definition 7.2 Absolute value

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depends_on definition 7.15 Absolute-value function ¶
depends_on definition 7.19 Infinite limits ¶
- depends_on lemma 7.67 The limit over real exponents ¶
  - depends_on proposition 7.68 The natural base ¶
    - depends_on remark 7.70 What the condition says ¶
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 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 definition 7.119 Envelope of a family ¶
      - depends_on proposition 7.120 The envelope touches every member it meets ¶
    - depends_on definition 7.126 Line and surface integrals ¶
      - depends_on corollary A.313 Stokes' theorem for an antisymmetric tensor field ¶
      - depends_on definition 30.17 Traction ¶
    - depends_on definition 7.127 Simple regions ¶
      - depends_on lemma A.512 The boundary strip is thin ¶
      - depends_on remark A.516 The hypotheses of the global form ¶
      - depends_on theorem 7.133 Gauss ¶
      - depends_on theorem 7.131 Green ¶
    - depends_on definition A.508 Primitive map ¶
      - depends_on lemma A.510 Every diffeomorphism factorises locally ¶ ↺
      - depends_on lemma A.509 Primitive maps have the substitution property ¶
    - depends_on definition A.116 Orthogonal curvilinear coordinates; scale factors ¶
      - depends_on definition A.125 Ellipsoidal coordinates ¶
      - depends_on definition A.118 Simple separation of the Helmholtz equation ¶
      - depends_on lemma A.117 Laplacian in orthogonal coordinates ¶
    - depends_on definition A.298 Half-space; smoothness on it ¶
      - depends_on definition A.299 Manifold with boundary ¶
    - depends_on definition 16.11 Admissible class; functional ¶
      - depends_on definition 16.59 Weierstrass excess function ¶
      - depends_on definition 16.12 The two norms; weak and strong neighbourhoods ¶
      - depends_on definition 16.15 Variation; the first variation ¶
      - depends_on proposition 16.68 Weierstrass' counterexample ¶
    - depends_on definition 10.1 Partial differential equation; order ¶
      - depends_on definition A.102 Normal Cauchy problem of order $k$ ¶
      - depends_on definition 10.2 Linear, semilinear, quasilinear ¶
      - depends_on definition 10.24 Well-posed problem ¶
    - depends_on definition 10.66 Harmonic function ¶
      - depends_on theorem A.135 Converse of the mean-value property ¶
      - depends_on theorem A.491 Dirichlet problem for the disc; quoted ¶
      - depends_on theorem 10.69 Mean-value property ¶
      - depends_on theorem 10.75 Weak maximum principle ¶
    - depends_on definition 9.30 Hyperbolic equilibrium ¶
      - depends_on theorem 9.32 Hartman–Grobman; quoted ¶
    - depends_on lemma A.520 A $C^{1}$ limit ¶
      - depends_on proposition A.521 Every derivative passes onto $g$ ¶
    - … 8 more
  - 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 9.144 The pendulum equation at finite amplitude ¶
    - depends_on proposition 9.22 The exponential and its derivative ¶
      - depends_on theorem 9.23 Solution of a constant-coefficient system ¶
    - depends_on theorem 9.8 Picard–Lindelöf ¶
      - depends_on corollary 32.5 Trajectories do not cross ¶
      - depends_on corollary 9.9 Linear equations: existence on the whole interval ¶
      - depends_on proposition 32.4 The flow is a one-parameter group ¶
      - depends_on proposition 10.16 Cauchy's characteristic strips ¶
      - depends_on proposition 20.7 Terminal speed and the approach to it ¶
      - depends_on proposition 9.11 Continuous dependence on the initial data ¶
      - depends_on proposition 9.142 The differential equation of the sine amplitude ¶
      - depends_on theorem A.74 Flow of a time-dependent vector field ¶
      - depends_on theorem 13.125 Existence, uniqueness and smoothness of the flow ¶
      - depends_on theorem 9.34 Poincaré–Bendixson; quoted ¶
  - depends_on proposition 7.105 Clairaut–Schwarz ¶
    - depends_on definition 10.4 The second-order operator ¶
      - depends_on definition 10.10 Characteristic surface ¶
