proposition 7.6 Algebra of limits

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:84 · p. 204

Rests on

depends_on definition 7.4 Convergence ¶
- depends_on definition 7.2 Absolute value ¶
depends_on proposition 7.3 Triangle inequality ¶
- depends_on definition 7.2 Absolute value ¶ ↺
- proves proof ch:05-real-analysis@proof-1 ¶
proves proof ch:05-real-analysis@proof-3 ¶

Supports

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 lemma 7.72 Addition theorems ¶
- depends_on proposition 7.79 The polygon recursion ¶
  - depends_on corollary 7.81 The decimal expansion of $\pi$ ¶
    - depends_on remark 7.88 $\pi$ elsewhere in this treatise ¶
  - depends_on proposition 7.83 Viète's product ¶
    - depends_on remark 7.84 One construction, two faces ¶
  - depends_on remark 7.82 What Archimedes proved ¶
  - depends_on remark 7.80 Why these are the polygon perimeters ¶
  - depends_on remark 7.84 One construction, two faces ¶ ↺
- depends_on proposition 7.77 Special values, periodicity, and the kernel ¶
  - depends_on lemma 7.78 Chord, arc, tangent ¶
    - depends_on proposition 7.79 The polygon recursion ¶ ↺
  - 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 7.79 The polygon recursion ¶ ↺
  - depends_on proposition 7.83 Viète's product ¶ ↺
  - depends_on proposition 125.2 Crystallographic restriction ¶
    - depends_on remark 125.3 What the theorem does and does not forbid ¶
  - depends_on remark 7.88 $\pi$ elsewhere in this treatise ¶ ↺
- 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 7.49 Cauchy product ¶
- 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 remark 7.70 What the condition says ¶ ↺
    - depends_on definition 7.62 Real powers ¶
      - depends_on definition 7.90 The golden ratio ¶
      - depends_on proposition 7.63 Laws of real powers ¶
      - depends_on proposition 7.89 Irrationality of square roots ¶
    - depends_on lemma 7.67 The limit over real exponents ¶ ↺
    - depends_on proposition 7.68 The natural base ¶ ↺
    - depends_on proposition 7.63 Laws of real powers ¶ ↺
  - depends_on proposition 7.63 Laws of real powers ¶ ↺
- depends_on lemma 7.71 Derivatives; the Pythagorean identity ¶
  - depends_on lemma 7.75 The first quadrant ¶
    - depends_on proposition 7.76 $\pi$ as the circle constant ¶
    - depends_on proposition 7.77 Special values, periodicity, and the kernel ¶ ↺
    - depends_on proposition 125.2 Crystallographic restriction ¶ ↺
  - depends_on lemma 7.78 Chord, arc, tangent ¶ ↺
  - depends_on lemma 9.136 Wallis integrals ¶
    - 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 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 7.31 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 proposition 16.6 The isoperimetric extremal is a circle ¶
      - depends_on theorem 16.41 Multiplier rule for pointwise constraints ¶
    - depends_on theorem 16.22 Euler–Lagrange ¶
      - depends_on definition 16.62 Field of extremals; slope function ¶
      - depends_on phenomenon 30.52 Euler buckling ¶
      - depends_on proposition 29.39 Steady precession ¶
      - depends_on theorem 16.35 System of Euler–Lagrange equations ¶ ↺
      - depends_on theorem 16.31 Natural boundary condition ¶
    - depends_on theorem 16.43 Euler's rule for integral constraints ¶
      - depends_on example 16.45 Two solved isoperimetric problems ¶
      - depends_on proposition 16.4 The hanging chain is a catenary ¶
      - depends_on proposition 16.6 The isoperimetric extremal is a circle ¶ ↺
  - 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 example 30.58 A steel ruler ¶
      - depends_on proposition 30.57 Bending waves are dispersive ¶
    - depends_on theorem 30.64 The Kirchhoff plate equation ¶
      - depends_on phenomenon 30.65 Chladni figures ¶
      - depends_on proposition 30.66 Chladni scaling ¶
      - depends_on remark 30.67 Germain, Kirchhoff, and the edge conditions ¶
- 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 lemma A.157 Two elementary identities ¶
      - depends_on proposition A.156 The Kruzhkov inequality ¶
      - depends_on theorem A.151 $L^{1}$ contraction on a cone ¶
    - depends_on definition 10.95 Weak solution of a conservation law ¶
      - depends_on definition 10.99 Entropy condition ¶
      - depends_on definition 10.101 Entropy pair; entropy inequality ¶
      - depends_on example 10.98 Weak solutions are not unique ¶
      - depends_on theorem A.151 $L^{1}$ contraction on a cone ¶ ↺
      - depends_on theorem 10.103 Uniqueness in the entropy class ¶
      - depends_on theorem 10.96 Rankine–Hugoniot condition ¶
    - 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 corollary 10.56 Domain of dependence and influence ¶
      - depends_on theorem 10.59 Kirchhoff'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 corollary 10.60 Huygens' principle in three space dimensions ¶
    - depends_on definition A.102 Normal Cauchy problem of order $k$ ¶
      - depends_on definition A.103 First-order quasilinear system with zero data ¶
      - depends_on lemma A.105 Formal reduction of a normal problem ¶
    - 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 theorem 7.43 Fundamental theorem of calculus, II ¶
    - 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 9.11 Continuous dependence on the initial data ¶
      - depends_on proposition 9.31 Lyapunov's first method: asymptotic stability ¶
    - 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 theorem A.513 Change of variables ¶
  - 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 corollary A.289 The zero set is a graph ¶
      - depends_on lemma A.290 $h$ is Lipschitz ¶
  - 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 corollary A.445 Displacing 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 theorem A.471 Spectral decomposition and completeness in $L^{2}_{r}$ ¶
  - 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 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 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 $k$ ¶
  - … 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 proposition 44.11 The boundary term, and the Gibbons–Hawking–York action ¶
    - depends_on lemma 44.7 Variation of inverse and determinant ¶
      - depends_on proposition 44.17 Stress–energy of the electromagnetic field ¶
      - depends_on theorem 44.9 Variation of the Einstein–Hilbert action ¶
    - 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 proposition 32.17 Classification of planar fixed points ¶
      - depends_on theorem 32.15 Hartman–Grobman, restated from Part II ¶
      - depends_on theorem 32.13 Linear stability ¶
    - depends_on lemma A.754 Exact depth expansion ¶
      - depends_on proposition A.755 The Boussinesq pair ¶
    - depends_on lemma A.187 Exponentiation ¶
    - depends_on lemma A.138 The flat exponential ¶
      - depends_on definition A.139 The standard radial mollifier ¶
      - depends_on lemma A.140 Properties of the mollifier ¶
    - depends_on lemma A.452 Young and Hölder ¶
      - depends_on lemma A.489 Courant–Lebesgue ¶
      - depends_on lemma A.463 Poincaré inequality; $B_{K}$ is an inner product ¶
      - depends_on lemma A.453 Uniform bound and uniform Hölder continuity ¶
    - depends_on lemma A.212 Second-order flatness ¶
      - depends_on lemma A.213 The three minima ¶
    - depends_on lemma A.209 Uniform third-order remainder ¶
      - depends_on corollary A.210 Uniform quadratic approximation on the $n^{-1/2}$ scale ¶
      - depends_on lemma A.211 Root-$n$ localisation ¶
    - 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 theorem 17.78 Causality implies dispersion relations ¶
      - depends_on theorem 17.93 Fundamental strip and 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 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 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 9.23 Solution of a constant-coefficient system ¶
      - depends_on corollary 9.25 Normal modes of a diagonalizable system ¶
      - depends_on proposition 9.87 The distinguished limit, and which end carries the layer ¶
      - depends_on proposition 9.31 Lyapunov's first method: asymptotic stability ¶ ↺
      - depends_on proposition 9.43 The unmodulated line, and the feet of the tongues ¶
      - depends_on theorem 9.32 Hartman–Grobman; quoted ¶
      - depends_on theorem 9.26 Structure of the solutions ¶
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 remark 7.70 What the condition says ¶ ↺
depends_on proposition 7.68 The natural base ¶ ↺
depends_on proposition 7.92 Continued fraction and the Fibonacci ratios ¶
- depends_on remark 7.95 Where $\varphi$ is used ¶
depends_on proposition 7.30 Leibniz rule ¶ ↺
depends_on proposition 7.83 Viète's product ¶ ↺
… 1 more

