multivariable calculus on time scales

Multivariable Calculus on Time Scales

Sinan Kapcak

Izmir University of Economics,Department of Mathematics, Sakarya Cad No:156

Izmir, TURKEYEmail: [email protected]

Unal Ufuktepe

Izmir University of Economics,Department of Mathematics, Sakarya Cad No:156

Izmir, TURKEYEmail: [email protected]

Abstract—Time scale analysis is the generalization of discreteand continuous analysis. In this research paper, we extend ourprevious study [6] and develop some important concepts; tan-gent planes and partial derivatives for multi-variable functionson time scales with Mathematica.

Keywords-partial delta derivative; time scale; mathematica;tangent plane;

I. INTRODUCTION

The theory time scales was initiated by Stefan Hilger

[5] in order to unify continuous and discrete analysis,

which has many potential for applications and has received

tremendous attention.

In this section and Section 2, we give some elements of

the time scales calculus for one variabe, and in Section 3

we briefly give some basics of multi-variable calculus on

time scales. We present some foundational definitions and

results, for more details, one can see [1], [3], [4].

Let T be a time scale which is nonempty closed subset

of real numbers. We define the forward and backward jumpoperators σ, ρ : T→ T respectively as follows:

σ(t) = inf{s ∈ T : s > t}, and

ρ(t) = sup{s ∈ T : s < t}.A point t ∈ T is called right-scattered, right-dense, left-

scattered, left-dense if σ(t) > t, σ(t) = t, ρ(t) < t, ρ(t) = tholds, respectively. The graininess μ : T→ [0,∞) is defined

by

μ(t) = σ(t)− t.

If T has left-scattered maximum then we define Tκ =

T−{maxT}, otherwise Tκ = T. If f is delta differentiable

at t ∈ Tκ then delta derivative of f is

fΔ(t) =

{lims→t

f(t)−f(s)t−s , μ(t) = 0;

f(σ(t))−f(t)σ(t)−t , μ(t) > 0.

We refer to the original work by Hilger [5] and the

recently appeared works can be found in [1], [2], [3].

II. THE TANGENT LINE WITH Mathematica

There are a few works on time scale analysis with

Mathematica. A.Yantır and U.Ufuktepe first defined

elements of time scales, and gave the delta derivative of

a function[7] for Mathematica, B.Balllard and B.Otegbade

from Baylor University created Time Scales Toolbox for

Matlab (2005), U.Ufuktepe and S.Kapcak improved their

previous works and got important symbolic results [6] for

Mathematica.

The codes of all Mathematica commands which have

written by the authors can be seen in the Appendix.

To describe a time scale in our Mathematica package,

we use a collection of three lists: a list of right-dense

left-scattered points, a list of left-dense right-scattered

points and a list of isolated points. For example, we

represent time scale T1 = [−2,−1] ∪ {2} consisting of one

closed interval and one isolated point by

In[1]:= T1={{-2},{-1},{2}};

We refer to [6] for more details about the symbolic and

numerical computations of jump operators, delta derivative

and delta integral as well as their visual representations.

In the rest of this section, we give the definition of

tangent line and visualize it by using our time scale

package.

We consider the geometric sense of delta differentiabil-

ity in simple variable functions on time scales. Consider

u = f(t) for t ∈ T. Let Γ be the curve represented by

the function f . Let t0 be a fixed point in Tκ. In this case

P0 = (t0, f(t0)) is a point on the curve.

Definition 1: A line � passing through the point P0 is

called the delta tangent line to the curve Γ at the point P0

if

a. � passes also through the point Pσ0 =

(σ(t0), f(σ(t0)).

2011 International Conference on Computational Science and Its Applications

978-0-7695-4404-5/11 $25.00 © 2011 IEEEDOI 10.1109/ICCSA.2011.28

41

https://www.researchgate.net/publication/245222750_Dynamic_Equations_on_Time_Scales?el=1_x_8&enrichId=rgreq-90a8f6c9-f55c-4775-8a87-5fe4fef071ba&enrichSource=Y292ZXJQYWdlOzI0MTE4NjY2MDtBUzoxNTY4NzU4MDcwMTA4MTZAMTQxNDQxMzUxNTQwMQ==

