proposition 9.22 The exponential and its derivative

open in the book · parts/02-mathematical-methods/07-odes-sturm-liouville.tex:741 · p. 281

Rests on

depends_on definition 9.21 Matrix exponential ¶
- depends_on definition 5.19 Norm ¶
  - depends_on definition 4.33 Vector space ¶
    - depends_on definition 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.4 Internal binary operation; magma ¶
    - depends_on definition 4.32 Field ¶
      - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.20 Monoid ¶
      - depends_on definition 4.30 Ring ¶
    - depends_on definition 4.31 Module ¶
      - depends_on definition 4.29 Action ¶
      - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.21 Group ¶
      - depends_on definition 4.30 Ring ¶ ↺
- depends_on proposition 5.125 prop:lin-matrix-algebra ¶
  - depends_on definition 5.121 Algebra ¶
    - depends_on definition 4.4 Internal binary operation; magma ¶ ↺
    - depends_on definition 4.33 Vector space ¶ ↺
  - depends_on definition 5.37 Linear transformation ¶
    - depends_on definition 4.33 Vector space ¶ ↺
  - depends_on equation 5.150 eq:lin-ctesesc2 ¶
  - depends_on equation 5.74 eq:lin-matrix-rep ¶
  - depends_on proposition 5.45 prop:lin-matrix-unique ¶
    - depends_on definition 5.15 Basis ¶
      - depends_on definition 5.12 Subspace generated by a set of vectors ¶
      - depends_on definition 5.14 Linear independence ¶
    - depends_on definition 5.37 Linear transformation ¶ ↺
    - depends_on equation 5.29 eq:lin-indepli2 ¶
    - depends_on equation 5.74 eq:lin-matrix-rep ¶ ↺
    - proves proof ch:03-linear-algebra-representations@proof-16 ¶
  - proves proof ch:03-linear-algebra-representations@proof-55 ¶
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 definition 7.2 Absolute value ¶
    - depends_on definition 7.9 Real function ¶
      - depends_on definition 3.43 Map ¶
  - 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 theorem A.40 Least-upper-bound property ¶
      - depends_on definition A.36 Cut ¶
      - depends_on definition A.38 Order on $\R$ ¶
      - proves proof app:A-long-proofs@proof-29 ¶
    - proves proof app:A-long-proofs@proof-33 ¶
  - depends_on definition 7.45 Series ¶
    - depends_on definition 7.2 Absolute value ¶ ↺
    - depends_on definition 7.4 Convergence ¶
      - depends_on definition 7.2 Absolute value ¶ ↺
  - depends_on theorem 7.8 Cauchy criterion ¶
    - depends_on corollary A.47 Cauchy completeness ¶
      - depends_on corollary A.46 Monotone convergence ¶ ↺
      - depends_on theorem A.40 Least-upper-bound property ¶ ↺
      - proves proof app:A-long-proofs@proof-34 ¶
    - depends_on definition 7.4 Convergence ¶ ↺
    - proves proof ch:05-real-analysis@proof-5 ¶
  - proves proof ch:05-real-analysis@proof-28 ¶
- proves proof ch:07-odes-sturm-liouville@proof-1 ¶
depends_on theorem 7.51 Termwise differentiation ¶
- depends_on proposition 7.47 Comparison; absolute convergence ¶ ↺
- depends_on proposition 7.27 Differentiable implies continuous ¶
  - depends_on definition 7.20 Continuity at a point ¶ ↺
  - 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 proposition 7.6 Algebra of limits ¶
    - depends_on definition 7.4 Convergence ¶ ↺
    - depends_on proposition 7.3 Triangle inequality ¶
      - depends_on definition 7.2 Absolute value ¶ ↺
      - proves proof ch:05-real-analysis@proof-1 ¶
    - proves proof ch:05-real-analysis@proof-3 ¶
  - proves proof ch:05-real-analysis@proof-11 ¶
- 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 proposition 7.6 Algebra of limits ¶ ↺
- depends_on theorem 7.50 Power series; radius of convergence ¶
  - depends_on axiom 7.1 Completeness of $\R$ ¶
  - depends_on proposition 7.47 Comparison; absolute convergence ¶ ↺
  - depends_on proposition 7.46 Geometric series ¶ ↺
  - proves proof ch:05-real-analysis@proof-31 ¶
- proves proof ch:05-real-analysis@proof-32 ¶
proves proof ch:07-odes-sturm-liouville@proof-11 ¶

Supports

depends_on theorem 9.23 Solution of a constant-coefficient system ¶
- depends_on corollary 9.25 Normal modes of a diagonalizable system ¶
  - depends_on corollary 9.27 Linear stability ¶
    - depends_on corollary 9.41 Hill's equation ¶
      - depends_on proposition 9.43 The unmodulated line, and the feet of the tongues ¶
    - depends_on proposition 9.31 Lyapunov's first method: asymptotic stability ¶
    - depends_on proposition 9.29 The planar classification ¶
    - depends_on theorem 9.39 Floquet ¶
      - depends_on corollary 9.41 Hill's equation ¶ ↺
  - depends_on proposition 9.29 The planar classification ¶ ↺
- depends_on proposition 9.87 The distinguished limit, and which end carries the layer ¶
  - depends_on proposition 9.88 The model problem, and the error of the composite ¶
- depends_on proposition 9.31 Lyapunov's first method: asymptotic stability ¶ ↺
- depends_on proposition 9.43 The unmodulated line, and the feet of the tongues ¶ ↺
- depends_on theorem 9.32 Hartman–Grobman; quoted ¶
  - depends_on theorem 32.15 Hartman–Grobman, restated from Part II ¶
    - depends_on definition 32.22 Bifurcation ¶
      - depends_on proposition 32.23 Saddle-node bifurcation ¶
      - depends_on proposition 32.24 Transcritical and pitchfork bifurcations ¶
      - depends_on theorem 32.25 Hopf bifurcation, quoted ¶
    - depends_on example 32.18 A linear centre that is really a stable focus ¶
- depends_on theorem 9.26 Structure of the solutions ¶
  - depends_on corollary 9.27 Linear stability ¶ ↺

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Matrix exponential	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:753
`depends_on`	→	Weierstrass $M$-test; uniform limits are continuous	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:753
`depends_on`	→	Termwise differentiation	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:753
`depends_on`	←	Solution of a constant-coefficient system	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:795
`proves`	←	ch:07-odes-sturm-liouville@proof-11	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:756