Mathematical Methods in Engineering and Physics: Overview
On January 31, 2011 we sent a document to John Wiley and Sons entitled "Textbook Proposal: Math Methods in Engineering and Physics." The proposal included two fully written chapters: one on Taylor Series and one on Partial Differential Equations. Wiley sent the proposal to professors of engineering and Physics, and when they got back a favorable response they offered us a contract.
The following text is adapted from that proposal, which is why it focuses on those two chapters. You are not Wiley, and we're not trying to get you to publish our book. So why do you care? Because this proposal served as our guide throughout the writing process, and it will give you a very good idea of how we designed our book to be different from existing books in this space.
We are writing a textbook for the "math methods" course for physics and engineering students. Sometimes referred to as "mathematical methods for physicists" or "applied mathematics for engineering," the math methods course introduces a range of mathematical techniques that students will need in later work.
One of our primary goals is to support the use of active learning. Research in engineering and physics education has shown that students learn more effectively when they are actively engaged. This research has been incorporated in a number of ways into textbooks at the introductory level, but has thus far had little impact on textbooks for higher level courses.
In addition, our book addresses some of the unique challenges presented by math methods courses. First, students in a math methods course are expected to learn many different mathematical topics in rapid succession, and then recall and apply them in higher level courses a year or more later. (Anyone who has taught upper level physics or engineering courses knows that this approach is not always highly successful.) Second, advances in computer technology are making some traditional math skills obsolete and some new skills essential. Finally, math methods is taught at levels ranging from sophomore to early graduate: the science and math prerequisites, the selection of topics, the level of sophistication at which they are taught, and the order of the topics all vary between institutions.
In addressing these issues we have been guided by the following principles. Each one is described in more detail below.
- Active learning: The textbook should provide active exercises that guide students through discovering key concepts on their own.
- Physical motivation: The mathematical ideas should be presented in the context of clear physical motivation.
- Use of computers: The focus should be on skills that are useful in the computer age. Those include activities that computers cannot do, such as setting up problems and interpreting answers, as well as intelligent and effective ways to use computers.
- Coverage of topics: Whenever possible, professors should be able to pick and choose among the topics and/or change the order of topics in the book.
- A reference as well as a textbook: The information should be organized in such a way that students will refer back to it in later years.
Active Learning
Consider a typical introduction to Maclaurin series.
- The textbook has a clear, carefully laid-out, rigorous explanation, which most students never read.
- The professor gives a lecture, and the students dutifully take notes. Passively listening, most students do not fully absorb the material.
- The professor assigns homework problems. In theory, these problems are intended to test and reinforce what the students have already learned. In practice, doing the homework is when the students actually learn the material. This may also be the only time most students look at the textbook—not to read the careful explanations, but to look for an easy "template" they can copy in doing the homework.
"If only those lazy students would take the time to read and listen!" But in fact, the students know intuitively what educational research has consistently demonstrated: people learn what they do, not what they hear and read. (See e.g. "Does Active Learning Work? A Review of the Research," Michael Prince, J. Engr. Education, 93(3), 223-231 (2004).) Consider, then, a different model.
- To set the stage, the students have previously gone through the motivating exercise "Vibrations in a Crystal." This establishes an important physical problem that can only be solved by means of a good approximation technique: see "Physical Motivation" below.
- The textbook section on Maclaurin series begins with a discovery exercise, "A Polynomial Equivalent of the Sine." This may be assigned as part of the previous homework set (on a different topic), or it may be done in class. (In the latter case only part of the exercise would be used, requiring about ten minutes of class time.) Either way, it is the students' first introduction to Maclaurin series, and they are introduced by finding one themselves.
- The lecture now serves as a follow-up, not an introduction. The professor's job is to summarize what the students have already done, to clarify the connection between the physical problem and the mathematical technique, and of course to answer questions.
- The homework problems then serve their intended purpose: practice, reinforcement, and deepening of a skill that has already been established.
The textbook offers many such exercises. "Motivating exercises" at the beginning of each chapter have the students set up a physical problem that shows the need for the math that follows. "Discovery exercises" at the beginning of most sections get the students to work out key ideas and techniques such as finding a Maclaurin series or solving a PDE using separation of variables. No professor will want to use all of them, so the explanations and problems don't rely on them. But when teachers begin to incorporate active learning with one or two topics, they generally find that the benefits are well worth the class time spent, and gradually use them more and more. Our textbook gives them an easy way to start, and supports them as they continue.
A more detailed discussion of Exercises is available at www.felderbooks.com/mathmethods/contents/exercises.html. There you can download any of the exercises from the book. You are welcome to use these exercises even if you do not use our textbook.
Scientists and engineers learn math to solve real problems. You wouldn't teach a second grader to add without saying something like "I have three apples and you have five..." You wouldn't introduce the derivative without saying, for example, "Find the velocity..."
