(************** Content-type: application/mathematica **************
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Notebook[{

Cell[CellGroupData[{
Cell["Multivariable Calculus", "Title",
  Evaluatable->False,
  TextAlignment->Center,
  TextJustification->0,
  AspectRatioFixed->True,
  Background->RGBColor[0, 1, 1]],

Cell["\<\
This notebook is by Steven Amgott.  Please send any questions or\
\[NonBreakingSpace]comments to samgott1@swarthmore.edu.  Feel free to use and \
distribute this notebook, but keep this author information in any copy you \
use or distribute.\
\>", "SmallText"],

Cell[TextData[{
  "Buttons to create many of the symbols in this notebook are available in \
the ",
  StyleBox["BasicInput Palette",
    FontColor->RGBColor[0, 0, 1]],
  ".  This palette is usually opened at the time ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " is started.  If it is not up on the screen, it can be displayed using the \
",
  StyleBox["Palette",
    FontColor->RGBColor[0, 0, 1]],
  " command in the ",
  StyleBox["File",
    FontColor->RGBColor[0, 0, 1]],
  " menu.  ",
  StyleBox["Take a brief look at this palette",
    FontColor->RGBColor[1, 0, 1]],
  " to see what symbols it contains before continuing with this notebook.  If \
the computer you are using has the additional  ",
  StyleBox["Multivariable ",
    FontColor->RGBColor[0, 0, 1]],
  "palette installed, you may also wish to look at it.\n\nTo open the closed \
group of cells for any of the sections or commands, you can ",
  StyleBox["double-click the bracket with a triangle at its bottom at the \
right edge of the cell",
    FontColor->RGBColor[1, 0, 1]],
  ".\n\nIf this is your first experience with ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  ", or if you need a brief refresher before continuing, you might first go \
through the ",
  Cell[BoxData[
      FormBox[
        ButtonBox[\(Introduction\ to\ Mathematica\),
          ButtonData:>{"Introduction_to_Mathematica.nb", None},
          ButtonStyle->"Hyperlink"], TraditionalForm]]],
  " notebook contained in the same folder as this notebook.  In particular, \
remember that to evaluate a cell you need to press the ",
  StyleBox["enter",
    FontColor->RGBColor[0, 0, 1]],
  " key on the number pad at the lower right side of the keyboard and NOT the \
",
  StyleBox["return",
    FontColor->RGBColor[0, 0, 1]],
  " key on a Macintosh or the ",
  StyleBox["Enter",
    FontColor->RGBColor[0, 0, 1]],
  " key next to the single and double quote marks on a Windows machine!  ",
  "(The latter only produces a line break.  You can, however, use the ",
  StyleBox["Shift",
    FontColor->RGBColor[0, 0, 1]],
  " and ",
  StyleBox["Enter",
    FontColor->RGBColor[0, 0, 1]],
  " keys on a Windows machine, or the ",
  StyleBox["shift",
    FontColor->RGBColor[0, 0, 1]],
  " and ",
  StyleBox["return",
    FontColor->RGBColor[0, 0, 1]],
  " keys on a Mac, simultaneously to evaluate a cell.  In particular, that is \
how you evaluate a cell on a notebook computer without a number pad.)"
}], "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[CellGroupData[{

Cell[TextData[{
  "What ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " can do"
}], "Section",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[{
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " can perform many of the basic operations of Multivariable Calculus.  Some \
of these (partial differentiation, iterated integration) use the same syntax \
as in the one variable case).  With the package ",
  StyleBox["\"Calculus",
    FontColor->RGBColor[1, 0, 0]],
  StyleBox["`",
    FontFamily->"Courier",
    FontWeight->"Bold",
    FontColor->RGBColor[1, 0, 0]],
  StyleBox["VectorAnalysis",
    FontColor->RGBColor[1, 0, 0]],
  StyleBox["`",
    FontFamily->"Courier",
    FontWeight->"Bold",
    FontColor->RGBColor[1, 0, 0]],
  StyleBox["\"",
    FontColor->RGBColor[1, 0, 0]],
  " loaded, ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " can also find gradients and Jacobians, calculate dot, cross, and triple \
scalar products, divergence, curl, and the Laplacian.  Of course, it can also \
do many different types of plots.  Examples of each of these commands are in \
the cells below."
}], "Text",
  Evaluatable->False,
  AspectRatioFixed->True]
}, Closed]],

