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Mathematical Methods

A Course of Mathematics for Engineers and Scientists

2nd Edition - January 1, 1970

Authors: Brian H. Chirgwin, Charles Plumpton

eBook ISBN:

9 7 8 - 1 - 4 8 3 1 - 8 1 9 3 - 6

Mathematical Methods is an introductory course on mathematical methods for students aiming for a first degree in engineering or science. Topics covered include differentiation and… Read more

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Mathematical Methods is an introductory course on mathematical methods for students aiming for a first degree in engineering or science. Topics covered include differentiation and integration and their applications; the geometry of two dimensions, and complex numbers. Statistics and probability are also discussed. Comprised of eight chapters, this volume begins with an introduction to fundamental concepts, including the roots of equations; elementary two-dimensional coordinate geometry; limits and continuity; inequalities and quadratic forms; mathematical induction; and convergence. The discussion then turns to the techniques of differentiation and integration and their applications; the geometry of two dimensions; and complex numbers and their roots, together with trigonometric expansions. The book concludes with a chapter on statistics and probability, paying particular attention to the properties of a frequency distribution; some special probability distributions; normal distribution and the error function; and some probability problems. This monograph is intended for students taking a course in engineering or science.

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