theorem 10.45 Symmetry of the Green's function

open in the book · parts/02-mathematical-methods/08-pdes.tex:1216 · p. 353

Rests on

depends_on definition 10.42 Green's function ¶
- depends_on definition 17.40 Transform and derivative of a distribution ¶
  - depends_on definition 17.38 Schwartz space and tempered distributions ¶
  - depends_on proposition 17.39 The transform preserves $\mathcal{S}$ ¶
    - depends_on definition 17.38 Schwartz space and tempered distributions ¶ ↺
    - depends_on proposition 17.30 Elementary properties ¶
      - depends_on definition 17.29 Fourier transform; the treatise convention ¶
      - depends_on equation 7.28 eq:ana-parts ¶
      - depends_on lemma 17.7 Riemann–Lebesgue ¶
      - proves proof ch:15-fourier-integral-transforms@proof-19 ¶
    - depends_on theorem 17.32 Fourier inversion ¶
      - depends_on definition 17.29 Fourier transform; the treatise convention ¶ ↺
      - depends_on equation 17.42 eq:ft-gaussian ¶
      - proves proof ch:15-fourier-integral-transforms@proof-20 ¶
    - proves proof ch:15-fourier-integral-transforms@proof-25 ¶
- depends_on definition 10.4 The second-order operator ¶
  - depends_on definition 10.3 Principal part and principal symbol ¶
    - depends_on definition 10.2 Linear, semilinear, quasilinear ¶
      - depends_on definition 10.1 Partial differential equation; order ¶
    - depends_on equation 10.2 eq:pde-multi-index ¶
  - depends_on proposition 7.105 Clairaut–Schwarz ¶
    - 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 theorem 7.35 Mean value theorem ¶
      - depends_on proposition 7.29 Linearity ¶
      - depends_on theorem 7.34 Rolle ¶
      - proves proof ch:05-real-analysis@proof-19 ¶
    - proves proof ch:05-real-analysis@proof-64 ¶
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 7.97 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-15 ¶

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type	direction	node	provenance	where
`depends_on`	→	Green's function	declared	parts/02-mathematical-methods/08-pdes.tex:1225
`depends_on`	→	Green's identities	declared	parts/02-mathematical-methods/08-pdes.tex:1225
`proves`	←	ch:08-pdes@proof-15	declared	parts/02-mathematical-methods/08-pdes.tex:1228