Investigating Student Understanding of Vector Calculus in Upper-Division Electricity and Magnetism: Construction and Determination of Differential Element in Non-Cartesian Coordinate Systems

Abstract

Differential length, area, and volume elements appear ubiquitously over the course of upper-division electricity and magnetism (E&M), used to sum the effects of or determine expressions for electric or magnetic fields. Given the plethora of tasks with spherical and cylindrical symmetry, non-Cartesian coordinates are commonly used, which include scaling factors as coefficients for the differential terms to account for the curvature of space. Furthermore, the application to vector fields means differential lengths and areas are vector quantities. So far, little of the education research in E&M has explored student understanding and construction of the non-Cartesian differential elements used in applications of vector calculus. This study contributes to the research base on the learning and teaching of these quantities. Following course observations of junior-level E&M, targeted investigations were conducted to categorize student understanding of the properties of these differentials as they are constructed in a coordinate system without a physics context and as they are determined within common physics tasks. In general, students did not have a strong understanding of the geometry of non-Cartesian coordinate systems. However, students who were able to construct differential area and volume elements as a product of differential lengths within a given coordinate system were more successful when applying vector calculus. The results of this study were used to develop preliminary instructional resources to aid in the teaching of this material. Lastly, this dissertation presents a theoretical model developed within the context of this study to describe students’ construction and interpretation of equations. The model joins existing theoretical frameworks: symbolic forms, used to describe students’ representational understanding of the structure of the constructed equation; and conceptual blending, which has been used to describe the ways in which students combine mathematics and physics knowledge when problem solving. In addition to providing a coherent picture for how the students in this study connect contextual information to symbolic representations, this model is broadly applicable as an analytical lens and allows for a detailed reinterpretation of similar analyses using these frameworks