The matrix representation of the delta shape operator on time scales

Abstract

In this work, we consider the delta shape operator of a surface parameterized by the product of two arbitrary time scales. In particular, we present a matrix representation of the delta shape operators with respect to partial delta derivatives.

Darboux curves on product time scales

“…If $$$ \phi $$$ is $$$ \Delta $$$‐ regular, that is, $$$ \frac{\partial \phi }{\Delta_1t}\times \frac{\partial \phi }{\Delta_1t}\ne 0 $$$ everywhere in $$$ {\mathbb{R}}^3 $$$, the normal vector field of $$$ \phi $$$ is defined by $$$$ \mathbf{U}=\frac{\alpha^{\Delta_1}\left({t}_1\right)\times \beta \left({t}_1\right)+{t}_2{\beta}^{\Delta_1}\left({t}_1\right)\times \beta \left({t}_1\right)}{\left\Vert {\alpha}^{\Delta_1}\left({t}_1\right)\times \beta \left({t}_1\right)+{t}_2{\beta}^{\Delta_1}\left({t}_1\right)\times \beta \left({t}_1\right)\right\Vert }. $$$$ (see literature 4,11–13,21 ). Here, we give some properties of Darboux curve with arc length and ruled surfaces on …”

“…Then, he extended well-known notions of discrete pseudospherical surfaces and smooth pseudospherical surfaces on more exotic domains. In 2010, Bohner and Guseinov 6 studied surfaces parametrized by time scale parameters and constructed an integral formula to compute area of surfaces on T. In fact, the main purpose of these studies is to unify the difference and differential geometries and to formulate the integrable geometry on T. The number of studies about the applications of differential geometry on T is gradually increasing (see literature [7][8][9][10][11][12][13][14] ).…”

“…In fact, the main purpose of these studies is to unify the difference and differential geometries and to formulate the integrable geometry on . The number of studies about the applications of differential geometry on is gradually increasing (see literature 7–14 ).…”

A certain class of surfaces on product time scales with interpretations from economics

“…Then, Samanci and Caliskan [35] considered the level curves and surfaces on time scales. Furthermore, Samanci [34] defined matrix presentation of delta shape operator on time scales.…”

“…Then, Samanci and Caliskan [35] studied level curves and surfaces by considering delta gradient functions on time scales in 2015. In 2016, Samanci [34] considered the delta shape operator of a surface parameterized by the product of two arbitrary time scales. All of these studies are related to generalizing some fundamental definitions and theorems in differential geometry by using time scale theory.…”