definition 17.38 Schwartz space and tempered distributions

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• depends_on definition 17.40 Transform and derivative of a distribution ¶
  • depends_on definition 10.42 Green's function ¶
    • depends_on definition 10.50 Retarded and advanced Green's functions ¶
    • depends_on example 10.49 The method of images ¶
    • depends_on proposition 28.24 Impulse response of the damped oscillator ¶
    • depends_on proposition 10.47 Bilinear expansion ¶
    • depends_on proposition 10.43 Representation of the solution ¶
    • depends_on theorem 10.45 Symmetry of the Green's function ¶
  • depends_on proposition 17.41 The identities physics uses ¶
    • depends_on proposition 17.90 Hankel transform as the axially symmetric Fourier transform ¶
    • depends_on proposition 17.52 Linear translation-invariant systems ¶
      • depends_on definition 28.25 Complex susceptibility ¶
      • depends_on proposition 17.61 Mean-square response of a damped resonator ¶
• depends_on definition 10.83 Weak solution ¶
  • depends_on definition 10.87 Weak form of an elliptic problem ¶
    • depends_on corollary 10.89 Existence for the elliptic problem ¶
    • depends_on proposition 10.91 Dirichlet principle ¶
    • depends_on theorem 10.88 Lax–Milgram ¶
      • depends_on corollary 10.89 Existence for the elliptic problem ¶ ↺
      • depends_on proposition 10.91 Dirichlet principle ¶ ↺
  • depends_on definition 10.85 Sobolev space ¶
    • depends_on definition A.450 The space $W^{1,r}(a,b)$ ¶
      • depends_on definition A.462 The weighted space and the energy space ¶
        • depends_on lemma A.463 Poincaré inequality; $B_{K}$ is an inner product ¶
      • depends_on lemma A.453 Uniform bound and uniform Hölder continuity ¶
        • depends_on theorem A.455 Compact embedding of $W^{1,r}(a,b)$ into the continuous functions ¶
      • depends_on lemma A.452 Young and Hölder ¶
        • depends_on lemma A.489 Courant–Lebesgue ¶
        • depends_on lemma A.463 Poincaré inequality; $B_{K}$ is an inner product ¶ ↺
        • depends_on lemma A.453 Uniform bound and uniform Hölder continuity ¶ ↺
    • depends_on definition 10.87 Weak form of an elliptic problem ¶ ↺
  • depends_on definition 10.95 Weak solution of a conservation law ¶
    • depends_on definition 10.99 Entropy condition ¶
      • depends_on definition 10.101 Entropy pair; entropy inequality ¶
        • depends_on definition A.150 Kruzhkov entropy pair ¶
        • depends_on proposition A.156 The Kruzhkov inequality ¶
        • depends_on proposition A.163 The viscous problem produces the entropy inequality ¶
        • depends_on proposition 10.102 Jump form of the entropy inequality ¶
        • depends_on theorem A.151 $L^{1}$ contraction on a cone ¶
        • depends_on theorem 10.103 Uniqueness in the entropy class ¶
    • depends_on definition 10.101 Entropy pair; entropy inequality ¶ ↺
    • depends_on example 10.98 Weak solutions are not unique ¶
    • depends_on theorem A.151 $L^{1}$ contraction on a cone ¶ ↺
    • depends_on theorem 10.103 Uniqueness in the entropy class ¶ ↺
    • depends_on theorem 10.96 Rankine–Hugoniot condition ¶
      • depends_on definition 10.99 Entropy condition ¶ ↺
      • depends_on example 10.97 Burgers shock ¶
      • depends_on example 10.98 Weak solutions are not unique ¶ ↺
      • depends_on proposition 10.102 Jump form of the entropy inequality ¶ ↺
  • depends_on proposition 10.84 Consistency ¶
• depends_on proposition A.522 The Newtonian potential inverts the Laplacian ¶
  • depends_on theorem A.523 Helmholtz decomposition: existence ¶
• depends_on proposition 17.52 Linear translation-invariant systems ¶ ↺
• depends_on proposition 17.42 Sokhotski–Plemelj ¶
  • depends_on proposition 105.22 The two-state effective generator ¶
    • depends_on proposition 105.23 Symmetry constraints on the mass matrix ¶
  • depends_on theorem 17.78 Causality implies dispersion relations ¶
    • depends_on remark 33.10 Fibre anelasticity, and why it biases one method and not the other ¶
    • depends_on remark 28.26 Causality is visible in the susceptibility ¶
    • depends_on remark 28.62 Neither velocity is a speed limit ¶
• depends_on proposition 17.39 The transform preserves $\mathcal{S}$ ¶
  • depends_on definition 17.40 Transform and derivative of a distribution ¶ ↺
• depends_on proposition 10.67 The Newtonian potential is the fundamental solution ¶
  • depends_on example 10.49 The method of images ¶ ↺
  • depends_on lemma A.94 Fundamental solution of $\pp_{\bar z}$ ¶
  • depends_on proposition A.522 The Newtonian potential inverts the Laplacian ¶ ↺
  • depends_on proposition 10.71 Green's representation formula ¶
  • depends_on remark 10.73 What compact support is doing, and the rate that replaces it ¶
  • depends_on theorem 7.137 Helmholtz decomposition ¶
    • depends_on remark A.527 The hypotheses, and what happens without them ¶
    • depends_on remark 30.46 Why the plane-wave route, and not the potentials ¶
    • depends_on remark 30.27 Two results this chapter borrows from Part II ¶
    • depends_on remark 10.73 What compact support is doing, and the rate that replaces it ¶ ↺
    • depends_on remark 10.74 Where the decomposition is used, and what it costs ¶