Forward Curvatures on Time Scales

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 ℝ³.

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, Δ-regular curve, and natural Δ-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 𝕋 is an arbitrary nonempty closed subset of the real numbers ℝ. The time scale 𝕋 is a complete metric space with the usual metric. We assume throughout that a time scale 𝕋 has the topology that it inherits from the real numbers with the standard topology.

For 𝑡∈𝕋 we define the forward jump operator 𝜎∶𝕋→𝕋 by 𝜎(𝑡)∶=inf{𝑠∈𝕋∶𝑠>𝑡},(2.1)

while the backward jump operator 𝜌∶𝕋→𝕋 is defined by𝜌(𝑡)∶=sup{𝑠∈𝕋∶𝑠<𝑡}.(2.2)

If 𝜎(𝑡)>𝑡, we say that 𝑡 is right-scattered, while, if 𝜌(𝑡)<𝑡, we say that 𝑡 is left-scattered. Also, if 𝜎(𝑡)=𝑡, then 𝑡 is called right-dense, and, if 𝜌(𝑡)=𝑡, then 𝑡 is called left-dense. The graininess function 𝜇∶𝕋→[0,∞) is defined by 𝜇(𝑡)∶=𝜎(𝑡)−𝑡.(2.3)

We introduce the set 𝕋^𝜅 which is derived from the time scale 𝕋 as follows. If 𝕋 has left-scattered maximum 𝑚, then 𝕋^𝜅=𝕋−{𝑚}, otherwise 𝕋^𝜅=𝕋. For 𝑎,𝑏∈𝕋 with 𝑎≤𝑏 we define the interval [𝑎,𝑏] in 𝕋 by [𝑎,𝑏]={𝑡∈𝕋∶𝑎≤𝑡≤𝑏}.(2.4)

We will let [𝑎,𝑏]^𝜅 denote [𝑎,𝜌(𝑏)] if 𝑏 is left-scattered and [𝑎,𝑏] if 𝑏 is left-dense.

Definition 2.1 (see [4]). Assume that 𝑓∶𝕋→ℝ is a function, and let 𝑡∈𝕋^𝜅. Then we define 𝑓^Δ(𝑡) to be the number (provided it exists) with the property that, given any 𝜀>0, there is a neighborhood 𝑈 of 𝑡 such that ||𝑓(𝜎(𝑡))−𝑓(𝑠)−𝑓^Δ(𝑡)[𝜎(𝑡)−𝑠]||≤𝜀||𝜎(𝑡)−𝑠||,∀𝑠∈𝑈.(2.5) We call 𝑓^Δ(𝑡) the delta (or Hilger) derivative of 𝑓 at 𝑡. Moreover, we say that 𝑓 is delta (or Hilger) differentiable on 𝕋^𝜅 provided 𝑓^Δ(𝑡) exists for all 𝑡∈𝕋^𝜅.

Theorem 2.2 (see [4]). For 𝑓∶ 𝕋→ℝ and 𝑡∈𝕋^𝜅 the following hold. (i)If 𝑓 is Δ-differentiable at 𝑡, then 𝑓 is continuous at 𝑡. (ii)If 𝑓 is continuous at 𝑡 and 𝑡 is right-scattered, then 𝑓 is Δ-differentiable at 𝑡 and 𝑓^Δ(𝑡)=𝑓(𝜎(𝑡))−𝑓(𝑡)𝜎(𝑡)−𝑡(2.6)(iii)If 𝑡 is right-dense, then 𝑓 is Δ-differentiable at 𝑡 if and only if the limit lim_𝑠→𝑡𝑓(𝑡)−𝑓(𝑠)𝑡−𝑠(2.7)     exists as a finite number. In this case 𝑓^Δ(𝑡) is equal to this limit. (iv)If 𝑓 is Δ-differentiable at 𝑡, then 𝑓(𝜎(𝑡))=𝑓(𝑡)+𝜇(𝑡)𝑓^Δ(𝑡).(2.8)

Theorem 2.3 (see [4]). If 𝑓,𝑔 is Δ-differentiable at 𝑡∈𝕋^𝜅, then (i)𝑓+𝑔 is Δ-differentiable at 𝑡 and (𝑓+𝑔)^Δ(𝑡)=𝑓^Δ(𝑡)+𝑔^Δ(𝑡).(2.9)(ii)For any constant 𝑐, 𝑐𝑓 is Δ-differentiable at 𝑡 and (𝑐𝑓)^Δ(𝑡)=𝑐𝑓^Δ(𝑡).(2.10)(iii)𝑓𝑔 is Δ-differentiable at 𝑡 and (𝑓𝑔)^Δ(𝑡)=𝑓^Δ(𝑡)𝑔(𝑡)+𝑓(𝜎(𝑡))𝑔^Δ(𝑡)=𝑓(𝑡)𝑔^Δ(𝑡)+𝑓^Δ(𝑡)𝑔(𝜎(𝑡)).(2.11)(iv)If 𝑔(𝑡)𝑔(𝜎(𝑡))≠0, then 𝑓/𝑔 is Δ-differentiable at 𝑡 and 𝑓𝑔^Δ(𝑡)=𝑓^Δ(𝑡)𝑔(𝑡)−𝑔^Δ(𝑡)𝑓(𝑡)𝑔(𝑡)𝑔(𝜎(𝑡)).(2.12)

