lemma A.518 The excised ball

open in the book · appendices/A-long-proofs.tex:25296 · p. 3045

Rests on

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 example 7.130 The two Jacobians this treatise uses ¶
- depends_on equation 7.131 eq:ana-nabla-sph-grad ¶
- depends_on theorem 7.129 Change of variables in a multiple integral ¶
  - depends_on corollary 7.113 Inverse function theorem ¶
    - 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 proposition 7.104 Chain rule in several variables ¶
      - depends_on definition 7.13 Composite function ¶
      - depends_on definition 7.99 Differentiability at a point ¶
      - depends_on definition 7.102 Differential of a function ¶
      - proves proof ch:05-real-analysis@proof-63 ¶
    - depends_on theorem 7.112 Implicit function theorem ¶
      - depends_on definition 7.99 Differentiability at a point ¶ ↺
      - depends_on proposition 7.104 Chain rule in several variables ¶ ↺
      - depends_on theorem 7.100 $C^{1}$ implies differentiable ¶
      - proves proof ch:05-real-analysis@prooflink-1 ¶
    - proves proof ch:05-real-analysis@prooflink-2 ¶
  - depends_on definition 7.125 Multiple integral ¶ ↺
  - 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 definition 7.39 Darboux sums and the definite integral ¶ ↺
      - depends_on theorem 7.24 Extreme value theorem ¶
      - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
      - proves proof ch:05-real-analysis@proof-23 ¶
    - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶ ↺
  - proves proof ch:05-real-analysis@prooflink-3 ¶
depends_on theorem 7.129 Change of variables in a multiple integral ¶ ↺
proves proof app:A-long-proofs@proof-309 ¶

Supports

depends_on proposition A.521 Every derivative passes onto $g$ ¶
- depends_on proposition A.524 Rate of decay ¶
- depends_on proposition A.522 The Newtonian potential inverts the Laplacian ¶
  - depends_on theorem A.523 Helmholtz decomposition: existence ¶
depends_on proposition A.519 The convolution exists ¶
- depends_on proposition A.524 Rate of decay ¶ ↺

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`	→	Multiple integral	declared	appendices/A-long-proofs.tex:25311
`depends_on`	→	The two Jacobians this treatise uses	declared	appendices/A-long-proofs.tex:25311
`depends_on`	→	Change of variables in a multiple integral	declared	appendices/A-long-proofs.tex:25311
`depends_on`	←	Every derivative passes onto $g$	declared	appendices/A-long-proofs.tex:25447
`depends_on`	←	The convolution exists	declared	appendices/A-long-proofs.tex:25355
`proves`	←	app:A-long-proofs@proof-309	declared	appendices/A-long-proofs.tex:25315