Schermerhorn & Thompson about differential volume elements

Benjamin P. Schermerhorn and John R. Thompson

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


Phys. Rev. Phys. Educ. Res. 15, 010112

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.

DOI: https://doi.org/10.1103/PhysRevPhysEducRes.15.010112

Schermerhorn and Thompson on differential length vectors

Benjamin P. Schermerhorn and John R. Thompson

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

Phys. Rev. Phys. Educ. Res. 15, 010111

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.

DOI: https://doi.org/10.1103/PhysRevPhysEducRes.15.010111

Schermerhorn Thompson on determining differential area elements

Student determination of differential area elements in upper-division physics

Benjamin P. Schermerhorn and John R. Thompson

Physics Education Research Conference Proceedings 2017

Given the significance of understanding differential area vectors in multivariable coordinate systems to the learning of electricity and magnetism (E&M), students in junior-level E&M were interviewed about E&M tasks involving integration over areas. In one task, students set up an integral for the magnetic flux through a square loop. A second task asked students to set up an integral to solve for the electric field from a circular sheet of charge. Analysis identified several treatments of the differential area: (1) a product of differential lengths, (2) a sum of differential lengths, (3) a product of a constant length with differential length in one direction, (4) a derivative of the expression for a given area, and (5) the full area.

Physics Education Research Conference 2017
Part of the PER Conference series
Cincinnati, OH: July 26-27, 2017
Pages 356-359

DOI: 10.1119/perc.2017.pr.084

Schermerhorn and Thompson on symbolic forms and differential length elements

Benjamin P. Schermerhorn and John R. Thompson

Students’ use of symbolic forms when constructing differential length elements

As part of an effort to examine students' understanding of the structure of non-Cartesian coordinate systems and the differential vector elements associated with these systems, students in junior-level electricity and magnetism (E&M) were interviewed in pairs. Students constructed differential length and volume elements for an unconventional spherical coordinate system. A symbolic forms analysis found that students invoked known as well as novel symbolic forms when building these vector expressions. Further analysis suggests that student difficulties were primarily conceptual rather than symbolic.

B. P. Schermerhorn and J. R. Thompson, Students’ use of symbolic forms when constructing differential length elements, 2016 PERC Proceedings [Sacramento, CA, July 20-21, 2016], edited by D. L. Jones, L. Ding, and A. Traxler, doi:10.1119/perc.2016.pr.073.