https://www.researchgate.net/publication/243774249_Dynamic_Equations_on_Time_Scales_An_Introduction_with_Application?el=1_x_8&enrichId=rgreq-90a8f6c9-f55c-4775-8a87-5fe4fef071ba&enrichSource=Y292ZXJQYWdlOzI0MTE4NjY2MDtBUzoxNTY4NzU4MDcwMTA4MTZAMTQxNDQxMzUxNTQwMQ==



https://www.researchgate.net/publication/232997902_An_introduction_to_complex_functions_on_products_of_two_time_scales?el=1_x_8&enrichId=rgreq-90a8f6c9-f55c-4775-8a87-5fe4fef071ba&enrichSource=Y292ZXJQYWdlOzI0MTE4NjY2MDtBUzoxNTY4NzU4MDcwMTA4MTZAMTQxNDQxMzUxNTQwMQ==

https://www.researchgate.net/publication/221435400_Mathematica_Applications_on_Time_Scales?el=1_x_8&enrichId=rgreq-90a8f6c9-f55c-4775-8a87-5fe4fef071ba&enrichSource=Y292ZXJQYWdlOzI0MTE4NjY2MDtBUzoxNTY4NzU4MDcwMTA4MTZAMTQxNDQxMzUxNTQwMQ==

https://www.researchgate.net/publication/257290127_Analysis_on_Measure_Chains_-_A_Unified_Approach_to_Continuous_and_Discrete_Calculus?el=1_x_8&enrichId=rgreq-90a8f6c9-f55c-4775-8a87-5fe4fef071ba&enrichSource=Y292ZXJQYWdlOzI0MTE4NjY2MDtBUzoxNTY4NzU4MDcwMTA4MTZAMTQxNDQxMzUxNTQwMQ==


b. if P0 is not an isolated point of the curve Γ, then

limP→P0, P �=P0

d(P, �)

d(P, P0)= 0,

where P is the moving point of the curve Γ, d(P, �)is the distance from the point P to the line �, and

d(P, P σ0 ) is the distance from the point P to the

point Pσ0 .[3].

Now, we plot the tangent line to the function

f : T1 → R, f(x) = x3 at the point x = −1 where

T1 is the time scale above:

In[2]:= TSTangentLine[T1,xˆ3,x,-1]

Out[2]= (see Figure 1)

�2 �1 1 2

�5

5

Figure 1. The tangent line to the curve f(x) = x3 at the point x = −1

III. FUNCTIONS WITH TWO VARIABLES ON TWO TIME

SCALES WITH MathematicaLet T1 and T2 be time scales. We define the product

T1 × T2 by

T1 × T2 = {(t, s) : t ∈ T1 and s ∈ T2}.Let σ1 and σ2 be two forward jump operators for T1 and T2,

respectively. Let ρ1 and ρ2 be two backward jump operators

for T1 and T2, respectively. We consider the surface S which

is defined by u = f(t, s) continuous function on T1 × T2:

the set of points

{(t, s, f(t, s)) : (t, s) ∈ T1 × T2}in three space.We refer to[4] for more details.

To get the visual representation of the product T1 × T2

by Mathematica, we first input T1 = [−1, 0] ∪ [1, 2] ∪ { 12}and T2 = [0, 1] ∪ [2, 3] ∪ [5, 6] ∪ {4, 8} as follows:

In[3]:= T1={{-1,1},{0,2},{1/2}};

In[4]:= T2={{0,2,5},{1,3,6},{4,8}};

We plot the set of points T1×T2 by using the command

In[5]:= DrawTimeScale3D[T1,T2]

Out[5]=(see Figure 2)

�1

0

1

2

TS1

0

2

4

6

8

TS2

�1.0

�0.5

0.0

0.5

1.0

Figure 2. T1 × T2

Let f : T1 × T2 → R such that f(t, s) = sin (t+ s).We plot this surface on T1 × T2 by using the command

TSPlot3D[T1,T2,f(t,s),t,s] where t ∈ T1,

s ∈ T2, as follows:

In[6]:= TSPlot3D[T1,T2,Sin[t+s],t,s]


IV. PARTIAL DIFFERENTIATION ON TIME SCALES

Let f : T1 × T2 → R be a function. The first order delta

derivatives of f at a point (t0, s0) ∈ Tκ1 ×T

κ2 are defined by

∂f(t0, s0)

Δ1t= lim

t→t0, t �=σ1(t0)

f(σ1(t0), s0)− f(t, s0)

σ1(t0)− t

and

∂f(t0, s0)

Δ2s= lim

s→s0, s �=σ2(s0)

f(t0, σ2(s0))− f(t0, s)

σ2(s0)− s.

