proposition 16.50 The second variation

Rests on

• 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 definition 7.9 Real function ¶
        • depends_on definition 3.43 Map ¶
    • depends_on proposition 7.27 Differentiable implies continuous ¶
      • 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 definition 7.26 Derivative of a function at a point ¶
        • depends_on definition 7.16 Limit ¶ ↺
        • depends_on definition 7.9 Real function ¶ ↺
      • depends_on proposition 7.6 Algebra of limits ¶
        • depends_on definition 7.4 Convergence ¶
        • depends_on proposition 7.3 Triangle inequality ¶
        • proves proof ch:05-real-analysis@proof-3 ¶
      • proves proof ch:05-real-analysis@proof-11 ¶
    • 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 definition 7.9 Real function ¶ ↺
    • 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 corollary 7.36 cor:ana-mvt-consequences ¶
        • depends_on theorem 7.35 Mean value theorem ¶
        • proves proof ch:05-real-analysis@proof-20 ¶
      • 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 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 7.24 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-26 ¶
• depends_on equation 16.14 eq:calcvar-gateaux ¶
• depends_on theorem 7.38 Taylor's theorem with Lagrange remainder ¶
  • depends_on proposition 7.30 Leibniz rule ¶ ↺
  • depends_on theorem 7.34 Rolle ¶
    • 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 ¶
        • depends_on definition 7.16 Limit ¶ ↺
        • depends_on equation 7.7 eq:ana-limit-left ¶ ↺
        • depends_on equation 7.5 eq:ana-limit-right ¶ ↺
        • proves proof ch:05-real-analysis@proof-6 ¶
      • proves proof ch:05-real-analysis@proof-17 ¶
    • depends_on theorem 7.24 Extreme value theorem ¶ ↺
    • proves proof ch:05-real-analysis@proof-18 ¶
  • proves proof ch:05-real-analysis@proof-22 ¶
• proves proof ch:14-calculus-of-variations@proof-25 ¶

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