Applied Engineering Mathematics

Brian Vick

• Publisher: CRC Press
• Year: 2020
• eBook ISBN: 9781000047653
• Edition: 1
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

Undergraduate engineering students need good mathematics skills. This textbook supports this need by placing a strong emphasis on visualization and the methods and tools needed across the whole of engineering. The visual approach is emphasized, and excessive proofs and derivations are avoided. The visual images explain and teach the mathematical methods. The book's website provides dynamic and interactive codes in Mathematica to accompany the examples for the reader to explore on their own with Mathematica or the free Computational Document Format player, and it provides access for instructors to a solutions manual.

• 
  
  

• Strongly emphasizes a visual approach to engineering mathematics

• 
  
  

• Written for years 2 to 4 of an engineering degree course

• 
  
  

• Website offers support with dynamic and interactive Mathematica code and instructor's solutions manual

Brian Vick is an associate professor at Virginia Tech in the United States and is a longtime teacher and researcher. His style has been developed from teaching a variety of engineering and mathematical courses in the areas of heat transfer, thermodynamics, engineering design, computer programming, numerical analysis, and system dynamics at both undergraduate and graduate levels.

eResource material is available for this title at www.crcpress.com/9780367432768.

Chapter 1

Overview

Chapter Objectives

This chapter is an introduction to and overview of the educational philosophy employed in this book. The physical applications and mathematical methods are briefly summarized.

Specific objectives and topics covered are

• Objectives
• Educational philosophy
• Physical processes: conservation laws, rate equations, and property relations
• Mathematical models: algebraic equations, ordinary differential equations, and partial differential equations
• Solution methods for algebraic and differential equations
• Software

1.1 Objectives

This is a comprehensive book consisting of a blend of fundamental physics, applied science, mathematical analysis, numerical computation, and critical thinking. It contains both theory and application, with the applications interwoven with the theory throughout the text. The emphasis is visual rather than procedural.

The specific goals consist of the following.

• Physical Processes
  To gain a fundamental understanding of the physical processes, fundamental principles, and mathematical formulations of physical problems.
• Mathematical Methods
  To learn mathematical techniques for solving model equations.
• Software
  To learn the use of software packages, such as Mathematica or MATLAB^®, to program solutions, perform calculations, and create graphics. Computational studies and graphics enhance insight into the effect of important parameters in addition to building a fundamental understanding of physical mechanisms.
• Insight and Critical Thinking
  To develop a sound foundation for problem-solving using critical thinking, interpretation, and reasoning skills. Applications are used extensively to foster insight and intuition.

1.2 Educational Philosophy

The following guiding principles are fundamental to learning;

• We learn by active participation, not by passive observation. That is, we learn by doing, not just by watching.
• A picture is worth a thousand words. The human brain is made to process visual information. More information can be assimilated in a few seconds by looking at graphics than by studying that same information for months from a printout of numerical values.
• We are all responsible for our own learning. You need to be self-motivated and have the desire to learn.

An important issue is knowledge versus information. There is an old saying:

Give me a fish and I’ll eat for a day. Teach me to fish and I’ll eat for a lifetime.

With today’s information explosion resulting from the internet, this old saying is more relevant than ever. Knowledge and fundamental reasoning skills are giving way to an unmanageable amount of information consisting of seemingly unconnected facts and figures. Hopefully, a greater emphasis can be placed on knowledge as opposed to just raw information. True progress requires a balance between raw information and basic knowledge.

In this same vein, too much coverage of material at the expense of depth of understanding can be the enemy of learning and leads to memorization and frustration. Depth of understanding is considered to be far more important than coverage of more topics.

In addition to developing an appreciation for and mastering knowledge of the fundamental physical principles and mathematical techniques, a major goal of this course is to develop basic learning skills and strategies. These include

• Reasoning and interpretive skills—the foundation of problem-solving
• Pattern recognition skills
• Adaptability—learning to recognize abstract concepts from specific applications and conversely, learning to apply abstract concepts to specific applications
• Thinking for yourself
• Motivation, enthusiasm, and passion

Regardless of your potential, there is no substitute for hard work. One must persist and struggle with difficult concepts until they are understood. Learning is a lifelong activity and is the key to success. Let’s make it fun and exciting!

1.3 Physical Processes

Observations of the physical world indicate that all processes are governed by a small number of fundamental principles. These are conservation principles, which are supplemented by rate equations and property relations. Together, they form a complete description of nature. Although there are only a handful of principles, there are countless applications and special cases. These principles are summarized in the following and are described in more detail in Chapters 2 and 3.

• The conservation laws are:
  • Conservation of mass: continuity
  • Conservation of mom...

Table of contents

• Cover
• Half-Title
• Title
• Copyright
• Contents
• Preface
• About the Author
• 1 Overview
• 2 Physical Processes
• 3 Modeling of Physical Processes
• 4 Calculus
• 5 Linear Algebra
• 6 Nonlinear Algebra: Root Finding
• 7 Introduction to Ordinary Differential Equations
• 8 Laplace Transforms
• 9 Numerical Solutions of Ordinary Differential Equations
• 10 First-Order Ordinary Differential Equations
• 11 Second-Order Ordinary Differential Equations
• Index