definition 3.43 Map

open in the book · parts/02-mathematical-methods/01-logic-sets.tex:1204 · p. 31

ground object -- no derivation owed

Rests on

depends_on definition 3.24 Quantifiers ¶
- depends_on definition 3.23 Predicate ¶
  - depends_on definition 3.1 Logical proposition ¶
  - depends_on definition 3.38 Ordered pair and Cartesian product ¶
depends_on definition 3.28 Set ¶

Supports

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 4.20 Monoid ¶
      - depends_on definition 4.30 Ring ¶
      - depends_on example 4.19 A two-element semigroup ¶
      - depends_on proposition 15.62 The complex line is not a semigroup algebra ¶
  - depends_on definition 4.28 Distributivity ¶
    - depends_on definition 4.34 Algebra ¶
    - depends_on definition 4.31 Module ¶
      - depends_on definition 4.33 Vector space ¶
    - 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 4.30 Ring ¶ ↺
      - depends_on definition 4.33 Vector space ¶ ↺
      - depends_on proposition 4.68 Direct product of abelian groups ¶
      - depends_on proposition 4.52 $S_3$ is a smallest non-abelian group ¶
      - depends_on proposition 4.41 prop:alg-zn-group ¶
    - depends_on definition 4.18 Semigroup ¶ ↺
    - depends_on definition 4.12 Substructure ¶
    - depends_on definition 5.121 Algebra ¶
      - depends_on definition 5.128 Derivation ¶
      - depends_on definition 5.124 Ideal ¶
      - depends_on definition 5.127 Lie algebra ¶
      - depends_on definition 5.122 Linear hull ¶
      - depends_on definition 5.123 Subalgebra ¶
      - depends_on definition 5.162 Representation of an algebra ¶
      - depends_on proposition 5.125 prop:lin-matrix-algebra ¶
  - depends_on definition 4.9 Neutral element ¶
    - depends_on definition 4.29 Action ¶
      - depends_on definition 4.31 Module ¶ ↺
    - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.32 Field ¶
      - depends_on definition 4.21 Group ¶
      - depends_on proposition 4.17 Trivial kernel of a homomorphism ¶
    - depends_on definition 4.15 Kernel ¶
      - depends_on proposition 4.17 Trivial kernel of a homomorphism ¶ ↺
    - 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 4.43 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 4.73 External 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 4.73 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 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 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 definition 9.140 Amplitude and the Jacobi elliptic functions ¶
      - depends_on proposition 7.61 The logarithm ¶
      - depends_on proposition 9.141 First properties ¶
      - depends_on proposition 9.19 Separation of variables ¶
  - depends_on proposition 7.31 Chain rule ¶
    - depends_on corollary 7.44 Substitution and integration by parts ¶
      - depends_on definition 8.8 Contour integral ¶
      - depends_on lemma 16.20 Mixed form ¶
      - depends_on lemma 9.136 Wallis integrals ¶
      - depends_on proposition 7.145 Delta as a limit; elementary properties ¶
      - depends_on proposition 16.50 The second variation ¶
      - depends_on proposition 5.95 The focal polar equation of a conic ¶
      - depends_on proposition 9.143 A cubic quadrature between two turning points ¶
      - depends_on proposition 9.144 The pendulum equation at finite amplitude ¶
      - depends_on theorem 16.38 Euler–Poisson equation ¶
    - depends_on lemma A.479 Conformal invariance of the Dirichlet integral ¶
      - depends_on lemma A.487 The normalisation costs nothing ¶
      - depends_on lemma A.485 Conformal automorphisms of the disc ¶
    - 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 definition 7.90 The golden ratio ¶
      - depends_on lemma 7.67 The limit over real exponents ¶
      - depends_on proposition 7.68 The natural base ¶
      - depends_on proposition 7.91 The golden ratio ¶
      - depends_on proposition 7.89 Irrationality of square roots ¶
    - depends_on proposition 16.26 Covariance under change of dependent variable ¶
    - depends_on proposition 10.15 Method of characteristics, first order ¶
      - depends_on definition 10.93 Scalar conservation law ¶
      - depends_on proposition 10.16 Cauchy's characteristic strips ¶
      - depends_on proposition 10.94 Gradient catastrophe ¶
      - depends_on remark 31.90 Why a compression wave must steepen ¶