Theorem 2.4 (chain rule [4]). Assume that 𝜈∶𝕋→ℝ is strictly increasing and 𝕋∶=𝜈(𝕋) is a time scale. Let 𝑤∶𝕋→ℝ. If 𝜈^Δ(𝑡) and 𝑤Δ(𝜈(𝑡)) exists for 𝑡∈𝕋^𝜅, then (𝑤∘𝜈)^Δ=𝑤Δ∘𝜈𝜈^Δ,(2.13) where one denotes the derivative on 𝕋 by Δ.

Theorem 2.5 (derivative of the inverse [4]). Assume that 𝜈∶ 𝕋→ℝ is strictly increasing and 𝕋∶=𝜈(𝕋) is a time scale. Then 1𝜈^Δ=𝜈⁻¹Δ∘𝜈(2.14) at points where 𝜈^Δ is different from zero.

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

Definition 2.7 (see [1]). A Δ-regular curve (or an arc of a Δ-regular curve) 𝛾 is defined as a mapping 𝛾∶[𝑎,𝑏]⊂𝕋→ℝ³ that is Δ-differentiable on [𝑎,𝑏]^𝜅 with rd-continuous Δ-derivatives and ‖‖𝛾^Δ(𝑡)‖‖≠0,for𝑡∈[𝑎,𝑏]^𝜅.(2.15)

Definition 2.8 (see [1]). Let 𝛾∶[𝑎,𝑏]⊂𝕋→ℝ³ be a curve, 𝑄₀ a point on 𝛾, and 𝐿 a line through 𝑄^𝜎0, where 𝑄₀=𝛾₁𝑡₀,𝛾₂𝑡₀,𝛾₃𝑡₀,𝑄^𝜎0=𝛾₁𝜎𝑡₀,𝛾₂𝜎𝑡₀,𝛾₃𝜎𝑡₀,𝑡₀∈[𝑎,𝑏]^𝜅.(2.16) Take on 𝛾 any point 𝑄. Denote by 𝑑 the distance of the point 𝑄 from the point 𝑄^𝜎0, and by 𝛿 the distance of 𝑄 from the line 𝐿. If 𝛿/𝑑→0 as 𝑄→𝑄₀,𝑄≠𝑄^𝜎0, then we say that 𝐿 is the forward tangent line to the curve 𝛾 at the point 𝑄₀.

Theorem 2.9 (see [1]). Every Δ-regular curve 𝛾∶[𝑎,𝑏]⊂𝕋→ℝ³ has at any point 𝑄₀=(𝛾₁(𝑡₀),𝛾₂(𝑡₀),𝛾₃(𝑡₀)),𝑡₀∈[𝑎,𝑏]^𝜅, the forward tangent line that has the vector 𝛾^Δ(𝑡₀) as its direction vector.

Definition 2.10 (see [1]). Let 𝛾 be a Δ-regular curve in ℝ³ given by the equation 𝛾=𝛾(𝑡),𝑡∈[𝑎,𝑏]⊂𝕋.(2.17) We define the function 𝑝 by 𝑝(𝑡)=^𝑡𝑎‖‖𝛾^Δ(𝑠)‖‖Δ𝑠,𝑡∈[𝑎,𝑏].(2.18) The variable 𝑝 can be used as a parameter for the curve 𝛾. Such a parametrization of a curve we call natural Δ-parametrization.

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

Definition 2.12 (see [2]). Let 𝛾 be a continuous curve with equation 𝛾(𝑡)=(𝛾₁(𝑡),𝛾₂(𝑡)). A partition of [𝑎,𝑏] is any finite ordered set 𝑃=𝑡₀,𝑡₁,…,𝑡_𝑛⊂[𝑎,𝑏],where𝑎=𝑡₀<𝑡₁<⋯<𝑡_𝑛=𝑏.(2.19) Let us set 𝑙(𝛾,𝑃)=^𝑛_𝑖=1𝛾₁𝑡_𝑖−𝛾₁𝑡_𝑖−1²+𝛾₂𝑡_𝑖−𝛾₂𝑡_𝑖−1².(2.20) The curve 𝛾 is rectifiable if sup_𝑃{𝑙(𝛾,𝑃)∶𝑃isapartitionof[𝑎,𝑏]}=∶𝑙(𝛾)<∞.(2.21) The nonnegative number 𝑙(𝛾) is called the length of the curve 𝛾. If the supremum does not exist, the curve is said to be nonrectifiable.

