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StyleBox[\(MATH\ 257\ Calculus\ III\t\t\t\tWeek\ of\ May\ 19, \ 2003\),
FontSize->14], "\n",
StyleBox[\(Lab\ 8 : \ Surfing\ Surfaces\),
"Title"], "\n",
StyleBox[\(Name\ 1\),
"Section"], "\n",
StyleBox[\(Name\ 2\),
"Section"], "\n",
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StyleBox["Make sure", "Text",
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StyleBox[" you execute all the commands in this lab as you read through it. \
Do this, or you will perish...", "Text"]
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FontSize->14],
Cell[TextData[StyleBox["Graphing 3D Surfaces",
FontWeight->"Bold",
FontColor->RGBColor[0, 0, 1]]], "Subtitle"],
Cell[TextData[{
"Graphing 3D surfaces in ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" is easy if you have a function of the form z=f(x,y). Just use the ",
StyleBox["Plot3D[]", "Input"],
" command, which works just like ",
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" except you have to specify the x ",
StyleBox["and",
FontWeight->"Bold"],
" y ranges.\n\nFor example, let's plot ",
StyleBox["f[x,y]=Cos[x*y] - Sin[x*y]",
FontWeight->"Bold"],
":"
}], "Text",
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Cell[BoxData[
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Cell[TextData[{
"Cool, but notice how chunky the graph is. To make it finer, use the ",
StyleBox["PlotPoints", "Input"],
" option:\n"
}], "Text",
FontSize->14],
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"\nWe can also make ",
StyleBox["traces",
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" for some constant k:"
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StyleBox["Question 1:",
FontWeight->"Bold",
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StyleBox[" In the above trace graph, which curve corresponds to the trace \
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We can also do the y-traces. Notice that for this surface that are \
notably similar to the x-traces.\
\>", "Text",
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StyleBox["Question 2:",
FontWeight->"Bold"],
" Notice how in the above contour plot the trace curves look like ... what? \
Hyperbolas? Get an equation for one of them using the ",
StyleBox["Solve[]", "Input"],
" command to solve f[x,y] = z when z=1. (Solve for either y or x.) Make a \
plot of the equation you get to see that it looks like one of the contours \
above."
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StyleBox["Question 3:",
FontWeight->"Bold"],
" For the following functions, make (i) a 3D plot, (ii) x- and y-traces, \
and (iii) a contour plot (z-trace). Be sure to ",
StyleBox["LABEL",
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" your images so we can tell what they are!",
"\n\t(a) ",
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"(b) ",
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StyleBox[Cell[BoxData[
\(TraditionalForm\`\[ExponentialE]\^\(-\(\(x\^2\ + \ y\^2\)\/3\)\)\)],
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Cell[TextData[StyleBox["Spherical Surfaces",
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FontColor->RGBColor[0, 0, 1]]], "Subtitle"],
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"Graphing surfaces made by equations in spherical coordinates takes a \
little bit of prep-work. ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" doesn't just do it all for you. It helps a lot if we define a function \
to ",
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StyleBox["DO THIS COMMAND:",
FontWeight->"Bold"]
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Cell[BoxData[
\(\(\(\[IndentingNewLine]\)\(sphrToRect[{\[Rho]_, \ \[Theta]_, \ \
\[Phi]_}] = {\[Rho]*Sin[\[Phi]]*Cos[\[Theta]], \ \[Rho]*Sin[\[Phi]]
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This takes a point in spherical coordinates (\[Rho], \[Theta], \
\[Phi]) and converts it to regular (x, y, z), like God intended!
Test it out: \
\>", "Text",
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Cell[BoxData[
\(\(\(\[IndentingNewLine]\)\(sphrToRect[{1, \ \[Pi]/6, \ \[Pi]/
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",
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FontWeight->"Bold"],
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\[Theta] and \[Phi] range over some values: 0 \[LessEqual] \[Theta] \
\[LessEqual] 2\[Pi] and 0 \[LessEqual] \[Phi] \[LessEqual] \[Pi]: We need \
to use the ",
StyleBox["ParametricPlot3D", "Input"],
" command to do this, however:",
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sphrToRect[{3, \[Theta], \[Phi]}], \ {\[Theta], \ 0, \
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But what happens if we change the range of, say, \[Theta]? Let's \
find out:\
\>", "Text",
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StyleBox["Question 4a:",
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" Make a good picture that shows what the spherical equation ",
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StyleBox[" ",
FontWeight->"Bold"],
" looks like. ",
StyleBox["Tip:",
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" you have to make \[Theta] range to some higher values, like 0 \
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to see all the cool stuff going on inside unless you change the range of \
\[Phi] as well. (Like, 1 \[LessEqual] \[Phi] \[LessEqual] \[Pi] or \
something like that.) Make sure your picture displays this. ",
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" describe what this surface is like in a sentance or two.\n"
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FontWeight->"Bold"],
". (Make a picture that shows what this surface does, and describe it in \
",
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FontWeight->"Bold"],
" as well.) Make sure to try some negative values of \[Phi] and see what's \
going on inside.",
"\n"
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title, your names, section number, and the questions and your answers\n\t\t\
(with any needed explanations, graphs, etc.).\n\t(2) \tSave this to your \
disk.\n\t(3)\tGo back to Blackboard and upload your final lab report to the \
",
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StyleBox["Note:",
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FontColor->RGBColor[0, 0, 1]],
StyleBox[" these graphic images can get BIG. Make sure you DELETE any \
images that you do not need for your answers. This will reduce the chances \
of something going wrong when you upload it to Blackboard.",
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FontWeight->"Bold"]
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you save this file from within Mathematica.
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