42












�1

0

1

2

TS1

�1

0

1

2

TS2

�1.0�0.50.00.5

1.0

Figure 3. The graph of f(t, s) = sin (t+ s) : t ∈ T1, s ∈ T2

To compute the partial delta derivative of

a given function at a given point, we use

the command PartialDeltaDerivative[{TS1,TS2},{t,s},f(t,s),var,p]. Here, the

partial delta derivative of f, a function of t ∈ TS1 and

s ∈ TS2, is taken with respect to var at the point p. Let

us get some results for the time scales T1 and T2 we have

given before, for a function f(t, s) = t2√s:

In[7]:= PartialDeltaDerivative[{T1,T2},{t,s},tˆ2 sˆ(1/2),t,{1,2}]

Out[7]= 2√2

In[8]:= PartialDeltaDerivative[{T1,T2},{t,s},tˆ2 sˆ(1/2),t,{1/2,3}]

Out[8]= 3√3

2

Now, let us give the definition of the tangent plane to a

surface defined on a product of two time scales.

Definition 2: A plane Ω0 passing through the point P0 =(t0, s0, f(t0, s0)) is called the delta tangent plane to the

surface S at the point P0 if

a. Ω0 passes also through the points

Pσ10 = (σ1(t0), s0, f(σ1(t0), s0)) and

Pσ20 = (t0, σ2(s0)f(t0, σ2(s0)));

b. if P0 is not an isolated point of the surface S , then

limP→P0, P �=P0

d(P,Ω0)

d(P, P0)= 0,

where P is the moving point of the surface S , d(P,Ω0) is

the distance from the point P to the plane Ω0, and d(P, P0)is the distance of the point P from the point P0.

The command DeltaTangentPlane[T1,T2,f(t,s),{t,s},p] plots the function f(t, s)

with t ∈ T1, s ∈ T2 at the point p. For the function

f(t, s) = sin(t + s), let us plot the tangent plane at the

point { 12 , 1}:

In[9]:= DeltaTangentPlane[T1,T2,Sin[t+s],{t,s},{1/2,1}]


�1 0 1 2TS1

�10

12

TS2

�1.0

�0.5

0.0

0.5

1.0

�1

0

1

2

TS1

�1

0

1

2

TS2

�1.0�0.5

0.0

0.5

1.0

Figure 4. The delta tangent plane to the surface f(t, s) = sin(t+ s) atthe point ( 1

2, 1) at two different view points

V. CONCLUSION

There have been two mathematical framework to model

the population dynamics; continuous time overlapping gen-

eration models (differential equations), and discrete (differ-

ence equations). But there are some complex population

systems such that the continuous time and the discrete

43

https://www.researchgate.net/publication/221435400_Mathematica_Applications_on_Time_Scales?el=1_x_8&enrichId=rgreq-90a8f6c9-f55c-4775-8a87-5fe4fef071ba&enrichSource=Y292ZXJQYWdlOzI0MTE4NjY2MDtBUzoxNTY4NzU4MDcwMTA4MTZAMTQxNDQxMzUxNTQwMQ==

time models may not really catch some of the properties

for certain specific population dynamics, in this case we

may need models on continuous-discrete-mixed time scales

(dynamic equations). Our future works will be focused on

the solutions of dynamic equations, system of dynamic

equations, and their orbits.

APPENDIX

QTimeScalenormal[TS_]:= Module[{intervals},intervals =Transpose[{Table[Sort[TS[[i]]],

{i, 2}][[1]],Table[Sort[TS[[i]]], {i, 2}][[2]]}];

checkdensepointforoneinterval[interval_, testpoint_] :=If[interval[[1]] < testpoint <interval[[2]], 1, 0];checkdensepointforTS[testpoint_] :=Function[checkdensepointforoneinterval[#, testpoint]] /@

intervals.Function[checkdensepointforoneinterval[#, testpoint]] /@intervals;

Which[Length[TS[[2]]] !=Length[TS[[1]]],Print["Length of rigth-denseleft-scattered and left-dense \

right-scattered points must be equal!"],Union[Sort[Flatten[TS]]] != Sort[Flatten[TS]],Print["Given intervals or isolatedpoints musn’t be intersected!"],Flatten[Transpose[{Sort[TS[[1]]], Sort[TS[[2]]]}]] !=Sort[Flatten[{TS[[1]], TS[[2]]}]],