Theorem 2.13 (see [2]). Let the functions 𝛾₁ and 𝛾₂ be continuous on [𝑎,𝑏] and Δ-differentiable on [𝑎,𝑏). If their Δ-derivatives 𝛾^Δ1 and 𝛾^Δ2 are bounded and Δ-integrable over [𝑎,𝑏), then the curve 𝛾(𝑡)=(𝛾₁(𝑡),𝛾₂(𝑡)) is rectifiable and its length 𝑙(𝛾) can be evaluated by the formula 𝑙(𝛾)=^𝑏𝑎𝛾^Δ1(𝑡)²+𝛾^Δ2(𝑡)²Δ𝑡.(2.22)

3. Forward Curvatures on Time Scales

It is easy to see that the notion of rectifiable curve in Definition 2.12 for ℝ² can be adapted to ℝ³.

Definition 3.1. Let 𝛾 be a continuous curve in ℝ³. Let 𝑃 be a partition of [𝑎,𝑏] as in (2.19), and set 𝑙(𝛾,𝑃)=^𝑛_𝑖=1⎷₃_𝑗=1𝛾_𝑗𝑡_𝑖−𝛾_𝑗𝑡_𝑖−1²,(3.1) where 𝛾(𝑡)=(𝛾₁(𝑡),𝛾₂(𝑡),𝛾₃(𝑡)). We say that curve 𝛾 is rectifiable if sup_𝑃{𝑙(𝛾,𝑃)∶𝑃isapartitionof[𝑎,𝑏]}=∶𝑙(𝛾)<∞.(3.2)   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 ℝ³ in the following lemma.

Lemma 3.2. Let the functions 𝛾₁,𝛾₂, and 𝛾₃ be continuous on [𝑎,𝑏] and Δ-differentiable on [𝑎,𝑏). If their Δ-derivatives 𝛾^Δ1,𝛾^Δ2,and𝛾^Δ3 are bounded and Δ-integrable over [a, b), then the curve 𝛾(𝑡)=(𝛾₁(𝑡),𝛾₂(𝑡),𝛾₃(𝑡)) is rectifiable and its length 𝑙(𝛾) can be evaluated by the formula 𝑙(𝛾)=^𝑏𝑎𝛾^Δ1(𝑡)²+𝛾^Δ2(𝑡)²+𝛾^Δ3(𝑡)²Δ𝑡.(3.3)

