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Notebook[{
Cell["Lab 3:  Graphs and Level Curves of Multivariable Functions", "Title",
  Background->RGBColor[0, 0, 1]],

Cell[TextData[{
  "Math 233\tFall 2002\t",
  StyleBox["Chapter 11, Section 1",
    FontFamily->"Arial",
    FontSize->16,
    FontWeight->"Bold"],
  "   "
}], "Subtitle",
  TextAlignment->Left,
  TextJustification->0],

Cell[TextData[StyleBox["",
  FontWeight->"Bold"]], "Subsubtitle"],

Cell[CellGroupData[{

Cell["Initialization Cells", "SmallText"],

Cell[BoxData[
    \(Clear[showsection, fxy, xrange, yrange, zrange, xsection]\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \(\(showsection[fxy_, xrange_, yrange_, zrange_, \ xsection_] := 
        CompoundExpression[\ 
          Clear[p3d, \ pln, \ 
            p2d], \(p3d = 
              Plot3D[fxy, xrange, yrange, BoxRatios -> {1, 1, 2}, 
                Axes \[Rule] True, AxesLabel \[Rule] {x, y, z}, 
                PlotRange \[Rule] {{xrange[\([2]\)], 
                      xrange[\([3]\)]}, {yrange[\([2]\)], 
                      yrange[\([3]\)]}, {zrange[\([2]\)], 
                      zrange[\([3]\)]}}, \ 
                DisplayFunction \[Rule] 
                  Identity\ \ ];\), \[IndentingNewLine]\(pln = 
              ParametricPlot3D[{xsection, x, y}, {x, \(-10\), 
                  10}, {y, \(-10\), 10}, Axes \[Rule] True, 
                AxesLabel \[Rule] {x, y, z}, 
                DisplayFunction \[Rule] Identity\ ];\), \n\ \ \ \ \ \(p2d = 
              Plot[fxy /. x \[Rule] xsection, yrange, Axes \[Rule] True, 
                AspectRatio \[Rule] 2, AxesLabel \[Rule] {y, z}, 
                PlotRange \[Rule] {{xrange[\([2]\)], 
                      xrange[\([3]\)]}, {yrange[\([2]\)], yrange[\([3]\)]}}, 
                DisplayFunction \[Rule] Identity];\), \(combo = 
              Show[{p3d, pln}, DisplayFunction \[Rule] Identity]\ ;\), 
          Show[GraphicsArray[{combo, p2d}], 
            DisplayFunction \[Rule] $DisplayFunction]];\)\)], "Input",
  InitializationCell->True]
}, Closed]],

Cell[CellGroupData[{

Cell[TextData[StyleBox["Instructions!",
  FontColor->RGBColor[1, 0, 0]]], "Section"],

Cell[TextData[{
  "Work in ",
  StyleBox["pairs",
    FontWeight->"Bold"],
  " today, going through the entire notebook.  Make sure that you answer ",
  StyleBox["all ",
    FontWeight->"Bold"],
  "questions posed.\nOn the You Try It, insert your own NEW values or \
functions.  This lab is mainly so that you get some experience learning to \
use ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " to explore multivariable functions and their graphs.  Take your time \
going through it; this is an opportunity to learn a little about what \
functions of various forms look like...\n\nThere won't be a formal write-up \
for this lab; the only thing you need to send me is a ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " notebook containing your function from the You Try It Part IV.\n"
}], "Subsubtitle"]
}, Closed]],

Cell[CellGroupData[{

Cell["Introduction", "Section"],

Cell[TextData[{
  "OBJECTIVE: Work on visualizing and graphing functions of 2 variables and \
to understand the information that you are seeing when you take a \
cross-section of a 3-D surface.  \nThe only ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " commands I want you to have a working knowledge of after today are the ",
  StyleBox["Plot3D",
    FontWeight->"Bold"],
  ", ",
  StyleBox["ContourPlot3D",
    FontWeight->"Bold"],
  ",",
  " and ",
  StyleBox["Table",
    FontWeight->"Bold"],
  " commands..."
}], "Text"],

