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. Program summary Program Title: Surf Program Files doi: http://dx.doi.org/10.17632/k6wp6ygcfm.1 Licensing provisions: CC0 1.0 Programming language: Maple 17 or later Nature of problem: Starting from a mathematical model in flat space, we aim to find the corresponding system in a given curved surface. This process can be applied in many areas of knowledge, such as fluid dynamics, quantum mechanics, diffusion on curved surfaces, biology, to name a few. Solution method: The Surf package determines several differential operators and metrics on a surface, such as Laplacian, gradient, mean square displacement, among others. This is done through the symbolic determination of differential operators and metrics, having a parameterized surface as input. Additional comments: We illustrate our methodology in Fig. 1. [GRAPHICS] (C) 2019 Elsevier B.V. All rights reserved.