definition 31.4 Material derivative

open in the book · parts/03-classical-mechanics/14-fluid-dynamics.tex:107 · p. 1042

ground object -- no derivation owed

Rests on

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 ¶
proves proof ch:14-fluid-dynamics@proof-1 ¶

Supports

depends_on definition 31.6 Streamline, pathline, streakline ¶
- depends_on proposition 31.7 The three coincide in steady flow ¶
depends_on lemma 31.11 Transport theorem for a material volume ¶
- depends_on theorem 31.12 Conservation of mass ¶
  - depends_on corollary 31.13 Incompressible flow ¶
    - depends_on corollary A.743 Two scalar functions instead of a tensor field ¶
      - depends_on lemma A.748 The second-order structure function ¶
      - depends_on lemma A.747 The third-order structure functions ¶
      - depends_on proposition A.746 The Kármán–Howarth equation in classical form ¶
    - depends_on example 31.15 The stream function of a plane flow ¶
      - depends_on definition A.715 Stokes stream function ¶
    - depends_on lemma A.744 Vanishing of the pressure–velocity correlation ¶
      - depends_on theorem A.745 Kármán–Howarth–Monin relation ¶
    - 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 lemma A.733 The circle with incidence and circulation ¶
      - 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 remark 31.14 When incompressibility is legitimate ¶
  - depends_on phenomenon 31.88 Sound travels at the adiabatic speed ¶
    - depends_on remark 31.89 The Mach number and the regimes it separates ¶
      - depends_on phenomenon 31.93 The Mach cone ¶
    - depends_on remark 31.90 Why a compression wave must steepen ¶
  - depends_on proposition 52.24 The p-mode cavity ¶
    - depends_on phenomenon 52.25 The Sun rings, and its interior can be inverted ¶
      - depends_on phenomenon 52.22 The solar-neutrino deficit, and its resolution ¶
  - depends_on theorem 31.91 Rankine–Hugoniot jump conditions ¶
    - depends_on phenomenon 31.93 The Mach cone ¶ ↺
    - depends_on remark 31.92 Entropy selects the physical branch ¶
- 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 remark 31.81 Where the threshold comes from, and what it costs ¶
  - 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 corollary A.799 The impressed pressure gradient ¶
    - depends_on lemma 31.67 Blasius' force formula ¶
      - depends_on theorem 31.68 Kutta–Joukowski ¶
    - depends_on phenomenon 31.25 Pressure falls where the flow speeds up ¶
      - depends_on example 31.26 The Venturi meter ¶
      - depends_on remark 31.27 What the 1738 source contains ¶
    - depends_on theorem A.707 D'Alembert's paradox ¶ ↺
    - depends_on theorem 31.74 The interface dispersion relation ¶ ↺
  - depends_on theorem 31.64 Kelvin's circulation theorem ¶
    - depends_on corollary 31.65 Lagrange: an irrotational flow stays irrotational ¶
      - depends_on remark 31.69 The Kutta condition, and where the circulation comes from ¶
    - depends_on remark 31.66 Only three things make circulation ¶
    - depends_on theorem 31.59 Helmholtz's vortex theorems ¶
      - depends_on corollary 31.65 Lagrange: an irrotational flow stays irrotational ¶ ↺
      - depends_on example 31.60 Vortex rings ¶
  - depends_on theorem 31.37 The Navier–Stokes equations ¶
    - depends_on definition A.741 The statistical setting ¶
    - depends_on definition A.760 The convecting layer ¶
      - depends_on proposition A.761 The Boussinesq equations ¶
    - depends_on definition 31.43 Creeping flow ¶
      - depends_on lemma A.716 The creeping-flow equation for $\psi$ ¶
      - depends_on proposition 31.44 Kinematic reversibility ¶
    - depends_on phenomenon 31.78 Convective onset at a computed threshold ¶
      - depends_on proposition A.761 The Boussinesq equations ¶ ↺
      - depends_on remark 31.79 Bénard's own cells, and Taylor's cylinders ¶
      - depends_on remark 31.81 Where the threshold comes from, and what it costs ¶ ↺
      - depends_on theorem A.767 Marginal stability of the conducting state ¶
    - depends_on phenomenon 31.41 The fourth-power law of pipe flow ¶
      - depends_on phenomenon 31.53 Transition to turbulence in a pipe ¶
      - depends_on remark 31.42 The entry length ¶
    - depends_on phenomenon 31.49 Dynamical similarity ¶
      - depends_on definition 31.51 The other similarity groups ¶
      - depends_on phenomenon 31.53 Transition to turbulence in a pipe ¶ ↺
      - depends_on remark 31.50 Reading the Reynolds number ¶
      - depends_on remark 31.52 Similarity is only as good as the control of the data ¶
      - depends_on remark 31.62 What the spacing calculation settles, and what it does not ¶
    - depends_on proposition A.761 The Boussinesq equations ¶ ↺
    - depends_on proposition 31.40 Circular Couette flow ¶
      - depends_on proposition 31.80 Rayleigh's circulation criterion ¶ ↺
      - depends_on remark 31.79 Bénard's own cells, and Taylor's cylinders ¶ ↺
    - depends_on proposition 31.39 Plane Couette flow ¶
      - depends_on proposition 31.40 Circular Couette flow ¶ ↺
    - depends_on remark 31.48 What is actually proved ¶
      - depends_on remark 31.87 What a theory of turbulence would have to do ¶
    - depends_on remark 31.38 Whose equations, and the Stokes hypothesis ¶
    - depends_on theorem A.745 Kármán–Howarth–Monin relation ¶ ↺
    - … 3 more
  - depends_on theorem 31.91 Rankine–Hugoniot jump conditions ¶ ↺
  - depends_on theorem 31.77 Rayleigh's inflexion-point criterion ¶

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`	→	eq:ana-nabla-gradient	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:114
`depends_on`	→	Chain rule in several variables	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:114
`depends_on`	←	Streamline, pathline, streakline	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:183
`depends_on`	←	Transport theorem for a material volume	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:291
`proves`	←	ch:14-fluid-dynamics@proof-1	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:117