theorem 7.133 Gauss

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:4488 · p. 258

Rests on

depends_on definition 7.127 Simple regions ¶
- depends_on definition 7.98 Functions of class $C^{1}$ ¶
  - depends_on definition 7.20 Continuity at a point ¶
    - depends_on definition 7.16 Limit ¶
      - depends_on definition 7.2 Absolute value ¶
      - depends_on definition 7.9 Real function ¶
    - depends_on equation 7.7 eq:ana-limit-left ¶
    - depends_on equation 7.5 eq:ana-limit-right ¶
  - depends_on definition 7.97 Partial derivative; gradient ¶
    - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 7.16 Limit ¶ ↺
      - depends_on definition 7.9 Real function ¶ ↺
    - depends_on definition 5.15 Basis ¶
      - depends_on definition 5.12 Subspace generated by a set of vectors ¶
      - depends_on definition 5.14 Linear independence ¶
- depends_on definition 6.9 Compact set ¶
  - depends_on definition 6.5 Open cover ¶
    - depends_on definition 6.2 Open set ¶
      - depends_on definition 6.1 Topological space ¶
    - depends_on equation 3.51 eq:set-indexed ¶
depends_on remark 7.128 What the derivations below take as given ¶
- depends_on definition 7.125 Multiple integral ¶
  - 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 $\R$ ¶ ↺
- depends_on theorem 7.40 Continuous functions are integrable ¶
  - depends_on definition 7.39 Darboux sums and the definite integral ¶ ↺
  - depends_on theorem 7.24 Extreme value theorem ¶
    - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
    - depends_on proposition 7.22 Sequential characterization ¶
      - depends_on definition 7.20 Continuity at a point ¶ ↺
      - depends_on definition 7.4 Convergence ¶
      - proves proof ch:05-real-analysis@proof-7 ¶
    - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - depends_on corollary A.45 Archimedean property and density of $\Q$ ¶
      - depends_on corollary A.46 Monotone convergence ¶
      - proves proof ch:05-real-analysis@proof-4 ¶
    - proves proof ch:05-real-analysis@proof-9 ¶
  - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
    - depends_on proposition 7.22 Sequential characterization ¶ ↺
    - depends_on theorem 7.7 Bolzano–Weierstrass ¶ ↺
    - proves proof ch:05-real-analysis@proof-10 ¶
  - proves proof ch:05-real-analysis@proof-23 ¶
- depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶ ↺
depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
- depends_on definition 7.41 Antiderivative ¶
  - depends_on corollary 7.36 cor:ana-mvt-consequences ¶
    - depends_on theorem 7.35 Mean value theorem ¶
      - depends_on proposition 7.29 Linearity ¶
      - depends_on theorem 7.34 Rolle ¶
      - proves proof ch:05-real-analysis@proof-19 ¶
    - proves proof ch:05-real-analysis@proof-20 ¶
  - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
- depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
  - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
  - depends_on theorem 7.40 Continuous functions are integrable ¶ ↺
  - depends_on theorem 7.24 Extreme value theorem ¶ ↺
  - proves proof ch:05-real-analysis@proof-24 ¶
- proves proof ch:05-real-analysis@proof-25 ¶
proves proof ch:05-real-analysis@proof-79 ¶

