Applications of fractional exterior differential in three-dimensional space. (English) Zbl 1039.58003
Summary: A brief survey of fractional calculus and fractional differential forms is firstly given. The fractional exterior transition to curvilinear coordinates at the origin is discussed and the two coordinate transformations for the fractional differentials for three-dimensional Cartesian coordinates to spherical and cylindrical coordinates are obtained, respectively. In particular, for \(v=m=1\), the usual exterior transformations, between the spherical coordinate and Cartesian coordinate, as well as the cylindrical coordinate and Cartesian coordinate, are found respectively, from the fractional exterior transformation.
MSC:
| 58A10 | Differential forms in global analysis |
Keywords:
fractional differential form; Cartesian coordinate; spherical coordinate; cylindrical coordinateReferences:
| [1] | Dold A, Eckmann B.Fractional Calculus and Its Aapplications[M]. Berlin: Springer-Verlag, 1975. |
| [2] | Kerner R. Z3-graded exterior differential calculus and gauge theories of higher order[J].Lett Math Phys, 1996,36(5):441–454. · Zbl 0852.58002 · doi:10.1007/BF00714408 |
| [3] | Dubois-Violette M. Generalized homologies ford N =0 and graded q-differential algebras [J].Contemp Math, 1998,219(1):69–79. · Zbl 1074.18510 |
| [4] | Coquereaux R. Differentials of higher order in noncommutative differential geometry[J].Lett Math Phys, 1997,42(3):241–259. · Zbl 0886.58005 · doi:10.1023/A:1007438020850 |
| [5] | Madore J.An Introduction to Noncommutative Differential Geometry and Its Applications[M]. Cambridge: Cambridge University Press, 1995. · Zbl 0842.58002 |
| [6] | Cottrill-Shepherd K, Naber M. Fractional differential forms[J].J Math Phys, 2001,42(5): 2203–2212. · Zbl 1011.58001 · doi:10.1063/1.1364688 |
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