theorem 7.43 Fundamental theorem of calculus, II

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:850 · p. 217

Rests on

depends_on definition 7.41 Antiderivative ¶
- depends_on corollary 7.36 cor:ana-mvt-consequences ¶
  - depends_on theorem 7.35 Mean value theorem ¶
    - depends_on proposition 7.29 Linearity ¶
      - depends_on definition 7.11 Sum of functions ¶
      - depends_on equation 7.14 eq:ana-derivh ¶
      - depends_on proposition 7.6 Algebra of limits ¶
      - proves proof ch:05-real-analysis@proof-13 ¶
    - depends_on theorem 7.34 Rolle ¶
      - depends_on lemma 7.33 Fermat: interior extremum ¶
      - depends_on theorem 7.24 Extreme value theorem ¶
      - proves proof ch:05-real-analysis@proof-18 ¶
    - proves proof ch:05-real-analysis@proof-19 ¶
  - proves proof ch:05-real-analysis@proof-20 ¶
- 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 7.9 Real function ¶ ↺
depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
- depends_on definition 7.26 Derivative of a function at a point ¶ ↺
- 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 theorem 7.25 Heine–Cantor: uniform continuity ¶
    - depends_on proposition 7.22 Sequential characterization ¶
      - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.4 Convergence ¶
      - 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 corollary A.46 Monotone convergence ¶
      - proves proof ch:05-real-analysis@proof-4 ¶
    - 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 ¶
proves proof ch:05-real-analysis@proof-25 ¶

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 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 corollary A.553 Quasi-periodic motion ¶
      - depends_on lemma A.549 The action is transitive ¶
      - depends_on proposition A.552 The component is a torus ¶
    - depends_on lemma A.567 The differential of the momentum map along the orbit ¶
      - depends_on theorem A.568 The radical is the isotropy orbit ¶
    - depends_on proposition 13.131 Commuting fields have commuting flows ¶
      - depends_on lemma A.548 The joint flow is a translation action on the level set ¶ ↺
      - depends_on proposition 13.132 Simultaneous straightening of commuting fields ¶
      - depends_on theorem 13.137 Commuting complete fields on a compact manifold ¶
    - 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 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.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 theorem A.729 Kármán's spacing ratio ¶
    - 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 proposition A.611 The constraint on $\Gamma$ ¶
  - depends_on theorem 8.19 Fundamental theorem of algebra ¶
    - depends_on corollary 5.72 Existence of an eigenvalue over $\C$ ¶
      - depends_on theorem 5.98 Primary decomposition ¶
      - depends_on theorem 5.79 Spectral theorem for a self-adjoint operator ¶
    - depends_on lemma 15.12 Artin decomposition ¶
      - depends_on corollary 15.11 Semisimplicity criterion ¶
      - depends_on theorem 15.13 Classification of $\mathfrak{g}_{A}$ by the algebra ¶
      - depends_on theorem 15.68 Positive Gram forms are positive sums of real characters ¶
    - 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 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 example 12.56 Multiplication by the coordinate: spectrum without eigenvectors ¶
      - depends_on theorem A.238 Spectral theorem, both forms ¶
      - depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶
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 ¶
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Edges

type	direction	node	provenance	where
`depends_on`	→	Antiderivative	declared	parts/02-mathematical-methods/05-real-analysis.tex:855
`depends_on`	→	Fundamental theorem of calculus, I	declared	parts/02-mathematical-methods/05-real-analysis.tex:855
`depends_on`	←	Substitution and integration by parts	declared	parts/02-mathematical-methods/05-real-analysis.tex:875
`depends_on`	←	du Bois-Reymond form	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:685
`depends_on`	←	Functions vanishing on a regular zero set	declared	parts/02-mathematical-methods/05-real-analysis.tex:3682
`depends_on`	←	Iterated integral inequality	declared	appendices/A-long-proofs.tex:4608
`depends_on`	←	A $C^{1}$ limit	declared	appendices/A-long-proofs.tex:25405
`depends_on`	←	Dirichlet	declared	appendices/A-long-proofs.tex:8958
`depends_on`	←	Counting identity	declared	appendices/A-long-proofs.tex:11221
`depends_on`	←	The computation in one chart	declared	appendices/A-long-proofs.tex:15282
`depends_on`	←	Irrationality of $\pi$	declared	parts/02-mathematical-methods/05-real-analysis.tex:2472
`depends_on`	←	Fundamental theorem for contours	declared	parts/02-mathematical-methods/06-complex-analysis.tex:215
`depends_on`	←	The identities physics uses	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:1611
`depends_on`	←	Initial- and final-value theorems	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2625
`depends_on`	←	The linear equation: integrating factor	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:616
`depends_on`	←	Gauss	declared	parts/02-mathematical-methods/05-real-analysis.tex:4495
`depends_on`	←	Green	declared	parts/02-mathematical-methods/05-real-analysis.tex:4333
`depends_on`	←	d'Alembert's formula	declared	parts/02-mathematical-methods/08-pdes.tex:1460
`proves`	←	ch:05-real-analysis@proof-25	declared	parts/02-mathematical-methods/05-real-analysis.tex:858