# Calculus (Book)

Calculus | |

Information | |
---|---|

Author | Tom Apostol |

Language | English |

Publisher | Wiley |

Publication Date | 16 January 1991 |

Pages | 666 |

ISBN-10 | 0471000051 |

ISBN-13 | 978-0471000051 |

The textbook * Calculus* by 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 second volume, not listed here, which covers multivariable topics and applications to subjects such as probability.

## Table of Contents[edit]

Chapter/Section # | Title | Page # |
---|---|---|

I. INTRODUCTION | ||

Part 1: Historical Introduction | ||

I 1.1 | The two basic concepts of calculus | 1 |

I 1.2 | Historical background | 2 |

I 1.3 | The method of exhaustion for the area of a parabolic segment | 3 |

*I 1.4 | Exercises | 8 |

I 1.5 | A critical analysis of the Archimedes' method | 8 |

I 1.6 | The approach to calculus to be used in this book | 10 |

Part 2: Some Basic Concepts of the Theory of Sets | ||

I 2.1 | Introduction to set theory | 11 |

I 2.2 | Notations for designating sets | 12 |

I 2.3 | Subsets | 12 |

I 2.4 | Unions, intersections, complements | 13 |

I 2.5 | Exercises | 15 |

Part 3: A set of Axioms for the Real-Number System | ||

I 3.1 | Introduction | 17 |

I 3.2 | The field axioms | 17 |

*I 3.3 | Exercises | 19 |

I 3.4 | The order axioms | 19 |

*I 3.5 | Exercises | 21 |

I 3.6 | Integers and rational numbers | 21 |

I 3.7 | Geometric interpretation of real numbers as points on a line | 22 |

I 3.8 | Upper bound of a set, maximum element, least upper bound (supremum) | 23 |

I 3.9 | The least-Upper-bound axiom (completeness axiom) | 25 |

I 3.10 | The Archimedean property of the real-number system | 25 |

I 3.11 | Fundamental properties of the supremum and infimum | 26 |

*I 3.12 | Exercises | 28 |

*I 3.13 | Existence of square roots of nonnegative real numbers | 29 |

*I 3.14 | Roots of higher order. Rational powers | 30 |

*I 3.15 | Representation of real numbers by decimals | 30 |

Part 4: Mathematical Induction, Summation Notation, and Related Topics | ||

I 4.1 | An example of a proof by mathematical induction | 32 |

I 4.2 | The principle of mathematical induction | 34 |

*I 4.3 | The well-ordering principle | 34 |

I 4.4 | Exercises | 35 |

*I 4.5 | Proof of the well-ordering principle | 37 |

I 4.6 | The summation notation | 37 |

I 4.7 | Exercises | 39 |

I 4.8 | Absolute values and the triangle inequality | 41 |

I 4.9 | Exercises | 43 |

*I 4.10 | Miscellaneous exercises involving induction | 44 |

1. THE CONCEPTS OF INTEGRAL CALCULUS | ||

1.1 | The basic ideas of Cartesian geometry | 48 |

1.2 | Functions. Informal description and examples | 50 |

*1.3 | Functions. Formal definition as a set of ordered pairs | 53 |

1.4 | More examples of real functions | 54 |

1.5 | Exercises | 56 |

1.6 | The concept of area as a set function | 57 |

1.7 | Exercises | 60 |

1.8 | Intervals and ordinate sets | 60 |

1.9 | Partitions and step functions | 61 |

1.10 | Sum and product of step functions | 63 |

1.11 | Exercises | 63 |

1.12 | The definition of the integral for step functions | 64 |

1.13 | Properties of the integral of a step function | 66 |

1.14 | Other notations for integrals | 69 |

1.15 | Exercises | 70 |

1.16 | The integral of more general functions | 72 |

1.17 | Upper and lower integrals | 74 |

1.18 | The area of an ordinate set expressed as an integral | 75 |

1.19 | Informal remarks on the theory and technique of integration | 75 |

1.20 | Monotonic and piecewise monotonic functions. Definitions and examples | 76 |

1.21 | Integrability of bounded monotonic functions | 77 |

1.22 | Calculation of the integral of a bounded monotonic function | 79 |

1.23 | Calculation of the integral [math]\displaystyle{ \int_0^b x^p dx }[/math] when [math]\displaystyle{ p }[/math] is a positive integer | 79 |