      - depends_on remark 23.3 What Part II owes this chapter ¶
    - depends_on theorem 16.29 Beltrami identity ¶
      - depends_on proposition 16.105 The optical invariant in a stratified medium ¶
      - depends_on proposition 16.39 Ostrogradsky instability ¶
    - depends_on theorem 10.8 The type is a coordinate invariant ¶
      - depends_on lemma A.90 Affine normalization ¶
      - depends_on proposition 10.13 Canonical form in two variables ¶
      - depends_on proposition 10.11 What a characteristic surface is ¶
  - depends_on proposition 7.104 Chain rule in several variables ¶
    - depends_on corollary 7.113 Inverse function theorem ¶
      - depends_on definition A.503 The substitution property ¶
      - depends_on lemma 7.116 Functions vanishing on a regular zero set ¶
      - depends_on lemma A.510 Every diffeomorphism factorises locally ¶
      - depends_on remark 7.114 Where the proof is, and why ¶
      - depends_on theorem 7.129 Change of variables in a multiple integral ¶
      - depends_on theorem 7.115 Constant rank ¶
    - depends_on corollary 7.110 Variable limits of integration ¶
    - depends_on corollary A.292 Inverse function theorem ¶
      - depends_on lemma A.549 The action is transitive ¶
      - depends_on lemma A.300 The boundary is well defined, and is a manifold ¶
      - depends_on proposition A.535 Existence of a slice ¶
      - depends_on proposition 22.32 Duality of the two brackets ¶
      - depends_on proposition 22.3 Invertibility of the Legendre map ¶
      - depends_on proposition 13.132 Simultaneous straightening of commuting fields ¶
      - depends_on remark 22.19 Which form exists ¶
      - depends_on theorem 13.63 Constant rank theorem ¶
    - depends_on definition 30.4 Material and spatial descriptions ¶
    - depends_on definition 31.4 Material derivative ¶
      - depends_on definition 31.6 Streamline, pathline, streakline ¶
      - depends_on lemma 31.11 Transport theorem for a material volume ¶
    - depends_on definition 22.26 Lagrange bracket ¶
    - depends_on definition 10.18 Complete integral; envelope ¶
      - depends_on proposition 10.19 An envelope of solutions is a solution ¶
    - 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 lemma A.307 Partition of unity on a compact manifold ¶
    - depends_on proposition 7.136 Properties of conservative fields ¶
      - depends_on example 31.15 The stream function of a plane flow ¶
      - depends_on proposition 31.28 Potential flow reduces to Laplace's equation ¶
      - depends_on theorem 16.63 Hilbert's invariant integral ¶
      - depends_on theorem 30.15 Compatibility is sufficient on a simply connected body ¶
    - 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 corollary 7.44 Substitution and integration by parts ¶ ↺
      - depends_on corollary 16.23 du Bois-Reymond form ¶
      - depends_on lemma 7.116 Functions vanishing on a regular zero set ¶ ↺
      - depends_on lemma A.72 Iterated integral inequality ¶
      - depends_on lemma A.520 A $C^{1}$ limit ¶
      - depends_on lemma A.172 Dirichlet ¶
      - depends_on lemma A.221 Counting identity ¶
      - depends_on lemma A.312 The computation in one chart ¶
      - depends_on proposition 7.85 Irrationality of $\pi$ ¶
      - depends_on proposition 8.10 Fundamental theorem for contours ¶
      - depends_on proposition 17.41 The identities physics uses ¶
      - depends_on proposition 17.73 Initial- and final-value theorems ¶
      - … 4 more
  - depends_on definition 7.97 Partial derivative; gradient ¶
    - depends_on definition 7.98 Functions of class $C^{1}$ ¶
      - depends_on corollary 7.113 Inverse function theorem ¶ ↺
      - depends_on definition 7.119 Envelope of a family ¶
      - depends_on definition 7.126 Line and surface integrals ¶
      - depends_on definition 7.127 Simple regions ¶
      - depends_on definition A.508 Primitive map ¶
      - depends_on definition A.116 Orthogonal curvilinear coordinates; scale factors ¶
      - depends_on definition A.298 Half-space; smoothness on it ¶
      - depends_on definition 16.11 Admissible class; functional ¶
      - depends_on definition 10.1 Partial differential equation; order ¶
      - depends_on definition 10.66 Harmonic function ¶
      - depends_on definition 9.30 Hyperbolic equilibrium ¶
      - depends_on lemma A.520 A $C^{1}$ limit ¶ ↺
      - … 8 more
    - depends_on definition 7.102 Differential of a function ¶
