remark 30.27 Two results this chapter borrows from Part II

Rests on

• 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 5.133 Isotropic Cartesian tensors of rank at most four ¶
  • depends_on lemma 5.33 Determinant through the Levi–Civita symbol ¶
    • depends_on definition 5.32 Levi–Civita symbol; cross product ¶
      • depends_on definition 4.48 Symmetric group ¶
        • depends_on definition 3.47 Bijective map ¶
      • depends_on definition 5.18 Inner product ¶
        • depends_on definition 4.33 Vector space ¶
      • depends_on definition 5.27 Orthonormal basis ¶
        • depends_on definition 5.26 Orthogonal basis ¶
        • depends_on definition 5.25 Unit vector ¶
    • depends_on equation 5.19 eq:lin-leibniz-det ¶
    • proves proof ch:03-linear-algebra-representations@proof-8 ¶
  • depends_on lemma 5.132 Parity constraint ¶
    • depends_on definition 5.131 Isotropic Cartesian tensor ¶
      • depends_on definition 5.32 Levi–Civita symbol; cross product ¶ ↺
      • depends_on equation 5.162 eq:rep-special-orthogonal ¶
    • depends_on equation 5.162 eq:rep-special-orthogonal ¶ ↺
    • proves proof ch:03-linear-algebra-representations@proof-57 ¶
  • depends_on proposition 5.130 An orthogonal transformation is an isometry ¶
    • depends_on definition 5.18 Inner product ¶ ↺
    • depends_on equation 5.44 eq:lin-norm-assoc ¶
    • depends_on proposition 5.49 Injective, surjective, invertible ¶
      • depends_on definition 5.39 Kernel, image, nullity, rank ¶
        • depends_on definition 5.37 Linear transformation ¶
        • depends_on definition 5.7 Vector subspace ¶
        • proves proof ch:03-linear-algebra-representations@proof-12 ¶
      • depends_on proposition 5.48 Existence, uniqueness, and linearity of the inverse ¶
        • depends_on definition 5.47 Inverse of a linear transformation ¶
        • depends_on definition 5.37 Linear transformation ¶ ↺
        • proves proof ch:03-linear-algebra-representations@proof-17 ¶
      • depends_on theorem 5.40 Rank–nullity ¶
        • depends_on definition 5.15 Basis ¶
        • depends_on definition 5.39 Kernel, image, nullity, rank ¶ ↺
        • depends_on lemma 5.38 Exchange and completion ¶
        • proves proof ch:03-linear-algebra-representations@proof-13 ¶
      • proves proof ch:03-linear-algebra-representations@proof-18 ¶
    • proves proof ch:03-linear-algebra-representations@proof-56 ¶
  • proves proof ch:03-linear-algebra-representations@proof-58 ¶

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