(* Content-type: application/mathematica *)

(*** Wolfram Notebook File ***)
(* http://www.wolfram.com/nb *)

(* CreatedBy='Mathematica 6.0' *)

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(* Beginning of Notebook Content *)
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    2, $CellContext`c$$ = 2 Pi, $CellContext`d$$ = 2 Pi, $CellContext`e$$ = 
    0, $CellContext`f$$ = 0, $CellContext`u$$ = 0, $CellContext`v$$ = 0, 
    Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, 
    Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 
    1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{
      Hold[
       Style["Torus Shape", 15, Bold]], Manipulate`Dump`ThisIsNotAControl}, {{
       Hold[$CellContext`a$$], 6, 
       "\!\(\*\nStyleBox[\"a\",\nFontColor->RGBColor[1, 0.5, 0]]\)"}, 0, 
      10}, {{
       Hold[$CellContext`b$$], 2, 
       "\!\(\*\nStyleBox[\"b\",\nFontColor->RGBColor[1, 0.5, 0]]\)"}, 0, 10}, {
      Hold[
       Style["Torus Cut", 15, Bold]], Manipulate`Dump`ThisIsNotAControl}, {{
       Hold[$CellContext`e$$], 0, ""}, 0, 2 Pi}, {{
       Hold[$CellContext`c$$], 2 Pi, ""}, 0, 2 Pi}, {{
       Hold[$CellContext`f$$], 0, ""}, 0, 2 Pi}, {{
       Hold[$CellContext`d$$], 2 Pi, ""}, 0, 2 Pi}, {
      Hold[
       Style["UV-Plane", 15, Bold]], Manipulate`Dump`ThisIsNotAControl}, {
      Hold[$CellContext`u$$], 0, 2 Pi}, {
      Hold[$CellContext`v$$], 0, 2 Pi}, {
      Hold[
       Dynamic[
        Show[
         Graphics[{Gray, 
           Rectangle[{0, 2 Pi}, {2 Pi, 0}]}], 
         Graphics[{
           RGBColor[1, 0, 0], 
           PointSize[0.04], 
           Point[{$CellContext`u$$, $CellContext`v$$}]}], 
         VectorFieldPlots`ListVectorFieldPlot[{{{$CellContext`u$$, \
$CellContext`v$$}, {0, 1}}}, ColorFunction -> (RGBColor[0, 0, 1]& )], 
         VectorFieldPlots`ListVectorFieldPlot[{{{$CellContext`u$$, \
$CellContext`v$$}, {1, 0}}}, ColorFunction -> (RGBColor[0, 1, 0]& )], Axes -> 
         True, AxesLabel -> {$CellContext`U, $CellContext`V}, 
         PlotRange -> {{0, 2 Pi + 1}, {0, 2 Pi + 1}}, 
         ImageSize -> {300, 350}]]], Manipulate`Dump`ThisIsNotAControl}}, 
    Typeset`size$$ = {702., {372., 377.}}, Typeset`update$$ = 0, 
    Typeset`initDone$$, Typeset`skipInitDone$$ = 
    False, $CellContext`a$1479$$ = 0, $CellContext`b$1480$$ = 
    0, $CellContext`e$1481$$ = 0, $CellContext`c$1482$$ = 
    0, $CellContext`f$1483$$ = 0, $CellContext`d$1484$$ = 
    0, $CellContext`u$1485$$ = 0, $CellContext`v$1486$$ = 0}, 
    DynamicBox[Manipulate`ManipulateBoxes[
     1, StandardForm, 
      "Variables" :> {$CellContext`a$$ = 6, $CellContext`b$$ = 
        2, $CellContext`c$$ = 2 Pi, $CellContext`d$$ = 2 Pi, $CellContext`e$$ = 
        0, $CellContext`f$$ = 0, $CellContext`u$$ = 0, $CellContext`v$$ = 0}, 
      "ControllerVariables" :> {
        Hold[$CellContext`a$$, $CellContext`a$1479$$, 0], 
        Hold[$CellContext`b$$, $CellContext`b$1480$$, 0], 
        Hold[$CellContext`e$$, $CellContext`e$1481$$, 0], 
        Hold[$CellContext`c$$, $CellContext`c$1482$$, 0], 
        Hold[$CellContext`f$$, $CellContext`f$1483$$, 0], 
        Hold[$CellContext`d$$, $CellContext`d$1484$$, 0], 
        Hold[$CellContext`u$$, $CellContext`u$1485$$, 0], 
        Hold[$CellContext`v$$, $CellContext`v$1486$$, 0]}, 
      "OtherVariables" :> {
       Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, 
        Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, 
        Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$,
         Typeset`skipInitDone$$}, 
      "Body" :> (Needs["VectorFieldPlots`"]; Column[{
          Labeled[
           Framed[
            Style[
            "r[u,v]={(\!\(\*\nStyleBox[\"a\",\nFontColor->RGBColor[1, 0.5, \
0]]\)+\!\(\*\nStyleBox[\"b\",\nFontColor->RGBColor[1, 0.5, \
0]]\)Cos[v])Cos[u],(\!\(\*\nStyleBox[\"a\",\nFontColor->RGBColor[1, 0.5, \
0]]\)+\!\(\*\nStyleBox[\"b\",\nFontColor->RGBColor[1, 0.5, \
0]]\)Cos[v])Sin[u],\!\(\*\nStyleBox[\"b\",\nFontColor->RGBColor[1, 0.5, \
0]]\)Sin[v]}", 20]], 
           Style["Torus", 20, Bold], Left], 
          Framed[
           Style[
           "\!\(\*\nStyleBox[SubscriptBox[\"r\", \"u\"],\n\
FontColor->RGBColor[0, 1, 0]]\)={-(a+bCos[v])Sin[u],Cos[u](a+bCos[v]),0}", 
            20]], 
          Framed[
           Style[
           "\!\(\*\nStyleBox[SubscriptBox[\"r\", \"v\"],\n\
FontColor->RGBColor[0, 0, 1]]\)= {-bCos[u]Sin[v],-bSin[u]Sin[v],bCos[v]}", 
            20]], 
          Labeled[
           Framed[
            Style[
            "\!\(\*OverscriptBox[\(n\), \
\(\[Rule]\)]\)={Cos[u]Cos[v],Cos[v]Sin[u],Sin[v]}", 20]], 
           Style[
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0]]\)\!\(\*\nStyleBox[\" \",\nFontColor->RGBColor[1, 0, 0]]\)\!\(\*\n\
StyleBox[\"Unit\",\nFontColor->RGBColor[1, 0, 0]]\)\!\(\*\nStyleBox[\" \",\n\
FontColor->RGBColor[1, 0, 0]]\)\!\(\*\nStyleBox[\"Normal\",\n\
FontColor->RGBColor[1, 0, 0]]\)", 20, Bold], Left], 
          "\!\(\*\nStyleBox[\"\[Copyright]\",\nFontSize->14]\)\!\(\*\n\
StyleBox[\" \",\nFontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\"N\",\n\
FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\".\",\n\
FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\" \",\n\
FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\"Bykov\",\n\
FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\",\",\n\
FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\" \",\n\
FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\"SJ\",\n\
FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\" \",\n\
FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\"Delta\",\n\
FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\" \",\n\
FontColor->RGBColor[0, 0, 1]]\)\!\(\*\nStyleBox[\"College\",\n\
FontColor->RGBColor[0, 0, 1]]\)", $CellContext`Torus = 
           ParametricPlot3D[{($CellContext`a$$ + $CellContext`b$$ 
                Cos[$CellContext`v$$]) 
              Cos[$CellContext`u$$], ($CellContext`a$$ + $CellContext`b$$ 
                Cos[$CellContext`v$$]) 
              Sin[$CellContext`u$$], $CellContext`b$$ 
              Sin[$CellContext`v$$]}, {$CellContext`u$$, $CellContext`e$$, \
$CellContext`c$$}, {$CellContext`v$$, $CellContext`f$$, $CellContext`d$$}, 
             PlotRange -> {{-($CellContext`a$$ + $CellContext`b$$) - 
                 1, $CellContext`a$$ + $CellContext`b$$} + 
               1, {-($CellContext`a$$ + $CellContext`b$$) - 
                1, $CellContext`a$$ + $CellContext`b$$ + 
                1}, {-$CellContext`b$$ - 1, $CellContext`b$$ + 1}}]; 
          Show[$CellContext`Torus, 
            Graphics3D[{
              RGBColor[1, 0, 0], 
              PointSize[0.02], 
              
