By: David Bressoud @dbressoud

Last month I wrote about the contributions to the June 2021 issue of ZDM – Mathematics Education from outside the United States. This month I want to reflect on one of the papers written inside the U.S.: “Teaching calculus with infinitesimals and differentials” by Robert Ely. Ely considers the various approaches toward differentials in single variable calculus and argues for treating them as infinitesimals.

There are several strong elements to his argument. One of the most powerful is that it facilitates thinking about relative rates of change of linked variables. Thus, given x, y, and t linked by the relationship  y = 5x^3 +  7 cos t, their differentials are related by dy = 15x^2 dx – sin t dt. Knowing how x and t change now easily conveys information on how y is changing. Implicit differentiation, so difficult for so many students, is greatly simplified.

Treating differentials as infinitesimals means that dy/dx really is a fraction. Integrals really are sums. This does make some sense. Beyond mathematicians, most of those who use integration are solving accumulation problems for which it is natural to think of the integral as a sum of products involving infinitesimal changes. Even Archimedes discovered the formula for the volume of a sphere by working with infinitesimally thin slices. That did not pass muster as an acceptable proof, but it provided the insight that could then lead to a proof.

Ely is not advocating a rigorous treatment of infinitesimals in single variable calculus, developing nonstandard analysis in its full complexity. But infinitesimals are appealing, and knowing that a logical foundation does exist eases one’s conscience in using them.

Today, the dominant approach to differentials in single variable calculus is to treat them as variables. Differentials are subject to the constraint that if y is a function of x, then the variable dy is proportional to dx, dy = f’(a) dx (Figure 1). This is how Dan Kennedy, Mike Boardman, and I define differentials in the latest edition of Calculus: Graphical, Numerical, Algebraic, written for AP Calculus students. This text is a direct descendant—from Thomas to Finney to Demana and Waites to Kennedy, myself and Boardman—of George Thomas’s Calculus and Analytic Geometry, maintaining the definition in Thomas’s first edition almost word for word.
I am not enamored of this definition. It provides no insight into why these particular variables are preceding by a d. A differential is not a completely arbitrary variable. It is useful only when it is kept very small. Nevertheless, this definition does serve to distinguish between the change in y, Delta y, and the differential of y, dy. It naturally leads to the observation that the easily calculated dy can serve as a useful approximation to Delta y.

One place where the dominant definition comes up short is helping students understand the roll of dx in an integral. To say that dx is a variable does not help students understand its role. I suspect that this is largely responsible for the common perception that the dx at the end of an integral is a notational appendix, simply a reminder of the variable of integration. I know from reading thousands of AP exams that students are prone to dropping it from their integrals. Failure to see that this differential carries an essential unit then contributes to student difficulties in recognizing that the units of an integral are not the same as the units of the integrand.

There is an alternate approach, developed by Pat Thompson and Mark Ashcroft with contributions from Fabio Milner for Project DIRACC (Developing and Investigating a Rigorous Approach to Conceptual Calculus). It seeks a middle way between infinitesimals and arbitrary variables. I talked briefly about this approach in my column of June 2017, “Re-imagining the Calculus Curriculum, II” and reproduced two of the important visuals earlier this year in “Thoughts on Advanced Placement Precalculus”. In Project DIRACC, the authors begin their introduction of differentials explaining that “the letter ‘d’ preceding a variable [means] that the variable's value ‘varies a little bit’.” If x and y are linked variables, linked by a functional relationship or an equation such as x^2 + xy + y^3 = 1, then given a fixed small change in x, denoted by Delta x, dx is a variable that is limited in magnitude to Delta x and that is proportional to dy.

That is almost the common approach. What makes it distinctive is the choice and use of Delta x. It needs to be small enough so that over an interval of length Delta x, “the function’s value varies at essentially a constant rate.” Over 1/1000th of a second, a car’s velocity can, for all practical purposes, be treated as constant (Figure 2). When we ask for the car’s velocity at a moment, we are actually asking for its velocity over a very short interval. The velocity is always a change in distance over a change in time. Since we often speak of a moment in time as a single value, Thompson and Ashcroft clarify that they are using “moment” to refer to any sufficiently short interval of the input variable.

How short is a moment? That depends. If the input is measured in lightyears, then a few meters may constitute a moment. But they also allow a moment to be an infinitesimal. In fact, they develop the notion of an accumulation function as a sum from some fixed starting point a up to the variable x of accumulations over infinitesimal moments. They thus embrace the approach to integration advocated by Ely.

