definition 7.4 Convergence

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:61 · p. 203

ground object -- no derivation owed

Rests on

depends_on definition 7.2 Absolute value ¶

Supports

depends_on definition 7.45 Series ¶
- depends_on definition 7.53 $\ee$ ¶
  - depends_on lemma A.72 Iterated integral inequality ¶
    - 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 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 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 proposition 7.68 The natural base ¶
    - depends_on remark 7.58 An identity, not a recipe ¶
      - depends_on remark 7.70 What the condition says ¶
- 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 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 7.63 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 example 11.30 Exponential lifetimes ¶ ↺
      - depends_on proposition 11.26 Product rule ¶
  - depends_on proposition 11.5 Transfer, linearity, monotonicity ¶
    - depends_on definition 11.11 Covariance and correlation ¶
      - depends_on corollary 11.15 Bounds on the correlation coefficient ¶
      - depends_on definition 11.16 Sample mean, variance and correlation ¶
      - depends_on lemma 11.63 Cauchy–Schwarz for random variables ¶
      - depends_on proposition 11.12 Bilinearity ¶
    - depends_on definition 11.6 Moments, variance, standard deviation ¶
      - depends_on definition 11.11 Covariance and correlation ¶ ↺
      - depends_on definition 11.16 Sample mean, variance and correlation ¶ ↺
      - depends_on example 11.18 Three standard discrete distributions ¶
      - depends_on lemma 11.63 Cauchy–Schwarz for random variables ¶ ↺
      - depends_on lemma 11.50 Markov and Chebyshev inequalities ¶
      - depends_on proposition 11.7 Variance identity ¶
    - 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 11.25 Joint density and independence ¶
      - depends_on definition 11.16 Sample mean, variance and correlation ¶ ↺
      - depends_on proposition 11.14 Independence implies zero covariance; the converse fails ¶
    - depends_on definition 11.8 Quantiles and median ¶
      - depends_on example 11.31 A length measurement ¶
      - depends_on proposition 11.10 What the mean and the median minimise ¶
  - depends_on definition 11.34 Measure; measure space ¶
    - depends_on definition 11.38 Measure-preserving map; invariant measure; ergodicity ¶
      - depends_on theorem 11.39 Birkhoff's pointwise ergodic theorem ¶
    - depends_on definition 11.35 Null set; almost everywhere ¶
      - depends_on definition 11.38 Measure-preserving map; invariant measure; ergodicity ¶ ↺
      - depends_on proposition 11.36 Monotonicity, countable subadditivity, and null sets ¶
    - 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 remark 7.95 Where $\varphi$ is used ¶
    - depends_on proposition 125.2 Crystallographic restriction ¶
      - depends_on remark 125.3 What the theorem does and does not forbid ¶
    - 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 remark 7.94 Three numbers, two kinds ¶ ↺
    - depends_on remark 7.87 Priority, and what irrationality does not give ¶
- depends_on proposition 7.52 Alternating series test ¶
  - depends_on lemma 7.73 $\cos 2 < 0$ ¶
    - depends_on definition 7.74 $\pi$ ¶
      - depends_on lemma 7.75 The first quadrant ¶
      - depends_on proposition 7.76 $\pi$ as the circle constant ¶
      - depends_on proposition 7.79 The polygon recursion ¶
      - depends_on proposition 7.77 Special values, periodicity, and the kernel ¶
      - depends_on proposition 125.2 Crystallographic restriction ¶ ↺
- 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 7.75 The first quadrant ¶ ↺
    - depends_on lemma 7.78 Chord, arc, tangent ¶
      - depends_on proposition 7.79 The polygon recursion ¶ ↺
    - depends_on lemma 9.136 Wallis integrals ¶
      - depends_on proposition 9.139 Series for the complete integral of the first kind ¶
    - depends_on proposition 7.76 $\pi$ as the circle constant ¶ ↺