Print["Given intervals are notproperly defined!"], \!$

\*UnderoverscriptBox[\(\[Sum]$, $j =1$, $Length[TS[\([$$3$$]$]]\)]\

(checkdensepointforTS[$TS[\([$$3$$\]$]\)[$[$$j$$]$]]\)\) != 0,

Print["Given intervals or isolatedpoints musn’t be \

intersected!!!"], Length[Flatten[TS]]== 0,Print["Time Scale can not be anempty set!"],Length[TS] != 3,Print["Length of the given list mustbe 3!"], 1 == 1, True

]]

QTimeScale[TS ]

:=If[(Length[TS] == 2 && Head[TS[[2]]]=== Symbol), True,

QTimeScalenormal[TS]]

checkelement[TS_, point_]:=Module[{criticalpoints, isolatedpoints,

RDLSpoints, LDRSpoints,intervals},criticalpoints = Sort[Flatten[TS]];isolatedpoints = Sort[TS[[3]]];RDLSpoints = Sort[TS[[1]]];LDRSpoints = Sort[TS[[2]]];intervals =Transpose[{Table[Sort[TS[[i]]],{i, 2}][[1]],

Table[Sort[TS[[i]]], {i, 2}][[2]]}];

checkdensepointforoneinterval[interval_, testpoint_] :=If[interval[[1]] < testpoint <interval[[2]], 1, 0];checkdensepointforTS[testpoint_] :=Function[checkdensepointforoneinterval[#, testpoint]] /@intervals.Function[

checkdensepointforoneinterval[#, testpoint]] /@intervals;

onpoint[point1_, point2_] := If[point1 == point2, 1, 0];checkRDLSpointforTS[testpoint_] :=Function[onpoint[testpoint, #]] /@

RDLSpoints.Function[onpoint[testpoint, #]] /@ RDLSpoints;

checkLDRSpointforTS[testpoint_] :=Function[onpoint[testpoint, #]] /@

LDRSpoints.Function[onpoint[testpoint, #]] /@ LDRSpoints;

checkisolatedpointforTS[testpoint_] :=Function[onpoint[testpoint, #]] /@

isolatedpoints.Function[onpoint[testpoint, #]] /@ isolatedpoints;

If[checkdensepointforTS[point] +checkRDLSpointforTS[point] +

checkLDRSpointforTS[point] +checkisolatedpointforTS[point] ==1,1, 0]

]

Sigmanormal[TS_, point_] :=

44

Module[{criticalpoints, isolatedpoints,RDLSpoints, LDRSpoints,intervals}, If[QTimeScale[TS] ===True,criticalpoints = Sort[Flatten[TS]];isolatedpoints = Sort[TS[[3]]];RDLSpoints = Sort[TS[[1]]];LDRSpoints = Sort[TS[[2]]];intervals =Transpose[{Table[Sort[TS[[i]]],{i, 2}][[1]],

Table[Sort[TS[[i]]], {i, 2}][[2]]}];

checkdensepointforoneinterval[interval_, testpoint_] :=If[interval[[1]] < testpoint <interval[[2]], 1, 0];

checkdensepointforTS[testpoint_] :=Function[checkdensepointforoneinterval[#, testpoint]] /@

intervals.Function[checkdensepointforoneinterval[#,testpoint]] /@ intervals;





checkisolatedpointforTS[testpoint_]:=Function[onpoint[testpoint, #]] /@


nextcriticalpoint[testpoint_] :=criticalpoints[[First[First[Position[criticalpoints,testpoint]]] + 1]];

Which[point == Last[criticalpoints],point,checkdensepointforTS[point] == 1,point,

checkRDLSpointforTS[point] == 1,point,checkLDRSpointforTS[point] == 1,nextcriticalpoint[point],checkisolatedpointforTS[point] == 1,nextcriticalpoint[point]]]

]

Sigmasq[an_, t_] := Module[{sol, b},If[Limit[an[[1]], an[[2]] -> Infinity]< an[[1]] /. an[[2]] -> 1,If[t === Limit[an[[1]], an[[2]] ->Infinity], t,If[(Reduce[an[[1]] == t && an[[2]]> 0, an[[2]], Integers] ===

False && Head[t] === Real),Print["Given time scale doesn’tcontain the point ", t, "."],If[t === (an[[1]] /. an[[2]] ->1), t,sol = Solve[an[[1]] == t, an[[2]],InverseFunctions -> True];b = an[[1]] /. an[[2]] ->(an[[2]] - 1);First[FullSimplify[b /. sol]]]]