      - depends_on proposition 7.104 Chain rule in several variables ¶ ↺
      - depends_on remark 7.103 rem:ana-differential-scalar ¶
    - depends_on definition 7.119 Envelope of a family ¶ ↺
    - depends_on definition A.70 Pullback ¶
      - depends_on definition 24.17 Symplectomorphism ¶
      - depends_on lemma A.75 Differentiating a pullback along a flow ¶
      - depends_on lemma A.71 The pullback is an algebra map commuting with $\dd$ ¶
    - depends_on definition 32.19 Lyapunov function ¶
      - depends_on theorem 32.20 Lyapunov's direct method ¶
    - depends_on definition 30.2 Representative volume element; continuum field ¶
      - depends_on definition 30.4 Material and spatial descriptions ¶ ↺
      - depends_on definition 30.17 Traction ¶
      - depends_on example 30.3 How small a piece of copper is still a continuum ¶
      - depends_on remark 30.5 Where the hypothesis fails, and how one can tell ¶
    - depends_on definition 22.29 Poisson bracket ¶
      - depends_on definition 26.20 Dirac bracket ¶
      - depends_on definition 26.12 First and second class ¶
      - depends_on definition 26.8 Total Hamiltonian ¶
      - depends_on definition 25.11 Generators of the Galilei transformations ¶
      - depends_on definition 25.6 Infinitesimal canonical transformation ¶
      - depends_on definition 25.2 Poisson algebra ¶
      - depends_on definition 24.36 Poisson manifold ¶
      - depends_on notation 26.28 Field systems ¶
      - depends_on notation 26.1 Weak and strong equality ¶
      - depends_on proposition 24.15 Poisson bracket from the symplectic form ¶
    - depends_on lemma A.73 Differentiation under the integral sign ¶
      - depends_on lemma A.77 Poincaré lemma, converse form ¶
  - 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 phenomenon 18.31 Uniform circular motion is accelerated motion ¶
      - depends_on proposition 18.32 Tangential and centripetal components ¶
      - depends_on remark 18.28 The radian carries no dimension ¶
    - 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 phenomenon 20.1 phen:parabolic-trajectory ¶
    - depends_on proposition 18.33 Intrinsic decomposition of the acceleration ¶
      - depends_on remark 18.34 The two ways an acceleration can be zero ¶
    - depends_on proposition 18.23 Transformation of velocity and acceleration ¶
      - depends_on phenomenon 18.26 Composition of velocities, and its limit ¶
  - 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 19.31 Energy ¶
      - depends_on theorem 19.38 Work–energy theorem ¶
    - depends_on definition 19.5 Linear momentum ¶
      - depends_on definition 27.2 Translational equilibrium ¶
      - depends_on definition 19.18 Angular momentum ¶
      - depends_on definition 19.51 Linear momentum of the centre of mass ¶
      - depends_on postulate 19.6 Newton's first law ¶
      - depends_on postulate 19.8 Newton's second law ¶
    - 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 remark 7.118 The multiplier rule and its functional counterpart ¶
    - depends_on proposition 16.16 Stationarity is necessary ¶
      - depends_on theorem 16.22 Euler–Lagrange ¶
      - depends_on theorem 16.51 Legendre's necessary condition ¶
    - depends_on theorem 7.34 Rolle ¶
      - depends_on theorem 7.37 Cauchy mean value theorem and l'Hôpital's rule ¶
      - depends_on theorem 7.35 Mean value theorem ¶
      - depends_on theorem 7.38 Taylor's theorem with Lagrange remainder ¶
    - 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 corollary 7.44 Substitution and integration by parts ¶ ↺
      - depends_on proposition 7.108 Euler's theorem on homogeneous functions ¶
      - depends_on proposition 7.111 Jacobi's formula, cofactor form ¶ ↺
      - depends_on proposition 7.122 Product identities of the nabla calculus ¶
      - depends_on proposition 5.2 Jacobi's formula, column form ¶
      - depends_on proposition 9.28 The scalar constant-coefficient equation ¶
      - depends_on theorem 7.38 Taylor's theorem with Lagrange remainder ¶ ↺
    - depends_on theorem 7.51 Termwise differentiation ¶
      - depends_on definition 7.74 $\pi$ ¶
      - depends_on lemma 7.93 A polynomial has at most $n$ roots ¶
      - depends_on lemma 7.71 Derivatives; the Pythagorean identity ¶
      - depends_on lemma 7.75 The first quadrant ¶
      - depends_on proposition 9.22 The exponential and its derivative ¶