Proof. We show that the curve 𝛾 is rectifiable. Let an arbitrary partition of [𝑎,𝑏] be of the form (2.19). Consider 𝑙(𝛾,𝑃) defined by (3.1). Applying to each of the functions 𝛾₁,𝛾₂, and 𝛾₃ the mean value theorem (see [5, Theorem 4.2]) on [𝑡_𝑖−1,𝑡_𝑖] for 𝑖∈{1,2,…,𝑛}, we get that there exist points 𝜉_𝑖,𝜉^′_𝑖,𝜂_𝑖,𝜂^′_𝑖 and 𝜁_𝑖,𝜁^′_𝑖 in [𝑡_𝑖−1,𝑡_𝑖) such that 𝛾𝛾^Δ1𝜉_𝑖𝑡_𝑖−𝑡_𝑖−1≤𝛾₁𝑡_𝑖−𝛾₁𝑡_𝑖−1≤𝛾^Δ1𝜉^′𝑖𝑡^𝑖−𝑡_𝑖−1,(3.4)_Δ2𝛾𝜂_𝑖𝑡_𝑖−𝑡_𝑖−1≤𝛾₂𝑡_𝑖−𝛾₂𝑡_𝑖−1≤𝛾^Δ2𝜂^′𝑖𝑡^𝑖−𝑡_𝑖−1,(3.5)_Δ3𝜁_𝑖𝑡_𝑖−𝑡_𝑖−1≤𝛾₃𝑡_𝑖−𝛾₃𝑡_𝑖−1≤𝛾^Δ3𝜁^′𝑖𝑡^𝑖−𝑡_𝑖−1.(3.6) From (3.4), (3.5), and (3.6) it follows that ||||𝛾𝛾₁₂𝑡_𝑖−𝛾₁𝑡_𝑖−1||≤𝐴_𝑖𝑡_𝑖−𝑡_𝑖−1,||𝛾₃𝑡_𝑖−𝛾₂𝑡_𝑖−1||≤𝐵_𝑖𝑡_𝑖−𝑡_𝑖−1,𝑡_𝑖−𝛾₃𝑡_𝑖−1||≤𝐶_𝑖𝑡_𝑖−𝑡_𝑖−1,(3.7)where 𝐴𝐵^𝑖_𝑖=max||𝛾^Δ1𝜉_𝑖||,|||𝛾^Δ1𝜉^′𝑖|||,𝐶_𝑖=max||𝛾^Δ2𝜂_𝑖||,|||𝛾^Δ2𝜂^′𝑖|||,=max||𝛾^Δ3𝜁_𝑖||,|||𝛾^Δ3𝜁^′𝑖|||.(3.8)By the assumption of the theorem, the derivatives 𝛾^Δ1,𝛾^Δ2,   and 𝛾^Δ3 are bounded on [𝑎,𝑏), so that there is a finite positive constant 𝑀 such that ||𝛾^Δ𝑗(𝑡)||≤𝑀,𝑗=1,2,3(3.9)for all 𝑡∈ [𝑎,𝑏). Thus ||𝛾_𝑗𝑡_𝑖−𝛾_𝑗𝑡_𝑖−1||≤𝑀𝑡_𝑖−𝑡_𝑖−1,𝑗=1,2,3(3.10)for all 𝑖∈{1,2,…,𝑛}, and we have from (3.1) 𝑙(𝛾,𝑃)≤𝑀√3^𝑛_𝑖=1𝑡_𝑖−𝑡_𝑖−1=𝑀√3(𝑏−𝑎),(3.11)so that we get that the curve 𝛾 is rectifiable.Now we prove the formula (3.3). Consider the Riemann Δ-sum 𝑆=^𝑛_𝑖=1𝛾^Δ1𝜉_𝑖²+𝛾^Δ2𝜉_𝑖²+𝛾^Δ3𝜉_𝑖²𝑡_𝑖−𝑡_𝑖−1(3.12)of the Δ-integrable function [𝛾^Δ1(𝑡)]²+[𝛾^Δ2(𝑡)]²+[𝛾^Δ3(𝑡)]², corresponding to the partition 𝑃 of [𝑎,𝑏] and the choice of intermediate points 𝜉_𝑖 defined in (3.4). For every 𝛿>0, there exists (see [6, Lemma 5.7]) at least one partition 𝑃 of [𝑎,𝑏] of the form (2.19) such that for each 𝑖∈{1,2,…,𝑛} either 𝑡_𝑖−𝑡_𝑖−1≤𝛿 or 𝑡_𝑖−𝑡_𝑖−1>𝛿 and 𝜎(𝑡_𝑖−1)=𝑡_i. Let us denote by 𝒫_𝛿([𝑎,𝑏]) the set of all such partitions 𝑃 of [𝑎,𝑏]. For an arbitrary 𝜀>0, there exists 𝛿>0 such that ||𝑙(𝛾,𝑃)−𝐼||<𝜀2∀𝑃∈𝒫^𝛿([𝑎,𝑏]),(3.13) where 𝐼=^𝑎𝑏𝛾^Δ1(𝑡)²+𝛾^Δ2(𝑡)²+𝛾^Δ3(𝑡)²Δ𝑡.