    - depends_on proposition 7.85 Irrationality of $\pi$ ¶ ↺
    - depends_on proposition 125.2 Crystallographic restriction ¶ ↺
    - 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 remark 7.80 Why these are the polygon perimeters ¶
  - 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 remark 7.84 One construction, two faces ¶
    - 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 9.139 Series for the complete integral of the first kind ¶ ↺
    - depends_on proposition 9.22 The exponential and its derivative ¶
      - depends_on theorem 9.23 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 7.48 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 A.786 From the half-strip to the half plane ¶
      - depends_on lemma A.787 From the potential strip to the half plane ¶
      - depends_on lemma A.785 The hodograph image ¶
    - depends_on lemma 7.72 Addition theorems ¶ ↺
    - depends_on proposition 7.85 Irrationality of $\pi$ ¶ ↺
    - 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 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 definition 8.22 Isolated singularities; residue ¶
      - depends_on theorem 8.16 Cauchy integral formula ¶
      - depends_on theorem 8.21 Laurent expansion ¶
      - depends_on theorem 8.24 Residue theorem ¶
    - depends_on lemma 17.14 Uniformly bounded sine sums ¶
      - depends_on lemma 17.15 A polynomial whose partial sum spikes at the origin ¶
  - 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 proposition 7.143 The Hausdorff dimension never exceeds the box 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 lemma A.291 $h$ is differentiable, with the stated derivative ¶
  - 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 A.609 The exponent is a quadratic polynomial ¶
    - 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 corollary 7.44 Substitution and integration by parts ¶
    - depends_on definition 8.8 Contour integral ¶
    - depends_on lemma 16.20 Mixed form ¶
      - depends_on theorem 16.35 System of Euler–Lagrange equations ¶
      - depends_on theorem 16.22 Euler–Lagrange ¶
      - depends_on theorem 16.43 Euler's rule for integral constraints ¶
    - 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 theorem 30.56 The beam equation ¶
      - depends_on theorem 30.64 The Kirchhoff plate equation ¶
  - depends_on lemma A.479 Conformal invariance of the Dirichlet integral ¶
    - depends_on lemma A.487 The normalisation costs nothing ¶
      - depends_on theorem A.492 Douglas' conformality theorem; quoted ¶
    - depends_on lemma A.485 Conformal automorphisms of the disc ¶
      - depends_on definition A.486 The normalised admissible class ¶
      - depends_on lemma A.487 The normalisation costs nothing ¶ ↺
  - 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 definition 10.93 Scalar conservation law ¶
      - depends_on definition A.150 Kruzhkov entropy pair ¶
      - depends_on definition 10.95 Weak solution of a conservation law ¶
      - depends_on example 10.97 Burgers shock ¶
      - depends_on proposition 10.94 Gradient catastrophe ¶
    - 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 lemma A.91 Smallness after rescaling ¶
    - depends_on proposition 10.13 Canonical form in two variables ¶
      - depends_on theorem A.83 Local canonical form for an elliptic operator with Hölder coefficients ¶
      - depends_on theorem 10.55 d'Alembert's formula ¶
    - depends_on proposition 10.11 What a characteristic surface is ¶
      - depends_on corollary 10.12 Characteristics of the three canonical operators ¶
      - depends_on definition A.102 Normal Cauchy problem of order $k$ ¶
      - depends_on theorem 10.22 Cauchy–Kovalevskaya ¶
- 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 9.144 The pendulum equation at finite amplitude ¶ ↺
  - 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 7.29 Linearity ¶
  - depends_on theorem 7.35 Mean value theorem ¶
