Mathematical Methods

Preface to the Second Edition

The Greek Alphabet

Chapter 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 Tables

Chapter 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 Differentials

Chapter 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 Integrals

Chapter 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 Curves

Chapter 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 Differentiation

Chapter 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 Integration

Chapter 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 + iy

Chapter 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 Problems

Appendix. the Digital Computer

A : 1 Introduction

A : 2 Steps in the Preparation of a Problem for Solution by Computer

A : 3 Flow Diagrams

Answers to Exercises

Index

Tables 7, 8 and 9