A praxeological investigation of divergence - exploring challenges of teaching and learning math-in-physics

DidMat seminar with Ricardo Karam and Anders Wolfsberg.

Abstract.

In this study a case of two undergraduate physics courses is investigated to identify and detail challenges of teaching and learning the notion of divergence of a vector field. On the basis of lecture observations and student interviews, a number of central challenges are indicated. 

Firstly, that of aligning two conceptually different perspectives on divergence - as a differential operator and as a measure of flux volume density. Due to the lack of arguments to relate these representations, two disconnected perspectives on divergence emerge: one instrumental for computations but unrelated to flux, and one relating divergence to flux but only intuitively and imprecisely so. 

Secondly, physical and geometrical interpretations relating divergence to flux, among other factors, are influential in effectively reducing the notion of divergence to that of net flux out of a volume in teaching.

Thirdly, portraying divergence in the context of fluid mechanics, albeit pedagogically pertinent, introduces a challenge of clarifying the conceptual distinction between the divergence of planar and spatial vector fields.

Lastly, Gauss' Law in differential form for electrostatics presents a need for an extended perspective on divergence from a vector differential operator to ”what the Divergence Theorem intuitively implies".