Some enumerations of non-trivial composition of the differential operations and the directional derivative

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
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 ℝⁿ (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

1. 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.
2. Balasubramanian, N. V., Lynn, J. W. & Sen Gupta, D. P. (1970). Differential Forms on Electromagnetic Networks, Butterworth & Co. Publ. Ltd, London.
3. Belyaev, A., Khesin, B. & Tabachnikov, S. (2012) Discrete spherical means of directional derivatives and Veronese maps, J. Geom. Phys., 62, 124–136.
4. Bendito, E., Carmona, A., Encinas, A. M. & Gesto, J. M. (2008) The curl of a weighted network, Appl. Anal. Discrete Math., 2, 241–254.
5. Chang, F. C. (2005) Matrix formulation of vector operations, Appl. Math. Comput., 170(2), 1135–1165.
6. Chang, F. C. (2012) Vector Operations Transform into Matrix Operations, IEEE Antennas Propag. Mag., 54(6), 161–175.
7. Katz, V. J. (1985) Differential forms – Cartan to De Rham, Arch. Hist. Exact Sci., 33, 321–336.
8. Kotiuga, P. R. (1989) Helicity functionals and metric invariance in three dimensions, IEEE Trans. Magn., 25, 2813–2815.
9. 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.
10. Lewis, G. N. (1910) On four-dimensional vector analysis, and its application in electrical theory, Proc. Am. Acad. Arts Sci., 46, 165–181.
11. 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/)
12. Malešević, B. J. (1998) Some combinatorial aspects of differential operation composition on the space  ℝⁿ , Univ. Beograd, Publ. Elektrotehn. Fak. Ser. Mat., 9, 29–33. (http://pefmath2.etf.rs/)
13. 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/)
14. 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/)
15. Myers, J. British Mathematical Olympiad 2008 − 2009, Round 1, Problem 1– Generalisation, preprint, accessed 31. Dec. 2008. (http://www.srcf.ucam.org/~jsm28/publications/)
16. Perot, B. J. & Zusi, J. C. (2014) Differential forms for scientists and engineers, J. Comput. Phys., 257, 1373–1393.
17. Schreiber, M. (1977) A Differential Forms: A Heuristic Introduction, Universitext series, Springer.
18. Sloane, N. J. A. (2015) The On -Line Encyclopedia of Integer Sequences, publ. electr. at http://oeis.org.
19. Sommerfeld, A. (1910) Zur Relativitatstheorie I: Vierdimensionale Vektoralgebra, Ann. Phys., 32, 749–776.
20. Sommerfeld, A. (1910). Zur Relativitatstheorie II: Vierdimensionale Vektoranalysis, Ann. Phys., 33, 649–689.
21. 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.
22. Weintraub, S. H. (2014). Differential Forms: Theory and Practice, 2nd Edition, Academic Press (Elsevier).
23. Von Westenholz, C. (1978). Differential Forms in Mathematical Physics, North Holland.
24. 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.