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                     CreatedBy='Mathematica 5.2'

                    Mathematica-Compatible Notebook

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Notebook[{
Cell["Visualizing Time-Dependent Scalar and Vector Fields", "Title"],

Cell[TextData[StyleBox["If you are viewing this notebook on the web, the \
animations will not appear in pdf format. See the html format for animations",
  
  FontSize->36,
  FontColor->RGBColor[1, 0, 0]]], "Subsubsection",
  Background->GrayLevel[0.900008]],

Cell["\<\
Note: This notebook will produce  animations but they each take \
some time to compute. And, if you save the notebook with the graphics, it \
will require 50MB or so of hard disk space.\
\>", "Subsubsection"],

Cell["\<\
You should definitely take the time to learn how to make your own \
animations because the results can be really instructive. You can even save \
animations in various formats and then import them in separate \
applications.\
\>", "Subsubsection"],

Cell[CellGroupData[{

Cell["Introduction: Making Animations", "Subtitle"],

Cell["\<\
An animation consists of a collection of images which are shown \
sequentially.  The most simple way to make an animation is to use \
Table[Plot[],{}] which will usually display a list of graphical objects, one \
at a time.  The images can be grouped within a single bracket; closed by \
clicking the ``container bracket''; and then initiating the animation with an \
item under the \"Cell\" menu.\
\>", "Section"],

Cell["\<\
Here are a few simple examples

These examples will work best if Cell->Cell Grouping is set to Automatic.
After all the images have finished displaying, click once the \
second-from-left bracket, and then click again to close it but leave it \
selected.
Display the animation using Cell->Animated Selected Graphics .\
\>", \
"Subsubtitle"],

Cell["A traveling wave", "Subsection"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(\(frequency\  = \ 3;\)\), "\[IndentingNewLine]", 
    \(Table[
      Plot[Evaluate[Sin[x - frequency\ t]], {x, \(-Pi\), \ Pi}], {t, 0, 2, 
        0.1}]\)}], "Input"],

Cell[GraphicsData["QuickTime", "\<\
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  AnimationCycleOffset->0,
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}, Open  ]],

Cell["Beats", "Subsection"],

Cell["\<\
The following example is not ideal--the y-axis will change making \
the spatial scale vary with time\
\>", "Section"],

Cell[CellGroupData[{

Cell[BoxData[{
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    \(\(frequency2\  = \ 2/3;\)\), "\[IndentingNewLine]", 
    \(\(k1\  = \ 1/2;\)\), "\[IndentingNewLine]", 
    \(\(k2\  = 4/7;\)\), "\[IndentingNewLine]", 
    \(Table[
      Plot[Evaluate[
          Sin[k1\ x - frequency1\ \ t]\  + \ 
            Sin[k2\ x - \ frequency2\ t]], {x, 0, 20  Pi}], {t, 0, 10, 
        0.25}]\)}], "Input"],

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Cell["\<\
This fixes the scale by calling the same PlotRange option for each \
plot. \
\>", "Section"],

Cell[CellGroupData[{

Cell[BoxData[{
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    \(\(frequency2\  = \ 2/3;\)\), "\[IndentingNewLine]", 
    \(\(k1\  = \ 1/2;\)\), "\[IndentingNewLine]", 
    \(\(k2\  = 4/7;\)\), "\[IndentingNewLine]", 
    \(Table[
      Plot[Evaluate[
          Sin[k1\ x - frequency1\ \ t]\  + \ 
            Sin[k2\ x - \ frequency2\ t]], {x, 0, 20  Pi}, 
        PlotRange -> {\(-2\), 2}], {t, 0, 10, 0.25}]\)}], "Input"],

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Cell["\<\
This is an alternative--and perhaps a bit more clever--way to \
produce a list of graphics, by creating the graphics first and then iterating \
over a list.\
\>", "Section"],

Cell["First the graphics list is computed and not displayed:", "Subsubsection"],

Cell[BoxData[
    \(\(GraphicsList\  = 
        Table[ParametricPlot[
            Evaluate[{Sin[t]\  + Sin[\ t\ a], Cos[t]\  + Cos[t\ a]}\ ], {t, 
              0, 2\ Pi}, AspectRatio -> 1, DisplayFunction -> Identity], {a, 
            0, \ 20}];\)\)], "Input"],

