proposition 16.6 The isoperimetric extremal is a circle

Rests on

• depends_on theorem 16.35 System of Euler–Lagrange equations ¶
  • 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 theorem 16.22 Euler–Lagrange ¶
    • depends_on equation 16.15 eq:calcvar-first-variation ¶
    • depends_on lemma 16.20 Mixed form ¶ ↺
    • depends_on proposition 16.16 Stationarity is necessary ¶
      • depends_on definition 16.15 Variation; the first variation ¶
        • depends_on definition 16.11 Admissible class; functional ¶
        • depends_on equation 7.177 eq:ana-functional-derivative ¶
      • depends_on definition 16.13 Weak and strong extrema ¶
        • depends_on definition 16.12 The two norms; weak and strong neighbourhoods ¶
      • 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 ¶
      • proves proof ch:14-calculus-of-variations@proof-6 ¶
    • proves proof ch:14-calculus-of-variations@proof-11 ¶
  • proves proof ch:14-calculus-of-variations@proof-19 ¶
• depends_on theorem 16.43 Euler's rule for integral constraints ¶
  • depends_on definition 16.15 Variation; the first variation ¶ ↺
  • depends_on lemma 16.20 Mixed form ¶ ↺
  • 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-3 ¶

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