Title: Essential Math Skills for Engineers
Author: Clayton R. Paul
Publisher: John Wiley, 2009
This book is a true surprise! In less than 250 pages it goes directly to the essence of engineering (regardless of the specialty – electrical, civil, or chemical) and provides the capability to understand, develop and use mathematical models.
The author asks two questions:
1) “What do engineers do?” Here the author’s answer: “Engineers develop and analyze mathematical models of physical systems for the purpose of designing those physical systems to perform a specific task.”
2) “Why do they do that?” And, again, the author’s answer: “Engineers do that so that they can develop insight into the behaviour of a physical system in order to construct mathematical models which, when implemented in a physical system, will accomplish certain design goals.”
The purpose of this brief book is not to talk about philosophy, but is to enumerate and discuss only those mathematical skills that engineers, EMC engineers included, 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. These are not 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 specialties (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 (and university professors). The majority of the math skills that a student will use on a frequent basis represent 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 occur. 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. 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.
The book is developed in nine chapters. Chapter 1 gives the reasons behind why this book was written and is supported by simple but effective engineering examples.
Chapter 2 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 3 discusses the solution of simultaneous, linear, and algebraic equations. Cramer’s rule, Gauss elimination, and basic matrix algebra are also described.
Chapter 4 is one of my two favorite chapters. It covers the solution of the most common form of ordinary differential equations: the linear, constant-coefficient, and 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. Here the author does his best in helping the reader to understand, in simple terms, what the equations and their solutions are; i.e. “trying to tell you” and why that makes sense. The author, thanks to his extensive didactical experience, provides simple “commonsense” explanations of the logical meaning of the equations and their solutions, as well as simple methods for obtaining their solutions.
In the last twenty years, discrete-time systems such as computers have become very common. These systems are described by differential equations. In Chapter 5 are discussed the most common type of differential equations: the linear, constant-coefficient, differential equation, and in Chapter 6 are discussed the solution of linear, constant-coefficient, partial differential equations.
The remaining three chapters discuss important solution topics for electrical engineers and in particular for those involved in EMC.
Chapter 7 offers a discussion on the Fourier series and Fourier transform.
Chapter 8 describes the Laplace transform method (direct and inverse) as a tool for solving ordinary as well as partial differential equations.
Chapter 9 briefly but effectively discusses the vector algebra and elementary vector calculus that is so important for dealing with electromagnetic fields. The geometrical and physical interpretations of the vector operators avoid the common risk in memorizing the equations and their solutions.
A personal note: This year I’ll include this textbook in the list of the recommended titles for my course of “Basic Electric Principles” (held during a student’s sophomore year in electrical engineering). I’m sure that my students will benefit from its content. EMC