theorem 5.133 Isotropic Cartesian tensors of rank at most four

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:5735 · p. 164

Rests on

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 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.32 Field ¶
      - depends_on definition 4.31 Module ¶
  - depends_on definition 5.27 Orthonormal basis ¶
    - depends_on definition 5.26 Orthogonal basis ¶
      - depends_on definition 5.15 Basis ¶
      - depends_on definition 5.24 Orthogonal vectors ¶
    - depends_on definition 5.25 Unit vector ¶
      - depends_on definition 5.19 Norm ¶
- 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 4.33 Vector space ¶ ↺
    - depends_on definition 5.7 Vector subspace ¶
      - depends_on definition 5.5 Linear combination ¶
    - 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 ¶ ↺
    - 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 ¶
      - depends_on definition 5.15 Basis ¶ ↺
      - depends_on definition 5.12 Subspace generated by a set of vectors ¶
      - depends_on definition 5.14 Linear independence ¶
      - proves proof ch:03-linear-algebra-representations@proof-11 ¶
    - 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 ¶

Supports

depends_on lemma A.742 Isotropic representation ¶
- depends_on corollary A.743 Two scalar functions instead of a tensor field ¶
  - depends_on lemma A.748 The second-order structure function ¶
    - depends_on theorem A.749 Kolmogorov's equation and the four-fifths law ¶
  - depends_on lemma A.747 The third-order structure functions ¶
    - depends_on theorem A.749 Kolmogorov's equation and the four-fifths law ¶ ↺
  - depends_on proposition A.746 The Kármán–Howarth equation in classical form ¶
- depends_on lemma A.747 The third-order structure functions ¶ ↺
- depends_on lemma A.744 Vanishing of the pressure–velocity correlation ¶
  - depends_on theorem A.745 Kármán–Howarth–Monin relation ¶
    - depends_on proposition A.746 The Kármán–Howarth equation in classical form ¶ ↺
    - depends_on theorem A.749 Kolmogorov's equation and the four-fifths law ¶ ↺
depends_on lemma A.671 The three products are independent ¶
- depends_on proposition A.673 Cubic tensors of rank four ¶
  - 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 ¶
depends_on lemma 30.26 Isotropic Cartesian tensors of rank four ¶
- 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 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 remark 30.36 How the moduli are actually measured ¶
  - depends_on remark 30.78 Anharmonicity: what the quadratic energy also suppresses ¶
depends_on remark A.677 The lower ranks, and why an isotropic solid is not piezoelectric ¶
depends_on remark 30.27 Two results this chapter borrows from Part II ¶
depends_on remark 13.7 What the classification is used for ¶

Neighborhood

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partly declared
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Edges

type	direction	node	provenance	where
`depends_on`	→	Determinant through the Levi–Civita symbol	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5750
`depends_on`	→	Parity constraint	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5750
`depends_on`	→	An orthogonal transformation is an isometry	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5750
`depends_on`	←	Isotropic representation	declared	appendices/A-long-proofs.tex:35984
`depends_on`	←	The three products are independent	declared	appendices/A-long-proofs.tex:32676
`depends_on`	←	Isotropic Cartesian tensors of rank four	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:813
`depends_on`	←	The lower ranks, and why an isotropic solid is not piezoelectric	declared	appendices/A-long-proofs.tex:32985
`depends_on`	←	Two results this chapter borrows from Part II	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:851
`depends_on`	←	What the classification is used for	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:625
`proves`	←	ch:03-linear-algebra-representations@proof-58	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5754