Personally, I believe that it is useful to allow students to think of the differential in an integral as an infinitesimal and, in fact, to encourage this as students are challenged to solve a variety of accumulation problems by first translating them into definite integrals. But I much prefer the approach to integration developed by Michael Oehrtman for his CLEAR Calculus (Coherent Labs to Enhance Accessible and Rigorous Calculus) at Oklahoma State. In Lab 12, he considers the problem of finding distance given an increasing velocity function. The point is that we can put upper and lower bounds on the total distance using the minimum and maximum velocities. As we take minimum and maximum velocities over shorter intervals, we can bring these bounds closer together, making their difference between the maximum and minimum possible distances as small as we wish. Although Oehrtman never uses the term, these are Darboux sums, a very intuitive and natural idea—it is how Newton explained integration—that provide a completely rigorous definition of the definite integral. I also explain this approach in the appendix to Calculus Reordered.

An Aside. I cannot resist adding that Riemann introduced his definition of the definite integral as a means of analyzing how discontinuous a function could be and still be integrable. It is ideally suited for that purpose. Otherwise, it is a terrible way of defining the definite integral that should have no place in first year calculus. Darboux translated Riemann’s work on integration into French and introduced his sums for the purpose of simplifying the definition of the definite integral.

So what is a differential? I admit that I prefer the modern understanding that sees them as a particular instance of a differential form. As I stated with italics in Second Year Calculus, “differential forms exist to be integrated,” (Bressoud, 1991, p. 79). Everything after the integral sign in a definite integral is actually a differential. It is the differential of any antiderivative. Differentials are operators acting on curves, surfaces, solids, or more general manifolds. But I do not say that to a general audience of students in the first year of calculus. Instead I tell them they are not ready for the actual definition, but here are some useful ways of thinking about them: as infinitesimals in the context of integration, as variables linked by a proportional relationship and bounded in magnitude by a sufficiently small Delta x in the context of differentiation.

Final Aside. My understanding of the differential was hard won. In my first year of graduate school I took the a course in algebraic topology. I had relished point-set topology as an undergraduate and thought this would be equally fun. It was a disaster. While I learned how to manipulate differential forms and construct short exact sequences well enough to pass the course, I had no idea what I was doing or why I was doing it. Most frustrating was that I finished the course without any understanding of what a differential form actually was.

Years later I taught an honors course on second year calculus and chose Advanced Calculus by Harold Edwards as the text, attracted by the way he drew on applications in physics to motivate the development. For my students, it was a bit too challenging, but I fell in love with that text. I wrote Second Year Calculus: from Celestial Mechanics to Special Relativity in an attempt to take what I had learned from Edwards and repackage it in a form suitable for good undergraduates in their second year of calculus. Even my repackaging is really too challenging for that level. The text has mostly been used for a junior course in vector calculus. But in the thirty years since it was published it has been gratifying to see this book picked up by doctors, engineers, and other professionals who studied calculus in college but never understood why it was important. Many have written to me to say that they found in my book what they felt they had missed.

I had known Freeman Dyson during my year at the Institute for Advanced Study and was inspired by his Gibbs lecture of 1972, “Missed Opportunities,” arguing for reigniting communication between physicists and mathematicians. I included a substantial portion of his address as an appendix to Second Year Calculus and sent Dyson a copy of the book. He wrote back to thank me and said that, for the first time in his life, he now understood what a differential is. How could I possibly embrace any other definition?

References

Bressoud, D.M. (1991). Second Year Calculus: from Celestial Mechanics to Special Relativity. New York, NY: Spring-Verlag.

Bressoud, D.M. (2019). Calculus Reordered: A history of the big ideas. Princeton, NJ: Princeton University Press

Demana, F., Waites, B., Kennedy, D., Bressoud, D., and Boardman, M. (2020). Calculus: Graphical, numerical, algebraic, 6th edition. Boston: Pearson.

Dyson, F. (1972). Missed Opportunities.  Bulletin of the American Mathematical Society, 78:5, 635–652.

Edwards, H.M. (1994). Advanced Calculus: A differential forms approach. Boston: Birkhäuser.

Ely, R. Teaching calculus with infinitesimals and differentials. (2021). ZDM Mathematics Education 53, 591–604. https://doi.org/10.1007/s11858-020-01194-2

Thomas, G.B., Jr. (1951). Calculus and Analytic Geometry. Cambridge, MA: Addison-Wesley.

Thompson, P.W., and Ashcroft, M. (2019). Calculus: Newton, Leibniz, and Robinson Meet Technology. http://patthompson.net/ThompsonCalc/index.html

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