definition 3.47 Bijective map

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Supports

• depends_on definition 4.44 Group isomorphism ¶
  • depends_on definition 4.45 Group automorphism ¶
    • depends_on proposition 4.72 The automorphism group ¶
      • depends_on definition 4.73 External semidirect product ¶
        • depends_on example 4.78 The Euclidean group ¶
        • depends_on example 4.79 The Poincaré group ¶
        • depends_on proposition 4.74 The semidirect product is a group ¶
        • depends_on proposition 18.20 The Galilean group ¶
        • depends_on theorem 4.76 Internal characterization of the semidirect product ¶
  • depends_on definition 5.143 Faithful representation ¶
  • depends_on proposition 4.70 Characterization of the direct product ¶
  • depends_on proposition 4.55 $S_3$ and the equilateral triangle ¶
  • depends_on theorem 4.56 Cayley's theorem ¶
• depends_on definition 4.48 Symmetric group ¶
  • depends_on definition 4.49 Permutation group ¶
    • depends_on proposition 4.50 Cardinal of the permutation group ¶
      • depends_on proposition 4.52 $S_3$ is a smallest non-abelian group ¶
    • depends_on proposition 4.53 The subgroups of $S_3$ ¶
    • depends_on theorem 4.56 Cayley's theorem ¶ ↺
  • depends_on definition 5.32 Levi–Civita symbol; cross product ¶
    • depends_on definition 5.131 Isotropic Cartesian tensor ¶
      • depends_on definition A.672 The octahedral rotation group ¶
        • depends_on proposition A.673 Cubic tensors of rank four ¶
      • depends_on lemma 5.132 Parity constraint ¶
        • depends_on theorem 5.133 Isotropic Cartesian tensors of rank at most four ¶
    • depends_on lemma 5.34 Contraction of two Levi–Civita symbols ¶
      • depends_on proposition 5.35 The identities of the vector algebra of $\R^{3}$ ¶
    • depends_on lemma 5.33 Determinant through the Levi–Civita symbol ¶
      • depends_on proposition 5.35 The identities of the vector algebra of $\R^{3}$ ¶ ↺
      • depends_on theorem 5.133 Isotropic Cartesian tensors of rank at most four ¶ ↺
  • depends_on definition 5.141 Representation of a group ¶
    • depends_on definition 5.146 Character ¶
    • depends_on definition 5.142 Dimension of a representation ¶
    • depends_on definition 5.145 Equivalent representations ¶
      • depends_on proposition 5.155 Averaging trick ¶
        • depends_on theorem 101.62 The mixing matrix is unitary ¶
      • depends_on theorem 5.160 Schur's second lemma ¶
        • depends_on lemma 25.39 The low-degree images are forced ¶
        • depends_on theorem 25.38 Groenewold–van Hove ¶
    • depends_on definition 5.143 Faithful representation ¶ ↺
    • depends_on definition 5.147 Invariant subspace ¶
      • depends_on definition 5.148 Irreducible representation ¶
        • depends_on proposition 5.161 Irreducible representations of an abelian group ¶
        • depends_on proposition 5.151 prop:rep-not-totally-reducible ¶
        • depends_on proposition 5.153 prop:rep-unitary-completely-reducible ¶
        • depends_on theorem 5.158 Schur's first lemma ¶
        • depends_on theorem 5.160 Schur's second lemma ¶ ↺
      • depends_on definition 5.149 Totally reducible representation ¶
        • depends_on proposition 5.151 prop:rep-not-totally-reducible ¶ ↺
        • depends_on proposition 5.153 prop:rep-unitary-completely-reducible ¶ ↺
      • depends_on lemma 5.150 Invariance of the orthogonal complement ¶
        • depends_on proposition 5.151 prop:rep-not-totally-reducible ¶ ↺
        • depends_on proposition 5.153 prop:rep-unitary-completely-reducible ¶ ↺
      • depends_on theorem 5.158 Schur's first lemma ¶ ↺
      • depends_on theorem 5.160 Schur's second lemma ¶ ↺
    • depends_on definition 5.144 Unitary representation ¶
      • depends_on lemma 5.150 Invariance of the orthogonal complement ¶ ↺
      • depends_on proposition 5.155 Averaging trick ¶ ↺
  • depends_on notation 4.51 Cycle notation ¶
  • depends_on proposition 4.72 The automorphism group ¶ ↺
• depends_on definition 7.14 Inverse function ¶
  • depends_on proposition 7.32 Derivative of the inverse function ¶
    • depends_on definition 9.140 Amplitude and the Jacobi elliptic functions ¶
      • depends_on proposition 9.143 A cubic quadrature between two turning points ¶