(3.14) From (3.4), (3.5), and (3.6) we get 00≤≤𝛾𝛾₁₂𝑡_𝑖−𝛾₁𝑡_𝑖−1−𝛾^Δ1𝜉_𝑖𝑡_𝑖−𝑡_𝑖−1≤𝛾^Δ1𝜉^′𝑖−𝛾^Δ1𝜉_𝑖𝑡_𝑖−𝑡_𝑖−1,0≤𝛾₃𝜂_𝑖−𝛾₂𝑡_𝑖−1−𝛾^Δ2𝜂_𝑖𝑡_𝑖−𝑡_𝑖−1≤𝛾^Δ2𝜂^′𝑖−𝛾^Δ2𝜂_𝑖𝑡_𝑖−𝑡_𝑖−1,𝜁_𝑖−𝛾₃𝑡_𝑖−1−𝛾^Δ3𝜁_𝑖𝑡_𝑖−𝑡_𝑖−1≤𝛾^Δ3𝜁^′𝑖−𝛾^Δ3𝜁_𝑖𝑡_𝑖−𝑡_𝑖−1,(3.15)and, consequently, 𝛾𝛾₁₂𝑡_𝑖−𝛾₁𝑡_𝑖−1=𝛾^Δ1𝜉_𝑖+𝛼_𝑖𝑡_𝑖−𝑡_𝑖−1,𝛾₃𝑡_𝑖−𝛾₂𝑡_𝑖−1=𝛾^Δ2𝜂_𝑖+𝛽_𝑖𝑡_𝑖−𝑡_𝑖−1,𝑡_𝑖−𝛾₃𝑡_𝑖−1=𝛾^Δ3𝜁_𝑖+𝜔_𝑖𝑡_𝑖−𝑡_𝑖−1,(3.16)where 00≤≤𝛼𝛽_𝑖𝑖≤𝛾^Δ1𝜉^′𝑖−𝛾^Δ1𝜉_𝑖≤𝑀_𝑖−𝑚_𝑖,0≤𝜔_𝑖≤𝛾^Δ2𝜂^′𝑖−𝛾^Δ2𝜂_𝑖≤𝑁_𝑖−𝑛_𝑖,≤𝛾^Δ3𝜁^′𝑖−𝛾^Δ3𝜁_𝑖≤𝑅_𝑖−𝑟_𝑖,(3.17)in which 𝑀_𝑖 and 𝑚_𝑖 are the supremum and infimum of 𝛾^Δ1 on [𝑡_𝑖−1,𝑡_𝑖) and 𝑁_𝑖, 𝑛_𝑖 and 𝑅_𝑖, 𝑟_𝑖 are corresponding numbers for 𝛾^Δ2 and 𝛾^Δ3, respectively. Using the inequality |||√𝑥²+𝑦²+𝑧²−𝑥²¹+𝑦²¹+𝑧²¹|||≤||𝑥−𝑥₁||+||𝑦−𝑦₁||+||𝑧−𝑧₁||(3.18)for 𝑥,𝑦,𝑧,𝑥₁,𝑦₁,𝑧₁∈ℝ, we obtain ||||𝛾^Δ1𝜉_𝑖+𝛼_𝑖²+𝛾^Δ2𝜂_𝑖+𝛽_𝑖²+𝛾^Δ3𝜁_𝑖+𝜔_𝑖²−𝛾^Δ1𝜉_𝑖²+𝛾^Δ2𝜉_𝑖²+𝛾^Δ3𝜉_𝑖²≤||𝛼_𝑖||||≤||𝛼_𝑖||+||𝛾^Δ2𝜂_𝑖+𝛽_𝑖−𝛾^Δ2𝜉_𝑖||+||𝛾^Δ3𝜁_𝑖+𝜔_𝑖−𝛾^Δ3𝜉_𝑖||≤𝑀_𝑖||+||𝛽_𝑖||+||𝜔_𝑖||+||𝛾^Δ2𝜂_𝑖−𝛾^Δ2𝜉_𝑖||+||𝛾^Δ3𝜁_𝑖−𝛾^Δ3𝜉_𝑖||−𝑚_𝑖+2𝑁_𝑖−𝑛_𝑖+2𝑅_𝑖−𝑟_𝑖.(3.19)Therefore, ||𝑙(𝛾,𝑃)−𝑆||=|||||^𝑛_𝑖=1𝛾^Δ1(𝜉)+𝛼_𝑖²+𝛾^Δ2𝜂_𝑖+𝛽_𝑖²+𝛾^Δ3𝜁_𝑖+𝜔_𝑖²𝑡^𝑖≤_𝑛−𝑡_𝑖−1|||||_𝑖=1𝑀_𝑖=𝑈𝛾_Δ1−𝑚_𝑖+2𝑁_𝑖−𝑛_𝑖+2𝑅_𝑖−𝑟_𝑖𝑡_𝑖−𝑡_𝑖−1+2𝑈,𝑃𝛾_Δ3−𝐿𝛾^Δ1,𝑃+2𝑈𝛾^Δ2,𝑃−𝐿𝛾^Δ2,𝑃,𝑃−𝐿𝛾^Δ3,𝑃,(3.20)where 𝑈 and 𝐿 denote the upper and lower Darboux Δ-sums, respectively. Since the functions [𝛾^Δ1]²+[𝛾^Δ2]²+[𝛾^Δ3]²,𝛾^Δ1,𝛾^Δ2 and 𝛾^Δ3 are Δ-integrable over [𝑎,𝑏), for arbitrary 𝜀>0, there exists 𝛿>0 such that ||𝑆−𝐼||<𝜀12,𝑈𝛾^Δ𝑗,𝑃−𝐿𝛾^Δ𝑗,𝑃<𝜀12,𝑗=1,2,3(3.21)for all 𝑃∈𝒫_𝛿([𝑎,𝑏]) (see [6, Theorem 5.9]) and the Riemann definition of Δ-integrability therein), where 𝐼 is defined by (3.14). Therefore, we get ||𝑙(Γ,𝑃)−𝐼||≤||𝑙(Γ,𝑃)−𝑆||+||𝑆−𝐼||<𝜀12+21𝜀2+21𝜀2+1𝜀2=2𝜀,(3.22)and so the validity of (3.13) is proved. On the other hand, among the partitions 𝑃 for which (3.13) is satisfied, there is a partition 𝑃 such that ||𝑙(𝛾,𝑃)−𝑙(𝛾)||<𝜀2.(3.23) Indeed, there is a partition 𝑃^∗ of [𝑎,𝑏] such that 0≤𝑙(𝛾)−𝑙𝛾,𝑃^∗<𝜀2.(3.24) Next, we refine the partition 𝑃^∗ adding to it new partition points so that we get a partition 𝑃 that belongs to 𝒫_𝛿([𝑎,𝑏]). Then by 𝑙(Γ,𝑃)≥𝑙(Γ,𝑃^∗), (3.24) yields 0≤𝑙(𝛾)−𝑙(𝛾,𝑃)<𝜀2,(3.25) so that (3.23) is shown. By (3.13) and (3.23), we get ||𝑙(Γ)−𝐼||<𝜀.(3.26)Since 𝜀>0 is arbitrary, we have 𝑙(Γ)=𝐼, and the proof is complete.

