theorem 16.43 Euler's rule for integral constraints

open in the book · parts/02-mathematical-methods/14-calculus-of-variations.tex:1279 · p. 645

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 lemma 16.20 Mixed form ¶
- 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 proposition 7.27 Differentiable implies continuous ¶
      - depends_on definition 7.20 Continuity at a point ¶ ↺
      - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
      - depends_on proposition 7.6 Algebra of limits ¶
      - 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 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 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 ¶
depends_on proposition 16.17 The first variation of an integral functional ¶
- depends_on equation 16.11 eq:calcvar-basic-functional ¶
- depends_on equation 16.14 eq:calcvar-gateaux ¶
- proves proof ch:14-calculus-of-variations@proof-7 ¶
proves proof ch:14-calculus-of-variations@proof-24 ¶

Supports

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
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:1292
`depends_on`	→	Mixed form	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1292
`depends_on`	→	The first variation of an integral functional	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1292
`depends_on`	←	Two solved isoperimetric problems	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1361
`depends_on`	←	The hanging chain is a catenary	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:178
`depends_on`	←	The isoperimetric extremal is a circle	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:225
`proves`	←	ch:14-calculus-of-variations@proof-24	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1296