remark 29.12 The coordinate singularity, and what it is not

open in the book · parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:320 · p. 980

remark -- no derivation owed by its kind

Rests on

depends_on proposition 14.31 $\SO(3,\R)$ is not simply connected ¶
- depends_on corollary 14.38 $\SU(2)$ is the universal cover ¶
  - depends_on proposition 14.33 $\SU(2)$ is the three-sphere ¶
    - depends_on definition 14.32 The special unitary group in two dimensions ¶
    - proves proof ch:12-lie-groups-fibre-bundles@proof-11 ¶
  - depends_on theorem 14.37 $\SU(2)$ is a two-to-one cover of $\SO(3,\R)$ ¶
    - depends_on proposition 14.30 Rodrigues formula; the exponential map is onto ¶
      - depends_on equation 14.28 eq:lie-expso2 ¶
      - depends_on equation 14.25 eq:lie-so2-matrix ¶
      - depends_on proposition 14.27 Generators of $\mathfrak{so}(3)$ ¶
      - proves proof ch:12-lie-groups-fibre-bundles@proof-9 ¶
    - depends_on proposition 14.36 Closed form of the exponential ¶
      - depends_on equation 14.28 eq:lie-expso2 ¶ ↺
      - depends_on proposition 14.35 Generators of $\mathfrak{su}(2)$ ¶
      - proves proof ch:12-lie-groups-fibre-bundles@proof-14 ¶
    - depends_on proposition 14.35 Generators of $\mathfrak{su}(2)$ ¶ ↺
    - depends_on proposition 14.33 $\SU(2)$ is the three-sphere ¶ ↺
    - proves proof ch:12-lie-groups-fibre-bundles@proof-15 ¶
  - proves proof app:A-long-proofs@proof-225 ¶
  - proves proof ch:12-lie-groups-fibre-bundles@prooflink-4 ¶
- depends_on definition 6.17 Simply connected space ¶
  - depends_on definition 6.6 Continuous map ¶
    - depends_on definition 3.53 Preimage ¶
    - depends_on definition 6.2 Open set ¶
      - depends_on definition 6.1 Topological space ¶
    - depends_on definition 6.1 Topological space ¶ ↺
  - depends_on definition 6.15 Path-connected space ¶
    - depends_on definition 6.6 Continuous map ¶ ↺
- depends_on proposition 14.30 Rodrigues formula; the exponential map is onto ¶ ↺
- proves proof ch:12-lie-groups-fibre-bundles@proof-10 ¶
depends_on proposition 29.11 Angular velocity in Euler angles ¶
- depends_on definition 29.4 Angular velocity ¶
  - depends_on definition 14.26 The rotation group ¶
  - depends_on lemma 14.62 The Levi-Civita symbol is invariant ¶
    - 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 proposition 13.93 Algebra of densities ¶
    - depends_on notation 14.1 not:lie-indices ¶
    - proves proof ch:12-lie-groups-fibre-bundles@proof-29 ¶
  - depends_on proposition 14.27 Generators of $\mathfrak{so}(3)$ ¶ ↺
- depends_on notation 29.9 The $z$–$x$–$z$ Euler angles used in this treatise ¶
  - depends_on definition 14.26 The rotation group ¶ ↺
  - depends_on theorem 29.3 Euler's rotation theorem ¶
    - depends_on proposition 14.30 Rodrigues formula; the exponential map is onto ¶ ↺
    - depends_on proposition 29.2 Six degrees of freedom ¶
      - depends_on definition 21.9 Degrees of freedom ¶
      - depends_on definition 29.1 Rigid body ¶
      - proves proof ch:12-rigid-body-rotating-frames@proof-1 ¶
    - proves proof ch:12-rigid-body-rotating-frames@proof-2 ¶
- proves proof ch:12-rigid-body-rotating-frames@proof-7 ¶

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type	direction	node	provenance	where
`depends_on`	→	$\SO(3,\R)$ is not simply connected	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:345
`depends_on`	→	Angular velocity in Euler angles	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:345