1.24 | The basic properties of the integral | 80 |

1.25 | Integration of polynomials | 81 |

1.26 | Exercises | 83 |

1.27 | Proofs of the basic properties of the integral | 84 |

2. SOME APPLICATIONS OF INTEGRATION | ||

2.1 | Introduction | 88 |

2.2 | The area of a region between two graphs expressed as an integral | 88 |

2.3 | Worked examples | 89 |

2.4 | Exercises | 94 |

2.5 | The trigonometric functions | 94 |

2.6 | Integration formulas for the sine and cosine | 94 |

2.7 | A geometric description of the sine and cosine functions | 94 |

2.8 | Exercises | 94 |

2.9 | Polar coordinates | 94 |

2.10 | The integral for area in polar coordinates | 94 |

2.11 | Exercises | 94 |

2.12 | Application of integration to the calculation of volume | 94 |

2.13 | Exercises | 94 |

2.14 | Application of integration to the calculation of work | 94 |

2.15 | Exercises | 94 |

2.16 | Average value of a function | 94 |

2.17 | Exercises | 94 |

2.18 | The integral as a function of the upper limit. Indefinite integrals | 94 |

2.19 | Exercises | 94 |

3. CONTINUOUS FUNCTIONS | ||

3.1 | Informal description of continuity | 126 |

3.2 | The definition of the limit of a function | 127 |

3.3 | The definition of continuity of a function | 130 |

3.4 | The basic limit theorems. More examples of continuous functions | 131 |

3.5 | Proofs of the basic limit theorems | 135 |

3.6 | Exercises | 138 |

3.7 | Composite functions and continuity | 140 |

3.8 | Exercises | 142 |

3.9 | Bolzano's theorem for continuous functions | 142 |

3.10 | The intermediate-value theorem for continuous functions | 144 |

3.11 | Exercises | 145 |

3.12 | The process of inversion | 146 |

3.13 | Properties of functions preserved by inversion | 147 |

3.14 | Inverses of piecewise monotonic functions | 148 |

3.15 | Exercises | 149 |

3.16 | The extreme-value theorem for continuous functions | 150 |

3.17 | The small-span theorem for continuous functions (uniform continuity) | 152 |

3.18 | The integrability theorem for continuous functions | 152 |

3.19 | Mean-value theorems for integrals of continuous functions | 154 |

3.20 | Exercises | 155 |

4. DIFFERENTIAL CALCULUS | ||

4.1 | Historical introduction | 156 |

4.2 | A problem involving velocity | 157 |

4.3 | The derivative of a function | 159 |

4.4 | Examples of derivatives | 161 |

4.5 | The algebra of derivatives | 164 |

4.6 | Exercises | 167 |

4.7 | Geometric interpretation of the derivative as a slope | 169 |

4.8 | Other notations for derivatives | 171 |

4.9 | Exercises | 173 |

4.10 | The chain rule for differentiating composite functions | 174 |

4.11 | Applications of the chain rule. Related rates and implicit differentiation | 176 |

4.12 | Exercises | 179 |

4.13 | Applications of the differentiation to extreme values of cuntions | 181 |

4.14 | The mean-value theorem for derivatives | 183 |

4.15 | Exercises | 186 |

4.16 | Applications of the mean-value theorem to geometric properties of functions | 187 |

4.17 | Second-derivative test for extrema | 188 |

4.18 | Curve sketching | 189 |

4.19 | Exercises | 191 |

4.20 | Worked examples of extremum problems | 191 |

4.21 | Exercises | 194 |

*4.22 | Partial derivatives | 196 |

*4.23 | Exercises | 201 |

5. THE RELATION BETWEEN INTEGRATION AND DIFFERENTIATION | ||

5.1 | The derivative of an indefinite integral. The first fundamental theorem of calculus | 202 |

5.2 | The zero-derivative theorem | 204 |

5.3 | Primitive functions and the second fundamental theorem of calculus | 205 |

5.4 | Properties of a function deduced from properties of its derivative | 207 |

5.5 | Exercises | 208 |

5.6 | The Leibniz notation for primitives | 210 |

5.7 | Integration by substitution | 212 |

5.8 | Exercises | 216 |

5.9 | Integration by parts | 217 |

5.10 | Exercises | 220 |

*5.11 | Miscellaneous review exercises | 222 |

6. THE LOGARITHM, THE EXPONENTIAL, AND THE INVERSE TRIGONOMETRIC FUNCTIONS | ||

6.1 | Introduction | 226 |

6.2 | Motivation for the definition of the natural logarithm as an integral | 227 |

6.3 | The definition of the logarithm. Basic properties | 229 |

6.4 | The graph of the natural logarithm | 230 |

6.5 | Consequences of the functional equation [math]\displaystyle{ L(ab) = L(a) + L(b) }[/math] | 230 |

