theorem 32.6 Evolution of phase volume

open in the book · parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:173 · p. 1083

Rests on

depends_on definition 32.3 Dynamical system, phase space, flow ¶
- depends_on definition 13.82 Vector field ¶
  - 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 13.45 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 ¶ ↺
- depends_on definition 9.1 Ordinary differential equation ¶
depends_on theorem 7.133 Gauss ¶
- depends_on definition 7.127 Simple regions ¶
  - depends_on definition 7.98 Functions of class $C^{1}$ ¶
    - 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 definition 7.97 Partial derivative; gradient ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 5.15 Basis ¶
  - depends_on definition 6.9 Compact set ¶
    - depends_on definition 6.5 Open cover ¶
      - depends_on definition 6.2 Open set ¶
      - depends_on equation 3.51 eq:set-indexed ¶
- depends_on remark 7.128 What the derivations below take as given ¶
  - depends_on definition 7.125 Multiple integral ¶
    - depends_on axiom 7.1 Completeness of $\R$ ¶
    - depends_on definition 7.39 Darboux sums and the definite integral ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
  - 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 axiom 7.1 Completeness of $\R$ ¶ ↺
      - depends_on proposition 7.22 Sequential characterization ¶
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - proves proof ch:05-real-analysis@proof-9 ¶
    - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
      - depends_on proposition 7.22 Sequential characterization ¶ ↺
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶ ↺
      - proves proof ch:05-real-analysis@proof-10 ¶
    - proves proof ch:05-real-analysis@proof-23 ¶
  - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶ ↺
- 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 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 theorem 7.24 Extreme value theorem ¶ ↺
    - proves proof ch:05-real-analysis@proof-24 ¶
  - proves proof ch:05-real-analysis@proof-25 ¶
- proves proof ch:05-real-analysis@proof-79 ¶
proves proof ch:15-nonlinear-dynamics-chaos@proof-3 ¶

Supports

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 remark A.771 What was assumed, and where it fails ¶
    - depends_on remark A.770 What the three variables and three parameters are ¶
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 ¶

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`	→	Dynamical system, phase space, flow	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:181
`depends_on`	→	Gauss	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:181
`depends_on`	←	Conservative and dissipative flows	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:202
`depends_on`	←	The Lorenz system	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:1887
`depends_on`	←	The Lorenz flow contracts volume	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:1935
`depends_on`	←	The exponents sum to the mean divergence	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:2267
`proves`	←	ch:15-nonlinear-dynamics-chaos@proof-3	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:184