theorem 31.68 Kutta–Joukowski

open in the book · parts/03-classical-mechanics/14-fluid-dynamics.tex:2147 · p. 1066

Rests on

depends_on lemma 31.67 Blasius' force formula ¶
- depends_on theorem 8.6 Cauchy–Riemann equations ¶
  - 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 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 5.15 Basis ¶
  - depends_on equation 8.4 eq:cpx-derivative ¶
  - proves proof ch:06-complex-analysis@proof-3 ¶
- depends_on theorem 31.24 Bernoulli ¶
  - depends_on equation 31.4 eq:fluid-lamb-identity ¶
  - depends_on theorem 31.22 Euler's equations of motion ¶
    - depends_on definition 31.21 Ideal fluid ¶
      - depends_on definition 31.1 Fluid ¶
    - depends_on lemma 31.11 Transport theorem for a material volume ¶
      - depends_on definition 31.4 Material derivative ¶
      - depends_on theorem 7.133 Gauss ¶
      - proves proof ch:14-fluid-dynamics@proof-4 ¶
    - depends_on theorem 30.21 Cauchy's equation of motion ¶
      - depends_on theorem 7.133 Gauss ¶ ↺
      - depends_on theorem 30.18 Cauchy: the stress tensor exists ¶
      - proves proof ch:13-continuum-elasticity@proof-9 ¶
    - proves proof ch:14-fluid-dynamics@proof-11 ¶
  - proves proof ch:14-fluid-dynamics@proof-12 ¶
- proves proof ch:14-fluid-dynamics@proof-31 ¶
depends_on theorem 8.24 Residue theorem ¶
- depends_on corollary 8.15 Deformation of contours ¶
  - depends_on definition 6.17 Simply connected space ¶
    - depends_on definition 6.6 Continuous map ¶
      - depends_on definition 3.53 Preimage ¶
      - depends_on definition 6.2 Open set ¶
      - depends_on definition 6.1 Topological space ¶
    - depends_on definition 6.15 Path-connected space ¶
      - depends_on definition 6.6 Continuous map ¶ ↺
  - depends_on theorem 8.12 Cauchy ¶
    - depends_on equation 8.5 eq:cpx-cr ¶
    - depends_on theorem 7.131 Green ¶
      - depends_on definition 7.127 Simple regions ¶
      - depends_on remark 7.128 What the derivations below take as given ¶
      - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      - proves proof ch:05-real-analysis@proof-77 ¶
    - proves proof ch:06-complex-analysis@proof-7 ¶
  - proves proof ch:06-complex-analysis@proof-9 ¶
- depends_on definition 8.22 Isolated singularities; residue ¶
  - depends_on lemma 8.11 The fundamental $2\pi\ii$ ¶
    - depends_on equation 8.6 eq:cpx-contour-integral ¶
    - depends_on proposition 8.4 Euler's formula ¶
      - depends_on equation 8.2 eq:cpx-exp-series ¶
      - depends_on proposition 7.47 Comparison; absolute convergence ¶
      - proves proof ch:06-complex-analysis@proof-2 ¶
    - proves proof ch:06-complex-analysis@proof-6 ¶
  - depends_on theorem 8.21 Laurent expansion ¶
    - depends_on corollary 8.15 Deformation of contours ¶ ↺
    - depends_on lemma 8.11 The fundamental $2\pi\ii$ ¶ ↺
    - depends_on theorem 8.16 Cauchy integral formula ¶
      - depends_on corollary 8.15 Deformation of contours ¶ ↺
      - depends_on lemma 8.11 The fundamental $2\pi\ii$ ¶ ↺
      - depends_on lemma 8.9 ML estimate ¶
      - proves proof ch:06-complex-analysis@proof-10 ¶
    - depends_on theorem 8.20 Taylor expansion ¶
      - depends_on equation 8.12 eq:cpx-cif ¶
      - depends_on equation 8.13 eq:cpx-derivative-formula ¶
      - depends_on lemma 8.9 ML estimate ¶ ↺
      - depends_on proposition 7.46 Geometric series ¶
      - proves proof ch:06-complex-analysis@proof-14 ¶
    - proves proof ch:06-complex-analysis@proof-15 ¶
- depends_on lemma 8.11 The fundamental $2\pi\ii$ ¶ ↺
- depends_on theorem 8.21 Laurent expansion ¶ ↺
- proves proof ch:06-complex-analysis@proof-17 ¶
proves proof ch:14-fluid-dynamics@proof-32 ¶

Supports

depends_on proposition 31.70 The lift-curve slope of a thin aerofoil ¶
depends_on remark 31.69 The Kutta condition, and where the circulation comes from ¶
depends_on theorem A.737 The lift-curve slope of the flat plate ¶

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
`cites`	→	Über die Konturen der Tragflächen der Drachenflieger	derived	parts/03-classical-mechanics/14-fluid-dynamics.tex:2157
`cites`	→	Auftriebskräfte in strömenden Flüssigkeiten	derived	parts/03-classical-mechanics/14-fluid-dynamics.tex:2157
`depends_on`	→	Blasius' force formula	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:2158
`depends_on`	→	Residue theorem	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:2158
`depends_on`	←	The lift-curve slope of a thin aerofoil	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:2241
`depends_on`	←	The Kutta condition, and where the circulation comes from	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:2222
`depends_on`	←	The lift-curve slope of the flat plate	declared	appendices/A-long-proofs.tex:35762
`proves`	←	ch:14-fluid-dynamics@proof-32	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:2161