],If[t === Limit[an[[1]], an[[2]] ->Infinity], t,If[(Reduce[an[[1]] == t && an[[2]]> 0, an[[2]], Integers] ===

False && Head[t] === Real),Print["Given time scale doesn’tcontain tthe point ", t, "."],sol = Solve[an[[1]] == t, an[[2]],InverseFunctions -> True];b = an[[1]] /. an[[2]] -> (an[[2]]+ 1);First[FullSimplify[b /. sol]]]

]]]

Sigma[TS ,point ]

:=If[Length[TS] == 2, Sigmasq[TS, point],Sigmanormal[TS, point]]

Rho[TS ,point ]

:=Module[{criticalpoints, isolatedpoints,

RDLSpoints, LDRSpoints,intervals}, If[QTimeScale[TS] ===True,criticalpoints = Sort[Flatten[TS]];isolatedpoints = Sort[TS[[3]]];RDLSpoints = Sort[TS[[1]]];LDRSpoints = Sort[TS[[2]]];intervals =Transpose[{Table[Sort[TS[[i]]],{i, 2}][[1]],

Table[Sort[TS[[i]]], {i, 2}][[2]]}];

45



checkdensepointforTS[testpoint_] :=Function[checkdensepointforoneinterval[#,testpoint]] /@intervals.Function[checkdensepointforoneinterval[#,testpoint]] /@ intervals;





checkisolatedpointforTS[testpoint_] :=Function[onpoint[testpoint, #]] /@


prevcriticalpoint[testpoint_] :=criticalpoints[[First[First[Position[criticalpoints,testpoint]]] - 1]];

Which[point == First[criticalpoints],point,checkdensepointforTS[point] == 1,point,checkLDRSpointforTS[point] == 1,point,checkRDLSpointforTS[point] == 1,prevcriticalpoint[point],checkisolatedpointforTS[point] == 1,prevcriticalpoint[point]]]

]

TSGraininess[TS_, point_] := Sigma[TS,point] - point

DeltaDerivativenormal[TS_, f_, x_,point_] :=Module[{criticalpoints, isolatedpoints,RDLSpoints, LDRSpoints,intervals},If[QTimeScale[TS] === True, If[checkelement[TS, point] == 1,criticalpoints = Sort[Flatten[TS]];isolatedpoints = Sort[TS[[3]]];

RDLSpoints = Sort[TS[[1]]];LDRSpoints = Sort[TS[[2]]];intervals =Transpose[{Table[Sort[TS[[i]]],{i, 2}][[1]],Table[Sort[TS[[i]]], {i, 2}][[2]]}];


checkdensepointforTS[testpoint_] :=Function[checkdensepointforoneinterval[#,testpoint]] /@intervals.Function[checkdensepointforoneinterval[#,testpoint]] /@ intervals;

onpoint[point1_, point2_] := If[point1 == point2, 1, 0];checkRDLSpointforTS[testpoint_] :=Function[onpoint[testpoint, #]] /@RDLSpoints.Function[onpoint[testpoint, #]] /@ RDLSpoints;

checkLDRSpointforTS[testpoint_] :=Function[onpoint[testpoint, #]] /@LDRSpoints.Function[onpoint[testpoint, #]] /@ LDRSpoints;

checkisolatedpointforTS[testpoint_]:=Function[onpoint[testpoint, #]] /@isolatedpoints.Function[onpoint[testpoint, #]] /@isolatedpoints;

nextcriticalpoint[testpoint_] :=criticalpoints[[First[First[Position[criticalpoints, testpoint]]]+ 1]];

If[Last[criticalpoints] == point,Print[point," is not an element of\!$\*SuperscriptBox[\"T\", \

\"\[Kappa]\"]$"],Which[checkdensepointforTS[point]== 1,Limit[(f - (f /. x -> point))/(x - point), x -> point],checkRDLSpointforTS[point] == 1,If[Limit[f, x -> point, Direction-> -1] != f /. x -> point,Print["Delta derivative at thepoint ", point," doesn’t exist."],

46


Limit[(f - (f /. x -> point))/(x - point), x -> point,Direction -> -1]],checkLDRSpointforTS[point] == 1,

If[Limit[f, x -> point, Direction-> 1] != f /. x -> point,Print["Delta derivative at thepoint ", point," doesn’t exist."], ((f /. x ->Sigma[TS, point]) - (f /.

x -> point))/(Sigma[TS,point] - point)],

checkisolatedpointforTS[point] ==1, ((f /. x -> Sigma[TS, point])- (f /. x -> point))/(Sigma[

TS, point] - point)]],Print[point, " is not an element ofTime Scale given!"]],

Print["Given list is not enteredproperly as a time scale."]]]

