Worked Examples in Mathematics for Scientists and Engineers

By G. Stephenson

This rich collection of fully worked problems in many areas of mathematics covers all the important subjects students are likely to encounter in their courses, from introductory to final-year undergraduate classes. Because lecture courses tend to focus on theory rather than examples, these exercises offer a valuable complement to classroom teachings, promoting the understanding of mathematical techniques and helping students prepare for exams. They will prove useful to undergraduates in mathematics; students in engineering, physics, and chemistry; and postgraduate scientists looking for a way to refresh their skills in specific topics.
The problems can supplement lecture notes and any conventional text. Starting with functions, inequalities, limits, differentiation, and integration, topics encompass integral inequalities, power series and convergence, complex variables, hyperbolic function, vector and matrix algebra, Laplace transforms, Fourier series, vector calculus, and many other subjects.

• Language: English
• Publisher: Dover Publications
• Release date: Oct 16, 2019
• ISBN: 9780486844619

Book preview

WORKED EXAMPLES

IN MATHEMATICS

for

Scientists & Engineers

WORKED EXAMPLES

IN MATHEMATICS

for

Scientists & Engineers

G. STEPHENSON

DOVER PUBLICATIONS, INC.

Mineola, New York

Copyright

Copyright © 1985 by G. Stephenson

All rights reserved.

Bibliographical Note

This Dover edition, first published in 2019, is an unabridged and corrected republication of the work originally printed by Longman Group Limited, Essex, England, in 1992 [first publication: 1985].

Library of Congress Cataloging-in-Publication Data

Names: Stephenson, G. (Geoffrey), 1927- author.

Title: Worked examples in mathematics for scientists and engineers / G. Stephenson.

Description: Dover edition. | Mineola, New York : Dover Publications, Inc., 2019. | Originally published: London : Longman, 1985. Republished: Essex, England : Longman, 1992. | Includes index.

Identifiers: LCCN 2019011725| ISBN 9780486837369 | ISBN 048683736X

Subjects: LCSH: Mathematics—Problems, exercises, etc. |

Science—Mathematics—Problems, exercises, etc. | Engineering mathematics—Problems, exercises, etc.

Classification: LCC QA43 .S79 2019 | DDC 510.76—dc23

LC record available at https://lccn.loc.gov/2019011725

Manufactured in the United States by LSC Communications

83736X01        2019

www.doverpublications.com

CONTENTS

Preface

1Functions

2Inequalities

3Limits

4Differentiation

5Integration

6Integral inequalities

7Power series and convergence

8Complex variable

9Hyperbolic functions

10 Partial differentiation

11 Stationary values of functions of two variables

12 Determinants and difference equations

13 Vector algebra

14 Matrix algebra

15 Line and double integrals

16 Ordinary differential equations

17 Special functions

18 Laplace transform

19 Fourier series

20 Vector calculus

21 Partial differential equations

22 Functions of matrices

23 Contour integration

24 Fourier transforms

25 Calculus of variations

26 Suffix notation

Index

PREFACE

It is widely accepted that worked examples have an important part to play in the teaching of mathematics, and that they lead to a greater understanding of essential mathematical ideas. Lecture courses usually tend to concentrate, if for no other reason than shortage of time, on the theory rather than examples. This collection of fully worked problems covers most of the topics met in ancillary mathematics courses, and used with a set of lecture notes or conventional textbook should greatly facilitate the understanding of mathematical techniques. It should also provide an effective means of revision for examinations.

The book is especially written for students in scientific disciplines who require mathematical skills, particularly engineers, physicists and chemists. Mathematics students studying mathematical methods as part of their Honours course should, however, find many of these problems of interest, and postgraduate scientists wishing to recall their knowledge of particular topics could well find this book a useful aid.

Most of the examples in this book have been taken from problem sheets I have set for students studying various disciplines at Imperial College. The origins of many of these problems are impossible to identify, and I am therefore grateful to any of my colleagues, past and present, who may have contributed at some time or other to their formulation. Other problems have been chosen from examination papers set for ancillary courses at Imperial College (London University) and have therefore been referred to by (L.U.) in the text. I am grateful to the University for permission to use these problems.; the solutions are, of course, my own responsibility.

I am grateful to Dr. Noel Baker and Norman Froment for reading the manuscript and making a number of valuable comments, and also to my many students who have attempted these problems and shown me more clearly where their difficulties occurred.

Finally it is a pleasure to thank the staff at Longman for their continued friendly cooperation and encouragement.

Imperial College, 1984.

1. FUNCTIONS

[Mappings, even and odd functions, periodic functions, discontinuities, modx, unit step function, [x]]

1.1Given f(x) = 2x² + x + 3, calculate the values at x = 1 of f(x – 1), f(x²), [f(x)]², f(f(x)). Show also that f(x) > 0.

f(x – 1) = 2(x – 1)² + (x — 1) + 3. At x = 1, f(x – 1) = f(0) = 3.

f(1²) = f(1) = 6.

[f(1)]² = 36.

f(f(x)) = 2(2x² + x + 3)² + (2x² + x + 3) + 3. At x = 1, therefore,

f(fwhich, for real x, is always positive.