Cell["\<\
And the list can be displayed later, with graphics directives added \
if wanted:\
\>", "Subsubsection"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Do[Show[GraphicsList[\([i]\)], \ 
        DisplayFunction -> $DisplayFunction], {i, 1, 
        Length[GraphicsList]}]\)], "Input"],

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Cell["Or coupled with other graphics directives:", "Subsubsection"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(\(RandomColors\  = \ 
        Table[Graphics[
            CMYKColor[0.5\  + \ 0.5  Sin[2\ Pi\ t]\ , 
              0.5\  + \ 0.5  Cos[2\ Pi\ t], \ 0.5, \ 0]], {t, 0, 1, 
            1/Length[GraphicsList]}];\)\), "\[IndentingNewLine]", 
    \(Do[Show[{RandomColors[\([i]\)], GraphicsList[\([i]\)]}, \ 
        AspectRatio -> 1, DisplayFunction -> $DisplayFunction], {i, 1, 
        Length[GraphicsList]}]\)}], "Input"],

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Cell[BoxData[""], "Input"],

Cell["A contrived example of a fluttering surface:", "Subsection"],

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  ImageMargins->{{0, 0}, {0, 0}},
  ImageRegion->{{0, 1}, {0, 1}}]
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}, Closed]],

Cell[CellGroupData[{

Cell["Cross Products", "Subtitle"],

Cell["Here is the built-in cross product between two vectors", "Text"],

Cell[BoxData[
    \(crossab\  = 
      Cross[{a\_1, a\_2, a\_3}\ \ , {b\_1, b\_2, b\_3}]\)], "Input"],

Cell["\<\
And, here is the standard visual way to do it by hand with the \
determinant.\
\>", "Text"],

Cell[BoxData[
    RowBox[{"detab", " ", "=", 
      RowBox[{"Det", "[", 
        RowBox[{"(", GridBox[{
              {"i", "j", "k"},
              {\(a\_1\), \(a\_2\), \(a\_3\)},
              {\(b\_1\), \(b\_2\), \(b\_3\)}
              },
            GridFrame->True,
            RowLines->True,
            ColumnLines->True], ")"}], "]"}]}]], "Input",
  CellTags->{
  "mtag:11:cross_products_example", "mtag:11:vector_products_example"}],

Cell["Pick out each of the compenents to create the vector", "Text"],

Cell[BoxData[
    \(testcrossab = {Coefficient[detab, i], \ Coefficient[detab, j], 
        Coefficient[detab, k]}\)], "Input"],

Cell["\<\
Check for equality between the old-fashioned way and Mathematica's \
built-in function\
\>", "Text"],

Cell[BoxData[
    \(testcrossab\  \[Equal] \ crossab\)], "Input"]
}, Closed]],

Cell[CellGroupData[{

Cell["Visualizing Curves as Time-Dependent Vectors", "Subtitle"],

Cell[BoxData[
    \(Clear[TimeVector]\)], "Input",
  CellTags->
  "mtag:11:derivatives_of_vector_quantities__examples_and_visualization"],

Cell["Create a trajectory of a point or a particle", "Text"],

Cell[BoxData[
    \(TimeVector\  = \ {Cos[4  \[Pi]\ t], Sin[8\ \[Pi]\ t], \ 
        Sin[2  \[Pi]\ t]}\)], "Input"],

Cell["\<\
ParametricPlot3D allows us to visualize the entire trajectory at \
once.\
\>", "Text"],

Cell[BoxData[
    \(ParametricPlot3D[TimeVector, {t, 0, 1}]\)], "Input",
  CellTags->"mtag:11:ParametricPlot3D[]"],

Cell["Here is another amusing curve", "Text"],

Cell[BoxData[
    \(DeltoidSpiral\  = \ {\((2\ Cos[\[Pi]\ t]\  + \ 
            Cos[2\ \[Pi]\ t])\), \((2\ Sin[\[Pi]\ t] - \ 
            Sin[2\ \[Pi]\ t])\), \ t/3}\)], "Input"],