DeltaDerivativesq[TS_, f_, x_, point_]:=If[(Limit[TS[[1]], TS[[2]] -> Infinity]===

point && ((Limit[TS[[1]], TS[[2]]-> Infinity]) < (TS[[1]] /.

TS[[2]] -> 1))), (D[f, x] /. x-> point),

FullSimplify[((f /. x -> Sigma[TS,point]) - (f /.

x -> point))/(Sigma[TS, point] -point)]]

DeltaDerivative[TS ,f ,x ,point ]

:=If[Length[TS] == 3,DeltaDerivativenormal[TS, f, x, point],DeltaDerivativesq[TS, f, x, point]]

DeltaIntegral[TS ,f ,x ,intrvll ]

:=Module[{SoFl, pnts, p1, p2, lastpnts,setsum, contsum, RI, RI2,Intervals, totalsum}, If[QTimeScale[TS] === True,If[checkelement[TS, intrvll[[1]]] +

checkelement[TS, intrvll[[2]]] == 2,SoFl = Sort[Flatten[TS]];pnts = Append[SoFl, intrvll[[1]]];pnts = Append[pnts, intrvll[[2]]];pnts = Union[Sort[pnts]];p1 = First[Position[pnts, intrvll[[1]]]];p2 = First[First[Position[pnts,intrvll[[2]]]]];

lastpnts = {};For[i = First[p1], i < p2, lastpnts =Append[lastpnts, pnts[[i]]];

i++];Intervals = Sort[Transpose[{TS[[1]],TS[[2]]}]];setsum = \!$

\*UnderoverscriptBox[\(\[Sum]$,$i = 1$, $Length[

lastpnts]$]$\((f /.x -> lastpnts[\([$$i$$]$])\)\ $(Sigma[TS,lastpnts[\([$$i$$]$]] -lastpnts[$[$$i$$]$])\)\)\);

RI = {};Contint[exint_] :=Which[intrvll[[1]] <= exint[[1]] <=exint[[2]] <= intrvll[[2]],Append[RI, exint],intrvll[[1]] <= exint[[1]] <=intrvll[[2]] <= exint[[2]],Append[RI, {exint[[1]],intrvll[[2]]}],exint[[1]] <= intrvll[[1]] <=exint[[2]] <= intrvll[[2]],Append[RI, {intrvll[[1]],exint[[2]]}],exint[[1]] <= intrvll[[1]] <=intrvll[[2]] <= exint[[2]],Append[RI, intrvll],intrvll[[1]] <= intrvll[[2]] <=exint[[1]] <= exint[[2]],Append[RI, {exint[[1]],exint[[1]]}],exint[[1]] <= exint[[2]] <=intrvll[[1]] <= intrvll[[2]],Append[RI, {exint[[1]],exint[[1]]}]];

RI2 = Function[First[#]] /@Function[Contint[#]] /@ Intervals;contsum = \!$

\*UnderoverscriptBox[\(\[Sum]$,$k = 1$, $Length[RI2]$]$\*SubsuperscriptBox[\(\[Integral]$,$\(RI2[\([$$k$$]$]\)[$[$\$1$$]$]\), $\(RI2[\([$$k$$]$]\)[$[$$2$$]$]\)]f \\[DifferentialD]x\)\);

totalsum = setsum + contsum;totalsum,Print["One or more end points ofgiven interval is/are not \

element(s) of given Time Scale!"]]]

