Project Vector Fields: A Visual Approach to Vector Calculus Integral Theorems

Vector fields are a central representation in physics education, used, for example, to describe electric fields or fluid flows. In addition to visualization through vector field diagrams and algebraic description via formulas, it is particularly the specific properties of vector fields that are relevant for physics. For instance, divergence provides information about sources and sinks of a vector field, while curl (rotation) offers insights into its vortices. These concepts also form essential components of various conservation laws, such as the continuity equation, which students encounter early in their physics studies. Recent research emphasizes the importance of a deep understanding of vector concepts for students' performance in the introductory phase of physics education. In this context, the divergence theorem (also known as Gauss's theorem) plays a central role in understanding the concept of divergence. Its counterpart for curl is the Stokes’ theorem. Both theorems connect various mathematical and physical concepts related to divergence and curl, making them key elements for a profound understanding of vector calculus relationships.

As current research shows, students rarely struggle with the calculation of divergence and curl; however, they often find conceptual explanations challenging. Yet it is precisely this conceptual understanding that is crucial for applications in physics. Since current instructional approaches are often heavily mathematical, there is a clear need for new explicit instructional methods that support students in developing a solid grasp of vector concepts. The goal of this project is therefore to design and study such interventions. In contrast to previous approaches, this project adopts a visual interpretation of divergence and curl using vector field diagrams, supported by interactive subject-specific teaching methods (e.g., in the form of simulations). The current version of a vector field simulation can be accessed via this link. In addition to traditional assessment methods, students’ eye movements will also be analyzed to gain insights into their cognitive processes during task completion.