A Course of Mathematics for Engineerings and Scientists

Preface to the Second Edition

Chapter 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 Formula

Chapter 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 Error

Chapter 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 Parameter

Chapter 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