              Point[{($CellContext`a$$ + $CellContext`b$$ 
                  Cos[$CellContext`v$$]) 
                Cos[$CellContext`u$$], ($CellContext`a$$ + $CellContext`b$$ 
                  Cos[$CellContext`v$$]) 
                Sin[$CellContext`u$$], $CellContext`b$$ 
                Sin[$CellContext`v$$]}]}], 
            
            VectorFieldPlots`ListVectorFieldPlot3D[{{{($CellContext`a$$ + \
$CellContext`b$$ Cos[$CellContext`v$$]) 
                Cos[$CellContext`u$$], ($CellContext`a$$ + $CellContext`b$$ 
                  Cos[$CellContext`v$$]) 
                Sin[$CellContext`u$$], $CellContext`b$$ 
                Sin[$CellContext`v$$]}, {(-($CellContext`a$$ + \
$CellContext`b$$ Cos[$CellContext`v$$])) Sin[$CellContext`u$$], 
                Cos[$CellContext`u$$] ($CellContext`a$$ + $CellContext`b$$ 
                  Cos[$CellContext`v$$]), 0}}}, 
             ColorFunction -> (RGBColor[0, 1, 0]& ), 
             VectorFieldPlots`VectorHeads -> True], 
            
            VectorFieldPlots`ListVectorFieldPlot3D[{{{($CellContext`a$$ + \
$CellContext`b$$ Cos[$CellContext`v$$]) 
                Cos[$CellContext`u$$], ($CellContext`a$$ + $CellContext`b$$ 
                  Cos[$CellContext`v$$]) 
                Sin[$CellContext`u$$], $CellContext`b$$ 
                Sin[$CellContext`v$$]}, {((-$CellContext`b$$) 
                 Cos[$CellContext`u$$]) 
                Sin[$CellContext`v$$], ((-$CellContext`b$$) 
                 Sin[$CellContext`u$$]) 
                Sin[$CellContext`v$$], $CellContext`b$$ 
                Cos[$CellContext`v$$]}}}, 
             ColorFunction -> (RGBColor[0, 0, 1]& ), 
             VectorFieldPlots`VectorHeads -> True], 
            
