theorem 31.74 The interface dispersion relation

open in the book · parts/03-classical-mechanics/14-fluid-dynamics.tex:2363 · p. 1068

Rests on

depends_on proposition 31.28 Potential flow reduces to Laplace's equation ¶
- 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 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 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 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 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 ¶
  - proves proof ch:05-real-analysis@proof-80 ¶
- proves proof ch:14-fluid-dynamics@proof-14 ¶
depends_on theorem 31.24 Bernoulli ¶
- depends_on equation 31.4 eq:fluid-lamb-identity ¶
- 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 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-12 ¶
proves proof ch:14-fluid-dynamics@proof-35 ¶

Supports

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`	→	Potential flow reduces to Laplace's equation	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:2382
`depends_on`	→	Bernoulli	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:2382
`depends_on`	←	Kelvin–Helmholtz instability	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:2488
`depends_on`	←	Rayleigh–Taylor instability	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:2457
`depends_on`	←	Gravity–capillary waves on a free surface	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:3217
`proves`	←	ch:14-fluid-dynamics@proof-35	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:2385