proposition 10.15 Method of characteristics, first order

open in the book · parts/02-mathematical-methods/08-pdes.tex:425 · p. 343

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 proposition 7.31 Chain rule ¶
- depends_on definition 7.13 Composite function ¶
  - depends_on definition 7.9 Real function ¶
    - depends_on definition 3.43 Map ¶
      - depends_on definition 3.24 Quantifiers ¶
      - depends_on definition 3.28 Set ¶
- 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 ¶
    - depends_on definition 7.4 Convergence ¶
      - depends_on definition 7.2 Absolute value ¶
    - depends_on proposition 7.3 Triangle inequality ¶
      - depends_on definition 7.2 Absolute value ¶ ↺
      - proves proof ch:05-real-analysis@proof-1 ¶
    - proves proof ch:05-real-analysis@proof-3 ¶
  - proves proof ch:05-real-analysis@proof-11 ¶
- depends_on proposition 7.6 Algebra of limits ¶ ↺
- proves proof ch:05-real-analysis@proof-15 ¶
proves proof ch:08-pdes@proof-5 ¶

Supports

depends_on definition 10.93 Scalar conservation law ¶
- depends_on definition A.150 Kruzhkov entropy pair ¶
  - depends_on lemma A.157 Two elementary identities ¶
    - depends_on proposition A.159 The doubled inequality ¶
      - depends_on proposition A.160 The cancellation ¶
    - depends_on proposition A.162 The limit $\delta\to0$ ¶
  - depends_on proposition A.156 The Kruzhkov inequality ¶
    - depends_on proposition A.159 The doubled inequality ¶ ↺
  - depends_on theorem A.151 $L^{1}$ contraction on a cone ¶
    - depends_on corollary A.152 Uniqueness ¶
- depends_on definition 10.95 Weak solution of a conservation law ¶
  - depends_on definition 10.99 Entropy condition ¶
    - depends_on definition 10.101 Entropy pair; entropy inequality ¶
      - depends_on definition A.150 Kruzhkov entropy pair ¶ ↺
      - depends_on proposition A.156 The Kruzhkov inequality ¶ ↺
      - depends_on proposition A.163 The viscous problem produces the entropy inequality ¶
      - depends_on proposition 10.102 Jump form of the entropy inequality ¶
      - depends_on theorem A.151 $L^{1}$ contraction on a cone ¶ ↺
      - depends_on theorem 10.103 Uniqueness in the entropy class ¶
  - depends_on definition 10.101 Entropy pair; entropy inequality ¶ ↺
  - depends_on example 10.98 Weak solutions are not unique ¶
  - depends_on theorem A.151 $L^{1}$ contraction on a cone ¶ ↺
  - depends_on theorem 10.103 Uniqueness in the entropy class ¶ ↺
  - depends_on theorem 10.96 Rankine–Hugoniot condition ¶
    - depends_on definition 10.99 Entropy condition ¶ ↺
    - depends_on example 10.97 Burgers shock ¶
    - depends_on example 10.98 Weak solutions are not unique ¶ ↺
    - depends_on proposition 10.102 Jump form of the entropy inequality ¶ ↺
- depends_on example 10.97 Burgers shock ¶ ↺
- depends_on proposition 10.94 Gradient catastrophe ¶
depends_on proposition 10.16 Cauchy's characteristic strips ¶
depends_on proposition 10.94 Gradient catastrophe ¶ ↺
depends_on remark 31.90 Why a compression wave must steepen ¶
depends_on remark 23.3 What Part II owes this chapter ¶

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`	→	Linear, semilinear, quasilinear	declared	parts/02-mathematical-methods/08-pdes.tex:435
`depends_on`	→	Chain rule	declared	parts/02-mathematical-methods/08-pdes.tex:435
`depends_on`	←	Scalar conservation law	declared	parts/02-mathematical-methods/08-pdes.tex:2460
`depends_on`	←	Cauchy's characteristic strips	declared	parts/02-mathematical-methods/08-pdes.tex:480
`depends_on`	←	Gradient catastrophe	declared	parts/02-mathematical-methods/08-pdes.tex:2479
`depends_on`	←	Why a compression wave must steepen	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:3114
`depends_on`	←	What Part II owes this chapter	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:127
`proves`	←	ch:08-pdes@proof-5	declared	parts/02-mathematical-methods/08-pdes.tex:438