Essential Math Skills for Engineers

Clayton R. Paul

• Publisher: Wiley-IEEE Press
• Year: 2011
• ISBN:
  9781118211106
• Edition: 1
• Topic: Computer Science
• Subtopic: Programming Algorithms
• Index: Computer Science

Just the math skills you need to excel in the study or practice of engineering

Good math skills are indispensable for all engineers regardless of their specialty, yet only a relatively small portion of the math that engineering students study in college mathematics courses is used on a frequent basis in the study or practice of engineering. That's why Essential Math Skills for Engineers focuses on only these few critically essential math skills that students need in order to advance in their engineering studies and excel in engineering practice.

Essential Math Skills for Engineers features concise, easy-to-follow explanations that quickly bring readers up to speed on all the essential core math skills used in the daily study and practice of engineering. These fundamental and essential skills are logically grouped into categories that make them easy to learn while also promoting their long-term retention. Among the key areas covered are:

• Algebra, geometry, trigonometry, complex arithmetic, and differential and integral calculus
  
  

• Simultaneous, linear, algebraic equations
  
  

• Linear, constant-coefficient, ordinary differential equations
  
  

• Linear, constant-coefficient, difference equations
  
  

• Linear, constant-coefficient, partial differential equations
  
  

• Fourier series and Fourier transform
  
  

• Laplace transform
  
  

• Mathematics of vectors
  
  

With the thorough understanding of essential math skills gained from this text, readers will have mastered a key component of the knowledge needed to become successful students of engineering. In addition, this text is highly recommended for practicing engineers who want to refresh their math skills in order to tackle problems in engineering with confidence.

1

What Do Engineers Do?

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

Students studying for a batchelor of science degree in one of the several engineering specialties (i.e., electrical, mechanical, civil, biomedical, environmental, aeronautical, etc.) are required to complete several courses in general mathematics. Prior to enrolling in a university, students are generally required to have completed high school courses covering algebra and trigonometry. At the author’s institution, which is fairly typical, students are required to complete a semester of differential calculus and a semester of integral calculus in the freshman year. In addition, a course in differential equations is required to be completed in the sophomore year. Many engineering programs also require a semester in the junior year on multivariable mathematics. All of this mathematical material represents a rather formidable amount of detail and can leave a student overwhelmed by its sheer volume. However, in a student’s everyday studies in engineering, as well as in professional engineering practice, the vast majority of those skills and concepts are never used or are used so infrequently that they are quickly forgotten. The majority of the math skills that students will use on a frequent basis represents only a small portion of the math topics studied in their math courses. It therefore makes sense for students to concentrate on and commit to memory and immediate usability only those skills that are used frequently, deferring those math topics that will be used infrequently to being “looked-up” when they arise. The healthy view of the role of mathematics in engineering is as a tool to understand the behavior of the particular engineering system being studied—in the same way that language is a means of communicating. If the student is distracted in a particular engineering course by struggling to remember 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 the majority of engineering practitioners.

Even though students do not use all these math skills on a frequent basis, they nevertheless benefit from being exposed to the majority of the mathematical concepts and mathematical sophistication they will study in their math courses; engineering study is also intended to be an “education.” Students who go on to graduate school and pursue advanced degrees in engineering will need more math skills and topics the farther they go. However, the success of students in a graduate engineering program will also rely primarily on their having a solid understanding of the undergraduate program, which requires a solid understanding of the basic mathematics covered in this book. But it is important to remember that the majority of undergraduate engineering students will not pursue a graduate engineering degree. Hence, both students and practitioners must firmly understand and be able easily to recall these few but critically important math skills if they are to be successful in their engineering studies and in their future engineering practice. Students tend to be “overwhelmed” by the vast body of mathematical knowledge they have encountered in their math courses and are not readily able to compartmentalize which skills are most important to commit to immediate memory. They have not completed their studies and are therefore not able to discern this for themselves. We teachers of engineering easily forget this important fact. We have been studying engineering for many years and it is abundantly clear which math skills are frequently encountered. Students do not possess our broad overall view of the situation.

It is for this reason that I have undertaken to write this brief textbook on the essential math skills required of engineers. I have been studying engineering for over 50 years and have been teaching the subject for over 40 years. Although the majority of that time was spent in academia, I have also spent a substantial portion of it involved in the concurrent practice of engineering in industry. It has become very clear to me after this lengthy experience that the primary attribute which determines whether a student will be successful in his or her engineering studies is the person’s ability for immediate recall and successful use of this small but important body of math skills. This observation also applies to engineering professionals.

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.

So the unique function of engineers is design. 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 the 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 you a competent engineer, but not being able to use those skills will handicap your ability to become a competent engineer.

Engineering systems and the design problems associated with them are much too complicated today (and will no doubt become increasingly so in the future) to be understood by one’s ordinary “life experiences.” In fact, if this were so, engineering companies would not be at the top of industry jobs paying the highest salaries. Many students today seem to believe that the digital computer will make engineering easy and obviate the need for them to learn mathematics. Numerous computer programs exist that can solve (give a numerical answer to) the complicated mathematical equations that describe engineering systems. If the design of engineering systems were that simple, industries could, perhaps, simply pay minimum wage to someone and train the person (in a very short time) to use that computer program. But that would be missing the point of what engineers do. These computer programs give a numerical “answer,” but they give little, if any, insight into how a particular engineering system behaves. To obt...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Dedication
• PREFACE
• 1: What Do Engineers Do?
• 2: Miscellaneous Math Skills
• 3: Solution of Simultaneous, Linear, Algebraic Equations
• 4: Solution of Linear, Constant-Coefficient, Ordinary Differential Equations
• 5: Solution of Linear, Constant-Coefficient, Difference Equations
• 6: Solution of Linear, Constant-Coefficient, Partial Differential Equations
• 7: The Fourier Series and Fourier Transform
• 8: The Laplace Transform
• 9: Mathematics of Vectors
• INDEX