theorem A.120 Stäckel and Robertson conditions

open in the book · appendices/A-long-proofs.tex:6829 · p. 2851

Rests on

depends_on definition A.118 Simple separation of the Helmholtz equation ¶
- 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 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 5.15 Basis ¶
  - 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 ¶
  - 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 definition 6.5 Open cover ¶
  - depends_on remark 7.128 What the derivations below take as given ¶
    - depends_on definition 7.125 Multiple integral ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶
      - depends_on definition 7.39 Darboux sums and the definite integral ¶
    - depends_on theorem 7.40 Continuous functions are integrable ¶
      - depends_on definition 7.39 Darboux sums and the definite integral ¶ ↺
      - depends_on theorem 7.24 Extreme value theorem ¶
      - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
      - proves proof ch:05-real-analysis@proof-23 ¶
    - 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 corollary 7.36 cor:ana-mvt-consequences ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
    - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
      - depends_on theorem 7.40 Continuous functions are integrable ¶ ↺
      - depends_on theorem 7.24 Extreme value theorem ¶ ↺
      - proves proof ch:05-real-analysis@proof-24 ¶
    - 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 ¶

Supports

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`	→	Simple separation of the Helmholtz equation	declared	appendices/A-long-proofs.tex:6846
`depends_on`	→	Cofactor identities	declared	appendices/A-long-proofs.tex:6846
`depends_on`	→	Laplacian in orthogonal coordinates	declared	appendices/A-long-proofs.tex:6846
`depends_on`	←	Cartesian	declared	appendices/A-long-proofs.tex:7014
`depends_on`	←	Circular cylindrical	declared	appendices/A-long-proofs.tex:7035
`depends_on`	←	Spherical	declared	appendices/A-long-proofs.tex:7064
`depends_on`	←	The ellipsoidal system separates	declared	appendices/A-long-proofs.tex:7232
`proves`	←	app:A-long-proofs@proof-76	declared	appendices/A-long-proofs.tex:6850
`proves`	←	app:A-long-proofs@proof-77	declared	appendices/A-long-proofs.tex:6897