Definition 3.3. The curve 𝛾∶[𝑎,𝑏]⊂𝕋→ℝ³ is given in the parametric form 𝛾(𝑡)=(𝛾₁(𝑡),𝛾₂(𝑡),𝛾₃(𝑡)). Let 𝑃₀∶=𝛾(𝑡₀),𝑃^𝜎0∶=𝛾(𝜎(𝑡₀)), for 𝑡₀∈[𝑎,𝑏]^𝜅2and 𝑃∶=𝛾(𝑡) for 𝑡∈[𝑎,𝑏]. We denote the angle between the forward tangent lines drawn to the curve at 𝑃₀ and 𝑃 by 𝜃_𝑃 and the arc length of the segment 𝑃𝑃₀ of the curve by |𝑃𝑃₀|. The forward curvature of 𝛾 at 𝑃₀ is defined as 𝜅=lim_{𝑃→𝑃𝜎0}𝜃_𝑃||𝑃𝑃₀||.(3.27)

Theorem 3.4. Let 𝛾∶[𝑎,𝑏]⊂𝕋→ℝ³ be a natural Δ-parametrized Δ-regular curve, there exists 𝛾^ΔΔ(𝑡₀) for 𝑡₀∈[𝑎,𝑏]^𝜅2. If 𝑡₀ is a right-dense point, then 𝜅=‖‖𝛾^ΔΔ𝑡₀‖‖.(3.28) If 𝑡₀ is a right-scattered point, then 𝜅=2𝜇𝑡₀arcsin𝜇𝑡⁰‖‖𝛾^ΔΔ𝑡₀‖‖2.(3.29)

Proof. Let 𝑃₀∶=𝛾(𝑡₀),𝑃^𝜎0∶=𝛾(𝜎(𝑡₀)), for a fix point 𝑡₀∈[𝑎,𝑏]^𝜅2and 𝑃∶=𝛾(𝑡) for 𝑡∈[𝑎,𝑏]. Let the unit vectors on the forward tangent lines to the curve at points 𝑃 and 𝑃₀ be 𝜏(𝑡) and 𝜏(𝑡₀), respectively, and assume that 𝜃(𝑡) is the angle between them. We denote 𝑙(𝑡)=|𝑃𝑃₀|. Then we have 𝑙(𝑡)=‖‖𝜏(^𝑡𝑡₀𝑡‖‖)𝛾−^Δ(𝜏𝑠)𝑡‖‖₀Δ𝑠=^𝑡𝑡₀Δ𝑠=𝑡−𝑡₀,(3.30)‖‖=2sin𝜃2(𝑡).(3.31) Assume that 𝑡₀ is a right-dense point. Then 𝑃→𝑃₀=𝑃^𝜎0 and 𝜃(𝑡)→0 as 𝑡→𝑡₀. Since lim_𝑡→𝑡0𝜃(𝑡)/𝑡−𝑡₀𝜃(𝑡)/𝑙(𝑡)=lim^𝑡→𝑡₀𝑙(𝑡)𝑡−𝑡₀=1,(3.32) we obtain lim_𝑡→𝑡0𝜃(𝑡)𝑡−𝑡₀=lim_𝑡→𝑡0𝜃(𝑡)𝑙(𝑡).(3.33) By using (3.31) we can write ‖‖‖‖𝜏(𝑡)−𝜏t₀𝑡−𝑡₀‖‖‖‖=2𝑡−𝑡₀sin𝜃(𝑡)2=sin𝜃((𝜃𝑡)(𝑡/)2/2)𝑡𝜃−(𝑡𝑡)₀.(3.34) Thus, we find 𝜅=lim_𝑡→𝑡0𝜃(𝑡)𝑙(𝑡)=‖‖𝛾^ΔΔ𝑡₀‖‖(3.35) as 𝑡→𝑡₀.
Assume that 𝑡₀ is a right-scattered point. Then 𝑃→𝑃^𝜎0 and 𝜃(𝑡)→𝜃(𝜎(𝑡₀)) as 𝑡→𝜎(𝑡₀). If the equality (3.31) is divided by 𝑡−𝑡₀, taking the limit as 𝑡→𝜎(𝑡₀), we have ‖‖𝛾^ΔΔ𝑡₀‖‖=‖‖‖‖𝜏𝜎𝑡⁰−𝜏𝑡₀𝜎𝑡₀−𝑡₀=2‖‖‖‖𝜎𝑡₀−𝑡₀sin𝜃𝜎𝑡⁰=22𝜇𝑡₀sin𝜃𝜎𝑡⁰2.(3.36) (3.16) On the other hand, we find 𝜅=lim_{𝑡→𝜎(𝑡0)}𝜃(𝑡)𝑙(𝑡)=𝜃𝜎𝑡⁰𝜎𝑡₀−𝑡₀=𝜃𝜎𝑡⁰𝜇𝑡₀.(3.37) It follows from (3.16) and (3.37) that 𝜅=2𝜇𝑡₀arcsin𝜇𝑡⁰‖‖𝛾^ΔΔ𝑡₀‖‖2.(3.38) This completes the proof.