      - depends_on definition 10.42 Green's function ¶
      - depends_on definition 10.6 Type of a second-order operator ¶
    - depends_on lemma A.75 Differentiating a pullback along a flow ¶
      - depends_on lemma A.77 Poincaré lemma, converse form ¶
    - depends_on lemma 22.20 The symplectic condition ¶
      - depends_on corollary 22.24 Invariance of the phase-space volume ¶
      - depends_on proposition 22.27 Properties of the Lagrange bracket ¶
      - depends_on proposition 22.30 Properties of the Poisson bracket ¶
      - depends_on remark 22.33 What the duality is good for ¶
      - depends_on remark 22.28 Where the content of the fundamental brackets lies ¶
      - depends_on theorem 22.22 Poincaré ¶
    - depends_on proposition 7.123 Second-order identities of the nabla calculus ¶
      - depends_on remark 7.124 The Laplacian of a vector field is a Cartesian notion ¶
      - depends_on theorem 7.137 Helmholtz decomposition ¶
      - depends_on theorem A.523 Helmholtz decomposition: existence ¶
      - depends_on theorem A.525 Helmholtz decomposition: uniqueness ¶
    - depends_on proposition 30.25 Twenty-one constants ¶
      - depends_on definition 30.37 Voigt notation ¶
      - depends_on proposition 30.38 The Cauchy relations ¶
      - depends_on theorem A.675 Three constants for a cubic crystal, two for an isotropic solid ¶
    - depends_on proposition 30.14 Saint-Venant compatibility is necessary ¶
      - depends_on theorem 30.15 Compatibility is sufficient on a simply connected body ¶
    - 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 definition 31.63 Circulation ¶
      - depends_on proposition 7.136 Properties of conservative fields ¶
    - depends_on theorem 7.106 Taylor's theorem in several variables ¶
      - depends_on remark 7.107 What the Hessian is for ¶
    - depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶
      - depends_on definition 13.153 Contractions ¶
      - depends_on example 21.74 Rindler coordinates ¶
      - depends_on lemma A.640 Variation of the integrated three-curvature ¶
      - depends_on lemma 44.8 Palatini identity ¶
      - depends_on proposition 44.44 Harmonic-gauge reduction ¶
      - depends_on proposition 13.155 Geodesic deviation; Jacobi equation ¶
      - depends_on proposition 13.140 Killing's equation ¶
      - depends_on proposition 13.162 Curvature induced on the quadric ¶
      - depends_on proposition 13.154 Symmetries of the curvature ¶
      - depends_on remark 30.16 Incompatibility is curvature ¶
      - depends_on theorem 13.156 Cartan structure equations ¶
    - … 2 more
  - depends_on proposition 7.145 Delta as a limit; elementary properties ¶
  - depends_on proposition 7.27 Differentiable implies continuous ¶
    - depends_on proposition 7.31 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 7.68 The natural base ¶ ↺
    - depends_on proposition 7.61 The logarithm ¶
      - depends_on corollary 7.69 $\ee$ is the unique self-reproducing base ¶
      - depends_on definition 7.62 Real powers ¶
      - depends_on lemma 7.67 The limit over real exponents ¶ ↺
      - depends_on proposition 7.68 The natural base ¶ ↺
      - depends_on proposition 7.63 Laws of real powers ¶ ↺
    - 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 proposition 7.22 Sequential characterization ¶
    - depends_on theorem 7.24 Extreme value theorem ¶
      - depends_on lemma A.221 Counting identity ¶
      - depends_on lemma 6.19 Continuous argument along a path ¶
      - depends_on theorem 7.40 Continuous functions are integrable ¶
      - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
      - depends_on theorem 7.34 Rolle ¶
    - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
      - depends_on lemma A.195 Riemann–Lebesgue, continuous compactly supported case ¶
      - depends_on lemma A.500 Graphs and $C^{1}$ images have zero content ¶
      - depends_on lemma A.499 What zero content buys ¶
      - depends_on lemma A.512 The boundary strip is thin ¶ ↺
