Each of us has taught vector calculus for many years: one of us (TD) as part of the second year calculus sequence in the mathematics department, and the other (CAM) as part of the third year physics course on mathematics methods. The latter course officially has the former as a prerequisite, although there is considerable overlap in the material covered. Part of the reason the physics course exists is to teach physics majors the “right” way to use this material, given that the mathematics department has failed to do so!

We are fortunate in having had an opportunity to compare notes frequently: we are long-standing collaborators in mathematical physics, and we are married to each other! Yet even so our conversations originally focused on what material was covered when, e.g., discussing which week was devoted to Stokes' Theorem, or whether to cover it before or after the Divergence Theorem. It literally took us years to discover that these theorems as taught by mathematicians bear little resemblance to the versions taught by physicists.

To make a long story short, we eventually decided to try to bridge this “vector calculus gap”; this instructor's guide is the main result. It contains an extensive discussion of the different ways in which mathematicians and other scientists, especially but not only physicists, view this material, as well as a detailed description of the group activities we developed to teach geometric visualization skills.

The activities do not depend on any particular textbook — our approach to vector calculus is not in any current text. Yet it is not sufficient to simply supplement a traditional vector course with our activities, as both our geometric approach and some of our notation may be unfamiliar to instructors and students alike. The first part of this guide therefore explains our approach in some detail, and can serve as the basis for minilectures relating our approach to what the students already know; some sections could also be given to the students as handouts.