lemma A.157 Two elementary identities

open in the book · appendices/A-long-proofs.tex:8314 · p. 2868

Rests on

depends_on definition A.150 Kruzhkov entropy pair ¶
- depends_on definition 10.93 Scalar conservation law ¶
  - 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 proposition 10.15 Method of characteristics, first order ¶
    - depends_on definition 10.2 Linear, semilinear, quasilinear ¶ ↺
    - depends_on proposition 7.31 Chain rule ¶
      - depends_on definition 7.13 Composite function ¶
      - depends_on proposition 7.27 Differentiable implies continuous ¶
      - depends_on proposition 7.6 Algebra of limits ¶
      - proves proof ch:05-real-analysis@proof-15 ¶
    - proves proof ch:08-pdes@proof-5 ¶
- depends_on definition 10.101 Entropy pair; entropy inequality ¶
  - depends_on definition 10.99 Entropy condition ¶
    - depends_on definition 10.95 Weak solution of a conservation law ¶
      - depends_on definition 10.93 Scalar conservation law ¶ ↺
      - depends_on definition 10.83 Weak solution ¶
    - depends_on theorem 10.96 Rankine–Hugoniot condition ¶
      - depends_on definition 10.95 Weak solution of a conservation law ¶ ↺
      - depends_on theorem 7.131 Green ¶
      - proves proof ch:08-pdes@proof-40 ¶
  - depends_on definition 10.95 Weak solution of a conservation law ¶ ↺
depends_on equation A.320 eq:app-kruzhkov-bounds ¶
proves proof app:A-long-proofs@proof-93 ¶

Supports

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.162 The limit $\delta\to0$ ¶ ↺

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`	→	Kruzhkov entropy pair	declared	appendices/A-long-proofs.tex:8328
`depends_on`	→	eq:app-kruzhkov-bounds	declared	appendices/A-long-proofs.tex:8328
`depends_on`	←	The doubled inequality	declared	appendices/A-long-proofs.tex:8373
`depends_on`	←	The limit $\delta\to0$	declared	appendices/A-long-proofs.tex:8488
`proves`	←	app:A-long-proofs@proof-93	declared	appendices/A-long-proofs.tex:8331