proposition 7.136 Properties of conservative fields

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:4597 · p. 259

Rests on

depends_on definition 7.135 Conservative field ¶
- depends_on equation 7.124 eq:ana-nabla-gradient ¶
depends_on proposition 7.104 Chain rule in several variables ¶
- depends_on definition 7.13 Composite function ¶
  - depends_on definition 7.9 Real function ¶
    - depends_on definition 3.43 Map ¶
      - depends_on definition 3.24 Quantifiers ¶
      - depends_on definition 3.28 Set ¶
- depends_on definition 7.99 Differentiability 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 definition 5.37 Linear transformation ¶
    - depends_on definition 4.33 Vector space ¶
      - depends_on definition 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.32 Field ¶
      - depends_on definition 4.31 Module ¶
  - depends_on equation 6.12 eq:top-euclidean-metric ¶
- depends_on definition 7.102 Differential of a function ¶
  - depends_on definition 7.99 Differentiability at a point ¶ ↺
  - 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 ¶
- proves proof ch:05-real-analysis@proof-63 ¶
depends_on theorem 7.132 Stokes ¶
- depends_on proposition 7.104 Chain rule in several variables ¶ ↺
- depends_on proposition 7.105 Clairaut–Schwarz ¶
  - 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 theorem 7.35 Mean value theorem ¶
    - depends_on proposition 7.29 Linearity ¶
      - depends_on definition 7.11 Sum of functions ¶
      - depends_on equation 7.14 eq:ana-derivh ¶
      - depends_on proposition 7.6 Algebra of limits ¶
      - proves proof ch:05-real-analysis@proof-13 ¶
    - depends_on theorem 7.34 Rolle ¶
      - depends_on lemma 7.33 Fermat: interior extremum ¶
      - depends_on theorem 7.24 Extreme value theorem ¶
      - proves proof ch:05-real-analysis@proof-18 ¶
    - proves proof ch:05-real-analysis@proof-19 ¶
  - proves proof ch:05-real-analysis@proof-64 ¶
- depends_on theorem 7.131 Green ¶
  - 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 7.97 Partial derivative; gradient ¶ ↺
    - depends_on definition 6.9 Compact set ¶
      - depends_on definition 6.5 Open cover ¶
  - 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 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 theorem 7.25 Heine–Cantor: uniform continuity ¶
      - 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 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 ¶
- proves proof ch:05-real-analysis@proof-78 ¶
proves proof ch:05-real-analysis@proof-80 ¶

Supports

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 lemma A.717 The radial solution ¶
      - depends_on theorem A.718 Stokes' drag law with its coefficient ¶
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 theorem A.737 The lift-curve slope of the flat plate ¶
- 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 phenomenon 31.96 The solitary wave ¶
    - depends_on phenomenon 31.95 The observed dispersion of water waves ¶
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 ¶

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`	→	Conservative field	declared	parts/02-mathematical-methods/05-real-analysis.tex:4614
`depends_on`	→	Chain rule in several variables	declared	parts/02-mathematical-methods/05-real-analysis.tex:4614
`depends_on`	→	Stokes	declared	parts/02-mathematical-methods/05-real-analysis.tex:4614
`depends_on`	←	The stream function of a plane flow	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:403
`depends_on`	←	Potential flow reduces to Laplace's equation	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:799
`depends_on`	←	Hilbert's invariant integral	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1881
`depends_on`	←	Compatibility is sufficient on a simply connected body	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:430
`proves`	←	ch:05-real-analysis@proof-80	declared	parts/02-mathematical-methods/05-real-analysis.tex:4618