theorem 7.137 Helmholtz decomposition

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:4652 · p. 260

Rests on

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 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 definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 5.15 Basis ¶
  - 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 6.9 Compact set ¶
    - depends_on remark 7.128 What the derivations below take as given ¶
      - depends_on definition 7.125 Multiple integral ¶
      - depends_on theorem 7.40 Continuous functions are integrable ¶
      - 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 theorem 7.42 Fundamental theorem of calculus, I ¶
      - proves proof ch:05-real-analysis@proof-25 ¶
    - 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 theorem 7.25 Heine–Cantor: uniform continuity ¶ ↺
- 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 ¶

Supports

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 ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
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Edges

type	direction	node	provenance	where
`depends_on`	→	Second-order identities of the nabla calculus	declared	parts/02-mathematical-methods/05-real-analysis.tex:4673
`depends_on`	→	The Newtonian potential is the fundamental solution	declared	parts/02-mathematical-methods/05-real-analysis.tex:4673
`depends_on`	→	Leibniz integral rule	declared	parts/02-mathematical-methods/05-real-analysis.tex:4673
`depends_on`	←	The hypotheses, and what happens without them	declared	appendices/A-long-proofs.tex:25777
`depends_on`	←	Why the plane-wave route, and not the potentials	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1604
`depends_on`	←	Two results this chapter borrows from Part II	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:851
`depends_on`	←	What compact support is doing, and the rate that replaces it	declared	parts/02-mathematical-methods/08-pdes.tex:2003
`depends_on`	←	Where the decomposition is used, and what it costs	declared	parts/02-mathematical-methods/08-pdes.tex:2026
`proves`	←	app:A-long-proofs@proof-317	declared	appendices/A-long-proofs.tex:25702
`proves`	←	ch:05-real-analysis@prooflink-4	declared	parts/02-mathematical-methods/05-real-analysis.tex:4677