definition 7.13 Composite function

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:188 · p. 205

ground object -- no derivation owed

Rests on

depends_on definition 7.9 Real function ¶
- depends_on definition 3.43 Map ¶
  - depends_on definition 3.24 Quantifiers ¶
    - depends_on definition 3.23 Predicate ¶
      - depends_on definition 3.1 Logical proposition ¶
      - depends_on definition 3.38 Ordered pair and Cartesian product ¶
  - depends_on definition 3.28 Set ¶

Supports

depends_on definition 7.14 Inverse function ¶
- depends_on proposition 7.32 Derivative of the inverse function ¶
  - depends_on definition 9.140 Amplitude and the Jacobi elliptic functions ¶
    - depends_on proposition 9.143 A cubic quadrature between two turning points ¶
    - depends_on proposition 9.141 First properties ¶
      - depends_on proposition 9.143 A cubic quadrature between two turning points ¶ ↺
      - depends_on proposition 9.142 The differential equation of the sine amplitude ¶
    - depends_on proposition 9.144 The pendulum equation at finite amplitude ¶
  - depends_on proposition 7.61 The logarithm ¶
    - depends_on corollary 7.69 $\ee$ is the unique self-reproducing base ¶
      - depends_on remark 7.70 What the condition says ¶
    - depends_on definition 7.62 Real powers ¶
      - depends_on definition 7.90 The golden ratio ¶
      - depends_on proposition 7.63 Laws of real powers ¶
      - depends_on proposition 7.89 Irrationality of square roots ¶
    - depends_on lemma 7.67 The limit over real exponents ¶
      - depends_on proposition 7.68 The natural base ¶
    - depends_on proposition 7.68 The natural base ¶ ↺
    - depends_on proposition 7.63 Laws of real powers ¶ ↺
  - depends_on proposition 9.141 First properties ¶ ↺
  - depends_on proposition 9.19 Separation of variables ¶
depends_on proposition 7.31 Chain rule ¶
- depends_on corollary 7.44 Substitution and integration by parts ¶
  - depends_on definition 8.8 Contour integral ¶
  - depends_on lemma 16.20 Mixed form ¶
    - depends_on theorem 16.35 System of Euler–Lagrange equations ¶
      - depends_on proposition 16.6 The isoperimetric extremal is a circle ¶
      - depends_on theorem 16.41 Multiplier rule for pointwise constraints ¶
    - depends_on theorem 16.22 Euler–Lagrange ¶
      - depends_on definition 16.62 Field of extremals; slope function ¶
      - depends_on phenomenon 30.52 Euler buckling ¶
      - depends_on proposition 29.39 Steady precession ¶
      - depends_on theorem 16.35 System of Euler–Lagrange equations ¶ ↺
      - depends_on theorem 16.31 Natural boundary condition ¶
    - depends_on theorem 16.43 Euler's rule for integral constraints ¶
      - depends_on example 16.45 Two solved isoperimetric problems ¶
      - depends_on proposition 16.4 The hanging chain is a catenary ¶
      - depends_on proposition 16.6 The isoperimetric extremal is a circle ¶ ↺
  - depends_on lemma 9.136 Wallis integrals ¶
    - depends_on proposition 9.139 Series for the complete integral of the first kind ¶
      - depends_on proposition 9.144 The pendulum equation at finite amplitude ¶ ↺
  - depends_on proposition 7.145 Delta as a limit; elementary properties ¶
  - depends_on proposition 16.50 The second variation ¶
  - depends_on proposition 5.95 The focal polar equation of a conic ¶
  - depends_on proposition 9.143 A cubic quadrature between two turning points ¶ ↺
  - depends_on proposition 9.144 The pendulum equation at finite amplitude ¶ ↺
  - depends_on theorem 16.38 Euler–Poisson equation ¶
    - depends_on theorem 30.56 The beam equation ¶
      - depends_on example 30.58 A steel ruler ¶
      - depends_on proposition 30.57 Bending waves are dispersive ¶
    - depends_on theorem 30.64 The Kirchhoff plate equation ¶
      - depends_on phenomenon 30.65 Chladni figures ¶
      - depends_on proposition 30.66 Chladni scaling ¶
      - depends_on remark 30.67 Germain, Kirchhoff, and the edge conditions ¶
- depends_on lemma A.479 Conformal invariance of the Dirichlet integral ¶
  - depends_on lemma A.487 The normalisation costs nothing ¶
    - depends_on theorem A.492 Douglas' conformality theorem; quoted ¶
