Engineering Mathematics: Volume 1

This book is the first volume of a two-volume text on mathematics for engineering students in universities and polytechnics, for use in the second and subsequent years of a first degree course. The text is primadly designed to assist engineedng undergraduates and their teachers, but we hope it may also prove of value to students of other disciplines that employ mathematics as a tool, to mathematicians who are interested in applications of their subject, and as a reference book for practising engineers and others. Volume J covers mathematical topics which most engineedng students are required to study; Volume 2 deals with more advanced subjects which are often available as options in the later stages of an undergraduate course. The text is based on courses in mathematics given by the authors to the engineedng students of the University of Nottingham. These courses have evolved over the last sixteen years, and have been developed in close consultation with our fellow teachers in the engineering departments of the University. In preparing the text, we have kept in mind the constraints imposed by the normal three or four year undergraduate course, and we believe that the choice of matedal in the two volumes is realistic in that respect. For completeness, some topics are pursued a little further than an engineedng mathematics lecture course would normally take them, but all the material and examples should be within the grasp of a competent engineering undergraduate student.

Table of Contents:
1. Ordinary Differential Equations.- 1.1 Introduction.- 1.2 Geometrical Interpretation of Solutions of Ordinary Differential Equations.- 1.3 First-order Equations.- 1.4 Linear Ordinary Differential Equations with Constant Coefficients. D Operator Notation.- 1.5 Solution of Homogeneous Linear Equations with Constant Coefficients.- 1.6 Theory of Damped Free Vibrations.- 1.7 Inhomogeneous Second-order Equations with Constant Coefficients.- 1.8 Theory of Forced Vibrations.- 1.9 Simultaneous Linear Differential Equations with Constant Coefficients.- 1.10 Euler’ s Equation 43 Problems 45 Bibliography.- 2. Fourier Series.- 2.1 Introduction.- 2.2 Derivation of the Fourier Series.- 2.3 Convergence of Fourier Series.- 2.4 Fourier Sine and Cosine Series.- 2.5 Integration and Differentiation of Fourier Series.- 2.6 Application of Fourier Series 80 Problems.- 3. Laplace Transforms.- 3.1 Introduction.- 3.2 Transforms of Derivatives.- 3.3 Step Function and Delta Function.- 3.4 Properties of the Laplace Transform.- 3.5 linear Ordinary Differential Equations.- 3.6 Difference and Integral Equations.- 3.7 Some Physical Problems.- 4. Partial Differentiation, with Applications.- 4.1 Basic Results.- 4.2 The Chain Rule and Taylor’s Theorem.- 4.3 Total Derivatives.- 4.4 Stationary Points.- 4.5 Further Applications 159 Problems 163 Bibliography.- 5. Multiple Integrals.- 5.1 Multiple Integrals and Ordinary Integrals.- 5.2 Evaluation of Double Integrals.- 5.3 Triple Integrals.- 5.4 Line Integrals.- 5.5 Surface Integrals 194 Problems 196 Bibliography.- 6. Vector Analysis.- 6.1 Introduction.- 6.2 Vector Functions of One Variable.- 6.3 Scalar and Vector Fields.- 6.4 The Divergence Theorem.- 6.5 Stokes’s Theorem.- 6.6 The Formulation of Partial Differential Equations.- 6.7 OrthogonalCurvilinear Coordinates 234 Problems 241 Bibliography.- 7. Partial Differential Equations.- 7.1 Introduction.- 7.2 The One-dimensional Wave Equation.- 7.3 The Method of Separation of Variables.- 7.4 The Wave Equation.- 7.5 The Heat Conduction and Diffusion Equation.- 7.6 Laplace’s Equation.- 7.7 Laplace’s Equation in Cylindrical and Spherical Polar Coordinates.- 7.8 Inhomogéneous Equations.- 7.9 General Second-order Equations 299 Problems 301 Bibliography.- 8. Linear Algebra — Theory.- 8.1 Systems of Linear Algebraic Equations. Matrix Notation.- 8.2 Elementary Operations of Matrix Algebra.- 8.3 Determinants.- 8.4 The Inverse of a Matrix.- 8.5 Orthogonal Matrices.- 8.6 Partitioned Matrices.- 8.7 Inhomogeneous Systems of Linear Equations.- 8.8 Homogeneous Systems of Linear Equations.- 8.9 Eigenvalues and Eigenvectors 347 Problems 356 Bibliography.- 9. Introduction to Numerical Analysis.- 9.1 Numerical Approximation.- 9.2 Evaluation of Formulae.- 9.3 Flow Diagrams or Charts.- 9.4 Solution of Single Algebraic and Transcendental Equations.- 10. Linear Algebra — Numerical Methods.- 10.1 Introduction.- 10.2 Direct Methods for the Solution of Linear Equations.- 10.3 Iterative Methods for the Solution of Linear Equations.- 10.4 Numerical Methods of Matrix Inversion.- 10.5 Eigenvalues and Eigenvectors 400 Problems 405 Bibliography.- 11. Finite Differences.- 11.1 Introduction.- 11.2 Finite Differences and Difference Tables.- 11.3 Interpolation.- 11.4 Numerical Integration.- 11.5 Numerical Differentiation 430 Problems 432 Bibliography.- 12. Elementary Statistics — Probability Theory.- 12.1 Introduction.- 12.2 Probability and Equi-likely Events.- 12.3 Probability and Relative Frequency.- 12.4 Probability and Set Theory.- 12.5 The Random Variable.- 12.6 Basic Variates.-12.7 Bivariate and Multivariate Probability Distributions.- 12.8 Simulation and Monte Carlo Methods.- Append.- Table A1: Laplace Transforms.- Table A2: The Standardized Normal Variate.- Answers to Exercises and Problems.