theorem 16.38 Euler–Poisson equation

open in the book · parts/02-mathematical-methods/14-calculus-of-variations.tex:1085 · p. 643

Rests on

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 definition 3.43 Map ¶
  - depends_on proposition 7.27 Differentiable implies continuous ¶
    - 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 7.26 Derivative of a function at a point ¶
      - depends_on definition 7.16 Limit ¶ ↺
      - depends_on definition 7.9 Real function ¶ ↺
    - depends_on proposition 7.6 Algebra of limits ¶
      - depends_on definition 7.4 Convergence ¶
      - depends_on proposition 7.3 Triangle inequality ¶
      - proves proof ch:05-real-analysis@proof-3 ¶
    - 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 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 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 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 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 equation 16.15 eq:calcvar-first-variation ¶
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 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 ¶
proves proof ch:14-calculus-of-variations@proof-21 ¶

Supports

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 ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
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Edges

type	direction	node	provenance	where
`depends_on`	→	Substitution and integration by parts	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1095
`depends_on`	→	eq:calcvar-first-variation	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1095
`depends_on`	→	Fundamental lemma	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1095
`depends_on`	←	The beam equation	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:2054
`depends_on`	←	The Kirchhoff plate equation	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:2293
`proves`	←	ch:14-calculus-of-variations@proof-21	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1099