Cell[CellGroupData[{

Cell["Technology Guidelines", "Subsection",
  CellDingbat->"\[LightBulb]"],

Cell[TextData[{
  StyleBox["NOTE:  If you have just finished a module, restart ",
    CellFrame->True,
    Background->None],
  StyleBox["Mathematica",
    CellFrame->True,
    FontSlant->"Italic",
    Background->None],
  StyleBox[" before executing a new module.\nTO OPEN CELLS, put your cursor \
on the right cell bracket and double click.",
    CellFrame->True,
    Background->None],
  "\nTO STOP AN EXECUTION\n\tSelect the ",
  StyleBox["Kernel",
    FontSlant->"Italic"],
  " pull-down menu and click on ",
  StyleBox["Abort Evaluation.\n",
    FontSlant->"Italic"],
  "ORDER OF EXECUTION\n\tExecute cells in the order given. Do not skip any \
Input cells within a given notebook.\nSAVING NOTEBOOKS\n\tYou can save \
anytime to any directory you choose, and it is wise to save often.\n\t\
However, before you do your final save, delete all your output by selecting \
the \n\t ",
  StyleBox["Delete All Output",
    FontSlant->"Italic"],
  " selection under the ",
  StyleBox["Kernel",
    FontSlant->"Italic"],
  " pull-down menu.\nEXPERIENCING MAJOR PROBLEMS\n\tSave if appropriate, and \
then shut down ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " and start it up again."
}], "Text"]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell["Part I:  Basic Functions and Plots", "Section"],

Cell["Let's define a function!", "Subsubsection"],

Cell["\<\
Here is how we define a multivariable function.  Note that the only \
difference between a single and multivariable function is that we define 2 \
independent variables!\
\>", "Text"],

Cell["\<\
Clear[f]
f[x_,y_]=10 Cos[x^2-y^2];\
\>", "Input",
  PageWidth->Infinity,
  FontFamily->"Courier New",
  FontSize->12,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

Cell["Let's evaluate this function when x=2 and y=1:", "Text"],

Cell["f[2,1]  ", "Input",
  PageWidth->Infinity,
  FontFamily->"Courier New",
  FontSize->12,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

Cell["\<\
Evaluate the function for the following (x,y) pairs:  (3,0) and (1,4) \
\>", "Text"],

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

Cell["Let's plot a function!", "Subsubsection"],

Cell[TextData[{
  "OK... so what does this thing look like?  \nWe use the ",
  StyleBox["Plot3D command",
    FontWeight->"Bold"],
  " to view functions of 2 variables:"
}], "Text"],

Cell[BoxData[
    \(\(?Plot3D\)\)], "Input"],

Cell[TextData[{
  "The important details are specifying the range of values that ",
  StyleBox["x",
    FontSlant->"Italic"],
  " and ",
  StyleBox["y",
    FontSlant->"Italic"],
  " will take on - everything else just makes the plot look nicer!",
  " "
}], "Text"],

Cell[BoxData[
    \(\(Plot3D[f[x, y], {x, \(-Pi\), Pi}, {y, \(-Pi\), Pi}, 
        AxesLabel \[Rule] {x, y, z}, 
        PlotLabel \[Rule] \ f[x, y]];\)\)], "Input"],

Cell[TextData[{
  "\nNotice how difficult it is to visualize details in the graph.  By \
default, ",
  StyleBox["Mathematica ",
    FontSlant->"Italic"],
  "selects 15  values from the ",
  StyleBox["x",
    FontSlant->"Italic"],
  "  domain and 15 ",
  StyleBox[" ",
    FontSlant->"Italic"],
  "values from the ",
  StyleBox["y",
    FontSlant->"Italic"],
  " domain.  It then evaluates the function for these 225 points (15x15), \
interpolates between evaluations, and graphs the data.  You can tell ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " to increase its sampling by using the ",
  StyleBox["PlotPoints",
    FontWeight->"Bold"],
  " option.  Increase the resolution of your graph by adding the following \
option to your ",
  StyleBox["Plot3D ",
    FontWeight->"Bold"],
  "command:  ",
  StyleBox["PlotPoints->30\n",
    FontWeight->"Bold"],
  "This will increase the number of values to sample for the domain to 30 for \
",
  StyleBox["x ",
    FontSlant->"Italic"],
  " and 30 for ",
  StyleBox["y",
    FontSlant->"Italic"],
  " (for a total of 900 points)... if you get too carried away, you can make \
",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " work really hard to graph your function!"
}], "Text"],

