Mathematical Methods in Engineering and Physics

For over 25 years I have been focusing my energy on developing and refining active learning courses based on findings in Physics Education Research. Consequently, I am delighted with the prospect of having instructors be able to use Felder and Felder's Mathematical Methods in Engineering and Physics for three major reasons:

  1. The writing is outstanding, clear and concise.
  2. It is impossible for students to master the math skills they'll need as professionals in a single course. Two of the major advantages of this new book are that it covers an extensive range of topics and the prerequisite knowledge to understand each topic are provided to students. Having a book that future professionals can use as a life-long reference is essential.
  3. The book incorporates active learning from a to z. For example, students can: (a) complete a set of motivational exercises before beginning a topic; (b) continue during class with collaborative learning exercises, and (c) culminate after class by completing a few of the many thought-provoking problems and skill exercises.

    ~ Priscilla Laws
       Research Professor of Physics, Dickinson College
       Leader of the Activity-Based Physics Group and author of Understanding Physics
       Recipient of the International Commission on Physics Education medal
       Voted by the AAPT as one of the 75 most influential physics researchers or educators in the past 75 years

We wish we could tell you in person all the reasons we think our book is great, but play this nine-minute video and maybe it will be almost the same. (WARNING: this video contains some fairly technical mathematical discussion.)

We wrote this book because we weren't happy with the existing texts for this course. They presented mathemathical techniques with little or no physical motivation or context. ("Now that we've seen how to prove a series converges, let's see what happens when a series has an x in it.") They didn't have enough problems, especially problems that would drive the students to master common techniques and deeply understand vital principles. And in a lot of cases they were just confusing.

Of course, we would say that our own book is clearer and better focused and better written than all the others. And we would actually believe all that, or we wouldn't have written it. So that doesn't tell you much, right?

That's why we're really hoping you'll find out for yourself.

Either way, here are a few key features we hope you will take note of.

Breadth and Depth of Problems The book has over 2800 problems. A typical section starts with a "walk-through" guiding the student step-by-step through a particular skill. This is followed by drill-and- practice problems reinforcing that skill. Other problems probe deeper into the math, explore different physical applications, and prove important results.

Math is Always Presented in a Physical Context This text always presents topics as "here is a real problem that someone might actually want to solve, and here is how this mathematical tool helps." Mathematicians start with a theorem, then prove it, and provide a context for it. This text, by contrast, always begins with a physical problem. The authors present tools for solving the problem and show you how to use those tools. Then, as the last step—often in a problem for the student, but sometimes in an explanation—they provide the proof.

Exercises/Active Learning Exercises are very different from problem sets. They are meant to step a student through a process, at the end of which the student has figured out a key mathematical idea on his/her own. "Active learning"—getting students engaged instead of just lecturing at them—has been shown to dramatically increase student comphrehension and retention, but it can be difficult and time-consuming for the teacher. This text is designed to help with that process. Click here for a more detailed description of "exercises" and free downloadable exercises you can use in your class today.

Computers You can use this text without ever mentioning computers—the Explanations don't require them, and the Exercises and Problems that do require computers are clearly marked so they can be skipped. But for people who want to use MatLab, Mathematica, Maple, and other software as tools for exploration, this text offers special problems.

I have taught this course for the last ten years using Boas and Arfken/Weber. I have always attempted to frame the course as a course that shows the nexus between math and physics, but I have come up short. However, this semester, Felder/Felder allows me to be more explicit in showing how the mathematics informs the physics.

As a specific example of how the book guides the pedagogy, I am planning to do an in-class group assignment asking students to work out all parts of problem 8.169 (p. 437). This problem does not involve huge amounts of computation; rather, it requires students to think through the physical meaning of the curl of a vector and motivates a derivation of Stokes' Law. This type of problem is particularly meaningful to me since on the first day of class, I ask the students (all of whom have taken multivariable calculus as a prerequisite) if they know what grad, div and curl are. And of course they know the formulae and how to compute them. Then I ask, 'Without resorting to formulae, tell me what grad, div and curl describe.' Some students will be able offer an answer for grad, but div and curl are just mathematical mysteries to them. Problems like this, which are well thought out and developmental in style, help students connect the physics to the math. And for physics majors, that is an important outcome of the class.

Congratulations on an excellent text. I think it will make a real difference in the lives of our physics students.

~ David B. Slavsky
   Director of the Loyola University Core Curriculum, former Director of the Loyola University Center for Science and Math Education
   Loyola University Chicago
This is the most well written math book I have ever had. I'm a high school physics teacher getting my masters so I can teach dual credit classes, and I haven't seen a lot of this stuff in 15 years. I wish my math books when I was an undergrad were this well written and made things as intuitive as this book does. I never understood calculus until we started applying it in physics classes; this book takes the time to apply it to real situations, and that makes the material easier to grasp.

~ Mike Sorola
I have read enough of the leading texts on math methods in engineering to be able to say that this is by far the most readable of them all—for the clarity of its explanations, the broad range of its examples, and its occasional flashes of humor. In addition, it is the only math methods text I have seen that is designed to be compatible with active learning and other research-proven pedagogical methods. (There is also something particularly appealing to me about the authors' last name, although I can't quite put my finger on what it is.) I only wish I had had this text to learn from when I took the course as a graduate student, and I wish even more that it had been available when I taught the course as an engineering faculty member.

~ Richard M. Felder
   Hoechst Celanese Professor Emeritus of Chemical Engineering
   North Carolina State University
   Winner, Inaugural Lifetime Achievement Award in Engineering Education, American Society for Engineering Education, 2012
   Coauthor, Teaching and Learning in STEM: A Practical Guide. Jossey-Bass, 2016
I have been using Felder and Felder in my Calculus III class for two years, and have found it to be a trusty guide to organizing lectures. The discovery exercises provide a good motivational exercise for each topic, if I think I need one; the material is complete, logically ordered, and broken up into short sections; the examples in the text are suitable for in-class demonstrations; and the problems, which are my favorite feature, are thorough and inventive, testing understanding, calculation, intuition, visualization, and use of analytical software. Students appreciate the conversational style and the focus on the practical meaning of the results and theorems.

~ Henry Rich
   Electrical engineer
   Multivariate Calculus and Linear Algebra teacher, Raleigh Charter High School