  Citation: Karim, A.O.; Shehata, E.M.; Cardoso, J.L. The Directional Derivative in General Quantum Calculus. Symmetry 2022, 14, 1766. https://doi.org/10.3390/ sym14091766 Academic Editor: Alexander Zaslavski, Calogero Vetro Received: 20 July 2022 Accepted: 21 August 2022 Published: 24 August 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). symmetry S S Article The Directional Derivative in General Quantum Calculus Avin O. Karim 1 , Enas M. Shehata 2, * and José Luis Cardoso 3 1 Department of Mathematical sciences, College of Basic Education, University of Sulaimani, Sulaimani P.O. Box 46, Kurdistan, Iraq 2 Department of Mathematics and Computer Science, Faculty of Science, Menoufia University, Shibin El-Kom 32511, Egypt 3 Escola de Ciências e Tecnologia, University of Trás-os-Montes e Alto Douro, 5000-801 Vila Real, Portugal * Correspondence: enas.soliman@science.menofia.edu.eg Abstract: In this paper, we define the β-partial derivative as well as the β-directional derivative of a multi-variable function based on the β-difference operator, D β , which is defined by D β f (t)= ( f ( β(t)) − f (t) ) / ( β(t) − t ) , where β is a strictly increasing continuous function. Some properties are proved. Furthermore, the β-gradient vector and the β-gradient directional derivative of a multi- variable function are introduced. Finally, we deduce the Hahn-partial and the Hahn-directional derivatives associated with the Hahn difference operator. Keywords: general quantum operator; β-partial derivative; β-directional derivative; β-gradient vector; Hahn-partial derivative; Hahn-directional derivative AMS (MOS) Subject Classification: 39A10; 39A13; 39A70; 47B39 1. Introduction and Preliminaries Calculus without limits is known as quantum calculus. Roughly speaking, quantum calculus is an approach to deal with non-differentiable functions. Similar to fractional calculus, it is related to the classical derivative operator. Although there is a common bond, they are very different branches of mathematics. The fractional branch consider functions that may have no derivative of first order but have fractional derivatives of orders less than one [1,2]; however, the quantum branch is particularly useful for modeling physical systems (see, for instance, [3–6]). Quantum calculus began with FH Jackson in the early twentieth century, but earlier, this type of calculus had been worked out by Euler and Jacobi. For the reader, we recom- mend the book by Kac and Cheung [7] for many fundamental features of quantum calculus. Quantum difference operators and their calculi have attracted many researchers in recent years due to their applications in different branches of mathematics as well as physics, such as orthogonal polynomials, basic hypergeometric series, variational calculus, combi- natorics, model physical and economical systems, quantum mechanics, and black holes (see, for example, [8–15]). As it usually happens in mathematics, the corresponding inverse operators (integral operators) are important directly related topics. There are plenty of quantum operators and associated calculi, such as forward, Jackson, Hahn, power quantum, and p, q-calculi (see [7,16,17]). Moreover, a symmetric quantum calculus was introduced in [18,19], based on the symmetric quantum difference operator, which is defined by D α,γ f (t)= f (t + α) − f (t − γ) α + γ , α, γ ∈ [0, ∞). (1) Note that in the case of γ = 0 in (1), we obtain the forward difference operator. In 2015, Hamza et al. [20] defined the general quantum difference operator denoted by D β as Symmetry 2022, 14, 1766. https://doi.org/10.3390/sym14091766 https://www.mdpi.com/journal/symmetry