theorem 16.31 Natural boundary condition

open in the book · parts/02-mathematical-methods/14-calculus-of-variations.tex:853 · p. 640

Rests on

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 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 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 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 16.13 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 ¶
proves proof ch:14-calculus-of-variations@proof-16 ¶

Supports

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 16.55 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-first-variation	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:861
`depends_on`	→	Fundamental lemma	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:861
`depends_on`	→	Euler–Lagrange	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:861
`depends_on`	←	Germain, Kirchhoff, and the edge conditions	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:2412
`depends_on`	←	Kirchhoff's edge conditions and the corner force	declared	appendices/A-long-proofs.tex:33593
`depends_on`	←	Transversality	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:891
`proves`	←	ch:14-calculus-of-variations@proof-16	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:865