definition 29.13 Inertia tensor, continuum form

open in the book · parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:352 · p. 980

ground object -- no derivation owed

Rests on

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 18.3 Particle ¶
    - depends_on definition 19.3 Mass ¶
      - depends_on definition 18.3 Particle ¶ ↺
      - depends_on postulate 19.1 Inertia ¶
  - depends_on definition 19.3 Mass ¶ ↺

Supports

depends_on definition 29.19 Inertia ellipsoid ¶
- 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 ¶
depends_on example 29.23 The polar moment of the Earth, and what it reveals ¶
- depends_on phenomenon 29.34 Free nutation of the Earth ¶
depends_on proposition 49.17 Magnetic-dipole spin-down ¶
depends_on proposition 29.14 Angular momentum and kinetic energy ¶
depends_on proposition 29.15 Positivity ¶
- depends_on definition 29.19 Inertia ellipsoid ¶ ↺
- depends_on theorem 29.16 Principal axes ¶
  - depends_on definition 29.19 Inertia ellipsoid ¶ ↺
  - depends_on definition 29.35 The Lagrange top ¶
  - depends_on definition 29.18 Classification of tops ¶
    - depends_on proposition 29.28 Free rotation of a symmetric body ¶
      - depends_on phenomenon 29.34 Free nutation of the Earth ¶ ↺
  - depends_on phenomenon 29.31 The intermediate-axis instability ¶
    - depends_on remark 29.32 What the linearization is entitled to conclude ¶ ↺
  - depends_on proposition 29.33 Dissipation drives a free body to its largest moment ¶
  - depends_on proposition 29.17 Triangle inequalities ¶
    - depends_on proposition 29.22 Perpendicular-axis theorem ¶
  - depends_on theorem 29.25 Euler's equations ¶
    - depends_on phenomenon 29.46 A fast spinning top does not fall ¶
      - depends_on phenomenon 29.49 Precession of the equinoxes ¶
    - depends_on phenomenon 29.31 The intermediate-axis instability ¶ ↺
    - depends_on proposition 29.26 The two integrals of torque-free motion ¶
      - depends_on proposition 29.33 Dissipation drives a free body to its largest moment ¶ ↺
      - depends_on proposition 29.43 The three integrable heavy tops ¶
      - depends_on remark 29.27 Euler's equations as a Lie–Poisson system ¶
      - depends_on remark 29.30 Polhode and herpolhode ¶ ↺
      - depends_on remark 24.42 The intermediate axis ¶
      - depends_on theorem 29.29 Poinsot's construction ¶ ↺
    - depends_on proposition 29.28 Free rotation of a symmetric body ¶ ↺
    - depends_on proposition 24.41 The free rigid body is a Lie–Poisson system ¶
      - depends_on remark 24.48 Reduction is what physicists do without saying so ¶
      - depends_on remark 24.42 The intermediate axis ¶ ↺
depends_on proposition 29.24 MacCullagh's formula ¶
- depends_on phenomenon 29.49 Precession of the equinoxes ¶ ↺
depends_on proposition 29.22 Perpendicular-axis theorem ¶ ↺
depends_on theorem 29.20 Parallel-axis theorem ¶
- depends_on corollary 29.21 The centre of mass minimizes the moment ¶
depends_on theorem 29.16 Principal axes ¶ ↺

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`	→	Multiple integral	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:373
`depends_on`	→	Tensor under orthogonal transformations	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:373
`depends_on`	→	Inertia tensor	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:373
`depends_on`	←	Inertia ellipsoid	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:535
`depends_on`	←	The polar moment of the Earth, and what it reveals	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:676
`depends_on`	←	Magnetic-dipole spin-down	declared	parts/05-general-relativity-cosmology/08-compact-stars.tex:1101
`depends_on`	←	Angular momentum and kinetic energy	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:392
`depends_on`	←	Positivity	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:441
`depends_on`	←	MacCullagh's formula	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:690
`depends_on`	←	Perpendicular-axis theorem	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:607
`depends_on`	←	Parallel-axis theorem	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:559
`depends_on`	←	Principal axes	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:466