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首页> 外文会议>Proceedings of the Fifth symposium on fractional differentiation and its applications >Fractional Geometric Calculus: Toward A Unied Mathematical Language for Physics and Engineering
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Fractional Geometric Calculus: Toward A Unied Mathematical Language for Physics and Engineering

机译：分数几何微积分：面向物理和工程学的统一数学语言

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摘要

This paper discuss the longstanding problems of fractional calculus such as too many definitions while lacking physical or geometrical meanings, and try to extend fractional calculus to any dimension. First, some different definitions of fractional derivatives, such as the Riemann-Liouville derivative, the Caputo derivative, Kolwankar's local derivative and Jumarie's modified Riemann-Liouville derivative, are discussed and conclude that the very reason for introducing fractional derivative is to study nondifferentiable functions. Then, a concise and essentially local definition of fractional derivative for one dimension function is introduced and its geometrical interpretation is given. Based on this simple definition, the fractional calculus is extended to any dimension and the Fractional Geometric Calculus is proposed. Geometric algebra provided an powerful mathematical framework in which the most advanced concepts modern physic, such as quantum mechanics, relativity, electromagnetism, etc., can be expressed in this framework graciously. At the other hand, recent developments in nonlinear science and complex system suggest that scaling, fractal structures, and nondifferentiable functions occur much more naturally and abundantly in formulations of physical theories. In this paper, the extended framework namely the Fractional Geometric Calculus is proposed naturally, which aims to give a unifying language for mathematics, physics and science of complexity of the 21st century.
机译：本文讨论了分数微积分的长期问题，例如定义过多而缺乏物理或几何意义，并试图将分数微积分扩展到任何维度。首先，讨论了分数导数的一些不同定义，例如Riemann-Liouville导数，Caputo导数，Kolwankar的局部导数和Jumarie的修饰的Riemann-Liouville导数，并得出结论，引入分数导数的根本原因是研究不可微函数。然后，引入一维函数的分数导数的简明且基本局部的定义，并给出其几何解释。基于这个简单的定义，分数阶微积分被扩展到任何维度，并提出了分数阶几何微积分。几何代数提供了一个强大的数学框架，其中可以优雅地表达现代物理的最先进概念，例如量子力学，相对论，电磁学等。另一方面，非线性科学和复杂系统的最新发展表明，在物理理论的表述中，标度，分形结构和不可微函数的出现更为自然和丰富。在本文中，自然提出了扩展框架，即分数几何微积分，旨在为21世纪的数学，物理学和复杂性科学提供统一的语言。

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