theorem 31.64 Kelvin's circulation theorem

open in the book · parts/03-classical-mechanics/14-fluid-dynamics.tex:2012 · p. 1064

Rests on

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 7.9 Real function ¶
    - depends_on definition 7.99 Differentiability at a point ¶
      - depends_on definition 7.16 Limit ¶
      - depends_on definition 5.37 Linear transformation ¶
      - depends_on equation 6.12 eq:top-euclidean-metric ¶
    - depends_on definition 7.102 Differential of a function ¶
      - depends_on definition 7.99 Differentiability at a point ¶ ↺
      - depends_on definition 7.97 Partial derivative; gradient ¶
    - 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 definition 7.16 Limit ¶ ↺
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
    - depends_on theorem 7.35 Mean value theorem ¶
      - depends_on proposition 7.29 Linearity ¶
      - depends_on theorem 7.34 Rolle ¶
      - proves proof ch:05-real-analysis@proof-19 ¶
    - proves proof ch:05-real-analysis@proof-64 ¶
  - depends_on theorem 7.131 Green ¶
    - depends_on definition 7.127 Simple regions ¶
      - depends_on definition 7.98 Functions of class $C^{1}$ ¶
      - depends_on definition 6.9 Compact set ¶
    - depends_on remark 7.128 What the derivations below take as given ¶
      - depends_on definition 7.125 Multiple integral ¶
      - depends_on theorem 7.40 Continuous functions are integrable ¶
      - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
    - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      - depends_on definition 7.41 Antiderivative ¶
      - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
      - proves proof ch:05-real-analysis@proof-25 ¶
    - 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 definition 30.17 Traction ¶
      - depends_on definition 13.5 Tensor under orthogonal transformations ¶
      - proves proof ch:13-continuum-elasticity@proof-7 ¶
- 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 ¶

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 remark 31.66 Only three things make circulation ¶
depends_on theorem 31.59 Helmholtz's vortex theorems ¶
- depends_on corollary 31.65 Lagrange: an irrotational flow stays irrotational ¶ ↺
- 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
`depends_on`	→	Circulation	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:2019
`depends_on`	→	Euler's equations of motion	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:2019
`depends_on`	←	Lagrange: an irrotational flow stays irrotational	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:2050
`depends_on`	←	Only three things make circulation	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:2084
`depends_on`	←	Helmholtz's vortex theorems	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:1896
`proves`	←	ch:14-fluid-dynamics@proof-29	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:2022