proposition 30.14 Saint-Venant compatibility is necessary

open in the book · parts/03-classical-mechanics/13-continuum-elasticity.tex:393 · p. 1011

Rests on

depends_on definition 30.6 Linear strain and infinitesimal rotation ¶
- depends_on definition 13.91 Symmetric and antisymmetric parts ¶
- depends_on equation 30.2 eq:elast-displacement ¶
depends_on proposition 7.105 Clairaut–Schwarz ¶
- depends_on definition 7.20 Continuity 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 3.43 Map ¶
  - 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 definition 7.11 Sum of functions ¶
      - depends_on definition 7.9 Real function ¶ ↺
    - depends_on equation 7.14 eq:ana-derivh ¶
    - 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-13 ¶
  - depends_on theorem 7.34 Rolle ¶
    - 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 ¶
    - 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 ¶
    - proves proof ch:05-real-analysis@proof-18 ¶
  - proves proof ch:05-real-analysis@proof-19 ¶
- proves proof ch:05-real-analysis@proof-64 ¶
proves proof ch:13-continuum-elasticity@proof-5 ¶

Supports

depends_on theorem 30.15 Compatibility is sufficient on a simply connected body ¶

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`	→	Linear strain and infinitesimal rotation	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:402
`depends_on`	→	Clairaut–Schwarz	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:402
`depends_on`	←	Compatibility is sufficient on a simply connected body	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:430
`proves`	←	ch:13-continuum-elasticity@proof-5	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:405