proposition A.128 The ellipsoidal system separates

open in the book · appendices/A-long-proofs.tex:7203 · p. 2855

Rests on

depends_on lemma A.126 Cartesian coordinates and scale factors ¶
- depends_on definition A.125 Ellipsoidal coordinates ¶
  - 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 7.97 Partial derivative; gradient ¶
    - depends_on equation 10.25 eq:pde-helmholtz-equation ¶
- depends_on equation A.263 eq:app-stackel-orthogonality ¶
- proves proof app:A-long-proofs@proof-78 ¶
depends_on lemma A.127 Lagrange's interpolation identity ¶
- depends_on equation A.287 eq:app-stackel-hi ¶
- proves proof app:A-long-proofs@proof-79 ¶
depends_on theorem A.120 Stäckel and Robertson conditions ¶
- depends_on definition A.118 Simple separation of the Helmholtz equation ¶
  - depends_on definition A.116 Orthogonal curvilinear coordinates; scale factors ¶ ↺
  - 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 ¶
    - proves proof ch:08-pdes@proof-9 ¶
- depends_on lemma A.119 Cofactor identities ¶
  - depends_on equation A.267 eq:app-stackel-nondegenerate ¶
  - proves proof app:A-long-proofs@proof-75 ¶
- depends_on lemma A.117 Laplacian in orthogonal coordinates ¶
  - depends_on definition A.116 Orthogonal curvilinear coordinates; scale factors ¶ ↺
  - depends_on theorem 7.133 Gauss ¶
    - depends_on definition 7.127 Simple regions ¶
      - depends_on definition 7.98 Functions of class $C^{1}$ ¶ ↺
      - depends_on definition 6.9 Compact set ¶
    - depends_on remark 7.128 What the derivations below take as given ¶
      - depends_on definition 7.125 Multiple integral ¶
      - depends_on theorem 7.40 Continuous functions are integrable ¶
      - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
    - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      - depends_on definition 7.41 Antiderivative ¶
      - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
      - proves proof ch:05-real-analysis@proof-25 ¶
    - proves proof ch:05-real-analysis@proof-79 ¶
  - proves proof app:A-long-proofs@proof-74 ¶
- proves proof app:A-long-proofs@proof-76 ¶
- proves proof app:A-long-proofs@proof-77 ¶
proves proof app:A-long-proofs@proof-80 ¶

Supports

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`	→	Cartesian coordinates and scale factors	declared	appendices/A-long-proofs.tex:7232
`depends_on`	→	Lagrange's interpolation identity	declared	appendices/A-long-proofs.tex:7232
`depends_on`	→	Stäckel and Robertson conditions	declared	appendices/A-long-proofs.tex:7232
`depends_on`	←	The count	declared	appendices/A-long-proofs.tex:7422
`proves`	←	app:A-long-proofs@proof-80	declared	appendices/A-long-proofs.tex:7236