theorem A.523 Helmholtz decomposition: existence

Rests on

• depends_on equation 7.144 eq:ana-curl-curl ¶
• 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 definition 7.2 Absolute value ¶
        • depends_on definition 7.9 Real function ¶
      • 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 definition 7.11 Sum of functions ¶
        • depends_on equation 7.14 eq:ana-derivh ¶
        • depends_on proposition 7.6 Algebra of limits ¶
        • proves proof ch:05-real-analysis@proof-13 ¶
      • depends_on theorem 7.34 Rolle ¶
        • depends_on lemma 7.33 Fermat: interior extremum ¶
        • depends_on theorem 7.24 Extreme value theorem ¶
        • proves proof ch:05-real-analysis@proof-18 ¶
      • 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 A.522 The Newtonian potential inverts the Laplacian ¶
  • depends_on definition 17.38 Schwartz space and tempered distributions ¶
  • depends_on proposition A.521 Every derivative passes onto $g$ ¶
    • depends_on lemma A.520 A $C^{1}$ limit ¶
      • 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 7.42 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 ¶
      • depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
        • depends_on definition 7.41 Antiderivative ¶
        • depends_on theorem 7.42 Fundamental theorem of calculus, I ¶ ↺
        • proves proof ch:05-real-analysis@proof-25 ¶
      • proves proof app:A-long-proofs@proof-311 ¶
    • depends_on lemma A.518 The excised ball ¶
      • 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 example 7.130 The two Jacobians this treatise uses ¶
        • depends_on equation 7.131 eq:ana-nabla-sph-grad ¶
        • depends_on theorem 7.129 Change of variables in a multiple integral ¶
      • depends_on theorem 7.129 Change of variables in a multiple integral ¶ ↺
      • proves proof app:A-long-proofs@proof-309 ¶
    • depends_on theorem 7.109 Leibniz integral rule ¶
      • depends_on definition 7.125 Multiple integral ¶ ↺
      • 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 7.35 Mean value theorem ¶ ↺
      • proves proof ch:05-real-analysis@proof-67 ¶
    • proves proof app:A-long-proofs@proof-312 ¶
  • 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 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:08-pdes@proof-14 ¶
    • proves proof ch:08-pdes@proof-27 ¶
  • proves proof app:A-long-proofs@proof-313 ¶
• proves proof app:A-long-proofs@proof-314 ¶

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