lemma 10.44 Green's identities

open in the book · parts/02-mathematical-methods/08-pdes.tex:1195 · p. 352

Rests on

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 definition 3.43 Map ¶
  - 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 5.5 Linear combination ¶
      - depends_on definition 5.7 Vector subspace ¶
      - proves proof ch:03-linear-algebra-representations@proof-3 ¶
    - depends_on definition 5.14 Linear independence ¶
      - depends_on definition 5.5 Linear combination ¶ ↺
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 definition 6.2 Open set ¶
      - depends_on equation 3.51 eq:set-indexed ¶
- 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 axiom 7.1 Completeness of $\R$ ¶ ↺
  - 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 axiom 7.1 Completeness of $\R$ ¶ ↺
      - depends_on proposition 7.22 Sequential characterization ¶
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - proves proof ch:05-real-analysis@proof-9 ¶
    - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
      - depends_on proposition 7.22 Sequential characterization ¶ ↺
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶ ↺
      - proves proof ch:05-real-analysis@proof-10 ¶
    - 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 theorem 7.35 Mean value theorem ¶
      - proves proof ch:05-real-analysis@proof-20 ¶
    - 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 ¶

Supports

depends_on proposition 10.71 Green's representation formula ¶
depends_on proposition 10.67 The Newtonian potential is the fundamental solution ¶
- 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 ¶
depends_on theorem 10.45 Symmetry of the Green's function ¶

Neighborhood

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partly declared
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Edges

type	direction	node	provenance	where
`depends_on`	→	Functions of class $C^{1}$	declared	parts/02-mathematical-methods/08-pdes.tex:1204
`depends_on`	→	Gauss	declared	parts/02-mathematical-methods/08-pdes.tex:1204
`depends_on`	←	Green's representation formula	declared	parts/02-mathematical-methods/08-pdes.tex:1913
`depends_on`	←	The Newtonian potential is the fundamental solution	declared	parts/02-mathematical-methods/08-pdes.tex:1798
`depends_on`	←	Symmetry of the Green's function	declared	parts/02-mathematical-methods/08-pdes.tex:1225
`proves`	←	ch:08-pdes@proof-14	declared	parts/02-mathematical-methods/08-pdes.tex:1207