theorem 30.56 The beam equation

open in the book · parts/03-classical-mechanics/13-continuum-elasticity.tex:2043 · p. 1030

Rests on

depends_on proposition 30.51 Moment and curvature ¶
- depends_on definition 30.31 The engineering moduli ¶
  - depends_on definition 30.20 Pressure and deviatoric stress ¶
    - depends_on proposition 30.19 The stress tensor is symmetric ¶
      - depends_on definition 30.17 Traction ¶
      - depends_on theorem 30.18 Cauchy: the stress tensor exists ¶
      - proves proof ch:13-continuum-elasticity@proof-8 ¶
    - depends_on theorem 5.79 Spectral theorem for a self-adjoint operator ¶
      - depends_on corollary 5.72 Existence of an eigenvalue over $\C$ ¶
      - depends_on definition 5.41 Adjoint ¶
      - depends_on theorem 5.40 Rank–nullity ¶
      - proves proof ch:03-linear-algebra-representations@proof-31 ¶
  - depends_on equation 30.21 eq:elast-hooke-isotropic ¶
- depends_on equation 30.21 eq:elast-hooke-isotropic ¶ ↺
- proves proof ch:13-continuum-elasticity@proof-27 ¶
depends_on theorem 16.38 Euler–Poisson equation ¶
- 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 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 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 ¶
proves proof ch:13-continuum-elasticity@proof-29 ¶

Supports

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 ¶

Neighborhood

Every logical edge within two steps of this node.

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partly declared
a check failed
not graded
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inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Moment and curvature	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:2054
`depends_on`	→	Euler–Poisson equation	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:2054
`depends_on`	←	A steel ruler	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:2144
`depends_on`	←	Bending waves are dispersive	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:2103
`proves`	←	ch:13-continuum-elasticity@proof-29	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:2057