Student Understanding of Constituent Derivatives of Divergence and Curl from Vector Field Diagrams

Vector calculus is central to several physics topics; derivatives of vector fields represent meaningful quantities. Previous research explored student understanding of divergence and curl of vector field diagrams. In order to examine how student understanding of constituent derivatives of divergence and curl relates to the determination of divergence and curl, we designed tasks providing a two-dimensional vector field representation and asking about the signs of the divergence and curl as well as their constituent derivatives at one or more locations. Written data were collected in Mathematical Methods in Physics courses, after relevant vector calculus instruction, at two universities. We observed different patterns of responses for the derivatives comprising the divergence and those comprising the curl. For the latter, many students struggled to identify which vector field component was described by a given derivative. Similar confusion was observed in responses to tasks in which the vector field had a single component.