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

Cell["Try it on another function!", "Subsubsection"],

Cell[TextData[{
  "Here is another function to try - it is pretty cool looking!  First define \
the function in ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " and then plot it.\n",
  Cell[BoxData[
      \(TraditionalForm\`f(x, y) = \(e\^\(-x\)\) \(cos(x\^2 + y\^2)\)\)]]
}], "Text"],

Cell[TextData[{
  "Note: Make sure that you pick an appropriate domain for ",
  StyleBox["x",
    FontSlant->"Italic"],
  "!"
}], "Commentary",
  FontColor->RGBColor[0, 0, 1]],

Cell["\<\
Clear[f]
f[x_,y_]=
Plot3D[\
\>", "Input",
  PageWidth->Infinity,
  FontFamily->"Courier New",
  FontSize->12,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

Cell[TextData[{
  "So you want a different view of this function?\nCopy your ",
  StyleBox["Plot3D",
    FontWeight->"Bold"],
  " command to the input box below and then click after your {y,lo,upper} \
specification.  Add a comma and then click the ",
  StyleBox["Input",
    FontWeight->"Bold"],
  " pull-down menu.  Select ",
  StyleBox["3-D ViewPoint Selecter",
    FontWeight->"Bold"],
  ".  You can orient the axes any way that you want to - pick one and click \
the ",
  StyleBox["Close Dialog Box",
    FontWeight->"Bold"],
  " button.  Then, execute your ",
  StyleBox["Plot3D ",
    FontWeight->"Bold"],
  "command."
}], "Text"],

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

Cell[CellGroupData[{

Cell["Part II: Cross Sections!", "Section"],

Cell[TextData[{
  "Let's look at what happens when we take slices of a 3-D surface.  We'll \
work with what is called a ",
  StyleBox["hyperbolic paraboloid ",
    FontSlant->"Italic"],
  "(AKA a saddle), which is a function of the form: \n",
  Cell[BoxData[
      \(TraditionalForm\`f(x, y) = x\^2 - y\^2\)]],
  ". "
}], "Text"],

Cell[BoxData[{
    \(Clear[x, y, hyp]\), "\[IndentingNewLine]", 
    \(hyp[x_, y_] = x^2 - y^2\)}], "Input"],

Cell[TextData[{
  "I've defined some more ugly ",
  StyleBox["Mathematica ",
    FontSlant->"Italic"],
  "code (again, don't worry about reproducing it, just use it!).  The command \
is  called ",
  StyleBox["showsection",
    FontWeight->"Bold"],
  ", and it plots the cross section of the surface as we take slices.  Here \
is an example:"
}], "Text"],

Cell[BoxData[
    \(\(showsection[
        hyp[x, y], {x, \(-3\), 3}, {y, \(-3\), 3}, {z, \(-9\), 9}, 
        1];\)\)], "Input",
  AnimationDisplayTime->0.2197],

Cell[TextData[{
  "Now,I'll create a table of graphs to animate.  \nWe'll use the ",
  StyleBox["Table",
    FontWeight->"Bold"],
  " command a lot, so let's look more closely at it. ",
  StyleBox["Table",
    FontWeight->"Bold"],
  " creates a table of output from whatever command you want.  I want a table \
of graphs, so I'll enclose my ",
  StyleBox["showsection",
    FontWeight->"Bold"],
  " command inside of the ",
  StyleBox["Table ",
    FontWeight->"Bold"],
  "command.  Instead of the value 1 that I have at the end of the ",
  StyleBox["showsection",
    FontWeight->"Bold"],
  " command above, I'll give a parameter, ",
  StyleBox["t",
    FontSlant->"Italic"],
  ", whose value will vary for each plot.  In this case, I want to plot \
sections as ",
  StyleBox["t",
    FontSlant->"Italic"],
  " goes from -2.5 to +2.5 in steps of 0.5 (so we'll have a graph for -2.5, \
-2.0, -1.5, etc.)\nGo ahead and execute this command, then animate it.",
  " "
}], "Text"],

