lemma A.712 The force is a far-field integral

Rests on

• 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 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 definition 7.97 Partial derivative; gradient ¶
        • depends_on definition 7.26 Derivative of a function at a point ¶
        • depends_on definition 5.15 Basis ¶
    • depends_on definition 6.9 Compact set ¶
      • depends_on definition 6.5 Open cover ¶
        • depends_on definition 6.2 Open set ¶
        • depends_on equation 3.51 eq:set-indexed ¶
  • 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 axiom 7.1 Completeness of $\R$ ¶ ↺
    • depends_on theorem 7.40 Continuous functions are integrable ¶
      • depends_on definition 7.39 Darboux sums and the definite integral ¶ ↺
      • depends_on theorem 7.24 Extreme value theorem ¶
        • depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
        • depends_on proposition 7.22 Sequential characterization ¶
        • depends_on theorem 7.7 Bolzano–Weierstrass ¶
        • proves proof ch:05-real-analysis@proof-9 ¶
      • 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 ¶
      • proves proof ch:05-real-analysis@proof-23 ¶
    • depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶ ↺
  • 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 theorem 7.35 Mean value theorem ¶
        • proves proof ch:05-real-analysis@proof-20 ¶
      • 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 ¶
• 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 definition 30.17 Traction ¶
        • depends_on definition 13.5 Tensor under orthogonal transformations ¶
        • proves proof ch:13-continuum-elasticity@proof-7 ¶
  • 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 ¶
        • depends_on definition 7.13 Composite function ¶
        • depends_on definition 7.99 Differentiability at a point ¶
        • depends_on definition 7.102 Differential of a function ¶
        • proves proof ch:05-real-analysis@proof-63 ¶
      • proves proof ch:14-fluid-dynamics@proof-1 ¶
    • depends_on theorem 7.133 Gauss ¶ ↺
    • proves proof ch:14-fluid-dynamics@proof-4 ¶
  • 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 app:A-long-proofs@proof-417 ¶

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