      - depends_on lemma A.73 Differentiation under the integral sign ¶
      - depends_on lemma A.176 Helly–Bray ¶
      - depends_on lemma A.488 Small chords cut off small arcs ¶
      - depends_on lemma 14.34 The $n$-sphere is simply connected for $n \ge 2$ ¶
      - depends_on lemma 6.19 Continuous argument along a path ¶ ↺
      - depends_on remark 7.128 What the derivations below take as given ¶
      - depends_on theorem 7.40 Continuous functions are integrable ¶ ↺
      - depends_on theorem 7.109 Leibniz integral rule ¶
      - … 1 more
  - 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 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 7.61 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 corollary 11.105 Trials factor at high significance ¶
      - depends_on example 11.107 Degrading a local five sigma ¶
    - depends_on theorem 29.37 Reduction to one degree of freedom ¶
      - depends_on phenomenon 29.38 Nutation of a released top ¶
      - depends_on proposition 29.40 The sleeping top ¶
      - depends_on remark 29.42 The closed form, and the mathematics it needs ¶
- 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 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 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 16.104 Snell's law of refraction ¶
  - depends_on proposition 7.27 Differentiable implies continuous ¶ ↺
  - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶ ↺
- 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 corollary 7.113 Inverse function theorem ¶ ↺
      - depends_on proposition 7.120 The envelope touches every member it meets ¶ ↺
      - depends_on proposition 7.117 Lagrange multipliers in finitely many variables ¶ ↺
      - depends_on remark 7.114 Where the proof is, and why ¶ ↺
    - depends_on theorem A.286 Implicit function theorem ¶
      - depends_on corollary A.292 Inverse function theorem ¶
      - depends_on corollary A.293 Solving one scalar equation for one coordinate ¶
      - depends_on lemma A.650 Flattening the constraints ¶
      - depends_on remark A.296 Both hypotheses are needed, and the conclusion is local ¶
      - depends_on remark 23.3 What Part II owes this chapter ¶
      - depends_on theorem 23.6 Jacobi ¶
      - depends_on theorem 13.63 Constant rank theorem ¶
      - depends_on theorem 13.62 Regular value theorem in codimension $k$ ¶
  - 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 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 lemma A.307 Partition of unity on a compact manifold ¶
      - depends_on definition A.308 Integral over the manifold ¶
depends_on definition 7.4 Convergence ¶
- depends_on definition 7.45 Series ¶
  - depends_on definition 7.53 $\ee$ ¶
    - depends_on lemma A.72 Iterated integral inequality ¶ ↺
    - depends_on lemma A.306 The standard smooth bump ¶ ↺
    - depends_on proposition 7.56 Irrationality ¶
      - depends_on remark 7.94 Three numbers, two kinds ¶
    - depends_on proposition 7.57 The limit definition ¶
      - depends_on lemma 7.67 The limit over real exponents ¶ ↺
      - depends_on remark 7.58 An identity, not a recipe ¶
  - depends_on definition 7.59 The real exponential ¶
    - depends_on lemma 7.60 Functional equation of the exponential ¶
      - depends_on proposition 7.68 The natural base ¶ ↺
      - depends_on proposition 7.61 The logarithm ¶ ↺
      - depends_on proposition 7.63 Laws of real powers ¶ ↺
  - depends_on definition 11.4 Expectation ¶
    - depends_on definition A.741 The statistical setting ¶
    - depends_on definition 11.23 Expectation, moments and quantiles in the continuous case ¶
      - depends_on example 11.30 Exponential lifetimes ¶
      - depends_on proposition 11.29 Gaussian moments ¶
      - depends_on proposition 11.27 Probability integral transform ¶
      - depends_on proposition 11.24 The algebra of the discrete case transfers ¶
    - depends_on proposition 11.5 Transfer, linearity, monotonicity ¶
      - depends_on definition 11.11 Covariance and correlation ¶
