Essential Math Skills for Engineers

CONTENTS

PREFACE

1 What Do Engineers Do?

2 Miscellaneous Math Skills

2.1 EQUATIONS OF LINES, PLANES, AND CIRCLES

2.2 AREAS AND VOLUMES OF COMMON SHAPES

2.3 ROOTS OF A QUADRATIC EQUATION

2.4 LOGARITHMS

2.5 REDUCTION OF FRACTIONS AND LOWEST COMMON DENOMINATORS

2.6 LONG DIVISION

2.7 TRIGONOMETRY

2.8 COMPLEX NUMBERS AND ALGEBRA, AND EULER’S IDENTITY

2.9 COMMON DERIVATIVES AND THEIR INTERPRETATION

2.10 COMMON INTEGRALS AND THEIR INTERPRETATION

2.11 NUMERICAL INTEGRATION

3 Solution of Simultaneous, Linear, Algebraic Equations

3.1 HOW TO IDENTIFY SIMULTANEOUS, LINEAR, ALGEBRAIC EQUATIONS

3.2 THE MEANING OF A SOLUTION

3.3 CRAMER’S RULE AND SYMBOLIC EQUATIONS

3.4 GAUSS ELIMINATION

3.5 MATRIX ALGEBRA

4 SOLUTION OF LINEAR, CONSTANT-COEFfiCIENT, ORDINARY DIFFERENTIAL EQUATIONS

4.1 HOW TO IDENTIFY LINEAR, CONSTANT-COEFfiCIENT, ORDINARY DIFFERENTIAL EQUATIONS

4.2 WHERE THEY ARISE: THE MEANING OF A SOLUTION

4.3 SOLUTION OF FIRST-ORDER EQUATIONS

4.4 SOLUTION OF SECOND-ORDER EQUATIONS

4.5 STABILITY OF THE SOLUTION

4.6 SOLUTION OF SIMULTANEOUS SETS OF ORDINARY DIFFERENTIAL EQUATIONS WITH THE DIFFERENTIAL OPERATOR

4.7 NUMERICAL (COMPUTER) SOLUTIONS

5 Solution of Linear, Constant-Coefficient, Difference Equations

5.1 WHERE DIFFERENCE EQUATIONS ARISE

5.2 HOW TO IDENTIFY LINEAR, CONSTANT-COEFFICIENT DIFFERENCE EQUATIONS

5.3 SOLUTION OF FIRST-ORDER EQUATIONS

5.4 SOLUTION OF SECOND-ORDER EQUATIONS

5.5 STABILITY OF THE SOLUTION

5.6 SOLUTION OF SIMULTANEOUS SETS OF DIFFERENCE EQUATIONS WITH THE DIFFERENCE OPERATOR

6 Solution of Linear, Constant-Coefficient, Partial Differential Equations

6.1 COMMON ENGINEERING PARTIAL DIFFERENTIAL EQUATIONS

6.2 THE LINEAR, CONSTANT-COEFFICIENT, PARTIAL DIFFERENTIAL EQUATION

6.3 THE METHOD OF SEPARATION OF VARIABLES

6.4 BOUNDARY CONDITIONS AND INITIAL CONDITIONS

6.5 NUMERICAL (COMPUTER) SOLUTIONS VIA FINITE DIFFERENCES: CONVERSION TO DIFFERENCE EQUATIONS

7 The Fourier Series and Fourier Transform

7.1 PERIODIC FUNCTIONS

7.2 THE FOURIER SERIES

7.3 THE FOURIER TRANSFORM

8 The Laplace Transform

8.1 TRANSFORMS OF IMPORTANT FUNCTIONS

8.2 USEFUL TRANSFORM PROPERTIES

8.3 TRANSFORMING DIFFERENTIAL EQUATIONS

8.4 OBTAINING THE INVERSE TRANSFORM USING PARTIAL FRACTION EXPANSIONS

9 Mathematics of Vectors

9.1 VECTORS AND COORDINATE SYSTEMS

9.2 THE LINE INTEGRAL

9.3 THE SURFACE INTEGRAL

9.4 DIVERGENCE

9.5 CURL

9.6 THE GRADIENT OF A SCALAR FIELD

INDEX

Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada.

