Calculus in the First Three Dimensions

by Sherman K. Stein

Overview

This introduction to calculus is designed for beginning college undergraduates majoring in mathematics as well as undergraduates pursuing other areas of science and engineering for whom calculus will be a vital tool.
The three-part treatment begins by exploring the core of the calculus, concentrating on three basic ideas: the definite integral, the derivative, and the fundamental theorem of calculus. Part Two takes up topics such as the maximum and minimum of a function, Taylor's series, partial derivatives, differentiation of vectors, and Green's theorem in the plane. Part Three, which contains no further mathematical development, applies the techniques developed earlier to significant problems in the natural, social, and physical sciences. Appendixes supplement the treatment, offering helpful information on the rudiments of analytic geometry, real numbers, and functions. Numerous examples and exercises appear throughout the text, and solutions to the problems are available as free downloads from the Dover website.

Product Details

ISBN-13:	9780486801148
Publisher:	Dover Publications
Publication date:	03/15/2016
Series:	Dover Books on Mathematics
Edition description:	First Edition, First
Pages:	640
Product dimensions:	6.10(w) x 9.00(h) x 1.30(d)

About the Author

Sherman K. Stein is Professor Emeritus of Mathematics at the University of California, Davis. He is the author of several other books on mathematical subjects, including Dover's Mathematics: The Man-Made Universe.

Table of Contents

Preface v

Part I The Core of the Calculus

1 The definite integral 3

1 Estimates in four problems 3

2 Precise answers to the four problems 9

3 The definite integral over an interval 14

4 The average of a function over an interval 24

5 The definite integral of a function over a set in the plane 29

6 The definite integral of a function over a set in three-dimensional space 37

7 Summary 40

2 The, derivative 41

1 Estimates in four problems 41

2 The derivative 49

3 Standard notations related to the derivative; the differential 58

4 A generalization of the derivative 64

5 Summary 65

3 Limits and continuous functions 66

1 The limit of a sequence 66

2 The limit of a function of a real variable 74

3 Continuous functions 81

4 Precise definitions of limits 86

5 Summary 91

4 The computation of derivatives 92

1 The derivatives of some basic functions 93

2 The derivatives of the sum, difference, product, and quotient of functions 98

3 The chain rule 102

4 The derivative of an inverse function 107

5 Summary 118

5 The law of the mean 119

1 Rolle's theorem 119

2 The law of the mean 122

3 Summary 131

6 The fundamental theorem of calculus 132

1 Various forms of the fundamental theorem of calculus 132

2 Proof of a special case of the fundamental theorem of calculus 139

3 A different view of the fundamental theorem of calculus 142

4 An alternative approach to the logarithm and exponential functions 149

5 The antiderivative 154

6 Summary 156

7 Computing antiderivatives 157

1 Some basic facts 158

2 The substitution technique 159

3 Integration by parts 163

4 The antidifferentiation of rational functions by partial fractions 167

5 Some special techniques 172

6 Summary and practice 176

8 Computing and applying definite integrals over intervals 180

1 Area 181

2 Volume (the cross-section and shell techniques) 189

3 The first moment 193

4 Arc length 203

5 Area of a surface of revolution 212

6 The higher moments of a function 218

7 Average value of a function 222

8 Improper integrals 227

9 Probability distribution and density 233

10 Summary and memory aids 240

9 Computing and applying definite integrals over plane and solid sets 243

2 The center of gravity of a flat object (lamina) 243

2 Computing ∫_Rf(P)dA by introducing rectangular coordinates in R 250

3 Computing ∫_Rf(P) dA by introducing polar coordinates in R 256

4 Coordinate systems in three dimensions and their volume elements 265

5 Computing ∫_Rf(P)dV by introducing coordinates in R 271

6 Summary 278

Part II Topics in the Calculus

10 The higher derivatives 281

1 The geometric significance of the sign of the second derivative 281

2 The significance in motion of the second derivative 287

3 The second derivative and the curvature of a curve 291

4 Information supplied by the higher derivatives 295

5 Summary 301

11 The maximum and minimum of a function 302

1 Maximum and minimum of f(x) 302

2 Maximum and minimum of f(x, y) 310

3 Maximum and minimum of ax + by + c; linear programming 314

4 Summary 316

12 Series 317

1 The nth term test, the integral test, and the alternating series test 318

2 The comparison test, the ratio test, and the absolute convergence test 323

3 The truncation error En 329

4 Power series 331

5 Summary 333

13 Taylor's series 336

1 Taylor's series in powers of x and in powers of x - a 337

2 R_n(x) in terms of a derivative; Newton's method 344

3 Summary 351

14 Estimating: the definite integral 353

15 Further applications of partial derivatives 363

1 The change δz and the differential dz 364

2 Higher partial derivatives and Taylor's series 372

3 Summary 376

16 Algebraic operations on vectors 378

1 The algebra of vectors 378

2 The dot product of two vectors 385

3 Directional derivatives and the gradient 390

4 Summary 398

17 The derivative of a vector function 399

1 The position and velocity vectors 400

2 The derivative of a vector function 403

3 The tangential and normal components of the acceleration vector 406

4 Summary 414

18 Curve integrals 415

1 The curve integral of a vector field (P,Q) 415

2 The curve integral of a gradient field ∇F 422

3 Other notations for curve integrals 426

4 Summary 429

19 Green's theorem in the plane 430

1 Green's theorem 430

2 Magnification in the plane: the Jacobian 438

3 The hyperbolic functions 445

4 Summary 450

20 The interchange of limits 451

Part III Further Applications of the Calculus

21 Growth in the natural world 469

22 Business management and economics 481

23 Psychology 489

24 Traffics 502

1 Preliminaries 502

2 The exponential (Poisson) model of random traffic 506

3 Cross traffic and the gap between cars 511

25 Rockets 519

1 The basic equation of rocket propulsion 520

2 Escape velocity 523

3 Orbit velocity 526

26 Gravity 531

1 Newton's laws and Kepler's laws 534

2 The gravitational attraction of a homogeneous sphere 539

A Analytic geometry 547

Analytic geometry in two dimensions 547

The distance formula 548

Line and slope 549

Analytic geometry in three dimensions 551

Polar coordinates 554

Conic sections in rectangular coordinates 556

Conic sections in polar coordinates 561

B The real numbers 565

The field axioms 565

The ordering axioms 566

Rational and irrational numbers 567

Completeness of the real numbers 569

C Functions 571

Inverse of a one-to-one correspondence 574

Special types of functions 575

Composite function 577

D Summation notation 579

Summation over i 579

Summation over several indices 581

E Length, area, and volume 583

F Limits and continuous functions (proofs) 589

Limits of functions of x 589

Limits of functions of x and y and of sequences 593

A function continuous throughout [0,1] but differentiable nowhere 594

G Partial-fraction representation of rational numbers 598

Partial-fraction representation of rational functions 599

H Short tables of functions 604

Some important derivatives 605

Some important antiderivatives 605

Index 607