definition 7.125 Multiple integral

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:4168 · p. 254

ground object -- no derivation owed

Rests on

depends_on axiom 7.1 Completeness of $\R$ ¶
depends_on definition 7.39 Darboux sums and the definite integral ¶
- depends_on axiom 7.1 Completeness of $\R$ ¶ ↺

Supports

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 definition 30.31 The engineering moduli ¶
      - depends_on definition 30.79 Maxwell viscoelasticity ¶
      - depends_on proposition 30.74 The maximum shear stress, and its indifference to pressure ¶
      - depends_on remark 30.75 Two criteria, one uncited ¶
  - depends_on theorem 30.18 Cauchy: the stress tensor exists ¶
    - depends_on definition 30.24 The elasticity tensor ¶
      - depends_on definition 30.37 Voigt notation ¶
      - depends_on phenomenon 30.28 Hooke's law ¶
      - depends_on proposition 30.25 Twenty-one constants ¶
      - depends_on proposition 30.38 The Cauchy relations ¶
      - depends_on remark 30.78 Anharmonicity: what the quadratic energy also suppresses ¶
      - depends_on theorem A.675 Three constants for a cubic crystal, two for an isotropic solid ¶
    - depends_on definition 31.1 Fluid ¶
      - depends_on definition 31.21 Ideal fluid ¶
      - depends_on definition 31.2 Knudsen number ¶
      - depends_on definition 31.32 Newtonian fluid; dynamic and kinematic viscosity ¶
      - depends_on proposition 31.16 Hydrostatic balance ¶
    - 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 proposition 30.76 Griffith's criterion ¶
    - depends_on theorem A.695 Boussinesq's point-force solution, quoted ¶
      - depends_on definition A.698 The Hertz contact problem ¶
      - depends_on lemma A.697 Surface displacement under a distributed pressure ¶
    - depends_on theorem 30.21 Cauchy's equation of motion ¶
      - depends_on definition 30.22 Boundary conditions of elastostatics ¶
      - depends_on proposition 30.23 Virtual work; the weak form ¶
      - depends_on proposition 31.16 Hydrostatic balance ¶ ↺
      - depends_on remark 30.69 Saint-Venant's principle, and its exact status ¶
      - depends_on theorem 30.40 Navier–Cauchy equations ¶
      - depends_on theorem 31.22 Euler's equations 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 definition 32.81 Correlation dimension ¶
      - depends_on example 32.80 The middle-thirds Cantor set ¶
    - depends_on proposition 7.143 The Hausdorff dimension never exceeds the box dimension ¶
    - depends_on remark 7.144 What is developed here, and what is not ¶
  - depends_on proposition 7.141 The critical exponent, and the Hausdorff dimension ¶
    - depends_on proposition 7.143 The Hausdorff dimension never exceeds the box dimension ¶ ↺
  - depends_on remark 7.144 What is developed here, and what is not ¶ ↺
- depends_on 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 theorem A.513 Change of variables ¶ ↺
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 corollary A.313 Stokes' theorem for an antisymmetric tensor field ¶ ↺
      - depends_on lemma A.555 The actions are well defined ¶
      - depends_on theorem 24.23 Poincaré–Cartan integral invariant ¶
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 proposition 29.28 Free rotation of a symmetric body ¶
    - depends_on phenomenon 29.31 The intermediate-axis instability ¶
      - depends_on remark 29.32 What the linearization is entitled to conclude ¶ ↺
    - depends_on proposition 29.33 Dissipation drives a free body to its largest moment ¶
    - depends_on proposition 29.17 Triangle inequalities ¶
      - depends_on proposition 29.22 Perpendicular-axis theorem ¶
    - depends_on theorem 29.25 Euler's equations ¶
      - depends_on phenomenon 29.46 A fast spinning top does not fall ¶
      - depends_on phenomenon 29.31 The intermediate-axis instability ¶ ↺
      - depends_on proposition 29.26 The two integrals of torque-free motion ¶
      - depends_on proposition 29.28 Free rotation of a symmetric body ¶ ↺
      - depends_on proposition 24.41 The free rigid body is a Lie–Poisson system ¶
- 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 example A.122 Cartesian ¶
      - depends_on example A.123 Circular cylindrical ¶
      - depends_on example A.124 Spherical ¶
      - depends_on proposition A.128 The ellipsoidal system separates ¶
  - 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 corollary 31.13 Incompressible flow ¶
      - depends_on phenomenon 31.88 Sound travels at the adiabatic speed ¶
      - depends_on proposition 52.24 The p-mode cavity ¶
      - depends_on theorem 31.91 Rankine–Hugoniot jump conditions ¶
    - 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 theorem 106.95 The cancellation condition ¶
  - 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 example 10.49 The method of images ¶
      - depends_on lemma A.94 Fundamental solution of $\pp_{\bar z}$ ¶
      - depends_on proposition A.522 The Newtonian potential inverts the Laplacian ¶ ↺
      - depends_on proposition 10.71 Green's representation formula ¶ ↺
      - depends_on remark 10.73 What compact support is doing, and the rate that replaces it ¶
      - depends_on theorem 7.137 Helmholtz decomposition ¶
    - 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 corollary 10.60 Huygens' principle in three space dimensions ¶
    - depends_on theorem 10.69 Mean-value property ¶
      - depends_on corollary A.145 Harmonic functions are smooth ¶
      - depends_on theorem A.135 Converse of the mean-value property ¶
      - depends_on theorem 10.77 Strong maximum principle for harmonic functions ¶
  - depends_on proposition 23.54 The geometrical amplitude diverges ¶
    - depends_on proposition 23.55 The fold, and the quarter-power law ¶
      - depends_on proposition 23.56 The phase jump at a fold ¶
      - depends_on remark 23.58 The caustics that are not turning surfaces ¶
    - 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 definition 44.36 Energy conditions ¶
      - depends_on proposition 44.17 Stress–energy of the electromagnetic field ¶
      - depends_on theorem 44.19 Covariant conservation of stress–energy ¶
      - depends_on theorem 44.9 Variation of the Einstein–Hilbert action ¶
    - 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 definition 32.62 The Lorenz system ¶ ↺
    - depends_on proposition 32.64 The Lorenz flow contracts volume ¶
      - depends_on phenomenon 32.68 Strange attractors ¶
    - 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 theorem 31.64 Kelvin's circulation theorem ¶
    - depends_on proposition 7.136 Properties of conservative fields ¶
      - depends_on example 31.15 The stream function of a plane flow ¶
      - depends_on proposition 31.28 Potential flow reduces to Laplace's equation ¶
      - depends_on theorem 16.63 Hilbert's invariant integral ¶
      - depends_on theorem 30.15 Compatibility is sufficient on a simply connected body ¶
  - depends_on theorem 8.12 Cauchy ¶
    - depends_on corollary 8.15 Deformation of contours ¶
      - depends_on lemma 106.1 The short-time kernel ¶
      - depends_on theorem 8.16 Cauchy integral formula ¶
      - depends_on theorem 8.21 Laurent expansion ¶
      - depends_on theorem 8.24 Residue theorem ¶
    - depends_on corollary 17.71 Residue evaluation and causality ¶
      - depends_on corollary 17.76 Heaviside's expansion theorem ¶
      - depends_on proposition 17.77 Poles and stability ¶
    - depends_on lemma 100.27 Wick rotation ¶
      - depends_on lemma 100.28 The $d$-dimensional loop integral ¶
    - depends_on proposition 106.14 The Feynman contour permits the rotation ¶
    - depends_on theorem 17.78 Causality implies dispersion relations ¶
      - depends_on remark 33.10 Fibre anelasticity, and why it biases one method and not the other ¶
      - depends_on remark 28.26 Causality is visible in the susceptibility ¶
      - depends_on remark 28.62 Neither velocity is a speed limit ¶
    - depends_on theorem 17.94 Mellin inversion ¶
      - depends_on proposition 17.96 Poles give asymptotics ¶
      - depends_on theorem 17.100 Moments diagonalise a convolution evolution ¶
  - 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 definition 10.101 Entropy pair; entropy inequality ¶
    - 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