No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750–8400, fax (978) 750–4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748–6011, fax (201) 748–6008, or online at http://www.wiley.com/go/permission.

Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations of warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.

For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762–2974, outside the United States at (317) 572–3993 or fax (317) 572–4002.

Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com.

Library of Congress Cataloging-in-Publication Data:

Paul, Clayton R.
Essential math skills for engineers / Clayton R. Paul.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-470-40502-4
1. Mathematics. 2. Engineering mathematics. I. Title.
QA37.3.P39 2009
620.001′51—dc22
2008042568

To the Humane and Compassionate Treatment of Animals

PREFACE

The purpose of this brief textbook is to enumerate and discuss only those mathematical skills that engineers use most often . These represent the math skills that engineering students and engineering practitioners must be able to recall immediately and understand thoroughly in order to become successful engineers. We will not stray into esoteric topics in mathematics. If a math topic is not encountered frequently in the daily study or practice of engineering, it is not discussed in this book. The goal of the book is to give the reader lasting and functional use of these essential mathematical skills and to categorize the skills into functional groups so as to promote their retention.

Students studying for a degree in one of the several engineering specialities (i.e., electrical, mechanical, civil, biomedical, environmental, aeronautical, etc.) are required to complete numerous courses on general mathematics. This material represents a rather formidable amount of mathematical detail and can leave a student overwhelmed by its sheer volume. However, in the student’s everyday studies in engineering, the vast majority of these skills and concepts are never used or are used so infrequently that they are quickly forgotten. This also applies to the engineering professional. The majority of the math skills that a student will use on a frequent basis represents only a small portion of all the math topics studied in math courses. If the student is distracted in a particular engineering course by struggling to remember these frequently encountered math skills, learning the particular engineering topic being studied will not take place. This book is intended to cover only those math skills used most frequently by engineering undergraduate students and engineering professionals. These few but critically important skills must be firmly understood and easily recalled by both students and practitioners if they are to be successful in their engineering studies and their future engineering practice.

I am frequently asked the question: “What do engineers do?” I have reduced my answer to the following succinct summary:

Engineers develop and analyze mathematical models of physical systems for the purpose of designing those physical systems to perform a specific task.

The next question that is asked is: “Why do they do that?” My answer to this question is:

Engineers do that so that they can develop insight into the behavior of a physical system in order to construct a mathematical model which when implemented in a physical system will accomplish certain design goals.

Hence, mathematics is at the heart of engineering design . This makes it abundantly clear that any student or practitioner of engineering must be fluent in the mathematical skills that they encounter frequently. Being able to use mathematical skills alone will not make a person a competent engineer, but not being able to use those skills will handicap their ability to become a competent engineer.

Chapter 1 covers miscellaneous math skills, such as writing the equation of a straight line, various formulas for the area and volume of common shapes, obtaining the roots of a quadratic equation, logarithms, reducing fractions via lowest common denominators, long division, trigonometry, complex numbers and algebra, common derivatives and integrals, and numerical integration.

Chapter 2 discusses the solution of simultaneous, linear, algebraic equations. Cramer’s rule, Gauss elimination, and basic matrix algebra are also described. Chapter 3 covers solution of the most common form of ordinary differential equations: the linear, constant-coefficient, ordinary differential equation. Although nonlinear and/or non-constant-coefficient differential equations are encountered in engineering, the linear, constant-coefficient differential equation is the type encountered most frequently.

In the last twenty years, discrete-time systems such as computers have become very common. These systems are described by difference equations. In Chapter 5 we discuss the most common type of difference equation: the linear, constant-coefficient, difference equation, and in Chapter 6 we discuss the solution of linear, constant-coefficient, partial differential equations.

In the remaining three chapters we discuss important specialized solution topics. The Fourier series and Fourier transform are discussed in Chapter 7 . In Chapter 8 we describe the Laplace transform method of solving ordinary as well as partial differential equations. Finally, in Chapter 9 we discuss briefly vector algebra and elementary vector calculus.

This text will provide readers with a compact but complete summary of all the mathematical skills that engineering students and professionals need in order to become successful engineers. Without a thorough and immediate understanding of these skills, one cannot expect to become a successful engineer.

Macon, Georgia

Clayton R. Paul