remark 23.21 The tensorial statement, and what Part II owes it

open in the book · parts/03-classical-mechanics/06-hamilton-jacobi.tex:664 · p. 830

remark -- no derivation owed by its kind

Rests on

depends_on proposition 23.20 Separability implies integrability ¶
- depends_on definition 23.13 Additive separability ¶
  - depends_on definition 23.5 Complete integral ¶
    - depends_on equation 23.2 eq:hj-equation ¶
  - depends_on equation 23.15 eq:hj-characteristic-equation ¶
- depends_on equation 22.44 eq:ham-poisson-fundamental ¶
- depends_on equation 22.45 eq:ham-poisson-invariance ¶
- proves proof ch:06-hamilton-jacobi@proof-7 ¶
depends_on proposition 13.140 Killing's equation ¶
- depends_on definition 13.139 Killing vector ¶
  - depends_on definition 13.120 Isometry ¶
    - depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶
      - depends_on definition 13.48 Differentiable manifold ¶
      - depends_on definition 13.84 Tensor ¶
  - depends_on equation 13.267 eq:mfd-lie-def ¶
- depends_on definition 13.149 Metric compatibility ¶
  - depends_on definition 13.145 Affine connection ¶
    - depends_on definition 13.90 Mixed tensor ¶
      - depends_on definition 13.89 Contravariant and covariant tensors ¶
      - depends_on definition 13.88 General coordinate transformation ¶
    - depends_on definition 13.82 Vector field ¶
      - depends_on definition 13.81 Vector on a manifold ¶
  - depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶ ↺
  - depends_on definition 13.146 Parallel transport and autoparallels ¶
    - depends_on definition 13.145 Affine connection ¶ ↺
- depends_on proposition 13.127 Component formulas ¶
  - depends_on definition 13.89 Contravariant and covariant tensors ¶ ↺
  - depends_on definition 13.90 Mixed tensor ¶ ↺
  - depends_on equation 13.267 eq:mfd-lie-def ¶ ↺
  - proves proof ch:11-manifolds-tensors-curvature@proof-28 ¶
- depends_on theorem 13.150 Levi-Civita connection and contorsion ¶
  - depends_on definition 13.145 Affine connection ¶ ↺
  - depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶ ↺
  - depends_on definition 13.149 Metric compatibility ¶ ↺
  - depends_on definition 13.147 Torsion ¶
    - depends_on definition 13.145 Affine connection ¶ ↺
  - proves proof ch:11-manifolds-tensors-curvature@proof-41 ¶
- depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶
  - depends_on definition 13.145 Affine connection ¶ ↺
  - depends_on definition 13.147 Torsion ¶ ↺
  - depends_on proposition 7.105 Clairaut–Schwarz ¶
    - 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 theorem 7.35 Mean value theorem ¶
      - depends_on proposition 7.29 Linearity ¶
      - depends_on theorem 7.34 Rolle ¶
      - proves proof ch:05-real-analysis@proof-19 ¶
    - proves proof ch:05-real-analysis@proof-64 ¶
  - proves proof ch:11-manifolds-tensors-curvature@proof-42 ¶
- proves proof ch:11-manifolds-tensors-curvature@proof-37 ¶

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type	direction	node	provenance	where
`depends_on`	→	Separability implies integrability	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:696
`depends_on`	→	Killing's equation	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:696