theorem 31.12 Conservation of mass

open in the book · parts/03-classical-mechanics/14-fluid-dynamics.tex:315 · p. 1044

Rests on

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 7.9 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 ¶
  - 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 7.20 Continuity at a point ¶
      - depends_on definition 7.97 Partial derivative; gradient ¶ ↺
    - depends_on definition 6.9 Compact set ¶
      - depends_on definition 6.5 Open cover ¶
  - depends_on remark 7.128 What the derivations below take as given ¶
    - depends_on definition 7.125 Multiple integral ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶
      - depends_on definition 7.39 Darboux sums and the definite integral ¶
    - depends_on theorem 7.40 Continuous functions are integrable ¶
      - depends_on definition 7.39 Darboux sums and the definite integral ¶ ↺
      - depends_on theorem 7.24 Extreme value theorem ¶
      - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
      - proves proof ch:05-real-analysis@proof-23 ¶
    - 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 corollary 7.36 cor:ana-mvt-consequences ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
    - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
      - depends_on theorem 7.40 Continuous functions are integrable ¶ ↺
      - depends_on theorem 7.24 Extreme value theorem ¶ ↺
      - proves proof ch:05-real-analysis@proof-24 ¶
    - 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 ¶

Supports

depends_on corollary 31.13 Incompressible flow ¶
- depends_on corollary A.743 Two scalar functions instead of a tensor field ¶
  - depends_on lemma A.748 The second-order structure function ¶
    - depends_on theorem A.749 Kolmogorov's equation and the four-fifths law ¶
  - depends_on lemma A.747 The third-order structure functions ¶
    - depends_on theorem A.749 Kolmogorov's equation and the four-fifths law ¶ ↺
  - depends_on proposition A.746 The Kármán–Howarth equation in classical form ¶
- depends_on example 31.15 The stream function of a plane flow ¶
  - depends_on definition A.715 Stokes stream function ¶
    - depends_on lemma A.716 The creeping-flow equation for $\psi$ ¶
      - depends_on lemma A.717 The radial solution ¶
- depends_on lemma A.744 Vanishing of the pressure–velocity correlation ¶
  - depends_on theorem A.745 Kármán–Howarth–Monin relation ¶
    - depends_on proposition A.746 The Kármán–Howarth equation in classical form ¶ ↺
    - depends_on theorem A.749 Kolmogorov's equation and the four-fifths law ¶ ↺
- depends_on proposition 31.28 Potential flow reduces to Laplace's equation ¶
  - 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 ¶
- depends_on remark 31.14 When incompressibility is legitimate ¶
depends_on phenomenon 31.88 Sound travels at the adiabatic speed ¶
- 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 ¶
depends_on proposition 52.24 The p-mode cavity ¶
- depends_on phenomenon 52.25 The Sun rings, and its interior can be inverted ¶
  - depends_on phenomenon 52.22 The solar-neutrino deficit, and its resolution ¶
depends_on theorem 31.91 Rankine–Hugoniot jump conditions ¶
- depends_on phenomenon 31.93 The Mach cone ¶ ↺
- depends_on remark 31.92 Entropy selects the physical branch ¶

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`	→	Transport theorem for a material volume	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:322
`depends_on`	←	Incompressible flow	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:350
`depends_on`	←	Sound travels at the adiabatic speed	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:3035
`depends_on`	←	The p-mode cavity	declared	parts/05-general-relativity-cosmology/11-stellar-structure-nucleosynthesis.tex:1883
`depends_on`	←	Rankine–Hugoniot jump conditions	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:3132
`proves`	←	ch:14-fluid-dynamics@proof-5	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:325