example 13.42 The sphere, the solid angle, and the pole

Rests on

• depends_on example 13.33 Sphere of radius $R$ ¶
• depends_on theorem 13.41 Holonomy equals the enclosed curvature; local Gauss–Bonnet ¶
  • depends_on equation 13.111 eq:mfd-egregium-liouville ¶
  • depends_on proposition 13.40 Parallel transport is an isometry of the tangent plane ¶
    • depends_on definition 13.21 First fundamental form ¶
      • depends_on definition 13.11 Arc length ¶
        • depends_on definition 13.8 Curve ¶
        • depends_on definition 13.9 Tangent vector to a curve ¶
      • depends_on definition 13.18 Surface ¶
    • depends_on equation 13.126 eq:mfd-campoparalelo ¶
    • depends_on equation 13.129 eq:mfd-geodesic-parallelism ¶
    • proves proof ch:11-manifolds-tensors-curvature@proof-6 ¶
  • depends_on theorem 7.131 Green ¶
    • depends_on definition 7.127 Simple regions ¶
      • depends_on definition 7.98 Functions of class $C^{1}$ ¶
        • depends_on definition 7.20 Continuity at a point ¶
        • depends_on definition 7.97 Partial derivative; gradient ¶
      • 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-77 ¶
  • proves proof ch:11-manifolds-tensors-curvature@proof-7 ¶

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