theorem 10.55 d'Alembert's formula

open in the book · parts/02-mathematical-methods/08-pdes.tex:1452 · p. 355

Rests on

depends_on proposition 10.13 Canonical form in two variables ¶
- 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-4 ¶
- proves proof ch:08-pdes@prooflink-1 ¶
depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
- depends_on definition 7.41 Antiderivative ¶
  - depends_on corollary 7.36 cor:ana-mvt-consequences ¶
    - depends_on theorem 7.35 Mean value theorem ¶ ↺
    - proves proof ch:05-real-analysis@proof-20 ¶
  - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
- depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
  - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
  - depends_on theorem 7.40 Continuous functions are integrable ¶
    - depends_on definition 7.39 Darboux sums and the definite integral ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶
    - depends_on theorem 7.24 Extreme value theorem ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
      - depends_on proposition 7.22 Sequential characterization ¶
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - proves proof ch:05-real-analysis@proof-9 ¶
    - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
      - depends_on proposition 7.22 Sequential characterization ¶ ↺
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶ ↺
      - proves proof ch:05-real-analysis@proof-10 ¶
    - proves proof ch:05-real-analysis@proof-23 ¶
  - depends_on theorem 7.24 Extreme value theorem ¶ ↺
  - proves proof ch:05-real-analysis@proof-24 ¶
- proves proof ch:05-real-analysis@proof-25 ¶
proves proof ch:08-pdes@proof-19 ¶

Supports

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

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Edges

type	direction	node	provenance	where
`depends_on`	→	Canonical form in two variables	declared	parts/02-mathematical-methods/08-pdes.tex:1460
`depends_on`	→	Fundamental theorem of calculus, II	declared	parts/02-mathematical-methods/08-pdes.tex:1460
`depends_on`	←	Domain of dependence and influence	declared	parts/02-mathematical-methods/08-pdes.tex:1506
`depends_on`	←	Kirchhoff's formula	declared	parts/02-mathematical-methods/08-pdes.tex:1620
`proves`	←	ch:08-pdes@proof-19	declared	parts/02-mathematical-methods/08-pdes.tex:1463