theorem 30.15 Compatibility is sufficient on a simply connected body

Rests on

• depends_on proposition 7.136 Properties of conservative fields ¶
  • depends_on definition 7.135 Conservative field ¶
    • depends_on equation 7.124 eq:ana-nabla-gradient ¶
  • depends_on proposition 7.104 Chain rule in several variables ¶
    • depends_on definition 7.13 Composite function ¶
      • depends_on definition 7.9 Real function ¶
        • depends_on definition 3.43 Map ¶
    • depends_on definition 7.99 Differentiability 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 5.37 Linear transformation ¶
        • depends_on definition 4.33 Vector space ¶
      • depends_on equation 6.12 eq:top-euclidean-metric ¶
    • depends_on definition 7.102 Differential of a function ¶
      • depends_on definition 7.99 Differentiability at a point ¶ ↺
      • depends_on definition 7.97 Partial derivative; gradient ¶
        • depends_on definition 7.26 Derivative of a function at a point ¶
        • depends_on definition 5.15 Basis ¶
    • proves proof ch:05-real-analysis@proof-63 ¶
  • depends_on theorem 7.132 Stokes ¶
    • depends_on proposition 7.104 Chain rule in several variables ¶ ↺
    • 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 ¶
    • 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 6.9 Compact set ¶
      • depends_on remark 7.128 What the derivations below take as given ¶
        • depends_on definition 7.125 Multiple integral ¶
        • depends_on theorem 7.40 Continuous functions are integrable ¶
        • 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 theorem 7.42 Fundamental theorem of calculus, I ¶
        • proves proof ch:05-real-analysis@proof-25 ¶
      • proves proof ch:05-real-analysis@proof-77 ¶
    • proves proof ch:05-real-analysis@proof-78 ¶
  • proves proof ch:05-real-analysis@proof-80 ¶
• depends_on proposition 30.14 Saint-Venant compatibility is necessary ¶
  • depends_on definition 30.6 Linear strain and infinitesimal rotation ¶
    • depends_on definition 13.91 Symmetric and antisymmetric parts ¶
    • depends_on equation 30.2 eq:elast-displacement ¶
  • depends_on proposition 7.105 Clairaut–Schwarz ¶ ↺
  • proves proof ch:13-continuum-elasticity@proof-5 ¶
• proves proof ch:13-continuum-elasticity@proof-6 ¶

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