6.6 | Logarithms referred to any positive base [math]\displaystyle{ b \ne 1 }[/math] | 232 |

6.7 | Differentiation and integration formulas involving logarithms | 233 |

6.8 | Logarithmic differentiation | 235 |

6.9 | Exercises | 236 |

6.10 | Polynomial approximations to the logarithm | 236 |

6.11 | Exercises | 242 |

6.12 | The exponential function | 242 |

6.13 | Exponentials expressed as powers of e | 242 |

6.14 | The definition of [math]\displaystyle{ e^x }[/math] for arbitrary real x | 244 |

6.15 | The definition of [math]\displaystyle{ a^x }[/math] for [math]\displaystyle{ a \gt 0 }[/math] and x real | 245 |

6.16 | Differentiation and integration formulas involving exponentials | 245 |

6.17 | Exercises | 248 |

6.18 | The hyperbolic functions | 251 |

6.19 | Exercises | 251 |

6.20 | Derivatives of inverse functions | 252 |

6.21 | Inverses of the trigonometric functions | 253 |

6.22 | Exercises | 256 |

6.23 | Integration by partial fractions | 258 |

6.24 | Integrals which can be transformed into integrals of rational functions | 264 |

6.25 | Exercises | 267 |

6.26 | Miscellaneous review exercises | 268 |

7. POLYNOMIAL APPROXIMATIONS TO FUNCTIONS | ||

7.1 | Introduction | 272 |

7.2 | The Taylor polynomials generated by a function | 273 |

7.3 | Calculus of Taylor polynomials | 275 |

7.4 | Exercises | 278 |

7.5 | Taylor's formula with remainder | 278 |

7.6 | Estimates for the error in Taylor's formula | 280 |

*7.7 | Other forms of the remainder in Taylor's formula | 283 |

7.8 | Exercises | 284 |

7.9 | Further remarks on the error in Taylor's formula. The o-notation | 286 |

7.10 | Applications to indeterminate forms | 289 |

7.11 | Exercises | 290 |

7.12 | L'Hopital's rule for the indeterminate form 0/0 | 292 |

7.13 | Exercises | 295 |

7.14 | The symbols [math]\displaystyle{ +\inf }[/math] and [math]\displaystyle{ -\inf }[/math]. Extension of L'Hopital's rule | 296 |

7.15 | Infinite limits | 298 |

7.16 | The behavior of log[math]\displaystyle{ x }[/math] and [math]\displaystyle{ e^x }[/math] for large [math]\displaystyle{ x }[/math] | 300 |

7.17 | Exercises | 303 |

8. INTRODUCTION TO DIFFERENTIAL EQUATIONS | ||

8.1 | Introduction | 305 |

8.2 | Terminology and notation | 306 |

8.3 | A first-order differential equation for the exponential function | 307 |

8.4 | First-order linear differential equations | 308 |

8.5 | Exercises | 311 |

8.6 | Some physical problems leading to first-order linear differential equations | 313 |

8.7 | Exercises | 319 |

8.8 | Linear equations of second order with constant coefficients | 322 |

8.9 | Existence of solutions of the equation [math]\displaystyle{ y^{''} + by = 0 }[/math] | 323 |

8.10 | Reduction of the general equation to the special case [math]\displaystyle{ y^{''} + by = 0 }[/math] | 324 |

8.11 | Uniqueness theorem for the equation [math]\displaystyle{ y^{''} + by = 0 }[/math] | 324 |

8.12 | Complete solution of the equation [math]\displaystyle{ y^{''} + by = 0 }[/math] | 326 |

8.13 | Complete solution of the equation [math]\displaystyle{ y^{''} + ay^' + by = 0 }[/math] | 326 |

8.14 | Exercises | 328 |

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 [math]\displaystyle{ y^{''} + ay^' + by = R }[/math] | 332 |

8.17 | Exercises | 333 |

8.18 | Examples of physical problems leading to linear second-order equations with constant coefficients | 334 |

8.19 | Exercises | 339 |

8.20 | Remarks concerning nonlinear differential equations | 339 |

8.21 | Integral curves and direction fields | 341 |

8.22 | Exercises | 344 |

8.23 | First-order separable equations | 345 |

8.24 | Exercises | 347 |

8.25 | Homogeneous first-order equations | 347 |

8.26 | Exercises | 350 |

8.27 | Some geometrical and physical problems leading to first-order equations | 351 |

