lemma 14.62 The Levi-Civita symbol is invariant

open in the book · parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2821 · p. 573

Rests on

depends_on equation 13.223 eq:mfd-epsilon-density ¶
depends_on example 13.94 The two densities this chapter uses ¶
- depends_on definition 13.92 Tensor density ¶
  - depends_on definition 13.89 Contravariant and covariant tensors ¶
    - depends_on definition 13.88 General coordinate transformation ¶
  - depends_on definition 13.88 General coordinate transformation ¶ ↺
- depends_on proposition 13.93 Algebra of densities ¶
  - depends_on definition 13.92 Tensor density ¶ ↺
  - proves proof ch:11-manifolds-tensors-curvature@proof-17 ¶
depends_on notation 14.1 not:lie-indices ¶
proves proof ch:12-lie-groups-fibre-bundles@proof-29 ¶

Supports

depends_on definition 29.4 Angular velocity ¶
- depends_on proposition 29.11 Angular velocity in Euler angles ¶
  - depends_on definition 29.35 The Lagrange top ¶
  - depends_on remark 29.12 The coordinate singularity, and what it is not ¶
- depends_on proposition 29.5 The velocity field of a rigid body ¶
  - depends_on proposition 29.14 Angular momentum and kinetic energy ¶
  - depends_on proposition 29.8 Chasles: the general displacement is a screw ¶
  - depends_on proposition 29.6 The angular velocity does not depend on the base point ¶
  - depends_on theorem 29.7 Transport theorem ¶
    - depends_on proposition 29.47 The gyrocompass ¶
    - depends_on theorem 29.25 Euler's equations ¶
      - depends_on phenomenon 29.46 A fast spinning top does not fall ¶
      - depends_on phenomenon 29.31 The intermediate-axis instability ¶
      - depends_on proposition 29.26 The two integrals of torque-free motion ¶
      - depends_on proposition 29.28 Free rotation of a symmetric body ¶
      - depends_on proposition 24.41 The free rigid body is a Lie–Poisson system ¶
    - depends_on theorem 29.51 Equation of motion in a rotating frame ¶
      - depends_on phenomenon 42.4 The lunar orbit is not polarized toward the Sun ¶
- depends_on theorem 29.7 Transport theorem ¶ ↺
depends_on lemma 14.64 Properties of $W$ ¶
- depends_on theorem 14.65 The two Casimir operators of the Poincaré algebra in $3+1$ dimensions ¶
  - depends_on example A.394 $D = 4$ ¶

Neighborhood

Every logical edge within two steps of this node.

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partly declared
a check failed
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Edges

type	direction	node	provenance	where
`depends_on`	→	eq:mfd-epsilon-density	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2828
`depends_on`	→	The two densities this chapter uses	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2828
`depends_on`	→	not:lie-indices	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2828
`depends_on`	←	Angular velocity	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:114
`depends_on`	←	Properties of $W$	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2857
`proves`	←	ch:12-lie-groups-fibre-bundles@proof-29	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2831