proposition 7.31 Chain rule

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:584 · p. 213

Rests on

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 3.23 Predicate ¶
    - depends_on definition 3.28 Set ¶
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-15 ¶

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 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 definition 7.90 The golden ratio ¶
  - depends_on proposition 7.91 The golden ratio ¶
    - depends_on proposition 7.92 Continued fraction and the Fibonacci ratios ¶
      - depends_on remark 7.95 Where $\varphi$ is used ¶
    - depends_on proposition 125.2 Crystallographic restriction ¶
      - depends_on remark 125.3 What the theorem does and does not forbid ¶
    - depends_on remark 7.95 Where $\varphi$ is used ¶ ↺
    - depends_on remark 7.94 Three numbers, two kinds ¶
- depends_on lemma 7.67 The limit over real exponents ¶
  - depends_on proposition 7.68 The natural base ¶
    - depends_on remark 7.70 What the condition says ¶
- depends_on proposition 7.68 The natural base ¶ ↺
- depends_on proposition 7.91 The golden ratio ¶ ↺
- depends_on proposition 7.89 Irrationality of square roots ¶
  - depends_on proposition 7.91 The golden ratio ¶ ↺
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.159 The doubled inequality ¶
      - depends_on proposition A.162 The limit $\delta\to0$ ¶
    - depends_on proposition A.156 The Kruzhkov inequality ¶
      - depends_on proposition A.159 The doubled inequality ¶ ↺
    - depends_on theorem A.151 $L^{1}$ contraction on a cone ¶
      - depends_on corollary A.152 Uniqueness ¶
  - 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 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 definition 10.99 Entropy condition ¶ ↺
      - depends_on example 10.97 Burgers shock ¶
      - depends_on example 10.98 Weak solutions are not unique ¶ ↺
      - depends_on proposition 10.102 Jump form of the entropy inequality ¶
  - 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 corollary 10.60 Huygens' principle in three space dimensions ¶
- 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.104 The recursion ¶
      - depends_on proposition A.110 Explicit solution of the majorant problem ¶
    - depends_on lemma A.105 Formal reduction of a normal problem ¶
  - depends_on theorem 10.22 Cauchy–Kovalevskaya ¶

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`	→	Composite function	declared	parts/02-mathematical-methods/05-real-analysis.tex:589
`depends_on`	→	Differentiable implies continuous	declared	parts/02-mathematical-methods/05-real-analysis.tex:589
`depends_on`	→	Algebra of limits	declared	parts/02-mathematical-methods/05-real-analysis.tex:589
`depends_on`	←	Substitution and integration by parts	declared	parts/02-mathematical-methods/05-real-analysis.tex:875
`depends_on`	←	Conformal invariance of the Dirichlet integral	declared	appendices/A-long-proofs.tex:23548
`depends_on`	←	Velocity and acceleration in plane polar coordinates	declared	parts/03-classical-mechanics/10-central-forces-statics.tex:419
`depends_on`	←	Jacobi's formula, cofactor form	declared	parts/02-mathematical-methods/05-real-analysis.tex:3440
`depends_on`	←	Laws of real powers	declared	parts/02-mathematical-methods/05-real-analysis.tex:1449
`depends_on`	←	Covariance under change of dependent variable	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:746
`depends_on`	←	Method of characteristics, first order	declared	parts/02-mathematical-methods/08-pdes.tex:435
`depends_on`	←	Beltrami identity	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:819
`depends_on`	←	The type is a coordinate invariant	declared	parts/02-mathematical-methods/08-pdes.tex:190
`proves`	←	ch:05-real-analysis@proof-15	declared	parts/02-mathematical-methods/05-real-analysis.tex:592