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Cell[CellGroupData[{
Cell["\<\
Problem 17.6 on
Explicit Surfaces\
\>", "Title",
Evaluatable->False,
AspectRatioFixed->True],
Cell["\<\
by
K. D. Stroyan
University of Iowa\
\>", "Subsubtitle",
Evaluatable->False,
AspectRatioFixed->True],
Cell["\<\
copyright 1997 by Academic Press, Inc. - All rights reserved.\
\>", "Text",
Evaluatable->False,
AspectRatioFixed->True,
FontFamily->"Times"],
Cell[CellGroupData[{
Cell[" Notebook Overview", "Section",
Evaluatable->False,
AspectRatioFixed->True],
Cell["\<\
This Mathematica Notebook is to help you make graphs of the functions given \
in text Problem 17.6 and followed up in Exercises 18.2.1, 2, 3, 4. We have \
started the first few to help you learn how to use the Plot3D[] function.\
\>", "Text",
Evaluatable->False],
Cell["\<\
Functions are defined as shown below. The underline after the variables \
indicates that it is a rule about x and y. Multiplication is indicated by a \
space. The variable xy is a single variable, but x y means x times y \
(note the space between x and y). Powers are denoted by the ^ symbol as \
shown in the first example below.\
\>", "Text",
Evaluatable->False],
Cell["\<\
Move the mouse so that the insert symbol is in the next cell, click, and then \
press the key at the far right of the keyboard (near the number pad.) \
This will command Mathematica to draw the explicit graph z = f(x,y) for \
the first function of the exercise. The range of the plot is: -3.2 < x < \
3.2 and -3.2 < y < 3.2. Draw the graph now. \
\>", "Text",
Evaluatable->False],
Cell[CellGroupData[{
Cell["References to the Text", "Subsection",
Evaluatable->False],
Cell["\<\
This NoteBook goes with Problem 17.6 and Exercises 18.2.1, 2, 3, 4.\
\>", "Text",
Evaluatable->False]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell["Problem 17.6", "Section",
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AspectRatioFixed->True],
Cell[TextData[{
" Use this NoteBook to plot all of the functions in this exercise. Compare \
the surface plots with their topographical maps from the ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" Notebook ContourPlots."
}], "Text",
Evaluatable->False],
Cell[CellGroupData[{
Cell["Function a", "Subsection",
Evaluatable->False],
Cell[" this computation:", "Text"],
Cell[BoxData[{
\(f[x_, y_] := x\ y\^2 + x\^3\ y\),
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"Input"]
}, Closed]],
Cell[CellGroupData[{
Cell["Function b", "Subsection",
Evaluatable->False],
Cell["\<\
The second function is defined below for you. Notice that Mathematica knows \
the sine and cosine functions, but that they must be typed with the notation \
Sin[x], Cos[y], that is, the first letter is capitalized and the argument \
goes between square brackets. The list of symbols \"sin(x)\" is treated \
simply as a new variable name. Mathematica does not recognize it as having \
anything to do with the sine function. It is also prejudiced against round \
brackets or parentheses ( and ) in functions, though you must use them in \
algebraic expressions such as\
\>", "Text"],
Cell["(x + y)/(x - y)", "Text"],
Cell["\<\
Move the mouse to the next cell, click, and command Mathematica to plot the \
surface. \
\>", "Text"],
Cell[BoxData[{
\(f[x_, y_] := sin\^3\ x + cos\^4\ y\ \ (*
Fix\ this\ function\ \(first!\)*) \),
\(Plot3D[f[x, y], {x, \(-3.2\), 3.2}, {y, \(-3.2\), 3.2}]\)}], "Input"]
}, Closed]],
Cell[CellGroupData[{
Cell["Function c", "Subsection",
Evaluatable->False],
Cell["\<\
\tHighlight the whole command cell above containing the function \
\>", "Text"],
Cell[TextData[Cell[BoxData[
\(f[x_, y_] := Sin[x]\^3 + Cos[y]\^4\)]]], "Text"],
Cell["\<\
(Not the cell with the surface you just drew, the one with the instructions \
to draw it.) You first put the mouse at the beginning of the cell before \
f[x_,y_], then press down the mouse button and hold it while you move to the \
end of the cell by ,3.2}]. This is called \"dragging\" the mouse. This \
will highlight the whole cell.\
\>", "Text"],
Cell["\<\
Once the cell is highlighted, copy it by holding down the key and \
