proposition 7.123 Second-order identities of the nabla calculus

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:4094 · p. 254

Rests on

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 definition 3.43 Map ¶
  - 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 definition 7.9 Real function ¶ ↺
    - depends_on equation 7.14 eq:ana-derivh ¶
    - depends_on proposition 7.6 Algebra of limits ¶
      - depends_on definition 7.4 Convergence ¶
      - depends_on proposition 7.3 Triangle inequality ¶
      - proves proof ch:05-real-analysis@proof-3 ¶
    - proves proof ch:05-real-analysis@proof-13 ¶
  - depends_on theorem 7.34 Rolle ¶
    - depends_on lemma 7.33 Fermat: interior extremum ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on proposition 7.18 Two-sided limit from one-sided limits ¶
      - proves proof ch:05-real-analysis@proof-17 ¶
    - depends_on theorem 7.24 Extreme value theorem ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶
      - depends_on proposition 7.22 Sequential characterization ¶
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - proves proof ch:05-real-analysis@proof-9 ¶
    - 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 ¶

Supports

depends_on remark 7.124 The Laplacian of a vector field is a Cartesian notion ¶
depends_on theorem 7.137 Helmholtz decomposition ¶
- 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 ¶
depends_on theorem A.523 Helmholtz decomposition: existence ¶
depends_on theorem A.525 Helmholtz decomposition: uniqueness ¶
- 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-nabla-laplacian	declared	parts/02-mathematical-methods/05-real-analysis.tex:4113
`depends_on`	→	Contraction of two Levi-Civita symbols	declared	parts/02-mathematical-methods/05-real-analysis.tex:4113
`depends_on`	→	Clairaut–Schwarz	declared	parts/02-mathematical-methods/05-real-analysis.tex:4113
`depends_on`	←	The Laplacian of a vector field is a Cartesian notion	declared	parts/02-mathematical-methods/05-real-analysis.tex:4157
`depends_on`	←	Helmholtz decomposition	declared	parts/02-mathematical-methods/05-real-analysis.tex:4673
`depends_on`	←	Helmholtz decomposition: existence	declared	appendices/A-long-proofs.tex:25546
`depends_on`	←	Helmholtz decomposition: uniqueness	declared	appendices/A-long-proofs.tex:25662
`proves`	←	ch:05-real-analysis@proof-76	declared	parts/02-mathematical-methods/05-real-analysis.tex:4117