Basic Mathematics for the Biological and Social Sciences - 1st Edition - ISBN: 9780080066646, 9781483136257

Basic Mathematics for the Biological and Social Sciences

1st Edition

Authors: F. H. C. Marriott
eBook ISBN: 9781483136257
Imprint: Pergamon
Published Date: 1st January 1970
Page Count: 342
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Basic Mathematics for the Biological and Social Sciences deals with the applications of basic mathematics in the biological and social sciences. Mathematical concepts that are discussed in this book include graphical methods, differentiation, trigonometrical or circular functions, limits and convergence, integration, vectors, and differential equations. The exponential function and related functions are also considered. This monograph is comprised of 11 chapters and begins with an overview of basic algebra, followed by an introduction to infinitesimal calculus, scalar and vector quantities, complex numbers, and the simplest types of differential equation. The use of graphs in the presentation of data is also described, along with limits and convergence, rules for differentiation, the exponential function, and maxima and minima. Techniques of integration, vectors and their derivatives, and simultaneous differential equations are explored as well. Examples from biology, economics and related subjects, probability theory, and physics are provided. This text will be a useful resource for mathematicians as well as biologists and social scientists interested in applying mathematics to their work.

Table of Contents



1. Basic Algebra

1.1. Symbols and Notation


The Sigma Notation


1.2. The Binomial Theorem

An Application to Probability Theory

1.3. Partial Fractions


2. Graphical Methods

2.1. Introduction

2.2. The Graphical Presentation of Data

2.3. Special Types of Graph

Changes of Scale

The Histogram, or Block Diagram

2.4. Cartesian Coordinates in Two Dimensions

2.5. Features of Plane Curves

Particular Points


Bounds and Impossible Areas

Maxima, Minima and Inflexions


2.6. The Conic Sections

2.7. Polar Coordinates

2.8. Three-dimensional Problems

2.9. Three-dimensional Polar Coordinates


3. Trigonometrical or Circular Functions

3.1. Definitions

3.2. Properties of the Trigonometrical Functions

3.3. The Graphs of the Trigonometrical Functions

3.4. The Inverse Functions

3.5. Applications of Trigonometric Functions


4. Limits and Convergence

4.1. The Idea of a Limit

4.2. Definition of a Limit

4.3. Series and Convergence

4.4. Some Important Limits

(i) A Theorem on Limits

(ii) Rational Functions

(iii) Rational Functions of x

(iv) The Limits of (sin x)/x . . . and (1—cos x)/x

4.5. The Importance of Limits


5. Differentiation (1)

5.1. Introduction

5.2. The Derivative

5.3. Standard Derivatives

5.4. Rules for Differentiation

5.5. Higher Derivatives

5.6. Differentials

5.7. Maxima and Minima

5.8. Small Errors

5.9. Newton's Method of Approximation


6. The Exponential Function and Related Functions

6.1. Introduction

6.2. Definition and Properties of the Exponential Function

6.3. The Natural Logarithm

6.4. The Hyperbolic Functions

6.5. Growth Curves


7. Differentiation (2)

7.1. Taylor's and Maclaurin's Series

7.2. Functions of Several Variables

7.3. Partial Derivatives

7.4. Small Errors

7.5. Maxima and Minima

7.6. Taylor's Theorem

7.7. Partial Differential Equations

7.8. Change of Variables

7.9. Maximum Subject to Constraints

7.10. Implicit Functions


8. Integration

8.1. Introduction

8.2. Area and the Definite Integral

8.3. The Indefinite Integral

8.4. Improper Integrals

8.5. Techniques of Integration

(i) Integration by Substitution

(ii) Rational Functions and Partial Fractions

(iii) Integration by Parts

8.6. Numerical Methods

(i) The Trapezium Rule

(ii) Simpson's Rule

8.7. Multiple Integrals

8.8. Miscellaneous Results

(i) Differentiation of Integrals

(ii) Stirling's Approximation

(iii) Mean Values and Centers of Mass


9. Vectors

9.1. Scalar and Vector Quantities

9.2. A Digression on Mechanics

9.3. Vectors in One Dimension

9.4. Vectors and Their Components

9.5. Derivatives of Vectors; Gradients

9.6. The Product of Two Vectors

9.7. The Line-integral


10. Complex Numbers

10.1. Introduction

10.2. Elementary Manipulation of Complex Numbers

Basic Rules

Modulus and Argument

The n-th Roots of a Complex Number

Roots of Polynomials

Complex Numbers and Vectors

10.3. Functions of Complex Variables

The Exponential Function

The Logarithmic Function

Trigonometric and Hyperbolic Functions

10.4. Applications of Complex Algebra

Differential Equations

Alternating Current Electricity


11. Differential Equations

11.1. Introduction

11.2. Arbitrary Constants and Initial Conditions

11.3. A First-order Differential Equation

11.4. Linear Equations with Constant Coefficients

11.5. The Complementary Function

11.6. Damped Oscillations

11.7. Particular Integrals

11.8. Forced Oscillations and Resonance

11.9. Simple First-Order Equations

(i) Variables Separable

(ii) Exact Derivatives

(iii) Linear Equations

11.10. Simultaneous Differential Equations

11.11. Numerical Methods


Appendix A. A Note on Definitions

Appendix B. Infinite Series and Convergence

Appendix C. Tables of the Exponential and Natural Logarithmic Function

Answers to Examples

Suggestions for Further Reading




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About the Author

F. H. C. Marriott

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