definition 13.82 Vector field

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3919 · p. 500

ground object -- no derivation owed

Rests on

depends_on definition 13.81 Vector on a manifold ¶
- depends_on definition 13.71 Differentiable curve ¶
  - depends_on definition 13.48 Differentiable manifold ¶
    - depends_on definition 13.47 Differentiable structure ¶
      - depends_on definition 13.44 Atlas ¶
- depends_on definition 13.45 Differentiable map on a topological space ¶
  - depends_on definition 13.44 Atlas ¶ ↺
  - depends_on definition 13.43 Coordinate system ¶
- depends_on definition 13.48 Differentiable manifold ¶ ↺

Supports

depends_on definition 32.3 Dynamical system, phase space, flow ¶
- depends_on definition 32.8 Attractor and basin ¶
  - depends_on definition 32.29 Limit cycle ¶
    - depends_on phenomenon 32.35 Self-sustained oscillation ¶
      - depends_on phenomenon 32.37 Frequency demultiplication and the first heard chaos ¶
  - depends_on definition 32.67 Strange attractor ¶
    - depends_on example 32.70 The Hénon map ¶
    - depends_on phenomenon 32.97 Turbulence sets in after a few transitions ¶
    - depends_on phenomenon 32.68 Strange attractors ¶
  - depends_on theorem 32.74 Oseledets, quoted ¶
    - depends_on phenomenon 32.72 Sensitive dependence on initial conditions ¶
      - depends_on definition 32.67 Strange attractor ¶ ↺
- depends_on definition 32.22 Bifurcation ¶
  - depends_on proposition 32.23 Saddle-node bifurcation ¶
    - depends_on proposition 32.95 Intermittency: the scaling of the laminar phase ¶
  - depends_on proposition 32.24 Transcritical and pitchfork bifurcations ¶
  - depends_on theorem 32.25 Hopf bifurcation, quoted ¶
    - depends_on proposition 32.26 The Hopf normal form ¶
- depends_on definition 32.10 Fixed point ¶
  - depends_on definition 32.12 Linearization ¶
    - depends_on proposition 32.17 Classification of planar fixed points ¶
      - depends_on example 32.18 A linear centre that is really a stable focus ¶
      - depends_on theorem 32.25 Hopf bifurcation, quoted ¶ ↺
    - depends_on theorem 32.15 Hartman–Grobman, restated from Part II ¶
      - depends_on definition 32.22 Bifurcation ¶ ↺
      - depends_on example 32.18 A linear centre that is really a stable focus ¶ ↺
    - depends_on theorem 32.13 Linear stability ¶
      - depends_on proposition 32.65 Fixed points of the Lorenz system and their stability ¶
      - depends_on proposition 32.23 Saddle-node bifurcation ¶ ↺
      - depends_on proposition 32.24 Transcritical and pitchfork bifurcations ¶ ↺
      - depends_on proposition 28.69 Which branch is stable, and where it ends ¶
  - depends_on definition 32.19 Lyapunov function ¶
    - depends_on theorem 32.20 Lyapunov's direct method ¶
      - depends_on example 32.21 The damped pendulum ¶
  - depends_on definition 32.11 Lyapunov stability ¶
    - depends_on theorem 32.20 Lyapunov's direct method ¶ ↺
- depends_on definition 32.41 The horseshoe map ¶
  - depends_on theorem 32.42 The horseshoe is a full shift, quoted ¶
    - depends_on example 32.80 The middle-thirds Cantor set ¶
    - depends_on proposition 32.43 What the shift does ¶
- depends_on definition 32.83 The logistic map ¶
- depends_on definition 32.38 Poincaré section and return map ¶
  - depends_on definition 32.91 The circle map ¶
    - depends_on proposition 32.93 The circle map loses invertibility at $K=1$ ¶
  - depends_on example 32.70 The Hénon map ¶ ↺
  - depends_on proposition 32.93 The circle map loses invertibility at $K=1$ ¶ ↺
  - depends_on proposition 32.95 Intermittency: the scaling of the laminar phase ¶ ↺
  - depends_on proposition 32.39 Stability of a periodic orbit ¶