      - depends_on definition 11.6 Moments, variance, standard deviation ¶
      - depends_on lemma 11.50 Markov and Chebyshev inequalities ¶
      - depends_on proposition 11.12 Bilinearity ¶
      - depends_on proposition 11.24 The algebra of the discrete case transfers ¶ ↺
      - depends_on proposition 11.7 Variance identity ¶
    - depends_on theorem A.169 Dominated convergence ¶
    - depends_on theorem A.170 Fubini's theorem, bounded case ¶
    - depends_on theorem A.223 Tonelli and dominated convergence, as used here ¶
  - depends_on definition 11.1 Discrete probability space ¶
    - depends_on definition 11.3 Discrete random variable ¶
      - depends_on definition 11.4 Expectation ¶ ↺
      - depends_on definition 11.13 Independence ¶
      - depends_on definition 11.8 Quantiles and median ¶
    - depends_on definition 11.34 Measure; measure space ¶
      - depends_on definition 11.38 Measure-preserving map; invariant measure; ergodicity ¶
      - depends_on definition 11.35 Null set; almost everywhere ¶
      - depends_on proposition 11.36 Monotonicity, countable subadditivity, and null sets ¶
    - depends_on definition 11.33 Sigma-algebra ¶
      - depends_on definition 11.34 Measure; measure space ¶ ↺
    - depends_on proposition 11.2 Elementary rules ¶
      - depends_on proposition 11.5 Transfer, linearity, monotonicity ¶ ↺
  - depends_on lemma 7.86 Terminating or repeating decimals are rational ¶
    - depends_on proposition 7.91 The golden ratio ¶
      - depends_on proposition 7.92 Continued fraction and the Fibonacci ratios ¶
      - depends_on proposition 125.2 Crystallographic restriction ¶
      - depends_on remark 7.95 Where $\varphi$ is used ¶
      - depends_on remark 7.94 Three numbers, two kinds ¶ ↺
    - depends_on proposition 7.85 Irrationality of $\pi$ ¶ ↺
  - depends_on proposition 7.52 Alternating series test ¶
    - depends_on lemma 7.73 $\cos 2 < 0$ ¶
      - depends_on definition 7.74 $\pi$ ¶ ↺
  - depends_on proposition 7.49 Cauchy product ¶
    - depends_on lemma 7.60 Functional equation of the exponential ¶ ↺
    - depends_on lemma 7.71 Derivatives; the Pythagorean identity ¶ ↺
    - depends_on lemma 7.72 Addition theorems ¶
      - depends_on proposition 7.79 The polygon recursion ¶
      - depends_on proposition 7.77 Special values, periodicity, and the kernel ¶
      - depends_on proposition 7.83 Viète's product ¶
      - depends_on proposition 125.2 Crystallographic restriction ¶ ↺
      - depends_on remark 7.88 $\pi$ elsewhere in this treatise ¶
      - depends_on remark 7.80 Why these are the polygon perimeters ¶
  - depends_on proposition 7.47 Comparison; absolute convergence ¶
    - depends_on lemma 9.6 Weierstrass $M$-test; uniform limits are continuous ¶ ↺
    - depends_on proposition 7.48 Ratio test ¶
      - depends_on definition 7.53 $\ee$ ¶ ↺
      - depends_on definition 7.59 The real exponential ¶ ↺
      - depends_on definition 8.3 Exponential, sine, cosine ¶
      - depends_on lemma 7.72 Addition theorems ¶ ↺
      - depends_on proposition 7.85 Irrationality of $\pi$ ¶ ↺
      - depends_on theorem 7.51 Termwise differentiation ¶ ↺
    - depends_on proposition 8.4 Euler's formula ¶
      - depends_on lemma A.786 From the half-strip to the half plane ¶
      - depends_on lemma 8.11 The fundamental $2\pi\ii$ ¶
      - depends_on lemma 17.14 Uniformly bounded sine sums ¶
    - depends_on theorem 7.50 Power series; radius of convergence ¶
      - depends_on definition 7.59 The real exponential ¶ ↺
      - depends_on definition 8.3 Exponential, sine, cosine ¶ ↺
      - depends_on theorem 7.51 Termwise differentiation ¶ ↺
    - depends_on theorem 7.51 Termwise differentiation ¶ ↺
  - depends_on proposition 7.46 Geometric series ¶
    - depends_on lemma 7.86 Terminating or repeating decimals are rational ¶ ↺
    - depends_on lemma 7.54 Truncation error ¶
      - depends_on corollary 7.55 The decimal expansion ¶
      - depends_on proposition 7.56 Irrationality ¶ ↺
      - depends_on remark 7.58 An identity, not a recipe ¶ ↺
