theorem 7.42 Fundamental theorem of calculus, I

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:828 · p. 216

Rests on

depends_on definition 7.26 Derivative of a function 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 definition 3.24 Quantifiers ¶
      - depends_on definition 3.28 Set ¶
- depends_on definition 7.9 Real function ¶ ↺
depends_on theorem 7.40 Continuous functions are integrable ¶
- depends_on definition 7.39 Darboux sums and the definite integral ¶
  - depends_on axiom 7.1 Completeness of $\R$ ¶
- depends_on theorem 7.24 Extreme value theorem ¶
  - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
  - depends_on proposition 7.22 Sequential characterization ¶
    - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.16 Limit ¶ ↺
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
    - depends_on definition 7.4 Convergence ¶
      - depends_on definition 7.2 Absolute value ¶ ↺
    - proves proof ch:05-real-analysis@proof-7 ¶
  - depends_on theorem 7.7 Bolzano–Weierstrass ¶
    - depends_on corollary A.45 Archimedean property and density of $\Q$ ¶
      - depends_on definition A.36 Cut ¶
      - depends_on theorem A.40 Least-upper-bound property ¶
      - proves proof app:A-long-proofs@proof-32 ¶
    - depends_on corollary A.46 Monotone convergence ¶
      - depends_on theorem A.40 Least-upper-bound property ¶ ↺
      - proves proof app:A-long-proofs@proof-33 ¶
    - proves proof ch:05-real-analysis@proof-4 ¶
  - proves proof ch:05-real-analysis@proof-9 ¶
- depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
  - depends_on proposition 7.22 Sequential characterization ¶ ↺
  - depends_on theorem 7.7 Bolzano–Weierstrass ¶ ↺
  - proves proof ch:05-real-analysis@proof-10 ¶
- proves proof ch:05-real-analysis@proof-23 ¶
depends_on theorem 7.24 Extreme value theorem ¶ ↺
proves proof ch:05-real-analysis@proof-24 ¶