  - depends_on lemma A.485 Conformal automorphisms of the disc ¶
    - depends_on definition A.486 The normalised admissible class ¶
    - depends_on lemma A.487 The normalisation costs nothing ¶ ↺
- depends_on lemma 27.15 Velocity and acceleration in plane polar coordinates ¶
- depends_on proposition 7.111 Jacobi's formula, cofactor form ¶
- depends_on proposition 7.63 Laws of real powers ¶ ↺
- depends_on proposition 16.26 Covariance under change of dependent variable ¶
- depends_on proposition 10.15 Method of characteristics, first order ¶
  - depends_on definition 10.93 Scalar conservation law ¶
    - depends_on definition A.150 Kruzhkov entropy pair ¶
      - depends_on lemma A.157 Two elementary identities ¶
      - depends_on proposition A.156 The Kruzhkov inequality ¶
      - depends_on theorem A.151 $L^{1}$ contraction on a cone ¶
    - depends_on definition 10.95 Weak solution of a conservation law ¶
      - depends_on definition 10.99 Entropy condition ¶
      - depends_on definition 10.101 Entropy pair; entropy inequality ¶
      - depends_on example 10.98 Weak solutions are not unique ¶
      - depends_on theorem A.151 $L^{1}$ contraction on a cone ¶ ↺
      - depends_on theorem 10.103 Uniqueness in the entropy class ¶
      - depends_on theorem 10.96 Rankine–Hugoniot condition ¶
    - depends_on example 10.97 Burgers shock ¶
    - depends_on proposition 10.94 Gradient catastrophe ¶
  - depends_on proposition 10.16 Cauchy's characteristic strips ¶
  - depends_on proposition 10.94 Gradient catastrophe ¶ ↺
  - depends_on remark 31.90 Why a compression wave must steepen ¶
  - depends_on remark 23.3 What Part II owes this chapter ¶
- depends_on theorem 16.29 Beltrami identity ¶
  - depends_on proposition 16.105 The optical invariant in a stratified medium ¶
  - depends_on proposition 16.39 Ostrogradsky instability ¶
- depends_on theorem 10.8 The type is a coordinate invariant ¶
  - depends_on lemma A.90 Affine normalization ¶
    - depends_on lemma A.91 Smallness after rescaling ¶
  - depends_on proposition 10.13 Canonical form in two variables ¶
    - depends_on theorem A.83 Local canonical form for an elliptic operator with Hölder coefficients ¶
    - depends_on theorem 10.55 d'Alembert's formula ¶
      - depends_on corollary 10.56 Domain of dependence and influence ¶
      - depends_on theorem 10.59 Kirchhoff's formula ¶
  - depends_on proposition 10.11 What a characteristic surface is ¶
    - depends_on corollary 10.12 Characteristics of the three canonical operators ¶
      - depends_on corollary 10.60 Huygens' principle in three space dimensions ¶
    - depends_on definition A.102 Normal Cauchy problem of order $k$ ¶
      - depends_on definition A.103 First-order quasilinear system with zero data ¶
      - depends_on lemma A.105 Formal reduction of a normal problem ¶
    - depends_on theorem 10.22 Cauchy–Kovalevskaya ¶
depends_on proposition 7.104 Chain rule in several variables ¶
- depends_on corollary 7.113 Inverse function theorem ¶
  - depends_on definition A.503 The substitution property ¶
    - depends_on lemma A.506 Locality ¶
      - depends_on proposition A.511 The substitution property is universal ¶
    - depends_on lemma A.504 Transitivity ¶
  - depends_on lemma 7.116 Functions vanishing on a regular zero set ¶
  - depends_on lemma A.510 Every diffeomorphism factorises locally ¶
    - depends_on proposition A.511 The substitution property is universal ¶ ↺
  - depends_on remark 7.114 Where the proof is, and why ¶
  - depends_on theorem 7.129 Change of variables in a multiple integral ¶
    - depends_on example 7.130 The two Jacobians this treatise uses ¶
      - depends_on lemma A.518 The excised ball ¶
    - depends_on lemma A.518 The excised ball ¶ ↺
    - depends_on proposition A.519 The convolution exists ¶
      - depends_on proposition A.524 Rate of decay ¶
  - depends_on theorem 7.115 Constant rank ¶
- depends_on corollary 7.110 Variable limits of integration ¶
- depends_on corollary A.292 Inverse function theorem ¶
  - depends_on lemma A.549 The action is transitive ¶
