theorem A.74 Flow of a time-dependent vector field

open in the book · appendices/A-long-proofs.tex:4668 · p. 2828

Rests on

depends_on corollary 9.9 Linear equations: existence on the whole interval ¶
- depends_on definition 9.2 Linear equation; homogeneity ¶
  - depends_on definition 9.1 Ordinary differential equation ¶
- depends_on remark 9.3 Normal form and first-order systems ¶
- depends_on theorem 9.8 Picard–Lindelöf ¶
  - depends_on definition 9.4 Lipschitz condition ¶
  - depends_on equation 9.3 eq:slt-ode-system ¶
  - depends_on lemma 9.6 Weierstrass $M$-test; uniform limits are continuous ¶
    - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.16 Limit ¶
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
    - depends_on proposition 7.47 Comparison; absolute convergence ¶
      - depends_on corollary A.46 Monotone convergence ¶
      - depends_on definition 7.45 Series ¶
      - depends_on theorem 7.8 Cauchy criterion ¶
      - proves proof ch:05-real-analysis@proof-28 ¶
    - proves proof ch:07-odes-sturm-liouville@proof-1 ¶
  - proves proof ch:07-odes-sturm-liouville@proof-2 ¶
- proves proof ch:07-odes-sturm-liouville@proof-3 ¶
depends_on lemma A.72 Iterated integral inequality ¶
- depends_on definition 7.53 $\ee$ ¶
  - depends_on definition 7.45 Series ¶ ↺
  - depends_on proposition 7.48 Ratio test ¶
    - depends_on proposition 7.47 Comparison; absolute convergence ¶ ↺
    - depends_on proposition 7.46 Geometric series ¶
      - depends_on corollary A.46 Monotone convergence ¶ ↺
      - depends_on definition 7.45 Series ¶ ↺
      - proves proof ch:05-real-analysis@proof-27 ¶
    - depends_on theorem 7.8 Cauchy criterion ¶ ↺
    - proves proof ch:05-real-analysis@proof-29 ¶
- depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
  - depends_on definition 7.41 Antiderivative ¶
    - depends_on corollary 7.36 cor:ana-mvt-consequences ¶
      - depends_on theorem 7.35 Mean value theorem ¶
      - proves proof ch:05-real-analysis@proof-20 ¶
    - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 7.16 Limit ¶ ↺
      - depends_on definition 7.9 Real function ¶
  - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
    - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
    - depends_on theorem 7.40 Continuous functions are integrable ¶
      - depends_on definition 7.39 Darboux sums and the definite integral ¶
      - depends_on theorem 7.24 Extreme value theorem ¶
      - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
      - proves proof ch:05-real-analysis@proof-23 ¶
    - depends_on theorem 7.24 Extreme value theorem ¶ ↺
    - proves proof ch:05-real-analysis@proof-24 ¶
  - proves proof ch:05-real-analysis@proof-25 ¶
- proves proof app:A-long-proofs@proof-52 ¶
depends_on theorem 9.8 Picard–Lindelöf ¶ ↺
proves proof app:A-long-proofs@proof-54 ¶

Supports

depends_on theorem 13.125 Existence, uniqueness and smoothness of the flow ¶
- depends_on lemma A.548 The joint flow is a translation action on the level set ¶
  - depends_on corollary A.553 Quasi-periodic motion ¶
  - depends_on lemma A.549 The action is transitive ¶
    - depends_on proposition A.552 The component is a torus ¶
      - depends_on corollary A.553 Quasi-periodic motion ¶ ↺
      - depends_on definition A.554 Actions on the torus ¶
      - depends_on proposition A.556 The Jacobian of the actions is the lattice matrix ¶
  - depends_on proposition A.552 The component is a torus ¶ ↺
- depends_on lemma A.567 The differential of the momentum map along the orbit ¶
  - depends_on theorem A.568 The radical is the isotropy orbit ¶
    - 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 proposition 13.131 Commuting fields have commuting flows ¶
  - depends_on lemma A.548 The joint flow is a translation action on the level set ¶ ↺
  - depends_on proposition 13.132 Simultaneous straightening of commuting fields ¶
    - depends_on theorem 13.133 Frobenius ¶
      - depends_on corollary 13.134 Frobenius for a Pfaffian system ¶
      - depends_on proposition A.547 The components of the level set are the leaves of an integrable distribution ¶
      - depends_on theorem A.545 Liouville–Arnold ¶
    - depends_on theorem 13.137 Commuting complete fields on a compact manifold ¶
  - depends_on theorem 13.137 Commuting complete fields on a compact manifold ¶ ↺
- depends_on proposition 13.132 Simultaneous straightening of commuting fields ¶ ↺
- depends_on theorem 13.137 Commuting complete fields on a compact manifold ¶ ↺

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Linear equations: existence on the whole interval	declared	appendices/A-long-proofs.tex:4682
`depends_on`	→	Iterated integral inequality	declared	appendices/A-long-proofs.tex:4682
`depends_on`	→	Picard–Lindelöf	declared	appendices/A-long-proofs.tex:4682
`depends_on`	←	Existence, uniqueness and smoothness of the flow	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5704
`proves`	←	app:A-long-proofs@proof-54	declared	appendices/A-long-proofs.tex:4686