    - depends_on phenomenon 32.87 Universal period doubling ¶
    - depends_on proposition 32.84 Fixed points and the first period doubling ¶
      - depends_on phenomenon 32.87 Universal period doubling ¶ ↺
- depends_on proposition 32.4 The flow is a one-parameter group ¶
  - depends_on corollary 32.5 Trajectories do not cross ¶
    - depends_on definition 32.29 Limit cycle ¶ ↺
    - depends_on definition 32.38 Poincaré section and return map ¶ ↺
    - depends_on theorem 32.30 Poincaré–Bendixson, restated from Part II ¶
      - depends_on corollary 32.32 No chaos in a planar flow ¶
- depends_on theorem 32.6 Evolution of phase volume ¶
  - depends_on definition 32.7 Conservative and dissipative flows ¶
    - depends_on definition 32.62 The Lorenz system ¶
      - depends_on theorem A.769 The Lorenz system ¶
  - depends_on definition 32.62 The Lorenz system ¶ ↺
  - depends_on proposition 32.64 The Lorenz flow contracts volume ¶
    - depends_on phenomenon 32.68 Strange attractors ¶ ↺
  - depends_on proposition 32.76 The exponents sum to the mean divergence ¶
depends_on definition 13.145 Affine connection ¶
- depends_on definition A.625 Four-dimensional extrinsic curvature ¶
  - depends_on lemma A.630 Codazzi equation ¶
    - depends_on theorem 44.42 The constraint equations ¶
      - depends_on theorem 44.45 Local existence and uniqueness; Choquet-Bruhat; imported ¶
      - depends_on theorem 44.46 Maximal globally hyperbolic development; Choquet-Bruhat and Geroch; imported ¶
  - depends_on lemma A.628 Gauss equation, arbitrary signature ¶
    - depends_on theorem A.632 The $3+1$ identity for the curvature scalar ¶
    - depends_on theorem 44.42 The constraint equations ¶ ↺
  - depends_on lemma A.626 Decomposition of $\nabla n$, and the acceleration ¶
    - depends_on lemma A.630 Codazzi equation ¶ ↺
    - depends_on lemma A.631 The normal–normal Ricci contraction ¶
      - depends_on theorem A.632 The $3+1$ identity for the curvature scalar ¶ ↺
- depends_on definition A.624 Projector and induced metric ¶
- depends_on definition 13.77 Curvature vector ¶
  - depends_on proposition 13.78 The curvature vector is orthogonal to the tangent ¶
    - depends_on theorem 13.79 Frenet–Serret equations, covariant form ¶
- depends_on definition 13.149 Metric compatibility ¶
  - depends_on definition 13.142 Killing tensor ¶
    - depends_on example 13.144 The two Killing tensors every metric carries ¶
    - depends_on theorem 13.143 Killing tensors and geodesic invariants ¶
  - depends_on example 13.144 The two Killing tensors every metric carries ¶ ↺
  - depends_on lemma 44.8 Palatini identity ¶
    - depends_on theorem 44.9 Variation of the Einstein–Hilbert action ¶
      - depends_on proposition 44.11 The boundary term, and the Gibbons–Hawking–York action ¶
      - depends_on proposition 44.30 The cosmological term ¶
  - depends_on proposition 44.30 The cosmological term ¶ ↺
  - depends_on proposition 13.78 The curvature vector is orthogonal to the tangent ¶ ↺
  - depends_on proposition 13.141 The invariant of a Killing vector along a geodesic ¶
    - depends_on phenomenon 45.52 The shadow ¶
      - depends_on phenomenon 50.4 A ring of light at event-horizon scale ¶
    - depends_on proposition 48.4 Peculiar momentum decays with the expansion ¶
    - depends_on proposition 45.6 Radial equation and effective potential ¶
      - depends_on proposition 45.10 Circular orbits and their energy ¶
      - depends_on proposition 45.8 Orbit equation ¶
      - depends_on proposition 45.21 Radial plunge: the cycloid ¶
    - depends_on theorem 13.143 Killing tensors and geodesic invariants ¶ ↺
  - depends_on proposition 13.140 Killing's equation ¶
