Physics students’ construction and checking of differential volume elements in an unconventional spherical coordinate system

In upper-division physics courses, students’ use of differential line, area, and volume elements and their facility with the various multivariable coordinate systems consistently go hand in hand. As part of an effort to investigate student understanding of the structure of non-Cartesian coordinate systems and the associated differential elements, we interviewed students (mostly in pairs) in junior-level electricity and magnetism courses at two universities. In a sequence of tasks, students were asked to construct a differential length vector and a differential volume element in an unconventional spherical coordinate system. None of the students were able to arrive at a correct differential length element initially. This work addresses the construction and checking of the volume element. Volume element construction occurred by either combining associated lengths, an attempt to determine sides of a differential cube, or mapping from the existing spherical coordinate system. Students who constructed volume elements from differential length components corrected their length element terms as a result of checking the volume element expression by integration. Other students who relied heavily on spherical coordinates displayed further difficulty connecting dimensionality and projection ideas to differential construction.

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 in (c) and (d), respectively.

Figure 2

Final differential volume constructed by Carol and Dan including incorrect trigonometric function.

Figure 3

(a) Incorrectly distributed length terms in a spherical differential volume written by Carol and Dan. (b) Unsuccessful attempt to reconstruct differential spherical length element by Greg and Harold.

Figure 4

Physical construction of a differential volume element by Rachel and Silas. (a) Beginning of volume construction using the single length component. (b) Final differential volume, where location of $dM$ has changed. Students do not connect the sides of the volume to length components.

Figure 5

Rachel and Silas directly compare coordinate systems and account for changes in variable placement as they construct a volume element.

Figure 6

Tyler’s determination of a schmerical differential volume by comparison to spherical. Tyler unknowingly uses mathematics conventions for spherical coordinates.

Figure 7

Elliot and Frank check their incorrect differential volume element by integrating.