            VectorFieldPlots`ListVectorFieldPlot3D[{{{($CellContext`a$$ + \
$CellContext`b$$ Cos[$CellContext`v$$]) 
                Cos[$CellContext`u$$], ($CellContext`a$$ + $CellContext`b$$ 
                  Cos[$CellContext`v$$]) 
                Sin[$CellContext`u$$], $CellContext`b$$ 
                Sin[$CellContext`v$$]}, {(Cos[$CellContext`u$$] 
                 Cos[$CellContext`v$$]) 
                Sign[$CellContext`b$$ ($CellContext`a$$ + $CellContext`b$$ 
                    Cos[$CellContext`v$$])], (Cos[$CellContext`v$$] 
                 Sign[$CellContext`b$$ ($CellContext`a$$ + $CellContext`b$$ 
                    Cos[$CellContext`v$$])]) Sin[$CellContext`u$$], 
                Sign[$CellContext`b$$ ($CellContext`a$$ + $CellContext`b$$ 
                    Cos[$CellContext`v$$])] Sin[$CellContext`v$$]}}}, 
             ColorFunction -> (RGBColor[1, 0, 0]& ), 
             VectorFieldPlots`VectorHeads -> True], 
            AxesLabel -> {$CellContext`X, $CellContext`Y, $CellContext`Z}, 
            ImageSize -> {600, 600}, ViewPoint -> {1.3, 2.4, 2.}]}, Center]), 
      "Specifications" :> {
        Style[
        "Torus Shape", 15, 
         Bold], {{$CellContext`a$$, 6, 
          "\!\(\*\nStyleBox[\"a\",\nFontColor->RGBColor[1, 0.5, 0]]\)"}, 0, 
         10}, {{$CellContext`b$$, 2, 
          "\!\(\*\nStyleBox[\"b\",\nFontColor->RGBColor[1, 0.5, 0]]\)"}, 0, 
         10}, Delimiter, 
        Style[
        "Torus Cut", 15, Bold], {{$CellContext`e$$, 0, ""}, 0, 2 
         Pi}, {{$CellContext`c$$, 2 Pi, ""}, 0, 2 Pi}, 
        Delimiter, {{$CellContext`f$$, 0, ""}, 0, 2 
         Pi}, {{$CellContext`d$$, 2 Pi, ""}, 0, 2 Pi}, Delimiter, 
        Style[
        "UV-Plane", 15, Bold], {$CellContext`u$$, 0, 2 Pi}, {$CellContext`v$$,
          0, 2 Pi}, 
        Dynamic[
         Show[
          Graphics[{Gray, 
            Rectangle[{0, 2 Pi}, {2 Pi, 0}]}], 
          Graphics[{
            RGBColor[1, 0, 0], 
            PointSize[0.04], 
            Point[{$CellContext`u$$, $CellContext`v$$}]}], 
          
          VectorFieldPlots`ListVectorFieldPlot[{{{$CellContext`u$$, \
$CellContext`v$$}, {0, 1}}}, ColorFunction -> (RGBColor[0, 0, 1]& )], 
          
          VectorFieldPlots`ListVectorFieldPlot[{{{$CellContext`u$$, \
$CellContext`v$$}, {1, 0}}}, ColorFunction -> (RGBColor[0, 1, 0]& )], Axes -> 
          True, AxesLabel -> {$CellContext`U, $CellContext`V}, 
          PlotRange -> {{0, 2 Pi + 1}, {0, 2 Pi + 1}}, 
          ImageSize -> {300, 350}]]}, "Options" :> {ControlPlacement -> Left},
       "DefaultOptions" :> {}],
     ImageSizeCache->{1049., {399., 404.}},
     SingleEvaluation->True],
    Deinitialization:>None,
    DynamicModuleValues:>{},
    Initialization:>({$CellContext`Torus = Graphics3D[
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           "]], {
         Axes -> True, PlotRange -> {{-8, 9}, {-9, 9}, {-3, 3}}, 
          PlotRangePadding -> {Automatic, Automatic, Automatic}}], 
       VectorFieldPlots`ListVectorFieldPlot3D[
         Pattern[VectorFieldPlots`Private`vects, {
           Repeated[{
             PatternTest[
              Blank[], VectorQ], 
             PatternTest[
              Blank[], VectorQ]}]}], 
         Pattern[VectorFieldPlots`Private`opts, 
          OptionsPattern[]]] := Condition[
         Module[{
          VectorFieldPlots`Private`maxsize, VectorFieldPlots`Private`scale, 
           VectorFieldPlots`Private`scalefunct, 
           VectorFieldPlots`Private`colorfunct, 
           VectorFieldPlots`Private`heads, VectorFieldPlots`Private`points, 
           VectorFieldPlots`Private`vectors, VectorFieldPlots`Private`mags, 
           VectorFieldPlots`Private`colors, 
           VectorFieldPlots`Private`scaledmag, 
           VectorFieldPlots`Private`allvecs, VectorFieldPlots`Private`vecs = 
           N[VectorFieldPlots`Private`vects]}, {
            VectorFieldPlots`Private`maxsize, VectorFieldPlots`Private`scale, 
