theorem 16.32 Transversality

open in the book · parts/02-mathematical-methods/14-calculus-of-variations.tex:884 · p. 641

Rests on

depends_on equation 16.20 eq:calcvar-euler-lagrange ¶
depends_on theorem 16.31 Natural boundary condition ¶
- depends_on equation 16.15 eq:calcvar-first-variation ¶
- depends_on lemma 16.18 Fundamental lemma ¶
  - 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 theorem 7.40 Continuous functions are integrable ¶ ↺
  - proves proof ch:14-calculus-of-variations@proof-8 ¶
- 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 proposition 7.30 Leibniz rule ¶
      - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      - 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 16.13 Weak and strong extrema ¶
      - depends_on definition 16.12 The two norms; weak and strong neighbourhoods ¶
    - depends_on lemma 7.33 Fermat: interior extremum ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on proposition 7.18 Two-sided limit from one-sided limits ¶
      - 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 ¶
- proves proof ch:14-calculus-of-variations@proof-16 ¶
proves proof ch:14-calculus-of-variations@proof-17 ¶

Supports

depends_on corollary 16.33 Weierstrass–Erdmann corner conditions ¶
- 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-euler-lagrange	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:891
`depends_on`	→	Natural boundary condition	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:891
`depends_on`	←	Weierstrass–Erdmann corner conditions	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:935
`proves`	←	ch:14-calculus-of-variations@proof-17	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:894