proposition A.521 Every derivative passes onto $g$

open in the book · appendices/A-long-proofs.tex:25436 · p. 3047

Rests on

depends_on lemma A.520 A $C^{1}$ limit ¶
- 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 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 definition 7.97 Partial derivative; gradient ¶
    - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 7.16 Limit ¶ ↺
      - depends_on definition 7.9 Real function ¶ ↺
    - depends_on definition 5.15 Basis ¶
      - depends_on definition 5.12 Subspace generated by a set of vectors ¶
      - depends_on definition 5.14 Linear independence ¶
- depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
  - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
  - depends_on theorem 7.40 Continuous functions are integrable ¶
    - depends_on definition 7.39 Darboux sums and the definite integral ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶
    - 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 ¶
    - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
      - depends_on proposition 7.22 Sequential characterization ¶ ↺
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶ ↺
      - proves proof ch:05-real-analysis@proof-10 ¶
    - proves proof ch:05-real-analysis@proof-23 ¶
  - depends_on theorem 7.24 Extreme value theorem ¶ ↺
  - proves proof ch:05-real-analysis@proof-24 ¶
- depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
  - depends_on definition 7.41 Antiderivative ¶
    - depends_on corollary 7.36 cor:ana-mvt-consequences ¶
      - depends_on theorem 7.35 Mean value theorem ¶
      - proves proof ch:05-real-analysis@proof-20 ¶
    - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
  - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶ ↺
  - proves proof ch:05-real-analysis@proof-25 ¶
- proves proof app:A-long-proofs@proof-311 ¶
depends_on lemma A.518 The excised ball ¶
- 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 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 proposition 7.104 Chain rule in several variables ¶
      - depends_on theorem 7.112 Implicit function theorem ¶
      - 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 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 ¶
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-312 ¶

Supports

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 ¶

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`	→	A $C^{1}$ limit	declared	appendices/A-long-proofs.tex:25447
`depends_on`	→	The excised ball	declared	appendices/A-long-proofs.tex:25447
`depends_on`	→	Leibniz integral rule	declared	appendices/A-long-proofs.tex:25447
`depends_on`	←	Rate of decay	declared	appendices/A-long-proofs.tex:25598
`depends_on`	←	The Newtonian potential inverts the Laplacian	declared	appendices/A-long-proofs.tex:25502
`proves`	←	app:A-long-proofs@proof-312	declared	appendices/A-long-proofs.tex:25451