             VectorFieldPlots`Private`scalefunct, 
             VectorFieldPlots`Private`colorfunct, 
             VectorFieldPlots`Private`heads} = 
           OptionValue[{
             VectorFieldPlots`MaxArrowLength, VectorFieldPlots`ScaleFactor, 
              VectorFieldPlots`ScaleFunction, ColorFunction, 
              VectorFieldPlots`VectorHeads}]; If[
            And[
             Not[
              NumberQ[VectorFieldPlots`Private`maxsize]], 
             VectorFieldPlots`Private`maxsize != None], 
            VectorFieldPlots`Private`maxsize = None, 
            VectorFieldPlots`Private`maxsize = 
            N[VectorFieldPlots`Private`maxsize]]; If[
            And[
             Not[
              NumberQ[VectorFieldPlots`Private`scale]], 
             VectorFieldPlots`Private`scale =!= Automatic], 
            VectorFieldPlots`Private`scale = Automatic, 
            VectorFieldPlots`Private`scale = 
            N[VectorFieldPlots`Private`scale]]; 
          VectorFieldPlots`Private`heads = 
           TrueQ[VectorFieldPlots`Private`heads]; 
          VectorFieldPlots`Private`vecs = 
           Cases[VectorFieldPlots`Private`vecs, {
              Blank[], 
              PatternTest[
               Blank[], VectorQ[#, NumberQ]& ]}]; {
            VectorFieldPlots`Private`points, 
             VectorFieldPlots`Private`vectors} = 
           Transpose[VectorFieldPlots`Private`vecs]; 
          VectorFieldPlots`Private`mags = 
           Map[VectorFieldPlots`Private`mag, 
             VectorFieldPlots`Private`vectors]; 
          If[VectorFieldPlots`Private`colorfunct == None, 
            VectorFieldPlots`Private`colorfunct = {}& ]; 
          If[Max[VectorFieldPlots`Private`mags - Min[
              VectorFieldPlots`Private`mags]] == 0, 
            VectorFieldPlots`Private`colors = 
            Map[VectorFieldPlots`Private`colorfunct, 
              Table[0, {
                Length[VectorFieldPlots`Private`mags]}]], 
            VectorFieldPlots`Private`colors = 
            Map[VectorFieldPlots`Private`colorfunct, (
               VectorFieldPlots`Private`mags - Min[
               VectorFieldPlots`Private`mags])/Max[
              VectorFieldPlots`Private`mags - Min[
               VectorFieldPlots`Private`mags]]]]; 
          If[VectorFieldPlots`Private`scalefunct =!= None, 
            VectorFieldPlots`Private`scaledmag = Map[If[# == 0, 0, 
                VectorFieldPlots`Private`scalefunct[#]]& , 
               VectorFieldPlots`Private`mags]; 
            VectorFieldPlots`Private`vectors = 
             MapThread[
              If[#2 == 0, {0, 0, 0}, (# #2)/#3]& , {
               VectorFieldPlots`Private`vectors, 
                VectorFieldPlots`Private`scaledmag, 
                VectorFieldPlots`Private`mags}]; 
            VectorFieldPlots`Private`mags = 
             VectorFieldPlots`Private`scaledmag]; 
          VectorFieldPlots`Private`allvecs = 
           Transpose[{
             VectorFieldPlots`Private`colors, VectorFieldPlots`Private`points,
               VectorFieldPlots`Private`vectors, 
              VectorFieldPlots`Private`mags}]; 
          If[VectorFieldPlots`Private`maxsize =!= None, 
            VectorFieldPlots`Private`allvecs = 
            Select[VectorFieldPlots`Private`allvecs, Part[#, 4] <= 
              VectorFieldPlots`Private`maxsize& ]]; 
          If[Max[VectorFieldPlots`Private`mags] != 0, 
            VectorFieldPlots`Private`scale = 
             VectorFieldPlots`Private`automatic[
               VectorFieldPlots`Private`scale, 
                Max[VectorFieldPlots`Private`mags]]/Max[
              VectorFieldPlots`Private`mags]; 
            VectorFieldPlots`Private`allvecs = Map[{
                Part[#, 1], 
                Part[#, 2], VectorFieldPlots`Private`scale Part[#, 3]}& , 
               VectorFieldPlots`Private`allvecs]]; Show[
            Graphics3D[
             Flatten[
              Apply[{#, 
                