47

]

TSPlottek[TS_, Fnct_, x_] :=Module[{Intervals, BPoints, IntervalPlot,AllIntervalPlot,AllBPoints},Intervals = Transpose[{Sort[TS[[1]]],Sort[TS[[2]]]}];BPoints = Flatten[TS];IntervalPlot[g_, t_, {min_, max_}] :=Plot[g, {t, min, max}];AllIntervalPlot = Function[IntervalPlot[Fnct, x, #]] /@ Intervals;BPointsPlot[g_, t_, iso_] :=Graphics[{{AbsolutePointSize[5],Point[{iso, g /. t -> iso}]}}];AllBPoints = Function[BPointsPlot[Fnct, x, #]] /@ BPoints;Show[AllIntervalPlot, AllBPoints,PlotRange -> All,AspectRatio -> Automatic, Axes ->True, AxesOrigin -> {0, 0}]]

TSPlotcok[TS_, Fnct_, x_] :=Show[Function[TSPlottek[TS, #, x]] /@Fnct]

TSPlot[TS ,Fnct ,x ]

:= If[Head[Fnct] === List, TSPlotcok[TS,Fnct, x],TSPlottek[TS, Fnct, x]]

TSPlot1[TS_, Fnct_, x_] :=Module[{Intervals, BPoints, IntervalPlot,AllIntervalPlot,AllBPoints},Intervals = Transpose[{Sort[TS[[1]]],Sort[TS[[2]]]}];BPoints = Flatten[TS];IntervalPlot[g_, t_, {min_, max_}] :=Plot[g, {t, min, max}, PlotStyle ->{Blue, Thickness[0.01]}];AllIntervalPlot = Function[IntervalPlot[Fnct, x, #]] /@Intervals;BPointsPlot[g_, t_, iso_] :=Graphics[{{AbsolutePointSize[5], Blue,

Point[{iso, g /. t -> iso}]}}];AllBPoints = Function[BPointsPlot[Fnct,x, #]] /@ BPoints;Show[AllIntervalPlot, AllBPoints,PlotRange -> All,Axes -> {True, False}]

]; DrawTimeScale[TS_] := TSPlot1[TS, 0, x]

TSTangentLine[TS ,f ,x ,pnt ]

:= Module[{m, n, aa, aaa, aaaa},m = DeltaDerivative[TS, f, x, pnt];n = -m pnt + f /. x -> pnt;aa = Plot[m x + n, {x, Min[Flatten[TS]], Max[Flatten[TS]]},PlotStyle -> Red];

aaa = TSPlot[TS, f, x];aaaa =Graphics[{{AbsolutePointSize[5], Red,

Point[{pnt, f /. x -> pnt}]}}];Show[aa, aaa, aaaa, AspectRatio ->Automatic, PlotRange -> All,Axes -> True, AxesOrigin -> {0, 0}]]

TSAreaPlot[TS_, func_, x_, intrvll_] :=Module[{aaa, AllIntervalPlott,Intervalss, listed}, If[QTimeScale[TS] === True, If[checkelement[TS, intrvll[[1]]] +checkelement[TS, intrvll[[2]]] ==

2,flatt = Flatten[TS];Intervalss = Transpose[{TS[[1]],TS[[2]]}];

rect[p_] :=Graphics[{Orange,Rectangle[{p, 0}, {Sigma[TS, p],func /. x -> p}]}];

aaa = TSPlot[TS, func, x];listed = (rect[#1] &) /@ flatt;IntervalPlott[g_, t_, {min_, max_}]:=Plot[g, {t, min, max}, Filling ->0, FillingStyle -> Orange];AllIntervalPlott =Function[IntervalPlott[func, x, #]]/@ Intervalss;

Show[{aaa, listed, AllIntervalPlott},AspectRatio -> Automatic,PlotRange -> All, Axes -> True,AxesOrigin -> {0, 0}],

Print["At least one point given isnot element of the time scale \

given!"]], Print["Time scale given is notdefined properly!"]

]]

PlotTxT[T1_, T2_, Funct_, x_, y_] :=

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Module[{IntT1, DPointT1, IntT2,DPointT2, Gixi, Gpxp, Gixp, Gpxi},IntT1 = Transpose[{T1[[1]], T1[[2]]}];DPointT1 = T1[[3]];IntT2 = Transpose[{T2[[1]], T2[[2]]}];DPointT2 = T2[[3]];ixi[int1_, int2_, func1_, x1_, y1_] :=Plot3D[func1, {x1, int1[[1]],int1[[2]]}, {y1, int2[[1]],

int2[[2]]}];allixi[intst1_, intst2_, func_, x2_,y2_] :=Table[ixi[intst1[[i]], intst2[[j]],func, x2, y2], {i, 1,

Length[intst1]}, {j, 1, Length[intst2]}];

pxp[point1_, point2_, funct_, x3_,y3_] :=ListPointPlot3D[{{point1, point2,

funct /. {x3 -> point1, y3 ->point2}}}];

allpxp[pointst1_, pointst2_, func_,x4_, y4_] :=Table[pxp[pointst1[[i]],pointst2[[j]], func, x4, y4], {i, 1,

