User:Maschen/Fractional differential forms

In mathematics, specifically differential geometry, fractional differential forms generalizes the standard differential forms and exterior calculus. The exterior derivative (used in differential forms) can be defined in terms of partial derivatives, and fractional partial derivatives of non-integer (or even complex) orders can be defined according to fractional calculus. So fractional partial derivatives lead to the definition of a fractional exterior derivative. All results naturally reduce to those of ordinary exterior calculus when the order of the coordinate differentials is set to 1.

Differential forms and exterior calculus are useful in because the formalism is coordinate-independent. Exterior calculus of differential forms give an alternative to coordinate-independent vector calculus.

They have been popularized by K. Cotrill-Shepherd and M. Naber, around the start of the third millennium (2001 - 2003).

Background
An n-form is:


 * $$\alpha = \alpha^{i_1 i_2 \cdots i_n} dx_{i_1} \wedge dx_{i_2} \wedge \cdots \wedge dx_{i_n}$$

where d denotes the exterior derivative defined as:


 * $$ d = dx_i \frac{\partial}{\partial x_i}$$

with the antisymmetric tensor property


 * $$ dx_i \wedge dx_j = - dx_j \wedge dx_i$$

and where dxi are coordinate differentials.

Fractional exterior derivative
A fractional exterior derivative can be defined as:


 * $$ dx = dx_i \frac{\partial^\lambda}{{[\partial (x_i - a_i)]}^\lambda}$$

where the partial derivatives have fractional order;


 * $$ \begin{align}

\frac{\partial^\lambda}{{[\partial (x_i - a_i)]}^\lambda}& = \frac{1}{\Gamma(-q)}\int_a^x  \, d\xi \frac{f(\xi)}{{(\xi-x)}^{q+1}} & \mathrm{Re}(q)<0 \\ & = \frac{1}{\Gamma(n-q)}\int_a^x d\xi \, \frac{f(\xi)}{{(\xi-x)}^{q-n+1}}& \mathrm{Re}(q) \geq 0 \end{align} $$

where n is an integer, q is a complex number, and


 * $$\Gamma(z) = \int_0^\infty dt \, e^{-t}t^{z-1}$$

is the Gamma function.

Fractional form spaces
This is found to generate new vector spaces of finite and infinite dimension; fractional form spaces.

The definitions of closed and exact forms can be extended to fractional form spaces with closure and integrability conditions.

Coordinate transformation and metric
When coordinates are needed, the coordinate transformation rules are different from those of the standard exterior calculus because of the properties of the fractional derivative.

Based on the coordinate transformation rules, the metric tensor for the fractional form spaces is given by;