proposition 10.13 Canonical form in two variables

open in the book · parts/02-mathematical-methods/08-pdes.tex:300 · p. 342

Rests on

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 definition 10.1 Partial differential equation; order ¶
    - 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 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 theorem 7.34 Rolle ¶
      - 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 ¶ ↺
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 5.37 Linear transformation ¶
    - depends_on definition 5.111 Symmetric and antisymmetric forms ¶
      - depends_on definition 5.110 Bilinear map ¶ ↺
      - depends_on equation 5.137 eq:lin-bilinear-matrix ¶
    - 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 definition 3.43 Map ¶
  - 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 definition 7.16 Limit ¶ ↺
      - depends_on definition 7.9 Real function ¶ ↺
    - depends_on proposition 7.6 Algebra of limits ¶
      - depends_on definition 7.4 Convergence ¶
      - depends_on proposition 7.3 Triangle inequality ¶
      - 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 ¶
- depends_on theorem 5.116 Sylvester's law of inertia ¶ ↺
- proves proof ch:08-pdes@proof-1 ¶
proves proof ch:08-pdes@proof-4 ¶
proves proof ch:08-pdes@prooflink-1 ¶

Supports

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 ¶

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`	→	Characteristic surface	declared	parts/02-mathematical-methods/08-pdes.tex:315
`depends_on`	→	The type is a coordinate invariant	declared	parts/02-mathematical-methods/08-pdes.tex:315
`depends_on`	←	Local canonical form for an elliptic operator with Hölder coefficients	declared	appendices/A-long-proofs.tex:5321
`depends_on`	←	d'Alembert's formula	declared	parts/02-mathematical-methods/08-pdes.tex:1460
`proves`	←	ch:08-pdes@proof-4	declared	parts/02-mathematical-methods/08-pdes.tex:318
`proves`	←	ch:08-pdes@prooflink-1	declared	parts/02-mathematical-methods/08-pdes.tex:422