Length[pointst1]}, {j, 1, Length[pointst2]}];

ixp[int_, point_, funct_, x5_, y5_] :=ParametricPlot3D[{x5, point, funct /.y5 -> point}, {x5, int[[1]],

int[[2]]}];allixp[intst1_, pointst2_, func_, x6_,y6_] :=Table[ixp[intst1[[i]], pointst2[[j]],func, x6, y6], {i, 1,

Length[intst1]}, {j, 1, Length[pointst2]}];

pxi[point_, int_, funct_, x7_, y7_] :=ParametricPlot3D[{point, y7, funct /.x7 -> point}, {y7, int[[1]],

int[[2]]}];allpxi[pointst1_, intst2_, func_, x8_,y8_] :=Table[pxi[pointst1[[i]], intst2[[j]],func, x8, y8], {i, 1,

Length[pointst1]}, {j, 1, Length[intst2]}];

Gixi = allixi[IntT1, IntT2, Funct, x,y];Gpxp = allpxp[DPointT1, DPointT2,Funct, x, y];Gixp = allixp[IntT1, DPointT2, Funct,x, y];Gpxi = allpxi[DPointT1, IntT2, Funct,

x, y];Show[{Gixi, Gpxp, Gixp, Gpxi},PlotRange -> All,AxesLabel -> {TS1, TS2}]]

DrawTimeScale3D[T1 ,T2 ]

:= PlotTxT[T1,T2, 0, x, y]

PartialDeltaDerivative[T12 ,var12 ,pfunct ,var ,p0 ]

:= Module[{pfunctt},Which[var === var12[[1]],pfunctt = pfunct /. {var12[[2]] ->p0[[2]]};

DeltaDerivative[T12[[1]], pfunctt,var, p0[[1]]],var === var12[[2]], pfunctt = pfunct/. {var12[[1]] -> p0[[1]]};DeltaDerivative[T12[[2]], pfunctt,var, p0[[2]]]]]

DeltaTangentPlane[T1 ,T2 ,TPfunct ,TPvars ,TPpoint ]

:= Module[{},denklem = ((TPfunct /. {TPvars[[1]]-> TPpoint[[1]],

TPvars[[2]] -> TPpoint[[2]]}) +(PartialDeltaDerivative[{T1,T2}, TPvars, TPfunct,

TPvars[[1]], {TPpoint[[1]],TPpoint[[2]]}]) (x -TPpoint[[1]]) + (PartialDeltaDerivative[{T1, T2},TPvars,

TPfunct, TPvars[[2]], {TPpoint[[1]], TPpoint[[2]]}])(y - TPpoint[[2]]));

tngntpln =Plot3D[denklem, {x, TPpoint[[1]] - 1,TPpoint[[1]] + 1}, {y,

TPpoint[[2]] - 1, TPpoint[[2]] + 1},ColorFunction -> "RustTones"];

yuzey = PlotTxT[T1, T2, TPfunct,TPvars[[1]], TPvars[[2]]];Show[{yuzey, tngntpln}]]

REFERENCES

[1] M. Bohner and A. Peterson, Dynamic Equations on TimeScales: An Introduction with Applications, Boston:Birkhauser, 2001.

[2] M. Bohner and A. Peterson, Advances in Dynamic Equationson Time Scales, Boston: Birkhauser, 2003.

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[3] M. Bohner, G. Sh. Guseinov, Patrial Differentiation on TimeScales, Dynamic Systems and Applications 13, 351–379, 2004.

[4] M. Bohner, G. Sh. Guseinov, An Introduction to ComplexFunctions on Product of Two Time Scales, J. Difference Equ.Appl. 2005.

[5] S. Hilger, Analysis on measure chains-a unified approach tocontinuous and discrete calculus, Results Math. 35, 18–56,1990.

[6] U. Ufuktepe, S. Kapcak, Unification of Analysis with Mathe-matica, Proceeding of the 2009 International Conference onComputational and Mathematical Methods in Science andEngineering, V 3, 1053–1063, 2009.

[7] A. Yantir, U. Ufuktepe, Mathematica Applications on TimeScales for Calculus, Lecture Notes in Computer Science, 529–537, 3482, 2005.

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multivariable calculus on time scales

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