    - depends_on corollary 7.36 cor:ana-mvt-consequences ¶
      - depends_on definition 7.41 Antiderivative ¶
      - depends_on lemma 7.67 The limit over real exponents ¶ ↺
      - depends_on lemma 7.75 The first quadrant ¶ ↺
      - depends_on lemma 7.78 Chord, arc, tangent ¶ ↺
      - depends_on lemma 9.10 Grönwall's inequality ¶
      - depends_on proposition 7.108 Euler's theorem on homogeneous functions ¶
      - depends_on proposition 7.61 The logarithm ¶ ↺
      - depends_on proposition 9.19 Separation of variables ¶ ↺
      - depends_on theorem 7.115 Constant rank ¶
    - 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 corollary A.514 Degeneracy on a negligible set ¶
      - depends_on lemma A.512 The boundary strip is thin ¶
    - depends_on lemma A.73 Differentiation under the integral sign ¶
      - depends_on lemma A.77 Poincaré lemma, converse form ¶
    - depends_on lemma A.287 The Newton map contracts ¶
      - depends_on lemma A.288 The iteration converges ¶ ↺
    - depends_on lemma A.290 $h$ is Lipschitz ¶ ↺
    - 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 A.226 The level factor ¶
    - depends_on lemma A.468 The kernel is well defined, symmetric and Lipschitz ¶
      - depends_on theorem A.470 $T$ is compact ¶
    - depends_on lemma 11.68 Jensen's inequality for the logarithm ¶
    - 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 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 ¶
    - … 2 more
- 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 lemma 44.10 Divergence theorem on $(M,g)$ ¶
      - depends_on lemma 44.7 Variation of inverse and determinant ¶
      - depends_on proposition 7.111 Jacobi's formula, cofactor form ¶ ↺
    - depends_on proposition 9.28 The scalar constant-coefficient equation ¶
      - depends_on proposition 9.88 The model problem, and the error of the composite ¶
    - depends_on theorem 7.38 Taylor's theorem with Lagrange remainder ¶
      - depends_on definition 32.57 The Hénon–Heiles Hamiltonian ¶
      - depends_on definition 32.12 Linearization ¶
      - depends_on lemma A.754 Exact depth expansion ¶
      - depends_on lemma A.187 Exponentiation ¶
      - depends_on lemma A.138 The flat exponential ¶
      - depends_on lemma A.452 Young and Hölder ¶
      - depends_on lemma A.212 Second-order flatness ¶
      - depends_on lemma A.209 Uniform third-order remainder ¶
      - depends_on phenomenon 28.13 Universality of small oscillations ¶
      - depends_on proposition 16.50 The second variation ¶ ↺
      - depends_on proposition 17.67 Holomorphy ¶
      - depends_on proposition 9.88 The model problem, and the error of the composite ¶ ↺
      - … 2 more
  - depends_on theorem 7.51 Termwise differentiation ¶ ↺
- 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 7.24 Extreme value theorem ¶
  - depends_on lemma A.221 Counting identity ¶
  - depends_on lemma 6.19 Continuous argument along a path ¶
    - depends_on definition 6.20 Winding number ¶
      - depends_on example 6.21 ex:top-winding-values ¶
      - depends_on lemma 6.22 Nearby loops wind alike ¶
    - depends_on lemma 6.22 Nearby loops wind alike ¶ ↺
  - depends_on theorem 7.40 Continuous functions are integrable ¶
    - depends_on corollary A.502 All iterated orders agree ¶
      - depends_on lemma A.507 Coordinate permutations ¶
    - 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 9.137 Elliptic integrals of the three kinds ¶
      - depends_on definition 9.140 Amplitude and the Jacobi elliptic functions ¶ ↺
      - depends_on proposition 9.139 Series for the complete integral of the first kind ¶ ↺
    - depends_on lemma A.501 Iterated integration over a box ¶
      - depends_on corollary A.502 All iterated orders agree ¶ ↺
      - depends_on lemma A.509 Primitive maps have the substitution property ¶
    - depends_on lemma 16.18 Fundamental lemma ¶
      - depends_on theorem A.688 The Kirchhoff plate equation ¶
      - depends_on theorem 16.36 Euler–Lagrange equations for several independent variables ¶
