definition 16.62 Field of extremals; slope function

open in the book · parts/02-mathematical-methods/14-calculus-of-variations.tex:1857 · p. 651

ground object -- no derivation owed

Rests on

depends_on definition 6.17 Simply connected space ¶
- depends_on definition 6.6 Continuous map ¶
  - depends_on definition 3.53 Preimage ¶
  - depends_on definition 6.2 Open set ¶
    - depends_on definition 6.1 Topological space ¶
      - depends_on definition 3.31 Empty set ¶
      - depends_on definition 3.35 Union, intersection, difference ¶
      - depends_on definition 3.30 Subset ¶
      - depends_on equation 3.51 eq:set-indexed ¶
  - depends_on definition 6.1 Topological space ¶ ↺
- depends_on definition 6.15 Path-connected space ¶
  - depends_on definition 6.6 Continuous map ¶ ↺
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 corollary 7.44 Substitution and integration by parts ¶
    - depends_on proposition 7.31 Chain rule ¶
      - depends_on definition 7.13 Composite function ¶
      - depends_on proposition 7.27 Differentiable implies continuous ¶
      - depends_on proposition 7.6 Algebra of limits ¶
      - proves proof ch:05-real-analysis@proof-15 ¶
    - depends_on proposition 7.30 Leibniz rule ¶
      - depends_on definition 7.12 Product of functions ¶
      - depends_on equation 7.14 eq:ana-derivh ¶
      - depends_on proposition 7.27 Differentiable implies continuous ¶ ↺
      - depends_on proposition 7.6 Algebra of limits ¶ ↺
      - proves proof ch:05-real-analysis@proof-14 ¶
    - 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-26 ¶
  - depends_on lemma 16.19 du Bois-Reymond ¶
    - depends_on equation 16.18 eq:calcvar-dbr-hyp ¶
    - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶ ↺
    - proves proof ch:14-calculus-of-variations@proof-9 ¶
  - proves proof ch:14-calculus-of-variations@proof-10 ¶
- depends_on proposition 16.16 Stationarity is necessary ¶
  - depends_on definition 16.15 Variation; the first variation ¶
    - depends_on definition 16.11 Admissible class; functional ¶
      - depends_on definition 7.98 Functions of class $C^{1}$ ¶
      - depends_on theorem 7.40 Continuous functions are integrable ¶
    - depends_on equation 7.177 eq:ana-functional-derivative ¶
  - depends_on definition 16.13 Weak and strong extrema ¶
    - depends_on definition 16.12 The two norms; weak and strong neighbourhoods ¶
      - depends_on definition 16.11 Admissible class; functional ¶ ↺
      - depends_on definition 6.24 Metric ¶
  - depends_on lemma 7.33 Fermat: interior extremum ¶
    - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 7.16 Limit ¶
      - depends_on definition 7.9 Real function ¶
    - depends_on proposition 7.18 Two-sided limit from one-sided limits ¶
      - depends_on definition 7.16 Limit ¶ ↺
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
      - proves proof ch:05-real-analysis@proof-6 ¶
    - proves proof ch:05-real-analysis@proof-17 ¶
  - proves proof ch:14-calculus-of-variations@proof-6 ¶
- proves proof ch:14-calculus-of-variations@proof-11 ¶

Supports

depends_on definition 23.52 Lagrangian family; caustic ¶
- depends_on proposition 23.54 The geometrical amplitude diverges ¶
  - depends_on proposition 23.55 The fold, and the quarter-power law ¶
    - depends_on proposition 23.56 The phase jump at a fold ¶
    - depends_on remark 23.58 The caustics that are not turning surfaces ¶
  - depends_on remark 23.59 What is observed ¶
- depends_on proposition 23.53 A caustic point is a conjugate point ¶
depends_on remark 23.8 The action as a function, not a functional ¶
depends_on theorem 16.63 Hilbert's invariant integral ¶
- depends_on remark 23.8 The action as a function, not a functional ¶ ↺
- depends_on theorem 16.64 Weierstrass' sufficient condition ¶
depends_on theorem 16.64 Weierstrass' sufficient condition ¶ ↺

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`	→	Simply connected space	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1866
`depends_on`	→	Euler–Lagrange	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1866
`depends_on`	←	Lagrangian family; caustic	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1790
`depends_on`	←	The action as a function, not a functional	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:296
`depends_on`	←	Hilbert's invariant integral	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1881
`depends_on`	←	Weierstrass' sufficient condition	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1938