theorem 10.59 Kirchhoff's formula

open in the book · parts/02-mathematical-methods/08-pdes.tex:1610 · p. 357

Rests on

depends_on lemma 10.58 Darboux's equation for spherical means ¶
- 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 theorem 7.133 Gauss ¶
  - depends_on definition 7.127 Simple regions ¶
    - depends_on definition 7.98 Functions of class $C^{1}$ ¶ ↺
    - depends_on definition 6.9 Compact set ¶
      - depends_on definition 6.5 Open cover ¶
  - depends_on remark 7.128 What the derivations below take as given ¶
    - depends_on definition 7.125 Multiple integral ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶
      - depends_on definition 7.39 Darboux sums and the definite integral ¶
    - depends_on theorem 7.40 Continuous functions are integrable ¶
      - depends_on definition 7.39 Darboux sums and the definite integral ¶ ↺
      - depends_on theorem 7.24 Extreme value theorem ¶
      - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
      - proves proof ch:05-real-analysis@proof-23 ¶
    - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶ ↺
  - 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 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 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:05-real-analysis@proof-79 ¶
- proves proof ch:08-pdes@proof-22 ¶
depends_on theorem 10.55 d'Alembert's formula ¶
- 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 proposition 7.105 Clairaut–Schwarz ¶
    - 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 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 ¶
    - 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 ¶ ↺
- proves proof ch:08-pdes@proof-19 ¶
proves proof ch:08-pdes@proof-23 ¶

Supports

depends_on corollary 10.60 Huygens' principle in three space dimensions ¶

Neighborhood

Every logical edge within two steps of this node.

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Edges

type	direction	node	provenance	where
`depends_on`	→	Darboux's equation for spherical means	declared	parts/02-mathematical-methods/08-pdes.tex:1620
`depends_on`	→	d'Alembert's formula	declared	parts/02-mathematical-methods/08-pdes.tex:1620
`depends_on`	←	Huygens' principle in three space dimensions	declared	parts/02-mathematical-methods/08-pdes.tex:1666
`proves`	←	ch:08-pdes@proof-23	declared	parts/02-mathematical-methods/08-pdes.tex:1623