A brief note of explanation concerning the origin of this book might prove of interest to the reader. Initially, it was envisaged as one volume in a series dealing with various aspects of physical processes in the chemical industry and it was intended that it should cover the mathematical techniques applied in the companion volumes. During the course of writing, however, it became increasingly evident that the contents could, with little modification, be of interest to a considerably wider readership than that for which the series was intended. It was therefore decided to publish the work in its own right.
The aim of the authors throughout has been to produce a text of wide range from which the student could derive the maximum benefit with the minimum of assistance from other sources. With this aim in view, the book starts with a revision course in basic algebra, geometry, and trigonometry, and subsequent chapters range over a wide field of mathematical techniques and their applications. Although the chapters are arranged in logical sequence, many are virtually self-contained and may be read in isolation. It is hoped that the inclusion of a large number of worked examples will materially assist the reader who is attempting to teach himself.
CONTENTS -
PREFACE -
1. REVIEW OF ELEMENTARY ALGEBRA -
Indices; Logarithms; Graphs-Solution of Linear Equations; Solution of Quadratic Equations in One Variable; Approximate Solution of Higher Order Equations; Solution of Exponential Equations -
2. REVIEW OF ELEMENTARY PLANE TRIGONOMETRY -
The Circular Functions; Radian Measure; Simple Applications of Trigonometry; Solution of General Triangles; Circular Functions of the Sum and Difference of Two Angles; Solution of Trigonometric Equations; The Inverse Circular Functions -
3. CO-ORDINATE PLANE GEOMETRY OF THE STRAIGHT LINE -
Cartesian and Polar Co-ordinates; Loci; Various Forms of the Equation of a Line; Distance from a Point to a Line; Angle between Two Lines; Point of Intersection of Two Lines; Pairs of Lines; Problems Involving Straight Lines -
4. CO-ORDINATE PLANE GEOMETRY OF THE CIRCLE -
Equations of a Circle; Equation of the Tangent to a Circle; Equation of the Normal to a Circle; Length of the Tangent from a Point to a Circle; Intersection of a Circle with a Line; Intersection of Two Circles; Problems Involving Circles and Lines -
5. THE BINOMIAL EXPANSION -
Permutations and Combinations; The Binomial Expansion for Positive Integral Index; Some Properties of the Binomial Coefficients; The Binomial Expansion for any Index; Validity of the Binomial Expansion for any Index; Applications of the Binomial Expansion -
6. PARTIAL FRACTIONS -
Rational Functions; Addition and Subtraction of Rational Functions ; Partial Fractions Viewed as an Inverse Process; The Denominator g(ⅹ) Containing Non-Repeated Linear Factors; The Denominator g(ⅹ) Containing Repeated Linear Factors; The Denominator g(x) Containing Non-Repeated Quadratic Factors; The Denominator g(x) Containing Repeated Quadratic Factors; The General Denominator Applications of Partial Fractions -
7. FUNCTIONS AND LIMITS -
Functional Relationships; Geometrical Representation of a Functional
Relationship; Values and Limits of a Function; The Derivative of a Function -
8. DIFFERENTIATION -
Differentiation from First Principles; Differentiation of x"-Differentiation of sin x and cos x; Differentiation of Sums, Differences, Products, and Quotients; Differentiation of Compound Circular Functions; Differentiation of a Function of a Function-Differentiation of log x; The Exponential and Hyperbolic Functions; Differentiation
of Inverse Circular and Inverse Hyperbolic Functions -
9. APPLICATIONS OF DIFFERENTIATION -
