lemma A.555 The actions are well defined

open in the book · appendices/A-long-proofs.tex:26797 · p. 3060

Rests on

depends_on definition A.554 Actions on the torus ¶
- depends_on equation 23.49 eq:hj-action-variable ¶
- depends_on equation 24.3 eq:sym-canonical-forms ¶
- depends_on proposition A.552 The component is a torus ¶
  - depends_on lemma A.548 The joint flow is a translation action on the level set ¶
    - depends_on lemma A.546 The Hamiltonian fields of the integrals ¶
      - depends_on equation A.844 eq:app-hj-liouville-arnold-involution ¶
      - depends_on equation 24.8 eq:sym-bracket-homomorphism ¶
      - depends_on equation 24.5 eq:sym-hamiltonian-field ¶
      - proves proof app:A-long-proofs@proof-330 ¶
    - depends_on proposition 13.131 Commuting fields have commuting flows ¶
      - depends_on equation 13.267 eq:mfd-lie-def ¶
      - depends_on equation 13.273 eq:mfd-lie-vector ¶
      - depends_on theorem 13.125 Existence, uniqueness and smoothness of the flow ¶
      - proves proof ch:11-manifolds-tensors-curvature@proof-31 ¶
    - depends_on theorem 13.125 Existence, uniqueness and smoothness of the flow ¶ ↺
    - proves proof app:A-long-proofs@proof-332 ¶
  - depends_on lemma A.549 The action is transitive ¶
    - depends_on corollary A.292 Inverse function theorem ¶
      - depends_on proposition 7.104 Chain rule in several variables ¶
      - depends_on theorem A.286 Implicit function theorem ¶
      - proves proof app:A-long-proofs@proof-186 ¶
    - depends_on lemma A.548 The joint flow is a translation action on the level set ¶ ↺
    - depends_on proposition A.547 The components of the level set are the leaves of an integrable distribution ¶
      - depends_on lemma A.546 The Hamiltonian fields of the integrals ¶ ↺
      - depends_on theorem 13.133 Frobenius ¶
      - depends_on theorem 13.59 Regular value theorem ¶
      - proves proof app:A-long-proofs@proof-331 ¶
    - proves proof app:A-long-proofs@proof-333 ¶
  - depends_on theorem A.550 Quoted: discrete subgroups of a real vector space ¶
    - depends_on definition 4.21 Group ¶
      - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.20 Monoid ¶
    - depends_on definition 6.9 Compact set ¶
      - depends_on definition 6.5 Open cover ¶
  - proves proof app:A-long-proofs@proof-334 ¶
depends_on proposition A.547 The components of the level set are the leaves of an integrable distribution ¶ ↺
depends_on theorem A.311 General Stokes theorem ¶
- depends_on definition A.309 The induced orientation of the boundary ¶
  - depends_on definition A.302 Orientation ¶
    - depends_on definition 13.44 Atlas ¶
      - depends_on definition 13.43 Coordinate system ¶
    - depends_on lemma A.301 Transformation of the top form ¶
      - depends_on definition 13.102 Wedge product ¶
      - depends_on equation 13.241 eq:mfd-antisimb ¶
      - depends_on proposition 13.100 The space of $k$-forms ¶
      - proves proof app:A-long-proofs@proof-189 ¶
  - depends_on lemma A.300 The boundary is well defined, and is a manifold ¶
    - depends_on corollary A.292 Inverse function theorem ¶ ↺
    - depends_on definition A.299 Manifold with boundary ¶
      - depends_on definition A.298 Half-space; smoothness on it ¶
      - depends_on definition 13.44 Atlas ¶ ↺
      - depends_on definition 13.48 Differentiable manifold ¶
    - proves proof app:A-long-proofs@proof-188 ¶
- depends_on definition A.308 Integral over the manifold ¶
  - depends_on lemma A.307 Partition of unity on a compact manifold ¶
    - depends_on definition A.299 Manifold with boundary ¶ ↺
    - depends_on definition 6.9 Compact set ¶ ↺
    - depends_on lemma A.306 The standard smooth bump ¶
      - depends_on definition 7.53 $\ee$ ¶
      - depends_on proposition 7.104 Chain rule in several variables ¶ ↺
      - depends_on theorem 7.37 Cauchy mean value theorem and l'Hôpital's rule ¶
      - proves proof app:A-long-proofs@proof-191 ¶
    - proves proof app:A-long-proofs@proof-192 ¶
  - depends_on lemma A.305 The chart integral is well defined ¶
    - depends_on definition A.304 Integral over a chart ¶
      - depends_on definition 7.125 Multiple integral ¶
      - depends_on definition A.302 Orientation ¶ ↺
      - depends_on proposition 13.100 The space of $k$-forms ¶ ↺
    - depends_on lemma A.301 Transformation of the top form ¶ ↺
    - depends_on theorem A.303 Change of variables for multiple integrals; quoted ¶
      - depends_on definition 7.125 Multiple integral ¶ ↺
      - depends_on remark 7.128 What the derivations below take as given ¶
    - proves proof app:A-long-proofs@proof-190 ¶
- depends_on definition 13.103 Exterior derivative ¶
  - depends_on definition 13.98 $k$-form ¶
    - depends_on definition 13.83 Covector ¶
      - depends_on definition 13.81 Vector on a manifold ¶
    - depends_on definition 13.84 Tensor ¶
      - depends_on definition 13.83 Covector ¶ ↺
      - depends_on definition 13.81 Vector on a manifold ¶ ↺
  - depends_on definition 13.102 Wedge product ¶ ↺
- proves proof app:A-long-proofs@proof-194 ¶
proves proof app:A-long-proofs@proof-336 ¶

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type	direction	node	provenance	where
`depends_on`	→	Actions on the torus	declared	appendices/A-long-proofs.tex:26802
`depends_on`	→	The components of the level set are the leaves of an integrable distribution	declared	appendices/A-long-proofs.tex:26802
`depends_on`	→	General Stokes theorem	declared	appendices/A-long-proofs.tex:26802
`proves`	←	app:A-long-proofs@proof-336	declared	appendices/A-long-proofs.tex:26806