lemma A.742 Isotropic representation

open in the book · appendices/A-long-proofs.tex:35960 · p. 3154

Rests on

depends_on definition 14.26 The rotation group ¶
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 app:A-long-proofs@proof-432 ¶

Supports

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 ¶ ↺

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	The rotation group	declared	appendices/A-long-proofs.tex:35984
`depends_on`	→	Isotropic Cartesian tensors of rank at most four	declared	appendices/A-long-proofs.tex:35984
`depends_on`	←	Two scalar functions instead of a tensor field	declared	appendices/A-long-proofs.tex:36046
`depends_on`	←	The third-order structure functions	declared	appendices/A-long-proofs.tex:36324
`depends_on`	←	Vanishing of the pressure–velocity correlation	declared	appendices/A-long-proofs.tex:36125
`proves`	←	app:A-long-proofs@proof-432	declared	appendices/A-long-proofs.tex:35987