Tangents and Normals to Curves; Estimation of Small Errors; Maclaurin and Taylor Series; Newton's Method for the Approximate Solution of Equations; Maxima and Minima -
10. INTEGRATION -
Integration as the Inverse of Differentiation; Special Methods of Integration; Integration Using Partial Fractions; Integration by Substitution; Integration by Parts; The Definite Integral; Reduction Formulae -
11. APPLICATIONS OF INTEGRATION -
Areas Bounded by Plane Curves; Mean Values; Arc Lengths of Plane Curves; Volumes of Solids of Revolution; Surfaces of Revolution; Centres of Gravity; Moments of Inertia; Hydrostatic Thrust and Centres of Pressure -
12. COMPLEX NUMBERS -
General Solution of the Quadratic Equation; The Argand Diagram; Addition,
Subtraction, Multiplication, and Division of Complex Numbers-The Polar Form of the Complex Number; Multiplication and Division in Polar Form; Powers and Roots of Complex Numbers; Functions of the Complex Variable -
13. ORDINARY LINEAR DIFFERENTIAL EQUATIONS -
Definition and Formulation of Differential Equations; Differential Equations Solvable by Direct Integration; First Order Linear Differential Equation Solvable by Use of Integrating Factor; Second Order Linear Differential Equations; The Complementary Function; The Particular Integral; Methods for Finding Particular Integrals; The Steady State Solution; Simultaneous Linear Differential Equations -
14 PARTIAL DIFFERENTIATION -
Functions of Two Variables; Partial Derivatives; The Total Differential; Applications of Partial Differentiation -
15. STATISTICAL METHOD -
Collective Properties; Variation; Populations and Samples; Universal Existence of Variation; Frequency Diagrams as a Simple Tool; Discrete and Continuous Data; Frequency Diagram for Discrete Data; Frequency Diagram for Continuous Data -
16. STANDARD NUMERICAL MEASURES FOR DESCRIBING FREQUENCY DIAGRAMS -
Properties of Frequency Diagrams; Measures of Location; Measures of Dispersion; Skewness -
17. PROBABILITY THEORY AND ITS APPLICATIONS -
Probability; Laws of Probability -
18. BINOMIAL AND POISSON DISTRIBUTIONS -
Binomial Distributions; Poisson Distribution; Cumulative Probabilities;
Poisson Distribution as Approximation to the Binomial -
19. NORMAL DISTRIBUTION -
Handling Data which Arise from Measurement; Normal Distribution
as Approximation to the Binomial; Normal Distribution as Approximation
to the Poisson -
20. CONTROL CHARTS -
Control Charts for Number Defective; Standard Error of the Average; Control Limits; Control Charts for Average and Range -
21. POPULATION AND SAMPLE -
The Sample Used to Predict the Population; Degrees of Freedom Distribution
of Averages; Confidence Limits; Single-sided and Double-sided Confidence Limits; Distribution of the Ratio of Two Variance Estimates -
22. DISTRIBUTION OF DIFFERENCES -
Differences between Averages; Confidence Limits on the Difference (ⅹ₁-ⅹ₂); Differences between Variances: Confidence Limits on the Ratio s½∕s½ -
23. TESTS OF SIGNIFICANCE -
The Nature of Tests of Significance; Comparison of Variances (F Test); Comparison of Averages (t Test); Relationship between Confidence Limits and Tests of Significance -
24. ANALYSIS OF VARIANCE -
A Simple Randomized Experiment; Use of Randomized Blocks; A Two-Factor Experiment; Interaction -
25. REGRESSION AND CORRELATION (LINEAR RELATIONSHIPS) -
Purpose of Linear Relationships; Scatter Diagrams; Three-Group Method; Calculation of the Best Straight Line; Variance of Errors of Prediction -
ANSWERS TO EXERCISES -
INDEX -
STATISTICAL TABLES -
1. Binomial Distribution -
2. Poisson Distribution -
3. Normal Distribution -
4. Number-Defective Control Limits from Poisson distribution -
5. Ratio of Range to Standard Deviation -
6. Control Limit Factors for Average Chart -
