theorem A.525 Helmholtz decomposition: uniqueness

open in the book · appendices/A-long-proofs.tex:25648 · p. 3049

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 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 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 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 ¶
      - depends_on definition 6.2 Open set ¶
      - depends_on equation 3.51 eq:set-indexed ¶
  - proves proof ch:04-topology@proof-2 ¶
- proves proof ch:08-pdes@proof-30 ¶
proves proof app:A-long-proofs@proof-316 ¶

Supports

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

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	eq:ana-curl-curl	declared	appendices/A-long-proofs.tex:25662
`depends_on`	→	Second-order identities of the nabla calculus	declared	appendices/A-long-proofs.tex:25662
`depends_on`	→	Weak maximum principle	declared	appendices/A-long-proofs.tex:25662
`depends_on`	←	The hypotheses, and what happens without them	declared	appendices/A-long-proofs.tex:25777
`proves`	←	app:A-long-proofs@proof-316	declared	appendices/A-long-proofs.tex:25666