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Notebook[{

Cell[CellGroupData[{
Cell["Solutions to Final Exam", "Title",
  TextAlignment->Center,
  TextJustification->0],

Cell["Physics 604", "Subtitle",
  TextAlignment->Center,
  TextJustification->0],

Cell["Dec. 15, 2003", "Subsubtitle",
  TextAlignment->Center,
  TextJustification->0],

Cell["\<\
This CLOSED BOOK exam is to be completed in 2 hours.  Choose THREE of the \
four problems \[LongDash] if you attempt all four, be sure to indicate \
clearly which three should be graded.  You must explain your reasoning to \
receive full credit.\
\>", "Text"],

Cell[CellGroupData[{

Cell["Initialization", "Subsubsection"],

Cell[BoxData[{
    \(\(ClearAll["\<Global`*\>"];\)\), "\n", 
    \(Off[General::spell, General::spell1]\)}], "Input",
  CellLabel->"In[58]:="],

Cell[BoxData[{
    \(\($DefaultFont = {"\<Times\>", 12};\)\), "\n", 
    \(\($TextStyle = {FontFamily \[Rule] "\<Times\>", FontSize \[Rule] 12, 
          FontSlant \[Rule] "\<Italic\>"};\)\)}], "Input",
  CellLabel->"In[60]:="],

Cell[BoxData[{
    \(\(Needs["\<Graphics`Master`\>"];\)\), "\n", 
    \(\(Needs["\<Utilities`Notation`\>"];\)\)}], "Input",
  CellLabel->"In[62]:="]
}, Open  ]],

Cell[CellGroupData[{

Cell["Problem 1 (points per part: 4, 4, 2)", "ExerciseTitle"],

Cell[TextData[{
  "Sound waves in a pipe of slowly varying cross section ",
  Cell[BoxData[
      \(TraditionalForm\`A[x]\)]],
  " satisfy the equation"
}], "ExerciseText"],

Cell[BoxData[
    \(TraditionalForm\`\(A[
                x]\/c\^2\) \[PartialD]\^2 \[Psi][x, t]\/\[PartialD]t\^2\  - \ \
\(\[PartialD]\/\[PartialD]x\) \((A[
                x] \[PartialD]\[Psi][x, t]\/\[PartialD]x)\)\  \[Equal] \ 
      0\)], "DisplayFormula"],

Cell[TextData[{
  "Consider an ",
  StyleBox["exponential horn",
    FontSlant->"Italic"],
  " with ",
  Cell[BoxData[
      \(TraditionalForm\`A[x] \[Proportional] \[ExponentialE]\^\(a\ x\)\)]],
  ".  "
}], "ExerciseText"],

Cell["a) Show that there exist travelling waves of the form", "ExerciseText"],

Cell[BoxData[
    \(TraditionalForm\`\[Psi][x, t]\  = \ \(\[ExponentialE]\^\(\(-b\)\ x\)\) 
        Cos[k\ x\  - \ \[Omega]\ t]\)], "DisplayFormula"],

Cell[TextData[{
  "where ",
  Cell[BoxData[
      \(TraditionalForm\`b\)]],
  " is independent of ",
  Cell[BoxData[
      \(TraditionalForm\`\[Omega]\)]],
  " and determine the dispersion relation ",
  Cell[BoxData[
      \(TraditionalForm\`k[\[Omega]]\)]],
  ".  "
}], "ExerciseText"],

Cell["\<\
b) Describe this solution.  What happens for low frequencies?  Develop and \
describe an alternative solution, if needed, for low frequencies.\
\>", "ExerciseText"],

Cell[TextData[{
  "c) Evaluate and sketch the phase and group velocities for propagating \
waves as functions of either ",
  Cell[BoxData[
      \(TraditionalForm\`k\)]],
  " or ",
  Cell[BoxData[
      \(TraditionalForm\`\[Omega]\)]],
  " (your choice).  "
}], "ExerciseText"],

Cell[CellGroupData[{

Cell["a)", "Subsubsection"],

Cell["\<\
We simply substitute the proposed solution into the wave equation to obtain\
\>", "Text"],

Cell[BoxData[
    \(TraditionalForm\`\(-\(\[Omega]\^2\/c\^2\)\) Exp[\((a - b)\) x] 
          Cos[k\ x\  - \[Omega]\ t]\  + \ \(\[PartialD]\/\[PartialD]x\) \((\ 
            Exp[\((a - b)\) x] \((b\ Cos[k\ x\  - \[Omega]\ t]\  + \ 
                  k\ Sin[k\ x\  - \[Omega]\ t])\)\ )\) \[Equal] \ 
      0\)], "DisplayFormula"],

