~~tJbER STUD;? ~lrI6~

Edward Batsche1et

Introduction to Mathematics for Life Scientists

Third Edition

With 227 Figures

Springer-Verlag Berlin Heidelberg New York 1979

Professor Dr. Edward Batschelet

Mathematisches Institut der UniversiHit Zurich, Switzerland

AMS Subject Classifications (1970) 92-01, 92A05, 98A35, 98A25, 60-01, 93-01, 40-01, 04-01, 15-01, 26-01, 26A06,

26A09, 26A12, 34-01, 35-01

ISBN-13: 978-3-540-09648-1 DOl: 10.1007/978-3-642-61869-7

e-ISBN-13: 978-3-642-61869-7

The first edition of this work was originally published by Springer-Verlag Berlin, Heidelberg in 1971 as volume 2 in the series Biomathematics, edited by K. Kricke­

berg, R. C. Lewontin, J. Neyman and M. Schreiber

Library of Congress Cataloging in Publication Data. Batscheiet, Edward. Introduction to mathematics for life

scientists. "Springer study edition." Bibliography: p. Includes index. 1. Biomathematics. I. Title. QH323.5.B37 1979b./510'.24'574./79-21113

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying, machine or similar means, and storage in data banks. Under § 54 of the German Copy­right Law where copies are made for other than private use, a fee is payabk: to the publisher, the amount

to the fee to be determined by agreement with the publisher.

Springer-Verlag Berlin Heidelberg New York a member of Springer, Science+Business Media

© by Springer-Verlag Berlin Heidelberg 1971, 1975, 1979

Typesetting: Briihlsche UniversiHitsdruckerei, GieBen

SPIN 114034632141/3111-54

9 8 7

Preface to the First Edition

A few decades ago mathematics played a modest role in life sciences. Today, however, a great variety of mathematical methods is applied in biology and medicine. Practically every mathematical procedure that is useful in physics, chemistry, engineering, and economics has also found an important application in the life sciences.

The past and present training of life scientists does by no means reflect this development. However, the impact ofthe fast growing number of applications of mathematical methods makes it indispensable that students in the life sciences are offered a basic training in mathematics, both on the undergraduate and the graduate level. This book is primarily designed as a textbook for an introductory course. Life scientists may also use it as a reference to find mathematical methods suitable to their research problems. Moreover, the book should be appropriate for self-teaching. It will also be a guide for teachers. Numerous references are included to assist the reader in his search for the pertinent literature.

Life scientists are hardly interested in going deeply into mathe­matics. Therefore, this course differs in many ways from a course offered to mathematicians. Each concept is introduced in an intuitive way. The reader is being kept informed why he is learning a particular method. The relevance of all procedures is proven by examples that have been selected from a wide area of research in the life sciences. It is not intended to distract the student of biology from his main field of activity and to train him as a competent mathematician. The aim is rather to prepare him for an understanding of the basic mathematical operations and to enable him to communicate successfully with a mathematician in case he needs his help.

Many illustrations and some historical notes are inserted to encourage the life scientist who is perhaps somewhat reluctant to be involved with the abstract side of mathematics. Most problems were tested in class. Sections and problems marked with an asterisk are not necessarily more difficult, but may be omitted on first reading.

The book avoids as much as possible the introduction of cookbook mathematics. This requires a somewhat broad presentation. As a conse­quence no attempt is made to comprise all mathematical methods that are important for life scientists. For instance, computer techniques and statistics are omitted. These two areas can only be presented in special

VI Preface

volumes. However, the reader will be prepared for an easier understanding of all topics that could not be covered in this book.

In the beginning I was encouraged to prepare the manuscript by Dr. Sidney R. Galler, SmithonianInstitution. Numerous friends supported the idea and gave me valuable advice and inspiration. I am unable to list all of them. I am very obliged to those biologists who read some chapters and offered valuable criticism and suggestions, especially to Dr. J. P. Hailman, University of Wisconsin, Dr. J. Hegmann and Dr. R. Milkman, both at the University of Iowa, Dr. W. M. Schleidt, University of Maryland.

