definition 3.28 Set

open in the book · parts/02-mathematical-methods/01-logic-sets.tex:752 · p. 25

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depends_on definition 4.1 Cartesian product ¶
- depends_on definition 4.2 Binary operation ¶
  - depends_on definition 4.3 Algebraic structure ¶
    - depends_on definition 4.6 Associativity ¶
      - depends_on definition 4.18 Semigroup ¶
    - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.34 Algebra ¶
      - depends_on definition 4.31 Module ¶
      - depends_on definition 4.30 Ring ¶
    - depends_on definition 4.4 Internal binary operation; magma ¶
      - depends_on definition 4.34 Algebra ¶ ↺
      - depends_on definition 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.18 Semigroup ¶ ↺
      - depends_on definition 4.12 Substructure ¶
      - depends_on definition 5.121 Algebra ¶
    - depends_on definition 4.9 Neutral element ¶
      - depends_on definition 4.29 Action ¶
      - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.15 Kernel ¶
      - depends_on definition 4.20 Monoid ¶
      - depends_on proposition 4.10 Uniqueness of the neutral element ¶
  - depends_on definition 4.4 Internal binary operation; magma ¶ ↺
- depends_on definition 4.65 Direct product of groups ¶
  - depends_on proposition 4.68 Direct product of abelian groups ¶
    - depends_on example 4.69 The Cayley table of $\Z_2\times\Z_4$ ¶
  - depends_on proposition 4.70 Characterization of the direct product ¶
  - depends_on proposition 4.66 prop:alg-direct-product-group ¶
  - depends_on proposition 4.77 When a semidirect product is direct ¶
    - depends_on example 4.78 The Euclidean group ¶
      - depends_on proposition 18.20 The Galilean 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 4.75 The two factors inside the semidirect product ¶
      - depends_on example 4.79 The Poincaré group ¶ ↺
      - depends_on proposition 4.77 When a semidirect product is direct ¶ ↺
  - depends_on proposition 18.20 The Galilean group ¶ ↺
  - depends_on theorem 4.76 Internal characterization of the semidirect product ¶
depends_on definition 3.43 Map ¶
- depends_on definition 4.2 Binary operation ¶ ↺
- depends_on definition 4.43 Group homomorphism ¶
  - 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 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 4.73 External semidirect product ¶ ↺
  - 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 5.160 Schur's second lemma ¶
    - depends_on definition 5.143 Faithful representation ¶ ↺
    - depends_on definition 5.147 Invariant subspace ¶
      - depends_on definition 5.148 Irreducible representation ¶
      - depends_on definition 5.149 Totally reducible representation ¶
      - depends_on lemma 5.150 Invariance of the orthogonal complement ¶
      - 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 proposition 4.42 When the homomorphisms form a group ¶
  - depends_on proposition 4.47 Image of the inverse element ¶
  - depends_on proposition 4.46 Image of the neutral element ¶
    - depends_on proposition 4.47 Image of the inverse element ¶ ↺
  - depends_on proposition 4.17 Trivial kernel of a homomorphism ¶
- depends_on definition 4.16 Image ¶
- depends_on definition 7.9 Real function ¶
  - depends_on definition 7.15 Absolute-value function ¶
  - depends_on definition 7.13 Composite function ¶
    - depends_on definition 7.14 Inverse function ¶
      - depends_on proposition 7.32 Derivative of the inverse function ¶
    - depends_on proposition 7.31 Chain rule ¶
      - depends_on corollary 7.44 Substitution and integration by parts ¶
      - depends_on lemma A.479 Conformal invariance of the Dirichlet integral ¶
      - depends_on lemma 27.15 Velocity and acceleration in plane polar coordinates ¶
      - depends_on proposition 7.111 Jacobi's formula, cofactor form ¶
      - depends_on proposition 7.63 Laws of real powers ¶
      - depends_on proposition 16.26 Covariance under change of dependent variable ¶
      - depends_on proposition 10.15 Method of characteristics, first order ¶
      - depends_on theorem 16.29 Beltrami identity ¶
      - depends_on theorem 10.8 The type is a coordinate invariant ¶
    - depends_on proposition 7.104 Chain rule in several variables ¶
      - depends_on corollary 7.113 Inverse function theorem ¶
      - depends_on corollary 7.110 Variable limits of integration ¶
      - depends_on corollary A.292 Inverse function theorem ¶
      - depends_on definition 30.4 Material and spatial descriptions ¶
      - depends_on definition 31.4 Material derivative ¶
      - depends_on definition 22.26 Lagrange bracket ¶
      - depends_on definition 10.18 Complete integral; envelope ¶
      - depends_on lemma A.510 Every diffeomorphism factorises locally ¶
      - depends_on lemma A.504 Transitivity ¶
      - depends_on lemma A.306 The standard smooth bump ¶
      - depends_on proposition 7.136 Properties of conservative fields ¶