Cell["or", "Text"],

Cell[BoxData[
    \(TraditionalForm\`\(-\(\[Omega]\^2\/c\^2\)\) 
          Cos[k\ x\  - \[Omega]\ t]\  + \ \((a - 
              b)\) \((b\ Cos[k\ x\  - \[Omega]\ t] + 
              k\ Sin[k\ x\  - \[Omega]\ t])\)\  + \ \((\(-k\)\ b\ Sin[
                k\ x\  - \[Omega]\ t]\  + \ \(k\^2\) 
              Cos[k\ x\  - \[Omega]\ t])\)\  \[Equal] \ 0\)], "DisplayFormula"],

Cell["\<\
Collecting the coefficients of the sine and cosine functions provides the two \
equations\
\>", "Text"],

Cell[BoxData[{
    \(TextForm\`\(-\(\[Omega]\^2\/c\^2\)\) + \((a - b)\) b\  + \ 
        k\^2\  \[Equal] \ 0\), "\[IndentingNewLine]", 
    \(TextForm\`\((a - b)\) k\  - \ k\ b\  \[Equal] \ 0\)}], "DisplayFormula"],

Cell["such that", "Text"],

Cell[BoxData[{
    \(TextForm\`b = a\/2\), "\[IndentingNewLine]", 
    \(TextForm\`\(-\(\[Omega]\^2\/c\^2\)\) + a\^2\/4 + k\^2\  \[Equal] \ 
        0\ \ \[DoubleLongRightArrow]\ \ \ k\  = \ \@\(\[Omega]\^2\/c\^2 - \
a\^2\/4\)\)}], "DisplayFormula"],

Cell[TextData[{
  "This solution is an attenuated travelling wave where the attenuation \
length, ",
  Cell[BoxData[
      \(TraditionalForm\`b\^\(-1\)\)]],
  ", is independent of frequency.  Hence, the balance between high and \
moderately low frequencies is preserved as the sound propagates.  "
}], "Text"]
}, Open  ]],

Cell[CellGroupData[{

Cell["b)", "Subsubsection"],

Cell[TextData[{
  "However, there is a low-frequency cut-off, ",
  Cell[BoxData[
      \(TraditionalForm\`\[Omega]\_c = \(a\ c/2 = b\ c\)\)]],
  ", below which waves do not propagate because ",
  Cell[BoxData[
      \(TraditionalForm\`k\)]],
  " becomes complex.  For very low frequencies, we substitute a trial \
solution of the form"
}], "Text"],

Cell[BoxData[
    \(TraditionalForm\`\[Psi][x, 
        t]\  = \ \(\[ExponentialE]\^\(\(-b\)\ x\)\) 
            Cos[\[Omega]\ t]\ \ \ \[DoubleLongRightArrow]\ \ \ \
\(-\(\[Omega]\^2\/c\^2\)\) + \ b(a - b)\  \[Equal] \ 0\)], "DisplayFormula"],

Cell["to obtain", "Text"],

Cell[BoxData[
    \(TraditionalForm\`b\  = \ 
      a\/2\  \[PlusMinus] \ \@\(a\^2\/4 - \[Omega]\^2\/c\^2\)\)], \
"DisplayFormula"],

Cell[TextData[{
  "for standing waves.  Both damping coefficients are positive and must be \
retained, but after some distance the smaller one will dominate and \
determines the maximum penetration of low-frequencies vibrations into the \
horn.   Notice that at ",
  Cell[BoxData[
      \(TraditionalForm\`\[Omega] \[Rule] 0\)]],
  " there is a static solution with ",
  Cell[BoxData[
      \(TraditionalForm\`b \[Rule] 0\)]],
  " that is not attenuated."
}], "Text"]
}, Open  ]],

Cell[CellGroupData[{

Cell["c)", "Subsubsection"],

Cell["The phase and group velocities are given by", "Text"],

Cell[BoxData[{
    \(TextForm\`v\_p = \(\[Omega]\/k\  = \ 
        c \@\( 1 + a\^2\/\(4  k\^2\)\)\)\), "\[IndentingNewLine]", 
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          k\ \[DifferentialD]k\ \ \[DoubleLongRightArrow]\ \ v\_g = \(\
\[DifferentialD]\[Omega]\/\[DifferentialD]k = 
        c\^2\/v\_p\)\)}], "DisplayFormula"],

Cell["such that", "Text"],

Cell[BoxData[
    \(TextForm\`\(v\_p\) v\_g = c\^2\)], "DisplayFormula"],

Cell[TextData[{
  "These velocities are sketched below as functions of ",
  Cell[BoxData[
      \(TraditionalForm\`k/a\)]],
  ".  Notice that ",
  Cell[BoxData[
      \(TraditionalForm\`v\_p\)]],
  " diverges and ",
  Cell[BoxData[
      \(TraditionalForm\`v\_g\)]],
  " vanishes at ",
  Cell[BoxData[
      \(TraditionalForm\`k = 0\)]],
  "."
}], "Text"],