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Edges

type	direction	node	provenance	where
`depends_on`	→	Convergence	declared	parts/02-mathematical-methods/05-real-analysis.tex:87
`depends_on`	→	Triangle inequality	declared	parts/02-mathematical-methods/05-real-analysis.tex:87
`depends_on`	←	The limit over real exponents	declared	parts/02-mathematical-methods/05-real-analysis.tex:1563
`depends_on`	←	Addition theorems	declared	parts/02-mathematical-methods/05-real-analysis.tex:1834
`depends_on`	←	Cauchy product	declared	parts/02-mathematical-methods/05-real-analysis.tex:953
`depends_on`	←	Chain rule	declared	parts/02-mathematical-methods/05-real-analysis.tex:589
`depends_on`	←	Derivative of the inverse function	declared	parts/02-mathematical-methods/05-real-analysis.tex:625
`depends_on`	←	Linearity	declared	parts/02-mathematical-methods/05-real-analysis.tex:548
`depends_on`	←	Differentiable implies continuous	declared	parts/02-mathematical-methods/05-real-analysis.tex:458
`depends_on`	←	The limit definition	declared	parts/02-mathematical-methods/05-real-analysis.tex:1256
`depends_on`	←	The natural base	declared	parts/02-mathematical-methods/05-real-analysis.tex:1628
`depends_on`	←	Continued fraction and the Fibonacci ratios	declared	parts/02-mathematical-methods/05-real-analysis.tex:2753
`depends_on`	←	Leibniz rule	declared	parts/02-mathematical-methods/05-real-analysis.tex:562
`depends_on`	←	Viète's product	declared	parts/02-mathematical-methods/05-real-analysis.tex:2412
`depends_on`	←	Termwise differentiation	declared	parts/02-mathematical-methods/05-real-analysis.tex:1018
`proves`	←	ch:05-real-analysis@proof-3	declared	parts/02-mathematical-methods/05-real-analysis.tex:90