Cell[BoxData[
    \(pp\  = 
      ParametricPlot3D[DeltoidSpiral, {t, \(-3\), 3}, 
        AxesLabel \[Rule] {"\<x\>", "\<y\>", "\<z\>"}]\)], "Input"],

Cell["Make the visualization easier to pick out", "Text"],

Cell[BoxData[
    \(Show[{Graphics3D[Thickness[0.01]], Graphics3D[Hue[1]], pp}]\)], "Input"],

Cell["\<\
This is the derivative of the function as a function of time, it \
should point along the tanget\
\>", "Text"],

Cell[BoxData[
    \(dDSt\  = \ D[DeltoidSpiral, t]\)], "Input"],

Cell[BoxData[
    \(ppdt = 
      ParametricPlot3D[dDSt, {t, \(-3\), 3}, 
        AxesLabel \[Rule] {"\<x\>", "\<y\>", "\<z\>"}]\)], "Input"],

Cell[BoxData[
    \(Show[{Graphics3D[Thickness[0.01]], Graphics3D[Hue[0.3]], 
        ppdt}]\)], "Input"],

Cell["\<\
To improve the visualization, it would be nice to see the \
time-development of the curve. Create functions to make graphics \
objects.\
\>", "Text"],

Cell[BoxData[
    \(ppdtlim[tl_] := {Graphics3D[Thickness[0.01]], Graphics3D[Hue[0.3]], 
        ParametricPlot3D[0.33*dDSt, {t, 0, tl}, 
          AxesLabel \[Rule] {"\<x\>", "\<y\>", "\<z\>"}, 
          Compiled \[Rule] False, 
          DisplayFunction \[Rule] Identity]}\)], "Input"],

Cell[BoxData[
    \(dtlim[tl_] := {Graphics3D[Thickness[0.01]], Graphics3D[Hue[1]], 
        ParametricPlot3D[DeltoidSpiral, {t, 0, tl}, 
          AxesLabel \[Rule] {"\<x\>", "\<y\>", "\<z\>"}, 
          Compiled \[Rule] False, 
          DisplayFunction \[Rule] Identity]}\)], "Input"],

Cell["Animate to see the curve and its tangent together", "Text"],

Cell[BoxData[
    \(\(TheGraphicsList = 
        Table[{ppdtlim[t], dtlim[t]}, {t,  .05, 3,  .05}];\)\)], "Input",
  CellTags->"mtag:11:animations__example_of_curves_in_3D"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Do[Show[TheGraphicsList[\([i]\)], 
        PlotRange \[Rule] {{\(-4.25\), 4.25}, {\(-4.25\), 4.25}, {0, 1}}, 
        PlotRegion \[Rule] {{0, 1}, {0, 1}}, SphericalRegion \[Rule] True, \ 
        Boxed \[Rule] True, BoxRatios \[Rule] {1, 1, 1}, 
        ViewCenter \[Rule] {0, 0, 0.5}, AspectRatio \[Rule] 1, 
        ViewPoint \[Rule] {1.2, \(-3\), 2}], {i, 1, 
        Length[TheGraphicsList]}]\)], "Input"],

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Cell[CellGroupData[{

Cell["\<\
Time Dependent Solution to the Diffusion Equation in the Plane with \
a Source at the Origin.\
\>", "Subtitle"],

Cell[BoxData[
    StyleBox[\(<< Graphics`\),
      FontWeight->"Bold"]], "Input"],

Cell["\<\
This is the solution to the time-dependent diffusion equation in \
the infinite plane for a point source located at the origin\
\>", "Text"],

Cell[BoxData[
    \(concentration\  = 
      Exp[\(-\((x^2\  + \ y^2)\)\)\/\(4  Diffusivity\ t\)]\/\(4\ Pi\ \
Diffusivity\ t\)\)], "Input",
  CellTags->
  "mtag:11:diffusion_equation_solution__example__2D_point_source_at_origin"],