      • depends_on proposition 9.141 First properties ¶
        • depends_on proposition 9.143 A cubic quadrature between two turning points ¶ ↺
        • depends_on proposition 9.142 The differential equation of the sine amplitude ¶
      • depends_on proposition 9.144 The pendulum equation at finite amplitude ¶
    • depends_on proposition 7.61 The logarithm ¶
      • depends_on corollary 7.69 $\ee$ is the unique self-reproducing base ¶
        • depends_on remark 7.70 What the condition says ¶
      • depends_on definition 7.62 Real powers ¶
        • depends_on definition 7.90 The golden ratio ¶
        • depends_on proposition 7.63 Laws of real powers ¶
        • depends_on proposition 7.89 Irrationality of square roots ¶
      • depends_on lemma 7.67 The limit over real exponents ¶
        • depends_on proposition 7.68 The natural base ¶
      • depends_on proposition 7.68 The natural base ¶ ↺
      • depends_on proposition 7.63 Laws of real powers ¶ ↺
    • depends_on proposition 9.141 First properties ¶ ↺
    • depends_on proposition 9.19 Separation of variables ¶
• depends_on definition 3.62 Equipotence ¶
  • depends_on definition 3.65 Countable set ¶
    • depends_on corollary 3.68 The rationals are countable ¶
      • depends_on corollary A.3 The continuum is the power set of the naturals ¶
      • depends_on corollary 3.70 cor:inf-irrationals-uncountable ¶
      • depends_on remark 7.94 Three numbers, two kinds ¶
    • depends_on definition 7.138 Set of measure zero ¶
      • depends_on definition 7.140 Hausdorff measure ¶
        • depends_on definition 7.142 Box-counting dimension ¶
        • depends_on proposition 7.141 The critical exponent, and the Hausdorff dimension ¶
        • depends_on remark 7.144 What is developed here, and what is not ¶
      • depends_on lemma 7.139 A countable union of null sets is null ¶
        • depends_on proposition 7.143 The Hausdorff dimension never exceeds the box dimension ¶
        • depends_on proposition 7.141 The critical exponent, and the Hausdorff dimension ¶ ↺
        • depends_on remark 7.94 Three numbers, two kinds ¶ ↺
      • depends_on proposition 7.141 The critical exponent, and the Hausdorff dimension ¶ ↺
      • depends_on remark 7.144 What is developed here, and what is not ¶ ↺
    • depends_on proposition 3.67 Countability of the plane of naturals ¶
      • depends_on corollary 3.68 The rationals are countable ¶ ↺
      • depends_on lemma 7.139 A countable union of null sets is null ¶ ↺
      • depends_on lemma A.6 Lindenbaum ¶
      • depends_on remark 7.94 Three numbers, two kinds ¶ ↺
    • depends_on theorem 3.69 Uncountability of the continuum ¶
      • depends_on corollary 3.70 cor:inf-irrationals-uncountable ¶ ↺
      • depends_on remark 7.94 Three numbers, two kinds ¶ ↺
  • depends_on proposition 3.63 prop:inf-equipotence-equivalence ¶
  • depends_on theorem 3.72 Cantor–Schröder–Bernstein ¶
• depends_on definition 3.51 Inverse map ¶
  • 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 definition 13.58 Hypersurface ¶
        • depends_on proposition A.534 Every orbit is an embedded copy of $G$ ¶
        • depends_on theorem 13.67 Quotient manifold theorem ¶
        • depends_on theorem 13.62 Regular value theorem in codimension $k$ ¶
      • 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.538 Submersions have smooth local sections ¶
        • depends_on proposition A.539 Universal property ¶
      • 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 A.537 The smooth structure on the orbit space ¶ ↺
    • 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 32.24 Transcritical and pitchfork bifurcations ¶
        • depends_on theorem 32.25 Hopf bifurcation, quoted ¶
      • 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 ¶ ↺
• depends_on definition 6.7 Homeomorphism ¶ ↺
• depends_on notation 4.51 Cycle notation ¶ ↺
• depends_on proposition 3.67 Countability of the plane of naturals ¶ ↺
• depends_on proposition 3.48 Characterization of bijectivity ¶
  • depends_on proposition 3.57 Composition of bijective maps ¶ ↺
  • depends_on theorem A.1 Cantor–Schröder–Bernstein ¶
    • depends_on corollary A.3 The continuum is the power set of the naturals ¶ ↺
• depends_on theorem 3.72 Cantor–Schröder–Bernstein ¶ ↺