Visualization of Orthonormal Triads in Cylindrical and Spherical Coordinates

Abstract

According to Committee on Programs for Advanced Study of Mathematics and Science in American High Schools [1] (Gollub et al. (eds.) in Learning and Understanding. Improving Advanced Study of Mathematics and Science in U.S. Hight Schools. National Academic Press, Washington, 2002), “the primary goal of advanced study in any discipline should be for students to achieve a deep conceptual understanding of the disciplines content”. It is undoubted that abstraction is one of the skills that teachers wish to improve in their students, but, how can teachers take advantage of technological resources, such as CAS or DGS, as help in their classes in undergraduate courses? One concept, whose importance is both theoretical and practical, corresponds to the coordinate transformation, in particular orthogonal coordinate systems. We can use trigonometric constructions to find the transformation equations, namely, the algorithm for transforming a Cartesian system into other coordinate system, as cylindrical or spherical coordinates. Not only, if we add the knowledge and some techniques from Linear Algebra, we can motivate new mathematical properties, but also we will increase considerably the abstract reasoning and symbolic calculation. We know that visualization helps intuitive understanding. Therefore, we propose using CAS and DGS to show how a triad, of basis vectors, is continuously changing direction, keeping the norm vector without change, and how this match visualization with the reasoning from theories of linear algebra.