definition 16.11 Admissible class; functional

open in the book · parts/02-mathematical-methods/14-calculus-of-variations.tex:372 · p. 635

ground object -- no derivation owed

Rests on

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 definition 7.2 Absolute value ¶
    - depends_on definition 7.9 Real function ¶
      - depends_on definition 3.43 Map ¶
  - 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 7.16 Limit ¶ ↺
    - depends_on definition 7.9 Real function ¶ ↺
  - depends_on definition 5.15 Basis ¶
    - depends_on definition 5.12 Subspace generated by a set of vectors ¶
      - depends_on definition 5.5 Linear combination ¶
      - depends_on definition 5.7 Vector subspace ¶
      - proves proof ch:03-linear-algebra-representations@proof-3 ¶
    - depends_on definition 5.14 Linear independence ¶
      - depends_on definition 5.5 Linear combination ¶ ↺
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 ¶
      - depends_on definition 7.2 Absolute value ¶ ↺
    - 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 definition A.36 Cut ¶
      - depends_on theorem A.40 Least-upper-bound property ¶
      - proves proof app:A-long-proofs@proof-32 ¶
    - depends_on corollary A.46 Monotone convergence ¶
      - depends_on theorem A.40 Least-upper-bound property ¶ ↺
      - proves proof app:A-long-proofs@proof-33 ¶
    - 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 ¶

Supports

depends_on definition 16.59 Weierstrass excess function ¶
depends_on definition 16.12 The two norms; weak and strong neighbourhoods ¶
- depends_on definition 16.13 Weak and strong extrema ¶
  - depends_on example 16.14 A weak minimum that is not strong ¶
  - depends_on proposition 16.16 Stationarity is necessary ¶
    - depends_on theorem 16.22 Euler–Lagrange ¶
      - depends_on definition 16.62 Field of extremals; slope function ¶
      - depends_on phenomenon 30.52 Euler buckling ¶
      - 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 theorem 16.51 Legendre's necessary condition ¶
      - depends_on corollary 16.65 Sufficiency for a weak minimum ¶
      - depends_on theorem 16.55 Jacobi's necessary condition ¶
  - depends_on theorem 16.60 Weierstrass' necessary condition ¶
depends_on definition 16.15 Variation; the first variation ¶
- depends_on definition 16.83 One-parameter transformation group; variational symmetry ¶
- depends_on lemma 16.18 Fundamental lemma ¶
  - 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 44.17 Stress–energy of the electromagnetic field ¶
      - depends_on theorem 44.19 Covariant conservation of stress–energy ¶
      - depends_on theorem 44.9 Variation of the Einstein–Hilbert action ¶
    - 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 theorem 30.64 The Kirchhoff plate equation ¶
      - depends_on phenomenon 30.65 Chladni figures ¶
      - depends_on proposition 30.66 Chladni scaling ¶
      - 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 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 ¶
- depends_on proposition 16.16 Stationarity is necessary ¶ ↺
- 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 ¶
depends_on proposition 16.68 Weierstrass' counterexample ¶

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`	→	Functions of class $C^{1}$	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:386
`depends_on`	→	Continuous functions are integrable	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:386
`depends_on`	←	Weierstrass excess function	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1751
`depends_on`	←	The two norms; weak and strong neighbourhoods	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:412
`depends_on`	←	Variation; the first variation	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:470
`depends_on`	←	Weierstrass' counterexample	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:2104