lemma 16.20 Mixed form

open in the book · parts/02-mathematical-methods/14-calculus-of-variations.tex:605 · p. 638

Rests on

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 definition 7.9 Real function ¶
      - depends_on definition 3.43 Map ¶
  - depends_on proposition 7.27 Differentiable implies continuous ¶
    - 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 7.26 Derivative of a function at a point ¶
      - depends_on definition 7.16 Limit ¶ ↺
      - depends_on definition 7.9 Real function ¶ ↺
    - depends_on proposition 7.6 Algebra of limits ¶
      - depends_on definition 7.4 Convergence ¶
      - depends_on proposition 7.3 Triangle inequality ¶
      - proves proof ch:05-real-analysis@proof-3 ¶
    - proves proof ch:05-real-analysis@proof-11 ¶
  - 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 definition 7.9 Real function ¶ ↺
  - 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 corollary 7.36 cor:ana-mvt-consequences ¶
      - depends_on theorem 7.35 Mean value theorem ¶
      - proves proof ch:05-real-analysis@proof-20 ¶
    - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
  - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
    - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
    - 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 theorem 7.24 Extreme value theorem ¶ ↺
    - proves proof ch:05-real-analysis@proof-24 ¶
  - 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 ¶

Supports

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 23.55 The fold, and the quarter-power law ¶
      - depends_on remark 23.59 What is observed ¶
    - 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 16.55 Jacobi's necessary condition ¶
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
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inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Substitution and integration by parts	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:612
`depends_on`	→	du Bois-Reymond	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:612
`depends_on`	←	System of Euler–Lagrange equations	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:993
`depends_on`	←	Euler–Lagrange	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:656
`depends_on`	←	Euler's rule for integral constraints	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1292
`proves`	←	ch:14-calculus-of-variations@proof-10	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:615