remark 31.90 Why a compression wave must steepen

Rests on

• depends_on phenomenon 31.88 Sound travels at the adiabatic speed ¶
  • depends_on theorem 31.12 Conservation of mass ¶
    • depends_on lemma 31.11 Transport theorem for a material volume ¶
      • depends_on definition 31.4 Material derivative ¶
        • depends_on equation 7.124 eq:ana-nabla-gradient ¶
        • depends_on proposition 7.104 Chain rule in several variables ¶
        • proves proof ch:14-fluid-dynamics@proof-1 ¶
      • depends_on theorem 7.133 Gauss ¶
        • 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-79 ¶
      • proves proof ch:14-fluid-dynamics@proof-4 ¶
    • proves proof ch:14-fluid-dynamics@proof-5 ¶
  • depends_on theorem 31.22 Euler's equations of motion ¶
    • depends_on definition 31.21 Ideal fluid ¶
      • depends_on definition 31.1 Fluid ¶
        • depends_on theorem 30.18 Cauchy: the stress tensor exists ¶
    • depends_on lemma 31.11 Transport theorem for a material volume ¶ ↺
    • depends_on theorem 30.21 Cauchy's equation of motion ¶
      • depends_on theorem 7.133 Gauss ¶ ↺
      • depends_on theorem 30.18 Cauchy: the stress tensor exists ¶ ↺
      • proves proof ch:13-continuum-elasticity@proof-9 ¶
    • proves proof ch:14-fluid-dynamics@proof-11 ¶
  • proves proof ch:14-fluid-dynamics@proof-44 ¶
• depends_on proposition 10.15 Method of characteristics, first order ¶
  • depends_on definition 10.2 Linear, semilinear, quasilinear ¶
    • depends_on definition 10.1 Partial differential equation; order ¶
      • 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 definition 6.2 Open set ¶
        • depends_on definition 6.1 Topological space ¶
  • 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 7.26 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 ¶
  • proves proof ch:08-pdes@proof-5 ¶

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