theorem 7.129 Change of variables in a multiple integral

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:4271 · p. 256

Rests on

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 7.16 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 proposition 7.104 Chain rule in several variables ¶
  - depends_on definition 7.13 Composite function ¶
    - depends_on definition 7.9 Real function ¶ ↺
  - depends_on definition 7.99 Differentiability at a point ¶
    - depends_on definition 7.16 Limit ¶ ↺
    - depends_on definition 5.37 Linear transformation ¶
      - depends_on definition 4.33 Vector space ¶
    - depends_on equation 6.12 eq:top-euclidean-metric ¶
  - depends_on definition 7.102 Differential of a function ¶
    - depends_on definition 7.99 Differentiability at a point ¶ ↺
    - depends_on definition 7.97 Partial derivative; gradient ¶ ↺
  - 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 ¶
    - depends_on definition 7.98 Functions of class $C^{1}$ ¶ ↺
    - depends_on definition 7.99 Differentiability 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-62 ¶
  - 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 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 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 axiom 7.1 Completeness of $\R$ ¶ ↺
    - depends_on proposition 7.22 Sequential characterization ¶
      - depends_on definition 7.20 Continuity at a point ¶ ↺
      - depends_on definition 7.4 Convergence ¶
      - proves proof ch:05-real-analysis@proof-7 ¶
    - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - depends_on corollary A.45 Archimedean property and density of $\Q$ ¶
      - depends_on corollary A.46 Monotone convergence ¶
      - proves proof ch:05-real-analysis@proof-4 ¶
    - 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 7.25 Heine–Cantor: uniform continuity ¶ ↺
proves proof ch:05-real-analysis@prooflink-3 ¶

Supports

depends_on example 7.130 The two Jacobians this treatise uses ¶
- depends_on lemma A.518 The excised ball ¶
  - 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 ¶ ↺
depends_on lemma A.518 The excised ball ¶ ↺
depends_on proposition A.519 The convolution exists ¶ ↺

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`	→	Inverse function theorem	declared	parts/02-mathematical-methods/05-real-analysis.tex:4290
`depends_on`	→	Multiple integral	declared	parts/02-mathematical-methods/05-real-analysis.tex:4290
`depends_on`	→	What the derivations below take as given	declared	parts/02-mathematical-methods/05-real-analysis.tex:4290
`depends_on`	←	The two Jacobians this treatise uses	declared	parts/02-mathematical-methods/05-real-analysis.tex:4322
`depends_on`	←	The excised ball	declared	appendices/A-long-proofs.tex:25311
`depends_on`	←	The convolution exists	declared	appendices/A-long-proofs.tex:25355
`proves`	←	ch:05-real-analysis@prooflink-3	declared	parts/02-mathematical-methods/05-real-analysis.tex:4294