remark 23.59 What is observed

open in the book · parts/03-classical-mechanics/06-hamilton-jacobi.tex:2074 · p. 847

remark -- no derivation owed by its kind

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depends_on proposition 23.57 The wave resolution at a turning surface ¶
- depends_on remark 23.51 Where the expansion breaks, and what is owed ¶
  - depends_on proposition 9.65 Validity criterion ¶
    - depends_on definition 9.64 Reduced de Broglie wavelength ¶
      - depends_on equation 9.60 eq:slt-wkb-normal-form ¶
    - depends_on equation 9.64 eq:slt-wkb-series ¶
    - 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 ¶
      - proves proof ch:07-odes-sturm-liouville@proof-31 ¶
    - proves proof ch:07-odes-sturm-liouville@proof-34 ¶
  - depends_on theorem 9.72 Connection formulae ¶
    - 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 ¶ ↺
      - proves proof ch:07-odes-sturm-liouville@proof-32 ¶
    - 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 ¶
- depends_on theorem 9.70 Asymptotics of the Airy functions ¶
  - depends_on definition 9.68 Airy functions ¶
  - depends_on proposition 9.69 The integral solves the Airy equation ¶
    - depends_on definition 9.68 Airy functions ¶ ↺
    - depends_on equation 9.76 eq:slt-wkb-airy-equation ¶ ↺
    - proves proof ch:07-odes-sturm-liouville@proof-35 ¶
  - proves proof ch:07-odes-sturm-liouville@proof-36 ¶
- depends_on theorem 9.72 Connection formulae ¶ ↺
- proves proof ch:06-hamilton-jacobi@proof-20 ¶
depends_on proposition 23.54 The geometrical amplitude diverges ¶
- depends_on definition 23.52 Lagrangian family; caustic ¶
  - depends_on definition 16.62 Field of extremals; slope function ¶
    - depends_on definition 6.17 Simply connected space ¶
      - depends_on definition 6.6 Continuous map ¶
      - depends_on definition 6.15 Path-connected space ¶
    - depends_on theorem 16.22 Euler–Lagrange ¶
      - depends_on equation 16.15 eq:calcvar-first-variation ¶
      - depends_on lemma 16.20 Mixed form ¶
      - depends_on proposition 16.16 Stationarity is necessary ¶
      - proves proof ch:14-calculus-of-variations@proof-11 ¶
  - depends_on definition 23.5 Complete integral ¶
    - depends_on equation 23.2 eq:hj-equation ¶
- depends_on equation 23.66 eq:hj-transport ¶
- 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:06-hamilton-jacobi@proof-17 ¶

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type	direction	node	provenance	where
`cites` (history)	→	On the intensity of light in the neighbourhood of a caustic	derived	parts/03-classical-mechanics/06-hamilton-jacobi.tex:2081
`cites` (context)	→	Gravitational microlensing by the galactic halo	derived	parts/03-classical-mechanics/06-hamilton-jacobi.tex:2090
`cites` (evidence)	→	0957 + 561 A, B: twin quasistellar objects or gravitational lens?	derived	parts/03-classical-mechanics/06-hamilton-jacobi.tex:2092
`depends_on`	→	The wave resolution at a turning surface	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:2096
`depends_on`	→	The geometrical amplitude diverges	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:2096