The more abstract the topic, the greater the need for concrete examples. Unfortunately, we see too many math methods textbooks that provide little motivation beyond "suppose you wanted to solve this equation." Students who see no purpose for math techniques are unlikely to be able to apply those techniques in context, or to retain them long after the final exam.
Should physical motivation be presented before, during, or after a particular mathematical technique is explained? Our answer, which is reflected in every topic we teach, is "all three."
- Before: Every chapter in the book begins with a "motivating exercise." As an example, we refer you to the "Vibrations in a Crystal" exercise, the first thing you will see in the book after the "overview" of the Taylor series chapter. The students actively walk through the derivation of a differential equation. (Modeling real-world scenarios with differential equations is one of the most important skills in the curriculum, and runs throughout the book.) They cannot solve the differential equation, so a substitution is offered—with no justification—"replace this complicated function with this simpler function." The students can solve this simplified differential equation and predict the frequency of oscillation of an atom in a crystal—and they do. But where did that approximation come from? The students are now primed for a chapter that will teach them how to approximate complicated functions with simpler ones.
- During: The explanations for mathematical techniques are presented whenever possible in the context of real world problems. As you look through the different PDE sections you will see waves on strings, heat distributions, electric potentials, and other familiar scenarios. It is not enough to say at the beginning "partial differential equations are used for all these topics"—each technique is presented through the lens of one of these important problems.
- After: The problems that follow each section are a mix of real-world examples and straight equations to solve. The goal is not to make each problem a new adventure in word problems. But when the students solve a partial differential equation, they should understand some of the many ways this equation could help them model important phenomena.
Use of Computers
Programs such as Mathematica, Matlab, Maple, Wolfram Alpha, and Excel form an indispensable part of the modern scientist's toolkit. To leave them out is to revert to the days of slide rules and log tables, leaving significant gaps—to be filled in at some unspecified future point—in the students' preparation for future classes and careers. On the other hand, overdependence on technology can leave students without a firm mathematical grounding; we see this every day as we watch our students punch problems like 13×10 into their TI-83s.
When textbook authors bring in technology as an afterthought, the results are uninspiring. Problems that really should be done on a computer are tediously worked out by hand. ("Let's find the determinant of this 4×4 matrix." "No, please, let's not.") Learning the software becomes one more tedious job, to be added to the already-long list of requirements.
But done correctly—with mathematical software woven into the structure of every chapter—the technology becomes a powerful tool for both the student and the professor. Numerical recipes and analytical methods work together to solve problems and enhance understanding.
As an example, consider one of the book's examples of using a Fourier transform to solve a PDE.
- We begin by analyzing the differential equation and discussing the kinds of behavior we expect. Our book emphasizes repeatedly that, while computers may help in solving equations, the job of the scientist or engineer is to interpret them.
- We then apply the method of transforms to solve the equation. For the most part, this is a manual process. Along the way, however, we highlight specific steps where a computer could help: "find the Fourier transform of this function" or "solve this ODE." (Both of these skills are taught carefully, both with and without computers, in their respective chapters. At this point in the text, however, they are merely stepping stones toward solving a PDE.)
- Finally, we calculate the limiting case of the resulting function (by hand), graph it (on a computer), and then discuss what the graph tells us about the behavior of the function.
You will see that combination of manual and technological techniques throughout the chapter on PDEs.
The problems for which students are expected to use mathematical software are written without regard for what software package they use. Instead of saying "Use DSolve to solve this equation" we say "Use a computer to solve this equation." This provides students and instructors with the flexibility to use any software, but it also means we can not include step by step instructions for any particular software package. (Even if we could, they would likely become outdated faster than we could write new editions.) If a professor chooses to assign computer problems then the students will need to learn how to use the software, either as part of the course or as a prerequisite to it. If mathematical software is not part of the course then these problems can simply be skipped.
Because of the wide range of topics and levels of math methods courses at different institutions, textbooks such as Boas, Arfken & Weber, and Kreyszig cover a range of topics that professors choose from in designing their curricula. Our textbook includes most or all of the topics covered in these books. We have surveyed professors a number of times to make sure we are including the topics that people want; we would love to hear from you if you have feedback on our table of contents as it currently stands.
Given this range of topics, it's important to keep the chapters as modular as possible. Nonetheless, certain dependencies are unavoidable; you can't learn how to solve partial differential equations if you've never seen a Fourier series. We therefore begin each chapter with a short bulleted list of prerequisite topics. We include references to where that material can be found in our book, but each chapter only relies on that basic knowledge, not on specifics of how we present it in earlier chapters.
In addition to teaching the students how to solve problems, a math methods textbook should serve as a reference. After the course is over, this book should occupy a prominent place on the student's bookshelf, to be turned to again and again during future courses.
The back of our book is organized to meet this goal, with tables of information and references back into the book for further details. A list of appendices can be found in our table of contents.