Cell[CellGroupData[{

Cell["Plots", "Section"],

Cell[CellGroupData[{

Cell["Plots of Functions", "Subsection"],

Cell[TextData[{
  "The command to draw graphs of functions of two variables is ",
  StyleBox["Plot3D",
    FontColor->RGBColor[1, 0, 0]],
  ".  Evaluating the next cell shows an example of its use."
}], "Text"],

Cell[BoxData[
    \(\(Plot3D[
        Sin[x] Cos[2\ y], {x, \(-2\), 2}, {y, \(-2\), 2}];\)\)], "Input"],

Cell[TextData[{
  "(As with the ",
  StyleBox["Plot",
    FontColor->RGBColor[1, 0, 0]],
  " command, the semicolon at the end of the line just keeps ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " from responding with an output cell containing only the text \
\"-SurfaceGraphics-.\")  The graph is a static figure, but you can change the \
viewpoint by with the ",
  StyleBox["3D Viewpoint Selector",
    FontColor->RGBColor[0, 0, 1]],
  ".  You use it by clicking in a Plot3D input cell (such as the one below) \
just before the final  ",
  StyleBox["]",
    FontColor->RGBColor[1, 0, 0]],
  ", typing a comma (",
  StyleBox[",",
    FontColor->RGBColor[1, 0, 0]],
  "), and then choosing Input...3D Viewpoint Selector.  In the resulting \
window, use your mouse to rotate the box any desired orientation, and click \
the ",
  StyleBox["Paste",
    FontColor->RGBColor[0, 0, 1]],
  " button.  ",
  StyleBox["Try this in the cell below",
    FontColor->RGBColor[1, 0, 1]],
  ".  You should get something that looks like ",
  StyleBox["Plot3D[Sin[x]Sin[2 y],{x,-2,2},{y,-2,2},ViewPoint\[Rule]{-2.756, \
-0.536, 1.889}]",
    FontColor->RGBColor[1, 0, 0]],
  " (but with different numbers after the word ",
  StyleBox["ViewPoint",
    FontColor->RGBColor[1, 0, 0]],
  ").  Then ",
  StyleBox["evaluate",
    FontColor->RGBColor[1, 0, 1]],
  " the cell to produce the graph from the new viewpoint."
}], "Text"],

Cell[BoxData[
    \(\(Plot3D[
        Sin[x] Cos[2\ y], {x, \(-2\), 2}, {y, \(-2\), 2}];\)\)], "Input"],

Cell[TextData[{
  "You can change the viewpoint once again by highlighting the part of the \
syntax the ",
  StyleBox["3D Viewpoint Selector",
    FontColor->RGBColor[0, 0, 1]],
  " pasted in, and then rotating the box again and pasting in a different \
location.  You may wish to try this."
}], "Text"],

Cell[TextData[{
  "There is another way to rotate the graph.  A new (experimental) package \
has been included in recent versions of ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " (but without documentation in the ",
  StyleBox["Help",
    FontColor->RGBColor[0, 0, 1]],
  " menu).  You can load this package by evaluating the cell below.  Check \
the output of the $Packages command to make sure ",
  StyleBox["RealTime3D",
    FontColor->RGBColor[0, 0, 1]],
  " is now loaded."
}], "Text"],

Cell[BoxData[{
    \(Needs["\<RealTime3D`\>"]\), "\[IndentingNewLine]", 
    \($Packages\)}], "Input"],

Cell[TextData[{
  "Once this package has been loaded, any additional 3D graphs you create can \
be rotated in real time.  (Note that graphs you created before loading the \
package still cannot be rotated this way.)  For instance, ",
  StyleBox["evaluate",
    FontColor->RGBColor[1, 0, 1]],
  " the syntax below to be able to rotate the graph from the beginning of \
this section.  Click and hold the (left) mouse button anywhere in the graph \
and move the mouse to rotate the graph.  When you release the mouse button, \
the graph \"freezes\" in the current position. "
}], "Text"],

