theorem 16.36 Euler–Lagrange equations for several independent variables

open in the book · parts/02-mathematical-methods/14-calculus-of-variations.tex:1007 · p. 642

Rests on

depends_on equation 16.31 eq:calcvar-field-functional ¶
depends_on lemma 16.18 Fundamental lemma ¶
- 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 7.97 Partial derivative; gradient ¶
    - 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 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 ¶
depends_on theorem 7.133 Gauss ¶
- 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 definition 6.5 Open cover ¶
      - depends_on definition 6.2 Open set ¶
      - depends_on equation 3.51 eq:set-indexed ¶
- depends_on remark 7.128 What the derivations below take as given ¶
  - depends_on definition 7.125 Multiple integral ¶
    - depends_on axiom 7.1 Completeness of $\R$ ¶
    - depends_on definition 7.39 Darboux sums and the definite 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 ¶
  - 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 definition 7.16 Limit ¶
      - depends_on definition 7.9 Real function ¶
  - 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 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-79 ¶
proves proof ch:14-calculus-of-variations@proof-20 ¶

Supports

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 ¶

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-field-functional	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1024
`depends_on`	→	Fundamental lemma	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1024
`depends_on`	→	Gauss	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1024
`depends_on`	←	Stress–energy tensor	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:245
`depends_on`	←	The wave equation of a stretched string	declared	parts/03-classical-mechanics/11-oscillations-waves.tex:1461
`proves`	←	ch:14-calculus-of-variations@proof-20	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1028