Supports

depends_on corollary 7.110 Variable limits of integration ¶
depends_on definition A.450 The space $W^{1,r}(a,b)$ ¶
- depends_on definition A.462 The weighted space and the energy space ¶
  - 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.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.453 Uniform bound and uniform Hölder continuity ¶ ↺
depends_on lemma A.520 A $C^{1}$ limit ¶
- depends_on proposition A.521 Every derivative passes onto $g$ ¶
  - depends_on proposition A.524 Rate of decay ¶
  - depends_on proposition A.522 The Newtonian potential inverts the Laplacian ¶
    - depends_on theorem A.523 Helmholtz decomposition: existence ¶
depends_on lemma A.440 Grönwall's inequality ¶
- depends_on theorem A.441 Continuous dependence on the data and on the initial point ¶
  - depends_on corollary A.445 Displacing the initial point ¶
depends_on lemma 16.19 du Bois-Reymond ¶
- 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 9.10 Grönwall's inequality ¶
- depends_on proposition 9.11 Continuous dependence on the initial data ¶
- depends_on proposition 9.31 Lyapunov's first method: asymptotic stability ¶
depends_on proposition 11.22 Properties of the distribution function ¶
- depends_on proposition 11.27 Probability integral transform ¶
depends_on proposition 9.18 The linear equation: integrating factor ¶
- 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 9.19 Separation of variables ¶
depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
- 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 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 corollary 16.23 du Bois-Reymond form ¶
  - depends_on corollary 16.33 Weierstrass–Erdmann corner conditions ¶
    - depends_on theorem 16.55 Jacobi's necessary condition ¶
      - depends_on proposition 23.53 A caustic point is a conjugate point ¶
      - depends_on remark 30.54 Other end conditions, and what buckling is variationally ¶ ↺
- depends_on lemma 7.116 Functions vanishing on a regular zero set ¶
- depends_on lemma A.72 Iterated integral inequality ¶
  - depends_on theorem A.74 Flow of a time-dependent vector field ¶
    - depends_on theorem 13.125 Existence, uniqueness and smoothness of the flow ¶
      - depends_on lemma A.548 The joint flow is a translation action on the level set ¶
      - depends_on lemma A.567 The differential of the momentum map along the orbit ¶
      - depends_on proposition 13.131 Commuting fields have commuting flows ¶
      - depends_on proposition 13.132 Simultaneous straightening of commuting fields ¶
      - depends_on theorem 13.137 Commuting complete fields on a compact manifold ¶
- depends_on lemma A.520 A $C^{1}$ limit ¶ ↺
- depends_on lemma A.172 Dirichlet ¶
  - depends_on lemma A.193 A cosine integral ¶
    - depends_on lemma A.194 The Fejér kernel and its transform ¶
      - depends_on theorem A.198 Esseen's smoothing inequality ¶
  - depends_on theorem A.177 Fourier inversion for a distribution function ¶
    - depends_on corollary A.178 The characteristic function determines the distribution ¶
      - depends_on lemma A.220 At each point, field and derivative are independent Gaussians ¶
    - depends_on lemma A.196 Inversion for a difference ¶
      - depends_on theorem A.198 Esseen's smoothing inequality ¶ ↺
- depends_on lemma A.221 Counting identity ¶
- depends_on lemma A.312 The computation in one chart ¶
- 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 8.10 Fundamental theorem for contours ¶
  - depends_on theorem 8.13 Goursat ¶
  - depends_on theorem 8.18 Liouville ¶
    - depends_on corollary A.723 Potential of one infinite row ¶
      - depends_on definition A.724 The staggered street ¶
      - depends_on proposition A.725 The street translates rigidly ¶
    - depends_on lemma A.608 An entire function with a quadratic bound on its real part ¶
      - depends_on proposition A.609 The exponent is a quadratic polynomial ¶
    - depends_on theorem 8.19 Fundamental theorem of algebra ¶
      - depends_on corollary 5.72 Existence of an eigenvalue over $\C$ ¶
      - depends_on lemma 15.12 Artin decomposition ¶
      - depends_on proposition 9.28 The scalar constant-coefficient equation ¶
    - depends_on theorem 12.54 The spectrum is compact and non-empty ¶
      - depends_on theorem 12.55 The spectrum of a self-adjoint operator is real ¶
- depends_on proposition 17.41 The identities physics uses ¶
  - depends_on proposition 17.90 Hankel transform as the axially symmetric Fourier transform ¶
  - depends_on proposition 17.52 Linear translation-invariant systems ¶
    - depends_on definition 28.25 Complex susceptibility ¶
    - depends_on proposition 17.61 Mean-square response of a damped resonator ¶
- depends_on proposition 17.73 Initial- and final-value theorems ¶
- … 4 more
depends_on theorem 10.52 Duhamel's principle ¶
- depends_on corollary 10.53 Duhamel for the wave equation ¶

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Derivative of a function at a point	declared	parts/02-mathematical-methods/05-real-analysis.tex:834
`depends_on`	→	Continuous functions are integrable	declared	parts/02-mathematical-methods/05-real-analysis.tex:834
`depends_on`	→	Extreme value theorem	declared	parts/02-mathematical-methods/05-real-analysis.tex:834
`depends_on`	←	Variable limits of integration	declared	parts/02-mathematical-methods/05-real-analysis.tex:3384
`depends_on`	←	The space $W^{1,r}(a,b)$	declared	appendices/A-long-proofs.tex:22181
`depends_on`	←	A $C^{1}$ limit	declared	appendices/A-long-proofs.tex:25405
`depends_on`	←	Grönwall's inequality	declared	appendices/A-long-proofs.tex:21604
`depends_on`	←	du Bois-Reymond	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:581
`depends_on`	←	Grönwall's inequality	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:402
`depends_on`	←	Properties of the distribution function	declared	parts/02-mathematical-methods/09-probability-statistics.tex:602
`depends_on`	←	The linear equation: integrating factor	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:616
`depends_on`	←	Separation of variables	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:656
`depends_on`	←	Fundamental theorem of calculus, II	declared	parts/02-mathematical-methods/05-real-analysis.tex:855
`depends_on`	←	Duhamel's principle	declared	parts/02-mathematical-methods/08-pdes.tex:1397
`proves`	←	ch:05-real-analysis@proof-24	declared	parts/02-mathematical-methods/05-real-analysis.tex:837