                VectorFieldPlots`Private`vector3D[#2, #3, 
                 VectorFieldPlots`Private`heads]}& , 
               VectorFieldPlots`Private`allvecs, {1}]], 
             FilterRules[
              Flatten[{VectorFieldPlots`Private`opts}], 
              Options[Graphics3D]]]]], Last[
           Dimensions[VectorFieldPlots`Private`vects]] === 3], 
       Options[VectorFieldPlots`ListVectorFieldPlot3D] = {
        AlignmentPoint -> Center, AspectRatio -> Automatic, Axes -> False, 
         AxesEdge -> Automatic, AxesLabel -> None, AxesStyle -> {}, 
         Background -> None, BaselinePosition -> Automatic, BaseStyle -> {}, 
         Boxed -> True, BoxRatios -> Automatic, BoxStyle -> {}, ColorFunction -> 
         None, ColorOutput -> Automatic, ContentSelectable -> Automatic, 
         ControllerLinking -> Automatic, ControllerMethod -> Automatic, 
         ControllerPath -> Automatic, DisplayFunction :> $DisplayFunction, 
         Epilog -> {}, FaceGrids -> None, FaceGridsStyle -> {}, FormatType :> 
         TraditionalForm, ImageMargins -> 0., ImagePadding -> All, ImageSize -> 
         Automatic, LabelStyle -> {}, Lighting -> Automatic, 
         VectorFieldPlots`MaxArrowLength -> None, Method -> Automatic, 
         PlotLabel -> None, PlotRange -> All, PlotRangePadding -> Automatic, 
         PlotRegion -> Automatic, PreserveImageOptions -> Automatic, 
         Prolog -> {}, RotationAction -> "Fit", VectorFieldPlots`ScaleFactor -> 
         Automatic, VectorFieldPlots`ScaleFunction -> None, SphericalRegion -> 
         False, Ticks -> Automatic, TicksStyle -> {}, 
         VectorFieldPlots`VectorHeads -> False, ViewAngle -> Automatic, 
         ViewCenter -> {1/2, 1/2, 1/2}, ViewMatrix -> Automatic, 
         ViewPoint -> {1.3, -2.4, 2.}, ViewRange -> All, ViewVector -> 
         Automatic, ViewVertical -> {0, 0, 1}}, 
       TagSet[VectorFieldPlots`ListVectorFieldPlot3D, 
        MessageName[VectorFieldPlots`ListVectorFieldPlot3D, "usage"], 
        "\!\(\*RowBox[{\"ListVectorFieldPlot3D\", \"[\", RowBox[{\"{\", \
RowBox[{RowBox[{\"{\", RowBox[{SubscriptBox[StyleBox[\"pt\", \"TI\"], \
StyleBox[\"1\", \"TR\"]], \",\", SubscriptBox[StyleBox[\"vec\", \"TI\"], \
StyleBox[\"1\", \"TR\"]]}], \"}\"}], \",\", RowBox[{\"{\", \
RowBox[{SubscriptBox[StyleBox[\"pt\", \"TI\"], StyleBox[\"2\", \"TR\"]], \
\",\", SubscriptBox[StyleBox[\"vec\", \"TI\"], StyleBox[\"2\", \"TR\"]]}], \
\"}\"}], \",\", StyleBox[\"\[Ellipsis]\", \"TR\"]}], \"}\"}], \"]\"}]\) \
generates a three-dimensional plot of the list of vectors \
\!\(\*RowBox[{SubscriptBox[StyleBox[\"vec\", \"TI\"], StyleBox[\"1\", \
\"TR\"]], \",\", \" \", SubscriptBox[StyleBox[\"vec\", \"TI\"], \
StyleBox[\"2\", \"TR\"]], \",\", \" \", StyleBox[\"\[Ellipsis]\", \
\"TR\"]}]\), with each vector based at a corresponding point \
\!\(\*RowBox[{SubscriptBox[StyleBox[\"pt\", \"TI\"], StyleBox[\"1\", \
\"TR\"]], \",\", \" \", SubscriptBox[StyleBox[\"pt\", \"TI\"], \
StyleBox[\"2\", \"TR\"]], \",\", \" \", StyleBox[\"\[Ellipsis]\", \
\"TR\"]}]\)."], 
       TagSet[VectorFieldPlots`MaxArrowLength, 
        MessageName[VectorFieldPlots`MaxArrowLength, "usage"], 
        "MaxArrowLength is an option for the vector field visualization \
functions that determines the longest vector to be drawn."], 
       TagSet[VectorFieldPlots`ScaleFactor, 
        MessageName[VectorFieldPlots`ScaleFactor, "usage"], 
        "ScaleFactor is an option for the vector field visualization \
functions that scales the vectors so that the longest vector displayed is of \
the length specified."], 
       TagSet[VectorFieldPlots`ScaleFunction, 
        MessageName[VectorFieldPlots`ScaleFunction, "usage"], 
        "ScaleFunction is an option for the vector field visualization \
functions that rescales each vector to a length determined by applying a pure \
function to the current length of that vector."], 
       TagSet[VectorFieldPlots`VectorHeads, 
        MessageName[VectorFieldPlots`VectorHeads, "usage"], 
        "VectorHeads is an option for the three-dimensional vector field \
functions, including VectorFieldPlot3D, that determines whether the vectors \
will be displayed with heads. "], VectorFieldPlots`Private`mag[
         Pattern[VectorFieldPlots`Private`a, 
          Blank[]]] := Sqrt[
         Apply[Plus, VectorFieldPlots`Private`a^2]], 
       VectorFieldPlots`Private`automatic[
         Pattern[VectorFieldPlots`Private`x, 
          Blank[]], 
         Pattern[VectorFieldPlots`Private`value, 
          Blank[]]] := 
       If[VectorFieldPlots`Private`x === Automatic, 
         VectorFieldPlots`Private`value, VectorFieldPlots`Private`x], 
       VectorFieldPlots`Private`vector3D[
         Pattern[VectorFieldPlots`Private`point, {
           Pattern[VectorFieldPlots`Private`x, 
            Blank[]], 
           Pattern[VectorFieldPlots`Private`y, 
            Blank[]], 
