theorem 24.23 Poincaré–Cartan integral invariant

Rests on

• depends_on equation 22.3 eq:ham-qdot ¶
• depends_on equation 24.3 eq:sym-canonical-forms ¶
• 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 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 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 definition 6.5 Open cover ¶
      • 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 ch:07-symplectic-geometry@proof-9 ¶

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