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

**Authors:** Xiong Wang

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 \emph{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.

**Comments:** 6 Pages.

**Download:** **PDF**

### Submission history

[v1] 2012-06-02 21:56:55

**Unique-IP document downloads:** 1480 times

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