definition 16.75 Rayleigh quotient

open in the book · parts/02-mathematical-methods/14-calculus-of-variations.tex:2337 · p. 657

ground object -- no derivation owed

Rests on

depends_on definition 9.54 Sturm–Liouville differential operator ¶
depends_on equation 16.42 eq:calcvar-sl-from-isoperimetric ¶

Supports

depends_on definition A.462 The weighted space and the energy space ¶
- depends_on lemma A.463 Poincaré inequality; $B_{K}$ is an inner product ¶
  - depends_on lemma A.464 $H_{E}$ is a Hilbert space ¶
  - depends_on lemma A.466 The homogeneous Dirichlet problem is trivial ¶
    - depends_on definition A.467 Green function of the shifted problem ¶
      - depends_on lemma A.468 The kernel is well defined, symmetric and Lipschitz ¶
      - depends_on theorem A.469 The Green operator inverts $L_{K}$ ¶
    - depends_on theorem A.469 The Green operator inverts $L_{K}$ ¶ ↺
  - depends_on theorem A.469 The Green operator inverts $L_{K}$ ¶ ↺
depends_on proposition 28.38 Rayleigh's bound on the lowest frequency ¶
depends_on theorem A.461 Completeness in the weighted and in the energy norm ¶

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`	→	Sturm–Liouville differential operator	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:2347
`depends_on`	→	eq:calcvar-sl-from-isoperimetric	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:2347
`depends_on`	←	The weighted space and the energy space	declared	appendices/A-long-proofs.tex:22830
`depends_on`	←	Rayleigh's bound on the lowest frequency	declared	parts/03-classical-mechanics/11-oscillations-waves.tex:1273
`depends_on`	←	Completeness in the weighted and in the energy norm	declared	appendices/A-long-proofs.tex:22801