phenomenon 31.88 Sound travels at the adiabatic speed

open in the book · parts/03-classical-mechanics/14-fluid-dynamics.tex:3019 · p. 1076

phenomenon-tested: no experiment declares a \tests edge to it and it is not listed in untested.txt

Rests on

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 ¶
      - 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 ¶
- proves proof ch:14-fluid-dynamics@proof-5 ¶
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 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-44 ¶

Supports

depends_on remark 31.89 The Mach number and the regimes it separates ¶
- depends_on phenomenon 31.93 The Mach cone ¶
depends_on remark 31.90 Why a compression wave must steepen ¶

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`	→	Sur la vitesse du son dans l'air et dans l'eau	derived	parts/03-classical-mechanics/14-fluid-dynamics.tex:3027
`cites`	→	Philosophiæ Naturalis Principia Mathematica	derived	parts/03-classical-mechanics/14-fluid-dynamics.tex:3026
`depends_on`	→	Conservation of mass	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:3035
`depends_on`	→	Euler's equations of motion	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:3035
`depends_on`	←	The Mach number and the regimes it separates	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:3093
`depends_on`	←	Why a compression wave must steepen	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:3114
`proves`	←	ch:14-fluid-dynamics@proof-44	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:3038