Abstract:

In pure mathematics, one deals with abstract objects which must be rigorously and well defined. This holds in particular for the basic set-theoretic notion of a "function", or "map". One possible definition may be attributed to Fraenkel (1922) and Skolem (1922) who saw a function as a specific kind of relation, where a relation is in turn defined as a set of ordered pairs.

However, in physics, engineering and other natural and applied sciences that rely on mathematical methods, the understanding of a functional relationship is often not as formal or abstract as this. This also reflects in the notation used in standard textbooks. For example, one may find for a physical law of inverse-distance, like the Coulomb potential u of a charged point particle, formulas like u(r)=-1/r, u=-1/r or u(x,y,z)=-1/r (omitting physical constants). Here, r denotes the distance to the point particle. Of course, the meaning of these formulas describing the same object "u" is immediate, and it may be interpreted as a function in one way or another, defined by the given formula. However, in our experience, much confusion arises among undergraduate students of physics and engineering at this point, for example in the context of differential operators in different coordinate systems. This confusion is based on the ambiguity of the precise mathematical meaning of u: It may be seen as a function of one variable (the distance r, due to rotational invariance) or three variables (the Cartesian coordinates (x,y,z)). From the mathematical viewpoint, the most elegant interpretation would probably be the differential geometric view where u is seen as a function on the manifold of Euclidean 3-space minus one point ("r=0"), in this case given on a specific chart in specific coordinates. However, neither this interpretation nor the basic formal definition of a function is very helpful to give students of physics or engineering a firm understanding of the function concept which also leverages practical insight for their particular field of study.

Based on these observations, our teaching experience as well as a student survey conducted by us, we will argue that a successful methodology for providing students of physics and engineering with a mathematically sound and practical understanding of a function must differ from teaching students of pure mathematics. We will also exemplify this view via a closer inspection of examples from multivariate calculus, applied to problems from electrical engineering. Finally, we will give practical hints on how to address the misconceptions among students that are the most common in this context.

Keywords:

mathematics education (for physicists and engineers), teaching experience.