proposition 31.28 Potential flow reduces to Laplace's equation

open in the book · parts/03-classical-mechanics/14-fluid-dynamics.tex:784 · p. 1050

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 equation 10.24 eq:pde-poisson-equation ¶
depends_on proposition 7.136 Properties of conservative fields ¶
- depends_on definition 7.135 Conservative field ¶
  - depends_on equation 7.124 eq:ana-nabla-gradient ¶ ↺
- depends_on proposition 7.104 Chain rule in several variables ¶ ↺
- depends_on theorem 7.132 Stokes ¶
  - depends_on proposition 7.104 Chain rule in several variables ¶ ↺
  - 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 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 ¶
- proves proof ch:05-real-analysis@proof-80 ¶
proves proof ch:14-fluid-dynamics@proof-14 ¶

Supports

depends_on example 31.29 Flow past a sphere ¶
depends_on lemma A.777 The tangential field above a bending plate ¶
- depends_on corollary A.778 The powder collects at the antinodes ¶
depends_on lemma A.733 The circle with incidence and circulation ¶
- depends_on lemma A.735 Transfer of the flow ¶
  - depends_on theorem A.737 The lift-curve slope of the flat plate ¶
depends_on proposition A.797 The outer limit loses a boundary condition ¶
depends_on proposition 31.70 The lift-curve slope of a thin aerofoil ¶
depends_on theorem A.707 D'Alembert's paradox ¶
depends_on theorem 31.74 The interface dispersion relation ¶
- depends_on corollary 31.76 Kelvin–Helmholtz instability ¶
- depends_on corollary 31.75 Rayleigh–Taylor instability ¶
- depends_on theorem 31.94 Gravity–capillary waves on a free surface ¶
  - depends_on phenomenon 31.96 The solitary wave ¶
  - depends_on phenomenon 31.95 The observed dispersion of water waves ¶

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	parts/03-classical-mechanics/14-fluid-dynamics.tex:799
`depends_on`	→	eq:pde-poisson-equation	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:799
`depends_on`	→	Properties of conservative fields	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:799
`depends_on`	←	Flow past a sphere	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:845
`depends_on`	←	The tangential field above a bending plate	declared	appendices/A-long-proofs.tex:38173
`depends_on`	←	The circle with incidence and circulation	declared	appendices/A-long-proofs.tex:35599
`depends_on`	←	The outer limit loses a boundary condition	declared	appendices/A-long-proofs.tex:39026
`depends_on`	←	The lift-curve slope of a thin aerofoil	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:2241
`depends_on`	←	D'Alembert's paradox	declared	appendices/A-long-proofs.tex:34260
`depends_on`	←	The interface dispersion relation	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:2382
`proves`	←	ch:14-fluid-dynamics@proof-14	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:803