Cell[BoxData[
    \(\(Table[
        showsection[
          hyp[x, y], {x, \(-3\), 3}, {y, \(-3\), 3}, {z, \(-9\), 9}, 
          t], {t, \(-2.5\), 2.5, 0.5}];\)\)], "Input",
  AnimationDisplayTime->0.2197],

Cell["You Try it!", "Subsubsection"],

Cell[TextData[{
  "Go ahead and try the showsection command on the following function:\n",
  Cell[BoxData[
      \(TraditionalForm\`f(x, y) = \(e\^\(-x\)\) \(cos(x\^2 + y\^2)\)\)]]
}], "Text"],

Cell[BoxData[{
    \(\ Clear[x, y, f]\), "\[IndentingNewLine]", 
    \(\(\(f[x_, y_]\)\(=\)\)\)}], "Input"],

Cell[BoxData[
    \(\(showsection[
        f[x, y], {x, \(-3\), 0}, {y, \(-3\), 3}, {z, \(-3\), 
          10}, \(-2\)];\)\)], "Input"],

Cell[TextData[{
  "Now create an animation... let ",
  StyleBox["t",
    FontSlant->"Italic"],
  " range from -3 to 0 (use a good stepsize)..."
}], "Text"],

Cell[BoxData[
    \(\(\(Table\)\([\)\(showsection[
        f[x, y], {x, \(-3\), 0}, {y, \(-3\), 3}, {z, \(-3\), 10}, 
        t], {t, \(?? ,??,??}];\)\)\)\)], "Input",
  AnimationDisplayTime->0.2197]
}, Closed]],

Cell[CellGroupData[{

Cell["Part III:  Graphing Cylinders and Quadric Surfaces", "Section"],

Cell[TextData[{
  "When you look at the equations for cylinders or quadric surfaces, notice \
that they are not in the standard form for graphing on most graphing \
calculators or software packages, because you cannot usually solve explicity \
for the ",
  StyleBox["z",
    FontSlant->"Italic"],
  " variable. To get around this, we use a special command called ",
  StyleBox["ContourPlot3D",
    FontWeight->"Bold"],
  " after we first read in a package to enable it to work.   "
}], "Text"],

Cell[BoxData[{
    RowBox[{\(Off[General::spell]\), " "}], "\n", \(Off[
      General::spell1]\), "\n", 
    RowBox[{"<<", 
      StyleBox[
        RowBox[{"Graphics", 
          StyleBox["`",
            "MB"], "ContourPlot3D", 
          StyleBox["`",
            "MB"]}]]}]}], "Input"],

Cell[TextData[{
  "You've already seen contour plots of functions of 2 variables.  We'll see \
today how to think of a function such as ",
  Cell[BoxData[
      \(TraditionalForm\`x\^2 + y2 - z\^2 = 1\)]],
  " as a function of 3 variables:  ",
  Cell[BoxData[
      \(TraditionalForm\`f(x, y, z) = x\^2 + y2 - z\^2 - 1\)]],
  " and we are interested in the ",
  StyleBox["level surface",
    FontWeight->"Bold"],
  " for when ",
  Cell[BoxData[
      \(TraditionalForm\`f(x, y, z) = 0\)]],
  ", which means that we want ",
  Cell[BoxData[
      \(TraditionalForm\`x\^2 + y2 - z\^2 - 1 = 0\)]],
  " (compare that to what we started with...).  \nLet's start with a simple \
case.  Suppose that you want to plot the 3-D cylinder ",
  Cell[BoxData[
      \(y\  = \ x\^2\)]],
  ". We think of this as ",
  Cell[BoxData[
      \(TraditionalForm\`f(x, y, z) = \(y - x\^2 = 0\)\)]],
  ", and we ask for a ",
  StyleBox["ContourPlot3D ",
    FontWeight->"Bold"],
  "of this function. The default is to plot only the contour when the \
function specified is 0. Note that we must specify the values of ",
  StyleBox["x, y,",
    FontSlant->"Italic"],
  " and ",
  StyleBox["z",
    FontSlant->"Italic"],
  " overwhich to extend our plot, even though ",
  StyleBox["z",
    FontSlant->"Italic"],
  " does not appear in the function itself."
}], "Text"],