Cell[BoxData[
    \(\(Plot3D[
        Sin[x] Cos[2\ y], {x, \(-2\), 2}, {y, \(-2\), 2}];\)\)], "Input"],

Cell[TextData[{
  "If you wish, you can change the region on which the graph is plotted, for \
instance, to zoom in.  All you need to do are to put in different values for \
the range on x and y.  The values in ",
  StyleBox["magenta",
    FontColor->RGBColor[1, 0, 1]],
  " below can be changed to do this.  ",
  StyleBox["Change them in the next cell in order to redraw the graph on the \
square -0.5 \[LessEqual] x \[LessEqual] 0.5 and -0.5 \[LessEqual] y \
\[LessEqual] 0.5, and then evaluate the cell to see the \"zoomed-in\" graph",
    FontColor->RGBColor[1, 0, 1]],
  "."
}], "Text"],

Cell[BoxData[
    RowBox[{
      RowBox[{"Plot3D", "[", 
        RowBox[{\(Sin[x] Cos[2\ y]\), ",", 
          RowBox[{"{", 
            RowBox[{"x", ",", 
              StyleBox[\(-2\),
                FontColor->RGBColor[1, 0, 1]], ",", 
              StyleBox["2",
                FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", 
          RowBox[{"{", 
            RowBox[{"y", ",", 
              StyleBox[\(-2\),
                FontColor->RGBColor[1, 0, 1]], ",", 
              StyleBox["2",
                FontColor->RGBColor[1, 0, 1]]}], "}"}]}], "]"}], 
      ";"}]], "Input"]
}, Closed]],

Cell[CellGroupData[{

Cell["Contour Plots", "Subsection"],

Cell[TextData[{
  "Another interesting type of plot, although not 3D, is a ",
  StyleBox["ContourPlot",
    FontColor->RGBColor[1, 0, 0]],
  ".  This plot shows graphs of the curves you get by slicing the surface at \
various heights.  (You may have seen maps of mountains and hills done this \
way.)  Here is the contour plot for the above function."
}], "Text"],

Cell[BoxData[
    \(\(ContourPlot[
        Sin[x] Cos[2\ y], {x, \(-2\), 2}, {y, \(-2\), 2}];\)\)], "Input"],

Cell[TextData[{
  "The shading reflects the height of the contour.  Darker is lower, while \
lighter is higher.  If you just want to see the contours without the shading, \
use the ",
  StyleBox["ContourShading\[Rule]False",
    FontColor->RGBColor[1, 0, 0]],
  " option."
}], "Text"],

Cell[BoxData[
    \(\(ContourPlot[Sin[x] Cos[2\ y], {x, \(-2\), 2}, {y, \(-2\), 2}, 
        ContourShading \[Rule] False];\)\)], "Input"]
}, Closed]],

Cell[CellGroupData[{

Cell["Parametric Plots", "Subsection"],

Cell[TextData[{
  "You can use ",
  StyleBox["ParametricPlot",
    FontColor->RGBColor[1, 0, 0]],
  " and ",
  StyleBox["ParametricPlot3D",
    FontColor->RGBColor[1, 0, 0]],
  " to plot curves.  Here is an ellipse done with ",
  StyleBox["ParametricPlot",
    FontColor->RGBColor[1, 0, 0]],
  "."
}], "Text"],

Cell[BoxData[
    \(\(ParametricPlot[{3\ Sin[t], 4\ Cos[t]}, {t, 0, 
          2\ \[Pi]}];\)\)], "Input"],

Cell[TextData[{
  "Notice that to get a good representation of a circle the ",
  StyleBox["AspectRatio",
    FontColor->RGBColor[1, 0, 0]],
  " option might be necessary.  Without this option the plot may look like an \
ellipse."
}], "Text"],

Cell[BoxData[
    \(\(ParametricPlot[{4\ Sin[t], 4\ Cos[t]}, {t, 0, 
          2\ \[Pi]}];\)\)], "Input"],

