proposition 30.25 Twenty-one constants

open in the book · parts/03-classical-mechanics/13-continuum-elasticity.tex:781 · p. 1016

Rests on

depends_on definition 30.24 The elasticity tensor ¶
- depends_on definition 30.6 Linear strain and infinitesimal rotation ¶
  - depends_on definition 13.91 Symmetric and antisymmetric parts ¶
  - depends_on equation 30.2 eq:elast-displacement ¶
- 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 ¶
depends_on proposition 7.105 Clairaut–Schwarz ¶
- 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 definition 3.43 Map ¶
  - depends_on equation 7.7 eq:ana-limit-left ¶
  - depends_on equation 7.5 eq:ana-limit-right ¶
- depends_on theorem 7.35 Mean value theorem ¶
  - depends_on proposition 7.29 Linearity ¶
    - depends_on definition 7.11 Sum of functions ¶
      - depends_on definition 7.9 Real function ¶ ↺
    - depends_on equation 7.14 eq:ana-derivh ¶
    - depends_on proposition 7.6 Algebra of limits ¶
      - depends_on definition 7.4 Convergence ¶
      - depends_on proposition 7.3 Triangle inequality ¶
      - proves proof ch:05-real-analysis@proof-3 ¶
    - proves proof ch:05-real-analysis@proof-13 ¶
  - depends_on theorem 7.34 Rolle ¶
    - depends_on lemma 7.33 Fermat: interior extremum ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on proposition 7.18 Two-sided limit from one-sided limits ¶
      - proves proof ch:05-real-analysis@proof-17 ¶
    - 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 ¶
    - proves proof ch:05-real-analysis@proof-18 ¶
  - proves proof ch:05-real-analysis@proof-19 ¶
- proves proof ch:05-real-analysis@proof-64 ¶
proves proof ch:13-continuum-elasticity@proof-11 ¶

Supports

depends_on definition 30.37 Voigt notation ¶
- depends_on remark 30.39 The rari-constant controversy, decided by measurement ¶
depends_on proposition 30.38 The Cauchy relations ¶
- depends_on remark 30.39 The rari-constant controversy, decided by measurement ¶ ↺
depends_on theorem A.675 Three constants for a cubic crystal, two for an isotropic solid ¶
- depends_on remark A.676 The third cubic constant is measured, and it is not small ¶

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`	→	The elasticity tensor	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:788
`depends_on`	→	Clairaut–Schwarz	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:788
`depends_on`	←	Voigt notation	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1240
`depends_on`	←	The Cauchy relations	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1271
`depends_on`	←	Three constants for a cubic crystal, two for an isotropic solid	declared	appendices/A-long-proofs.tex:32876
`proves`	←	ch:13-continuum-elasticity@proof-11	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:791