theorem 7.131 Green

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:4325 · p. 256

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-77 ¶

Supports

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 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 proposition 7.136 Properties of conservative fields ¶
  - depends_on example 31.15 The stream function of a plane flow ¶
    - depends_on definition A.715 Stokes stream function ¶
      - depends_on lemma A.716 The creeping-flow equation for $\psi$ ¶
  - 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 corollary A.778 The powder collects at the antinodes ¶
    - depends_on lemma A.733 The circle with incidence and circulation ¶
      - depends_on lemma A.735 Transfer of the flow ¶
    - 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 corollary 31.76 Kelvin–Helmholtz instability ¶
      - depends_on corollary 31.75 Rayleigh–Taylor instability ¶
      - depends_on theorem 31.94 Gravity–capillary waves on a free surface ¶
  - 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 phenomenon 106.5 The Aharonov–Bohm effect ¶
      - depends_on theorem 106.16 Feynman–Kac ¶
  - depends_on theorem 8.16 Cauchy integral formula ¶
    - depends_on theorem 8.17 Derivatives of all orders; Cauchy estimates ¶
      - depends_on theorem 8.18 Liouville ¶
    - depends_on theorem 8.21 Laurent expansion ¶
      - depends_on definition 8.22 Isolated singularities; residue ¶
      - depends_on theorem 8.24 Residue theorem ¶
  - 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 106.45 One-loop effective potential ¶
    - depends_on theorem 100.35 One-loop electron self-energy ¶
      - depends_on phenomenon 100.66 The Lamb shift ¶
    - depends_on theorem 100.31 One-loop vacuum polarization ¶
      - depends_on phenomenon 100.69 Light scatters off light ¶
- 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 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 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 ¶ ↺

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`	→	Simple regions	declared	parts/02-mathematical-methods/05-real-analysis.tex:4333
`depends_on`	→	What the derivations below take as given	declared	parts/02-mathematical-methods/05-real-analysis.tex:4333
`depends_on`	→	Fundamental theorem of calculus, II	declared	parts/02-mathematical-methods/05-real-analysis.tex:4333
`depends_on`	←	Bendixson's negative criterion	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:747
`depends_on`	←	Bendixson–Dulac negative criterion	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1300
`depends_on`	←	Stokes	declared	parts/02-mathematical-methods/05-real-analysis.tex:4409
`depends_on`	←	Cauchy	declared	parts/02-mathematical-methods/06-complex-analysis.tex:267
`depends_on`	←	Holonomy equals the enclosed curvature; local Gauss–Bonnet	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2040
`depends_on`	←	Rankine–Hugoniot condition	declared	parts/02-mathematical-methods/08-pdes.tex:2523
`proves`	←	ch:05-real-analysis@proof-77	declared	parts/02-mathematical-methods/05-real-analysis.tex:4337