Mathematical Methods

Preface to the Second EditionThe Greek AlphabetChapter I. Introductory Concepts 1 : 1 Functional Notation and Fundamental Definitions 1 : 2 the Roots Of Equations 1 : 3 Elementary Two-Dimensional Coordinate Geometry 1 : 4 Limits and Continuity 1 : 5 Orders Of Magnitude 1 : 6 Inequalities and Quadratic Forms 1 : 7 Mathematical Induction 1 : 8 the Concept Of Convergence 1 : 9 Tabulated Functions-Difference TablesChapter II. The Technique Of Differentiation 2 : 1 Differentiation from First Principles 2 : 2 the Rules of Differentiation 2 : 3 Repeated Differentiation 2 : 4 Exponentials, Logarithms and Hyperbolic Functions 2 : 5 Inverse Functions 2 : 6 Differentiation of Equations 2 : 7 Leibniz's Theorem on Repeated Differentiation 2 : 8 Elementary Partial Differentiation 2 : 9 DifferentialsChapter III. The Technique of Integration 3 : 1 Definitions and Standard Forms 3 : 2 The Definite Integral as the Limit of a Sum 3 : 3 Elementary Rules and Examples 3 : 4 Integration by Substitution 3 : 5 Integration by Parts 3 : 6 Partial Fractions 3 : 7 Integration of Rational Functions 3 : 8 Miscellaneous Methods 3 : 9 Reduction Formulae 3 : 10 The Convergence of Integrals 3 : 11 Functions Defined by IntegralsChapter IV. Geometry of Two Dimensions 4 : 1 Introduction 4 : 2 Gradient, Tangent and Normal 4 : 3 Points Of Inflexion 4 : 4 the Arc Length of a Curve 4 : 5 Curvature 4 : 6 Envelopes 4 : 7 the Loaded Cable: Intrinsic Equations 4 : 8 Polar Coordinates 4 : 9 Curve Sketching 4 : 10 Translation and Rotation of Axes 4 : 11 the Area of a Triangle 4 : 12 the General Equation Of the Second Degree 219 4 : 13 the Properties of the Ellipse 4 : 14 the Properties of the Hyperbola 4 : 15 the Properties of the Parabola 4 : 16 the Polar Equation of a Conic 4 : 17 Some Special CurvesChapter V. Applications of Differentiation 5 : 1 Convergence of Series 5 : 2 Inequalities 5 : 3 The Mean-Value Theorem and Linear Approximations 5 : 4 Taylor's and Maclaurin's Theorems 5 : 5 Expansions in Power Series 5 : 6 Maxima and Minima 5 : 7 Small Increments and Proportional Errors 5 : 8 Kinematics 5 : 9 Approximate Solution Of Equations 5 : 10 Symbolic Operators 5 : 11 Interpolation 5 : 12 Lagrangian Interpolation 5 : 13 Numerical DifferentiationChapter VI. Applications of Integration 352 6 : 1 Introduction-The Area Bounded by a Plane Curve 6 : 2 Volumes and Surfaces of Revolution 6 : 3 Polar Coordinates 6 : 4 First Moments 6 : 5 The Theorems of Pappus 6 : 6 Second Moments-Moments of Inertia 6 : 7 Applications to Hydrostatics 6 : 8 Inequalities Involving Integrals 6 : 9 Numerical IntegrationChapter VII. Complex Numbers 7 : 1 Introduction- The Argand Diagram 7 : 2 De Moivre's Theorem 7 : 3 Multiplication and Division on the Argand Diagram 7 : 4 the Roots Of Complex Numbers 7 : 5 Trigonometric Expansions 7 : 6 Functions of x + iyChapter Viii. Statistics and Probability 8 : 1 Introduction 8 : 2 the Properties of a Frequency Distribution 8 : 3 Introduction to the Theory Of Probability 8 : 4 Some Special Probability Distributions 8 : 5 the Normal Distribution and the Error Function 8 : 6 Further Probability ProblemsAppendix. the Digital Computer A : 1 Introduction A : 2 Steps in the Preparation of a Problem for Solution by Computer A : 3 Flow DiagramsAnswers to ExercisesIndexTables 7, 8 and 9