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  ImageRangeCache->{{{0, 359}, {221.375, 0}} -> {-1.05237, -0.648943, \
0.0073285, 0.0073285}, {{8.875, 350.063}, {216.063, 5.25}} -> {-1.4312, \
0.0865052, 0.0241145, 0.00817521}, {{230.625, 333}, {59.5, 16.875}} -> \
{-7.67618, -9.67027, 0.0410565, 0.0739016}, {{236.125, 266.563}, {56.4375, \
39.5}} -> {-9.12981, -21.4997, 0.0478916, 0.15498}, {{236.125, 266.563}, \
{36.8125, 19.875}} -> {-9.12981, -23.9329, 0.0478916, 0.15498}}]
}, Open  ]]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Problem 2 (10 points)", "ExerciseTitle"],

Cell["Use the Laplace transform to solve the initial-value problem", \
"ExerciseText"],

Cell[BoxData[
    \(TextForm\`\(y\^\[Prime]\)[
          t]\  + \ \[Integral]\_0\%t 
              y[\[Tau]] \[DifferentialD]\[Tau]\  \[Equal] \ \[ExponentialE]\^\
\(-t\)\ , \ y[0] \[Equal] 1\)], "DisplayFormula"],

Cell[CellGroupData[{

Cell["solution", "Subsubsection"],

Cell["\<\
The Laplace transform of this integro-differential equation reads\
\>", "Text"],

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\ y\&~\  = \ \(s \((s + 2)\)\)\/\(\((s + 1)\) \((s\^2 + 1)\)\)\)], \
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The solution transform is resolved into partial fractions according to\
\>", "Text"],

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            A \((s\^2 + 1)\) + B \((s + 1)\) \((s - \[ImaginaryI])\) + 
        C \((s + 1)\) \((s + \[ImaginaryI])\) \[Equal] 
      s\^2 + 2  s\)], "DisplayFormula"],

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Cell[BoxData[
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        0\ \ \[DoubleLongRightArrow]\ \ \ A\  = \ \(\(2  \[ImaginaryI]\)\/\(1 \
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\[ImaginaryI]\)\/\(4  \[ImaginaryI]\) = \(\(\ \)\(3 - \
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\[ImaginaryI])\)\^2\  = \ \(-\(1\/2\)\)\)\)\), "\[IndentingNewLine]", 
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\[ImaginaryI])\)\)\/\(4 \((1 - \[ImaginaryI])\)\)\  = \ \(\(\((4 - 2  \
\[ImaginaryI])\) \((1 + \[ImaginaryI])\)\)\/8\  = \ \(3 + \
\[ImaginaryI]\)\/4\)\)\)\)}], "DisplayFormula"],

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Cell[CellGroupData[{

Cell["Problem 3 (10 points)", "ExerciseTitle"],

Cell["Consider an integral of the form", "ExerciseText"],

Cell[BoxData[
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  FontFamily->"Times New Roman"],

Cell[TextData[{
  "where ",
  Cell[BoxData[
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  " is well-behaved on the positive real axis but is not symmetric with \
respect to the sign of ",
  Cell[BoxData[
      \(TraditionalForm\`x\)]],
  ", such that the integration interval cannot be extended for use with a \
great semicircle.  If the corresponding ",
  Cell[BoxData[
      \(TraditionalForm\`g[z]\)]],
  " is a meromorphic function that decreases sufficiently rapidly for ",
  Cell[BoxData[
      \(TraditionalForm\`z \[Rule] \[Infinity]\)]],
  " and is sufficiently small at the origin, one can often use ",
  Cell[BoxData[
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  " contour of the form shown below.  Apply this method for the following \
integral."
}], "ExerciseText"],

Cell[BoxData[
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Cell["\<\
For this integral one could resolve the integrand into partial fractions \
instead, but to obtain full credit you must employ the method described \
above, which can  be useful for more complicated problems that are not \
suitable for partial-fraction decomposition.  Be sure to justify neglect of \
any portions of the contour that do not contribute.\
\>", "ExerciseText"],

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Cell["Let", "Text"],

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  "For large ",
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Cell["Problem 4 (5 points per part)", "ExerciseTitle"],

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  "a) Develop an expansion for the electrostatic potential ",
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appropriate integral over ",
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Cell["The Laplacian takes the form", "Text"],