  - 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 definition A.462 The weighted space and the energy space ¶
      - depends_on lemma A.453 Uniform bound and uniform Hölder continuity ¶
      - depends_on lemma A.452 Young and Hölder ¶
    - depends_on lemma A.520 A $C^{1}$ limit ¶ ↺
    - depends_on lemma A.440 Grönwall's inequality ¶
      - depends_on theorem A.441 Continuous dependence on the data and on the initial point ¶
    - depends_on lemma 16.19 du Bois-Reymond ¶
      - depends_on lemma 16.20 Mixed form ¶ ↺
    - depends_on lemma 9.10 Grönwall's inequality ¶
      - depends_on proposition 9.11 Continuous dependence on the initial data ¶
      - depends_on proposition 9.31 Lyapunov's first method: asymptotic stability ¶
    - depends_on proposition 11.22 Properties of the distribution function ¶
      - depends_on proposition 11.27 Probability integral transform ¶
    - depends_on proposition 9.18 The linear equation: integrating factor ¶
      - depends_on proposition 9.87 The distinguished limit, and which end carries the layer ¶
    - 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 corollary 10.53 Duhamel for the wave equation ¶
- 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 9.139 Series for the complete integral of the first kind ¶
      - depends_on proposition 9.22 The exponential and its derivative ¶ ↺
      - depends_on theorem 9.8 Picard–Lindelöf ¶
    - depends_on proposition 7.105 Clairaut–Schwarz ¶
      - depends_on definition 10.4 The second-order operator ¶
      - depends_on lemma A.75 Differentiating a pullback along a flow ¶ ↺
      - depends_on lemma 22.20 The symplectic condition ¶
      - depends_on proposition 7.123 Second-order identities of the nabla calculus ¶
      - depends_on proposition 30.25 Twenty-one constants ¶
      - depends_on proposition 30.14 Saint-Venant compatibility is necessary ¶
      - depends_on proposition 22.30 Properties of the Poisson bracket ¶
      - depends_on proposition 13.148 prop:mfd-torsion-tensor ¶
      - depends_on proposition 10.16 Cauchy's characteristic strips ¶ ↺
      - depends_on theorem 7.132 Stokes ¶
      - depends_on theorem 7.106 Taylor's theorem in several variables ¶
      - depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶
      - … 2 more
    - 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 theorem 7.24 Extreme value theorem ¶
      - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
    - 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 7.74 $\pi$ ¶ ↺
      - depends_on lemma 7.75 The first quadrant ¶ ↺
      - depends_on lemma 6.19 Continuous argument along a path ¶
      - depends_on proposition 7.32 Derivative of the inverse function ¶ ↺
      - depends_on proposition 7.61 The logarithm ¶ ↺
      - depends_on proposition 7.76 $\pi$ as the circle constant ¶
      - depends_on theorem A.443 Sturm separation theorem ¶
      - depends_on theorem 11.101 Davies' bound ¶
      - depends_on theorem 29.37 Reduction to one degree of freedom ¶
  - 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 lemma A.291 $h$ is differentiable, with the stated derivative ¶ ↺
      - depends_on remark 7.101 Partial derivatives alone do not suffice ¶ ↺
      - depends_on theorem 7.112 Implicit function theorem ¶
      - depends_on theorem A.286 Implicit function theorem ¶
    - 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 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 definition 32.81 Correlation dimension ¶
      - depends_on example 32.80 The middle-thirds Cantor set ¶
    - 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 30.49 The Rayleigh speed of a Poisson solid ¶
    - depends_on proposition 30.23 Virtual work; the weak form ¶
      - depends_on phenomenon 30.70 Hertzian contact ¶
      - depends_on proposition 30.34 Minimum of the potential energy ¶
      - depends_on remark 30.69 Saint-Venant's principle, and its exact status ¶
  - 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 proposition A.280 Direct-integral form of the spectral theorem ¶
  - depends_on definition 12.2 Hilbert space ¶
    - depends_on definition A.278 Countably Hilbert nuclear space ¶
      - depends_on theorem A.281 Nuclear spaces embed by Hilbert–Schmidt maps; quoted ¶
    - depends_on definition 12.32 Separable Hilbert space ¶ ↺