      - depends_on theorem 16.38 Euler–Poisson equation ¶ ↺
      - depends_on theorem 16.41 Multiplier rule for pointwise constraints ¶
      - depends_on theorem 16.31 Natural boundary condition ¶
      - depends_on theorem 25.19 Hamilton's equations from the first-order action ¶
      - depends_on theorem 25.22 Faddeev–Jackiw equations and brackets ¶
    - depends_on remark 7.128 What the derivations below take as given ¶
      - depends_on lemma A.312 The computation in one chart ¶
      - depends_on theorem 7.129 Change of variables in a multiple integral ¶
      - depends_on theorem 7.133 Gauss ¶
      - depends_on theorem 7.131 Green ¶
      - depends_on theorem A.303 Change of variables for multiple integrals; quoted ¶
    - 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 theorem 7.42 Fundamental theorem of calculus, I ¶ ↺
  - depends_on theorem 7.34 Rolle ¶
    - 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 theorem 7.35 Mean value theorem ¶ ↺
    - depends_on theorem 7.38 Taylor's theorem with Lagrange remainder ¶ ↺
- depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
  - depends_on lemma A.195 Riemann–Lebesgue, continuous compactly supported case ¶
    - depends_on lemma A.196 Inversion for a difference ¶
      - depends_on theorem A.198 Esseen's smoothing inequality ¶
  - depends_on lemma A.500 Graphs and $C^{1}$ images have zero content ¶ ↺
  - depends_on lemma A.499 What zero content buys ¶
    - depends_on corollary A.514 Degeneracy on a negligible set ¶ ↺
    - depends_on lemma A.500 Graphs and $C^{1}$ images have zero content ¶ ↺
    - depends_on theorem A.513 Change of variables ¶
      - depends_on corollary A.514 Degeneracy on a negligible set ¶ ↺
      - depends_on remark A.516 The hypotheses of the global form ¶
  - 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 corollary A.181 The converse ¶
  - depends_on lemma A.488 Small chords cut off small arcs ¶
    - depends_on theorem A.490 Equicontinuity of the normalised class ¶
  - 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 ¶
    - depends_on corollary 7.110 Variable limits of integration ¶ ↺
    - depends_on lemma 7.116 Functions vanishing on a regular zero set ¶
    - depends_on lemma A.607 A zero-free entire function is an exponential ¶
      - depends_on proposition A.609 The exponent is a quadratic polynomial ¶ ↺
    - depends_on lemma A.599 One variable, complex coefficient ¶
      - depends_on lemma A.601 Several variables, complex symmetric matrix ¶
    - depends_on proposition A.524 Rate of decay ¶
    - depends_on proposition A.521 Every derivative passes onto $g$ ¶
      - depends_on proposition A.524 Rate of decay ¶ ↺
      - depends_on proposition A.522 The Newtonian potential inverts the Laplacian ¶
    - depends_on theorem 7.137 Helmholtz decomposition ¶
      - depends_on remark A.527 The hypotheses, and what happens without them ¶
      - depends_on remark 30.46 Why the plane-wave route, and not the potentials ¶
      - depends_on remark 30.27 Two results this chapter borrows from Part II ¶
      - depends_on remark 10.73 What compact support is doing, and the rate that replaces it ¶
      - depends_on remark 10.74 Where the decomposition is used, and what it costs ¶
  - … 1 more
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.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 ¶ ↺

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Edges

type	direction	node	provenance	where
`depends_on`	→	Absolute value	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`	←	Algebra of limits	declared	parts/02-mathematical-methods/05-real-analysis.tex:87
`depends_on`	←	Uniqueness of the limit	declared	parts/02-mathematical-methods/05-real-analysis.tex:73
`depends_on`	←	Sequential characterization	declared	parts/02-mathematical-methods/05-real-analysis.tex:353
`depends_on`	←	Cauchy criterion	declared	parts/02-mathematical-methods/05-real-analysis.tex:135