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.

Topics
Coordinate transformations
1.
R.
Kerner
, “
Z3-graded exterior differential calculus and gauge theories of higher order
,”
Lett. Math. Phys.
36
,
441
–
454
(
1996
).
Google Scholar
Crossref
Search ADS
 
2.
M.
Dubois-Violette
, “
Generalized homologies for dN=0 and graded q-differential algebras
,”
Contemp. Math.
219
,
69
–
79
(
1998
).
Google Scholar
Crossref
Search ADS
 
3.
R.
Coquereaux
, “
Differentials of higher order in noncommutative differential geometry
,”
Lett. Math. Phys.
42
,
241
–
259
(
1997
).
Google Scholar
Crossref
Search ADS
 
4.
J. Madore, An Introduction to Noncomutative Differential Geometry and its Physical Applications (Cambridge University Press, Cambridge, 1995).
5.
Harley Flanders, Differential Forms with Applications to the Physical Sciences (Dover, New York, 1989).
6.
D. Lovelock and H. Rund, Tensors, Differential Forms, and Variational Principles (Dover, New York, 1989).
7.
K. B. Oldham and J. Spanier, The Fractional Calculus (Academic, New York, 1974).
8.
K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations (Wiley, New York, 1993).
9.
I. Podlubny, Fractional Differential Equations (Academic, New York, 1999).
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© 2001 American Institute of Physics.
2001
American Institute of Physics
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