Cell["With the option you get the true shape of the circle.", "Text"],

Cell[BoxData[
    \(\(ParametricPlot[{4\ Sin[t], 4\ Cos[t]}, {t, 0, 2\ \[Pi]}, 
        AspectRatio \[Rule] Automatic];\)\)], "Input"],

Cell[TextData[{
  "This plots a spiral in three-dimensional space using ",
  StyleBox["ParametricPlot3D",
    FontColor->RGBColor[1, 0, 0]],
  "."
}], "Text"],

Cell[BoxData[
    \(\(ParametricPlot3D[{7\ Cos[u], 7\ Sin[u], u}, {u, 0, 
          8  \[Pi]}];\)\)], "Input"],

Cell[TextData[{
  "You can also use ",
  StyleBox["ParametricPlot3D",
    FontColor->RGBColor[1, 0, 0]],
  " to plot surfaces.  Here is a paraboloid."
}], "Text"],

Cell[BoxData[
    \(\(ParametricPlot3D[{2\ r\ Cos[\[Theta]], 2\ r\ Sin[\[Theta]], 
          r\^2}, {r, 0, 4}, {\[Theta], 0, 2\ \[Pi]}];\)\)], "Input"],

Cell[TextData[{
  "Notice the advantages to plotting the paraboloid using ",
  StyleBox["ParametricPlot3D",
    FontColor->RGBColor[1, 0, 0]],
  ", rather than using ",
  StyleBox["Plot3D",
    FontColor->RGBColor[1, 0, 0]],
  "."
}], "Text"],

Cell[BoxData[
    \(\(Plot3D[
        1\/4\ x\^2 + 1\/4\ y\^2, {x, \(-8\), 8}, {y, \(-8\), 
          8}];\)\)], "Input"]
}, Closed]],

Cell[CellGroupData[{

Cell["Vector Field Plots", "Subsection"],

Cell[TextData[{
  "In order to use the commands in this section, the package ",
  StyleBox["\"Graphics",
    FontColor->RGBColor[1, 0, 0]],
  StyleBox["`",
    FontFamily->"Courier",
    FontWeight->"Bold",
    FontColor->RGBColor[1, 0, 0]],
  StyleBox["PlotField",
    FontColor->RGBColor[1, 0, 0]],
  StyleBox["`",
    FontFamily->"Courier",
    FontWeight->"Bold",
    FontColor->RGBColor[1, 0, 0]],
  StyleBox["\"",
    FontColor->RGBColor[1, 0, 0]],
  StyleBox[" ",
    FontColor->RGBColor[0.499992, 0.0658274, 0.17203]],
  "(for two dimensional fields) and/or the package ",
  StyleBox["\"Graphics",
    FontColor->RGBColor[1, 0, 0]],
  StyleBox["`",
    FontFamily->"Courier",
    FontWeight->"Bold",
    FontColor->RGBColor[1, 0, 0]],
  StyleBox["PlotField3D",
    FontColor->RGBColor[1, 0, 0]],
  StyleBox["`",
    FontFamily->"Courier",
    FontWeight->"Bold",
    FontColor->RGBColor[1, 0, 0]],
  StyleBox["\"",
    FontColor->RGBColor[1, 0, 0]],
  " (for three dimensional fields) must be loaded.  ",
  StyleBox["Evaluate the next cell to load the first of these packages.",
    FontColor->RGBColor[1, 0, 1]],
  StyleBox["  ",
    FontColor->RGBColor[1, 0.0721599, 0.542763]],
  "You will have to load the package during any session in which you plan to \
use these commands.  ",
  StyleBox["$Packages",
    FontColor->RGBColor[1, 0, 0]],
  " displays the current list of loaded packages."
}], "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[BoxData[{
    \(Needs["\<Graphics`PlotField`\>"]\), "\[IndentingNewLine]", 
    \($Packages\)}], "Input"],

Cell["\<\
Using this package you can display gradient fields of functions of two \
variables.\
\>", "Text"],