I gratefully. acknowledge the encouragement and considerable sup­port which I received by Dr. Eugene Lukacs, Director of the Statistical Laboratory at Catholic University, Washington, D.C. Some of the more difficult illustrations were made by Mr. C. H. Reinecke with finan­cial support by the Office of Naval Research. I also enjoyed the advice by Dr. V. Ziswiler. The text was carefully typewritten by Mrs. Amelia Miller and Mrs. Phyllis Spath elf for whose patience I wish to express my gratitude. Stylistic, grammatical errors, and other shortcomings were corrected by Dr. Inge F. Christensen and Dr. Maren Brown with great care. I am also indebted to my wife and to Mrs. Eva Minzloff for proof­reading and to the staff of the Springer-Verlag for the careful edition.

I would appreciate it if the readers would draw my attention to errors, obscurities and misprints that might still be present in print.

Zurich, October 1971 Edward Batschelet

Preface to the Second Edition Many users of the first edition complained that the problem section

was not large enough. For this reason, numerous problems, both solved and unsolved, were added to the second edition. They are listed at the end of each chapter, but numbered according to sections to facilitate the assignment of problems. At the end of the book the solutions for the odd numbered problems are given.

To make the book self-contained, an appendix with ten numerical tables was added. Chapter 9 was enlarged by a section on methods of integration and Chapter 14 by four sections on determinants and related topics. Many parts of the book were updated and provided with new references. Further, 28 illustrations were added. As a result of the alterations, the size of the book has been enlarged by about a hundred pages.

Many scholars, too many to be listed here, have kindly given me their advice. In addition, lowe particular thanks to Mrs. Alice Peters, Springer-Verlag, for her most valuable recommendations. Dr. Joan Davis edited the text for stylistic errors, and Dr. Armand Wyler carefully checked the problem section. New illustrations were drawn, the manu­script typed, proofs read and reread by Mrs. R. Boller, Mrs. C. Heinzer, Mrs. B. Henop, and by my wife. I had also the unfailing cooperation of the staff of Springer-Verlag. They all deserve my warmest thanks for their help and patience.

Readers are again requested to draw my attention to errors, obscurities and misprints.

Zurich, June 1975 Edward Batschelet

Preface to the Third Edition Since the second edition appeared in 1975, the units of the "Systeme

international" (SI-units) became legal. To comply with this sysh:m, [ converted calories into Joules, dynes into Newtons, units of pressure into Pascal, and the unit of viscosity, poise, into Pascalsecond. I also used the opportunity to eliminate some misprints and obscurities. I should like to thank many readers for their help, especially Dr. Tor Gulliksen, University of Oslo, for his substantial contributions.

University of Zurich July, 1979

Edward Batschelet

Contents Chapter 1. Real Numbers 1.1 Introduction . . . . . . . . . 1.2 Classification and Measurement . 1.3 A Problem with Percentages . . 1.4 Proper and Improper Use of Percentages 1.5 Algebraic Laws . 1.6 Relative Numbers 1.7 Inequalities. 1.8 Mean Values 1.9 Summation. 1.10 Powers 1.11 Fractional Powers . 1.12 Calculations with Approximate Numbers

*1.13 An Application . 1.14 Survey. . . . .

Problems for Solution

Chapter 2. Sets and Symbolic Logic 2.1 "New Mathematics" . . 2.2 Sets . . ... . . . .. 2.3 Notations and Symbols 2.4 Variable Members. 2.5 Complementary Set 2.6 The Union . . . 2.7 The Intersection. .

*2.8 Symbolic Logic . . *2.9 Negation and Implication *2.10 Boolean Algebra

Problems for Solution . . . .

Chapter 3. Relations and Functions 3.1 Introduction 3.2 Product Sets 3.3 Relations. 3.4 Functions .

