lemma A.77 Poincaré lemma, converse form

Rests on

• depends_on definition A.76 Star-shaped set ¶
  • depends_on definition 6.26 Open ball; metric topology ¶
    • depends_on definition 6.24 Metric ¶
      • depends_on definition 3.43 Map ¶
        • depends_on definition 3.24 Quantifiers ¶
        • depends_on definition 3.28 Set ¶
      • depends_on definition 3.38 Ordered pair and Cartesian product ¶
    • depends_on definition 6.2 Open set ¶
      • depends_on definition 6.1 Topological space ¶
        • depends_on definition 3.31 Empty set ¶
        • depends_on definition 3.35 Union, intersection, difference ¶
        • depends_on definition 3.30 Subset ¶
        • depends_on equation 3.51 eq:set-indexed ¶
    • depends_on definition 6.1 Topological space ¶ ↺
• depends_on lemma A.73 Differentiation under the integral sign ¶
  • 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 theorem 7.25 Heine–Cantor: uniform continuity ¶
    • depends_on proposition 7.22 Sequential characterization ¶
      • 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 7.4 Convergence ¶
        • depends_on definition 7.2 Absolute value ¶ ↺
      • proves proof ch:05-real-analysis@proof-7 ¶
    • depends_on theorem 7.7 Bolzano–Weierstrass ¶
      • depends_on corollary A.45 Archimedean property and density of $\Q$ ¶
        • depends_on definition A.36 Cut ¶
        • depends_on theorem A.40 Least-upper-bound property ¶
        • proves proof app:A-long-proofs@proof-32 ¶
      • depends_on corollary A.46 Monotone convergence ¶
        • depends_on theorem A.40 Least-upper-bound property ¶ ↺
        • proves proof app:A-long-proofs@proof-33 ¶
      • proves proof ch:05-real-analysis@proof-4 ¶
    • proves proof ch:05-real-analysis@proof-10 ¶
  • 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 app:A-long-proofs@proof-53 ¶
• depends_on lemma A.75 Differentiating a pullback along a flow ¶
  • depends_on definition A.70 Pullback ¶
    • depends_on definition 7.97 Partial derivative; gradient ¶ ↺
    • 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 13.45 Differentiable map on a topological space ¶
        • 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 theorem 7.35 Mean value theorem ¶ ↺
    • 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 ¶
• depends_on proposition 13.128 Cartan's magic formula ¶
  • depends_on definition 13.103 Exterior derivative ¶
    • depends_on definition 13.98 $k$-form ¶ ↺
    • depends_on definition 13.102 Wedge product ¶
      • depends_on definition 13.98 $k$-form ¶ ↺
      • depends_on equation 13.239 eq:mfd-dual-basis ¶
  • depends_on equation 13.246 eq:mfd-leibniz-forms ¶
  • depends_on equation 13.245 eq:mfd-nilpotency ¶
  • depends_on proposition 13.127 Component formulas ¶ ↺
  • proves proof ch:11-manifolds-tensors-curvature@proof-29 ¶
• proves proof app:A-long-proofs@proof-56 ¶

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