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首页> 外文期刊>International journal of theoretical physics, group theory, and nonlinear optics >An Approach to Fractional Differential Geometry and Fractal Space-Time via Fractional Calculus for Non-Differentiable Functions
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An Approach to Fractional Differential Geometry and Fractal Space-Time via Fractional Calculus for Non-Differentiable Functions

机译：非可微分函数分数微积分的分数微分几何和分形空间的方法

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In this paper, one will show how fractional differential derived from fractional difference provides a basis to expand a theory of fractional differential geometry which would apply to non-differentiable manifolds. One of the keys to the approach is our (controversial to some readers) fractional derivative Leibniz chain rule which applies to non-differentiable functions only. One begins with standard background on fractional calculus via fractional difference, one defines fractional differentiable manifolds, and then one switches to the arc length of non- differentiable curves, for which several models are proposed, what is quite right so since one deals with non-differentiable functions. Then one considers radius of curvature for fractional curves, one examines what happens with covariant derivative of fractional order and one introduces fractional velocity and fractional acceleration to obtain the fundamental forms and area on non-differentiable manifolds. Then one comes across geodesics on fractional manifolds therefore one arrives at a Minkowski geodesic on fractal space time. Everywhere in the paper one has to keep in mind that fractional derivatives for non-differentiable functions are not commutative. The framework is quite suitable to expand a theory of fractional white noise calculus.
机译：在本文中，一个将展示源于分数差异的分数差异提供了扩展分数差分几何形状的基础，这适用于非微弱歧管。该方法的一个钥匙是我们（对某些读者的争议）分数衍生利用leibniz链规则，仅适用于非可分子功能。一个首先通过分数差异的分数微积分开始标准背景，一个定义分数微分歧管，然后一个开关到非可分子曲线的电弧长度，提出了几种模型，所以自那样的是非可怜的功能。然后，一个考虑分数曲线的曲率半径，一检测分数顺序的协调衍生物发生的情况，并且引入了分数速度和分数加速，以获得非微弱歧管上的基本形式和区域。然后，在分数歧管上跨越大测地仪，因此一个到达分形空间时间的Minkowski GeodeSic。本文到处都是必须记住，非可微分功能的分数衍生物不会进行换向。该框架非常适合扩展分数白噪声微积分理论。

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