    - depends_on example 13.144 The two Killing tensors every metric carries ¶ ↺
    - depends_on proposition 14.69 The de~Sitter algebras are isometry algebras ¶
    - depends_on proposition 14.59 The Killing fields of a flat pseudo-Euclidean space ¶
      - depends_on corollary A.344 The algebras $\mathfrak{so}(p,q)$ are semisimple ¶
      - depends_on proposition 14.69 The de~Sitter algebras are isometry algebras ¶ ↺
    - depends_on proposition 13.141 The invariant of a Killing vector along a geodesic ¶ ↺
    - depends_on remark 23.21 The tensorial statement, and what Part II owes it ¶
    - depends_on theorem 13.160 Maximal symmetry forces constant curvature ¶
      - depends_on proposition 44.31 de Sitter geometry ¶
      - depends_on proposition 15.31 Radius and cosmological constant ¶
      - depends_on theorem 48.3 The Robertson–Walker metric ¶
  - depends_on proposition 13.154 Symmetries of the curvature ¶
    - depends_on definition 13.111 Linearised curvature of a symmetric field ¶
      - depends_on theorem 13.112 Symmetric analogue of the converse Poincaré lemma ¶
    - depends_on theorem 44.23 Contracted Bianchi identity ¶
      - depends_on corollary 44.24 Conservation is an integrability condition ¶
      - depends_on proposition 44.41 Four of the ten equations are constraints ¶
      - depends_on proposition 44.30 The cosmological term ¶ ↺
      - depends_on theorem 44.33 Lovelock's uniqueness theorem; imported ¶
    - depends_on theorem 13.160 Maximal symmetry forces constant curvature ¶ ↺
  - depends_on theorem 44.23 Contracted Bianchi identity ¶ ↺
  - depends_on theorem 13.79 Frenet–Serret equations, covariant form ¶ ↺
  - depends_on theorem 13.150 Levi-Civita connection and contorsion ¶
    - depends_on definition 13.77 Curvature vector ¶ ↺
    - depends_on lemma 44.8 Palatini identity ¶ ↺
    - depends_on proposition 44.44 Harmonic-gauge reduction ¶
      - depends_on theorem 44.45 Local existence and uniqueness; Choquet-Bruhat; imported ¶ ↺
    - depends_on proposition 21.73 Free motion in flat spacetime, any coordinates ¶
    - depends_on proposition 13.140 Killing's equation ¶ ↺
    - depends_on proposition 13.158 The connection determined by vielbein and torsion ¶
      - depends_on proposition 44.13 First-order variation in vielbein-form variables ¶
      - depends_on theorem 43.3 Equivalence of the two variable sets ¶
    - depends_on proposition 13.154 Symmetries of the curvature ¶ ↺
    - depends_on theorem 45.1 Schwarzschild solution ¶
      - depends_on phenomenon 50.4 A ring of light at event-horizon scale ¶ ↺
      - depends_on phenomenon 50.5 Black-hole masses from gravitational waves ¶
      - depends_on phenomenon 45.52 The shadow ¶ ↺
      - depends_on proposition 45.14 Isotropic coordinates ¶
      - depends_on proposition 45.43 Static acceleration and surface gravity ¶
      - depends_on proposition 45.16 Curvature at the horizon and at the centre ¶
      - depends_on proposition 45.19 Kruskal–Szekeres coordinates ¶
      - depends_on proposition 45.18 The one-way membrane ¶
      - depends_on proposition 45.6 Radial equation and effective potential ¶ ↺
      - depends_on proposition 45.37 Reissner–Nordström solution ¶
      - depends_on theorem 45.4 Jebsen–Birkhoff ¶
      - depends_on theorem 45.39 Israel; imported ¶
      - … 1 more
- depends_on definition 13.146 Parallel transport and autoparallels ¶
  - depends_on definition 13.149 Metric compatibility ¶ ↺
  - depends_on proposition 13.155 Geodesic deviation; Jacobi equation ¶
  - depends_on proposition 13.141 The invariant of a Killing vector along a geodesic ¶ ↺