Cell[TextData[{
  "Note: The Boxed->False option on the command turns off the usual box that \
we have around 3-D plots in ",
  StyleBox["Mathematica.  ",
    FontSlant->"Italic"],
  "The ",
  StyleBox["AspectRatio->Automatic",
    FontWeight->"Bold"],
  " doesn't do much for us with this plot, but helps insure that level curves \
that should be circles show up as circles (instead of ellipses)."
}], "Commentary",
  FontColor->RGBColor[0, 0, 1]],

Cell[BoxData[{
    \(Clear[x, y, z, function]\), "\n", 
    \(\(function = y - x\^2;\)\), "\n", 
    \(\(ContourPlot3D[
        function, {x, \(-3\), 3}, \ {y, \(-2\), 2}, \ {z, \(-4\), 4}, 
        Axes -> True, AxesLabel -> {x, y, z}, Boxed -> False, 
        AspectRatio -> Automatic];\)\)}], "Input"],

Cell["Let's try another cylinder.", "Text"],

Cell[BoxData[{
    \(\(function = \ y\^2\  - 4\ z\^2\  - 1;\)\), "\n", 
    \(\(ContourPlot3D[\ 
        function, {x, \(-2\), 2}, \ {y, \(-2\), 2}, \ {z, \(-2\), 2}, 
        Axes -> True, AxesLabel -> {x, y, z}, Boxed -> False, 
        AspectRatio -> Automatic];\)\)}], "Input"],

Cell[BoxData[
    \(TextForm\`Next\ we\ will\ plot\ the\ quadric\ surface, \ 
    z\^2 - \((2  x\^2 + 3  y\^2)\) = \ 0. \)], "Text"],

Cell[BoxData[{
    \(\(function = z\^2 - \((2  x\^2 + 3  y\^2)\);\)\), "\n", 
    \(\(ContourPlot3D[
        function, {x, \(-3\), 3}, \ {y, \(-2\), 2}, \ {z, \(-4\), 4}, 
        Axes -> True, AxesLabel -> {x, y, z}, Boxed -> False, 
        AspectRatio -> Automatic];\)\)}], "Input"],

Cell["Here's an ellipsoid.", "Text"],

Cell[BoxData[{
    \(\(function = x^2\  + 4  y^2\  + \ z^2\  - 4;\)\), "\n", 
    \(\(ContourPlot3D[\ \ function, {x, \(-2\), 2}, \ {y, \(-1\), 
          1}, \ {z, \(-2\), 2}, Axes -> True, AxesLabel -> {x, y, z}, 
        Boxed -> False, AspectRatio -> Automatic];\)\)}], "Input"]
}, Closed]],

Cell[CellGroupData[{

Cell["You Try It: Part III", "Section"],

Cell[TextData[{
  "To plot any other surfaces we've talked about,  go back to any of the \
plots and alter the functions as you wish.  Note that some 3-D plots take \
quite a bit of time to plot. If you find you are waiting too long, you can \
always pull down the Kernel menu and select Abort Evaluation. \nTo \
experiment, replace the function in red with another function of ",
  Cell[BoxData[
      \(TraditionalForm\`x, \ y\)]],
  ", and ",
  Cell[BoxData[
      \(TraditionalForm\`z\)]],
  " - why not use the function you are assigned to analyze?  Be certain to \
use correct terminology. "
}], "Text"],

