Mathematics for Engineers and Scientists

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1 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.

2 Contents Preface to the first edition Preface to the fourth edition xi xiii 1 Introduction to Sets and Numbers Sets and algebra Integers, rationals and arithmetic laws Absolute value of a real number Mathematical induction Cartesian geometry Polar coordinates 38 Problems 41 2 Variables, Functions, and Mappings Variables and functions Inverse functions Some special functions Curves and parameters Functions of several real variables 67 Problems 73 3 Sequences, Limits, and Continuity Sequences Limits of sequences The number e Limits of functions continuity Functions of several variables limits, continuity A useful connecting theorem Asymptotes 110 Problems Complex Numbers and Vectors Introductory ideas Basic algebraic rules for complex numbers Complex numbers as vectors 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.

3 vi / CONTENTS 4.5 Roots of complex numbers Introduction to space vectors Scalar and vector products Geometrical applications Applications to mechanics 168 Problems 172 Differentiation of Functions of One or More Real Variables 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 Exponential, Hyperbolic, and Logarithmic Functions The exponential function Differentiation of functions involving the exponential function The logarithmic function Hyperbolic functions Exponential function with a complex argument 298 Problems 302 Fundamentals of Integration Definite integrals and areas Integration of arbitrary continuous functions Integral inequalities The definite integral as a function of its upper limit-indefinite integral Differentiation of an integral containing a parameter Other geometrical applications of definite integrals Centre of mass and moment of inertia Double integrals 346 Problems 353

$5 Integration of rational functions partial fractions 379 8.6 Other special techniques of Integration 386 8.$

4 CONTENTS / vii 8 Systematic Integration Integration of elementary functions Integration by Substitution Integration by parts Reduction formulae Integration of rational functions partial fractions Other special techniques of Integration Integration by means of tables 390 Problems Matrices and Linear Transformations Introductory ideas Matrix algebra Determinants Linear dependence and linear independence Inverse and adjoint matrix Matrix functions of a Single variable Solution of Systems of linear equations Eigenvalues and eigenvectors Matrix interpretation of change of variables in partial differentiation Linear transformations Applications of matrices and linear transformations 449 Problems Functions ofa Complex Variable Curves and regions Function of a complex variable, limits, continuity and differentiability Conformal mapping 476 Problems Sealars, Vectors, and Fields Curves in space Antiderivatives and integrals of vector functions Some applications Fields, gradient, and directional derivative 515 Problems Series, Taylor's Theorem and its Uses Series Power series 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.

5 viii / CONTENTS 12.4 Application oftaylor's theorem Applications of the generalized mean value theorem 565 Problems Differential Equations and Geometry Introductory ideas Possible physical origin of some equations Arbitrary constants and initial conditions Properties of Solutions isoclines Orthogonal trajectories 609 Problems First Order Differential Equations Equations with separable variables Homogeneous equations Exact equations The linear equation of first Order Direct deductions and comparison theorems 628 Problems Higher Order Differential Equations Linear equations with constant coefficients homogeneous case Linear equations with constant coefficients inhomogeneous case Variation of parameters Oscillatory Solutions Coupled oscillations and normal modes Systems of first order equations Two point boundary value problems Laplace transform Applications of the Laplace transform 691 Problems FourierSeries Introductory ideas Convergence of Fourier series Different forms of Fourier series 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.

6 CONTENTS / ix 17 Numerical Analysis Errors Solution of linear equations Interpolation Numerical Integration Solution of polynomial and transcendental equations Numerical Solutions of differential equations Determination of eigenvalues and eigenvectors 783 Problems Probability and Statistics Probability, discrete distributions and moments Continuous distributions normal distribution Mean and variance of sum of random variables Statistics inference drawn from observations Linear regression 834 Problems 836 Answers 841 Reference Lists Index 871

Mathematics for Chemists

Mathematics for Chemists MATHEMATICS FOR CHEMISTS D. M. Hirst Department of Molecular Sciences, university of Warwick, Coventry M D. M. Hirst 1976 All rights reserved. No part of this publication may be