definition 7.41 Antiderivative

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:811 · p. 216

ground object -- no derivation owed

Rests on

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 definition 7.9 Real function ¶
    - depends_on equation 7.14 eq:ana-derivh ¶
    - depends_on proposition 7.6 Algebra of limits ¶
      - depends_on definition 7.4 Convergence ¶
      - depends_on proposition 7.3 Triangle inequality ¶
      - proves proof ch:05-real-analysis@proof-3 ¶
    - proves proof ch:05-real-analysis@proof-13 ¶
  - depends_on theorem 7.34 Rolle ¶
    - depends_on lemma 7.33 Fermat: interior extremum ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on proposition 7.18 Two-sided limit from one-sided limits ¶
      - proves proof ch:05-real-analysis@proof-17 ¶
    - 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 theorem 7.7 Bolzano–Weierstrass ¶
      - proves proof ch:05-real-analysis@proof-9 ¶
    - 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 ¶ ↺

Supports

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 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 theorem 16.22 Euler–Lagrange ¶
      - depends_on definition 16.62 Field of extremals; slope function ¶
      - depends_on phenomenon 30.52 Euler buckling ¶
      - 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 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 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 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 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

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Edges

type	direction	node	provenance	where
`depends_on`	→	cor:ana-mvt-consequences	declared	parts/02-mathematical-methods/05-real-analysis.tex:823
`depends_on`	→	Derivative of a function at a point	declared	parts/02-mathematical-methods/05-real-analysis.tex:823
`depends_on`	←	Fundamental theorem of calculus, II	declared	parts/02-mathematical-methods/05-real-analysis.tex:855