proposition 14.30 Rodrigues formula; the exponential map is onto

open in the book · parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1136 · p. 554

Rests on

depends_on equation 14.28 eq:lie-expso2 ¶
depends_on equation 14.25 eq:lie-so2-matrix ¶
depends_on proposition 14.27 Generators of $\mathfrak{so}(3)$ ¶
- depends_on definition 14.26 The rotation group ¶
- proves proof ch:12-lie-groups-fibre-bundles@proof-8 ¶
proves proof ch:12-lie-groups-fibre-bundles@proof-9 ¶

Supports

depends_on proposition 14.31 $\SO(3,\R)$ is not simply connected ¶
- depends_on remark 29.12 The coordinate singularity, and what it is not ¶
depends_on theorem 14.37 $\SU(2)$ is a two-to-one cover of $\SO(3,\R)$ ¶
- depends_on corollary 14.38 $\SU(2)$ is the universal cover ¶
  - depends_on proposition 14.31 $\SO(3,\R)$ is not simply connected ¶ ↺
  - depends_on proposition 14.45 Which representations descend to $\SO(3,\R)$ ¶
- depends_on lemma A.369 $\Phi$ is a two-sheeted covering ¶
depends_on theorem 29.3 Euler's rotation theorem ¶
- depends_on notation 29.9 The $z$–$x$–$z$ Euler angles used in this treatise ¶
  - 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 remark 29.10 Other conventions, and why the choice must be stated ¶
- depends_on proposition 125.2 Crystallographic restriction ¶
  - depends_on remark 125.3 What the theorem does and does not forbid ¶
- depends_on proposition 29.8 Chasles: the general displacement is a screw ¶

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`	→	eq:lie-expso2	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1153
`depends_on`	→	eq:lie-so2-matrix	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1153
`depends_on`	→	Generators of $\mathfrak{so}(3)$	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1153
`depends_on`	←	$\SO(3,\R)$ is not simply connected	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1199
`depends_on`	←	$\SU(2)$ is a two-to-one cover of $\SO(3,\R)$	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1451
`depends_on`	←	Euler's rotation theorem	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:82
`proves`	←	ch:12-lie-groups-fibre-bundles@proof-9	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1156