proposition 28.42 The wave equation of a stretched string

Rests on

• depends_on equation 28.46 eq:osc-wave-eq ¶
• depends_on theorem 16.36 Euler–Lagrange equations for several independent variables ¶
  • depends_on equation 16.31 eq:calcvar-field-functional ¶
  • depends_on lemma 16.18 Fundamental lemma ¶
    • 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 theorem 7.40 Continuous functions are integrable ¶
      • depends_on equation 7.177 eq:ana-functional-derivative ¶
    • depends_on theorem 7.40 Continuous functions are integrable ¶ ↺
    • proves proof ch:14-calculus-of-variations@proof-8 ¶
  • depends_on theorem 7.133 Gauss ¶
    • 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 definition 6.5 Open cover ¶
    • depends_on remark 7.128 What the derivations below take as given ¶
      • depends_on definition 7.125 Multiple integral ¶
        • depends_on axiom 7.1 Completeness of $\R$ ¶
        • depends_on definition 7.39 Darboux sums and the definite integral ¶
      • depends_on theorem 7.40 Continuous functions are integrable ¶ ↺
      • depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
        • depends_on proposition 7.22 Sequential characterization ¶
        • depends_on theorem 7.7 Bolzano–Weierstrass ¶
        • proves proof ch:05-real-analysis@proof-10 ¶
    • 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 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 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-79 ¶
  • proves proof ch:14-calculus-of-variations@proof-20 ¶
• proves proof ch:11-oscillations-waves@proof-22 ¶

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