definition A.118 Simple separation of the Helmholtz equation

open in the book · appendices/A-long-proofs.tex:6757 · p. 2851

ground object -- no derivation owed

Rests on

depends_on definition A.116 Orthogonal curvilinear coordinates; scale factors ¶
- 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 definition 7.2 Absolute value ¶
      - depends_on definition 7.9 Real function ¶
    - 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 7.16 Limit ¶ ↺
      - depends_on definition 7.9 Real function ¶ ↺
    - 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 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 6.2 Open set ¶
      - depends_on definition 6.1 Topological space ¶
- proves proof ch:08-pdes@proof-9 ¶

Supports

depends_on theorem A.120 Stäckel and Robertson conditions ¶
- depends_on example A.122 Cartesian ¶
- depends_on example A.123 Circular cylindrical ¶
- depends_on example A.124 Spherical ¶
- depends_on proposition A.128 The ellipsoidal system separates ¶
  - depends_on proposition A.132 The count ¶

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`	→	Orthogonal curvilinear coordinates; scale factors	declared	appendices/A-long-proofs.tex:6781
`depends_on`	→	The separation constant	declared	appendices/A-long-proofs.tex:6781
`depends_on`	←	Stäckel and Robertson conditions	declared	appendices/A-long-proofs.tex:6846