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{{Stub}}
{{InfoboxBook
{{InfoboxBook
|title=Calculus
|title=Calculus
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|isbn13=978-0471000051
|isbn13=978-0471000051
}}
}}
The textbook '''''Calculus''''' by [https://en.wikipedia.org/wiki/Tom_M._Apostol Tom Apostol] introduces calculus.  
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The textbook [https://simeioseismathimatikwn.files.wordpress.com/2013/03/apostol-calculusi.pdf '''''Calculus'''''] by [https://en.wikipedia.org/wiki/Tom_M._Apostol Tom Apostol] introduces calculus. It provides a rigorous treatment of theory and application, in addition to the historical context of its topics. It should be noted that there is a [https://archive.org/details/calculus-tom-m.-apostol-calculus-volume-2-2nd-edition-proper-2-1975-wiley-sons-libgen.lc/Apostol%20T.%20M.%20-%20Calculus%20vol%20II%20%281967%29/ second volume], not listed here, which covers multivariable topics and applications to subjects such as probability.


== Table of Contents ==
== Table of Contents ==
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| 1.2 || Functions. Informal description and examples || 50
| 1.2 || Functions. Informal description and examples || 50
|-
|-
| 1.3 || Functions. Formal definition as a set of ordered pairs || 53
| <nowiki>*</nowiki>1.3 || Functions. Formal definition as a set of ordered pairs || 53
|-
|-
| 1.4 || More examples of real functions || 54
| 1.4 || More examples of real functions || 54
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| 1.22 || Calculation of the integral of a bounded monotonic function || 79
| 1.22 || Calculation of the integral of a bounded monotonic function || 79
|-
|-
| 1.23 || Calculation of the integral \(\int_0^b x^p dx\) when \(p\) is a positive integer || 79
| 1.23 || Calculation of the integral <math>\int_0^b x^p dx</math> when <math>p</math> is a positive integer || 79
|-
|-
| 1.24 || The basic properties of the integral || 80
| 1.24 || The basic properties of the integral || 80
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| 4.21 || Exercises || 194
| 4.21 || Exercises || 194
|-
|-
| 4.22 || Partial derivatives || 196
| <nowiki>*</nowiki>4.22 || Partial derivatives || 196
|-
|-
| 4.23 || Exercises || 201
| <nowiki>*</nowiki>4.23 || Exercises || 201
|-
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! colspan="3" | 5. THE RELATION BETWEEN INTEGRATION AND DIFFERENTIATION
! colspan="3" | 5. THE RELATION BETWEEN INTEGRATION AND DIFFERENTIATION
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| 5.10 || Exercises || 220
| 5.10 || Exercises || 220
|-
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| 5.11 || Miscellaneous review exercises || 222
| <nowiki>*</nowiki>5.11 || Miscellaneous review exercises || 222
|-
|-
! colspan="3" | 6. THE LOGARITHM, THE EXPONENTIAL, AND THE INVERSE TRIGONOMETRIC FUNCTIONS
! colspan="3" | 6. THE LOGARITHM, THE EXPONENTIAL, AND THE INVERSE TRIGONOMETRIC FUNCTIONS
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| 6.4 || The graph of the natural logarithm || 230
| 6.4 || The graph of the natural logarithm || 230
|-
|-
| 6.5 || Consequences of the functional equation \(L(ab) = L(a) + L(b)\) || 230
| 6.5 || Consequences of the functional equation <math>L(ab) = L(a) + L(b)</math> || 230
|-
|-
| 6.6 || Logarithms referred to any positive base \(b \ne 1\) || 232
| 6.6 || Logarithms referred to any positive base <math>b \ne 1</math> || 232
|-
|-
| 6.7 || Differentiation and integration formulas involving logarithms || 233