8.28 | Miscellaneous review exercises | 355 |

9. COMPLEX NUMBERS | ||

9.1 | Historical introduction | 358 |

9.2 | Definitions and field properties | 358 |

9.3 | The complex numbers as an extension of the real numbers | 360 |

9.4 | The imaginary unit [math]\displaystyle{ i }[/math] | 361 |

9.5 | Geometric interpretation. Modulus and argument | 362 |

9.6 | Exercises | 365 |

9.7 | Complex exponentials | 366 |

9.8 | Complex-valued functions | 368 |

9.9 | Examples of differentiation and integration formulas | 369 |

9.10 | Exercises | 371 |

10. SEQUENCES, INFINITE SERIES, IMPROPER INTEGRALS | ||

10.1 | Zeno's paradox | 374 |

10.2 | Sequences | 378 |

10.3 | Monotonic sequences of real numbers | 381 |

10.4 | Exercises | 382 |

10.5 | Infinite series | 383 |

10.6 | The linearity property of convergent series | 385 |

10.7 | Telescoping series | 386 |

10.8 | The geometric series | 388 |

10.9 | Exercises | 391 |

*10.10 | Exercises on decimal expansions | 393 |

10.11 | Tests for convergence | 394 |

10.12 | Comparison tests for series of nonnegative terms | 394 |

10.13 | The integral test | 397 |

10.14 | Exercises | 398 |

10.15 | The root test and the ratio test for series of nonnegative terms | 399 |

10.16 | Exercises | 402 |

10.17 | Alternating series | 403 |

10.18 | Conditional and absolute convergence | 406 |

10.19 | The convergence tests of Dirichlet and Abel | 407 |

10.20 | Exercises | 409 |

*10.21 | Rearrangements of series | 411 |

10.22 | Miscellaneous review exercises | 414 |

10.23 | Improper integrals | 416 |

10.24 | Exercises | 420 |

11. SEQUENCES AND SERIES OF FUNCTIONS | ||

11.1 | Pointwise convergence of sequences of functions | 422 |

11.2 | Uniform convergence of sequences of functions | 423 |

11.3 | Uniform convergence and continuity | 424 |

11.4 | Uniform convergence and integration | 425 |

11.5 | A sufficient condition for uniform convergence | 427 |

11.6 | Power series. Circle of convergence | 428 |

11.7 | Exercises | 430 |

11.8 | Properties of functions represented by real power series | 431 |

11.9 | The Taylor's series generated by a function | 434 |

11.10 | A sufficient condition for convergence of a Taylor's series | 435 |

11.11 | Power-series expansions for the exponential and trigonometric functions | 435 |

*11.12 | Bernstein's theorem | 437 |

11.13 | Exercises | 438 |

11.14 | Power series and differential equations | 439 |

11.15 | The binomial series | 441 |

11.16 | Exercises | 443 |

12. VECTOR ALGEBRA | ||

12.1 | Historical introduction | 445 |

12.2 | The vector space of n-tuples of real numbers | 446 |

12.3 | Geometric interpretation for [math]\displaystyle{ n \leq 3 }[/math] | 448 |

12.4 | Exercises | 450 |

12.5 | The dot product | 451 |

12.6 | Length or norm of a vector | 453 |

12.7 | Orthogonality of vectors | 455 |

12.8 | Exercises | 456 |

12.9 | Projections. Angle between vectors in n-space | 457 |

12.10 | The unit coordinate vectors | 458 |

12.11 | Exercises | 460 |

12.12 | The linear span of a finite set of vectors | 462 |

12.13 | Linear independence | 463 |

12.14 | Bases | 466 |

12.15 | Exercises | 467 |

12.16 | The vector space [math]\displaystyle{ V_N(C) }[/math] of n-tuples of complex numbers | 468 |

12.17 | Exercises | 470 |

13. APPLICATIONS OF VECTOR ALGEBRA TO ANALYTIC GEOMETRY | ||

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 |

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 |

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 |

15.6 | Subspaces of a linear space | 556 |

15.7 | Dependent and independent sets in a linear space | 557 |

15.8 | Bases and dimension | 559 |

15.9 | Exercises | 560 |

15.10 | Inner products, Euclidean spaces, norms | 561 |

15.11 | Orthogonality in a Euclidean space | 564 |

15.12 | Exercises | 566 |

15.13 | Construction of orthogonal sets. The Gram-Schmidt process | 568 |

15.14 | Orthogonal complements. Projections | 572 |

15.15 | Best approximation of elements in a Euclidean space by elements in a finite-dimensional subspace | 574 |

15.16 | Exercises | 576 |

16. LINEAR TRANSFORMATIONS AND MATRICES | ||

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 |

Answers to exercises | 617 | |

Index | 657 |