Cell[BoxData[
    RowBox[{
      RowBox[{"PlotGradientField", "[", 
        RowBox[{\(3  x + 12  y - x\^3 - y\^3\), ",", 
          RowBox[{"{", 
            RowBox[{"x", 
              StyleBox[",",
                FontColor->GrayLevel[0]], \(-2\), ",", "2"}], "}"}], ",", 
          RowBox[{"{", 
            RowBox[{"y", 
              StyleBox[",",
                FontColor->GrayLevel[0]], \(-3\), ",", "3"}], "}"}]}], "]"}], 
      ";"}]], "Input"],

Cell["\<\
There are, of course, tons of customization options.  You can find \
information on this by evaluating the following cell.\
\>", "Text"],

Cell[BoxData[
    \(?? PlotGradientField\)], "Input"],

Cell["You can also directly plot vector fields.", "Text"],

Cell[BoxData[
    \(\(PlotVectorField[{\(-y\), x}, {x, \(-2\), 2}, {y, \(-2\), 
          2}];\)\)], "Input"],

Cell[TextData[{
  "One way to get more information on this is to highlight the words \
PlotVectorField in the previous cell using your mouse, and then click the \
menu item ",
  StyleBox["Help...Find Selected Function",
    FontColor->RGBColor[0, 0, 1]],
  ".  Do so, then click the link to ",
  StyleBox["Graphics`PlotField`",
    FontColor->RGBColor[0, 0, 1]],
  " under Standard Packages, or the section number link under the ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " Book."
}], "Text"],

Cell["\<\
To work in three dimensions you need to load the second of the packages.\
\>", "Text"],

Cell[BoxData[{
    \(Needs["\<Graphics`PlotField3D`\>"]\), "\[IndentingNewLine]", 
    \($Packages\)}], "Input"],

Cell["\<\
You can then display gradient fields of functions of three variables\
\>", "Text"],

Cell[BoxData[
    \(\(PlotGradientField3D[
        x\ y\ z, {x, \(-1\), 1}, {y, \(-1\), 1}, {z, \(-1\), 1}, 
        VectorHeads \[Rule] True];\)\)], "Input"],

Cell["or other arbitrary vector fields with three components.", "Text"],

Cell[BoxData[
    \(\(PlotVectorField3D[{x, y, z}, {x, 0, 2}, {y, 0, 2}, {z, 0, 2}, 
        PlotPoints \[Rule] 5, VectorHeads \[Rule] True];\)\)], "Input"],

Cell["\<\
Notice the options in these cells.  The default is to draw without arrows, as \
you can see by evaluating the following cell, which displays the default \
options.\
\>", "Text"],

Cell[BoxData[
    \(Options[PlotVectorField3D]\)], "Input"]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell["Partial Derivatives", "Section",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[{
  "The basic syntax for the command to find partial derivatives is \
essentially the same as for one variable derivatives.    You can use the \
palette symbols ",
  Cell[BoxData[
      \(TraditionalForm\`\[PartialD]\_\[Placeholder]\ \[Placeholder]\)],
    FontColor->RGBColor[1, 0, 0]],
  " or ",
  Cell[BoxData[
      \(TraditionalForm\`\[PartialD]\_\(\[Placeholder], \[Placeholder]\)\
\[Placeholder]\)],
    FontColor->RGBColor[1, 0, 0]],
  ", where the subscript boxes are filled by the variables and the final box \
is filled by the function.   As you will see in the examples below, you can \
use more than two variables.  You can also use the older syntax ",
  StyleBox["D[f[x1,x2,x3,...,xn],xi]",
    FontColor->RGBColor[1, 0, 0]],
  ".  You can use any variable names, as in the examples below.  Make sure to \
leave a ",
  StyleBox["space",
    FontColor->RGBColor[1, 0, 0]],
  " or put a ",
  StyleBox["*",
    FontColor->RGBColor[1, 0, 0]],
  " between letters that are being multiplied (such as the use of ",
  StyleBox["x y",
    FontColor->RGBColor[1, 0, 0]],
  ", instead of ",
  StyleBox["xy",
    FontColor->RGBColor[1, 0, 0]],
  StyleBox[",",
    FontColor->RGBColor[0, 0, 1]],
  " in the following example).    ",
  StyleBox["Evaluate each input cell below by clicking anywhere in the cell, \
and pressing the ",
    FontColor->RGBColor[1, 0, 1]],
  StyleBox["enter",
    FontColor->RGBColor[0, 0, 1]],
  StyleBox[" key",
    FontColor->RGBColor[1, 0, 1]],
  "."
}], "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[BoxData[
    \(\[PartialD]\_x Sin[x\ y]\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(\[PartialD]\_y Sin[x\ y]\)], "Input",
  AspectRatioFixed->True],

