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Concepts covered in introductory electricity and magnetism such as electric and magnetic field vectors, solenoids, and electromagnetic waves are difficult concepts for students to visualize. Part of this difficulty may be due to the representation of three-dimensional objects on the two-dimensional planes of course textbooks and classroom whiteboards. The use of two-dimensional platforms limits the visualization of phenomena such as the vector field of a point charge or test charges traveling in the three-dimensional space of an electric field. In addition, working in two dimensions may add to students’ difficulties orienting their body correctly to use the right-hand rule when determining the direction of a magnetic field. These difficulties in visualization may limit the conceptual understanding of these fundamental topics. To promote conceptual understanding of electromagnetism we are cyclically developing and researching three spatial computing 3D environments covering electric fields, magnetic fields and electromagnetic waves. Each environment will be developed and tested in both augmented and virtual reality. The first of our environments, the electric field, has been built and tested in augmented reality (AR) with introductory physics students in the Fall 2023 semester. Our study is currently in phase IV of the National Science Foundation’s Design and Development Cycle. Data collected during phase II is being analyzed to support revision to the environment as well as data collection protocols. This article will outline findings from qualitative data gathered during the AR experience as well as during student post interviews following participation in the electric field space. These findings are characterized and then responded to with recommendations for the design team regarding content and testing procedures. In what follows, we first present a framework listing current knowledge regarding students' difficulties learning electric fields and how these guided our design of this electric field augmented reality environment. We next present themes that emerged from discussions during the experience as well as the post interviews. We conclude with suggestions to inform our second round of environmental design. 

The velocity–vorticity formulation of the 3D Navier–Stokes equations was recently found to give excellent numerical results for flows with strong rotation. In this work, we propose a new regularization of the 3D Navier–Stokes equations, which we call the 3D velocity–vorticity-Voigt (VVV) model, with a Voigt regularization term added to momentum equation in velocity–vorticity form, but with no regularizing term in the vorticity equation. We prove global well-posedness and regularity of this model under periodic boundary conditions. We prove convergence of the model's velocity and vorticity to their counterparts in the 3D Navier–Stokes equations as the Voigt modeling parameter tends to zero. We prove that the curl of the model's velocity converges to the model vorticity (which is solved for directly), as the Voigt modeling parameter tends to zero. Finally, we provide a criterion for finite-time blow-up of the 3D Navier–Stokes equations based on this inviscid regularization.