theorem A.513 Change of variables

open in the book · appendices/A-long-proofs.tex:25036 · p. 3043

Rests on

depends_on lemma A.499 What zero content buys ¶
- 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 definition A.498 Zero content ¶
  - depends_on definition 7.125 Multiple integral ¶ ↺
  - depends_on definition 6.9 Compact set ¶
    - depends_on definition 6.5 Open cover ¶
      - depends_on definition 6.2 Open set ¶
      - depends_on equation 3.51 eq:set-indexed ¶
- depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
  - depends_on proposition 7.22 Sequential characterization ¶
    - 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 7.4 Convergence ¶
      - depends_on definition 7.2 Absolute value ¶
    - 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 definition A.36 Cut ¶
      - depends_on theorem A.40 Least-upper-bound property ¶
      - proves proof app:A-long-proofs@proof-32 ¶
    - depends_on corollary A.46 Monotone convergence ¶
      - depends_on theorem A.40 Least-upper-bound property ¶ ↺
      - proves proof app:A-long-proofs@proof-33 ¶
    - proves proof ch:05-real-analysis@proof-4 ¶
  - proves proof ch:05-real-analysis@proof-10 ¶
- proves proof app:A-long-proofs@proof-295 ¶
depends_on lemma A.512 The boundary strip is thin ¶
- depends_on definition 7.127 Simple regions ¶
  - 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 definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 5.15 Basis ¶
  - depends_on definition 6.9 Compact set ¶ ↺
- depends_on lemma A.500 Graphs and $C^{1}$ images have zero content ¶
  - depends_on lemma A.499 What zero content buys ¶ ↺
  - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶ ↺
  - depends_on theorem 7.35 Mean value theorem ¶
    - depends_on proposition 7.29 Linearity ¶
      - depends_on definition 7.11 Sum of functions ¶
      - depends_on equation 7.14 eq:ana-derivh ¶
      - depends_on proposition 7.6 Algebra of limits ¶
      - proves proof ch:05-real-analysis@proof-13 ¶
    - depends_on theorem 7.34 Rolle ¶
      - depends_on lemma 7.33 Fermat: interior extremum ¶
      - depends_on theorem 7.24 Extreme value theorem ¶
      - proves proof ch:05-real-analysis@proof-18 ¶
    - proves proof ch:05-real-analysis@proof-19 ¶
  - proves proof app:A-long-proofs@proof-296 ¶
- depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶ ↺
- proves proof app:A-long-proofs@proof-306 ¶
depends_on proposition A.511 The substitution property is universal ¶
- depends_on lemma A.510 Every diffeomorphism factorises locally ¶
  - 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 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 A.508 Primitive map ¶
    - depends_on definition 7.98 Functions of class $C^{1}$ ¶ ↺
    - depends_on equation 5.19 eq:lin-leibniz-det ¶
  - depends_on proposition 7.104 Chain rule in several variables ¶ ↺
  - proves proof app:A-long-proofs@proof-304 ¶
- depends_on lemma A.506 Locality ¶
  - depends_on definition A.503 The substitution property ¶
    - depends_on corollary 7.113 Inverse function theorem ¶ ↺
    - depends_on definition 7.125 Multiple integral ¶ ↺
  - depends_on definition 6.9 Compact set ¶ ↺
  - depends_on lemma A.505 A continuous partition of unity ¶
    - depends_on definition 6.9 Compact set ¶ ↺
    - depends_on definition 6.6 Continuous map ¶
      - depends_on definition 3.53 Preimage ¶
      - depends_on definition 6.2 Open set ¶ ↺
      - depends_on definition 6.1 Topological space ¶
    - depends_on definition 6.24 Metric ¶
      - depends_on definition 3.43 Map ¶
      - depends_on definition 3.38 Ordered pair and Cartesian product ¶
    - proves proof app:A-long-proofs@proof-300 ¶
  - proves proof app:A-long-proofs@proof-301 ¶
- depends_on lemma A.509 Primitive maps have the substitution property ¶
  - depends_on definition A.508 Primitive map ¶ ↺
  - depends_on equation 7.27 eq:ana-substitution ¶
  - depends_on lemma A.501 Iterated integration over a box ¶
    - depends_on definition 7.39 Darboux sums and the definite integral ¶ ↺
    - 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 ¶
    - proves proof app:A-long-proofs@proof-297 ¶
  - proves proof app:A-long-proofs@proof-303 ¶
- proves proof app:A-long-proofs@proof-305 ¶
proves proof app:A-long-proofs@proof-307 ¶

Supports

depends_on corollary A.514 Degeneracy on a negligible set ¶
depends_on remark A.516 The hypotheses of the global form ¶

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`	→	What zero content buys	declared	appendices/A-long-proofs.tex:25046
`depends_on`	→	The boundary strip is thin	declared	appendices/A-long-proofs.tex:25046
`depends_on`	→	The substitution property is universal	declared	appendices/A-long-proofs.tex:25046
`depends_on`	←	Degeneracy on a negligible set	declared	appendices/A-long-proofs.tex:25108
`depends_on`	←	The hypotheses of the global form	declared	appendices/A-long-proofs.tex:25196
`proves`	←	app:A-long-proofs@proof-307	declared	appendices/A-long-proofs.tex:25050