Cell[BoxData[
    RowBox[{
      RowBox[{"function", "=", 
        StyleBox[\(z\^3 - 2  x\^2 + 3  y\^2\),
          FontColor->RGBColor[1, 0, 0]]}], ";", 
      "\n", \(ContourPlot3D[
        function, \n\ \ \ {x, \(-3\), 3}, \ {y, \(-2\), 2}, \ {z, \(-4\), 
          4}, \n\ \ \ \ \ Axes -> True, AxesLabel -> {x, y, z}, 
        Boxed -> False, AspectRatio -> Automatic]\), ";"}]], "Input"]
}, Closed]],

Cell[CellGroupData[{

Cell["Part IV: More Surfaces, Contours, and Level Curves", "Section"],

Cell[TextData[{
  "In Part IV, we'll look at other surfaces that are easier to represent in \
",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " - those that we can explicitly solve for ",
  StyleBox["z",
    FontSlant->"Italic"],
  ".  \nThe following code gives a set of ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " commands that can be used as a template for defining and plotting \
surfaces in three-dimensional space. You can get more information on these \
commands and their options in the ",
  StyleBox["Help",
    FontWeight->"Bold"],
  " menu. You can insert several new functions of your own, each of a \
different type, to develop a visualization of different surfaces. Try some \
new options for the plots as well."
}], "Text"],

Cell[TextData[{
  "Note: Notice that I've used the ",
  StyleBox["Show",
    FontWeight->"Bold"],
  " command here to present the 3D plot from a different perspective.  Nice \
to have a couple of views simultaneously..."
}], "Commentary",
  FontColor->RGBColor[0, 0, 1]],

Cell[BoxData[{
    \(Clear[x, y, z, f]\), "\n", 
    \(f[x_, y_] = 
      Exp[\(-\((x\^2 + y\^2)\)\)/
            8] \((\((Cos[x])\)\^2 + \((Sin[y])\)\^2)\)\), "\n", 
    \(\(xmin = \(-5\);\)\), "\n", 
    \(\(xmax = \(+5\);\)\), "\n", 
    \(\(ymin = \(-\[Pi]\);\)\), "\n", 
    \(\(ymax = \[Pi];\)\), "\n", 
    \(\(pict = 
        Plot3D[f[x, y], {x, xmin, xmax}, {y, ymin, ymax}, 
          PlotPoints \[Rule] 40, AxesLabel -> {x, y, z}];\)\), "\n", 
    \(\(Show[pict, ViewPoint \[Rule] {0, \(-2\), 0}];\)\)}], "Input"],

Cell[TextData[{
  "You can change the viewpoint manually in the last command,or you can \
highlight the ",
  StyleBox["ViewPoint ",
    FontWeight->"Bold"],
  "command and then select the",
  StyleBox["3D ViewPoint Selector",
    FontWeight->"Bold"],
  " button under the ",
  StyleBox["Input ",
    FontWeight->"Bold"],
  "heading at the top.  Move the box around to select a new viewpoint and \
then ",
  StyleBox["paste",
    FontWeight->"Bold"],
  " that viewpoint into your code."
}], "Text",
  FontColor->RGBColor[0, 0, 1]],

Cell["Here are the level curves for the function.  ", "Text"],

Cell[BoxData[{
    \(\(ContourPlot[f[x, y], {x, xmin, xmax}, {y, ymin, ymax}, 
        PlotPoints \[Rule] 40, Axes -> True, 
        AxesLabel -> {x, y}];\)\), "\n", 
    \(\(ContourPlot[f[x, y], {x, xmin, xmax}, {y, ymin, ymax}, 
        ContourShading \[Rule] False, PlotPoints \[Rule] 40, Axes -> True, 
        AxesLabel -> {x, y}];\)\)}], "Input"],

Cell[TextData[StyleBox["Question 1:\t What is the difference between the two \
plots below, and what does light to dark signify in the first plot?",
  FontWeight->"Bold"]], "Text",
  Background->RGBColor[0, 1, 1]],

Cell[TextData[{
  "Next, look at the curves formed when you hold ",
  StyleBox["x",
    FontSlant->"Italic"],
  " or ",
  StyleBox["y",
    FontSlant->"Italic"],
  " constant."
}], "Text"],

Cell[TextData[{
  "Note: The ",
  StyleBox["Table ",
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