A Course of Mathematics for Engineerings and Scientists

Preface to the Second EditionChapter I. First Order Differential Equations 1:1 Introduction-Formation of Differential Equations 1:3 Separable and Homogeneous Types 1:4 First Order Linear Equations-Bernoulli's Equation 1:5 Miscellaneous First Order Types 1:6 Orthogonal Trajectories and Geometrical Applications 1:7 Application to Dynamics-Resisted Motion 1:8 Other Applications 1:9 Graphical Methods 1:10 Picard's Method 1:11 The Taylor Series Method 1:12 Step-By-Step Methods: 1. The Runge-Kutta Method 2. Predictor-Corrector Formula 1:13 The Use of Difference FormulaChapter II. Linear Differential Equations 2:1 Introduction and General Principles 2:2 Linear Differential Equations with Constant Coefficients-The Operator D 2:2a The Complementary Function 2:2b The Operator D 2:2c The Particular Integral 2:2d The Particular Integral by Trial 2:3 Homogeneous Linear Differential Equations 2:4 Simultaneous Linear Differential Equations with Constant Coefficients 2:5 Special Methods For The Solution of Differential Equations of the Second Order 2:6 Separable Partial Differential Equations 2:7 Simple Applications to Particle Dynamics 2:8 Applications to Electric Circuits 2:9 Linear Difference Equations 2:10 Step-By-Step Methods for Second and Higher Order Differential Equations 2:11 Predictor-Corrector Methods 2:12 Build-Up of ErrorChapter III. Linear Equations, Matrices and Determinants 3:1 Introduction 3:2 Linear Equations in Two Unknowns 3:3 Matrix Notation 3:4 The Determinant of a Matrix 3:5 Matrix Algebra 3:6 Some Properties of Determinants 3:7 The Solution of Linear Equations 3:8 The Use of Triangular Matrices 3:9 Singular Matrices 3:10 Linear Dependence 3:11 Eigenvalues and Eigenvectors Chapter IV. Vector Algebra and Coordinate Geometry of Three Dimensions 4:1 The Concept of a Vector 4:2 Cartesian Coordinates and Elements 4:3 The Definitions of Vectors and Scalars 4:4 The Addition and Subtraction of Vectors 4:5 The Scalar Product 4:6 The Vector Product 4:7 Triple Products 4:8 Surfaces in General 4:9 Special Characteristics of Surfaces 4:10 The Sphere 4:11 The Standard Equation of a Quadric Surface 4:12 Curves in Space 4:13 Differentiation and Integration of Vectors with Respect to a Scalar ParameterChapter V. Partial Differentiation 5:1 Continuity and Partial Derivatives 5:2 Applications of the Mean Value Theorem 5:3 Differentiation Under the Integral Sign 5:4 Taylor's Theorem 5:5 Tangent Plane and Normal to a Surface 5:6 Maxima and Minima 5:7 Conditional Stationary Points 5:8 The Method of Least Squares 5:9 Change of Variable 5:10 Jacobians Chapter VI. Multiple Integrals 6:1 Area Integrals 6:2 Volume Integrals 6:3 Change in the Order of Integration in Repeated Integrals 6:4 Change of Variables in Multiple Integrals 6:5 Applications of Multiple Integrals 6:6 Some Special Integrals: Gamma and Beta Functions Answers to the Exercises Index