           Pattern[VectorFieldPlots`Private`z, 
            Blank[]]}], 
         Pattern[VectorFieldPlots`Private`grad, {
           Pattern[VectorFieldPlots`Private`dx, 
            Blank[]], 
           Pattern[VectorFieldPlots`Private`dy, 
            Blank[]], 
           Pattern[VectorFieldPlots`Private`dz, 
            Blank[]]}], False] := 
       Line[{VectorFieldPlots`Private`point, VectorFieldPlots`Private`point + 
          VectorFieldPlots`Private`grad}], VectorFieldPlots`Private`vector3D[
         Pattern[VectorFieldPlots`Private`point, {
           Pattern[VectorFieldPlots`Private`x, 
            Blank[]], 
           Pattern[VectorFieldPlots`Private`y, 
            Blank[]], 
           Pattern[VectorFieldPlots`Private`z, 
            Blank[]]}], 
         Pattern[VectorFieldPlots`Private`grad, {
           Pattern[VectorFieldPlots`Private`dx, 
            Blank[]], 
           Pattern[VectorFieldPlots`Private`dy, 
            Blank[]], 
           Pattern[VectorFieldPlots`Private`dz, 
            Blank[]]}], True] := Condition[
         Point[{
          VectorFieldPlots`Private`x, VectorFieldPlots`Private`y, 
           VectorFieldPlots`Private`z}], 
         VectorFieldPlots`Private`grad == {0, 0, 0}], 
       VectorFieldPlots`Private`vector3D[
         Pattern[VectorFieldPlots`Private`point, {
           Pattern[VectorFieldPlots`Private`x, 
            Blank[]], 
           Pattern[VectorFieldPlots`Private`y, 
            Blank[]], 
           Pattern[VectorFieldPlots`Private`z, 
            Blank[]]}], 
         Pattern[VectorFieldPlots`Private`grad, {
           Pattern[VectorFieldPlots`Private`dx, 
            Blank[]], 
           Pattern[VectorFieldPlots`Private`dy, 
            Blank[]], 
           Pattern[VectorFieldPlots`Private`dz, 
            Blank[]]}], True] := 
       Module[{VectorFieldPlots`Private`endpoint, 
          VectorFieldPlots`Private`perp, VectorFieldPlots`Private`perpm, 
          VectorFieldPlots`Private`offsetPoint, 
          VectorFieldPlots`Private`arrowA, VectorFieldPlots`Private`arrowB, 
          VectorFieldPlots`Private`arrowC, VectorFieldPlots`Private`arrowD}, 
         VectorFieldPlots`Private`endpoint = 
          VectorFieldPlots`Private`point + VectorFieldPlots`Private`grad; 
         VectorFieldPlots`Private`perp = 
          VectorFieldPlots`Private`cross3[
           VectorFieldPlots`Private`grad, {0, 0, 1}]; 
         VectorFieldPlots`Private`perpm = 
          VectorFieldPlots`Private`mag[VectorFieldPlots`Private`perp]; 
         If[VectorFieldPlots`Private`perpm == 0, 
           VectorFieldPlots`Private`perp = 
            VectorFieldPlots`Private`cross3[
             VectorFieldPlots`Private`grad, {0, 1, 0}]; 
           VectorFieldPlots`Private`perpm = 
            VectorFieldPlots`Private`mag[VectorFieldPlots`Private`perp]]; 
         VectorFieldPlots`Private`perp = (VectorFieldPlots`Private`perp 
            VectorFieldPlots`Private`mag[VectorFieldPlots`Private`grad])/(7 
           VectorFieldPlots`Private`perpm); 
         VectorFieldPlots`Private`offsetPoint = 
          VectorFieldPlots`Private`point + (4 VectorFieldPlots`Private`grad)/
            5; VectorFieldPlots`Private`arrowA = 
          VectorFieldPlots`Private`offsetPoint + 
           VectorFieldPlots`Private`perp; 
         VectorFieldPlots`Private`perp = 
          VectorFieldPlots`Private`cross3[
           VectorFieldPlots`Private`grad, VectorFieldPlots`Private`perp]; 
         VectorFieldPlots`Private`perp = (VectorFieldPlots`Private`perp 
            VectorFieldPlots`Private`mag[VectorFieldPlots`Private`grad])/(7 
           VectorFieldPlots`Private`mag[VectorFieldPlots`Private`perp]); 
         VectorFieldPlots`Private`arrowB = 
          VectorFieldPlots`Private`offsetPoint + 
           VectorFieldPlots`Private`perp; 
         VectorFieldPlots`Private`perp = 
          VectorFieldPlots`Private`cross3[
           VectorFieldPlots`Private`grad, VectorFieldPlots`Private`perp]; 
         VectorFieldPlots`Private`perp = (VectorFieldPlots`Private`perp 
            VectorFieldPlots`Private`mag[VectorFieldPlots`Private`grad])/(7 
           VectorFieldPlots`Private`mag[VectorFieldPlots`Private`perp]); 
         VectorFieldPlots`Private`arrowC = 
          VectorFieldPlots`Private`offsetPoint + 
           VectorFieldPlots`Private`perp; 
         VectorFieldPlots`Private`perp = 
          VectorFieldPlots`Private`cross3[
           VectorFieldPlots`Private`grad, VectorFieldPlots`Private`perp]; 
         VectorFieldPlots`Private`perp = (VectorFieldPlots`Private`perp 
            VectorFieldPlots`Private`mag[VectorFieldPlots`Private`grad])/(7 
           VectorFieldPlots`Private`mag[VectorFieldPlots`Private`perp]); 
         VectorFieldPlots`Private`arrowD = 
          VectorFieldPlots`Private`offsetPoint + 
           VectorFieldPlots`Private`perp; {
           
