proposition 31.16 Hydrostatic balance

open in the book · parts/03-classical-mechanics/14-fluid-dynamics.tex:408 · p. 1045

Rests on

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 7.126 Line and surface integrals ¶
      - depends_on definition 7.98 Functions of class $C^{1}$ ¶
      - depends_on definition 7.125 Multiple integral ¶
    - depends_on definition 30.2 Representative volume element; continuum field ¶
      - depends_on definition 7.97 Partial derivative; gradient ¶
  - depends_on definition 13.5 Tensor under orthogonal transformations ¶
    - depends_on definition 13.4 Orthogonal coordinate transformation ¶
      - depends_on definition 13.3 Orthogonal coordinate system ¶
  - proves proof ch:13-continuum-elasticity@proof-7 ¶
depends_on theorem 30.21 Cauchy's equation of motion ¶
- 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 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 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 ¶
- 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-7 ¶

Supports

depends_on example 31.19 The barometric distribution ¶
depends_on phenomenon 49.8 The white-dwarf luminosity function has an edge ¶
depends_on phenomenon 31.18 Archimedes' principle ¶
- depends_on example 31.46 Terminal velocity, and Millikan's droplets ¶
depends_on proposition 52.1 Hydrostatic equilibrium of a self-gravitating sphere ¶
- depends_on phenomenon 52.12 The mass–luminosity relation ¶
  - depends_on phenomenon 52.17 The Hertzsprung–Russell diagram ¶
- depends_on proposition 49.3 Lane–Emden form of hydrostatic equilibrium; restated from the stellar-structure chapter ¶
  - depends_on theorem 49.7 The Chandrasekhar mass ¶
    - depends_on phenomenon 49.6 A maximum mass for a cold degenerate star ¶
- depends_on theorem 52.2 Virial theorem for a self-gravitating sphere ¶
  - depends_on phenomenon 52.4 The Sun has shone for billions of years ¶
    - depends_on phenomenon 52.23 The Sun's energy budget audited reaction by reaction ¶
  - depends_on proposition 52.18 Schönberg–Chandrasekhar limit ¶

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`	→	Fluid	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:416
`depends_on`	→	Cauchy's equation of motion	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:416
`depends_on`	←	The barometric distribution	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:526
`depends_on`	←	The white-dwarf luminosity function has an edge	declared	parts/05-general-relativity-cosmology/08-compact-stars.tex:524
`depends_on`	←	Archimedes' principle	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:477
`depends_on`	←	Hydrostatic equilibrium of a self-gravitating sphere	declared	parts/05-general-relativity-cosmology/11-stellar-structure-nucleosynthesis.tex:55
`proves`	←	ch:14-fluid-dynamics@proof-7	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:419