definition 13.4 Orthogonal coordinate transformation

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:201 · p. 451

ground object -- no derivation owed

Rests on

depends_on definition 13.3 Orthogonal coordinate system ¶

Supports

depends_on definition 13.5 Tensor under orthogonal transformations ¶
- 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 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 proposition 29.24 MacCullagh's formula ¶
      - depends_on phenomenon 29.49 Precession of the equinoxes ¶
    - 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 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 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 definition 30.20 Pressure and deviatoric stress ¶
      - depends_on definition 30.31 The engineering moduli ¶
      - depends_on definition 30.79 Maxwell viscoelasticity ¶
      - depends_on proposition 30.74 The maximum shear stress, and its indifference to pressure ¶
      - 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 definition A.698 The Hertz contact problem ¶
      - depends_on lemma A.700 The displacement produced by an elliptic pressure ¶
      - depends_on proposition A.701 The two-body compliance rule ¶
      - depends_on remark A.699 The third condition is not decoration ¶
      - depends_on theorem A.702 Hertz's solution ¶
    - depends_on lemma A.697 Surface displacement under a distributed pressure ¶
      - depends_on lemma A.700 The displacement produced by an elliptic pressure ¶ ↺
      - depends_on proposition A.701 The two-body compliance rule ¶ ↺
  - 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 ¶ ↺

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Edges

type	direction	node	provenance	where
`depends_on`	→	Orthogonal coordinate system	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:211
`depends_on`	←	Tensor under orthogonal transformations	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:312