7. Control Limit Factors for Range Chart -
8. Percentage Points of t Distribution -
9. Estimation of Standard Deviation from Average Range -
10. Percentage Points of F Distribution -
CONTENTS - 1. REVIEW OF ELEMENTARY ALGEBRA - Indices; Logarithms; Graphs-Solution of Linear Equations; Solution of Quadratic Equations in One Variable; Approximate Solution of Higher Order Equations; Solution of Exponential Equations - 2. REVIEW OF ELEMENTARY PLANE TRIGONOMETRY - The Circular Functions; Radian Measure; Simple Applications of Trigonometry; Solution of General Triangles; Circular Functions of the Sum and Difference of Two Angles; Solution of Trigonometric Equations; The Inverse Circular Functions - 3. CO-ORDINATE PLANE GEOMETRY OF THE STRAIGHT LINE - Cartesian and Polar Co-ordinates; Loci; Various Forms of the Equation of a Line; Distance from a Point to a Line; Angle between Two Lines; Point of Intersection of Two Lines; Pairs of Lines; Problems Involving Straight Lines - 4. CO-ORDINATE PLANE GEOMETRY OF THE CIRCLE - Equations of a Circle; Equation of the Tangent to a Circle; Equation of the Normal to a Circle; Length of the Tangent from a Point to a Circle; Intersection of a Circle with a Line; Intersection of Two Circles; Problems Involving Circles and Lines - 5. THE BINOMIAL EXPANSION - Permutations and Combinations; The Binomial Expansion for Positive Integral Index; Some Properties of the Binomial Coefficients; The Binomial Expansion for any Index; Validity of the Binomial Expansion for any Index; Applications of the Binomial Expansion - 6. PARTIAL FRACTIONS - Rational Functions; Addition and Subtraction of Rational Functions; Partial Fractions Viewed as an Inverse Process; The Denominator g(ⅹ) Containing Non-Repeated Linear Factors; The Denominator g(ⅹ) Containing Repeated Linear Factors; The Denominator g(x) Containing Non-Repeated Quadratic Factors; The Denominator g(x) Containing Repeated Quadratic Factors; The General Denominator Applications of Partial Fractions - 7. FUNCTIONS AND LIMITS - Functional Relationships; Geometrical Representation of a Functional Relationship; Values and Limits of a Function; The Derivative of a Function - 8. DIFFERENTIATION - Differentiation from First Principles; Differentiation of x"-Differentiation of sin x and cos x; Differentiation of Sums, Differences, Products, and Quotients; Differentiation of Compound Circular Functions; Differentiation of a Function of a Function-Differentiation of log x; The Exponential and Hyperbolic Functions; Differentiation of Inverse Circular and Inverse Hyperbolic Functions - 9. APPLICATIONS OF DIFFERENTIATION - Tangents and Normals to Curves; Estimation of Small Errors; Maclaurin and Taylor Series; Newton's Method for the Approximate Solution of Equations; Maxima and Minima - 10. INTEGRATION - Integration as the Inverse of Differentiation; Special Methods of Integration; Integration Using Partial Fractions; Integration by Substitution; Integration by Parts; The Definite Integral; Reduction Formulae - 11. APPLICATIONS OF INTEGRATION - Areas Bounded by Plane Curves; Mean Values; Arc Lengths of Plane Curves; Volumes of Solids of Revolution; Surfaces of Revolution; Centres of Gravity; Moments of Inertia; Hydrostatic Thrust and Centres of Pressure - 12. COMPLEX NUMBERS - 13. ORDINARY LINEAR DIFFERENTIAL EQUATIONS - 14 PARTIAL DIFFERENTIATION - 15. STATISTICAL METHOD - 16. STANDARD NUMERICAL MEASURES FOR DESCRIBING FREQUENCY DIAGRAMS - 17. PROBABILITY THEORY AND ITS APPLICATIONS - 18. BINOMIAL AND POISSON DISTRIBUTIONS - 19. NORMAL DISTRIBUTION - 20. CONTROL CHARTS - 21. POPULATION AND SAMPLE - 22. DISTRIBUTION OF DIFFERENCES - 23. TESTS OF SIGNIFICANCE - 24. ANALYSIS OF VARIANCE - 25. REGRESSION AND CORRELATION (LINEAR RELATIONSHIPS) -