proposition 7.30 Leibniz rule

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:557 · p. 213

Rests on

depends_on definition 7.12 Product of functions ¶
- 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 3.28 Set ¶
depends_on equation 7.14 eq:ana-derivh ¶
depends_on proposition 7.27 Differentiable implies continuous ¶
- depends_on definition 7.20 Continuity 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 equation 7.7 eq:ana-limit-left ¶
  - depends_on equation 7.5 eq:ana-limit-right ¶
- 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 proposition 7.6 Algebra of limits ¶
  - depends_on definition 7.4 Convergence ¶
    - depends_on definition 7.2 Absolute value ¶ ↺
  - depends_on proposition 7.3 Triangle inequality ¶
    - depends_on definition 7.2 Absolute value ¶ ↺
    - proves proof ch:05-real-analysis@proof-1 ¶
  - proves proof ch:05-real-analysis@proof-3 ¶
- proves proof ch:05-real-analysis@proof-11 ¶
depends_on proposition 7.6 Algebra of limits ¶ ↺
proves proof ch:05-real-analysis@proof-14 ¶

Supports

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 proposition 21.56 The two prescriptions and their difference ¶
  - depends_on theorem 16.22 Euler–Lagrange ¶
    - depends_on definition 16.62 Field of extremals; slope function ¶
      - depends_on definition 23.52 Lagrangian family; caustic ¶
      - depends_on remark 23.8 The action as a function, not a functional ¶
      - depends_on theorem 16.63 Hilbert's invariant integral ¶
      - depends_on theorem 16.64 Weierstrass' sufficient condition ¶
    - depends_on phenomenon 30.52 Euler buckling ¶
      - depends_on example 30.53 A steel rod buckles at a twentieth of its crushing load ¶
      - depends_on remark 30.54 Other end conditions, and what buckling is variationally ¶
      - depends_on remark 30.55 Beyond the critical load: the elastica ¶
      - depends_on remark 30.68 Shells carry load in a different way ¶
    - 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 remark 30.67 Germain, Kirchhoff, and the edge conditions ¶
      - depends_on theorem A.691 Kirchhoff's edge conditions and the corner force ¶
      - depends_on theorem 16.32 Transversality ¶
  - 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 example 30.58 A steel ruler ¶ ↺
      - depends_on remark 30.59 Where Euler–Bernoulli fails ¶
  - 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 phenomenon 30.65 Chladni figures ¶ ↺
    - 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 44.31 de Sitter geometry ¶
- 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 proposition 32.26 The Hopf normal form ¶
  - depends_on theorem 32.15 Hartman–Grobman, restated from Part II ¶
    - depends_on definition 32.22 Bifurcation ¶
      - depends_on proposition 32.23 Saddle-node bifurcation ¶
      - depends_on proposition 32.24 Transcritical and pitchfork bifurcations ¶
      - depends_on theorem 32.25 Hopf bifurcation, quoted ¶ ↺
    - 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 proposition A.143 Reproduction identity ¶
    - depends_on lemma A.140 Properties of the mollifier ¶
      - depends_on proposition A.143 Reproduction identity ¶ ↺
      - depends_on proposition A.142 The mollification is smooth ¶
  - 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 definition A.467 Green function of the shifted problem ¶
      - depends_on theorem A.469 The Green operator inverts $L_{K}$ ¶
    - 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 theorem A.456 Convexity in $q$ implies weak lower semicontinuity ¶
- 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 17.96 Poles give asymptotics ¶
      - depends_on theorem 17.100 Moments diagonalise a convolution evolution ¶
- depends_on proposition 9.88 The model problem, and the error of the composite ¶ ↺
- … 2 more

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Edges

type	direction	node	provenance	where
`depends_on`	→	Product of functions	declared	parts/02-mathematical-methods/05-real-analysis.tex:562
`depends_on`	→	eq:ana-derivh	declared	parts/02-mathematical-methods/05-real-analysis.tex:562
`depends_on`	→	Differentiable implies continuous	declared	parts/02-mathematical-methods/05-real-analysis.tex:562
`depends_on`	→	Algebra of limits	declared	parts/02-mathematical-methods/05-real-analysis.tex:562
`depends_on`	←	Substitution and integration by parts	declared	parts/02-mathematical-methods/05-real-analysis.tex:875
`depends_on`	←	Euler's theorem on homogeneous functions	declared	parts/02-mathematical-methods/05-real-analysis.tex:3274
`depends_on`	←	Jacobi's formula, cofactor form	declared	parts/02-mathematical-methods/05-real-analysis.tex:3440
`depends_on`	←	Product identities of the nabla calculus	declared	parts/02-mathematical-methods/05-real-analysis.tex:4035
`depends_on`	←	Jacobi's formula, column form	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:430
`depends_on`	←	The scalar constant-coefficient equation	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1024
`depends_on`	←	Taylor's theorem with Lagrange remainder	declared	parts/02-mathematical-methods/05-real-analysis.tex:742
`proves`	←	ch:05-real-analysis@proof-14	declared	parts/02-mathematical-methods/05-real-analysis.tex:565