1 1 4 6 7

10 13 14 15 17 20 22 24 25

26

36 36 37 40 40 41 42 45 48 50

55

59 59 62 65

x Contents

3.5 A Special Linear Function . 3.6 The General Linear Function

*3.7 Linear Relations

Problems for Solution . . . . .

Chapter 4. The Power Function and Related Functions 4.1 Definitions. . . . . . . . . 4.2 Examples of Power Functions . 4.3 Polynomials . 4.4 Differences. . . . . 4.5 An Application . . . 4.6 Quadratic Equations .

Problems for Solution . .

Chapter 5. Periodic Functions 5.1 Introduction and Definition. 5.2 Angles. . . . . . 5.3 Polar Coordinates. . . . . 5.4 Sine and Cosine. . . . . . 5.5 Conversion of Polar Coordinates 5.6 Right Triangles . . . . 5.7 Trigonometric Relations . .

*5.8 Polar Graphs. . . . . . . *5.9 Trigonometric Polynomials .

Problems for Solution . . . . .

Chapter 6. Exponential and Logarithmic Functions I 6.1 Sequences . . . . . . . 6.2 The Exponential Function . 6.3 Inverse Functions . . . . . 6.4 The Logarithmic Functions . 6.5 Applications

*6.6 Scaling. . . . . *6.7 Spirals. . . . .

Problems for Solution

Chapter 7. Graphical Methods 7.1 Nonlinear Scales . . . 7.2 Semilogarithmic Plot 7.3 Double-Logarithmic Plot.

70 74 78

82

90 91 97 99

101 104

106

110 111 114 115 119 123 129 130 132

139

143 146 148 152 155 157 162

165

170 173 176

Contents XI

7.4 Triangular Charts . 179 *7.5 Nomography. . . 183 *7.6 Pictorial Views . .188

Problems for Solution . 195

Chapter 8. Limits 8.1 Limits of Sequences .202 8.2 Some Special Limits .209 8.3 Series . . . . . . .212 8.4 Limits of Functions .217

*8.5 The Fibonacci Sequence . 224

Problems for Solution . . . . 228

Chapter 9. Differential and Integral Calculus 9.1 Growth Rates. . . .234 9.2 Differentiation . . .242 9.3 The Antiderivative . .251 9.4 Integrals. . . . . . 253 9.5 Integration. . . . .259 9.6 The Second Derivative . . 265 9.7 Extremes. . . . . . . . 272 9.8 Mean of a Continuous Function. .281 9.9 Small Changes . . . . . .286

*9.10 Techniques of Integration .289

Problems for Solution . . . . .291

Chapter 10. Exponential and Logarithmic Functions II

10.1 Introduction. . . . 301 to.2 Integral of 11x . . . . . . . 302 10.3 Properties of lnx. . . . . . 304 10.4 The Inverse Function oflnx .306 10.5 The General Defmition of a Power . 308 10.6 Relationship between Natural and Common Logarithms. .310 10.7 Differentiation and Integration. .311 to.8 Some Limits. . . . . . . . . . . . .313 10.9 Applications. . . . . . . . . . . . .314 10.10 Approximations and Series Expansions .320

*10.11 Hyperbolic Functions. .324

Problems for Solution . . . . . . . . . . . 327

XII Contents

Chapter 11. Ordinary Differential Equations 11.1 Introduction. . . . . . . . . . . 334 11.2 Geometric Interpretation . . . . . . 335 11.3 The Differential Equation y' = ay. . .336 11.4 The Differential Equation y' = ay+ b .346 11.5 The Differential Equation y' =ay2+ by+ c . .352 11.6 The Differential Equation dy/dx=k· yjx . .359 11.7 A System of Linear Differential Equations . . 361 11.8 A System of Nonlinear Differential Equations . 369

*11.9 Classification of Differential Equations .373

Problems for Solution . . . . . . . . . . . . . . 375

Chapter 12. Functions of Two or More Independent Variables 12.1 Introduction. . . . .381 12.2 Partial Derivatives . . . . . .384 12.3 Maxima and Minima . . . . .387