      - depends_on proposition 7.120 The envelope touches every member it meets ¶
      - … 9 more
  - depends_on definition 7.10 Constant function ¶
    - depends_on proposition 7.28 Derivative of a constant function ¶
  - depends_on definition 7.26 Derivative of a function at a point ¶
    - depends_on definition 7.41 Antiderivative ¶
      - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
    - depends_on definition 7.97 Partial derivative; gradient ¶
      - depends_on definition 7.98 Functions of class $C^{1}$ ¶
      - depends_on definition 7.102 Differential of a function ¶
      - depends_on definition 7.119 Envelope of a family ¶
      - depends_on definition A.70 Pullback ¶
      - depends_on definition 32.19 Lyapunov function ¶
      - depends_on definition 30.2 Representative volume element; continuum field ¶
      - depends_on definition 22.29 Poisson bracket ¶
      - depends_on lemma A.73 Differentiation under the integral sign ¶
    - depends_on definition 18.11 Acceleration ¶
      - depends_on definition 18.27 Angular position, velocity and acceleration ¶
      - depends_on lemma 27.15 Velocity and acceleration in plane polar coordinates ¶ ↺
      - depends_on phenomenon 18.31 Uniform circular motion is accelerated motion ¶
      - depends_on phenomenon 18.12 Uniformly accelerated motion ¶
      - depends_on proposition 18.33 Intrinsic decomposition of the acceleration ¶
      - depends_on proposition 18.23 Transformation of velocity and acceleration ¶
    - depends_on definition 18.10 Velocity ¶
      - depends_on definition 18.11 Acceleration ¶ ↺
      - depends_on definition 18.27 Angular position, velocity and acceleration ¶ ↺
      - depends_on definition 18.29 Angular velocity vector ¶
      - depends_on definition 19.29 Kinetic energy ¶
      - depends_on definition 19.5 Linear momentum ¶
      - depends_on lemma 27.15 Velocity and acceleration in plane polar coordinates ¶ ↺
      - depends_on proposition 18.23 Transformation of velocity and acceleration ¶ ↺
    - depends_on lemma 7.33 Fermat: interior extremum ¶
      - depends_on proposition 7.117 Lagrange multipliers in finitely many variables ¶
      - depends_on proposition 16.16 Stationarity is necessary ¶
      - depends_on theorem 7.34 Rolle ¶
      - depends_on theorem 16.104 Snell's law of refraction ¶
    - depends_on proposition 7.27 Differentiable implies continuous ¶
      - depends_on proposition 7.31 Chain rule ¶ ↺
      - depends_on proposition 7.68 The natural base ¶
      - depends_on proposition 7.61 The logarithm ¶
      - depends_on proposition 7.30 Leibniz rule ¶
      - depends_on theorem 7.51 Termwise differentiation ¶
    - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
      - depends_on corollary 7.110 Variable limits of integration ¶ ↺
      - depends_on definition A.450 The space $W^{1,r}(a,b)$ ¶
      - depends_on lemma A.520 A $C^{1}$ limit ¶
      - depends_on lemma A.440 Grönwall's inequality ¶
      - depends_on lemma 16.19 du Bois-Reymond ¶
      - depends_on lemma 9.10 Grönwall's inequality ¶
      - depends_on proposition 11.22 Properties of the distribution function ¶
      - depends_on proposition 9.18 The linear equation: integrating factor ¶
      - depends_on proposition 9.19 Separation of variables ¶
      - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶ ↺
      - depends_on theorem 10.52 Duhamel's principle ¶
  - depends_on definition 7.16 Limit ¶
    - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.98 Functions of class $C^{1}$ ¶ ↺
      - depends_on lemma 9.6 Weierstrass $M$-test; uniform limits are continuous ¶
      - depends_on proposition 7.105 Clairaut–Schwarz ¶
      - depends_on proposition 7.145 Delta as a limit; elementary properties ¶
      - depends_on proposition 7.27 Differentiable implies continuous ¶ ↺
      - depends_on proposition 7.22 Sequential characterization ¶
      - depends_on remark 7.21 rem:ana-discontinuities ¶
      - depends_on remark 7.101 Partial derivatives alone do not suffice ¶
      - depends_on theorem 7.23 Intermediate value theorem ¶
    - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
    - depends_on definition 7.99 Differentiability at a point ¶
      - depends_on definition 7.102 Differential of a function ¶ ↺
      - depends_on lemma A.291 $h$ is differentiable, with the stated derivative ¶
      - depends_on proposition 7.104 Chain rule in several variables ¶ ↺
      - depends_on theorem 7.100 $C^{1}$ implies differentiable ¶
      - depends_on theorem 7.112 Implicit function theorem ¶
      - depends_on theorem A.286 Implicit function theorem ¶
    - depends_on definition 7.19 Infinite limits ¶
      - depends_on lemma 7.67 The limit over real exponents ¶
    - depends_on definition 9.86 Outer and inner expansions; matching ¶
    - depends_on proposition 7.18 Two-sided limit from one-sided limits ¶
      - depends_on lemma 7.33 Fermat: interior extremum ¶ ↺
    - depends_on remark 7.21 rem:ana-discontinuities ¶ ↺
