theorem 7.51 Termwise differentiation

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:1008 · p. 218

Rests on

depends_on proposition 7.47 Comparison; absolute convergence ¶
- depends_on corollary A.46 Monotone convergence ¶
  - depends_on theorem A.40 Least-upper-bound property ¶
    - depends_on definition A.36 Cut ¶
    - depends_on definition A.38 Order on $\R$ ¶
      - depends_on definition A.36 Cut ¶ ↺
    - proves proof app:A-long-proofs@proof-29 ¶
  - proves proof app:A-long-proofs@proof-33 ¶
- depends_on definition 7.45 Series ¶
  - depends_on definition 7.2 Absolute value ¶
  - depends_on definition 7.4 Convergence ¶
    - depends_on definition 7.2 Absolute value ¶ ↺
- depends_on theorem 7.8 Cauchy criterion ¶
  - depends_on corollary A.47 Cauchy completeness ¶
    - depends_on corollary A.46 Monotone convergence ¶ ↺
    - depends_on theorem A.40 Least-upper-bound property ¶ ↺
    - proves proof app:A-long-proofs@proof-34 ¶
  - depends_on definition 7.4 Convergence ¶ ↺
  - proves proof ch:05-real-analysis@proof-5 ¶
- proves proof ch:05-real-analysis@proof-28 ¶
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 definition 3.43 Map ¶
  - 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 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 7.48 Ratio test ¶
- depends_on proposition 7.47 Comparison; absolute convergence ¶ ↺
- depends_on proposition 7.46 Geometric series ¶
  - depends_on corollary A.46 Monotone convergence ¶ ↺
  - depends_on definition 7.45 Series ¶ ↺
  - proves proof ch:05-real-analysis@proof-27 ¶
- depends_on theorem 7.8 Cauchy criterion ¶ ↺
- proves proof ch:05-real-analysis@proof-29 ¶
depends_on proposition 7.6 Algebra of limits ¶ ↺
depends_on theorem 7.50 Power series; radius of convergence ¶
- depends_on axiom 7.1 Completeness of $\R$ ¶
- depends_on proposition 7.47 Comparison; absolute convergence ¶ ↺
- depends_on proposition 7.46 Geometric series ¶ ↺
- proves proof ch:05-real-analysis@proof-31 ¶
proves proof ch:05-real-analysis@proof-32 ¶

Supports

depends_on definition 7.74 $\pi$ ¶
- depends_on lemma 7.75 The first quadrant ¶
  - depends_on proposition 7.76 $\pi$ as the circle constant ¶
  - depends_on proposition 7.77 Special values, periodicity, and the kernel ¶
    - depends_on lemma 7.78 Chord, arc, tangent ¶
      - depends_on proposition 7.79 The polygon recursion ¶
    - depends_on proposition 7.85 Irrationality of $\pi$ ¶
      - depends_on remark 7.94 Three numbers, two kinds ¶
      - depends_on remark 7.87 Priority, and what irrationality does not give ¶
    - depends_on proposition 7.79 The polygon recursion ¶ ↺
    - depends_on proposition 7.83 Viète's product ¶
      - depends_on remark 7.84 One construction, two faces ¶
    - depends_on proposition 125.2 Crystallographic restriction ¶
      - depends_on remark 125.3 What the theorem does and does not forbid ¶
    - depends_on remark 7.88 $\pi$ elsewhere in this treatise ¶
  - depends_on proposition 125.2 Crystallographic restriction ¶ ↺
- depends_on proposition 7.76 $\pi$ as the circle constant ¶ ↺
- depends_on proposition 7.79 The polygon recursion ¶ ↺
- depends_on proposition 7.77 Special values, periodicity, and the kernel ¶ ↺
- depends_on proposition 125.2 Crystallographic restriction ¶ ↺
depends_on lemma 7.93 A polynomial has at most $n$ roots ¶
depends_on lemma 7.71 Derivatives; the Pythagorean identity ¶
- depends_on lemma 7.75 The first quadrant ¶ ↺
- depends_on lemma 7.78 Chord, arc, tangent ¶ ↺
- 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 7.76 $\pi$ as the circle constant ¶ ↺
- depends_on proposition 7.85 Irrationality of $\pi$ ¶ ↺
- depends_on proposition 125.2 Crystallographic restriction ¶ ↺
- 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 remark 7.80 Why these are the polygon perimeters ¶
depends_on lemma 7.75 The first quadrant ¶ ↺
depends_on proposition 9.22 The exponential and its derivative ¶
- depends_on theorem 9.23 Solution of a constant-coefficient system ¶
  - depends_on corollary 9.25 Normal modes of a diagonalizable system ¶
    - depends_on corollary 9.27 Linear stability ¶
      - depends_on corollary 9.41 Hill's equation ¶
      - depends_on proposition 9.31 Lyapunov's first method: asymptotic stability ¶
      - depends_on proposition 9.29 The planar classification ¶
      - depends_on theorem 9.39 Floquet ¶
    - depends_on proposition 9.29 The planar classification ¶ ↺
  - depends_on proposition 9.87 The distinguished limit, and which end carries the layer ¶
    - depends_on proposition 9.88 The model problem, and the error of the composite ¶
  - depends_on proposition 9.31 Lyapunov's first method: asymptotic stability ¶ ↺
  - depends_on proposition 9.43 The unmodulated line, and the feet of the tongues ¶
  - depends_on theorem 9.32 Hartman–Grobman; 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 9.26 Structure of the solutions ¶
    - depends_on corollary 9.27 Linear stability ¶ ↺

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`	→	Comparison; absolute convergence	declared	parts/02-mathematical-methods/05-real-analysis.tex:1018
`depends_on`	→	Differentiable implies continuous	declared	parts/02-mathematical-methods/05-real-analysis.tex:1018
`depends_on`	→	Ratio test	declared	parts/02-mathematical-methods/05-real-analysis.tex:1018
`depends_on`	→	Algebra of limits	declared	parts/02-mathematical-methods/05-real-analysis.tex:1018
`depends_on`	→	Power series; radius of convergence	declared	parts/02-mathematical-methods/05-real-analysis.tex:1018
`depends_on`	←	$\pi$	declared	parts/02-mathematical-methods/05-real-analysis.tex:1947
`depends_on`	←	A polynomial has at most $n$ roots	declared	parts/02-mathematical-methods/05-real-analysis.tex:2790
`depends_on`	←	Derivatives; the Pythagorean identity	declared	parts/02-mathematical-methods/05-real-analysis.tex:1765
`depends_on`	←	The first quadrant	declared	parts/02-mathematical-methods/05-real-analysis.tex:1976
`depends_on`	←	The exponential and its derivative	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:753
`proves`	←	ch:05-real-analysis@proof-32	declared	parts/02-mathematical-methods/05-real-analysis.tex:1021