remark A.527 The hypotheses, and what happens without them

Rests on

• depends_on equation 7.126 eq:ana-nabla-curl ¶
• 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 ¶
        • 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 ¶
  • 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 A.525 Helmholtz decomposition: uniqueness ¶
  • depends_on equation 7.144 eq:ana-curl-curl ¶
  • depends_on proposition 7.123 Second-order identities of the nabla calculus ¶ ↺
  • depends_on theorem 10.75 Weak maximum principle ¶
    • depends_on definition 10.66 Harmonic function ¶
      • depends_on definition 7.98 Functions of class $C^{1}$ ¶ ↺
      • depends_on equation 10.24 eq:pde-poisson-equation ¶
    • depends_on theorem 6.11 Heine–Borel on $\R$ ¶
      • depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
      • depends_on definition 6.9 Compact set ¶
        • depends_on definition 6.5 Open cover ¶
      • proves proof ch:04-topology@proof-2 ¶
    • proves proof ch:08-pdes@proof-30 ¶
  • proves proof app:A-long-proofs@proof-316 ¶

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