    - depends_on remark 7.17 rem:ana-limit-punctured ¶
    - depends_on theorem 7.37 Cauchy mean value theorem and l'Hôpital's rule ¶
      - depends_on lemma A.306 The standard smooth bump ¶ ↺
  - depends_on definition 7.12 Product of functions ¶
    - depends_on proposition 7.30 Leibniz rule ¶ ↺
  - depends_on definition 7.11 Sum of functions ¶
    - depends_on proposition 7.29 Linearity ¶
      - depends_on theorem 7.35 Mean value theorem ¶
- depends_on definition 6.24 Metric ¶
  - depends_on definition 7.140 Hausdorff measure ¶
    - depends_on definition 7.142 Box-counting dimension ¶
      - depends_on definition 32.79 Box-counting dimension, restated from Part II ¶
      - depends_on proposition 7.143 The Hausdorff dimension never exceeds the box dimension ¶
      - depends_on remark 7.144 What is developed here, and what is not ¶
    - depends_on proposition 7.141 The critical exponent, and the Hausdorff dimension ¶
      - depends_on proposition 7.143 The Hausdorff dimension never exceeds the box dimension ¶ ↺
    - depends_on remark 7.144 What is developed here, and what is not ¶ ↺
  - depends_on definition 16.12 The two norms; weak and strong neighbourhoods ¶
    - depends_on definition 16.13 Weak and strong extrema ¶
      - depends_on example 16.14 A weak minimum that is not strong ¶
      - depends_on proposition 16.16 Stationarity is necessary ¶ ↺
      - depends_on theorem 16.60 Weierstrass' necessary condition ¶
  - depends_on definition 5.22 Metric associated with a norm ¶
  - depends_on definition 10.24 Well-posed problem ¶
    - depends_on corollary A.152 Uniqueness ¶
    - depends_on corollary 10.76 Uniqueness and stability for the Dirichlet problem ¶
    - depends_on corollary 10.81 Uniqueness for the heat equation ¶
    - depends_on definition 30.22 Boundary conditions of elastostatics ¶
      - depends_on proposition 30.50 Love waves need a slow surface layer ¶
      - depends_on proposition 30.48 Rayleigh's secular equation ¶
      - depends_on proposition 30.23 Virtual work; the weak form ¶
    - depends_on proposition 10.25 The backward heat problem is ill posed ¶
    - depends_on theorem 44.45 Local existence and uniqueness; Choquet-Bruhat; imported ¶
      - depends_on theorem 44.46 Maximal globally hyperbolic development; Choquet-Bruhat and Geroch; imported ¶
  - depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
    - depends_on corollary 12.24 Separable spaces have countable orthonormal families ¶
      - depends_on theorem 12.33 Every separable Hilbert space is $\ell^{2}$ ¶
    - depends_on definition 12.32 Separable Hilbert space ¶
      - depends_on lemma A.250 Decomposition into cyclic subspaces ¶
    - depends_on definition 12.2 Hilbert space ¶
      - depends_on definition A.278 Countably Hilbert nuclear space ¶
      - depends_on definition 12.32 Separable Hilbert space ¶ ↺
      - depends_on definition 12.94 Tensor product of Hilbert spaces ¶
      - depends_on definition 10.85 Sobolev space ¶
      - depends_on example 12.9 The sequence space $\ell^{2}$ ¶
      - depends_on example 12.11 The function space $L^{2}$ ¶
      - depends_on lemma A.254 Riemann integral of a continuous curve ¶
      - depends_on proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶
      - depends_on proposition 12.28 Convergence criterion for orthogonal series ¶
      - depends_on proposition 12.51 The three cases are exclusive and exhaustive ¶
      - depends_on theorem 16.70 The direct method ¶
      - depends_on theorem 12.14 Closest point in a closed convex set ¶
      - … 1 more
    - depends_on definition 6.29 Sequential compactness ¶
      - depends_on lemma 6.30 Lebesgue number ¶
      - depends_on theorem 6.31 Compactness and sequential compactness ¶
    - depends_on proposition 12.8 Absolutely convergent series test ¶
      - depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶
      - depends_on theorem 12.12 Riesz–Fischer ¶
  - depends_on definition 6.26 Open ball; metric topology ¶
    - depends_on definition A.76 Star-shaped set ¶
      - depends_on lemma A.77 Poincaré lemma, converse form ¶
    - depends_on lemma 6.30 Lebesgue number ¶ ↺
    - depends_on proposition 6.28 $\varepsilon$–$\delta$ characterization ¶
      - depends_on proposition 12.36 Boundedness is continuity ¶
  - depends_on definition 6.29 Sequential compactness ¶ ↺
  - depends_on lemma A.505 A continuous partition of unity ¶
    - depends_on lemma A.506 Locality ¶
      - depends_on proposition A.511 The substitution property is universal ¶
depends_on theorem 3.75 Russell, 1901 ¶

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Edges

type	direction	node	provenance	where
`depends_on`	←	Cartesian product	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:53
`depends_on`	←	Map	declared	parts/02-mathematical-methods/01-logic-sets.tex:1215
`depends_on`	←	Russell, 1901	declared	parts/02-mathematical-methods/01-logic-sets.tex:1926