lemma 16.19 du Bois-Reymond

open in the book · parts/02-mathematical-methods/14-calculus-of-variations.tex:574 · p. 637

Rests on

depends_on equation 16.18 eq:calcvar-dbr-hyp ¶
depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
- 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 3.43 Map ¶
  - depends_on definition 7.9 Real function ¶ ↺
- 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 definition 7.20 Continuity at a point ¶
      - depends_on definition 7.4 Convergence ¶
      - proves proof ch:05-real-analysis@proof-7 ¶
    - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - depends_on corollary A.45 Archimedean property and density of $\Q$ ¶
      - depends_on corollary A.46 Monotone convergence ¶
      - proves proof ch:05-real-analysis@proof-4 ¶
    - 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 theorem 7.24 Extreme value theorem ¶ ↺
- proves proof ch:05-real-analysis@proof-24 ¶
proves proof ch:14-calculus-of-variations@proof-9 ¶

Supports

depends_on lemma 16.20 Mixed form ¶
- 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 16.22 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 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 16.43 Euler's rule for integral constraints ¶
  - depends_on example 16.45 Two solved isoperimetric problems ¶
  - depends_on proposition 16.4 The hanging chain is a catenary ¶
  - depends_on proposition 16.6 The isoperimetric extremal is a circle ¶ ↺

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-dbr-hyp	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:581
`depends_on`	→	Fundamental theorem of calculus, I	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:581
`depends_on`	←	Mixed form	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:612
`proves`	←	ch:14-calculus-of-variations@proof-9	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:584