proposition A.673 Cubic tensors of rank four

open in the book · appendices/A-long-proofs.tex:32720 · p. 3120

Rests on

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 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 5.162 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 ¶
      - depends_on definition 5.39 Kernel, image, nullity, rank ¶
      - depends_on proposition 5.48 Existence, uniqueness, and linearity of the inverse ¶
      - depends_on theorem 5.40 Rank–nullity ¶
      - 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 app:A-long-proofs@proof-401 ¶
proves proof app:A-long-proofs@proof-402 ¶

Supports

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.

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partly declared
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declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	The octahedral rotation group	declared	appendices/A-long-proofs.tex:32744
`depends_on`	→	The three products are independent	declared	appendices/A-long-proofs.tex:32744
`depends_on`	←	Three constants for a cubic crystal, two for an isotropic solid	declared	appendices/A-long-proofs.tex:32876
`proves`	←	app:A-long-proofs@proof-402	declared	appendices/A-long-proofs.tex:32748