NSF Proposal Summary DUE-0088901
1/01–7/03; $112,513

Tevian Dray & Corinne A. Manogue

There is a “vector calculus gap” between the way vector calculus is usually taught by mathematicians and the way it is used by other scientists. This material is essential for physicists and some engineers due to its central role in the description of electricity and magnetism. It is the goal of this proposal to bridge this gap.

The vector calculus gap goes much deeper than a difference in emphasis. Ask a physicist or engineer what topics should be covered in vector calculus, and the answer will pretty much agree with the existing syllabus used by mathematicians. But the traditional language used by mathematicians to teach this material is so different from the way it is used in applications that students are often unable to translate.

A major part of the problem is the traditional mathematics emphasis on Cartesian coordinates to describe vectors as triples of numbers, rather than emphasizing that vectors are arrows in space. This leads to the all-important dot and cross products being memorized as algebraic formulas, rather than statements about projections and areas, respectively. It is hardly surprising that many students are then barely able to compute line and surface integrals, or the divergence and curl of a vector field, let alone understand their geometric interpretation.

The traditional approach has one big advantage: It provides a single framework for handling quite general problems, the classic example being problems involving a paraboloid. But most practical applications, including virtually all at the undergraduate level, fall into a small number of special cases, such as those with spherical or cylindrical symmetry. There are no paraboloids in undergraduate physics! Problems with a high degree of symmetry become much more intuitive when the computations are not only done in appropriate coordinates, but also using a vector basis adapted to those coordinates. This emphasizes the geometry of the particular problem, rather than a brute force algebraic computation which many students fail to find illuminating.

We propose to develop supplemental materials, especially small group activities, which emphasize the geometry of highly symmetric situations, some of which are intended for use with an otherwise traditional vector calculus course, and some of which are intended for use in a new, upper-division physics course on related material. Such activities will introduce students to the types of problems – and methods of solution – which they will encounter in their chosen specialization, while at the same time increasing their understanding of traditional vector calculus and its applications, thus bridging the gap.

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