definition 10.10 Characteristic surface

open in the book · parts/02-mathematical-methods/08-pdes.tex:227 · p. 341

ground object -- no derivation owed

Rests on

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 ¶
depends_on definition 10.3 Principal part and principal symbol ¶ ↺

Supports

depends_on proposition 10.13 Canonical form in two variables ¶
- depends_on theorem A.83 Local canonical form for an elliptic operator with Hölder coefficients ¶
- depends_on theorem 10.55 d'Alembert's formula ¶
  - depends_on corollary 10.56 Domain of dependence and influence ¶
  - depends_on theorem 10.59 Kirchhoff's formula ¶
    - depends_on corollary 10.60 Huygens' principle in three space dimensions ¶
depends_on proposition 10.11 What a characteristic surface is ¶
- depends_on corollary 10.12 Characteristics of the three canonical operators ¶
  - depends_on corollary 10.60 Huygens' principle in three space dimensions ¶ ↺
- depends_on definition A.102 Normal Cauchy problem of order $k$ ¶
  - depends_on definition A.103 First-order quasilinear system with zero data ¶
    - depends_on lemma A.104 The recursion ¶
      - depends_on lemma A.109 Domination ¶
      - depends_on lemma A.105 Formal reduction of a normal problem ¶
    - depends_on proposition A.110 Explicit solution of the majorant problem ¶
      - depends_on theorem A.112 Cauchy–Kovalevskaya, first-order form ¶
  - depends_on lemma A.105 Formal reduction of a normal problem ¶ ↺
- depends_on theorem 10.22 Cauchy–Kovalevskaya ¶

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`	→	The second-order operator	declared	parts/02-mathematical-methods/08-pdes.tex:235
`depends_on`	→	Principal part and principal symbol	declared	parts/02-mathematical-methods/08-pdes.tex:235
`depends_on`	←	Canonical form in two variables	declared	parts/02-mathematical-methods/08-pdes.tex:315
`depends_on`	←	What a characteristic surface is	declared	parts/02-mathematical-methods/08-pdes.tex:245