remark 10.73 What compact support is doing, and the rate that replaces it

open in the book · parts/02-mathematical-methods/08-pdes.tex:1975 · p. 361

remark -- no derivation owed by its kind

Rests on

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 7.16 Limit ¶
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
    - 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 7.137 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 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 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

Nothing declares a dependency on this node yet.

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`	→	The Newtonian potential is the fundamental solution	declared	parts/02-mathematical-methods/08-pdes.tex:2003
`depends_on`	→	Helmholtz decomposition	declared	parts/02-mathematical-methods/08-pdes.tex:2003