proposition 52.24 The p-mode cavity

open in the book · parts/05-general-relativity-cosmology/11-stellar-structure-nucleosynthesis.tex:1857 · p. 1408

Rests on

depends_on equation 52.20 eq:stellar-brunt ¶
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 ¶
depends_on theorem 9.78 Semiclassical quantisation of a well ¶
- depends_on corollary 9.62 The semiclassical wave ¶
  - depends_on equation 9.62 eq:slt-wkb-ansatz ¶
  - depends_on equation 9.61 eq:slt-wkb-kappa ¶
  - depends_on proposition 9.61 Eikonal and transport equations ¶
    - depends_on equation 9.64 eq:slt-wkb-series ¶
    - depends_on proposition 9.60 Exact Riccati form ¶
      - depends_on equation 9.62 eq:slt-wkb-ansatz ¶ ↺
      - depends_on equation 9.60 eq:slt-wkb-normal-form ¶
      - proves proof ch:07-odes-sturm-liouville@proof-30 ¶
    - proves proof ch:07-odes-sturm-liouville@proof-31 ¶
  - proves proof ch:07-odes-sturm-liouville@proof-32 ¶
- depends_on equation 9.59 eq:slt-wkb-schrodinger ¶
- depends_on theorem 9.72 Connection formulae ¶
  - depends_on corollary 9.62 The semiclassical wave ¶ ↺
  - depends_on equation 9.79 eq:slt-wkb-airy-ai ¶
  - depends_on equation 9.80 eq:slt-wkb-airy-bi ¶
  - depends_on equation 9.76 eq:slt-wkb-airy-equation ¶
  - depends_on equation 9.74 eq:slt-wkb-airy-scale ¶
  - proves proof ch:07-odes-sturm-liouville@proof-37 ¶
- proves proof ch:07-odes-sturm-liouville@proof-40 ¶
proves proof ch:11-stellar-structure-nucleosynthesis@proof-20 ¶

Supports

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 ¶

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:stellar-brunt	declared	parts/05-general-relativity-cosmology/11-stellar-structure-nucleosynthesis.tex:1883
`depends_on`	→	Conservation of mass	declared	parts/05-general-relativity-cosmology/11-stellar-structure-nucleosynthesis.tex:1883
`depends_on`	→	Euler's equations of motion	declared	parts/05-general-relativity-cosmology/11-stellar-structure-nucleosynthesis.tex:1883
`depends_on`	→	Semiclassical quantisation of a well	declared	parts/05-general-relativity-cosmology/11-stellar-structure-nucleosynthesis.tex:1883
`depends_on`	←	The Sun rings, and its interior can be inverted	declared	parts/05-general-relativity-cosmology/11-stellar-structure-nucleosynthesis.tex:1963
`proves`	←	ch:11-stellar-structure-nucleosynthesis@proof-20	declared	parts/05-general-relativity-cosmology/11-stellar-structure-nucleosynthesis.tex:1887