lemma 30.26 Isotropic Cartesian tensors of rank four

open in the book · parts/03-classical-mechanics/13-continuum-elasticity.tex:804 · p. 1016

Rests on

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 ¶
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 definition 4.48 Symmetric group ¶
      - depends_on definition 3.47 Bijective map ¶
    - depends_on definition 5.18 Inner product ¶
      - depends_on definition 4.33 Vector space ¶
    - depends_on definition 5.27 Orthonormal basis ¶
      - depends_on definition 5.26 Orthogonal basis ¶
      - depends_on definition 5.25 Unit vector ¶
  - 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 definition 5.32 Levi–Civita symbol; cross product ¶ ↺
    - depends_on equation 5.162 eq:rep-special-orthogonal ¶
  - 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 ¶
    - depends_on definition 5.39 Kernel, image, nullity, rank ¶
      - depends_on definition 5.37 Linear transformation ¶
      - depends_on definition 5.7 Vector subspace ¶
      - proves proof ch:03-linear-algebra-representations@proof-12 ¶
    - depends_on proposition 5.48 Existence, uniqueness, and linearity of the inverse ¶
      - depends_on definition 5.47 Inverse of a linear transformation ¶
      - depends_on definition 5.37 Linear transformation ¶ ↺
      - proves proof ch:03-linear-algebra-representations@proof-17 ¶
    - depends_on theorem 5.40 Rank–nullity ¶
      - depends_on definition 5.15 Basis ¶
      - depends_on definition 5.39 Kernel, image, nullity, rank ¶ ↺
      - depends_on lemma 5.38 Exchange and completion ¶
      - proves proof ch:03-linear-algebra-representations@proof-13 ¶
    - proves proof ch:03-linear-algebra-representations@proof-18 ¶
  - proves proof ch:03-linear-algebra-representations@proof-56 ¶
- proves proof ch:03-linear-algebra-representations@proof-58 ¶
proves proof ch:13-continuum-elasticity@prooflink-1 ¶

Supports

depends_on phenomenon 30.28 Hooke's law ¶
- depends_on phenomenon 30.32 Poisson contraction ¶
  - depends_on proposition 30.33 Conversion among the moduli ¶
    - depends_on definition 30.63 Flexural rigidity ¶
      - depends_on proposition A.684 The plate energy ¶
      - depends_on proposition A.682 The plane-stress law ¶
      - depends_on proposition 30.66 Chladni scaling ¶
      - depends_on remark A.704 $E^{*}$ in two roles, and one trap ¶
      - depends_on remark A.686 Why the operator comes out as $\nabla^{4}$ ¶
      - depends_on remark A.683 Where the factor $1-\nu^{2}$ comes from ¶
      - depends_on remark 30.68 Shells carry load in a different way ¶
      - depends_on theorem 30.64 The Kirchhoff plate equation ¶
    - depends_on example 30.35 From single crystals to the isotropic aggregate ¶
    - depends_on proposition A.682 The plane-stress law ¶ ↺
    - depends_on proposition 30.44 The speed ratio is fixed by Poisson's ratio ¶
      - depends_on example 30.45 Elastic waves in steel, and the distance to a source ¶
    - depends_on remark 30.36 How the moduli are actually measured ¶
- depends_on remark 30.78 Anharmonicity: what the quadratic energy also suppresses ¶

Neighborhood

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partly declared
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Edges

type	direction	node	provenance	where
`depends_on`	→	Tensor under orthogonal transformations	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:813
`depends_on`	→	Isotropic Cartesian tensors of rank at most four	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:813
`depends_on`	←	Hooke's law	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:868
`proves`	←	ch:13-continuum-elasticity@prooflink-1	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:816