proposition 13.128 Cartan's magic formula

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5820 · p. 523

Rests on

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 13.71 Differentiable curve ¶
      - depends_on definition 13.45 Differentiable map on a topological space ¶
      - depends_on definition 13.48 Differentiable 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 ¶
  - depends_on definition 13.98 $k$-form ¶ ↺
  - depends_on equation 13.239 eq:mfd-dual-basis ¶
depends_on equation 13.246 eq:mfd-leibniz-forms ¶
depends_on equation 13.245 eq:mfd-nilpotency ¶
depends_on proposition 13.127 Component formulas ¶
- depends_on definition 13.89 Contravariant and covariant tensors ¶
  - depends_on definition 13.88 General coordinate transformation ¶
- depends_on definition 13.90 Mixed tensor ¶
  - depends_on definition 13.89 Contravariant and covariant tensors ¶ ↺
  - depends_on definition 13.88 General coordinate transformation ¶ ↺
- depends_on equation 13.267 eq:mfd-lie-def ¶
- proves proof ch:11-manifolds-tensors-curvature@proof-28 ¶
proves proof ch:11-manifolds-tensors-curvature@proof-29 ¶

Supports

depends_on lemma A.77 Poincaré lemma, converse form ¶
depends_on proposition 13.129 Commutation of Lie derivative and interior product ¶
depends_on theorem 24.16 The Hamiltonian flow preserves the symplectic form ¶
- depends_on proposition A.570 Existence and uniqueness of the reduced form ¶
  - depends_on proposition A.571 Invariant Hamiltonians descend with their flows ¶
- depends_on remark 24.35 What non-squeezing does and does not say about nature ¶
- depends_on theorem 24.21 Liouville ¶
  - depends_on example 24.29 A volume-preserving squeeze ¶
  - depends_on proposition 32.40 A Hamiltonian section preserves area ¶
  - depends_on remark 22.25 This is Liouville's theorem ¶
  - depends_on remark 24.22 Liouville's theorem is the foundation of statistical mechanics ¶
  - depends_on theorem 23.31 Liouville–Arnold ¶
    - depends_on proposition 24.56 The Bohr–Sommerfeld condition is a triviality condition on the prequantum holonomy ¶
      - depends_on remark 24.57 The half-integer, and the honest status of the construction ¶
    - depends_on remark 23.33 Integrability is exceptional ¶
    - depends_on remark 23.32 What that proof imports ¶
    - depends_on theorem 32.51 Kolmogorov–Arnold–Moser, quoted ¶
      - depends_on phenomenon 32.58 Regular and chaotic motion coexist ¶
      - depends_on phenomenon 32.59 The solar system is chaotic ¶
      - depends_on remark 23.33 Integrability is exceptional ¶ ↺
    - depends_on theorem 32.47 Poincaré's non-integrability, quoted ¶
  - depends_on theorem 25.57 The Moyal equation, and its classical limit ¶
    - depends_on proposition 25.59 Homodyne tomography inverts a Radon transform ¶
  - depends_on theorem 24.24 Poincaré recurrence ¶
    - depends_on remark 24.26 Recurrence and irreversibility ¶
    - depends_on remark 24.25 What the recurrence proof assumes about volume ¶

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`	→	Exterior derivative	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5829
`depends_on`	→	eq:mfd-leibniz-forms	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5829
`depends_on`	→	eq:mfd-nilpotency	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5829
`depends_on`	→	Component formulas	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5829
`depends_on`	←	Poincaré lemma, converse form	declared	appendices/A-long-proofs.tex:4947
`depends_on`	←	Commutation of Lie derivative and interior product	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5859
`depends_on`	←	The Hamiltonian flow preserves the symplectic form	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:499
`proves`	←	ch:11-manifolds-tensors-curvature@proof-29	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5832