Ivana Jovović and Branko Malešević

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 23, 2017, Number 1, Pages 28—38

**Download full paper: PDF, 179 Kb**

## Details

### Authors and affiliations

Ivana Jovović

* University of Belgrade, Faculty of Electrical Engineering,
Bulevar Kralja Aleksandra 73, 11000 Belgrade, Serbia
*

Branko Malešević

* University of Belgrade, Faculty of Electrical Engineering,
Bulevar Kralja Aleksandra 73, 11000 Belgrade, Serbia
*

### Abstract

This paper deals with some enumerations of the higher order non-trivial compositions of the differential operations and the directional derivative in the space ℝ* ^{n}* (

*n*≥ 3). One new enumeration of the higher order non-trivial compositions is obtained.

### Keywords

- Div
- Grad
- Curl
- Directional derivative
- Differential forms
- Fibonacci numbers
- Compositions of the differential operations

### AMS Classification

- 58A10
- 47B33
- 05C30

### References

- Arnold, D. N., Falk, R. S. & Winther, R. (2010) Finite element exterior calculus: from Hodge theory to numerical stability, Bull. Amer. Math. Soc., 47, 281–354.
- Balasubramanian, N. V., Lynn, J. W. & Sen Gupta, D. P. (1970). Differential Forms on Electromagnetic Networks, Butterworth & Co. Publ. Ltd, London.
- Belyaev, A., Khesin, B. & Tabachnikov, S. (2012) Discrete spherical means of directional derivatives and Veronese maps, J. Geom. Phys., 62, 124–136.
- Bendito, E., Carmona, A., Encinas, A. M. & Gesto, J. M. (2008) The curl of a weighted network, Appl. Anal. Discrete Math., 2, 241–254.
- Chang, F. C. (2005) Matrix formulation of vector operations, Appl. Math. Comput., 170(2), 1135–1165.
- Chang, F. C. (2012) Vector Operations Transform into Matrix Operations, IEEE Antennas Propag. Mag., 54(6), 161–175.
- Katz, V. J. (1985) Differential forms – Cartan to De Rham, Arch. Hist. Exact Sci., 33, 321–336.
- Kotiuga, P. R. (1989) Helicity functionals and metric invariance in three dimensions, IEEE Trans. Magn., 25, 2813–2815.
- Kraft, C. (1911) Eine Identitat in der Vierdimensionalen Vektoranalysis und deren Anwendung in der Elektrodynamik, Bulletin international de l’Academie des sciences de Cracovie – Serie A, 537–541.
- Lewis, G. N. (1910) On four-dimensional vector analysis, and its application in electrical theory, Proc. Am. Acad. Arts Sci., 46, 165–181.
- Malešević, B. J. (1996) A note on higher-order differential operations, Univ. Beograd, Publ. Elektrotehn. Fak. Ser. Mat., 7, 105–109. (http://pefmath2.etf.rs/)
- Malešević, B. J. (1998) Some combinatorial aspects of differential operation composition on the space ℝ
, Univ. Beograd, Publ. Elektrotehn. Fak. Ser. Mat., 9, 29–33. (http://pefmath2.etf.rs/)^{n} - Malešević, B. J. (2006) Some combinatorial aspects of the composition of a set of function, Novi Sad J. Math., 36, 3–9. (http://www.dmi.uns.ac.rs/NSJOM/)
- Malešević, B. J. & Jovović, I. V. (2007) The compositions of differential operations and Gateaux directional derivative, J. Integer Seq., 10, 1–11. (http://www.cs.uwaterloo.ca/journals/JIS/)
- Myers, J. British Mathematical Olympiad 2008 − 2009, Round 1, Problem 1– Generalisation, preprint, accessed 31. Dec. 2008. (http://www.srcf.ucam.org/~jsm28/publications/)
- Perot, B. J. & Zusi, J. C. (2014) Differential forms for scientists and engineers, J. Comput. Phys., 257, 1373–1393.
- Schreiber, M. (1977) A Differential Forms: A Heuristic Introduction, Universitext series, Springer.
- Sloane, N. J. A. (2015) The On -Line Encyclopedia of Integer Sequences, publ. electr. at http://oeis.org.
- Sommerfeld, A. (1910) Zur Relativitatstheorie I: Vierdimensionale Vektoralgebra, Ann. Phys., 32, 749–776.
- Sommerfeld, A. (1910). Zur Relativitatstheorie II: Vierdimensionale Vektoranalysis, Ann. Phys., 33, 649–689.
- Walter, S. A., Breaking in the 4-vectors: the four-dimensional movement in gravitation, 1905–1910, in The Genesis of General Relativity, Vol. 3: Theories of gravitation in the twilight of classical physics, Part I, Jurgen Renn (ed.), Springer, 2007, pp. 193–252.
- Weintraub, S. H. (2014). Differential Forms: Theory and Practice, 2nd Edition, Academic Press (Elsevier).
- Von Westenholz, C. (1978). Differential Forms in Mathematical Physics, North Holland.
- Wilson, E. B. & Lewis, G. N. (1912). The space – time manifold of relativity: the non-Euclidean geometry of mechanics and electromagnetics, Proc. Am. Acad. Arts Sci., 48, 389–507.

## Related papers

## Cite this paper

Jovović, I. & Malešević, B. (2017). Some enumerations of non-trivial composition of the differential operations and the directional derivative. Notes on Number Theory and Discrete Mathematics, 23(1), 28-38.