definition 16.13 Weak and strong extrema

open in the book · parts/02-mathematical-methods/14-calculus-of-variations.tex:415 · p. 635

ground object -- no derivation owed

Rests on

depends_on definition 16.12 The two norms; weak and strong neighbourhoods ¶
- depends_on definition 16.11 Admissible class; functional ¶
  - depends_on definition 7.98 Functions of class $C^{1}$ ¶
    - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.16 Limit ¶
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
    - depends_on definition 7.97 Partial derivative; gradient ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 5.15 Basis ¶
  - depends_on theorem 7.40 Continuous functions are integrable ¶
    - depends_on definition 7.39 Darboux sums and the definite integral ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶
    - depends_on theorem 7.24 Extreme value theorem ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
      - depends_on proposition 7.22 Sequential characterization ¶
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - proves proof ch:05-real-analysis@proof-9 ¶
    - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
      - depends_on proposition 7.22 Sequential characterization ¶ ↺
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶ ↺
      - proves proof ch:05-real-analysis@proof-10 ¶
    - proves proof ch:05-real-analysis@proof-23 ¶
- depends_on definition 6.24 Metric ¶
  - depends_on definition 3.43 Map ¶
    - depends_on definition 3.24 Quantifiers ¶
      - depends_on definition 3.23 Predicate ¶
    - depends_on definition 3.28 Set ¶
  - depends_on definition 3.38 Ordered pair and Cartesian product ¶

Supports

depends_on example 16.14 A weak minimum that is not strong ¶
depends_on proposition 16.16 Stationarity is necessary ¶
- depends_on theorem 16.22 Euler–Lagrange ¶
  - depends_on definition 16.62 Field of extremals; slope function ¶
    - depends_on definition 23.52 Lagrangian family; caustic ¶
      - depends_on proposition 23.54 The geometrical amplitude diverges ¶
      - 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 ¶ ↺
  - depends_on phenomenon 30.52 Euler buckling ¶
    - depends_on example 30.53 A steel rod buckles at a twentieth of its crushing load ¶
    - depends_on remark 30.54 Other end conditions, and what buckling is variationally ¶
    - depends_on remark 30.55 Beyond the critical load: the elastica ¶
    - depends_on remark 30.68 Shells carry load in a different way ¶
  - depends_on proposition 29.39 Steady precession ¶
  - depends_on theorem 16.35 System of Euler–Lagrange equations ¶
    - depends_on proposition 16.6 The isoperimetric extremal is a circle ¶
    - depends_on theorem 16.41 Multiplier rule for pointwise constraints ¶
      - depends_on proposition 21.56 The two prescriptions and their difference ¶
  - depends_on theorem 16.31 Natural boundary condition ¶
    - depends_on remark 30.67 Germain, Kirchhoff, and the edge conditions ¶
    - depends_on theorem A.691 Kirchhoff's edge conditions and the corner force ¶
      - depends_on remark A.692 The corner force is real, and you can feel it ¶
    - depends_on theorem 16.32 Transversality ¶
      - depends_on corollary 16.33 Weierstrass–Erdmann corner conditions ¶
- depends_on theorem 16.51 Legendre's necessary condition ¶
  - depends_on corollary 16.65 Sufficiency for a weak minimum ¶
  - depends_on theorem 16.55 Jacobi's necessary condition ¶
    - depends_on proposition 23.53 A caustic point is a conjugate point ¶ ↺
    - depends_on remark 30.54 Other end conditions, and what buckling is variationally ¶ ↺
depends_on theorem 16.60 Weierstrass' necessary 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`	→	The two norms; weak and strong neighbourhoods	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:421
`depends_on`	←	A weak minimum that is not strong	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:451
`depends_on`	←	Stationarity is necessary	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:478
`depends_on`	←	Weierstrass' necessary condition	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1761