Physics students’ construction of differential length vectors in an unconventional spherical coordinate system

Vector calculus and multivariable coordinate systems play a large role in the understanding and calculation of much of the physics in upper-division electricity and magnetism. Differential vector elements represent one key mathematical piece of students’ use of vector calculus. In an effort to examine students’ understanding of non-Cartesian differential length elements, students in junior-level electricity and magnetism were interviewed in pairs and asked to construct a differential length vector for an unconventional spherical coordinate system. One aspect of this study identified symbolic forms invoked by students when building these vector expressions, some previously identified and some novel, given the vector calculus context. Analysis also highlighted several common ideas in students’ concept images of a non-Cartesian differential length vector as they determined their expressions. As no interview initially resulted in the construction of an appropriate differential, analysis addresses the role of the evoked concept images and symbolic forms on students’ performance.

Figure 1

(a) Conventional (physics) spherical coordinates; (b) an unconventional spherical coordinate system given to students, for which they were to construct differential length and volume elements. The correct elements for each system are shown in (c) and (d), respectively.

Figure 2

Beginning stages of construction for Carol and Dan showing the coupling of the parts of a whole and magnitude direction symbolic forms.

Figure 3

Perry’s and Quinn’s writing of the differential length elements as a sum of three terms, depicting the parts of a whole structure despite writing out each component explicitly.

Figure 4

Rachel’s and Silas’s final expression for their differential length vector.

Figure 5

Greg and Harold before (a) and after (b) recognizing the need to include unit vectors.

Figure 6

(a) Carol and Dan incorrectly incorporating the idea of a differential by writing “$d$” before their $\widehat{\mathit{\beta }}$ term. (b) Frank attempting to account for the arc length of a small angle and forcibly insert both a differential and trigonometric function into his expression.

Figure 7

Elliot and Frank constructing the $\beta$ component of the differential length. Initially they leave space to write the needed coefficient and unit vector. After discussion they include a coefficient lacking the projection term $\cos \alpha$.

Figure 8

Adam before and after using reasoning about including dimensionality to fill the coefficient template of the $\widehat{\mathit{\alpha }}$ component.