lemma A.312 The computation in one chart

Rests on

• depends_on definition 13.103 Exterior derivative ¶
  • depends_on definition 13.98 $k$-form ¶
    • 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 13.84 Tensor ¶
      • depends_on definition 13.83 Covector ¶ ↺
      • depends_on definition 13.81 Vector on a manifold ¶ ↺
  • 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 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 axiom 7.1 Completeness of $\R$ ¶ ↺
  • 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 axiom 7.1 Completeness of $\R$ ¶ ↺
      • depends_on proposition 7.22 Sequential characterization ¶
        • depends_on definition 7.20 Continuity at a point ¶
        • depends_on definition 7.4 Convergence ¶
        • 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 corollary A.46 Monotone convergence ¶
        • proves proof ch:05-real-analysis@proof-4 ¶
      • proves proof ch:05-real-analysis@proof-9 ¶
    • depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
      • depends_on proposition 7.22 Sequential characterization ¶ ↺
      • depends_on theorem 7.7 Bolzano–Weierstrass ¶ ↺
      • proves proof ch:05-real-analysis@proof-10 ¶
    • 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 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-20 ¶
    • 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 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 app:A-long-proofs@proof-193 ¶

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