definition 7.39 Darboux sums and the definite integral

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Rests on

depends_on axiom 7.1 Completeness of $\R$ ¶

Supports

depends_on definition 7.125 Multiple integral ¶
- 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 proposition 30.19 The stress tensor is symmetric ¶
      - depends_on definition 30.20 Pressure and deviatoric stress ¶
    - depends_on theorem 30.18 Cauchy: the stress tensor exists ¶
      - depends_on definition 30.24 The elasticity tensor ¶
      - depends_on definition 31.1 Fluid ¶
      - depends_on proposition 30.19 The stress tensor is symmetric ¶ ↺
      - depends_on proposition 30.74 The maximum shear stress, and its indifference to pressure ¶
      - depends_on remark 30.72 Stress concentration ¶
      - depends_on theorem A.695 Boussinesq's point-force solution, quoted ¶
      - depends_on theorem 30.21 Cauchy's equation of motion ¶
- depends_on definition 7.138 Set of measure zero ¶
  - depends_on definition 7.140 Hausdorff measure ¶
    - depends_on definition 7.142 Box-counting dimension ¶
      - depends_on definition 32.79 Box-counting dimension, restated from Part II ¶
      - depends_on proposition 7.143 The Hausdorff dimension never exceeds the box dimension ¶
      - depends_on remark 7.144 What is developed here, and what is not ¶
    - depends_on proposition 7.141 The critical exponent, and the Hausdorff dimension ¶
      - depends_on proposition 7.143 The Hausdorff dimension never exceeds the box dimension ¶ ↺
    - depends_on remark 7.144 What is developed here, and what is not ¶ ↺
  - depends_on 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 proposition 7.141 The critical exponent, and the Hausdorff dimension ¶ ↺
  - depends_on remark 7.144 What is developed here, and what is not ¶ ↺
- depends_on definition A.498 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 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 corollary A.514 Degeneracy on a negligible set ¶ ↺
      - depends_on remark A.516 The hypotheses of the global form ¶
- depends_on definition A.503 The substitution property ¶
  - 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 ¶ ↺
  - depends_on lemma A.504 Transitivity ¶
- depends_on definition A.304 Integral over a chart ¶
  - depends_on lemma A.305 The chart integral is well defined ¶
    - depends_on definition A.308 Integral over the manifold ¶
      - depends_on theorem A.311 General Stokes theorem ¶
- depends_on definition 29.13 Inertia tensor, continuum form ¶
  - depends_on definition 29.19 Inertia ellipsoid ¶
    - depends_on theorem 29.29 Poinsot's construction ¶
      - depends_on remark 29.32 What the linearization is entitled to conclude ¶
      - depends_on remark 29.30 Polhode and herpolhode ¶
  - depends_on example 29.23 The polar moment of the Earth, and what it reveals ¶
    - depends_on phenomenon 29.34 Free nutation of the Earth ¶
  - depends_on proposition 49.17 Magnetic-dipole spin-down ¶
  - depends_on proposition 29.14 Angular momentum and kinetic energy ¶
  - depends_on proposition 29.15 Positivity ¶
    - depends_on definition 29.19 Inertia ellipsoid ¶ ↺
    - depends_on theorem 29.16 Principal axes ¶
      - depends_on definition 29.19 Inertia ellipsoid ¶ ↺
      - depends_on definition 29.35 The Lagrange top ¶
      - depends_on definition 29.18 Classification of tops ¶
      - depends_on phenomenon 29.31 The intermediate-axis instability ¶
      - depends_on proposition 29.33 Dissipation drives a free body to its largest moment ¶
      - depends_on proposition 29.17 Triangle inequalities ¶
      - depends_on theorem 29.25 Euler's equations ¶
  - depends_on proposition 29.24 MacCullagh's formula ¶
    - depends_on phenomenon 29.49 Precession of the equinoxes ¶
      - depends_on proposition 29.50 The principal nutation ¶
  - depends_on proposition 29.22 Perpendicular-axis theorem ¶
  - depends_on theorem 29.20 Parallel-axis theorem ¶
    - depends_on corollary 29.21 The centre of mass minimizes the moment ¶