Supports

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 proposition A.132 The count ¶
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 corollary A.743 Two scalar functions instead of a tensor field ¶
      - depends_on lemma A.748 The second-order structure function ¶
      - depends_on lemma A.747 The third-order structure functions ¶
      - depends_on proposition A.746 The Kármán–Howarth equation in classical form ¶
    - depends_on example 31.15 The stream function of a plane flow ¶
      - depends_on definition A.715 Stokes stream function ¶
    - depends_on lemma A.744 Vanishing of the pressure–velocity correlation ¶
      - depends_on theorem A.745 Kármán–Howarth–Monin relation ¶
    - depends_on proposition 31.28 Potential flow reduces to Laplace's equation ¶
      - depends_on example 31.29 Flow past a sphere ¶
      - depends_on lemma A.777 The tangential field above a bending plate ¶
      - depends_on lemma A.733 The circle with incidence and circulation ¶
      - depends_on proposition A.797 The outer limit loses a boundary condition ¶
      - depends_on proposition 31.70 The lift-curve slope of a thin aerofoil ¶
      - depends_on theorem A.707 D'Alembert's paradox ¶
      - depends_on theorem 31.74 The interface dispersion relation ¶
    - depends_on remark 31.14 When incompressibility is legitimate ¶
  - depends_on phenomenon 31.88 Sound travels at the adiabatic speed ¶
    - depends_on remark 31.89 The Mach number and the regimes it separates ¶
      - depends_on phenomenon 31.93 The Mach cone ¶
    - depends_on remark 31.90 Why a compression wave must steepen ¶
  - depends_on proposition 52.24 The p-mode cavity ¶
    - depends_on phenomenon 52.25 The Sun rings, and its interior can be inverted ¶
      - depends_on phenomenon 52.22 The solar-neutrino deficit, and its resolution ¶
  - depends_on theorem 31.91 Rankine–Hugoniot jump conditions ¶
    - depends_on phenomenon 31.93 The Mach cone ¶ ↺
    - depends_on remark 31.92 Entropy selects the physical branch ¶
- depends_on theorem 31.22 Euler's equations of motion ¶
  - depends_on lemma A.712 The force is a far-field integral ¶ ↺
  - depends_on phenomenon 31.88 Sound travels at the adiabatic speed ¶ ↺
  - depends_on proposition A.797 The outer limit loses a boundary condition ¶ ↺
  - depends_on proposition 31.80 Rayleigh's circulation criterion ¶
    - depends_on remark 31.81 Where the threshold comes from, and what it costs ¶
  - depends_on proposition 52.24 The p-mode cavity ¶ ↺
  - depends_on remark 31.23 Boundary conditions, and what the ideal model omits ¶
  - depends_on theorem 31.24 Bernoulli ¶
    - depends_on corollary A.799 The impressed pressure gradient ¶
    - depends_on lemma 31.67 Blasius' force formula ¶
      - depends_on theorem 31.68 Kutta–Joukowski ¶
    - depends_on phenomenon 31.25 Pressure falls where the flow speeds up ¶
      - depends_on example 31.26 The Venturi meter ¶
      - depends_on remark 31.27 What the 1738 source contains ¶
    - depends_on theorem A.707 D'Alembert's paradox ¶ ↺
    - depends_on theorem 31.74 The interface dispersion relation ¶ ↺
  - depends_on theorem 31.64 Kelvin's circulation theorem ¶
    - depends_on corollary 31.65 Lagrange: an irrotational flow stays irrotational ¶
      - depends_on remark 31.69 The Kutta condition, and where the circulation comes from ¶
    - depends_on remark 31.66 Only three things make circulation ¶
    - depends_on theorem 31.59 Helmholtz's vortex theorems ¶
      - depends_on corollary 31.65 Lagrange: an irrotational flow stays irrotational ¶ ↺
      - depends_on example 31.60 Vortex rings ¶
  - depends_on theorem 31.37 The Navier–Stokes equations ¶
    - depends_on definition A.741 The statistical setting ¶
    - depends_on definition A.760 The convecting layer ¶
      - depends_on proposition A.761 The Boussinesq equations ¶
    - depends_on definition 31.43 Creeping flow ¶
      - depends_on lemma A.716 The creeping-flow equation for $\psi$ ¶
      - depends_on proposition 31.44 Kinematic reversibility ¶
    - depends_on phenomenon 31.78 Convective onset at a computed threshold ¶
      - depends_on proposition A.761 The Boussinesq equations ¶ ↺
      - depends_on remark 31.79 Bénard's own cells, and Taylor's cylinders ¶
      - depends_on remark 31.81 Where the threshold comes from, and what it costs ¶ ↺
      - depends_on theorem A.767 Marginal stability of the conducting state ¶
    - depends_on phenomenon 31.41 The fourth-power law of pipe flow ¶
      - depends_on phenomenon 31.53 Transition to turbulence in a pipe ¶
      - depends_on remark 31.42 The entry length ¶
    - depends_on phenomenon 31.49 Dynamical similarity ¶
      - depends_on definition 31.51 The other similarity groups ¶
      - depends_on phenomenon 31.53 Transition to turbulence in a pipe ¶ ↺
      - depends_on remark 31.50 Reading the Reynolds number ¶
      - depends_on remark 31.52 Similarity is only as good as the control of the data ¶
      - depends_on remark 31.62 What the spacing calculation settles, and what it does not ¶
    - depends_on proposition A.761 The Boussinesq equations ¶ ↺
    - depends_on proposition 31.40 Circular Couette flow ¶
      - depends_on proposition 31.80 Rayleigh's circulation criterion ¶ ↺
      - depends_on remark 31.79 Bénard's own cells, and Taylor's cylinders ¶ ↺
    - depends_on proposition 31.39 Plane Couette flow ¶
      - depends_on proposition 31.40 Circular Couette flow ¶ ↺
    - depends_on remark 31.48 What is actually proved ¶
      - depends_on remark 31.87 What a theory of turbulence would have to do ¶
    - depends_on remark 31.38 Whose equations, and the Stokes hypothesis ¶
    - depends_on theorem A.745 Kármán–Howarth–Monin relation ¶ ↺
    - … 3 more
  - depends_on theorem 31.91 Rankine–Hugoniot jump conditions ¶ ↺
  - depends_on theorem 31.77 Rayleigh's inflexion-point criterion ¶
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 proposition 104.53 One generation is anomaly free ¶
    - depends_on proposition 106.96 The Standard Model cancels, and only with three colours ¶
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 theorem A.523 Helmholtz decomposition: existence ¶
  - 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 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 ¶
- 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 corollary A.146 Locally uniform limits of harmonic functions ¶
    - depends_on corollary A.145 Harmonic functions are smooth ¶ ↺
  - 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 44.38 The electromagnetic field satisfies NEC, WEC, SEC and DEC ¶
    - depends_on proposition 44.37 Energy conditions for the perfect fluid ¶
    - depends_on proposition 44.39 SEC is the attraction condition ¶
  - depends_on proposition 44.17 Stress–energy of the electromagnetic field ¶
    - depends_on proposition 44.38 The electromagnetic field satisfies NEC, WEC, SEC and DEC ¶ ↺
  - depends_on theorem 44.19 Covariant conservation of stress–energy ¶
    - depends_on proposition 44.20 Perfect-fluid equations of motion ¶
      - depends_on corollary 44.21 Dust flows on geodesics ¶
  - depends_on theorem 44.9 Variation of the Einstein–Hilbert action ¶
    - depends_on proposition 44.11 The boundary term, and the Gibbons–Hawking–York action ¶ ↺
    - depends_on proposition 44.30 The cosmological term ¶
      - depends_on proposition 44.31 de Sitter geometry ¶
- 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 theorem A.769 The Lorenz system ¶
      - depends_on remark A.771 What was assumed, and where it fails ¶
      - depends_on remark A.770 What the three variables and three parameters are ¶
- 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

