`Abstract and Applied AnalysisVolume 2011 (2011), Article ID 805948, 14 pageshttp://dx.doi.org/10.1155/2011/805948`
Research Article

## Forward Curvatures on Time Scales

Department of Mathematics, Faculty of Arts and Sciences, Usak University, 1 Eylul Campus, 64200, Usak, Turkey

Received 14 December 2010; Accepted 31 March 2011

Copyright © 2011 M. S. Seyyidoglu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We also introduce forward curvature of a curve and give some formulas to calculate forward curvature of a curve on time scales which may be an arbitrary closed subsets of the set of all real numbers. We also introduce the length of a curve parametrized by a time scale parameter in .

#### 1. Introduction

The study of dynamic equations on time scales is an area of mathematics that recently has received a lot of attention. The calculus on time scales has been introduced in order to unify the theories of continuous and discrete processes and in order to extend those theories to a more general class of so-called dynamic equations.

In recent years there have been a few research activities concerning the application of differential geometry on time scales. In [1] Guseinov and Ozyılmaz have defined the notions of forward tangent line, -regular curve, and natural -parametrization. Furthermore, in [2] Bohner and Guseinov, have introduced the concept of a curve parametrized by a time scale parameter and they have given integral formulas for computation of its length in plane. They have established a version of the classical Green formula suitable to time scales. In [3] Ozyılmaz has introduced the directional derivative according to the vector fields.

The general idea of this paper is to study forward curvature of curves where in the parametric equations the parameter varies in a time scale. We present the “differential” part of classical differential geometry on time scale calculus. The new results generalize the well known formulas stated in classical differential geometry. We illustrate our results by applying them to various kinds of time scales.

#### 2. Basic Definitions

A time scale is an arbitrary nonempty closed subset of the real numbers . The time scale is a complete metric space with the usual metric. We assume throughout that a time scale has the topology that it inherits from the real numbers with the standard topology.

For we define the forward jump operator by

while the backward jump operator is defined by

If , we say that is right-scattered, while, if , we say that is left-scattered. Also, if , then is called right-dense, and, if , then is called left-dense. The graininess function is defined by

We introduce the set which is derived from the time scale as follows. If has left-scattered maximum , then , otherwise . For with we define the interval in by

We will let denote if is left-scattered and if is left-dense.

Definition 2.1 (see [4]). Assume that is a function, and let . Then we define to be the number (provided it exists) with the property that, given any , there is a neighborhood of such that We call the delta (or Hilger) derivative of at . Moreover, we say that is delta (or Hilger) differentiable on provided exists for all .

Theorem 2.2 (see [4]). For and the following hold. (i)If is -differentiable at , then is continuous at . (ii)If is continuous at and is right-scattered, then is -differentiable at and (iii)If is right-dense, then is -differentiable at if and only if the limit     exists as a finite number. In this case is equal to this limit. (iv)If is -differentiable at , then

Theorem 2.3 (see [4]). If is -differentiable at , then (i) is -differentiable at and (ii)For any constant , is -differentiable at and (iii) is -differentiable at and (iv)If , then is -differentiable at and

Theorem 2.4 (chain rule [4]). Assume that is strictly increasing and is a time scale. Let . If and exists for , then where one denotes the derivative on by .

Theorem 2.5 (derivative of the inverse [4]). Assume that is strictly increasing and is a time scale. Then at points where is different from zero.

Definition 2.6 (see [4]). A function is called rd-continuous provided it is continuous at right-dense points in and its left-sided limits exist (finite) at left-dense points in .

Definition 2.7 (see [1]). A -regular curve (or an arc of a -regular curve) is defined as a mapping that is -differentiable on with rd-continuous -derivatives and

Definition 2.8 (see [1]). Let be a curve, a point on , and a line through , where Take on any point . Denote by the distance of the point from the point , and by the distance of from the line . If as , then we say that is the forward tangent line to the curve at the point .

Theorem 2.9 (see [1]). Every -regular curve has at any point , the forward tangent line that has the vector as its direction vector.

Definition 2.10 (see [1]). Let be a -regular curve in given by the equation We define the function by The variable can be used as a parameter for the curve . Such a parametrization of a curve we call natural -parametrization.

Theorem 2.11 (see [1]). In the case of natural -parametrization of the curve the forward tangent vector is a unit vector.

Definition 2.12 (see [2]). Let be a continuous curve with equation . A partition of is any finite ordered set Let us set The curve is rectifiable if The nonnegative number is called the length of the curve . If the supremum does not exist, the curve is said to be nonrectifiable.

Theorem 2.13 (see [2]). Let the functions and be continuous on and -differentiable on . If their -derivatives and are bounded and -integrable over , then the curve is rectifiable and its length can be evaluated by the formula

#### 3. Forward Curvatures on Time Scales

It is easy to see that the notion of rectifiable curve in Definition 2.12 for can be adapted to .

Definition 3.1. Let be a continuous curve in . Let be a partition of as in (2.19), and set where . We say that curve is rectifiable if   In Theorem 2.13, the length of the curve in plane is given. We introduce the length of a curve parametrized by a time scale parameter in in the following lemma.