Theorem 3.5. Let 𝛾∶[𝑎,𝑏]⊂𝕋→ℝ³ be a curve with arbitrary parameters that is second Δ-differentiable on [𝑎,𝑏]^𝜅2 and ‖𝛾^Δ(𝑡)‖≠0 for all 𝑡∈[𝑎,𝑏]^𝜅2. Moreover assume that the function ℎ∶[𝑎,𝑏]^𝜅2→ℝ defined by ℎ(𝑡)∶=⟨𝛾^Δ(𝑡),𝛾^Δ(𝑡)⟩ is strictly increasing. The forward curvature of 𝛾 at the right-dense point 𝑡∈[𝑎,𝑏]^𝜅2 is 𝜅=‖‖𝛾^Δ(𝑡)×𝛾^ΔΔ(𝑡)‖‖‖‖𝛾^Δ(𝑡)‖‖³.(3.39)

Proof. The function 𝑝∶[𝑎,𝑏]→𝑝([𝑎,𝑏])=𝕋 defined by 𝑠=𝑝(𝑡)= ∫^𝑡𝑎‖𝛾^Δ(𝑣)‖Δ𝑣 is continuous and strictly increasing. Therefore 𝕋 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 𝜎 and Δ respectively. Since the curve 𝛾∘𝑝⁻¹ is a natural Δ-parametrized, to find the curvature to the 𝛾 at the point 𝑡, it is sufficient to find that the curvature to the 𝛾∘𝑝⁻¹ at the point 𝑠. 𝑝⁻¹Δ(𝑠)=1𝑝𝑝^Δ(𝑡)=‖‖𝛾^Δ1(𝑡)₋₁‖‖,Δ𝜎(𝑠)=1𝑝𝑝^Δ𝑝⁻¹𝜎(𝑠)=𝑝^Δ(𝜎1(𝑡))=‖‖𝛾^Δ(𝜎1(𝑡))‖‖=‖‖𝛾^Δ1(𝑡)₋₁‖‖,ΔΔ(𝑠)=−𝑝^ΔΔ(𝑡)𝑝^Δ(𝑡)²𝑝^Δ(𝜎(𝑡))=−‖‖𝛾^Δ(𝑡)‖‖^Δ‖‖𝛾^Δ(𝑡)‖‖³.(3.40)
By using (3.40), one can easily find that ‖‖‖𝛾∘𝑝₋₁ΔΔ(𝑠)‖‖‖==‖‖‖‖‖𝛾‖‖^Δ𝛾(𝑡)𝑝⁻¹ΔΔ(𝑠)+𝛾^ΔΔ(𝑡)𝑝⁻¹Δ(𝑠)𝑝⁻¹Δ𝜎(𝑠)_Δ‖‖‖(𝑡)‖‖𝛾^ΔΔ(𝑡)−‖‖𝛾^Δ(𝑡)‖‖^Δ𝛾^Δ(𝑡)‖‖‖‖𝛾^Δ(𝑡)‖‖³.(3.41) As 𝑡∈[𝑎,𝑏]^𝜅2 is a right-dense point and the function ℎ(𝑡)∶=⟨𝛾^Δ(𝑡),𝛾^Δ(𝑡)⟩ is strictly increasing, we have ‖‖𝛾^Δ(𝑡)‖‖^Δ=𝛾^Δ(𝑡),𝛾^ΔΔ(𝑡)‖‖𝛾^Δ(𝑡)‖‖.(3.42) Substituting (3.42) into (3.41), we obtain 𝜅=‖‖𝛾^Δ(𝑡)×𝛾^ΔΔ(𝑡)×𝛾^Δ(𝑡)‖‖‖‖𝛾^Δ(𝑡)‖‖⁴=‖‖𝛾^Δ(𝑡)×𝛾^ΔΔ(𝑡)‖‖‖‖𝛾^Δ(𝑡)‖‖sin(𝜋/2)‖‖𝛾^Δ(𝑡)‖‖⁴=‖‖𝛾^Δ(𝑡)×𝛾^ΔΔ(𝑡)‖‖‖‖𝛾^Δ(𝑡)‖‖³.(3.43) The proof is complete.

Remark 3.6. It is easy to see that, for the case 𝕋=ℝ, the results (3.28) and (3.39) generalize the following formulas stated in classical differential geometry: 𝜅=‖‖𝛾^′′𝑡₀‖‖,𝜅=‖‖𝛾^′(𝑡)×𝛾^′′(𝑡)‖‖‖‖𝛾^′(𝑡)‖‖³.(3.44)

Theorem 3.7. Let 𝛾∶[𝑎,𝑏]⊂𝕋→ℝ³ be a curve with arbitrary parameters that is second Δ differentiable on [𝑎,𝑏]^𝜅2and has continuous Δ derivative on [𝑎,𝑏]^𝜅 and ‖𝛾^Δ(𝑡)‖≠0 for all 𝑡∈[𝑎,𝑏]^𝜅. Moreover assume that the function ℎ∶[𝑎,𝑏]^𝜅→ℝ defined by ℎ(𝑡)∶=⟨𝛾^Δ(𝑡),𝛾^Δ(𝑡)⟩ is strictly increasing. The forward curvature of 𝛾 at the right-scattered point 𝑡∈[𝑎,𝑏]^𝜅2 is 𝜅=2𝜇(𝑡)‖‖𝛾^Δ(𝑡)‖‖arcsin𝜇(2𝑡𝑞)₂𝑞¹,(3.45) where 𝑞₁=‖‖‖‖𝛾^Δ(𝑡)‖‖‖‖𝑞𝛾^Δ(𝑡)‖‖+‖‖𝛾^Δ(𝜎(𝑡))‖‖𝛾^ΔΔ(𝑡)−𝛾^ΔΔ(𝑡),𝛾^Δ(𝑡)+𝛾^Δ(𝜎(𝑡))𝛾^Δ(𝑡)₂‖‖,=‖‖𝛾^Δ(𝑡)‖‖‖‖𝛾^Δ(𝜎(𝑡))‖‖‖‖𝛾^Δ(𝑡)‖‖+‖‖𝛾^Δ(𝜎(𝑡))‖‖.(3.46)