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Simple regions	declared	parts/02-mathematical-methods/05-real-analysis.tex:4495
`depends_on`	→	What the derivations below take as given	declared	parts/02-mathematical-methods/05-real-analysis.tex:4495
`depends_on`	→	Fundamental theorem of calculus, II	declared	parts/02-mathematical-methods/05-real-analysis.tex:4495
`depends_on`	←	The disturbance flux vanishes	declared	appendices/A-long-proofs.tex:34295
`depends_on`	←	The force is a far-field integral	declared	appendices/A-long-proofs.tex:34418
`depends_on`	←	Laplacian in orthogonal coordinates	declared	appendices/A-long-proofs.tex:6719
`depends_on`	←	Divergence theorem on $(M,g)$	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:467
`depends_on`	←	Transport theorem for a material volume	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:291
`depends_on`	←	A shift of a divergent integral leaves a surface term	declared	parts/11-qft-standard-model/08-path-integral-quantization.tex:4110
`depends_on`	←	Green's identities	declared	parts/02-mathematical-methods/08-pdes.tex:1204
`depends_on`	←	Darboux's equation for spherical means	declared	parts/02-mathematical-methods/08-pdes.tex:1585
`depends_on`	←	The geometrical amplitude diverges	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1840
`depends_on`	←	Energy in a backward cone	declared	parts/02-mathematical-methods/08-pdes.tex:1536
`depends_on`	←	Euler–Lagrange equations for several independent variables	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1024
`depends_on`	←	Evolution of phase volume	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:181
`depends_on`	←	Cauchy's equation of motion	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:668
`proves`	←	ch:05-real-analysis@proof-79	declared	parts/02-mathematical-methods/05-real-analysis.tex:4499