  - depends_on theorem 29.16 Principal axes ¶ ↺
- 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 9.143 A cubic quadrature between two turning points ¶
    - 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 proposition 9.144 The pendulum equation at finite amplitude ¶
  - 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 lemma A.501 Iterated integration over a box ¶
  - depends_on corollary A.502 All iterated orders agree ¶
    - depends_on lemma A.507 Coordinate permutations ¶
  - depends_on lemma A.509 Primitive maps have the substitution property ¶
    - depends_on proposition A.511 The substitution property is universal ¶ ↺
- depends_on lemma A.499 What zero content buys ¶ ↺
- depends_on lemma A.518 The excised ball ¶
  - 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 A.523 Helmholtz decomposition: existence ¶
  - depends_on proposition A.519 The convolution exists ¶
    - depends_on proposition A.524 Rate of decay ¶ ↺
- 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 example 7.130 The two Jacobians this treatise uses ¶
      - depends_on lemma A.518 The excised ball ¶ ↺
    - depends_on lemma A.518 The excised ball ¶ ↺
    - depends_on proposition A.519 The convolution exists ¶ ↺
  - depends_on theorem 7.133 Gauss ¶
    - depends_on lemma A.708 The disturbance flux vanishes ¶
      - depends_on corollary A.711 The disturbance is a dipole at leading order ¶
    - depends_on lemma A.712 The force is a far-field integral ¶
    - depends_on lemma A.117 Laplacian in orthogonal coordinates ¶
      - depends_on theorem A.120 Stäckel and Robertson conditions ¶
    - 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 31.11 Transport theorem for a material volume ¶
      - depends_on theorem 31.12 Conservation of mass ¶
      - depends_on theorem 31.22 Euler's equations of motion ¶
    - depends_on lemma 106.86 A shift of a divergent integral leaves a surface term ¶
      - depends_on proposition 106.88 The obstruction ¶
    - depends_on lemma 10.44 Green's identities ¶
      - depends_on proposition 10.71 Green's representation formula ¶
      - depends_on proposition 10.67 The Newtonian potential is the fundamental solution ¶
      - depends_on theorem 10.45 Symmetry of the Green's function ¶
    - depends_on lemma 10.58 Darboux's equation for spherical means ¶
      - depends_on theorem 10.59 Kirchhoff's formula ¶
      - depends_on theorem 10.69 Mean-value property ¶
    - depends_on proposition 23.54 The geometrical amplitude diverges ¶
      - depends_on proposition 23.55 The fold, and the quarter-power law ¶
      - depends_on remark 23.59 What is observed ¶
    - depends_on proposition 10.57 Energy in a backward cone ¶
      - depends_on proposition 28.63 Energy density, flux, and the conservation law ¶
    - depends_on theorem 16.36 Euler–Lagrange equations for several independent variables ¶
      - depends_on definition 44.6 Stress–energy tensor ¶
      - depends_on proposition 28.42 The wave equation of a stretched string ¶
    - depends_on theorem 32.6 Evolution of phase volume ¶
      - depends_on definition 32.7 Conservative and dissipative flows ¶
      - depends_on definition 32.62 The Lorenz system ¶
      - depends_on proposition 32.64 The Lorenz flow contracts volume ¶
      - depends_on proposition 32.76 The exponents sum to the mean divergence ¶
    - … 1 more
  - depends_on theorem 7.131 Green ¶
    - depends_on proposition 32.33 Bendixson's negative criterion ¶
    - depends_on proposition 9.36 Bendixson–Dulac negative criterion ¶
    - depends_on theorem 7.132 Stokes ¶
      - depends_on definition 31.63 Circulation ¶
      - depends_on proposition 7.136 Properties of conservative fields ¶
    - depends_on theorem 8.12 Cauchy ¶
      - depends_on corollary 8.15 Deformation of contours ¶
      - depends_on corollary 17.71 Residue evaluation and causality ¶
      - depends_on lemma 100.27 Wick rotation ¶
      - depends_on proposition 106.14 The Feynman contour permits the rotation ¶
