Fractional Geometric Calculus: Toward A Unified Mathematical Language for Physics and Engineering ?

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.