proposition 23.55 The fold, and the quarter-power law

open in the book · parts/03-classical-mechanics/06-hamilton-jacobi.tex:1876 · p. 845

Rests on

depends_on equation 23.86 eq:hj-caustic-jacobian ¶
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 ¶
proves proof ch:06-hamilton-jacobi@proof-18 ¶

Supports

depends_on proposition 23.56 The phase jump at a fold ¶
depends_on remark 23.58 The caustics that are not turning surfaces ¶

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:hj-caustic-jacobian	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1891
`depends_on`	→	The geometrical amplitude diverges	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1891
`depends_on`	←	The phase jump at a fold	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1910
`depends_on`	←	The caustics that are not turning surfaces	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1987
`proves`	←	ch:06-hamilton-jacobi@proof-18	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1894