definition A.102 Normal Cauchy problem of order $k$

open in the book · appendices/A-long-proofs.tex:6111 · p. 2843

ground object -- no derivation owed

Rests on

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 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 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 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 3.8 Conjunction ¶
      - depends_on definition 3.9 Disjunction ¶
      - depends_on definition 3.7 Negation ¶
    - depends_on definition 3.30 Subset ¶
    - depends_on equation 3.51 eq:set-indexed ¶
depends_on proposition 10.11 What a characteristic surface is ¶
- depends_on definition 10.10 Characteristic surface ¶
  - 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 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 theorem 7.35 Mean value theorem ¶
      - proves proof ch:05-real-analysis@proof-64 ¶
  - depends_on definition 10.3 Principal part and principal symbol ¶ ↺
- depends_on theorem 10.8 The type is a coordinate invariant ¶
  - depends_on definition 10.6 Type of a second-order operator ¶
    - depends_on definition 10.4 The second-order operator ¶ ↺
    - depends_on theorem 5.116 Sylvester's law of inertia ¶
      - depends_on definition 5.110 Bilinear map ¶
      - depends_on definition 5.111 Symmetric and antisymmetric forms ¶
      - depends_on equation 5.139 eq:lin-bilinear-congruence ¶
      - proves proof ch:03-linear-algebra-representations@proof-53 ¶
  - depends_on proposition 7.31 Chain rule ¶
    - depends_on definition 7.13 Composite function ¶
      - depends_on definition 7.9 Real function ¶ ↺
    - depends_on proposition 7.27 Differentiable implies continuous ¶
      - depends_on definition 7.20 Continuity at a point ¶ ↺
      - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
      - depends_on proposition 7.6 Algebra of limits ¶
      - proves proof ch:05-real-analysis@proof-11 ¶
    - depends_on proposition 7.6 Algebra of limits ¶ ↺
    - proves proof ch:05-real-analysis@proof-15 ¶
  - depends_on theorem 5.116 Sylvester's law of inertia ¶ ↺
  - proves proof ch:08-pdes@proof-1 ¶
- proves proof ch:08-pdes@proof-2 ¶

Supports

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 theorem A.112 Cauchy–Kovalevskaya, first-order form ¶
  - 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 ¶ ↺

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`	→	Partial differential equation; order	declared	appendices/A-long-proofs.tex:6121
`depends_on`	→	What a characteristic surface is	declared	appendices/A-long-proofs.tex:6121
`depends_on`	←	First-order quasilinear system with zero data	declared	appendices/A-long-proofs.tex:6135
`depends_on`	←	Formal reduction of a normal problem	declared	appendices/A-long-proofs.tex:6212