definition 3.51 Inverse map

open in the book · parts/02-mathematical-methods/01-logic-sets.tex:1331 · p. 32

ground object -- no derivation owed

Rests on

depends_on definition 3.47 Bijective map ¶
depends_on definition 3.49 Composition map ¶
depends_on definition 3.50 Identity map ¶

Supports

depends_on definition 6.7 Homeomorphism ¶
- depends_on definition 13.50 Diffeomorphism ¶
- depends_on definition 13.53 Immersion, submersion, embedding ¶
  - depends_on definition 13.54 Embedded submanifold ¶
    - depends_on corollary 13.64 The image of a constant-rank map, locally ¶
    - depends_on definition 13.130 Distribution; involutive; integrable ¶
      - depends_on corollary 13.134 Frobenius for a Pfaffian system ¶
      - depends_on theorem 13.133 Frobenius ¶
    - depends_on definition 13.58 Hypersurface ¶
      - depends_on example 13.61 The $2$-sphere in $\R^{3}$ ¶
      - depends_on theorem 13.59 Regular value theorem ¶
    - depends_on proposition A.534 Every orbit is an embedded copy of $G$ ¶
    - depends_on theorem 13.67 Quotient manifold theorem ¶
      - depends_on example 13.68 Why each hypothesis is there ¶
      - depends_on proposition A.566 The isotropy group acts, and the quotient is smooth ¶
      - depends_on theorem A.563 Marsden–Weinstein reduction ¶
    - depends_on theorem 13.62 Regular value theorem in codimension $k$ ¶
      - depends_on corollary 13.64 The image of a constant-rank map, locally ¶ ↺
  - depends_on example 13.55 An injective immersion that is not an embedding ¶
  - depends_on lemma A.533 The orbit map has constant rank $d$ ¶
    - depends_on proposition A.534 Every orbit is an embedded copy of $G$ ¶ ↺
    - depends_on proposition A.535 Existence of a slice ¶
      - depends_on lemma A.536 A slice is a chart domain downstairs ¶
      - depends_on remark A.542 Where each hypothesis is spent ¶
  - depends_on lemma A.538 Submersions have smooth local sections ¶
    - depends_on proposition A.539 Universal property ¶
      - depends_on corollary A.540 Uniqueness of the smooth structure ¶
  - depends_on proposition A.534 Every orbit is an embedded copy of $G$ ¶ ↺
  - depends_on theorem A.537 The smooth structure on the orbit space ¶
- depends_on lemma A.536 A slice is a chart domain downstairs ¶ ↺
- depends_on theorem 32.15 Hartman–Grobman, restated from Part II ¶
  - depends_on definition 32.22 Bifurcation ¶
    - depends_on proposition 32.23 Saddle-node bifurcation ¶
      - depends_on proposition 32.95 Intermittency: the scaling of the laminar phase ¶
    - depends_on proposition 32.24 Transcritical and pitchfork bifurcations ¶
    - depends_on theorem 32.25 Hopf bifurcation, quoted ¶
      - depends_on proposition 32.26 The Hopf normal form ¶
  - depends_on example 32.18 A linear centre that is really a stable focus ¶
- depends_on theorem 9.32 Hartman–Grobman; quoted ¶
  - depends_on theorem 32.15 Hartman–Grobman, restated from Part II ¶ ↺
depends_on notation 4.13 The opposite map is not the inverse map ¶
depends_on proposition 3.63 prop:inf-equipotence-equivalence ¶
depends_on proposition 3.57 Composition of bijective maps ¶
- depends_on proposition 3.63 prop:inf-equipotence-equivalence ¶ ↺

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`	→	Bijective map	declared	parts/02-mathematical-methods/01-logic-sets.tex:1336
`depends_on`	→	Composition map	declared	parts/02-mathematical-methods/01-logic-sets.tex:1336
`depends_on`	→	Identity map	declared	parts/02-mathematical-methods/01-logic-sets.tex:1336
`depends_on`	←	Homeomorphism	declared	parts/02-mathematical-methods/04-topology.tex:96
`depends_on`	←	The opposite map is not the inverse map	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:328
`depends_on`	←	prop:inf-equipotence-equivalence	declared	parts/02-mathematical-methods/01-logic-sets.tex:1637
`depends_on`	←	Composition of bijective maps	declared	parts/02-mathematical-methods/01-logic-sets.tex:1435