ ¶
- depends_on lemma A.520 A $C^{1}$ limit ¶
  - 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 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 5.15 Basis ¶
  - 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 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 7.24 Extreme value theorem ¶ ↺
    - proves proof ch:05-real-analysis@proof-24 ¶
  - 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 ¶ ↺
    - proves proof ch:05-real-analysis@proof-25 ¶
  - proves proof app:A-long-proofs@proof-311 ¶
- depends_on lemma A.518 The excised ball ¶
  - 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 example 7.130 The two Jacobians this treatise uses ¶
    - depends_on equation 7.131 eq:ana-nabla-sph-grad ¶
    - depends_on theorem 7.129 Change of variables in a multiple integral ¶
      - depends_on corollary 7.113 Inverse function theorem ¶
      - depends_on definition 7.125 Multiple integral ¶ ↺
      - depends_on remark 7.128 What the derivations below take as given ¶
      - proves proof ch:05-real-analysis@prooflink-3 ¶
  - depends_on theorem 7.129 Change of variables in a multiple integral ¶ ↺
  - proves proof app:A-long-proofs@proof-309 ¶
- depends_on theorem 7.109 Leibniz integral rule ¶
  - depends_on definition 7.125 Multiple integral ¶ ↺
  - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶ ↺
  - depends_on theorem 7.35 Mean value theorem ¶
    - depends_on proposition 7.29 Linearity ¶
      - depends_on definition 7.11 Sum of functions ¶
      - depends_on equation 7.14 eq:ana-derivh ¶
      - depends_on proposition 7.6 Algebra of limits ¶
      - proves proof ch:05-real-analysis@proof-13 ¶
    - depends_on theorem 7.34 Rolle ¶
      - depends_on lemma 7.33 Fermat: interior extremum ¶
      - depends_on theorem 7.24 Extreme value theorem ¶ ↺
      - proves proof ch:05-real-analysis@proof-18 ¶
    - proves proof ch:05-real-analysis@proof-19 ¶
  - proves proof ch:05-real-analysis@proof-67 ¶
- proves proof app:A-long-proofs@proof-312 ¶
depends_on proposition 10.67 The Newtonian potential is the fundamental solution ¶
- 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 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 remark 7.128 What the derivations below take as given ¶ ↺
    - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶ ↺
    - proves proof ch:05-real-analysis@proof-79 ¶
  - proves proof ch:08-pdes@proof-14 ¶
- proves proof ch:08-pdes@proof-27 ¶
proves proof app:A-long-proofs@proof-313 ¶

Supports

depends_on theorem A.523 Helmholtz decomposition: existence ¶

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Schwartz space and tempered distributions	declared	appendices/A-long-proofs.tex:25502
`depends_on`	→	Every derivative passes onto $g$	declared	appendices/A-long-proofs.tex:25502
`depends_on`	→	The Newtonian potential is the fundamental solution	declared	appendices/A-long-proofs.tex:25502
`depends_on`	←	Helmholtz decomposition: existence	declared	appendices/A-long-proofs.tex:25546
`proves`	←	app:A-long-proofs@proof-313	declared	appendices/A-long-proofs.tex:25506