definition 13.5 Tensor under orthogonal transformations

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:305 · p. 453

ground object -- no derivation owed

Rests on

depends_on definition 13.4 Orthogonal coordinate transformation ¶
- depends_on definition 13.3 Orthogonal coordinate system ¶

Supports

depends_on definition 19.42 Inertia tensor ¶
- depends_on definition 29.13 Inertia tensor, continuum form ¶
  - depends_on definition 29.19 Inertia ellipsoid ¶
    - depends_on theorem 29.29 Poinsot's construction ¶
      - depends_on remark 29.32 What the linearization is entitled to conclude ¶
      - depends_on remark 29.30 Polhode and herpolhode ¶
  - depends_on example 29.23 The polar moment of the Earth, and what it reveals ¶
    - depends_on phenomenon 29.34 Free nutation of the Earth ¶
  - depends_on proposition 49.17 Magnetic-dipole spin-down ¶
  - depends_on proposition 29.14 Angular momentum and kinetic energy ¶
  - depends_on proposition 29.15 Positivity ¶
    - depends_on definition 29.19 Inertia ellipsoid ¶ ↺
    - depends_on theorem 29.16 Principal axes ¶
      - depends_on definition 29.19 Inertia ellipsoid ¶ ↺
      - depends_on definition 29.35 The Lagrange top ¶
      - depends_on definition 29.18 Classification of tops ¶
      - depends_on phenomenon 29.31 The intermediate-axis instability ¶
      - depends_on proposition 29.33 Dissipation drives a free body to its largest moment ¶
      - depends_on proposition 29.17 Triangle inequalities ¶
      - depends_on theorem 29.25 Euler's equations ¶
  - depends_on proposition 29.24 MacCullagh's formula ¶
    - depends_on phenomenon 29.49 Precession of the equinoxes ¶
      - depends_on proposition 29.50 The principal nutation ¶
  - depends_on proposition 29.22 Perpendicular-axis theorem ¶
  - depends_on theorem 29.20 Parallel-axis theorem ¶
    - depends_on corollary 29.21 The centre of mass minimizes the moment ¶
  - depends_on theorem 29.16 Principal axes ¶ ↺
- depends_on example 19.45 The uniform solid sphere ¶
- depends_on proposition 19.43 The moment of inertia is a quadratic form in the axis ¶
depends_on definition 29.13 Inertia tensor, continuum form ¶ ↺
depends_on lemma 30.26 Isotropic Cartesian tensors of rank four ¶
- depends_on phenomenon 30.28 Hooke's law ¶
  - depends_on phenomenon 30.32 Poisson contraction ¶
    - depends_on proposition 30.33 Conversion among the moduli ¶
      - depends_on definition 30.63 Flexural rigidity ¶
      - depends_on example 30.35 From single crystals to the isotropic aggregate ¶
      - depends_on proposition A.682 The plane-stress law ¶
      - depends_on proposition 30.44 The speed ratio is fixed by Poisson's ratio ¶
      - depends_on remark 30.36 How the moduli are actually measured ¶
  - depends_on remark 30.78 Anharmonicity: what the quadratic energy also suppresses ¶
depends_on notation 30.1 Strain and displacement share a letter ¶
depends_on proposition 30.7 Strain is a Cartesian tensor of rank two ¶
depends_on proposition 13.96 The cross product is an axial vector ¶
depends_on proposition 13.6 Levi-Civita identities in three dimensions ¶
- depends_on proposition 13.96 The cross product is an axial vector ¶ ↺
- depends_on proposition 13.95 The elementary cross-product identities ¶
depends_on proposition 19.43 The moment of inertia is a quadratic form in the axis ¶ ↺
depends_on remark 13.7 What the classification is used for ¶
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 proposition 30.25 Twenty-one constants ¶
    - depends_on definition 30.37 Voigt notation ¶ ↺
    - depends_on proposition 30.38 The Cauchy relations ¶
      - depends_on remark 30.39 The rari-constant controversy, decided by measurement ¶ ↺
    - depends_on theorem A.675 Three constants for a cubic crystal, two for an isotropic solid ¶
      - depends_on remark A.676 The third cubic constant is measured, and it is not small ¶
  - 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 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 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 theorem 31.64 Kelvin's circulation theorem ¶
      - depends_on theorem 31.37 The Navier–Stokes equations ¶
      - depends_on theorem 31.91 Rankine–Hugoniot jump conditions ¶
      - depends_on theorem 31.77 Rayleigh's inflexion-point criterion ¶
  - 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 theorem A.718 Stokes' drag law with its coefficient ¶
      - depends_on theorem 31.37 The Navier–Stokes equations ¶ ↺
  - 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 example 31.46 Terminal velocity, and Millikan's droplets ¶
    - depends_on proposition 52.1 Hydrostatic equilibrium of a self-gravitating sphere ¶
      - depends_on phenomenon 52.12 The mass–luminosity relation ¶
      - depends_on proposition 49.3 Lane–Emden form of hydrostatic equilibrium; restated from the stellar-structure chapter ¶
      - depends_on theorem 52.2 Virial theorem for a self-gravitating sphere ¶
- 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 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 30.49 The Rayleigh speed of a Poisson solid ¶
    - depends_on proposition 30.23 Virtual work; the weak form ¶
      - depends_on phenomenon 30.70 Hertzian contact ¶ ↺
      - depends_on proposition 30.34 Minimum of the potential energy ¶
      - depends_on remark 30.69 Saint-Venant's principle, and its exact status ¶
  - 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 phenomenon 30.47 The three seismic phases ¶
      - depends_on remark 30.46 Why the plane-wave route, and not the potentials ¶
    - depends_on proposition 30.41 Energy balance and its flux ¶
      - depends_on theorem 30.42 Uniqueness in elastodynamics ¶
    - 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

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Edges

type	direction	node	provenance	where
`depends_on`	→	Orthogonal coordinate transformation	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:312
`depends_on`	←	Inertia tensor	declared	parts/03-classical-mechanics/02-newtonian-dynamics.tex:1239
`depends_on`	←	Inertia tensor, continuum form	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:373
`depends_on`	←	Isotropic Cartesian tensors of rank four	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:813
`depends_on`	←	Strain and displacement share a letter	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:58
`depends_on`	←	Strain is a Cartesian tensor of rank two	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:215
`depends_on`	←	The cross product is an axial vector	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:4700
`depends_on`	←	Levi-Civita identities in three dimensions	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:520
`depends_on`	←	The moment of inertia is a quadratic form in the axis	declared	parts/03-classical-mechanics/02-newtonian-dynamics.tex:1252
`depends_on`	←	What the classification is used for	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:625
`depends_on`	←	Cauchy: the stress tensor exists	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:560