definition 9.37 Fundamental matrix

open in the book · parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1333 · p. 287

ground object -- no derivation owed

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 theorem 9.13 Dimension of the solution space ¶
- depends_on corollary 9.9 Linear equations: existence on the whole interval ¶ ↺
- depends_on definition 9.2 Linear equation; homogeneity ¶ ↺
- proves proof ch:07-odes-sturm-liouville@proof-6 ¶

Supports

depends_on proposition 9.38 Liouville's formula ¶
- depends_on corollary 9.41 Hill's equation ¶
  - depends_on proposition 9.43 The unmodulated line, and the feet of the tongues ¶
- depends_on theorem 9.39 Floquet ¶
  - depends_on corollary 9.41 Hill's equation ¶ ↺
depends_on theorem 9.39 Floquet ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
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Edges

type	direction	node	provenance	where
`depends_on`	→	Linear equations: existence on the whole interval	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1344
`depends_on`	→	Dimension of the solution space	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1344
`depends_on`	←	Liouville's formula	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1354
`depends_on`	←	Floquet	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1397