definition 7.126 Line and surface integrals

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:4182 · p. 255

ground object -- no derivation owed

Rests on

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 definition 3.43 Map ¶
  - 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 5.5 Linear combination ¶
      - depends_on definition 5.7 Vector subspace ¶
      - proves proof ch:03-linear-algebra-representations@proof-3 ¶
    - depends_on definition 5.14 Linear independence ¶
      - depends_on definition 5.5 Linear combination ¶ ↺
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 $\R$ ¶ ↺

Supports

depends_on corollary A.313 Stokes' theorem for an antisymmetric tensor field ¶
depends_on definition 30.17 Traction ¶
- depends_on proposition 30.19 The stress tensor is symmetric ¶
  - depends_on definition 30.20 Pressure and deviatoric stress ¶
    - depends_on definition 30.31 The engineering moduli ¶
      - depends_on phenomenon 30.70 Hertzian contact ¶
      - depends_on phenomenon 30.60 Coulomb's law of torsion ¶
      - depends_on proposition 30.33 Conversion among the moduli ¶
      - depends_on proposition 30.51 Moment and curvature ¶
      - depends_on remark A.676 The third cubic constant is measured, and it is not small ¶
      - depends_on theorem A.695 Boussinesq's point-force solution, quoted ¶
    - depends_on definition 30.79 Maxwell viscoelasticity ¶
      - depends_on proposition 30.80 A Maxwell material is a solid or a fluid depending on how long you watch ¶
    - depends_on proposition 30.74 The maximum shear stress, and its indifference to pressure ¶
      - depends_on phenomenon 30.73 Yield and brittle fracture ¶
      - depends_on remark A.705 The dangerous stress is below the surface ¶
      - depends_on remark 30.75 Two criteria, one uncited ¶
    - depends_on remark 30.75 Two criteria, one uncited ¶ ↺
- depends_on theorem 30.18 Cauchy: the stress tensor exists ¶
  - depends_on definition 30.24 The elasticity tensor ¶
    - depends_on definition 30.37 Voigt notation ¶
      - depends_on remark 30.39 The rari-constant controversy, decided by measurement ¶
    - depends_on phenomenon 30.28 Hooke's law ¶
      - depends_on phenomenon 30.32 Poisson contraction ¶
      - depends_on remark 30.78 Anharmonicity: what the quadratic energy also suppresses ¶
    - depends_on proposition 30.25 Twenty-one constants ¶
      - depends_on definition 30.37 Voigt notation ¶ ↺
      - depends_on proposition 30.38 The Cauchy relations ¶
      - depends_on theorem A.675 Three constants for a cubic crystal, two for an isotropic solid ¶
    - depends_on proposition 30.38 The Cauchy relations ¶ ↺
    - depends_on remark 30.78 Anharmonicity: what the quadratic energy also suppresses ¶ ↺
    - depends_on theorem A.675 Three constants for a cubic crystal, two for an isotropic solid ¶ ↺
  - depends_on definition 31.1 Fluid ¶
    - depends_on definition 31.21 Ideal fluid ¶
      - depends_on theorem 31.22 Euler's equations of motion ¶
    - depends_on definition 31.2 Knudsen number ¶
      - depends_on remark 31.3 When the continuum picture is allowed ¶
    - depends_on definition 31.32 Newtonian fluid; dynamic and kinematic viscosity ¶
      - depends_on remark 31.35 Non-Newtonian fluids are out of scope ¶
      - depends_on remark 31.33 Reading the two viscosities ¶
      - depends_on theorem 31.36 The Newtonian constitutive relation ¶
    - depends_on proposition 31.16 Hydrostatic balance ¶
      - depends_on example 31.19 The barometric distribution ¶
      - depends_on phenomenon 49.8 The white-dwarf luminosity function has an edge ¶
      - depends_on phenomenon 31.18 Archimedes' principle ¶
      - depends_on proposition 52.1 Hydrostatic equilibrium of a self-gravitating sphere ¶
  - depends_on proposition 30.19 The stress tensor is symmetric ¶ ↺
  - depends_on proposition 30.74 The maximum shear stress, and its indifference to pressure ¶ ↺
  - depends_on remark 30.72 Stress concentration ¶
    - depends_on proposition 30.76 Griffith's criterion ¶
      - depends_on example 30.77 Why glass is weak, and thin glass is not ¶
      - depends_on phenomenon 30.73 Yield and brittle fracture ¶ ↺
  - depends_on theorem A.695 Boussinesq's point-force solution, quoted ¶ ↺
  - depends_on theorem 30.21 Cauchy's equation of motion ¶
    - depends_on definition 30.22 Boundary conditions of elastostatics ¶
      - depends_on proposition 30.50 Love waves need a slow surface layer ¶
      - depends_on proposition 30.48 Rayleigh's secular equation ¶
      - depends_on proposition 30.23 Virtual work; the weak form ¶
    - depends_on proposition 30.23 Virtual work; the weak form ¶ ↺
    - depends_on proposition 31.16 Hydrostatic balance ¶ ↺
    - depends_on remark 30.69 Saint-Venant's principle, and its exact status ¶
    - depends_on theorem 30.40 Navier–Cauchy equations ¶
      - depends_on phenomenon 30.43 Two bulk wave speeds ¶
      - depends_on proposition 30.41 Energy balance and its flux ¶
      - depends_on proposition 30.50 Love waves need a slow surface layer ¶ ↺
      - depends_on proposition 30.48 Rayleigh's secular equation ¶ ↺
      - depends_on remark 30.46 Why the plane-wave route, and not the potentials ¶
    - depends_on theorem 31.22 Euler's equations of motion ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

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partly declared
a check failed
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Edges

type	direction	node	provenance	where
`depends_on`	→	Functions of class $C^{1}$	declared	parts/02-mathematical-methods/05-real-analysis.tex:4215
`depends_on`	→	Multiple integral	declared	parts/02-mathematical-methods/05-real-analysis.tex:4215
`depends_on`	←	Stokes' theorem for an antisymmetric tensor field	declared	appendices/A-long-proofs.tex:15431
`depends_on`	←	Traction	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:543