lemma A.747 The third-order structure functions

open in the book · appendices/A-long-proofs.tex:36314 · p. 3157

Rests on

depends_on corollary A.743 Two scalar functions instead of a tensor field ¶
- depends_on corollary 31.13 Incompressible flow ¶
  - depends_on theorem 31.12 Conservation of mass ¶
    - depends_on lemma 31.11 Transport theorem for a material volume ¶
      - depends_on definition 31.4 Material derivative ¶
      - depends_on theorem 7.133 Gauss ¶
      - proves proof ch:14-fluid-dynamics@proof-4 ¶
    - proves proof ch:14-fluid-dynamics@proof-5 ¶
  - proves proof ch:14-fluid-dynamics@proof-6 ¶
- depends_on lemma A.742 Isotropic representation ¶
  - 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 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-432 ¶
- proves proof app:A-long-proofs@proof-433 ¶
depends_on lemma A.742 Isotropic representation ¶ ↺
proves proof app:A-long-proofs@proof-437 ¶

Supports

depends_on theorem A.749 Kolmogorov's equation and the four-fifths law ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Two scalar functions instead of a tensor field	declared	appendices/A-long-proofs.tex:36324
`depends_on`	→	Isotropic representation	declared	appendices/A-long-proofs.tex:36324
`depends_on`	←	Kolmogorov's equation and the four-fifths law	declared	appendices/A-long-proofs.tex:36422
`proves`	←	app:A-long-proofs@proof-437	declared	appendices/A-long-proofs.tex:36328