proposition 30.9 Principal strains

open in the book · parts/03-classical-mechanics/13-continuum-elasticity.tex:267 · p. 1010

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 theorem 5.79 Spectral theorem for a self-adjoint operator ¶
- depends_on corollary 5.72 Existence of an eigenvalue over $\C$ ¶
  - depends_on theorem 8.19 Fundamental theorem of algebra ¶
    - depends_on theorem 8.18 Liouville ¶
      - depends_on proposition 8.10 Fundamental theorem for contours ¶
      - depends_on theorem 8.17 Derivatives of all orders; Cauchy estimates ¶
      - proves proof ch:06-complex-analysis@proof-12 ¶
    - proves proof ch:06-complex-analysis@proof-13 ¶
  - depends_on theorem 5.71 The eigenvalues are the roots of the characteristic polynomial ¶
    - depends_on definition 5.70 Characteristic polynomial ¶
      - depends_on definition 5.56 Endomorphism ¶
      - depends_on equation 5.19 eq:lin-leibniz-det ¶
      - depends_on equation 5.74 eq:lin-matrix-rep ¶
      - proves proof ch:03-linear-algebra-representations@proof-25 ¶
    - depends_on definition 5.69 Eigenvector, eigenvalue, eigenspace ¶
      - depends_on definition 5.56 Endomorphism ¶ ↺
      - depends_on definition 5.39 Kernel, image, nullity, rank ¶
    - depends_on proposition 5.49 Injective, surjective, invertible ¶
      - depends_on definition 5.39 Kernel, image, nullity, rank ¶ ↺
      - depends_on proposition 5.48 Existence, uniqueness, and linearity of the inverse ¶
      - depends_on theorem 5.40 Rank–nullity ¶
      - proves proof ch:03-linear-algebra-representations@proof-18 ¶
    - proves proof ch:03-linear-algebra-representations@proof-26 ¶
  - proves proof ch:03-linear-algebra-representations@proof-27 ¶
- depends_on definition 5.41 Adjoint ¶
  - depends_on definition 5.18 Inner product ¶
    - depends_on definition 4.33 Vector space ¶
      - depends_on definition 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.32 Field ¶
      - depends_on definition 4.31 Module ¶
  - depends_on definition 5.37 Linear transformation ¶
    - depends_on definition 4.33 Vector space ¶ ↺
- depends_on theorem 5.40 Rank–nullity ¶ ↺
- proves proof ch:03-linear-algebra-representations@proof-31 ¶
proves proof ch:13-continuum-elasticity@proof-3 ¶

Supports

depends_on example 30.10 Simple shear and pure shear are the same strain ¶

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:273
`depends_on`	→	Spectral theorem for a self-adjoint operator	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:273
`depends_on`	←	Simple shear and pure shear are the same strain	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:313
`proves`	←	ch:13-continuum-elasticity@proof-3	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:276