| 6.7 || Differentiation and integration formulas involving logarithms || 233
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| 6.13 || Exponentials expressed as powers of e || 242
| 6.13 || Exponentials expressed as powers of e || 242
|-
|-
| 6.14 || The definition of \(e^x\) for arbitrary real x || 244
| 6.14 || The definition of <math>e^x</math> for arbitrary real x || 244
|-
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| 6.15 || The definition of \(a^x\) for \(a > 0\) and x real || 245
| 6.15 || The definition of <math>a^x</math> for <math>a > 0</math> and x real || 245
|-
|-
| 6.16 || Differentiation and integration formulas involving exponentials || 245
| 6.16 || Differentiation and integration formulas involving exponentials || 245
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| 7.6 || Estimates for the error in Taylor's formula || 280
| 7.6 || Estimates for the error in Taylor's formula || 280
|-
|-
| 7.7 || Other forms of the remainder in Taylor's formula || 283
| <nowiki>*</nowiki>7.7 || Other forms of the remainder in Taylor's formula || 283
|-
|-
| 7.8 || Exercises || 284
| 7.8 || Exercises || 284
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| 7.13 || Exercises || 295
| 7.13 || Exercises || 295
|-
|-
| 7.14 || The symbols \(+\inf\) and \(-\inf\). Extension of L'Hopital's rule || 296
| 7.14 || The symbols <math>+\inf</math> and <math>-\inf</math>. Extension of L'Hopital's rule || 296
|-
|-
| 7.15 || Infinite limits || 298
| 7.15 || Infinite limits || 298
|-
|-
| 7.16 || The behavior of log\(x\) and \(e^x\) for large \(x\) || 300
| 7.16 || The behavior of log<math>x</math> and <math>e^x</math> for large <math>x</math> || 300
|-
|-
| 7.17 || Exercises || 303
| 7.17 || Exercises || 303
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| 8.8 || Linear equations of second order with constant coefficients || 322
| 8.8 || Linear equations of second order with constant coefficients || 322
|-
|-
| 8.9 || Existence of solutions of the equation \(y^{''} + by = 0\) || 323
| 8.9 || Existence of solutions of the equation <math>y^{''} + by = 0</math> || 323
|-
|-
| 8.10 || Reduction of the general equation to the special case \(y^{''} + by = 0\) || 324
| 8.10 || Reduction of the general equation to the special case <math>y^{''} + by = 0</math> || 324
|-
|-
| 8.11 || Uniqueness theorem for the equation \(y^{''} + by = 0\) || 324
| 8.11 || Uniqueness theorem for the equation <math>y^{''} + by = 0</math> || 324
|-
|-
| 8.12 || Complete solution of the equation \(y^{''} + by = 0\) || 326
| 8.12 || Complete solution of the equation <math>y^{''} + by = 0</math> || 326
|-
|-
| 8.13 || Complete solution of the equation \(y^{''} + ay^' + by = 0\) || 326
| 8.13 || Complete solution of the equation <math>y^{''} + ay^{'} + by = 0</math> || 326
|-
|-
| 8.14 || Exercises || 328
| 8.14 || Exercises || 328
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| 8.15 || Nonhomogeneous linear equations of second order with constant coefficients || 329
| 8.15 || Nonhomogeneous linear equations of second order with constant coefficients || 329
|-
|-
| 8.16 || Special methods for determining a particular solution of the nonhomogeneous equation \(y^{''} + ay^' + by = R\) || 332
| 8.16 || Special methods for determining a particular solution of the nonhomogeneous equation <math>y^{''} + ay^{'} + by = R</math> || 332
|-
|-
| 8.17 || Exercises || 333
| 8.17 || Exercises || 333
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| 9.3 || The complex numbers as an extension of the real numbers || 360
