Engineering Mathematics

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.- 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 Orthogonal Curvilinear 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 Inhomogeneous 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.- Problems.- 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.- Problems.- Append.- Table A1: Laplace Transforms.- Table A2: The Standardized Normal Variate.- Answers to Exercises and Problems.