Mathematics for Engineers and Scientists Fourth edition ALAN JEFFREY University of Newcastle-upon-Tyne B CHAPMAN & HALL University and Professional Division London New York Tokyo Melbourne Madras
Contents Preface to the first edition Preface to the fourth edition xi xiii 1 Introduction to Sets and Numbers 1 1.1 Sets and algebra 1 1.2 Integers, rationals and arithmetic laws 10 1.3 Absolute value of a real number 19 1.4 Mathematical induction 21 1.5 Cartesian geometry 27 1.6 Polar coordinates 38 Problems 41 2 Variables, Functions, and Mappings 45 2.1 Variables and functions 45 2.2 Inverse functions 51 2.3 Some special functions 57 2.4 Curves and parameters 63 2.5 Functions of several real variables 67 Problems 73 3 Sequences, Limits, and Continuity 77 3.1 Sequences 77 3.2 Limits of sequences 84 3.3 The number e 91 3.4 Limits of functions continuity 95 3.5 Functions of several variables limits, continuity 102 3.6 A useful connecting theorem 106 3.7 Asymptotes 110 Problems 113 4 Complex Numbers and Vectors 119 4.1 Introductory ideas 119 4.2 Basic algebraic rules for complex numbers 122 4.3 Complex numbers as vectors 128 4.4 Modulus argument form of complex numbers 132
vi / CONTENTS 4.5 Roots of complex numbers 137 4.6 Introduction to space vectors 139 4.7 Scalar and vector products 152 4.8 Geometrical applications 162 4.9 Applications to mechanics 168 Problems 172 Differentiation of Functions of One or More Real Variables 180 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 Problems The derivative Rules of differentiation Some important consequences of differentiability Higher derivatives applications Partial differentiation Total differential Envelopes The chain rule and its consequences Change of variable Implicit functions Higher order partial derivatives 180 192 201 222 229 233 240 244 247 251 257 264 Exponential, Hyperbolic, and Logarithmic Functions 276 6.1 The exponential function 276 6.2 Differentiation of functions involving the exponential function 283 6.3 The logarithmic function 287 6.4 Hyperbolic functions 292 6.5 Exponential function with a complex argument 298 Problems 302 Fundamentals of Integration 306 7.1 Definite integrals and areas 306 7.2 Integration of arbitrary continuous functions 316 7.3 Integral inequalities 324 7.4 The definite integral as a function of its upper limit-indefinite integral 326 7.5 Differentiation of an integral containing a parameter 330 7.6 Other geometrical applications of definite integrals 333 7.7 Centre of mass and moment of inertia 339 7.8 Double integrals 346 Problems 353
CONTENTS / vii 8 Systematic Integration 358 8.1 Integration of elementary functions 358 8.2 Integration by Substitution 362 8.3 Integration by parts 373 8.4 Reduction formulae 375 8.5 Integration of rational functions partial fractions 379 8.6 Other special techniques of Integration 386 8.7 Integration by means of tables 390 Problems 392 9 Matrices and Linear Transformations 397 9.1 Introductory ideas 397 9.2 Matrix algebra 406 9.3 Determinants 415 9.4 Linear dependence and linear independence 423 9.5 Inverse and adjoint matrix 426 9.6 Matrix functions of a Single variable 429 9.7 Solution of Systems of linear equations 433 9.8 Eigenvalues and eigenvectors 441 9.9 Matrix interpretation of change of variables in partial differentiation 445 9.10 Linear transformations 447 9.11 Applications of matrices and linear transformations 449 Problems 457 10 Functions ofa Complex Variable 465 10.1 Curves and regions 465 10.2 Function of a complex variable, limits, continuity and differentiability 469 10.3 Conformal mapping 476 Problems 487 11 Sealars, Vectors, and Fields 491 11.1 Curves in space 491 11.2 Antiderivatives and integrals of vector functions 504 11.3 Some applications 509 11.4 Fields, gradient, and directional derivative 515 Problems 519 12 Series, Taylor's Theorem and its Uses 524 12.1 Series 524 12.2 Power series 541 12.3 Taylor's theorem 549
viii / CONTENTS 12.4 Application oftaylor's theorem 563 12.5 Applications of the generalized mean value theorem 565 Problems 580 13 Differential Equations and Geometry 587 13.1 Introductory ideas 587 13.2 Possible physical origin of some equations 590 13.3 Arbitrary constants and initial conditions 593 13.4 Properties of Solutions isoclines 596 13.5 Orthogonal trajectories 609 Problems 610 14 First Order Differential Equations 613 14.1 Equations with separable variables 613 14.2 Homogeneous equations 618 14.3 Exact equations 620 14.4 The linear equation of first Order 624 14.5 Direct deductions and comparison theorems 628 Problems 632 15 Higher Order Differential Equations 636 15.1 Linear equations with constant coefficients homogeneous case 637 15.2 Linear equations with constant coefficients inhomogeneous case 645 15.3 Variation of parameters 656 15.4 Oscillatory Solutions 661 15.5 Coupled oscillations and normal modes 664 15.6 Systems of first order equations 670 15.7 Two point boundary value problems 671 15.8 Laplace transform 674 15.9 Applications of the Laplace transform 691 Problems 702 16 FourierSeries 709 16.1 Introductory ideas 710 16.2 Convergence of Fourier series 724 16.3 Different forms of Fourier series 731 16.4 Differentiation and integration 739 Problems 745
CONTENTS / ix 17 Numerical Analysis 748 17.1 Errors 748 17.2 Solution of linear equations 751 17.3 Interpolation 754 17.4 Numerical Integration 757 17.5 Solution of polynomial and transcendental equations 768 17.6 Numerical Solutions of differential equations 775 17.7 Determination of eigenvalues and eigenvectors 783 Problems 788 18 Probability and Statistics 793 18.1 Probability, discrete distributions and moments 794 18.2 Continuous distributions normal distribution 813 18.3 Mean and variance of sum of random variables 822 18.4 Statistics inference drawn from observations 823 18.5 Linear regression 834 Problems 836 Answers 841 Reference Lists 1-4 860 Index 871