  - depends_on remark 29.62 The same rate read as a holonomy ¶
  - depends_on theorem 13.143 Killing tensors and geodesic invariants ¶ ↺
- depends_on definition 13.147 Torsion ¶
  - depends_on proposition 13.148 prop:mfd-torsion-tensor ¶
  - depends_on theorem 13.150 Levi-Civita connection and contorsion ¶ ↺
  - depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶
    - depends_on definition 13.153 Contractions ¶
      - depends_on definition 44.2 Einstein tensor ¶
      - depends_on lemma A.627 The dictionary between $\mathcal{K},\gamma$ and $K,h$ ¶
      - depends_on lemma A.631 The normal–normal Ricci contraction ¶ ↺
      - depends_on lemma A.640 Variation of the integrated three-curvature ¶
      - depends_on proposition 15.31 Radius and cosmological constant ¶ ↺
      - depends_on theorem A.49 thm:app-eh-equivalence ¶
      - depends_on theorem 13.160 Maximal symmetry forces constant curvature ¶ ↺
    - depends_on example 21.74 Rindler coordinates ¶
    - depends_on lemma A.640 Variation of the integrated three-curvature ¶ ↺
    - depends_on lemma 44.8 Palatini identity ¶ ↺
    - depends_on proposition 44.44 Harmonic-gauge reduction ¶ ↺
    - depends_on proposition 13.155 Geodesic deviation; Jacobi equation ¶ ↺
    - depends_on proposition 13.140 Killing's equation ¶ ↺
    - depends_on proposition 13.162 Curvature induced on the quadric ¶
      - depends_on theorem 48.3 The Robertson–Walker metric ¶ ↺
    - depends_on proposition 13.154 Symmetries of the curvature ¶ ↺
    - depends_on remark 30.16 Incompatibility is curvature ¶
    - depends_on theorem 13.156 Cartan structure equations ¶
      - depends_on proposition 44.13 First-order variation in vielbein-form variables ¶ ↺
      - depends_on proposition 13.157 Bianchi identities ¶
      - depends_on proposition 13.158 The connection determined by vielbein and torsion ¶ ↺
  - depends_on theorem 13.156 Cartan structure equations ¶ ↺
- depends_on theorem 13.150 Levi-Civita connection and contorsion ¶ ↺
- depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶ ↺
depends_on definition 13.130 Distribution; involutive; integrable ¶
- depends_on corollary 13.134 Frobenius for a Pfaffian system ¶
- 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 lemma A.549 The action is transitive ¶
      - depends_on proposition A.552 The component is a torus ¶
    - depends_on lemma A.555 The actions are well defined ¶
  - depends_on theorem A.545 Liouville–Arnold ¶
depends_on definition 13.124 Integral curve; complete vector field ¶
- 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 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 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 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 ¶ ↺
depends_on example 13.86 A curve ¶
depends_on example 13.85 Euclidean space $E_{3}$ ¶
depends_on example 13.87 The $2$-sphere ¶

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Vector on a manifold	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3922
`depends_on`	←	Dynamical system, phase space, flow	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:119
`depends_on`	←	Affine connection	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6507
`depends_on`	←	Distribution; involutive; integrable	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5929
`depends_on`	←	Integral curve; complete vector field	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5681
`depends_on`	←	A curve	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:4143
`depends_on`	←	Euclidean space $E_{3}$	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:4120
`depends_on`	←	The $2$-sphere	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:4203