Cell["Let the diffusivity be 1 for visualization purposes", "Subsubsection"],

Cell[BoxData[
    \(\(Diffusivity = 1;\)\)], "Input"],

Cell["\<\
Visualize the solution by plotting the concentration as  function \
of x and y, and animate as  function of time.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Table[
      Plot3D[Evaluate[concentration], {x, \(-2\), 2}, {y, \(-2\), 2}, 
        PlotPoints \[Rule] 40, PlotRange \[Rule] {0, 4}], {t, 
        0.0125,  .25,  .0125}]\)], "Input",
  CellTags->"mtag:11:MoviePlot3D[]"],

Cell[GraphicsData["QuickTime", "\<\
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Cell["\<\
Another visualization scheme: plot the isoconcentration lines as  \
function of x and y, and animate as function of time.  This is like the \
temporal evolution of a topographical map.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Table[
      ContourPlot[Evaluate[concentration], {x, \(-2\), 2}, {y, \(-2\), 2}, 
        PlotPoints \[Rule] 40, PlotRange \[Rule] {0, 0.5}, 
        ColorFunction \[Rule] \((Hue[1 - \ 0.75  #] &)\)], {t, 0.0125, 
        1,  .025}]\)], "Input",
  CellTags->"mtag:11:MovieContourPlot[]"],

Cell[GraphicsData["QuickTime", "\<\
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Cell["\<\
Flux is a vector that points in the direction of the flow  and is a \
measure of how much is flowing per unit time. This is Fick's first law.\
\>", \
"Text"],

Cell[BoxData[
    \(<< Graphics`PlotField`\)], "Input"],

Cell[BoxData[
    \(flux\  = \ {\(-D[concentration, x]\), \(-D[concentration, 
            y]\)}\)], "Input"],

Cell[BoxData[
    \(This\ is\ an\ example\ of\ a\ time - 
      dependent\ vector\ field\ \(j\&\[LongRightArrow]\) \((x, y, 
          t)\)\)], "Subsubsection"],

Cell["\<\
An example of the vector field as a function of position at time = \
0.8\
\>", "Text"],

Cell[BoxData[
    \(PlotVectorField[flux /. t \[Rule] 0.8, {x, \(-2\), 2}, {y, \(-2\), 2}, 
      PlotPoints \[Rule] 20, 
      ColorFunction \[Rule] \((Hue[1 - 0.75  #] &)\)]\)], "Input",
  CellTags->{
  "mtag:11:PlotVectorField[]", "mtag:11:ScaleFunction__in_PlotVectorField[]"}],\


Cell["\<\
An example of the vector field as a function of position at time = \
0.2, but with more control over the appearance of the arrows.\
\>", "Text"],

Cell[BoxData[
    \(PlotVectorField[
      flux /. {t \[Rule] 0.2}, {x, \(-2\), 2}, {y, \(-2\), 2}, 
      PlotPoints \[Rule] 21, \ Frame -> True, 
      ScaleFunction -> \((10.0\ # &)\), \ MaxArrowLength -> 50, \ 
      ScaleFactor -> None, 
      ColorFunction \[Rule] \((Hue[1 - 0.75  #] &)\)]\)], "Input"],

Cell["\<\
To create an animation, we can make a table of graphics objects. To \
see animations, group all the graphics cells together, and then animate using \
\"Animate Selected Graphics\" from the Cell menu.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Table[
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        PlotPoints \[Rule] 19, ScaleFunction -> \((100.0\ # &)\), \ 
        MaxArrowLength -> 10, \ ScaleFactor -> None, 
        ColorFunction \[Rule] \((Hue[1 - 0.75  #] &)\)], {t, 0.01, 
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  CellTags->"mtag:11:Animate_and_PlotVectorField[]"],

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  CellTags->"mtag:11:Animate_and_PlotVectorField[]"]
}, Open  ]],

Cell["\<\
To see the evolution of only the x component of the flux, we could \
use a dot product and then animate.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Table[
      ContourPlot[flux . {1, 0}, {x, \(-2\), 2}, {y, \(-2\), 2}, 
        PlotPoints \[Rule] 40, PlotRange \[Rule] {0, 0.5}, 
        ColorFunction \[Rule] \((Hue[1 - 0.75  #] &)\)], {t, 0.01, 
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