Development of Interactive 3D Learning Objects for Visualizing Vector Calculus Concepts in Matlab

Abstract

Vector calculus concepts such as vector fields, divergence, and curl are often difficult for students to understand because they involve abstract multidimensional structures and spatial interpretation processes. In traditional instructional approaches, these concepts are generally presented through symbolic notation and static visual representations, which may limit the interpretation of the dynamic behavior of three-dimensional mathematical structures. This study addresses the design, development, and expert evaluation of a Matlab-based interactive learning object for the three-dimensional examination of vector calculus concepts. The proposed system integrates symbolic computation, numerical evaluation, and three-dimensional graphical visualization within a single computational environment. Using a selected vector field example, the study examines vector direction, magnitude, divergence, curl, and parameter-dependent graphical transformations. In addition, two independent experts evaluated the developed learning object in terms of mathematical accuracy, visual clarity, usability, reproducibility, and instructional suitability. The overall expert-review mean was 4.50/5.00, and both experts evaluated the learning object as suitable in its current form. However, this study is not an effectiveness study in which learning outcomes are experimentally measured. Therefore, the impact of the proposed learning object on students' learning outcomes should be further examined through future user studies, pilot implementations, and classroom-based experimental research.