    - 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 lemma A.288 The iteration converges ¶
      - depends_on corollary A.289 The zero set is a graph ¶
      - depends_on lemma A.290 $h$ is Lipschitz ¶
    - depends_on proposition 7.48 Ratio test ¶ ↺
    - depends_on theorem 7.50 Power series; radius of convergence ¶ ↺
    - depends_on theorem 8.20 Taylor expansion ¶
      - depends_on lemma A.608 An entire function with a quadratic bound on its real part ¶
      - depends_on proposition 8.23 Residue at a simple pole ¶
      - depends_on theorem A.598 Hudson ¶
      - depends_on theorem 8.21 Laurent expansion ¶
- depends_on proposition 7.6 Algebra of limits ¶
  - depends_on lemma 7.67 The limit over real exponents ¶ ↺
  - depends_on lemma 7.72 Addition theorems ¶ ↺
  - depends_on proposition 7.49 Cauchy product ¶ ↺
  - depends_on proposition 7.31 Chain rule ¶ ↺
  - depends_on proposition 7.32 Derivative of the inverse function ¶ ↺
  - depends_on proposition 7.29 Linearity ¶
    - depends_on theorem 7.35 Mean value theorem ¶
      - depends_on corollary 7.36 cor:ana-mvt-consequences ¶
      - depends_on lemma 7.93 A polynomial has at most $n$ roots ¶ ↺
      - depends_on lemma A.500 Graphs and $C^{1}$ images have zero content ¶ ↺
      - depends_on lemma A.73 Differentiation under the integral sign ¶ ↺
      - depends_on lemma A.287 The Newton map contracts ¶
      - depends_on lemma A.290 $h$ is Lipschitz ¶ ↺
      - depends_on lemma A.440 Grönwall's inequality ¶
      - depends_on lemma A.226 The level factor ¶
      - depends_on lemma A.468 The kernel is well defined, symmetric and Lipschitz ¶
      - depends_on lemma 11.68 Jensen's inequality for the logarithm ¶
      - depends_on proposition 7.105 Clairaut–Schwarz ¶ ↺
      - depends_on theorem 7.100 $C^{1}$ implies differentiable ¶ ↺
      - … 2 more
  - depends_on proposition 7.27 Differentiable implies continuous ¶ ↺
  - depends_on proposition 7.57 The limit definition ¶ ↺
  - depends_on proposition 7.68 The natural base ¶ ↺
  - depends_on proposition 7.92 Continued fraction and the Fibonacci ratios ¶ ↺
  - depends_on proposition 7.30 Leibniz rule ¶ ↺
  - depends_on proposition 7.83 Viète's product ¶ ↺
  - … 1 more
- depends_on proposition 7.5 Uniqueness of the limit ¶
- depends_on proposition 7.22 Sequential characterization ¶ ↺
- depends_on theorem 7.8 Cauchy criterion ¶
  - depends_on lemma A.288 The iteration converges ¶ ↺
  - depends_on lemma A.454 Arzelà–Ascoli on an interval ¶
    - depends_on theorem A.490 Equicontinuity of the normalised class ¶
    - depends_on theorem A.470 $T$ is compact ¶
      - depends_on theorem A.471 Spectral decomposition and completeness in $L^{2}_{r}$ ¶
    - depends_on theorem A.455 Compact embedding of $W^{1,r}(a,b)$ into the continuous functions ¶
      - depends_on theorem A.456 Convexity in $q$ implies weak lower semicontinuity ¶
  - depends_on proposition 7.47 Comparison; absolute convergence ¶ ↺
  - depends_on proposition 7.85 Irrationality of $\pi$ ¶ ↺
  - depends_on proposition 7.48 Ratio test ¶ ↺
  - depends_on proposition A.519 The convolution exists ¶
    - depends_on proposition A.524 Rate of decay ¶
depends_on definition 7.45 Series ¶ ↺
depends_on proposition 7.3 Triangle inequality ¶
- depends_on proposition 7.6 Algebra of limits ¶ ↺
- depends_on proposition 7.5 Uniqueness of the limit ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
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declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	←	Absolute-value function	declared	parts/02-mathematical-methods/05-real-analysis.tex:211
`depends_on`	←	Infinite limits	declared	parts/02-mathematical-methods/05-real-analysis.tex:315
`depends_on`	←	Limit	declared	parts/02-mathematical-methods/05-real-analysis.tex:229
`depends_on`	←	Convergence	declared	parts/02-mathematical-methods/05-real-analysis.tex:68
`depends_on`	←	Series	declared	parts/02-mathematical-methods/05-real-analysis.tex:893
`depends_on`	←	Triangle inequality	declared	parts/02-mathematical-methods/05-real-analysis.tex:50