theorem 30.40 Navier–Cauchy equations

open in the book · parts/03-classical-mechanics/13-continuum-elasticity.tex:1343 · p. 1022

Rests on

depends_on equation 30.21 eq:elast-hooke-isotropic ¶
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 7.20 Continuity at a point ¶
      - depends_on definition 7.97 Partial derivative; gradient ¶
    - 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 ¶
- 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 7.98 Functions of class $C^{1}$ ¶ ↺
      - depends_on definition 7.125 Multiple integral ¶ ↺
    - depends_on definition 30.2 Representative volume element; continuum field ¶
      - depends_on definition 7.97 Partial derivative; gradient ¶ ↺
  - depends_on definition 13.5 Tensor under orthogonal transformations ¶
    - depends_on definition 13.4 Orthogonal coordinate transformation ¶
      - depends_on definition 13.3 Orthogonal coordinate system ¶
  - proves proof ch:13-continuum-elasticity@proof-7 ¶
- proves proof ch:13-continuum-elasticity@proof-9 ¶
proves proof ch:13-continuum-elasticity@proof-18 ¶

Supports

depends_on phenomenon 30.43 Two bulk wave speeds ¶
- depends_on phenomenon 30.47 The three seismic phases ¶
- depends_on remark 30.46 Why the plane-wave route, and not the potentials ¶
depends_on proposition 30.41 Energy balance and its flux ¶
- depends_on theorem 30.42 Uniqueness in elastodynamics ¶
depends_on proposition 30.50 Love waves need a slow surface layer ¶
depends_on proposition 30.48 Rayleigh's secular equation ¶
- depends_on proposition 30.49 The Rayleigh speed of a Poisson solid ¶
  - depends_on phenomenon 30.47 The three seismic phases ¶ ↺
depends_on remark 30.46 Why the plane-wave route, and not the potentials ¶ ↺

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`	→	eq:elast-hooke-isotropic	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1351
`depends_on`	→	Cauchy's equation of motion	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1351
`depends_on`	←	Two bulk wave speeds	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1461
`depends_on`	←	Energy balance and its flux	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1388
`depends_on`	←	Love waves need a slow surface layer	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1827
`depends_on`	←	Rayleigh's secular equation	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1680
`depends_on`	←	Why the plane-wave route, and not the potentials	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1604
`proves`	←	ch:13-continuum-elasticity@proof-18	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1354