pressing the key. You won't see anything happen, but move the mouse so \
the insert bar is in the blank window just below here, click, and then press \
+ which is the \"paste\" command. This should give you a copy \
of the previous comand window. Edit it so you have the function \
\>", "Text"],
Cell["f(x,y) = Sin[x*y]*Cos[x*y]", "Text"],
Cell[TextData[{
"(Fix the syntax so that ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" can understand you!)\nand plot command"
}], "Text"],
Cell["Plot3D[f[x,y], {x,-2,2}, {y,-2,2}]", "Text"],
Cell["\<\
Then press and draw the graph of the third function of the \
homework.\
\>", "Text"],
Cell["\<\
f[x_,y_] := Sin[x*y]*Cos[x*y]
Plot3D[???\
\>", "Input"]
}, Closed]],
Cell[CellGroupData[{
Cell["A Function with a Square Root", "Subsection",
Evaluatable->False],
Cell["\<\
The square root function is used in the next cell. You type it with \"Sqrt[ \
]\" or the radical symbol from the Basic Input Palette. Graph the function \
in the next cell now and explain the error messages. (Type your explanation \
in the blank text cell below the graph.)\
\>", "Text"],
Cell[BoxData[{
\(f[x_, y_] := \@\(1 - x\^2 - y\^2\)\),
\(Plot3D[f[x, y], {x, \(-1\), 1}, {y, \(-1\), 1}]\)}], "Input"],
Cell["Type your explanation here:", "Text"]
}, Closed]],
Cell[CellGroupData[{
Cell["Function d", "Subsection",
Evaluatable->False],
Cell["\<\
Now copy the cell above into the next command cell and edit it to plot the \
function\
\>", "Text"],
Cell[TextData[{
"f(x,y) = SquareRoot(1 - x^2 - y^2) ",
StyleBox[".",
FontVariations->{"CompatibilityType"->"Superscript"}],
" cos 2\[Pi]xy"
}], "Text"],
Cell[TextData[{
"Don't forget that space (or * ) means times... and ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" uses [.] for functions and Sin,..."
}], "Text",
Evaluatable->False]
}, Closed]],
Cell[CellGroupData[{
Cell["Remaining Functions", "Subsection",
Evaluatable->False],
Cell["Finish the assignment with the functions:", "Text",
Evaluatable->False],
Cell[TextData[{
"e) ",
Cell[BoxData[
\(z = \@\(x\^2 + Sin[y\^2]\)\)]]
}], "Text",
Evaluatable->False],
Cell[TextData[{
"f) ",
Cell[BoxData[
\(z = 1\/\@\(x\^2 + y\^2\)\)]]
}], "Text",
Evaluatable->False],
Cell[TextData[{
"g) ",
Cell[BoxData[
\(z = \((2\ x\^2 + 3\ y\^2)\)\ Exp[\(-\((x\^2 + y\^2)\)\)]\)]],
"\n\t or z = ",
Cell[BoxData[
\(\((2\ x\^2 + 3\ y\^2)\)\ \[ExponentialE]\^\(-\((x\^2 + y\^2)\)\)\)]]
}], "Text",
Evaluatable->False],
Cell[TextData[{
"h) ",
Cell[BoxData[
\(z = 2\ x\ y\ Exp[\(-\((x\^2\/2 + y\^2\/8)\)\)]\)]]
}], "Text",
Evaluatable->False],
Cell[TextData[{
"i) ",
Cell[BoxData[
\(z = ArcTan[x\^2\ y\^3]\)]]
}], "Text",
Evaluatable->False],
Cell[TextData[{
"j) ",
Cell[BoxData[
\(z = Log[x\^2 + y\^4]\)]]
}], "Text",
Evaluatable->False],
Cell["\<\
Simply copy, paste and edit as needed. Be careful to plot your functions \
over interesting domains of the independent variables.\
\>", "Text",
Evaluatable->False],
Cell[CellGroupData[{
Cell["Function ?", "Subsubsection",
Evaluatable->False],
Cell[BoxData[""], "Input"]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell["Special Message", "Subsection",
Evaluatable->False],
Cell["\<\
Save this NoteBook on your account, but leave it open and also open the \
ContourPlots NoteBook. The second part of this assignment is to compare your \
explicit surfaces with their contour plots.\
\>", "Text",
Evaluatable->False]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell[" Exercise 18.2.3", "Section",
Evaluatable->False,
AspectRatioFixed->True],
Cell["\<\
For each of the functions in Exercise 18.2.1 - plotted above, find one X or \
(x,y) point where the graph has a horizontal tangent plane, if there is one. \
Use this NoteBook to help see where to look for the solution of the \
appropriate condition on the partial derivatives you computed in Exercise \
18.2.2 using the PartialD NoteBook.\
\>", "Text",
Evaluatable->False]
}, Closed]]
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*)
(***********************************************************************
End of Mathematica Notebook file.
***********************************************************************)