proposition 30.34 Minimum of the potential energy

open in the book · parts/03-classical-mechanics/13-continuum-elasticity.tex:1063 · p. 1019

Rests on

depends_on proposition 30.30 Stability bounds ¶
- depends_on equation 30.23 eq:elast-energy-density ¶
- depends_on theorem 5.116 Sylvester's law of inertia ¶
  - depends_on definition 5.110 Bilinear map ¶
    - depends_on definition 5.37 Linear transformation ¶
      - depends_on definition 4.33 Vector space ¶
  - depends_on definition 5.111 Symmetric and antisymmetric forms ¶
    - depends_on definition 5.110 Bilinear map ¶ ↺
    - depends_on equation 5.137 eq:lin-bilinear-matrix ¶
  - depends_on equation 5.139 eq:lin-bilinear-congruence ¶
  - proves proof ch:03-linear-algebra-representations@proof-53 ¶
- proves proof ch:13-continuum-elasticity@proof-13 ¶
depends_on proposition 30.23 Virtual work; the weak form ¶
- depends_on definition 30.22 Boundary conditions of elastostatics ¶
  - depends_on definition 10.24 Well-posed problem ¶
    - depends_on definition 10.1 Partial differential equation; order ¶
      - depends_on definition 7.98 Functions of class $C^{1}$ ¶
      - depends_on definition 6.2 Open set ¶
    - depends_on definition 6.24 Metric ¶
      - depends_on definition 3.43 Map ¶
      - depends_on definition 3.38 Ordered pair and Cartesian product ¶
  - depends_on theorem 30.21 Cauchy's equation of motion ¶
    - depends_on theorem 7.133 Gauss ¶
      - depends_on definition 7.127 Simple regions ¶
      - depends_on remark 7.128 What the derivations below take as given ¶
      - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      - proves proof ch:05-real-analysis@proof-79 ¶
    - depends_on theorem 30.18 Cauchy: the stress tensor exists ¶
      - depends_on definition 30.17 Traction ¶
      - depends_on definition 13.5 Tensor under orthogonal transformations ¶
      - proves proof ch:13-continuum-elasticity@proof-7 ¶
    - proves proof ch:13-continuum-elasticity@proof-9 ¶
- depends_on theorem 30.21 Cauchy's equation of motion ¶ ↺
- proves proof ch:13-continuum-elasticity@proof-10 ¶
proves proof ch:13-continuum-elasticity@proof-16 ¶

Supports

depends_on example 30.35 From single crystals to the isotropic aggregate ¶

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`	→	Stability bounds	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1073
`depends_on`	→	Virtual work; the weak form	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1073
`depends_on`	←	From single crystals to the isotropic aggregate	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1191
`proves`	←	ch:13-continuum-elasticity@proof-16	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1076