Lemma 3.2. Let the functions , and be continuous on and -differentiable on . If their -derivatives are bounded and -integrable over [a, b), then the curve is rectifiable and its length can be evaluated by the formula

Proof. We show that the curve is rectifiable. Let an arbitrary partition of be of the form (2.19). Consider defined by (3.1). Applying to each of the functions , and the mean value theorem (see [5, Theorem 4.2]) on for , we get that there exist points and in such that From (3.4), (3.5), and (3.6) it follows that where By the assumption of the theorem, the derivatives ,   and are bounded on , so that there is a finite positive constant such that for all . Thus for all , and we have from (3.1) so that we get that the curve is rectifiable.Now we prove the formula (3.3). Consider the Riemann -sum of the -integrable function , corresponding to the partition of and the choice of intermediate points defined in (3.4). For every , there exists (see [6, Lemma 5.7]) at least one partition of of the form (2.19) such that for each either or and . Let us denote by the set of all such partitions of . For an arbitrary , there exists such that where From (3.4), (3.5), and (3.6) we get and, consequently, where in which and are the supremum and infimum of on and , and , are corresponding numbers for and , respectively. Using the inequality for , we obtain Therefore, where and denote the upper and lower Darboux -sums, respectively. Since the functions and are -integrable over , for arbitrary , there exists such that for all (see [6, Theorem 5.9]) and the Riemann definition of -integrability therein), where is defined by (3.14). Therefore, we get and so the validity of (3.13) is proved. On the other hand, among the partitions for which (3.13) is satisfied, there is a partition such that Indeed, there is a partition of such that Next, we refine the partition adding to it new partition points so that we get a partition that belongs to . Then by , (3.24) yields so that (3.23) is shown. By (3.13) and (3.23), we get Since is arbitrary, we have , and the proof is complete.

Definition 3.3. The curve is given in the parametric form . Let , for and for . We denote the angle between the forward tangent lines drawn to the curve at and by and the arc length of the segment of the curve by . The forward curvature of at is defined as

Theorem 3.4. Let be a natural -parametrized -regular curve, there exists for . If is a right-dense point, then If is a right-scattered point, then

Proof. Let , for a fix point and for . Let the unit vectors on the forward tangent lines to the curve at points and be and , respectively, and assume that is the angle between them. We denote . Then we have Assume that is a right-dense point. Then and as . Since we obtain By using (3.31) we can write Thus, we find as .
Assume that is a right-scattered point. Then and as . If the equality (3.31) is divided by , taking the limit as , we have (3.16) On the other hand, we find It follows from (3.16) and (3.37) that This completes the proof.

Theorem 3.5. Let be a curve with arbitrary parameters that is second -differentiable on and for all . Moreover assume that the function defined by is strictly increasing. The forward curvature of at the right-dense point is

Proof. The function defined by is continuous and strictly increasing. Therefore will be a time scale (see [1, pages 560-561]). The forward jump operator and the derivative operator on this time scale, will be denoted by and respectively. Since the curve is a natural -parametrized, to find the curvature to the at the point , it is sufficient to find that the curvature to the at the point .
By using (3.40), one can easily find that As is a right-dense point and the function is strictly increasing, we have Substituting (3.42) into (3.41), we obtain The proof is complete.

Remark 3.6. It is easy to see that, for the case , the results (3.28) and (3.39) generalize the following formulas stated in classical differential geometry:

Theorem 3.7. Let be a curve with arbitrary parameters that is second differentiable on and has continuous derivative on and for all . Moreover assume that the function defined by is strictly increasing. The forward curvature of at the right-scattered point is where

Proof. Let be a right-scattered point. In this case we have By using (3.41) and (3.48), we find Substituting (3.49) into (3.29), we obtain

Example 3.8. Let and , .The curve satisfies the conditions of Theorem 3.4 for the case . In this case we have , and Since every point of is right-scattered point, the curvature of at any point is This value is the angle between the line through , and the line through , .

Example 3.9. Assume that , is a non--natural parametrized curve. The only right-dense point of the time scale is , and the other points of the time scale are right-scattered. The forward jump operator and the graininess function are Furthermore we have and, from Theorem 3.5, the forward curvature of the curve at the point is For the right-scattered point , by using Theorem 3.7, we find This value is the ratio of the angle between the line through the points , and the line through the points , and the distance between the points and .

#### Acknowledgment

The authors wish to express their sincere thanks to the referee for his encouraging attitude and valuable suggestions in the review process of the work.

#### References

1. G. Sh. Guseinov and E. Ozyılmaz, “Tangent lines of generalized regular curves parametrized by time scales,” Turkish Journal of Mathematics, vol. 25, no. 4, pp. 553–562, 2001.
2. M. Bohner and G. Sh. Guseinov, “Line integrals and Green's formula on time scales,” Journal of Mathematical Analysis and Applications, vol. 326, no. 2, pp. 1124–1141, 2007.
3. E. Ozyılmaz, “Directional derivative of vector field and regular curves on time scales,” Applied Mathematics and Mechanics (English Edition), vol. 27, no. 10, pp. 1349–1360, 2006.
4. M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An introduction with Application, Birkhauser, Boston, Mass, USA, 2001.
5. M. Bohner and G. Sh. Guseinov, “Partial differentiation on time scales,” Dynamic Systems and Applications, vol. 13, no. 3–4, pp. 351–379, 2004.
6. M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, Mass, USA, 2003.