           Line[{VectorFieldPlots`Private`point, 
             VectorFieldPlots`Private`endpoint}], 
           
           Line[{VectorFieldPlots`Private`arrowA, 
             VectorFieldPlots`Private`endpoint, 
             VectorFieldPlots`Private`arrowC}], 
           
           Line[{VectorFieldPlots`Private`arrowB, 
             VectorFieldPlots`Private`endpoint, 
             VectorFieldPlots`Private`arrowD}], 
           
           Line[{VectorFieldPlots`Private`arrowA, 
             VectorFieldPlots`Private`arrowB, VectorFieldPlots`Private`arrowC,
              VectorFieldPlots`Private`arrowD, 
             VectorFieldPlots`Private`arrowA}]}], 
       VectorFieldPlots`Private`cross3[{
          Pattern[VectorFieldPlots`Private`a1, 
           Blank[]], 
          Pattern[VectorFieldPlots`Private`a2, 
           Blank[]], 
          Pattern[VectorFieldPlots`Private`a3, 
           Blank[]]}, {
          Pattern[VectorFieldPlots`Private`b1, 
           Blank[]], 
          Pattern[VectorFieldPlots`Private`b2, 
           Blank[]], 
          Pattern[VectorFieldPlots`Private`b3, 
           Blank[]]}] := {-(VectorFieldPlots`Private`a3 
           VectorFieldPlots`Private`b2) + 
         VectorFieldPlots`Private`a2 VectorFieldPlots`Private`b3, 
         VectorFieldPlots`Private`a3 VectorFieldPlots`Private`b1 - 
         VectorFieldPlots`Private`a1 
         VectorFieldPlots`Private`b3, -(VectorFieldPlots`Private`a2 
           VectorFieldPlots`Private`b1) + 
         VectorFieldPlots`Private`a1 
          VectorFieldPlots`Private`b2}, $DisplayFunction = Identity, 
       VectorFieldPlots`ListVectorFieldPlot[
         Pattern[VectorFieldPlots`Private`vects, {
           Repeated[{{
              Blank[], 
              Blank[]}, {
              Blank[], 
              Blank[]}}]}], 
         PatternTest[
          Pattern[VectorFieldPlots`Private`opts, 
           BlankNullSequence[]], OptionQ]] := 
       Module[{VectorFieldPlots`Private`maxsize, 
          VectorFieldPlots`Private`scale, VectorFieldPlots`Private`scalefunct,
           VectorFieldPlots`Private`colorfunct, 
          VectorFieldPlots`Private`points, VectorFieldPlots`Private`vectors, 
          VectorFieldPlots`Private`colors, VectorFieldPlots`Private`mags, 
          VectorFieldPlots`Private`scaledmag, 
          VectorFieldPlots`Private`allvecs, VectorFieldPlots`Private`vecs = 
          N[VectorFieldPlots`Private`vects], 
          VectorFieldPlots`Private`arropts}, {
           VectorFieldPlots`Private`maxsize, VectorFieldPlots`Private`scale, 
            VectorFieldPlots`Private`scalefunct, 
            VectorFieldPlots`Private`colorfunct} = 
          ReplaceAll[{
            VectorFieldPlots`MaxArrowLength, VectorFieldPlots`ScaleFactor, 
             VectorFieldPlots`ScaleFunction, ColorFunction}, 
            Flatten[{VectorFieldPlots`Private`opts, 
              Options[VectorFieldPlots`ListVectorFieldPlot]}]]; 
         VectorFieldPlots`Private`vecs = 
          Cases[VectorFieldPlots`Private`vecs, {{
              PatternTest[
               Blank[], VectorFieldPlots`Private`numberQ], 
              PatternTest[
               Blank[], VectorFieldPlots`Private`numberQ]}, {
              PatternTest[
               Blank[], VectorFieldPlots`Private`numberQ], 
              PatternTest[
               Blank[], VectorFieldPlots`Private`numberQ]}}, Infinity]; {
           VectorFieldPlots`Private`points, VectorFieldPlots`Private`vectors} = 
          Transpose[VectorFieldPlots`Private`vecs]; 
         VectorFieldPlots`Private`mags = 
          Map[
           VectorFieldPlots`Private`magnitude, 
            VectorFieldPlots`Private`vectors]; 
         If[VectorFieldPlots`Private`colorfunct == None, 
           VectorFieldPlots`Private`colorfunct = {}& ]; If[
           Apply[Equal, VectorFieldPlots`Private`mags], 
           VectorFieldPlots`Private`colors = Table[
             Evaluate[
              VectorFieldPlots`Private`colorfunct[0]], {
              Length[VectorFieldPlots`Private`mags]}], 
           VectorFieldPlots`Private`colors = 
           Map[VectorFieldPlots`Private`colorfunct, (
              VectorFieldPlots`Private`mags - Min[
              VectorFieldPlots`Private`mags])/Max[
             VectorFieldPlots`Private`mags - Min[
              VectorFieldPlots`Private`mags]]]]; 
         If[VectorFieldPlots`Private`scalefunct =!= None, 
           VectorFieldPlots`Private`scaledmag = Map[If[# == 0, 0, 
               VectorFieldPlots`Private`scalefunct[#]]& , 
              VectorFieldPlots`Private`mags]; {
             VectorFieldPlots`Private`vectors, VectorFieldPlots`Private`mags} = 
            Transpose[
              MapThread[If[
                Or[#3 == 0, 
                 Not[
                  VectorFieldPlots`Private`numberQ[#2]]], {{0, 0}, 
                 0}, {(# #2)/#3, #2}]& , {
               VectorFieldPlots`Private`vectors, 
                VectorFieldPlots`Private`scaledmag, 
                VectorFieldPlots`Private`mags}]]]; 
         VectorFieldPlots`Private`allvecs = 
          Transpose[{
            VectorFieldPlots`Private`colors, VectorFieldPlots`Private`points, 
             VectorFieldPlots`Private`vectors, 
             VectorFieldPlots`Private`mags}]; If[
           VectorFieldPlots`Private`numberQ[VectorFieldPlots`Private`maxsize],
            VectorFieldPlots`Private`allvecs = 
           Select[VectorFieldPlots`Private`allvecs, Part[#, 4] <= 
             N[VectorFieldPlots`Private`maxsize]& ]]; If[
           VectorFieldPlots`Private`numberQ[VectorFieldPlots`Private`scale], 
           VectorFieldPlots`Private`scale = 
           VectorFieldPlots`Private`scale/Max[VectorFieldPlots`Private`mags], 
           VectorFieldPlots`Private`scale = 1]; 
         VectorFieldPlots`Private`arropts = 
          VectorFieldPlots`Private`getoldarrowopts[
            Flatten[{VectorFieldPlots`Private`opts, 
              Options[VectorFieldPlots`ListVectorFieldPlot]}]]; 
         If[VectorFieldPlots`Private`arropts =!= {}, Needs["Graphics`Arrow`"]; 
           VectorFieldPlots`Private`allvecs = Apply[Flatten[{#, 
                
