lemma A.744 Vanishing of the pressure–velocity correlation

open in the book · appendices/A-long-proofs.tex:36120 · p. 3155

Rests on

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 equation 7.124 eq:ana-nabla-gradient ¶
      - depends_on proposition 7.104 Chain rule in several variables ¶
      - proves proof ch:14-fluid-dynamics@proof-1 ¶
    - depends_on theorem 7.133 Gauss ¶
      - depends_on definition 7.127 Simple regions ¶
      - depends_on remark 7.128 What the derivations below take as given ¶
      - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      - proves proof ch:05-real-analysis@proof-79 ¶
    - 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 definition 4.48 Symmetric group ¶
      - depends_on definition 5.18 Inner product ¶
      - depends_on definition 5.27 Orthonormal basis ¶
    - 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 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-432 ¶
proves proof app:A-long-proofs@proof-434 ¶

Supports

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

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`	→	Incompressible flow	declared	appendices/A-long-proofs.tex:36125
`depends_on`	→	Isotropic representation	declared	appendices/A-long-proofs.tex:36125
`depends_on`	←	Kármán–Howarth–Monin relation	declared	appendices/A-long-proofs.tex:36166
`proves`	←	app:A-long-proofs@proof-434	declared	appendices/A-long-proofs.tex:36128