definition 29.19 Inertia ellipsoid

open in the book · parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:528 · p. 982

ground object -- no derivation owed

Rests on

depends_on definition 29.13 Inertia tensor, continuum form ¶
- depends_on definition 7.125 Multiple integral ¶
  - depends_on axiom 7.1 Completeness of $\R$ ¶
  - depends_on definition 7.39 Darboux sums and the definite integral ¶
    - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
- depends_on definition 13.5 Tensor under orthogonal transformations ¶
  - depends_on definition 13.4 Orthogonal coordinate transformation ¶
    - depends_on definition 13.3 Orthogonal coordinate system ¶
- depends_on definition 19.42 Inertia tensor ¶
  - depends_on definition 13.5 Tensor under orthogonal transformations ¶ ↺
  - depends_on definition 19.41 Moment of inertia about an axis ¶
    - depends_on definition 19.39 Body ¶
      - depends_on definition 18.4 Body ¶
      - depends_on definition 19.3 Mass ¶
    - depends_on definition 19.3 Mass ¶ ↺
depends_on proposition 29.15 Positivity ¶
- depends_on definition 29.13 Inertia tensor, continuum form ¶ ↺
- proves proof ch:12-rigid-body-rotating-frames@proof-9 ¶
depends_on theorem 29.16 Principal axes ¶
- depends_on definition 29.13 Inertia tensor, continuum form ¶ ↺
- depends_on proposition 29.15 Positivity ¶ ↺
- 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 ¶
      - 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 5.69 Eigenvector, eigenvalue, eigenspace ¶
      - depends_on proposition 5.49 Injective, surjective, invertible ¶
      - 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 5.37 Linear transformation ¶
      - depends_on definition 4.33 Vector space ¶ ↺
  - depends_on theorem 5.40 Rank–nullity ¶
    - depends_on definition 5.15 Basis ¶
      - depends_on definition 5.12 Subspace generated by a set of vectors ¶
      - depends_on definition 5.14 Linear independence ¶
    - depends_on definition 5.39 Kernel, image, nullity, rank ¶
      - depends_on definition 5.37 Linear transformation ¶ ↺
      - depends_on definition 5.7 Vector subspace ¶
      - proves proof ch:03-linear-algebra-representations@proof-12 ¶
    - depends_on lemma 5.38 Exchange and completion ¶
      - depends_on definition 5.15 Basis ¶ ↺
      - depends_on definition 5.12 Subspace generated by a set of vectors ¶ ↺
      - depends_on definition 5.14 Linear independence ¶ ↺
      - proves proof ch:03-linear-algebra-representations@proof-11 ¶
    - proves proof ch:03-linear-algebra-representations@proof-13 ¶
  - proves proof ch:03-linear-algebra-representations@proof-31 ¶
- proves proof ch:12-rigid-body-rotating-frames@proof-10 ¶

Supports

depends_on theorem 29.29 Poinsot's construction ¶
- depends_on remark 29.32 What the linearization is entitled to conclude ¶
- depends_on remark 29.30 Polhode and herpolhode ¶

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`	→	Inertia tensor, continuum form	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:535
`depends_on`	→	Positivity	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:535
`depends_on`	→	Principal axes	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:535
`depends_on`	←	Poinsot's construction	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:911