                Arrow[#2, #2 + VectorFieldPlots`Private`scale #3, 
                 VectorFieldPlots`Private`arropts, Graphics`Arrow`HeadScaling -> 
                 Automatic, Graphics`Arrow`HeadLength -> 0.02]}]& , 
              VectorFieldPlots`Private`allvecs, {1}], 
           VectorFieldPlots`Private`allvecs = Apply[Flatten[{#, 
               If[VectorFieldPlots`Private`scale #3 == {0., 0.}, 
                Point[#2], 
                Arrow[{#2, #2 + VectorFieldPlots`Private`scale #3}]]}]& , 
             VectorFieldPlots`Private`allvecs, {1}]]; Graphics[{
            Thickness[Small], 
            Arrowheads[0.02], VectorFieldPlots`Private`allvecs}, 
           FilterRules[
            Flatten[{VectorFieldPlots`Private`opts, 
              Options[VectorFieldPlots`ListVectorFieldPlot]}], 
            Options[Graphics]]]], VectorFieldPlots`ListVectorFieldPlot[
         PatternTest[
          Pattern[VectorFieldPlots`Private`vects, 
           Blank[List]], TensorRank[#] === 3& ], 
         Pattern[VectorFieldPlots`Private`opts, 
          BlankNullSequence[]]] := VectorFieldPlots`ListVectorFieldPlot[
         Flatten[
          MapIndexed[{
            Reverse[#2], #}& , 
           Reverse[VectorFieldPlots`Private`vects], {2}], 1], 
         VectorFieldPlots`Private`opts], VectorFieldPlots`ListVectorFieldPlot[
         Pattern[VectorFieldPlots`Private`v, 
          Blank[]], 
         BlankNullSequence[]] := Condition[Null, Message[
           MessageName[VectorFieldPlots`ListVectorFieldPlot, "lpvf"]]; False],
        Options[VectorFieldPlots`ListVectorFieldPlot] := {
        AlignmentPoint -> Center, AspectRatio -> Automatic, Axes -> False, 
         AxesLabel -> None, AxesOrigin -> Automatic, AxesStyle -> {}, 
         Background -> None, BaselinePosition -> Automatic, BaseStyle -> {}, 
         ColorFunction -> None, ColorOutput -> Automatic, ContentSelectable -> 
         Automatic, Epilog -> {}, Frame -> False, FrameLabel -> None, 
         FrameStyle -> {}, FrameTicks -> Automatic, FrameTicksStyle -> {}, 
         GridLines -> None, GridLinesStyle -> {}, ImageMargins -> 0., 
         ImagePadding -> All, ImageSize -> Automatic, LabelStyle -> {}, 
         VectorFieldPlots`MaxArrowLength -> None, Method -> Automatic, 
         PlotLabel -> None, PlotRange -> All, PlotRangeClipping -> False, 
         PlotRangePadding -> Automatic, PlotRegion -> Automatic, 
         PreserveImageOptions -> Automatic, Prolog -> {}, RotateLabel -> True,
          VectorFieldPlots`ScaleFactor -> Automatic, 
         VectorFieldPlots`ScaleFunction -> None, Ticks -> Automatic, 
         TicksStyle -> {}, DisplayFunction :> $DisplayFunction, FormatType :> 
         TraditionalForm}, 
       TagSet[VectorFieldPlots`ListVectorFieldPlot, 
        MessageName[VectorFieldPlots`ListVectorFieldPlot, "lpvf"], 
        "ListVectorFieldPlot requires a rectangular array of vectors or a \
list of {base, vector} pairs."], 
       TagSet[VectorFieldPlots`ListVectorFieldPlot, 
        MessageName[VectorFieldPlots`ListVectorFieldPlot, "usage"], 
        "\!\(\*RowBox[{\"ListVectorFieldPlot\", \"[\", RowBox[{\"{\", \
RowBox[{RowBox[{\"{\", RowBox[{RowBox[{\"{\", \
RowBox[{SubscriptBox[StyleBox[\"x\", \"TI\"], StyleBox[\"11\", \"TR\"]], \
\",\", SubscriptBox[StyleBox[\"y\", \"TI\"], StyleBox[\"11\", \"TR\"]]}], \"}\
\"}], \",\", RowBox[{\"{\", RowBox[{SubscriptBox[StyleBox[\"x\", \"TI\"], \
StyleBox[\"12\", \"TR\"]], \",\", SubscriptBox[StyleBox[\"y\", \"TI\"], \
StyleBox[\"12\", \"TR\"]]}], \"}\"}], \",\", StyleBox[\"\[Ellipsis]\", \
\"TR\"]}], \"}\"}], \",\", StyleBox[\"\[Ellipsis]\", \"TR\"]}], \"}\"}], \
\"]\"}]\) generates a plot of the vector field corresponding to the array of \
vectors \!\(\*RowBox[{\"{\", RowBox[{RowBox[{\"{\", RowBox[{RowBox[{\"{\", \
RowBox[{SubscriptBox[StyleBox[\"x\", \"TI\"], StyleBox[\"11\", \"TR\"]], \
\",\", SubscriptBox[StyleBox[\"y\", \"TI\"], StyleBox[\"11\", \"TR\"]]}], \"}\
\"}], \",\", StyleBox[\"\[Ellipsis]\", \"TR\"]}], \"}\"}], \",\", StyleBox[\"\
\[Ellipsis]\", \"TR\"]}], \"}\"}]\).\n\!\(\*RowBox[{\"ListVectorFieldPlot\", \
\"[\", RowBox[{\"{\", RowBox[{RowBox[{\"{\", \
RowBox[{SubscriptBox[StyleBox[\"pt\", \"TI\"], StyleBox[\"1\", \"TR\"]], \
\",\", SubscriptBox[StyleBox[\"vec\", \"TI\"], StyleBox[\"1\", \"TR\"]]}], \
\"}\"}], \",\", RowBox[{\"{\", RowBox[{SubscriptBox[StyleBox[\"pt\", \"TI\"], \
StyleBox[\"2\", \"TR\"]], \",\", SubscriptBox[StyleBox[\"vec\", \"TI\"], \
StyleBox[\"2\", \"TR\"]]}], \"}\"}], \",\", StyleBox[\"\[Ellipsis]\", \
\"TR\"]}], \"}\"}], \"]\"}]\) generates a plot of a list of vectors, each \
based at the corresponding point."], VectorFieldPlots`Private`numberQ[
         Pattern[VectorFieldPlots`Private`n, 
          Blank[]]] := NumberQ[
         N[VectorFieldPlots`Private`n]], VectorFieldPlots`Private`magnitude[
         Pattern[VectorFieldPlots`Private`v, 
          Blank[List]]] := Sqrt[
         Dot[VectorFieldPlots`Private`v, VectorFieldPlots`Private`v]], 
       VectorFieldPlots`Private`getoldarrowopts[
         Pattern[VectorFieldPlots`Private`opts, 
          Blank[]]] := 
       Module[{VectorFieldPlots`Private`selopts}, 
         VectorFieldPlots`Private`selopts = 
          Select[VectorFieldPlots`Private`opts, 
            MemberQ[{
             "HeadScaling", "HeadLength", "HeadCenter", "HeadWidth", 
              "HeadShape", "ZeroShape"}, 
             SymbolName[
              First[#]]]& ]; Map[ReplaceAll[
             SymbolName[
              First[#]], {
             "HeadScaling" -> Graphics`Arrow`HeadScaling, "HeadLength" -> 
              Graphics`Arrow`HeadLength, "HeadCenter" -> 
              Graphics`Arrow`HeadCenter, "HeadWidth" -> 
              Graphics`Arrow`HeadWidth, "HeadShape" -> 
              Graphics`Arrow`HeadShape, "ZeroShape" -> 
              Graphics`Arrow`ZeroShape}] -> Part[#, 2]& , 
           VectorFieldPlots`Private`selopts]], $CellContext`V[
         Pattern[$CellContext`t, 
          Blank[]]] := Derivative[1][$CellContext`r][$CellContext`t], 
       Attributes[Derivative] = {NHoldAll, ReadProtected}, $CellContext`r[
         Pattern[$CellContext`t, 
          Blank[]]] := {
         Sin[$CellContext`t], 
         Cos[$CellContext`t], 
         Sin[2 $CellContext`t]}}; Typeset`initDone$$ = True),
    SynchronousInitialization->True,
    UnsavedVariables:>{Typeset`initDone$$},
    UntrackedVariables:>{Typeset`size$$}], "Manipulate",
   Deployed->True,
   StripOnInput->False],
  Manipulate`InterpretManipulate[1]]], "Output",
 CellChangeTimes->{{3.438288260046875*^9, 3.438288270421875*^9}, 
   3.438288371609375*^9, {3.438288610515625*^9, 3.438288633609375*^9}, {
   3.438288675921875*^9, 3.4382886986875*^9}}]
},
WindowSize->{1272, 903},
WindowMargins->{{0, Automatic}, {Automatic, 0}},
DockedCells->FEPrivate`If[
  FEPrivate`SameQ[FEPrivate`$ProductIDName, "MathematicaPlayer"], 
  FEPrivate`Join[{
    Cell[
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      GraphicsBox[
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