Fractional differential forms.(English)Zbl 1011.58001

Summary: 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.

MSC:

 58A10 Differential forms in global analysis 26A33 Fractional derivatives and integrals
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References:

 [1] Kerner, Lett. Math. Phys. 36 pp 441– (1996) [2] Dubois-Violette, Contemp. Math. 219 pp 69– (1998) [3] Coquereaux, Lett. Math. Phys. 42 pp 241– (1997) [4] J. Madore,An Introduction to Noncomutative Differential Geometry and its Physical Applications(Cambridge University Press, Cambridge, 1995). · Zbl 0842.58002 [5] Harley Flanders,Differential Forms with Applications to the Physical Sciences(Dover, New York, 1989). · Zbl 0733.53002 [6] D. Lovelock and H. Rund,Tensors, Differential Forms, and Variational Principles(Dover, New York, 1989). · Zbl 0308.53008 [7] K. B. Oldham and J. Spanier,The Fractional Calculus(Academic, New York, 1974). · Zbl 0292.26011 [8] K. S. Miller and B. Ross,An Introduction to the Fractional Calculus and Fractional Differential Equations(Wiley, New York, 1993). · Zbl 0789.26002 [9] I. Podlubny,Fractional Differential Equations(Academic, New York, 1999). · Zbl 0924.34008
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