theorem 31.59 Helmholtz's vortex theorems

open in the book · parts/03-classical-mechanics/14-fluid-dynamics.tex:1880 · p. 1063

Rests on

depends_on theorem 31.64 Kelvin's circulation theorem ¶
- depends_on definition 31.63 Circulation ¶
  - depends_on definition 31.8 Rate of strain and vorticity ¶
    - depends_on definition 13.91 Symmetric and antisymmetric parts ¶
    - depends_on equation 7.126 eq:ana-nabla-curl ¶
  - depends_on theorem 7.132 Stokes ¶
    - depends_on proposition 7.104 Chain rule in several variables ¶
      - depends_on definition 7.13 Composite function ¶
      - depends_on definition 7.99 Differentiability at a point ¶
      - depends_on definition 7.102 Differential of a function ¶
      - proves proof ch:05-real-analysis@proof-63 ¶
    - depends_on proposition 7.105 Clairaut–Schwarz ¶
      - depends_on definition 7.20 Continuity at a point ¶
      - depends_on theorem 7.35 Mean value theorem ¶
      - proves proof ch:05-real-analysis@proof-64 ¶
    - depends_on theorem 7.131 Green ¶
      - 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-77 ¶
    - proves proof ch:05-real-analysis@proof-78 ¶
- 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 theorem 30.18 Cauchy: the stress tensor exists ¶
  - 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 ¶
  - 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-29 ¶
depends_on theorem 31.58 The vorticity transport equation ¶
- depends_on equation 31.5 eq:fluid-curl-cross ¶
- 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 ¶ ↺
  - proves proof ch:14-fluid-dynamics@proof-18 ¶
- proves proof ch:14-fluid-dynamics@proof-27 ¶
proves proof ch:14-fluid-dynamics@proof-28 ¶

Supports

depends_on corollary 31.65 Lagrange: an irrotational flow stays irrotational ¶
- depends_on remark 31.69 The Kutta condition, and where the circulation comes from ¶
depends_on example 31.60 Vortex rings ¶

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
`cites`	→	Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen	derived	parts/03-classical-mechanics/14-fluid-dynamics.tex:1895
`depends_on`	→	Kelvin's circulation theorem	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:1896
`depends_on`	→	The vorticity transport equation	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:1896
`depends_on`	←	Lagrange: an irrotational flow stays irrotational	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:2050
`depends_on`	←	Vortex rings	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:1938
`proves`	←	ch:14-fluid-dynamics@proof-28	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:1899