theorem 16.63 Hilbert's invariant integral

open in the book · parts/02-mathematical-methods/14-calculus-of-variations.tex:1869 · p. 652

Rests on

depends_on definition 16.62 Field of extremals; slope function ¶
- depends_on definition 6.17 Simply connected space ¶
  - depends_on definition 6.6 Continuous map ¶
    - depends_on definition 3.53 Preimage ¶
    - depends_on definition 6.2 Open set ¶
      - depends_on definition 6.1 Topological space ¶
    - depends_on definition 6.1 Topological space ¶ ↺
  - depends_on definition 6.15 Path-connected space ¶
    - depends_on definition 6.6 Continuous map ¶ ↺
- 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 proposition 7.30 Leibniz rule ¶
      - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      - 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 16.11 Admissible class; functional ¶
      - depends_on equation 7.177 eq:ana-functional-derivative ¶
    - depends_on definition 16.13 Weak and strong extrema ¶
      - depends_on definition 16.12 The two norms; weak and strong neighbourhoods ¶
    - depends_on lemma 7.33 Fermat: interior extremum ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on proposition 7.18 Two-sided limit from one-sided limits ¶
      - 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 ¶
depends_on equation 16.20 eq:calcvar-euler-lagrange ¶
depends_on proposition 7.136 Properties of conservative fields ¶
- depends_on definition 7.135 Conservative field ¶
  - depends_on equation 7.124 eq:ana-nabla-gradient ¶
- depends_on proposition 7.104 Chain rule in several variables ¶
  - depends_on definition 7.13 Composite function ¶
    - depends_on definition 7.9 Real function ¶
      - depends_on definition 3.43 Map ¶
  - depends_on definition 7.99 Differentiability 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 5.37 Linear transformation ¶
      - depends_on definition 4.33 Vector space ¶
    - depends_on equation 6.12 eq:top-euclidean-metric ¶
  - depends_on definition 7.102 Differential of a function ¶
    - depends_on definition 7.99 Differentiability at a point ¶ ↺
    - 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 ¶
  - proves proof ch:05-real-analysis@proof-63 ¶
- depends_on theorem 7.132 Stokes ¶
  - depends_on proposition 7.104 Chain rule in several variables ¶ ↺
  - depends_on proposition 7.105 Clairaut–Schwarz ¶
    - 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 theorem 7.35 Mean value theorem ¶
      - depends_on proposition 7.29 Linearity ¶
      - depends_on theorem 7.34 Rolle ¶
      - proves proof ch:05-real-analysis@proof-19 ¶
    - proves proof ch:05-real-analysis@proof-64 ¶
  - depends_on theorem 7.131 Green ¶
    - depends_on definition 7.127 Simple regions ¶
      - depends_on definition 7.98 Functions of class $C^{1}$ ¶
      - depends_on definition 6.9 Compact set ¶
    - depends_on remark 7.128 What the derivations below take as given ¶
      - depends_on definition 7.125 Multiple integral ¶
      - depends_on theorem 7.40 Continuous functions are integrable ¶
      - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
    - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶ ↺
    - proves proof ch:05-real-analysis@proof-77 ¶
  - proves proof ch:05-real-analysis@proof-78 ¶
- proves proof ch:05-real-analysis@proof-80 ¶
proves proof ch:14-calculus-of-variations@proof-30 ¶

Supports

depends_on remark 23.8 The action as a function, not a functional ¶
depends_on theorem 16.64 Weierstrass' sufficient condition ¶

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`	→	Field of extremals; slope function	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1881
`depends_on`	→	eq:calcvar-euler-lagrange	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1881
`depends_on`	→	Properties of conservative fields	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1881
`depends_on`	←	The action as a function, not a functional	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:296
`depends_on`	←	Weierstrass' sufficient condition	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1938
`proves`	←	ch:14-calculus-of-variations@proof-30	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1885