theorem A.675 Three constants for a cubic crystal, two for an isotropic solid

open in the book · appendices/A-long-proofs.tex:32857 · p. 3121

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 A.673 Cubic tensors of rank four ¶
- depends_on definition A.672 The octahedral rotation group ¶
  - depends_on definition 14.26 The rotation group ¶
  - depends_on definition 5.131 Isotropic Cartesian tensor ¶
    - depends_on definition 5.32 Levi–Civita symbol; cross product ¶
      - depends_on definition 4.48 Symmetric group ¶
      - depends_on definition 5.18 Inner product ¶
      - depends_on definition 5.27 Orthonormal basis ¶
    - depends_on equation 5.162 eq:rep-special-orthogonal ¶
- depends_on lemma A.671 The three products are independent ¶
  - depends_on equation 5.168 eq:rep-isotropic-4 ¶
  - depends_on theorem 5.133 Isotropic Cartesian tensors of rank at most four ¶
    - depends_on lemma 5.33 Determinant through the Levi–Civita symbol ¶
      - depends_on definition 5.32 Levi–Civita symbol; cross product ¶ ↺
      - depends_on equation 5.19 eq:lin-leibniz-det ¶
      - proves proof ch:03-linear-algebra-representations@proof-8 ¶
    - depends_on lemma 5.132 Parity constraint ¶
      - depends_on definition 5.131 Isotropic Cartesian tensor ¶ ↺
      - depends_on equation 5.162 eq:rep-special-orthogonal ¶ ↺
      - proves proof ch:03-linear-algebra-representations@proof-57 ¶
    - depends_on proposition 5.130 An orthogonal transformation is an isometry ¶
      - depends_on definition 5.18 Inner product ¶ ↺
      - depends_on equation 5.44 eq:lin-norm-assoc ¶
      - depends_on proposition 5.49 Injective, surjective, invertible ¶
      - proves proof ch:03-linear-algebra-representations@proof-56 ¶
    - proves proof ch:03-linear-algebra-representations@proof-58 ¶
  - proves proof app:A-long-proofs@proof-401 ¶
- proves proof app:A-long-proofs@proof-402 ¶
depends_on proposition 30.25 Twenty-one constants ¶
- depends_on definition 30.24 The elasticity tensor ¶ ↺
- 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 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 equation 7.14 eq:ana-derivh ¶
      - depends_on proposition 7.6 Algebra of limits ¶
      - proves proof ch:05-real-analysis@proof-13 ¶
    - depends_on theorem 7.34 Rolle ¶
      - depends_on lemma 7.33 Fermat: interior extremum ¶
      - depends_on theorem 7.24 Extreme value theorem ¶
      - 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 ¶
proves proof app:A-long-proofs@proof-403 ¶

Supports

depends_on remark A.676 The third cubic constant is measured, and it is not small ¶

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	The elasticity tensor	declared	appendices/A-long-proofs.tex:32876
`depends_on`	→	Cubic tensors of rank four	declared	appendices/A-long-proofs.tex:32876
`depends_on`	→	Twenty-one constants	declared	appendices/A-long-proofs.tex:32876
`depends_on`	←	The third cubic constant is measured, and it is not small	declared	appendices/A-long-proofs.tex:32962
`proves`	←	app:A-long-proofs@proof-403	declared	appendices/A-long-proofs.tex:32880