Focus on Calculus

A Physicist's View of Teaching Calculus

Ralph Baierlein, Wesleyan University

What topics in calculus would physicists like to see their students learn? A lot of topics, of course, but the need to economize in this article forces me to economize in what I might ask of a calculus teacher, who is strapped for time, if not for space. Here are some highlights--- some areas of especial concern to the study of physics.

Exponentials and logarithms

That the derivative of an exponential function is proportional to the function itself is the most important property of those functions. A physicist would like to start with that property, as displayed here:

Numerical exploration with base (b=2) yields a coefficient of bX that is less than 1; trying (b = 10) yields a coefficient greater than 1. In between 2 and 10 there ought to be a number that yields 1, and thereby e enters the scene. Then it is a matter of small, relatively easy steps to develop the topic and to finish with ln y as the integral of one-over-x dx. That logarithmic relation always puzzles students, and so it is best to place it last, not at the start, where it might derail the entire development.

Expansions and approximations

Almost every ``exactly-solved'' problem in physics is based on some initial approximation. To be sure, solutions have become famous as exact solutions to nonlinear differential equations, but those equations themselves are merely approximations to more fundamental equations. Physics students need to become handy with the Taylor expansion and the binomial expansion. Their level of expertise should enable them to apply those expansions to functions like

Limiting behavior

Physicists often need to examine the limiting behavior of some function or solution. They need to know what happens when something goes to zero or pi/2 or infinity or .... Indeterminate forms appear often, and so L'Hospital's rule is an essential part of a physicist's repertoire. It is a good idea to link that rule to a Taylor expansion, separately, of the numerator and denominator of a quotient. Though less general, the expansion is easier for a student to appreciate and, when it may be applied, can give one more information about the quotient.

``Modeling'' and ODE's

How to translate from a verbal context to ordinary differential equations is a skill that physics students need to learn. The spring term of a typical introductory physics course will derive for the students the following equations for an electrical circuit:

The first equation relates the charge, Q, on a capacitor to the current, I, through a resistor and to a battery's voltage. The second equation says that the current alters the charge on the capacitor. Students need to learn that they can eliminate the current, I, between the two equations, get a single differential equation for the charge, Q, namely,

and then solve that equation.

Newton's second law of motion, that force equals mass times acceleration, generates its own many differential equations, now second order in time because the acceleration enters. In general, students need familiarity with three aspects of ordinary differential equations:

* How to solve linear ODE's that have constant coefficients.

* How to solve (some) first order ODE's by separation of variables, followed by integration.

* How to use Euler's method for numerical integration.

Multivariable Calculus

For the last highlight, let me stretch to multivariable calculus. In the broad range of disciplines from chemistry to astrophysics, functions defined over all space dominate the research scene, rather than trajectories or surfaces in space. The functions may be scalars or vectors, as illustrated below in some fundamental equations from electromagnetism:

The electric field, E, expressed as the gradient of a scalar function, phi, and the time derivative of a vector potential, A. The divergence of the electric field is determined by the density, rho, of the electric charge. The curl of the electric field is proportional to a time derivative of the magnetic field, B. The vector operations of grad, div, and curl play essential roles, and students need a good geometric sense for what they mean. Loosely,

grad: maximal spatial rate of change (with direction);

div: surface integral per unit volume; and

curl: maximal line integral per unit area (pointing along normal).

These operations should be more than a mess of partial derivatives.

The transition from a geometric, coordinate-free definition to an explicit representation will surely be made first to Cartesian coordinates. To really understand how areas and lengths enter, however, students need to see---in addition---the transition to either cylindrical or spherical coordinates.

It would be good to rearrange the order of presentation in multivariable calculus so that grad, div, and curl come by mid-semester and then get some regular reinforcement. Those operations (and their combination in the Laplacian, div grad) deserve a prominent place in multivariable calculus for the physical sciences.

You may say to yourself, it's all well and good for him to advocate teaching these highlights, but where will I find the time? I do have some suggestions for items that one can soft pedal or even eliminate, but they would have to come in another article. For now, let me close by saying that the movement to revitalize calculus is making real progress, and I wish you all well.