proposition 30.41 Energy balance and its flux

open in the book · parts/03-classical-mechanics/13-continuum-elasticity.tex:1377 · p. 1022

Rests on

depends_on equation 30.23 eq:elast-energy-density ¶
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 ¶
  - 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 ¶
- proves proof ch:13-continuum-elasticity@proof-18 ¶
proves proof ch:13-continuum-elasticity@proof-19 ¶

Supports

depends_on theorem 30.42 Uniqueness in elastodynamics ¶

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-energy-density	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1388
`depends_on`	→	Navier–Cauchy equations	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1388
`depends_on`	←	Uniqueness in elastodynamics	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1416
`proves`	←	ch:13-continuum-elasticity@proof-19	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1391