1.2find g(f(x)) and f(g(x)).

Hence g(f(x)) = f(g(x)). (N.B. This is not necessarily so for other forms of f and g. For example, if f(x) = ex, g(y) = sin y, then f(g(x)) = esin x, whereas g(f(x)) = sin (ex).)

1.3Show that the function inverse to (x + 1)/(x – 1) is the same function.

If f(x) = (x + 1)/(x – 1) – y, say, then x = f–1(y) = (y + 1)/(y – 1). Hence f–1(x) = (x + 1)/(x – 1), which is the function f(x).

1.4Determine the range of the function f(x) = x² + 1 corresponding to each of the x

The graph of f(x) is shown in Fig. 1.

Fig. 1

(a)f(x) lies between 1 (when x = 0) and 2 (when |x| = 1). Hence range is 1 ≤ f ≤ 2. Mapping is many: 1 since if x = ±a, where a ≤ 1, we find the same values of f(x). Mapping is, in fact, 2:1.

(b)f(2) = 5, f(4) = 17. Function is strictly monotonic increasing in 2 ≤ x ≤ 4. Range is 5 ≤ f ≤ 17. For each x in this domain there is only one value of f. Hence mapping is 1:1.

1.5Mod x (=|x|) is defined by

For x ≥ 0, x – |x| = 0. For x < 0, x – |x| = 2x (see Fig. 2).

Fig. 2

1.6Show that the equation x² + 2y³ = 3 determines y as a single-valued function of x, where x and y are real numbers.

The cube root of a real number (positive or negative) is unique. Hence y is a single-valued function.

1.7Determine whether the following functions are even, odd or neither:

(a)x³ + 6x,

(b)x² + 2 sin x,

(c) ex² cos 3x.

Even if f(–x) = f(x), odd if f(–x) = –f(x).

(a)f(x) = x³ + 6x, f(–x) = –x³ – 6x = –f(x). Hence odd.

(b)f(x) = x² + 2 sin x, f(–x) = x² – 2 sin x. Neither even nor odd.

(c)f(x) = ex² cos 3x, f(–x) = ex² cos 3x = f(x). Hence even.

1.8as the sum of an even and an odd function, and state the range of values of x for which the function is defined.

The first term is an odd function of x, and the second an even function. The function exists for x² – 1 > 0 and x = 1.

1.9Find which of the following functions are periodic and state their periods:

(a)cos 2 x,

(c)|sin x|.

f(x) is periodic with period T if f(x + nT) = f(x) for all x (n = 0, 1, 2, . . .).

(a)cos 2(x + nT) = cos 2x if T = π. Hence periodic; period π.

(b)sin x/x is a sine curve with an amplitude decreasing as 1/x (see Fig. 3). Hence not periodic.

(c)|sin x| = |sin (x + nT)| if T = π (see Fig. 4). Periodic; period π.

1.10Sketch the graphs of f(x) = xH(x), where H(x) is the unit step function, and of f(x) = H(x) – H(x – c) + H(x – 2c) – · · ·, where 0 ≤ x < ∞, and c is a positive constant.

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Hence f(x) = xH(x) = 0 for x < 0, and equals x for x ≥ 0 (see Fig. 5). The graph of f(x) = H(x) – H(x – c) + H(x – 2c)– · · · is shown in Fig. 6.

1.11A function f(x) is such that for x < 0, f(x) = x², and for x ≥ 0, f(x) = x. Represent f(x) in terms of x, x² and H(x).

Consider x² + H(x)(x – x²). Then for x < 0 this expression is x². For x ≥ 0, expression is x. Hence f(x) = x² + H(x)(x – x²).

1.12At what points are the following functions discontinuous:

(c) cosec x.

(a)Function has an infinite discontinuity at x = 1.

(b)Function has infinite discontinuities when x² – 4 = 0, i.e. at x = ±2.

(c)cosec x = 1/sin x. Now sin x = 0 at x = nπ, where n = 0, ±1, ±2, . . . . Hence discontinuities at all these values.

1.13The function [x] is defined as the integer part of x. Sketch the graphs of [x], and of x–[x].

[0.4] = 0, [1.2] = 1, [–0.5] = [–1 + 0.5] = –1, and so on (see Fig. 7). x – [x] = x if 0 < x < 1, x – [x] = x – 1 if 1 < x < 2, and so on (see Fig. 8). The period of this function is unity.

Fig. 7

Fig. 8

1.14If f(f(x)) = x² – 2, verify that two solutions are

and

Consider the first solution. Then

Similarly for the second solution.

2. INEQUALITIES

[Arithmetic, geometric and harmonic means, induction methods, Stirling’s formula]

2.1Find the ranges of real values of x which satisfy the in-equalities

(a)x² + 8 < 2x,

(a) We require x² – 2x + 8 < 0. But x² – 2x + 8 = (x – l)² + 7, which is always positive. Hence no real x satisfies this inequality.

(b)For (3x + 4)/(x – 6) > –1 we have (4x – 2)/(x – 6) > 0, which is satisfied