lemma 16.18 Fundamental lemma

open in the book · parts/02-mathematical-methods/14-calculus-of-variations.tex:537 · p. 637

Rests on

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 7.16 Limit ¶
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
    - depends_on definition 7.97 Partial derivative; gradient ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 5.15 Basis ¶
  - 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 theorem 7.7 Bolzano–Weierstrass ¶
      - 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 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 ¶

Supports

depends_on theorem A.688 The Kirchhoff plate equation ¶
- depends_on remark A.690 Poisson's count, and why it cannot be right ¶
- 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 16.36 Euler–Lagrange equations for several independent variables ¶
- depends_on definition 44.6 Stress–energy tensor ¶
  - depends_on definition 44.36 Energy conditions ¶
    - depends_on proposition 44.38 The electromagnetic field satisfies NEC, WEC, SEC and DEC ¶
    - depends_on proposition 44.37 Energy conditions for the perfect fluid ¶
    - depends_on proposition 44.39 SEC is the attraction condition ¶
  - depends_on proposition 44.17 Stress–energy of the electromagnetic field ¶
    - depends_on proposition 44.38 The electromagnetic field satisfies NEC, WEC, SEC and DEC ¶ ↺
  - depends_on theorem 44.19 Covariant conservation of stress–energy ¶
    - depends_on proposition 44.20 Perfect-fluid equations of motion ¶
      - depends_on corollary 44.21 Dust flows on geodesics ¶
  - depends_on theorem 44.9 Variation of the Einstein–Hilbert action ¶
    - depends_on proposition 44.11 The boundary term, and the Gibbons–Hawking–York action ¶
    - depends_on proposition 44.30 The cosmological term ¶
      - depends_on proposition 44.31 de Sitter geometry ¶
- depends_on proposition 28.42 The wave equation of a stretched string ¶
depends_on theorem 16.38 Euler–Poisson equation ¶
- depends_on theorem 30.56 The beam equation ¶
  - depends_on example 30.58 A steel ruler ¶
  - depends_on proposition 30.57 Bending waves are dispersive ¶
    - depends_on example 30.58 A steel ruler ¶ ↺
    - depends_on remark 30.59 Where Euler–Bernoulli fails ¶
- depends_on theorem 30.64 The Kirchhoff plate equation ¶
  - depends_on phenomenon 30.65 Chladni figures ¶
  - depends_on proposition 30.66 Chladni scaling ¶
    - depends_on phenomenon 30.65 Chladni figures ¶ ↺
  - depends_on remark 30.67 Germain, Kirchhoff, and the edge conditions ¶
depends_on theorem 16.41 Multiplier rule for pointwise constraints ¶
- depends_on proposition 21.56 The two prescriptions and their difference ¶
depends_on theorem 16.31 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 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 ¶
depends_on theorem 25.19 Hamilton's equations from the first-order action ¶
depends_on theorem 25.22 Faddeev–Jackiw equations and brackets ¶
- depends_on example 25.24 A charged particle in a strong magnetic field ¶

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`	→	Variation; the first variation	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:544
`depends_on`	→	Continuous functions are integrable	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:544
`depends_on`	←	The Kirchhoff plate equation	declared	appendices/A-long-proofs.tex:33446
`depends_on`	←	Euler–Lagrange equations for several independent variables	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1024
`depends_on`	←	Euler–Poisson equation	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1095
`depends_on`	←	Multiplier rule for pointwise constraints	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1208
`depends_on`	←	Natural boundary condition	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:861
`depends_on`	←	Hamilton's equations from the first-order action	declared	parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:748
`depends_on`	←	Faddeev–Jackiw equations and brackets	declared	parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:863
`proves`	←	ch:14-calculus-of-variations@proof-8	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:547