theorem 10.52 Duhamel's principle

open in the book · parts/02-mathematical-methods/08-pdes.tex:1386 · p. 354

Rests on

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 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 definition 6.2 Open set ¶
    - depends_on definition 6.1 Topological space ¶
      - depends_on definition 3.31 Empty set ¶
      - depends_on definition 3.35 Union, intersection, difference ¶
      - depends_on definition 3.30 Subset ¶
      - depends_on equation 3.51 eq:set-indexed ¶
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 axiom 7.1 Completeness of $\R$ ¶
  - 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 definition 7.20 Continuity at a point ¶ ↺
      - depends_on definition 7.4 Convergence ¶
      - proves proof ch:05-real-analysis@proof-7 ¶
    - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - depends_on corollary A.45 Archimedean property and density of $\Q$ ¶
      - depends_on corollary A.46 Monotone convergence ¶
      - proves proof ch:05-real-analysis@proof-4 ¶
    - 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 7.24 Extreme value theorem ¶ ↺
- proves proof ch:05-real-analysis@proof-24 ¶
proves proof ch:08-pdes@proof-17 ¶

Supports

depends_on corollary 10.53 Duhamel for the wave equation ¶

Neighborhood

Every logical edge within two steps of this node.

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partly declared
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Edges

type	direction	node	provenance	where
`depends_on`	→	Linear, semilinear, quasilinear	declared	parts/02-mathematical-methods/08-pdes.tex:1397
`depends_on`	→	Fundamental theorem of calculus, I	declared	parts/02-mathematical-methods/08-pdes.tex:1397
`depends_on`	←	Duhamel for the wave equation	declared	parts/02-mathematical-methods/08-pdes.tex:1422
`proves`	←	ch:08-pdes@proof-17	declared	parts/02-mathematical-methods/08-pdes.tex:1400