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Completeness of $\R$	declared	parts/02-mathematical-methods/05-real-analysis.tex:4179
`depends_on`	→	Darboux sums and the definite integral	declared	parts/02-mathematical-methods/05-real-analysis.tex:4179
`depends_on`	←	Line and surface integrals	declared	parts/02-mathematical-methods/05-real-analysis.tex:4215
`depends_on`	←	Set of measure zero	declared	parts/02-mathematical-methods/05-real-analysis.tex:4725
`depends_on`	←	Zero content	declared	appendices/A-long-proofs.tex:24462
`depends_on`	←	The substitution property	declared	appendices/A-long-proofs.tex:24675
`depends_on`	←	Integral over a chart	declared	appendices/A-long-proofs.tex:15032
`depends_on`	←	Inertia tensor, continuum form	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:373
`depends_on`	←	Elliptic integrals of the three kinds	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:5250
`depends_on`	←	Iterated integration over a box	declared	appendices/A-long-proofs.tex:24598
`depends_on`	←	What zero content buys	declared	appendices/A-long-proofs.tex:24487
`depends_on`	←	The excised ball	declared	appendices/A-long-proofs.tex:25311
`depends_on`	←	What the derivations below take as given	declared	parts/02-mathematical-methods/05-real-analysis.tex:4261
`depends_on`	←	Change of variables in a multiple integral	declared	parts/02-mathematical-methods/05-real-analysis.tex:4290
`depends_on`	←	Leibniz integral rule	declared	parts/02-mathematical-methods/05-real-analysis.tex:3333
`depends_on`	←	Change of variables for multiple integrals; quoted	declared	appendices/A-long-proofs.tex:15017
`depends_on`	←	Poincaré	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:738