theorem 16.64 Weierstrass' sufficient condition

Rests on

• depends_on definition 16.62 Field of extremals; slope function ¶
  • depends_on definition 6.17 Simply connected space ¶
    • depends_on definition 6.6 Continuous map ¶
      • depends_on definition 3.53 Preimage ¶
      • depends_on definition 6.2 Open set ¶
        • depends_on definition 6.1 Topological space ¶
      • depends_on definition 6.1 Topological space ¶ ↺
    • depends_on definition 6.15 Path-connected space ¶
      • depends_on definition 6.6 Continuous map ¶ ↺
  • 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 corollary 7.44 Substitution and integration by parts ¶
        • depends_on proposition 7.31 Chain rule ¶
        • depends_on proposition 7.30 Leibniz rule ¶
        • depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
        • 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 16.16 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 ¶
• depends_on equation 16.51 eq:calcvar-excess ¶
• depends_on theorem 16.63 Hilbert's invariant integral ¶
  • depends_on definition 16.62 Field of extremals; slope function ¶ ↺
  • depends_on equation 16.20 eq:calcvar-euler-lagrange ¶
  • 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 7.99 Differentiability at a point ¶
        • depends_on definition 7.16 Limit ¶
        • depends_on definition 5.37 Linear transformation ¶
        • 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 ¶
      • 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 theorem 7.35 Mean value theorem ¶
        • proves proof ch:05-real-analysis@proof-64 ¶
      • depends_on theorem 7.131 Green ¶
        • depends_on definition 7.127 Simple regions ¶
        • depends_on remark 7.128 What the derivations below take as given ¶
        • depends_on theorem 7.43 Fundamental theorem of calculus, II ¶ ↺
        • proves proof ch:05-real-analysis@proof-77 ¶
      • proves proof ch:05-real-analysis@proof-78 ¶
    • proves proof ch:05-real-analysis@proof-80 ¶
  • proves proof ch:14-calculus-of-variations@proof-30 ¶
• proves proof ch:14-calculus-of-variations@proof-31 ¶

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