Cell[TextData[
"The first output from the following cell is the function, the second the \
partial derivative of this function with respect to t, the third the partial \
with respect to u, and the fourth the partial with respect to v."], "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[BoxData[{
    \(Clear[h, t, u, v]\), "\[IndentingNewLine]", 
    \(h[t_, u_, v_] = t\^2\ E\^\(t\ u\ v\) + u\^3\), "\n", 
    \(\[PartialD]\_t h[t, u, v]\), "\n", 
    \(\[PartialD]\_u h[t, u, v]\), "\n", 
    \(\[PartialD]\_v h[t, u, v]\)}], "Input",
  AspectRatioFixed->True],

Cell[TextData[{
  "You can find higher order derivatives with respect to one of the variables \
by including the order in the command.  For example, to find the 2nd \
derivative of ",
  StyleBox["x Cos[x y]",
    FontColor->RGBColor[1, 0, 0]],
  " with respect to ",
  StyleBox["x",
    FontColor->RGBColor[1, 0, 0]],
  " twice, use: "
}], "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[BoxData[
    \(\[PartialD]\_{x, 2}\((x\ Cos[x\ y])\)\)], "Input",
  AspectRatioFixed->True],

Cell[TextData[{
  "Be sure to put the braces ",
  StyleBox["{ }",
    FontColor->RGBColor[1, 0, 0]],
  " around the variable and the order."
}], "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell["\<\
You can find mixed higher order partials by listing the variables with which \
you wish to differentiate after the expression.  For example, the next cell \
finds the third order partial derivative with respect to x twice, and z once.\
\
\>", "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[BoxData[
    \(\[PartialD]\_\(x, x, z\)\((z\ \[ExponentialE]\^\(x\ y\ z\))\)\)], \
"Input",
  AspectRatioFixed->True],

Cell["This can also be done by", "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[BoxData[
    \(\[PartialD]\_\({x, 2}, z\)\((z\ \[ExponentialE]\^\(x\ y\ z\))\)\)], \
"Input",
  AspectRatioFixed->True],

Cell["and, of course, by", "Text"],

Cell[BoxData[
    \(D[z\ E^\((x\ y\ z)\), {x, x, z}]\)], "Input"],

Cell["as in the older version 2 syntax.", "Text"],

Cell[TextData[{
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " assumes the order of the variables is unimportant.  For example, it will \
give the same answer to ",
  StyleBox["D[f[x,y],x,y]",
    FontColor->RGBColor[1, 0, 0]],
  " and ",
  StyleBox["D[f[x,y],y,x]",
    FontColor->RGBColor[1, 0, 0]],
  ".  As you may know from class, this is true for \"nice\" (but not all) \
functions."
}], "Text",
  Evaluatable->False,
  AspectRatioFixed->True]
}, Closed]],

Cell[CellGroupData[{

Cell["Iterated Integrals", "Section",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[{
  "In order to evaluate iterated integrals, you simply \"nest\" one variable \
integral commands.  (The additional ",
  StyleBox["Multivariable",
    FontColor->RGBColor[0, 0, 1]],
  " palette has buttons to quickly create the \"nested\" integrals for double \
and triple iterated integrals.  Look in ",
  StyleBox["File...Palettes",
    FontColor->RGBColor[0, 0, 1]],
  " to see if it is on your computer.)  For example, the following cell \
evaluates the iterated integral of the function ",
  Cell[BoxData[
      FormBox[
        SuperscriptBox[
          StyleBox["x",
            FontSlant->"Plain"], "2"], TraditionalForm]]],
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