    - depends_on proposition A.552 The component is a torus ¶
      - depends_on corollary A.553 Quasi-periodic motion ¶
      - depends_on definition A.554 Actions on the torus ¶
      - depends_on proposition A.556 The Jacobian of the actions is the lattice matrix ¶
  - depends_on lemma A.300 The boundary is well defined, and is a manifold ¶
    - depends_on definition A.309 The induced orientation of the boundary ¶
      - depends_on theorem A.311 General Stokes theorem ¶
  - depends_on proposition A.535 Existence of a slice ¶
    - depends_on lemma A.536 A slice is a chart domain downstairs ¶
      - depends_on theorem A.537 The smooth structure on the orbit space ¶
    - depends_on remark A.542 Where each hypothesis is spent ¶
  - depends_on proposition 22.32 Duality of the two brackets ¶
    - depends_on remark 22.33 What the duality is good for ¶
  - depends_on proposition 22.3 Invertibility of the Legendre map ¶
    - depends_on definition 26.3 Singular Lagrangian ¶
      - depends_on proposition 26.5 Primary constraints ¶
    - depends_on remark 22.4 When the Legendre map is singular ¶
  - depends_on proposition 13.132 Simultaneous straightening of commuting fields ¶
    - depends_on theorem 13.133 Frobenius ¶
      - depends_on corollary 13.134 Frobenius for a Pfaffian system ¶
      - depends_on proposition A.547 The components of the level set are the leaves of an integrable distribution ¶
      - depends_on theorem A.545 Liouville–Arnold ¶
    - depends_on theorem 13.137 Commuting complete fields on a compact manifold ¶
  - depends_on remark 22.19 Which form exists ¶
  - depends_on theorem 13.63 Constant rank theorem ¶
    - depends_on corollary 13.64 The image of a constant-rank map, locally ¶
    - depends_on lemma A.533 The orbit map has constant rank $d$ ¶
      - depends_on proposition A.534 Every orbit is an embedded copy of $G$ ¶
      - depends_on proposition A.535 Existence of a slice ¶ ↺
    - depends_on lemma A.538 Submersions have smooth local sections ¶
      - depends_on proposition A.539 Universal property ¶
    - depends_on theorem 13.67 Quotient manifold theorem ¶
      - depends_on example 13.68 Why each hypothesis is there ¶
      - depends_on proposition A.566 The isotropy group acts, and the quotient is smooth ¶
      - depends_on theorem A.563 Marsden–Weinstein reduction ¶
- depends_on definition 30.4 Material and spatial descriptions ¶
- depends_on definition 31.4 Material derivative ¶
  - 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 phenomenon 31.88 Sound travels at the adiabatic speed ¶
      - depends_on proposition 52.24 The p-mode cavity ¶
      - depends_on theorem 31.91 Rankine–Hugoniot jump conditions ¶
    - 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 22.26 Lagrange bracket ¶
- depends_on definition 10.18 Complete integral; envelope ¶
  - depends_on proposition 10.19 An envelope of solutions is a solution ¶
- depends_on lemma A.510 Every diffeomorphism factorises locally ¶ ↺
- depends_on lemma A.504 Transitivity ¶ ↺
- depends_on lemma A.306 The standard smooth bump ¶
  - depends_on lemma A.307 Partition of unity on a compact manifold ¶
    - depends_on definition A.308 Integral over the manifold ¶
      - depends_on theorem A.311 General Stokes theorem ¶ ↺
- depends_on proposition 7.136 Properties of conservative fields ¶
  - 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 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 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 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 ¶
- depends_on proposition 7.120 The envelope touches every member it meets ¶
- … 9 more

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Edges

type	direction	node	provenance	where
`depends_on`	→	Real function	declared	parts/02-mathematical-methods/05-real-analysis.tex:195
`depends_on`	←	Inverse function	declared	parts/02-mathematical-methods/05-real-analysis.tex:205
`depends_on`	←	Chain rule	declared	parts/02-mathematical-methods/05-real-analysis.tex:589
`depends_on`	←	Chain rule in several variables	declared	parts/02-mathematical-methods/05-real-analysis.tex:3105