Fractional Differential Forms

TL;DR

This paper introduces a fractional exterior calculus by generalizing derivatives to fractional orders, creating new vector spaces of differential forms with modified properties and transformation rules, reducing to standard calculus when fractional order is one.

Contribution

It defines fractional exterior derivatives and form spaces, extending classical exterior calculus to fractional orders with new transformation and metric rules.

Findings

Fractional exterior derivatives generate new vector spaces of differential forms.

Transformation rules differ from classical calculus due to fractional derivatives.

Results reduce to standard exterior calculus when fractional order is one.

Abstract

A generalization of exterior calculus is considered by allowing the partial derivatives in the exterior derivative to assume fractional orders. That is, a fractional exterior derivative is defined. This is found to generate new vector spaces of finite and infinite dimension, fractional differential form spaces. The definitions of closed and exact forms are extended to the new fractional form spaces with closure and integrability conditions worked out for a special case. Coordinate transformation rules are also computed. The transformation rules are different from those of the standard exterior calculus due to the properties of the fractional derivative. The metric for the fractional form spaces is given, based on the coordinate transformation rules. All results are found to reduce to those of standard exterior calculus when the order of the coordinate differentials is set to one.