*12.4 Partial Differential Equations .392

Problems for Solution . . . . . . .398

Chapter 13. Probability

13.1 Introduction. . . . . . . .401 13.2 Events . . . . . . . . . .402 13.3 The Concept of Probability .405 13.4 The Axioms of Probability Theory .409 13.5 Conditional Probabilities .412 13.6 The Multiplication Rule. .419 13.7 Counting . . . . . . .423 13.8 Binomial Distribution .431 13.9 Random Variables . . .439 13.10 The Poisson Distribution .446 13.11 Continuous Distributions .452

Problems for Solution . . . . .463

Chapter 14. Matrices and Vectors

14.1 Notations. . . .475 14.2 Matrix Algebra .477 14.3 Applications. . .485 14.4 Vectors in Space .495 14.5 Applications .502 14.6 Determinants .511

Contents

14.7 Inverse of a Matrix ... . 14.8 Linear Dependence ... . 14.9 Eigenvalues and Eigenvectors

Problems for Solution . . . .

Chapter 15. Complex Numbers 15.1 Introduction 15.2 The Complex Plane. . . 15.3 Algebraic Operations . . 15.4 Exponential Functions of Complex Variables . 15.5 Quadratic Equations 15.6 Oscillations . .

Problems for Solution .

Appendix (Tables A to K)

Solutions to Odd Numbered Problems

References . . . . . .

Author and Subject Index

XIII

.518

.521

.527

.531

.547

.548

.551 .554 .559 .560

.569

.572

.587

.610

.623

Index of Symbols a=l=b a is not equal to b 12 a~b a is approximately equal to b 287 b>a b is greater than a 11 b~a b is greater than or equal to a 12 a<b a is less than b 11 a~b a is less than or equal to b 12 lal absolute value of a 11

n

L Xk summation of Xk' k ranging from 1 to n 16 k=l

1 n

X= - L xk arithmetic mean of Xl' X2 , """' xn 16 n k=l

{Xl' X2 , """' Xn} set containing Xl' X2, """' xn 37 {xla<x<b} set of all X between a and b 40 xeA X is element of set A 38 x¢A X does not belong to A 38 ACB A is subset of B 38 B)A B contains A 38 0 empty set, null set 37 IN set of natural numbers 37 IR set of real numbers 38 A complementary set 41 AuB union of A and B 41 AnB intersection of A and B 43 A\B subtraction of sets 44 AxB product set, Cartesian product 60 (x,y) ordered pair 60 PVQ PorQ 46 PAQ PandQ 47 ...,P not P, negation 48 P=Q P implies Q 49 x~y X is mapped into y 67 an--. 00 an diverges to infinity 203 an--.A an converges to A 205 lim an limit of an as n tends to infinity 203 n-+ ao

xjxo X tends to Xo from lower values 220

x!xo X tends to Xo from higher values 220

Index of Symbols XV

yocx y is proportional to x 71 logx common logarithm of x 153, 311 lnx natural logarithm of x 303 expx=ex exponential function of x 307 L1y difference of y 74,99 L12y second difference of y 99 L1y/L1x difference quotient 238 f'(x) = dy/dx derivative 240

f"(x)= d2y dx2 second derivative 266

f = oz x ox partial derivative 384

[a,b] interval from a to b 258 b

J f(x) dx definite integral 254,261 a

J f(x) dx indefinite integral 262 bx small change 287 n! n factorial 10

(~) binomial coefficient 426

P(A) probability of event A 406 P(AIB) conditional probability 415 P(X=k) probability that X takes on k 441 E(X) expectation (mean) of X 442 Var(X) variance of X 444

A= (: ~) matrix 475

I: ~1=detA determinant 511

aik double subscript 476 A' transpose of matrix 483 lal absolute value of vector 498, 501 z=x+iy complex number 548 Rez real part of z 548 Imz imaginary part of z 548 z=x-iy complex conjugate 552