theorem 31.24 Bernoulli

open in the book · parts/03-classical-mechanics/14-fluid-dynamics.tex:644 · p. 1048

Rests on

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 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 ¶
      - 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 ¶
    - proves proof ch:14-fluid-dynamics@proof-1 ¶
  - depends_on theorem 7.133 Gauss ¶
    - 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-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-12 ¶

Supports

depends_on corollary A.799 The impressed pressure gradient ¶
depends_on lemma 31.67 Blasius' force formula ¶
- depends_on theorem 31.68 Kutta–Joukowski ¶
  - depends_on proposition 31.70 The lift-curve slope of a thin aerofoil ¶
  - depends_on remark 31.69 The Kutta condition, and where the circulation comes from ¶
  - depends_on theorem A.737 The lift-curve slope of the flat plate ¶
depends_on phenomenon 31.25 Pressure falls where the flow speeds up ¶
- depends_on example 31.26 The Venturi meter ¶
- depends_on remark 31.27 What the 1738 source contains ¶
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`	→	eq:fluid-lamb-identity	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:660
`depends_on`	→	Euler's equations of motion	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:660
`depends_on`	←	The impressed pressure gradient	declared	appendices/A-long-proofs.tex:39173
`depends_on`	←	Blasius' force formula	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:2114
`depends_on`	←	Pressure falls where the flow speeds up	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:711
`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-12	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:663