      - depends_on theorem 17.78 Causality implies dispersion relations ¶
      - depends_on theorem 17.94 Mellin inversion ¶
    - depends_on theorem 13.41 Holonomy equals the enclosed curvature; local Gauss–Bonnet ¶
      - depends_on example 13.42 The sphere, the solid angle, and the pole ¶
    - depends_on theorem 10.96 Rankine–Hugoniot condition ¶
      - depends_on definition 10.99 Entropy condition ¶
      - depends_on example 10.97 Burgers shock ¶
      - depends_on example 10.98 Weak solutions are not unique ¶
      - depends_on proposition 10.102 Jump form of the entropy inequality ¶
  - depends_on theorem A.303 Change of variables for multiple integrals; quoted ¶
    - depends_on lemma A.305 The chart integral is well defined ¶ ↺
- depends_on theorem 7.129 Change of variables in a multiple integral ¶ ↺
- … 3 more
depends_on definition 11.20 Density ¶
- depends_on definition 11.21 Distribution function ¶
  - depends_on definition A.174 Tightness ¶
    - depends_on lemma A.175 Tightness makes the Helly limit proper ¶
    - depends_on lemma A.180 Tightness ¶
  - depends_on lemma 11.52 Slutsky ¶
    - depends_on theorem A.206 Cramér–Wold, and continuous mapping ¶
      - depends_on lemma A.208 The three averages ¶
      - depends_on theorem A.214 Wilks, $k$ parameters and $r$ constraints ¶
    - depends_on theorem 11.67 Asymptotics of the MLE ¶
      - depends_on proposition 11.72 Nuisance parameters cost information ¶
      - depends_on theorem 11.88 Wilks ¶
    - depends_on theorem 11.88 Wilks ¶ ↺
  - depends_on proposition 11.22 Properties of the distribution function ¶
    - depends_on proposition 11.27 Probability integral transform ¶
  - depends_on theorem A.168 Lévy's continuity theorem ¶
    - depends_on corollary A.181 The converse ¶
  - depends_on theorem A.173 Helly's selection theorem ¶
    - depends_on lemma A.176 Helly–Bray ¶
      - depends_on corollary A.181 The converse ¶ ↺
    - depends_on lemma A.175 Tightness makes the Helly limit proper ¶ ↺
  - depends_on theorem A.206 Cramér–Wold, and continuous mapping ¶ ↺
- 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 definition 11.28 Gaussian distribution ¶
  - depends_on definition A.218 The hypotheses ¶
    - depends_on lemma A.220 At each point, field and derivative are independent Gaussians ¶
    - depends_on theorem A.219 Rice's formula ¶
      - depends_on corollary A.227 The stationary case ¶
  - depends_on lemma A.226 The level factor ¶
  - depends_on lemma A.225 The mean positive part ¶
  - depends_on proposition 11.29 Gaussian moments ¶ ↺
  - depends_on theorem A.192 Berry–Esseen ¶
  - depends_on theorem A.183 Lindeberg–Feller ¶
    - depends_on corollary A.188 Lindeberg–Lévy is the special case ¶
    - depends_on corollary A.189 Lyapunov's condition ¶
  - depends_on theorem 11.102 Rice's formula ¶
    - depends_on theorem 11.103 Level dependence of the mean upcrossing count ¶
      - depends_on corollary 11.105 Trials factor at high significance ¶
      - depends_on example 11.107 Degrading a local five sigma ¶
- depends_on definition 11.25 Joint density and independence ¶
  - depends_on proposition 11.26 Product rule ¶ ↺
depends_on lemma A.501 Iterated integration over a box ¶ ↺
depends_on lemma 17.7 Riemann–Lebesgue ¶
- depends_on corollary 17.8 Localisation ¶
- depends_on proposition 17.10 Smoothness and coefficient decay ¶
  - depends_on corollary 17.11 Uniform convergence for $C^{1}$ functions ¶
- depends_on proposition 17.30 Elementary properties ¶
  - depends_on example 17.31 The three standard pairs ¶
  - depends_on proposition 17.39 The transform preserves $\mathcal{S}$ ¶
    - depends_on definition 17.40 Transform and derivative of a distribution ¶
      - depends_on definition 10.42 Green's function ¶
      - depends_on proposition 17.41 The identities physics uses ¶
  - depends_on theorem 17.43 Bandwidth theorem ¶
    - depends_on corollary 17.44 Duration and bandwidth in SI ¶
    - depends_on remark 17.46 The quantum reading, in SI ¶
- depends_on theorem 17.9 Dirichlet ¶
