tailieunhanh - Tangent lines of generalized regular curves parametrized by time scales

In this paper a generalization of the notion of regular curve is introduced. For such curves the concept of tangent line is investigated. In order to unify continuous and discrete analysis, by Aulbach and Hilger was introduced the concept of time scale (or measure chain) and the theory of calculus on time scales. | Turk J Math 25 (2001) , 553 – 562. ¨ ITAK ˙ c TUB Tangent Lines of Generalized Regular Curves Parametrized by Time Scales ¨ Gusein Sh. Guseinov and Emin Ozyılmaz Abstract In this paper a generalization of the notion of regular curve is introduced. For such curves the concept of tangent line is investigated. Key Words: Time scale, delta derivative, generalized regular curve,tangent line. 1. Introduction In order to unify continuous and discrete analysis, by Aulbach and Hilger [5,10] was introduced the concept of time scale (or measure chain) and the theory of calculus on time scales. This theory has recently received a lot of attention and has proved to be useful in the mathematical modeling of several important dynamic processes. As a result the theory of dynamic systems on time scales is developed [7,12]. The general idea in this paper is to study curves where in the parametric equations the parameter varies in a so –called time scale, which may be an arbitrary closed subset of the set of all real numbers. So our intention is to use as the “differential” part of classical Differential Geometry the time scales calculus. AMS Subject Classfication: 53A04 , 39A10 553 ¨ GUSEINOV, OZYILMAZ 2. Preliminaries from the Time Scales Calculus For an introduction to the theory of calculus on time scales we refer to the original works by Aulbach and Hilger [5,10] and to the recently appeared works [1-4,6,7,9,11,12]. To meet the requirements in the next sections here we introduce the basic notions and notations connected to time scales analysis. A time scale ( or measure chain ) T is an arbitrary nonempty closed subset of the real numbers R. The time scale T is a complete metric space with the metric d (t1 , t2 ) = |t1 − t2 |. For t ∈ T we define the forward jump operator σ : T−→T by σ (t) = inf {s ∈ T : s > t} while the backward jump operator ρ : T−→T is defined by ρ (t) = sup {s ∈ T : s t , we say that t is right-scattered, while if ρ (t) 0 there is a .