remark 7.128 What the derivations below take as given

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:4238 · p. 255

remark -- no derivation owed by its kind

Rests on

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 7.16 Limit ¶
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
    - depends_on definition 7.4 Convergence ¶
      - depends_on definition 7.2 Absolute value ¶
    - 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 definition A.36 Cut ¶
      - depends_on theorem A.40 Least-upper-bound property ¶
      - proves proof app:A-long-proofs@proof-32 ¶
    - depends_on corollary A.46 Monotone convergence ¶
      - depends_on theorem A.40 Least-upper-bound property ¶ ↺
      - proves proof app:A-long-proofs@proof-33 ¶
    - 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 ¶ ↺

Supports

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 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 proposition A.519 The convolution exists ¶
      - depends_on proposition A.524 Rate of decay ¶ ↺
- 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 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 example 31.15 The stream function of a plane flow ¶
      - depends_on lemma A.744 Vanishing of the pressure–velocity correlation ¶
      - depends_on proposition 31.28 Potential flow reduces to Laplace's equation ¶
      - 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 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 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 phenomenon 31.25 Pressure falls where the flow speeds up ¶
      - 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 31.66 Only three things make circulation ¶
      - depends_on theorem 31.59 Helmholtz's vortex theorems ¶
    - 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 definition 31.43 Creeping flow ¶
      - depends_on phenomenon 31.78 Convective onset at a computed threshold ¶
      - depends_on phenomenon 31.41 The fourth-power law of pipe flow ¶
      - depends_on phenomenon 31.49 Dynamical similarity ¶
      - depends_on proposition A.761 The Boussinesq equations ¶
      - depends_on proposition 31.40 Circular Couette flow ¶
      - depends_on proposition 31.39 Plane Couette flow ¶
      - depends_on remark 31.48 What is actually proved ¶
      - 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 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 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 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 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 remark 23.8 The action as a function, not a functional ¶
      - depends_on theorem 16.64 Weierstrass' sufficient condition ¶
    - 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 106.3 Feynman's sum over histories ¶
    - depends_on theorem 8.16 Cauchy integral formula ¶
      - depends_on theorem 8.17 Derivatives of all orders; Cauchy estimates ¶
      - depends_on theorem 8.21 Laurent expansion ¶
    - 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 example 8.25 A real integral by residues ¶
      - depends_on proposition 17.96 Poles give asymptotics ¶
      - depends_on proposition 17.61 Mean-square response of a damped resonator ¶
      - depends_on theorem 31.68 Kutta–Joukowski ¶
      - depends_on theorem 9.26 Structure of the solutions ¶
  - depends_on corollary 17.71 Residue evaluation and causality ¶ ↺
  - depends_on lemma 100.27 Wick rotation ¶
    - depends_on lemma 100.28 The $d$-dimensional loop integral ¶
      - depends_on proposition 106.45 One-loop effective potential ¶
      - depends_on theorem 100.35 One-loop electron self-energy ¶
      - depends_on theorem 100.31 One-loop vacuum polarization ¶
  - 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 corollary 17.101 Coupled channels ¶
- 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 definition A.150 Kruzhkov entropy pair ¶
      - depends_on proposition A.156 The Kruzhkov inequality ¶
      - depends_on proposition A.163 The viscous problem produces the entropy inequality ¶
      - depends_on proposition 10.102 Jump form of the entropy inequality ¶
      - depends_on theorem A.151 $L^{1}$ contraction on a cone ¶
      - depends_on theorem 10.103 Uniqueness in the entropy class ¶
  - 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 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 ¶

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Multiple integral	declared	parts/02-mathematical-methods/05-real-analysis.tex:4261
`depends_on`	→	Continuous functions are integrable	declared	parts/02-mathematical-methods/05-real-analysis.tex:4261
`depends_on`	→	Heine–Cantor: uniform continuity	declared	parts/02-mathematical-methods/05-real-analysis.tex:4261
`depends_on`	←	The computation in one chart	declared	appendices/A-long-proofs.tex:15282
`depends_on`	←	Change of variables in a multiple integral	declared	parts/02-mathematical-methods/05-real-analysis.tex:4290
`depends_on`	←	Gauss	declared	parts/02-mathematical-methods/05-real-analysis.tex:4495
`depends_on`	←	Green	declared	parts/02-mathematical-methods/05-real-analysis.tex:4333
`depends_on`	←	Change of variables for multiple integrals; quoted	declared	appendices/A-long-proofs.tex:15017