theorem 16.51 Legendre's necessary condition

open in the book · parts/02-mathematical-methods/14-calculus-of-variations.tex:1520 · p. 648

Rests on

depends_on equation 16.45 eq:calcvar-second-variation ¶
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 definition 7.20 Continuity at a point ¶
      - depends_on definition 7.97 Partial derivative; gradient ¶
    - 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 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 definition 3.43 Map ¶
      - depends_on definition 3.38 Ordered pair and Cartesian product ¶
- 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 7.2 Absolute value ¶
      - depends_on definition 7.9 Real function ¶
    - 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-26 ¶

Supports

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 ¶

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:calcvar-second-variation	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1528
`depends_on`	→	Stationarity is necessary	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1528
`depends_on`	←	Sufficiency for a weak minimum	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1971
`depends_on`	←	Jacobi's necessary condition	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1624
`proves`	←	ch:14-calculus-of-variations@proof-26	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1531