proposition 30.48 Rayleigh's secular equation

open in the book · parts/03-classical-mechanics/13-continuum-elasticity.tex:1665 · p. 1025

Rests on

depends_on definition 30.22 Boundary conditions of elastostatics ¶
- depends_on definition 10.24 Well-posed problem ¶
  - 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 7.97 Partial derivative; gradient ¶
    - depends_on definition 6.2 Open set ¶
      - depends_on definition 6.1 Topological space ¶
  - depends_on definition 6.24 Metric ¶
    - depends_on definition 3.43 Map ¶
      - depends_on definition 3.24 Quantifiers ¶
      - depends_on definition 3.28 Set ¶
    - depends_on definition 3.38 Ordered pair and Cartesian product ¶
- depends_on theorem 30.21 Cauchy's equation of motion ¶
  - 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 remark 7.128 What the derivations below take as given ¶
      - depends_on definition 7.125 Multiple integral ¶
      - depends_on theorem 7.40 Continuous functions are integrable ¶
      - 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 theorem 7.42 Fundamental theorem of calculus, I ¶
      - proves proof ch:05-real-analysis@proof-25 ¶
    - proves proof ch:05-real-analysis@proof-79 ¶
  - depends_on theorem 30.18 Cauchy: the stress tensor exists ¶
    - depends_on definition 30.17 Traction ¶
      - depends_on definition 7.126 Line and surface integrals ¶
      - depends_on definition 30.2 Representative volume element; continuum field ¶
    - depends_on definition 13.5 Tensor under orthogonal transformations ¶
      - depends_on definition 13.4 Orthogonal coordinate transformation ¶
    - proves proof ch:13-continuum-elasticity@proof-7 ¶
  - proves proof ch:13-continuum-elasticity@proof-9 ¶
depends_on theorem 30.40 Navier–Cauchy equations ¶
- depends_on equation 30.21 eq:elast-hooke-isotropic ¶
- depends_on theorem 30.21 Cauchy's equation of motion ¶ ↺
- proves proof ch:13-continuum-elasticity@proof-18 ¶
proves proof ch:13-continuum-elasticity@proof-24 ¶

Supports

depends_on proposition 30.49 The Rayleigh speed of a Poisson solid ¶
- depends_on phenomenon 30.47 The three seismic phases ¶

Neighborhood

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Edges

type	direction	node	provenance	where
`cites`	→	On Waves Propagated along the Plane Surface of an Elastic Solid	derived	parts/03-classical-mechanics/13-continuum-elasticity.tex:1678
`depends_on`	→	Boundary conditions of elastostatics	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1680
`depends_on`	→	Navier–Cauchy equations	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1680
`depends_on`	←	The Rayleigh speed of a Poisson solid	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1780
`proves`	←	ch:13-continuum-elasticity@proof-24	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1683