Interacting With Ordinary Differential Equations

Guest post by Stephen Kennedy (Carleton College), AMS/MAA Press Acquisitions

Reprinted with permission from MAA Books Beat

I often find myself in conversations about the future of the textbook. As I hope you are aware, both the MAA and the AMS have vibrant undergraduate textbook series. Each organization, within the past decade, independently decided that we had a role to play in delivering high-quality, reasonably priced undergraduate textbooks. In each case, the organizations were reacting to the predatory practices of commercial publishers in the math textbook market. We want to deliver textbooks that are useful to you and your students at prices that don’t make you wince. The merger of the MAA and AMS book programs brought a renewed emphasis on that goal.

In these conversations about the future of the textbook, I tend to be the conservative voice. I think there’s something different about reading mathematics that makes print a better medium. For most of my non-mathematical reading I have converted to electronic media. But when I want to understand some mathematical writing, I print it out. Partly it’s the ability to scribble in the margins, mostly it’s about being able to quickly flip back and forth and compare passages on different pages. But this is a debate for another time (although I’m happy to hear your thoughts by email), here I want to tell you about a new MAA textbook that just might be an example of the future of textbooks.

Sandy Saperstone has been teaching differential equations for decades at George Mason University. As with all of us who teach that course, both his content and delivery method have changed radically over the course of those decades. There is more emphasis on qualitative techniques, much more visualization, more, and more diverse, modeling examples. Mostly of course this is the result of easy access to computation. (There is also an effect on the course content due to the explosion of interest in dynamical systems, but that too is, in part, a result of better computational tools.) Sandy’s computational tool of choice is Mathematica. He uses it for mechanical manipulation to find explicit solutions (DSolve), numerical solutions (NDSolve), and, of course and most importantly, for visualization. He plots explicit and numerical solutions, draws direction fields, and the explores the effects of changing parameters and initial conditions.

As the years went on his lecture notes came to include more and more bits of Mathematica code and interactive graphics. Eventually he put them online so his students could interact with the Mathematica and, after some years of polishing, they are now published as Interacting with Ordinary Differential Equations, an online interactive textbook in the MAA Textbook series. (Co-author Max Saperstone is Sandy’s son, Max is responsible for much of the html and Mathematica coding). The “book” is actually a sequence of webpages, roughly one webpage per day or two of Sandy’s course. And the interactivity is at two levels. There is beautiful exposition that contains clickable links that open up to expand the details of an argument or computation. On first reading these can be ignored to get the big picture, students can go back on second or third reading and open these links up to go deeper. The more exciting interactivity is live computation. The pages contain scores, maybe hundreds, of interactive Mathematica cells. Students can manipulate sliders to vary parameters or initial conditions or watch movies of solution curves being generating. Differential equations are about motion, now your textbook can illustrate the motion!

The figures below show some examples. Figure 1a is from an interact that exhibits a saddle-node bifurcation in a model of ocean circulation. As salinity increases (Figure 1b)—salinity is the value in the slider—the equilibria at k1 and k2 merge and annihilate one another. The figure illustrates the slope field and several solution curves for two different values of salinity, one on either side of the bifurcation. Of course, in the book, the reader can drag the slider and watch the evolution. Figure 2 shows a phase portrait for the Van der Pol oscillator and, separately, a plot of x(t). One sees clearly the converging spirals and consequent periodic solutions trapped between them.

It is important to note that the user does not need to have Mathematica on her device. All the computations are done in the cloud seamlessly from the point of view of the reader. The reader experiences the book as if she is reading a webpage; she moves a slider or enters a parameter value and the webpage calls up the Mathematica engine and returns the result of the computation.

This is the future of textbooks, or maybe a future of textbooks. Your textbook now answers your questions interactively. If you wonder what would happen if you changed something in an example—just do it and the book shows you!

Fig 1a                                                         Fig 1b

I should point out that, for technical reasons, what AMS is selling here is six months access to the website. So, students should be aware that their access will eventually be cut off. I suppose I should also point out that, just in case you are not ready for the future to be here this semester, AMS-MAA have three other truly excellent ODE textbooks: Differential Equations: From Calculus to Dynamical Systems by Anne Noonburg, Lectures on Differential Equations by Phil Korman, and Differential Equations: Technique, Theory, and Applications by Barbara MacCluer, Paul Bourdon, and Thomas Kriete. If you happen to be teaching differential equations this year, we’ve got you covered.

Fig 2

To learn more and request exam copy access, please click here.

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