INTEGRATED ENGINEERING

MATHEMATICS

A COMPREHENSIVE REVIEW AND ADVANCED MATERIAL FOR GRADUATE STUDENTS

AND PROFESSIONALS

VLADIMIR SOLOVJOV

BRIGHAM YOUNG UNIVERSITY

PROVO, UTAH

2017

“… and what is the use of a book,” thought Alice, “without pictures?”

ME EN – 505

Applied Engineering Mathematics

“… mathematics contains some very ingenious inventions which

can serve just as well to satisfy the curious as to make all arts and crafts easier and to lessen man’s work.”

René Descartes

PREFACE

Engineering as a discipline has matured in a mathematical sense during the past 50 years. Typical engineering

curricula today include a sequence of differential and integral calculus, linear algebra, vector calculus, numerical

methods, and ordinary differential equations. Graduate programs further this sequence with forays into complex

variables, abstract algebra, statistics, variational calculus and partial differential equations. Often, undergraduate

engineering programs require that students take the math sequence early in their undergraduate career to prepare

them for the engineering courses. The math sequence is therefore completed by the start of the junior year. During

the junior and senior year, though engineering students encounter math in the engineering courses, their

development of skills and abilities in the language of math is put on hold. Consequently, as the students prepare for

graduate school, entrance exams, qualifying exams, etc., it has been at least two years since most of them would

have had a math course.

Career changes can have a similar effect. If an engineer, physicist or scientist has been in a professional position for

an extended period of time and then desires to change into a new position requiring different skills mathematically,

it often becomes a daunting task to retool in preparation for applying for the position as well as performing in the

position should it be attained.

This book has been prepared to help the senior level undergraduate student as well as the professional engineer,

physicist or scientist to retool, sharpen, or refocus their mathematical skills in preparation for graduate school,

exams, or career changes. It is also intended to function as a reference for these same individuals. The book has been

organized by topic area so that sections of the book can be studied selectively based on need. Also the sections have

been prepared assuming that the student already has a typical engineering background in mathematics.

The material presented reviews the basic engineering fundamental of mathematics on a broad front with respect to

all the topics related to engineering. Consequently, the student will increase his or hers breadth as well as depth in

engineering mathematics. Exercises are presented to help the students practice and refine their skills.

Modern engineers are required to know how to utilize computers in the solution of mathematical problems. This

involves symbolic solutions, numerical calculations, visualization and presentation of results. There are several

commercially available software packages focused on supporting these needs, examples of these are: MAPLE,

MATHCAD, MATLAB, MATHEMATICA, etc. This book provides exercises and examples to help students learn

how to take advantage of these software tools. The sections in the book have been prepared using MAPLE as the

example software tool. These portions of the book are designed to demonstrate methods of use and limits of

software tools. It is not the intention of the author that this book be used to teach MAPLE or any other software tool.

All problems can be solved without the use of the software tools.

The hope of the author is that this book can aid engineers, physicists and scientists improve their math

understandings and skills throughout their careers and therefore improve their professional experiences. It has been

an exciting process to prepare this material with the prospect of helping fellow professionals.

DO

LEARN REVIEW

DO

After the completion of this course the student

- should recall the general concepts and technique of engineering mathematics

- should be able to apply them for the solution of a wide range of problems

- should know how to use computer software (Maple) for the solution of mathematical problems

- should have a general view of engineering mathematics and see the connection between the

different parts (integrate the knowledge of different parts of mathematics)

- should understand how mathematics is applied in engineering and science

“The best way to explain it is to do it,” said Dodo.

Objective: To assist students who are going to graduate studies and professional engineers to recall, review and enhance their knowledge of the basics of engineering mathematics needed for better transition to graduate level classes and retooling for professional work and research.

The General Concept: • The present manual is intended for:

- general review of basic topics on engineering mathematics which were studied by students in undergraduate level math classes;

but review with some advancement on higher levels to see the bigger picture;

- fix the gaps in previous studies; - can be used as a textbook for classes or for independent study; - can be used as a reference handbook source on basic math.

• It is assumed that the majority of students have already studied this

material, but the presentation takes into account that they

- can have different levels of preparation; some can come from other colleges with different programs in mathematics where some topics were not included in the course;

- because of significant time gap between math classes some material can be completely forgotten, so most topics should be presented in a complete form as if for new students;

- to not overload the text, proofs of results and theorems in most cases are omitted in the core text and presented in the appendix;

• General overview of engineering math: unification of notations and better

connection between different parts (like calculus and linear algebra), so students can see better the interrelations and applications.

• It should be remembered that this manual is for engineers not for pure

mathematicians - we follow an engineering style of presentation: clear, visual, practical (with more attention to applications), with fully developed results but still rigorous and not oversimplified.

• It is written with the intention that students can read from any place,

however the consequence of presentation should be logical and with increase in level and rigorousness; examples and exercises should help to see the connection between topics.

Topics in the manual are presented generally in the following manner: • Objectives what student should know after completion of this section • Concepts what it is about, where does it occur (for faster recall) • Definitions definitions, main results, facts, equations, formulas • Problems typical problems, approach for solution • Examples a range of example problems • Theorems a collection of all pertinent theorems • Maple example application of commercial software • Questions comprehensive questions for review and understanding • Exercises exercises with hints, answers and complete solutions • Tests self-assessment tests

N. Bogdanov-Belsky “ Verbal counting in the village school of S.A. Rachinsky”, 1895