definition 10.42 Green's function

open in the book · parts/02-mathematical-methods/08-pdes.tex:1150 · p. 352

ground object -- no derivation owed

Rests on

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 ¶
      - depends_on definition 7.39 Darboux sums and the definite integral ¶
      - proves proof ch:15-fourier-integral-transforms@proof-5 ¶
    - 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 definition 7.98 Functions of class $C^{1}$ ¶
      - depends_on definition 6.2 Open set ¶
  - 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 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 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-64 ¶

Supports

depends_on definition 10.50 Retarded and advanced Green's functions ¶
depends_on example 10.49 The method of images ¶
depends_on proposition 28.24 Impulse response of the damped oscillator ¶
depends_on proposition 10.47 Bilinear expansion ¶
depends_on proposition 10.43 Representation of the solution ¶
depends_on theorem 10.45 Symmetry of the Green's function ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Transform and derivative of a distribution	declared	parts/02-mathematical-methods/08-pdes.tex:1159
`depends_on`	→	The second-order operator	declared	parts/02-mathematical-methods/08-pdes.tex:1159
`depends_on`	←	Retarded and advanced Green's functions	declared	parts/02-mathematical-methods/08-pdes.tex:1338
`depends_on`	←	The method of images	declared	parts/02-mathematical-methods/08-pdes.tex:1320
`depends_on`	←	Impulse response of the damped oscillator	declared	parts/03-classical-mechanics/11-oscillations-waves.tex:731
`depends_on`	←	Bilinear expansion	declared	parts/02-mathematical-methods/08-pdes.tex:1273
`depends_on`	←	Representation of the solution	declared	parts/02-mathematical-methods/08-pdes.tex:1171
`depends_on`	←	Symmetry of the Green's function	declared	parts/02-mathematical-methods/08-pdes.tex:1225