Surfing on curved surfaces—The Maple Package Surf
Curved structures are ubiquitous in nature, particularly in applied mathematics. The systems where they are present include membranes, interfaces, curved spacetime, fluid mechanics, etc. In contrast, the cumbersome aspects of some calculations, sometimes make the use of differential–geometric tools to model systems in curved space, infeasible for practical analytical purposes. In this work, we introduce Surf, a Maple package for differential geometry of surfaces, with functions to ease interdisciplinary modeling in curved systems. The main idea is to have a parameterized surface as input and, as output, a model system on curved surface. The usual operators for a given surface are implemented, so that an arbitrary model can be immediately mapped from flat to curved space. The simplicity of our approach is illustrated with some applications in a variety of interdisciplinary problems: diffusion on a curved space, quantum mechanics on surfaces, and modeling the trajectory of a C. elegans worm. We hope that this package will contribute for easing the implementation of mathematical modeling on surfaces in general.