Proof. Let 𝑡∈[𝑎,𝑏]^𝜅2 be a right-scattered point. In this case we have 𝜇(𝑠)===𝑝𝜇𝜎(((𝜎𝑠𝑡))(𝑡‖‖−)𝛾)_Δ𝑠−𝑝(𝑡)‖‖𝛾(𝑡)‖‖,(3.47)_Δ(𝑡)‖‖^Δ=𝛾^ΔΔ(𝑡),𝛾^Δ(𝑡)+𝛾^Δ(𝜎(𝑡))‖‖𝛾^Δ(𝑡)‖‖+‖‖𝛾^Δ(𝜎(𝑡))‖‖.(3.48) By using (3.41) and (3.48), we find ‖‖‖𝛾∘𝑝₋₁ΔΔ(𝑠)‖‖‖=𝑞¹‖‖𝛾^Δ(𝑡)‖‖²‖‖𝛾^Δ(𝜎(𝑡))‖‖‖‖𝛾^Δ(𝑡)‖‖+‖‖𝛾^Δ(𝜎(𝑡))‖‖.(3.49) Substituting (3.49) into (3.29), we obtain 𝜅=2𝜇(𝑡)‖‖𝛾^Δ(𝑡)‖‖arcsin𝜇(2𝑡𝑞)₂𝑞¹.(3.50)

Example 3.8. Let 𝕋=ℤ and 𝛾∶𝕋→ℝ³, 𝛾(𝑡)=𝛽(cos𝑡,sin𝑡,𝑡).The curve 𝛾 satisfies the conditions of Theorem 3.4 for the case 𝛽=1/√3−2cos1. In this case we have 𝜎(𝑡)=𝑡+1,𝜇(𝑡)=1, and 𝛾𝛾^Δ(𝑡)=𝛽(cos(𝑡+1)−cos𝑡,sin(𝑡+1)−sin𝑡,1),_ΔΔ(𝑡)=𝛽(cos(𝑡+2)−2cos(𝑡+1)+cos𝑡,sin(𝑡+2)−2sin(𝑡+1)+sin𝑡,0).(3.51) Since every point of 𝕋 is right-scattered point, the curvature of 𝛾 at any point 𝑡 is 𝜅=2𝜇(𝑡)arcsin𝜇(𝑡)‖‖𝛾^ΔΔ(𝑡)‖‖=2arcsin2𝛽sin₂221.(3.52) This value is the angle between the line through 𝛾(𝑡), 𝛾(𝑡+1) and the line through 𝛾(𝑡+1), 𝛾(𝑡+2).

Example 3.9. Assume that 𝛾∶𝕋={1,1/2,1/3,…}∪{0}→ℝ³,𝛾(𝑡)=(𝑡,𝑡²,0), is a non-Δ-natural parametrized curve. The only right-dense point of the time scale 𝕋 is 𝑡=0, and the other points of the time scale are right-scattered. The forward jump operator and the graininess function are 𝜇𝜎((𝑡𝑡))==⎧⎪⎨⎪⎩⎧⎪⎨⎪⎩11,𝑡−𝑡₂𝑡,𝑡𝑡≠=11,,01,−𝑡,𝑡𝑡≠=11,.(3.53) Furthermore we have 𝛾^Δ(𝑡)=1,𝑡²−2𝑡𝛾𝑡−1,0,for𝑡≠1_ΔΔ(𝑡)=0,2𝑡−22𝑡−1,0,for𝑡≠1,𝑡≠12,(3.54) and, from Theorem 3.5, the forward curvature of the curve 𝛾 at the point 𝑡=0 is 𝜅(0)=‖‖𝛾^Δ(0)×𝛾^ΔΔ(0)‖‖‖‖𝛾^Δ(0)‖‖³=2.(3.55) For the right-scattered point 𝑡=1/3, by using Theorem 3.7, we find 𝜅13=√7621arcsin⎛⎜⎜⎜⎝22√793−6₄√7933√13+√61⎞⎟⎟⎟⎠.(3.56) This value is the ratio of the angle between the line through the points 𝛾(1/3), 𝛾(1/2) and the line through the points 𝛾(1/2), 𝛾(1) and the distance between the points 𝛾(1/3) and 𝛾(1/2).

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.