theorem 10.34 Separable systems for the Helmholtz operator

open in the book · parts/02-mathematical-methods/08-pdes.tex:944 · p. 349

Rests on

depends_on equation 10.25 eq:pde-helmholtz-equation ¶
depends_on lemma 10.28 The separation constant ¶
- depends_on definition 10.2 Linear, semilinear, quasilinear ¶
  - depends_on definition 10.1 Partial differential equation; order ¶
    - depends_on definition 7.98 Functions of class $C^{1}$ ¶
      - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.97 Partial derivative; gradient ¶
    - depends_on definition 6.2 Open set ¶
      - depends_on definition 6.1 Topological space ¶
- proves proof ch:08-pdes@proof-9 ¶
proves proof app:A-long-proofs@proof-82 ¶
proves proof ch:08-pdes@prooflink-3 ¶

Supports

depends_on remark 23.19 The same conditions govern the Helmholtz equation ¶

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`	→	eq:pde-helmholtz-equation	declared	parts/02-mathematical-methods/08-pdes.tex:955
`depends_on`	→	The separation constant	declared	parts/02-mathematical-methods/08-pdes.tex:955
`depends_on`	←	The same conditions govern the Helmholtz equation	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:617
`proves`	←	app:A-long-proofs@proof-82	declared	appendices/A-long-proofs.tex:7457
`proves`	←	ch:08-pdes@prooflink-3	declared	parts/02-mathematical-methods/08-pdes.tex:958