Bridging the Vector Calculus Gap

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Mathematical Sciences (21) Physics (13)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. 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 dot and cross products being memorized as algebraic formulas, rather than statements about projections and areas, respectively. The traditional approach has the one big advantage of providing a single framework for handling quite general problems. 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. Problems with a high degree of symmetry become much more intuitive when the computations are done in appropriate coordinates, using a vector basis adapted to those coordinates. This emphasizes the geometry of the problem, rather than a brute force algebraic computation.This project is developing 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 introduce students to the types of problems and methods of solution which they encounter in their chosen specialization, while at the same time increasing their understanding of traditional vector calculus and its applications, thus bridging the vector calculus gap.