remark 30.46 Why the plane-wave route, and not the potentials

Rests on

• depends_on phenomenon 30.43 Two bulk wave speeds ¶
  • depends_on proposition 30.30 Stability bounds ¶
    • depends_on equation 30.23 eq:elast-energy-density ¶
    • depends_on theorem 5.116 Sylvester's law of inertia ¶
      • depends_on definition 5.110 Bilinear map ¶
        • depends_on definition 5.37 Linear transformation ¶
      • depends_on definition 5.111 Symmetric and antisymmetric forms ¶
        • depends_on definition 5.110 Bilinear map ¶ ↺
        • depends_on equation 5.137 eq:lin-bilinear-matrix ¶
      • depends_on equation 5.139 eq:lin-bilinear-congruence ¶
      • proves proof ch:03-linear-algebra-representations@proof-53 ¶
    • proves proof ch:13-continuum-elasticity@proof-13 ¶
  • depends_on theorem 30.40 Navier–Cauchy equations ¶
    • depends_on equation 30.21 eq:elast-hooke-isotropic ¶
    • depends_on theorem 30.21 Cauchy's equation of motion ¶
      • 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 ¶
      • depends_on theorem 30.18 Cauchy: the stress tensor exists ¶
        • depends_on definition 30.17 Traction ¶
        • depends_on definition 13.5 Tensor under orthogonal transformations ¶
        • proves proof ch:13-continuum-elasticity@proof-7 ¶
      • proves proof ch:13-continuum-elasticity@proof-9 ¶
    • proves proof ch:13-continuum-elasticity@proof-18 ¶
  • proves proof ch:13-continuum-elasticity@proof-21 ¶
• depends_on theorem 7.137 Helmholtz decomposition ¶
  • depends_on proposition 7.123 Second-order identities of the nabla calculus ¶
    • depends_on equation 7.127 eq:ana-nabla-laplacian ¶
    • depends_on lemma 7.121 Contraction of two Levi-Civita symbols ¶
      • depends_on equation 7.134 eq:ana-epsilon-curl ¶
      • proves proof ch:05-real-analysis@proof-74 ¶
    • depends_on proposition 7.105 Clairaut–Schwarz ¶
      • 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 theorem 7.35 Mean value theorem ¶
        • depends_on proposition 7.29 Linearity ¶
        • depends_on theorem 7.34 Rolle ¶
        • proves proof ch:05-real-analysis@proof-19 ¶
      • proves proof ch:05-real-analysis@proof-64 ¶
    • proves proof ch:05-real-analysis@proof-76 ¶
  • depends_on proposition 10.67 The Newtonian potential is the fundamental solution ¶
    • depends_on definition 17.38 Schwartz space and tempered distributions ¶
    • depends_on lemma 10.44 Green's identities ¶
      • 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 theorem 7.133 Gauss ¶ ↺
      • proves proof ch:08-pdes@proof-14 ¶
    • proves proof ch:08-pdes@proof-27 ¶
  • depends_on theorem 7.109 Leibniz integral rule ¶
    • 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 axiom 7.1 Completeness of $\R$ ¶ ↺
    • depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
      • depends_on proposition 7.22 Sequential characterization ¶
        • depends_on definition 7.20 Continuity at a point ¶ ↺
        • depends_on definition 7.4 Convergence ¶
        • proves proof ch:05-real-analysis@proof-7 ¶
      • depends_on theorem 7.7 Bolzano–Weierstrass ¶
        • depends_on corollary A.45 Archimedean property and density of $\Q$ ¶
        • depends_on corollary A.46 Monotone convergence ¶
        • proves proof ch:05-real-analysis@proof-4 ¶
      • proves proof ch:05-real-analysis@proof-10 ¶
    • depends_on theorem 7.35 Mean value theorem ¶ ↺
    • proves proof ch:05-real-analysis@proof-67 ¶
  • proves proof app:A-long-proofs@proof-317 ¶
  • proves proof ch:05-real-analysis@prooflink-4 ¶
• depends_on theorem 30.40 Navier–Cauchy equations ¶ ↺

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