  - depends_on corollary 17.11 Uniform convergence for $C^{1}$ functions ¶ ↺
  - depends_on theorem 17.47 Poisson summation ¶
    - depends_on corollary 17.81 Sampling periodises the spectrum; aliasing ¶
    - depends_on corollary 17.49 Lattice sums and the reciprocal lattice ¶
      - depends_on definition 17.25 Multiple Fourier series ¶
depends_on theorem 7.40 Continuous functions are integrable ¶
- depends_on corollary A.502 All iterated orders agree ¶ ↺
- 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 16.13 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 16.15 Variation; the first variation ¶
    - depends_on definition 16.83 One-parameter transformation group; variational symmetry ¶
    - 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 proposition 16.16 Stationarity is necessary ¶ ↺
    - 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 proposition 16.68 Weierstrass' counterexample ¶
- depends_on definition 9.137 Elliptic integrals of the three kinds ¶ ↺
- depends_on lemma A.501 Iterated integration over a box ¶ ↺
- depends_on lemma 16.18 Fundamental lemma ¶ ↺
- depends_on remark 7.128 What the derivations below take as given ¶ ↺
- 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.463 Poincaré inequality; $B_{K}$ is an inner product ¶
    - depends_on lemma A.453 Uniform bound and uniform Hölder continuity ¶
      - depends_on theorem A.455 Compact embedding of $W^{1,r}(a,b)$ into the continuous functions ¶
    - 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.520 A $C^{1}$ limit ¶
    - depends_on proposition A.521 Every derivative passes onto $g$ ¶ ↺
  - 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 16.19 du Bois-Reymond ¶
    - 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 9.10 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 9.18 The linear equation: integrating factor ¶
    - depends_on proposition 9.87 The distinguished limit, and which end carries the layer ¶
      - depends_on proposition 9.88 The model problem, and the error of the composite ¶
  - depends_on proposition 9.19 Separation of variables ¶
  - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
    - 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 corollary 16.23 du Bois-Reymond form ¶
      - depends_on corollary 16.33 Weierstrass–Erdmann corner conditions ¶
    - depends_on lemma 7.116 Functions vanishing on a regular zero set ¶
    - depends_on lemma A.72 Iterated integral inequality ¶
      - depends_on theorem A.74 Flow of a time-dependent vector field ¶
    - depends_on lemma A.520 A $C^{1}$ limit ¶ ↺
    - depends_on lemma A.172 Dirichlet ¶
      - depends_on lemma A.193 A cosine integral ¶
      - depends_on theorem A.177 Fourier inversion for a distribution function ¶
    - 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 remark 7.94 Three numbers, two kinds ¶ ↺
      - depends_on remark 7.87 Priority, and what irrationality does not give ¶
    - depends_on proposition 8.10 Fundamental theorem for contours ¶
      - depends_on theorem 8.13 Goursat ¶
      - depends_on theorem 8.18 Liouville ¶
    - depends_on proposition 17.41 The identities physics uses ¶ ↺
    - depends_on proposition 17.73 Initial- and final-value theorems ¶
    - … 4 more
  - depends_on theorem 10.52 Duhamel's principle ¶
    - depends_on corollary 10.53 Duhamel for the wave equation ¶

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type	direction	node	provenance	where
`depends_on`	→	Completeness of $\R$	declared	parts/02-mathematical-methods/05-real-analysis.tex:782
`depends_on`	←	Multiple integral	declared	parts/02-mathematical-methods/05-real-analysis.tex:4179
`depends_on`	←	Density	declared	parts/02-mathematical-methods/09-probability-statistics.tex:582
`depends_on`	←	Iterated integration over a box	declared	appendices/A-long-proofs.tex:24598
`depends_on`	←	Riemann–Lebesgue	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:341
`depends_on`	←	Continuous functions are integrable	declared	parts/02-mathematical-methods/05-real-analysis.tex:788