definition 7.12 Product of functions

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:178 · 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 proposition 7.30 Leibniz 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 proposition 7.108 Euler's theorem on homogeneous functions ¶
- depends_on proposition 7.111 Jacobi's formula, cofactor form ¶
- depends_on proposition 7.122 Product identities of the nabla calculus ¶
- depends_on proposition 5.2 Jacobi's formula, column form ¶
  - depends_on lemma 44.10 Divergence theorem on $(M,g)$ ¶
    - depends_on proposition 44.11 The boundary term, and the Gibbons–Hawking–York action ¶
  - depends_on lemma 44.7 Variation of inverse and determinant ¶
    - depends_on proposition 44.17 Stress–energy of the electromagnetic field ¶
      - depends_on proposition 44.38 The electromagnetic field satisfies NEC, WEC, SEC and DEC ¶
    - depends_on theorem 44.9 Variation of the Einstein–Hilbert action ¶
      - depends_on proposition 44.11 The boundary term, and the Gibbons–Hawking–York action ¶ ↺
      - depends_on proposition 44.30 The cosmological term ¶
  - depends_on proposition 7.111 Jacobi's formula, cofactor form ¶ ↺
- depends_on proposition 9.28 The scalar constant-coefficient equation ¶
  - depends_on proposition 9.88 The model problem, and the error of the composite ¶
- depends_on theorem 7.38 Taylor's theorem with Lagrange remainder ¶
  - depends_on definition 32.57 The Hénon–Heiles Hamiltonian ¶
  - depends_on definition 32.12 Linearization ¶
    - depends_on proposition 32.17 Classification of planar fixed points ¶
      - depends_on example 32.18 A linear centre that is really a stable focus ¶
      - depends_on theorem 32.25 Hopf bifurcation, quoted ¶
    - depends_on theorem 32.15 Hartman–Grobman, restated from Part II ¶
      - depends_on definition 32.22 Bifurcation ¶
      - depends_on example 32.18 A linear centre that is really a stable focus ¶ ↺
    - depends_on theorem 32.13 Linear stability ¶
      - depends_on proposition 32.65 Fixed points of the Lorenz system and their stability ¶
      - depends_on proposition 32.23 Saddle-node bifurcation ¶
      - depends_on proposition 32.24 Transcritical and pitchfork bifurcations ¶
      - depends_on proposition 28.69 Which branch is stable, and where it ends ¶
  - depends_on lemma A.754 Exact depth expansion ¶
    - depends_on proposition A.755 The Boussinesq pair ¶
      - depends_on theorem A.756 The Korteweg–de Vries equation ¶
  - depends_on lemma A.187 Exponentiation ¶
  - depends_on lemma A.138 The flat exponential ¶
    - depends_on definition A.139 The standard radial mollifier ¶
      - depends_on definition A.141 Mollification ¶
      - depends_on lemma A.140 Properties of the mollifier ¶
    - depends_on lemma A.140 Properties of the mollifier ¶ ↺
  - depends_on lemma A.452 Young and Hölder ¶
    - depends_on lemma A.489 Courant–Lebesgue ¶
      - depends_on theorem A.490 Equicontinuity of the normalised class ¶
    - depends_on lemma A.463 Poincaré inequality; $B_{K}$ is an inner product ¶
      - depends_on lemma A.464 $H_{E}$ is a Hilbert space ¶
      - depends_on lemma A.466 The homogeneous Dirichlet problem is trivial ¶
      - depends_on theorem A.469 The Green operator inverts $L_{K}$ ¶
    - depends_on lemma A.453 Uniform bound and uniform Hölder continuity ¶
      - depends_on theorem A.455 Compact embedding of $W^{1,r}(a,b)$ into the continuous functions ¶
  - depends_on lemma A.212 Second-order flatness ¶
    - depends_on lemma A.213 The three minima ¶
      - depends_on theorem A.214 Wilks, $k$ parameters and $r$ constraints ¶
  - depends_on lemma A.209 Uniform third-order remainder ¶
    - depends_on corollary A.210 Uniform quadratic approximation on the $n^{-1/2}$ scale ¶
      - depends_on lemma A.213 The three minima ¶ ↺
    - depends_on lemma A.211 Root-$n$ localisation ¶
      - depends_on lemma A.213 The three minima ¶ ↺
  - depends_on phenomenon 28.13 Universality of small oscillations ¶
  - depends_on proposition 16.50 The second variation ¶ ↺
  - depends_on proposition 17.67 Holomorphy ¶
    - depends_on theorem 17.78 Causality implies dispersion relations ¶
      - depends_on remark 33.10 Fibre anelasticity, and why it biases one method and not the other ¶
      - depends_on remark 28.26 Causality is visible in the susceptibility ¶
      - depends_on remark 28.62 Neither velocity is a speed limit ¶
    - depends_on theorem 17.93 Fundamental strip and holomorphy ¶
      - depends_on theorem 17.94 Mellin inversion ¶
  - depends_on proposition 9.88 The model problem, and the error of the composite ¶ ↺
  - … 2 more

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`	→	Real function	declared	parts/02-mathematical-methods/05-real-analysis.tex:185
`depends_on`	←	Leibniz rule	declared	parts/02-mathematical-methods/05-real-analysis.tex:562