    - depends_on definition 12.94 Tensor product of Hilbert spaces ¶
      - depends_on definition 12.99 Product and entangled vectors ¶
      - depends_on example 12.98 Two particles in three-dimensional space ¶
      - depends_on proposition 12.95 The tensor inner product is well defined and positive definite ¶
    - depends_on definition 10.85 Sobolev space ¶
      - depends_on definition A.450 The space $W^{1,r}(a,b)$ ¶ ↺
      - depends_on definition 10.87 Weak form of an elliptic problem ¶
    - depends_on example 12.9 The sequence space $\ell^{2}$ ¶
      - depends_on definition 12.84 External direct sum ¶
      - depends_on proposition 12.10 $\ell^{2}$ is complete ¶
    - depends_on example 12.11 The function space $L^{2}$ ¶
      - depends_on example 12.81 Momentum on a finite interval: a circle of self-adjoint momenta ¶
      - depends_on example 12.56 Multiplication by the coordinate: spectrum without eigenvectors ¶
      - depends_on example 12.25 Orthogonal polynomials ¶
      - depends_on example 12.110 The Schrödinger system ¶
      - depends_on example 12.104 The Schwartz triple ¶
      - depends_on example 12.98 Two particles in three-dimensional space ¶ ↺
      - depends_on proposition 12.102 Neither plane waves nor deltas are in $L^{2}$ ¶
      - depends_on theorem 12.12 Riesz–Fischer ¶
    - depends_on lemma A.254 Riemann integral of a continuous curve ¶
      - depends_on lemma A.255 Smoothed vectors lie in the domain ¶
      - depends_on lemma A.257 Integrated form of the equation of motion ¶
      - depends_on lemma A.583 Absolutely convergent operator-valued integrals ¶
    - depends_on proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶
      - depends_on corollary 12.5 Continuity of the norm and of orthogonality ¶
      - depends_on lemma A.606 $F$ is entire, and is the overlap in disguise ¶
      - depends_on lemma A.254 Riemann integral of a continuous curve ¶ ↺
      - depends_on lemma A.583 Absolutely convergent operator-valued integrals ¶ ↺
      - depends_on proposition 12.102 Neither plane waves nor deltas are in $L^{2}$ ¶ ↺
      - depends_on theorem 25.53 Properties of the Wigner function ¶
    - depends_on proposition 12.28 Convergence criterion for orthogonal series ¶
      - depends_on theorem 12.30 Completeness, expansion, Parseval ¶
    - depends_on proposition 12.51 The three cases are exclusive and exhaustive ¶
    - depends_on theorem 16.70 The direct method ¶
      - depends_on lemma 16.72 Convexity implies weak lower semicontinuity ¶
      - depends_on theorem A.477 Douglas; Radó ¶
      - depends_on theorem A.448 Tonelli ¶
      - depends_on theorem 16.81 Douglas; Radó ¶
      - depends_on theorem 16.73 Tonelli ¶
    - depends_on theorem 12.14 Closest point in a closed convex set ¶
      - depends_on theorem 12.18 Projection theorem ¶
    - … 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 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 lemma A.240 Spectral mapping for polynomials ¶
      - depends_on proposition 12.65 Exponential of a bounded self-adjoint operator ¶
      - depends_on proposition 12.61 Uniqueness of the continuous functional calculus ¶
      - depends_on proposition 12.39 Algebra of the adjoint; the $C^{\ast}$ identity ¶
      - depends_on proposition 12.52 Neumann series; the spectrum is bounded ¶
      - depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶
      - depends_on theorem 12.75 The canonical commutation relation admits no bounded solution ¶
    - 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 theorem 12.74 Hellinger–Toeplitz ¶
- 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 remark A.516 The hypotheses of the global form ¶
      - depends_on theorem A.513 Change of variables ¶

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`	→	Quantifiers	declared	parts/02-mathematical-methods/01-logic-sets.tex:1215
`depends_on`	→	Set	declared	parts/02-mathematical-methods/01-logic-sets.tex:1215
`depends_on`	←	Binary operation	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:74
`depends_on`	←	Group homomorphism	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:1487
`depends_on`	←	Image	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:428
`depends_on`	←	Real function	declared	parts/02-mathematical-methods/05-real-analysis.tex:156
`depends_on`	←	Metric	declared	parts/02-mathematical-methods/04-topology.tex:500