| 9.3 || The complex numbers as an extension of the real numbers || 360
|-
|-
| 9.4 || The imaginary unit \(i\) || 361
| 9.4 || The imaginary unit <math>i</math> || 361
|-
|-
| 9.5 || Geometric interpretation. Modulus and argument || 362
| 9.5 || Geometric interpretation. Modulus and argument || 362
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| 10.9 || Exercises || 391
| 10.9 || Exercises || 391
|-
|-
| 10.10 || Exercises on decimal expansions || 393
| <nowiki>*</nowiki>10.10 || Exercises on decimal expansions || 393
|-
|-
| 10.11 || Tests for convergence || 394
| 10.11 || Tests for convergence || 394
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| 10.20 || Exercises || 409
| 10.20 || Exercises || 409
|-
|-
| 10.21 || Rearrangements of series || 411
| <nowiki>*</nowiki>10.21 || Rearrangements of series || 411
|-
|-
| 10.22 || Miscellaneous review exercises || 414
| 10.22 || Miscellaneous review exercises || 414
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| 11.11 || Power-series expansions for the exponential and trigonometric functions || 435
| 11.11 || Power-series expansions for the exponential and trigonometric functions || 435
|-
|-
| 11.12 || Bernstein's theorem || 437
| <nowiki>*</nowiki>11.12 || Bernstein's theorem || 437
|-
|-
| 11.13 || Exercises || 438
| 11.13 || Exercises || 438
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| 12.2 || The vector space of n-tuples of real numbers || 446
| 12.2 || The vector space of n-tuples of real numbers || 446
|-
|-
| 12.3 || Geometric interpretation for \(n \leq 3\) || 448
| 12.3 || Geometric interpretation for <math>n \leq 3</math> || 448
|-
|-
| 12.4 || Exercises || 450
| 12.4 || Exercises || 450
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| 12.15 || Exercises || 467
| 12.15 || Exercises || 467
|-
|-
| 12.16 || The vector space \(V_N(C)\) of n-tuples of complex numbers || 468
| 12.16 || The vector space <math>V_N(C)</math> of n-tuples of complex numbers || 468
|-
|-
| 12.17 || Exercises || 470
| 12.17 || Exercises || 470
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! colspan="3" | 13. APPLICATIONS OF VECTOR ALGEBRA TO ANALYTIC GEOMETRY
! colspan="3" | 13. APPLICATIONS OF VECTOR ALGEBRA TO ANALYTIC GEOMETRY
|-
|-
| 1 || Introduction || 471
| 13.1 || Introduction || 471
|-
| 13.2 || Lines in n-space || 472
|-
| 13.3 || Some simple properties of straight lines || 473
|-
| 13.4 || Lines and vector-valued functions || 474
|-
| 13.5 || Exercises || 477
|-
| 13.6 || Planes in Euclidean n-space || 478
|-
| 13.7 || Planes and vector-valued functions || 481
|-
| 13.8 || Exercises || 482
|-
| 13.9 || The cross product || 483
|-
| 13.10 || The cross product expressed as a determinant || 486
|-
| 13.11 || Exercises || 487
|-
| 13.12 || The scalar triple product || 488
|-
| 13.13 || Cramer's rule for solving a system of three linear equations || 490
|-
| 13.14 || Exercises || 491
|-
| 13.15 || Normal vectors to planes || 493
|-
| 13.16 || Linear Cartesian equations for planes || 494
|-
| 13.17 || Exercises || 496
|-
| 13.18 || The conic sections || 497
|-
| 13.19 || Eccentricity of conic sections || 500
|-
| 13.20 || Polar equations for conic sections || 501
|-
| 13.21 || Exercises || 503
|-
| 13.22 || Conic sections symmetric about the origin || 504
|-
| 13.23 || Cartesian equations for the conic sections || 505
|-
| 13.24 || Exercises || 508
|-
| 13.25 || Miscellaneous exercises on conic sections || 509
|-
! colspan="3" | 14. CALCULUS OF VECTOR-VALUED FUNCTIONS
|-
| 14.1 || Vector-valued functions of a real variable || 512
|-
| 14.2 || Algebraic operations. Components || 512
