lemma A.75 Differentiating a pullback along a flow

open in the book · appendices/A-long-proofs.tex:4844 · p. 2830

Rests on

depends_on definition A.70 Pullback ¶
- 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 7.2 Absolute value ¶
      - depends_on definition 7.9 Real function ¶
    - 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 5.5 Linear combination ¶
      - depends_on definition 5.7 Vector subspace ¶
      - proves proof ch:03-linear-algebra-representations@proof-3 ¶
    - depends_on definition 5.14 Linear independence ¶
      - depends_on definition 5.5 Linear combination ¶ ↺
- depends_on definition 13.83 Covector ¶
  - depends_on definition 13.81 Vector on a manifold ¶
    - depends_on definition 13.71 Differentiable curve ¶
      - depends_on definition 13.48 Differentiable manifold ¶
    - depends_on definition 13.45 Differentiable map on a topological space ¶
      - depends_on definition 13.44 Atlas ¶
      - depends_on definition 13.43 Coordinate system ¶
    - depends_on definition 13.48 Differentiable manifold ¶ ↺
- depends_on definition 13.98 $k$-form ¶
  - depends_on definition 13.83 Covector ¶ ↺
  - depends_on definition 13.84 Tensor ¶
    - depends_on definition 13.83 Covector ¶ ↺
    - depends_on definition 13.81 Vector on a manifold ¶ ↺
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 definition 7.11 Sum of functions ¶
      - depends_on definition 7.9 Real function ¶ ↺
    - depends_on equation 7.14 eq:ana-derivh ¶
    - depends_on proposition 7.6 Algebra of limits ¶
      - depends_on definition 7.4 Convergence ¶
      - depends_on proposition 7.3 Triangle inequality ¶
      - proves proof ch:05-real-analysis@proof-3 ¶
    - proves proof ch:05-real-analysis@proof-13 ¶
  - depends_on theorem 7.34 Rolle ¶
    - depends_on lemma 7.33 Fermat: interior extremum ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
      - depends_on proposition 7.18 Two-sided limit from one-sided limits ¶
      - proves proof ch:05-real-analysis@proof-17 ¶
    - depends_on theorem 7.24 Extreme value theorem ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶
      - depends_on proposition 7.22 Sequential characterization ¶
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - proves proof ch:05-real-analysis@proof-9 ¶
    - proves proof ch:05-real-analysis@proof-18 ¶
  - proves proof ch:05-real-analysis@proof-19 ¶
- proves proof ch:05-real-analysis@proof-64 ¶
depends_on proposition 13.127 Component formulas ¶
- depends_on definition 13.89 Contravariant and covariant tensors ¶
  - depends_on definition 13.88 General coordinate transformation ¶
- 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 equation 13.267 eq:mfd-lie-def ¶
- proves proof ch:11-manifolds-tensors-curvature@proof-28 ¶
proves proof app:A-long-proofs@proof-55 ¶

Supports

depends_on lemma A.77 Poincaré lemma, converse form ¶

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`	→	Pullback	declared	appendices/A-long-proofs.tex:4866
`depends_on`	→	Clairaut–Schwarz	declared	appendices/A-long-proofs.tex:4866
`depends_on`	→	Component formulas	declared	appendices/A-long-proofs.tex:4866
`depends_on`	←	Poincaré lemma, converse form	declared	appendices/A-long-proofs.tex:4947
`proves`	←	app:A-long-proofs@proof-55	declared	appendices/A-long-proofs.tex:4870