proposition A.746 The Kármán–Howarth equation in classical form

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 theorem A.745 Kármán–Howarth–Monin relation ¶
  • depends_on lemma A.744 Vanishing of the pressure–velocity correlation ¶
    • depends_on corollary 31.13 Incompressible flow ¶ ↺
    • depends_on lemma A.742 Isotropic representation ¶ ↺
    • proves proof app:A-long-proofs@proof-434 ¶
  • depends_on theorem 31.37 The Navier–Stokes equations ¶
    • depends_on theorem 31.36 The Newtonian constitutive relation ¶
      • depends_on definition 31.32 Newtonian fluid; dynamic and kinematic viscosity ¶
        • depends_on definition 31.1 Fluid ¶
        • depends_on definition 31.8 Rate of strain and vorticity ¶
      • depends_on proposition 31.9 Cauchy–Stokes decomposition ¶
        • depends_on definition 31.8 Rate of strain and vorticity ¶ ↺
        • proves proof ch:14-fluid-dynamics@proof-3 ¶
      • proves proof ch:14-fluid-dynamics@proof-17 ¶
    • depends_on theorem 31.22 Euler's equations of motion ¶
      • depends_on definition 31.21 Ideal fluid ¶
        • depends_on definition 31.1 Fluid ¶ ↺
      • depends_on lemma 31.11 Transport theorem for a material volume ¶ ↺
      • depends_on theorem 30.21 Cauchy's equation of motion ¶
        • depends_on theorem 7.133 Gauss ¶ ↺
        • depends_on theorem 30.18 Cauchy: the stress tensor exists ¶
        • proves proof ch:13-continuum-elasticity@proof-9 ¶
      • proves proof ch:14-fluid-dynamics@proof-11 ¶
    • proves proof ch:14-fluid-dynamics@proof-18 ¶
  • proves proof app:A-long-proofs@proof-435 ¶
• proves proof app:A-long-proofs@proof-436 ¶

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