|-
| 14.3 || Limits, derivatives, and integrals || 513
|-
| 14.4 || Exercises || 516
|-
| 14.5 || Applications to curves. Tangency || 517
|-
| 14.6 || Applications to curvilinear motion. Velocity, speed, and acceleration || 520
|-
| 14.7 || Exercises || 524
|-
| 14.8 || The unit tangent, the principal normal, and the osculating plane of a curve || 525
|-
| 14.9 || Exercises || 528
|-
| 14.10 || The definition of arc length || 529
|-
| 14.11 || Additivity of arc length || 532
|-
| 14.12 || The arc-length function || 533
|-
| 14.13 || Exercises || 535
|-
| 14.14 || Curvature of a curve || 536
|-
| 14.15 || Exercises || 538
|-
| 14.16 || Velocity and acceleration in polar coordinates || 540
|-
| 14.17 || Plane motion with radial acceleration || 542
|-
| 14.18 || Cylindrical coordinates || 543
|-
| 14.19 || Exercises || 543
|-
| 14.20 || Applications to planetary motion || 545
|-
| 14.21 || Miscellaneous review exercises || 549
|-
! colspan="3" | 15. LINEAR SPACES
|-
| 15.1 || Introduction || 551
|-
| 15.2 || The definition of a linear space || 551
|-
| 15.3 || Examples of linear spaces || 552
|-
| 15.4 || Elementary consequences of the axioms || 554
|-
| 15.5 || Exercises || 555
|-
|-
| 2 || Lines in n-space || 472
| 15.6 || Subspaces of a linear space || 556
|-
|-
! colspan="3" | Chapter 16: Induction and Summations
| 15.7 || Dependent and independent sets in a linear space || 557
|-
|-
| 1 || Induction || 383
| 15.8 || Bases and dimension || 559
|-
|-
| 2 || Summations || 388
| 15.9 || Exercises || 560
|-
|-
| 3 || Geometric series || 396
| 15.10 || Inner products, Euclidean spaces, norms || 561
|-
|-
! colspan="3" | Chapter 17: Determinants
| 15.11 || Orthogonality in a Euclidean space || 564
|-
|-
| 1 || Matrices || 401
| 15.12 || Exercises || 566
|-
|-
| 2 || Determinants of order 2 || 406
| 15.13 || Construction of orthogonal sets. The Gram-Schmidt process || 568
|-
|-
| 3 || Properties of 2 x 2 determinants || 409
| 15.14 || Orthogonal complements. Projections || 572
|-
|-
| 4 || Determinants of order 3 || 414
| 15.15 || Best approximation of elements in a Euclidean space by elements in a finite-dimensional subspace || 574
|-
|-
| 5 || Properties of 3 x 3 determinants || 418
| 15.16 || Exercises || 576
|-
|-
| 6 || Cramer's Rule || 424
! colspan="3" | 16. LINEAR TRANSFORMATIONS AND MATRICES
|-
|-
! colspan="2" | Index || 429
| 16.1 || Linear transformations || 578
|-
| 16.2 || Null space and range || 579
|-
| 16.3 || Nullity and rank || 581
|-
| 16.4 || Exercises || 582
|-
| 16.5 || Algebraic operations on linear transformations || 583
|-
| 16.6 || Inverses || 585
|-
| 16.7 || One-to-one linear transformations || 587
|-
| 16.8 || Exercises || 589
|-
| 16.9 || Linear transformations with prescribed values || 590
|-
| 16.10 || Matrix representations of linear transformations || 591
|-
| 16.11 || Construction of a matrix representation in diagonal form || 594
|-
| 16.12 || Exercises || 596
|-
| 16.13 || Linear spaces of matrices || 597
|-
| 16.14 || Isomorphism between linear transformations and matrices || 599
|-
| 16.15 || Multiplication of matrices || 600
|-
| 16.16 || Exercises || 603
|-
| 16.17 || Systems of linear equations || 605
|-
| 16.18 || Computation techniques || 607
|-
| 16.19 || Inverses of square matrices || 611
|-
| 16.20 || Exercises || 613
|-
| 16.21 || Miscellaneous exercises on matrices || 614
|-
! colspan="2" | Answers to exercises || 617
|-
! colspan="2" | Index || 657
|-
|-
|}
|}


[[Category:Mathematics]]
[[Category:Mathematics]]
{{Stub}}
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