proposition 16.4 The hanging chain is a catenary

Rests on

• depends_on equation 16.25 eq:calcvar-beltrami ¶
• depends_on equation 16.3 eq:calcvar-catenary-energy ¶
• depends_on theorem 16.43 Euler's rule for integral constraints ¶
  • depends_on definition 16.15 Variation; the first variation ¶
    • depends_on definition 16.11 Admissible class; functional ¶
      • 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 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 equation 7.177 eq:ana-functional-derivative ¶
  • depends_on lemma 16.20 Mixed form ¶
    • depends_on corollary 7.44 Substitution and integration by parts ¶
      • depends_on proposition 7.31 Chain rule ¶
        • depends_on definition 7.13 Composite function ¶
        • depends_on proposition 7.27 Differentiable implies continuous ¶
        • depends_on proposition 7.6 Algebra of limits ¶
        • proves proof ch:05-real-analysis@proof-15 ¶
      • depends_on proposition 7.30 Leibniz rule ¶
        • depends_on definition 7.12 Product of functions ¶
        • depends_on equation 7.14 eq:ana-derivh ¶
        • depends_on proposition 7.27 Differentiable implies continuous ¶ ↺
        • depends_on proposition 7.6 Algebra of limits ¶ ↺
        • proves proof ch:05-real-analysis@proof-14 ¶
      • 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-26 ¶
    • depends_on lemma 16.19 du Bois-Reymond ¶
      • depends_on equation 16.18 eq:calcvar-dbr-hyp ¶
      • depends_on theorem 7.42 Fundamental theorem of calculus, I ¶ ↺
      • proves proof ch:14-calculus-of-variations@proof-9 ¶
    • proves proof ch:14-calculus-of-variations@proof-10 ¶
  • depends_on proposition 16.17 The first variation of an integral functional ¶
    • depends_on equation 16.11 eq:calcvar-basic-functional ¶
    • depends_on equation 16.14 eq:calcvar-gateaux ¶
    • proves proof ch:14-calculus-of-variations@proof-7 ¶
  • proves proof ch:14-calculus-of-variations@proof-24 ¶
• proves proof ch:14-calculus-of-variations@proof-2 ¶

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