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",
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    CellFrameLabelMargins->6,
    InputAutoReplacements->{"TeX"->StyleBox[ 
            RowBox[ {"T", 
              AdjustmentBox[ 
              "E", BoxMargins -> {{-.074999999999999997, \
-.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], 
        "LaTeX"->StyleBox[ 
            RowBox[ {"L", 
              StyleBox[ 
                AdjustmentBox[ 
                "A", BoxMargins -> {{-.35999999999999999, \
-.10000000000000001}, {0, 0}}, BoxBaselineShift -> -.20000000000000001], 
                FontSize -> Smaller], "T", 
              AdjustmentBox[ 
              "E", BoxMargins -> {{-.074999999999999997, \
-.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], 
        "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica"},
    CounterIncrements->"Subsection",
    CounterAssignments->{{"Subsubsection", 0}},
    FontFamily->"Helvetica",
    FontSize->15,
    FontWeight->"Bold"],
  
  Cell[StyleData["Subsection", "Printout"],
    CellMargins->{{2, 4}, {2, 18}},
    FontSize->12]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["Subsubsection"],
    CellMargins->{{10, 4}, {2, 18}},
    CellElementSpacings->{"ClosedGroupTopMargin"->12},
    CellGroupingRules->{"SectionGrouping", 50},
    PageBreakBelow->False,
    InputAutoReplacements->{"TeX"->StyleBox[ 
            RowBox[ {"T", 
              AdjustmentBox[ 
              "E", BoxMargins -> {{-.074999999999999997, \
-.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], 
        "LaTeX"->StyleBox[ 
            RowBox[ {"L", 
              StyleBox[ 
                AdjustmentBox[ 
                "A", BoxMargins -> {{-.35999999999999999, \
-.10000000000000001}, {0, 0}}, BoxBaselineShift -> -.20000000000000001], 
                FontSize -> Smaller], "T", 
              AdjustmentBox[ 
              "E", BoxMargins -> {{-.074999999999999997, \
-.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], 
        "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica"},
    CounterIncrements->"Subsubsection",
    FontFamily->"Helvetica",
    FontWeight->"Bold"],
  
  Cell[StyleData["Subsubsection", "Printout"],
    CellMargins->{{2, 4}, {2, 18}},
    FontSize->10]
  }, Closed]]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell["Text", "Subsection"],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["Text"],
    CellMargins->{{10, 4}, {0, 8}},
    InputAutoReplacements->{"TeX"->StyleBox[ 
            RowBox[ {"T", 
              AdjustmentBox[ 
              "E", BoxMargins -> {{-.074999999999999997, \
-.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], 
        "LaTeX"->StyleBox[ 
            RowBox[ {"L", 
              StyleBox[ 
                AdjustmentBox[ 
                "A", BoxMargins -> {{-.35999999999999999, \
-.10000000000000001}, {0, 0}}, BoxBaselineShift -> -.20000000000000001], 
                FontSize -> Smaller], "T", 
              AdjustmentBox[ 
              "E", BoxMargins -> {{-.074999999999999997, \
-.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], 
        "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica"},
    Hyphenation->True,
    ParagraphSpacing->{0, 8},
    CounterIncrements->"Text"],
  
  Cell[StyleData["Text", "Printout"],
    CellMargins->{{2, 4}, {0, 8}},
    FontSize->10]
  }, Open  ]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["MathCaption"],
    CellFrame->{{4, 0}, {0, 0}},
    CellMargins->{{47, 62}, {0, 14}},
    CellFrameMargins->5,
    CellFrameColor->RGBColor[0, 0.2, 1],
    Hyphenation->True,
    LineSpacing->{1, 1},
    ParagraphSpacing->{0, 8},
    FontColor->RGBColor[0, 0, 0.6]],
  
  Cell[StyleData["MathCaption", "Printout"],
    CellMargins->{{34, 62}, {0, 14}},
    CellFrameColor->GrayLevel[0.700008],
    FontSize->10,
    FontColor->GrayLevel[0]]
  }, Closed]]
  }, Open  ]],
  
  Cell[CellGroupData[{
  
  Cell["Input/Output", "Subsection"],
  
  Cell["\<\
The cells in this section define styles used for input and output to the \
kernel.  Be careful when modifying, renaming, or removing these styles, \
because the front end associates special meanings with these style names.    \
\
\>", "Text"],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["Input"],
    CellMargins->{{56, 4}, {3, 9}},
    Evaluatable->True,
    CellGroupingRules->"InputGrouping",
    PageBreakWithin->False,
    GroupPageBreakWithin->False,
    CellLabelMargins->{{21, Inherited}, {Inherited, Inherited}},
    DefaultFormatType->DefaultInputFormatType,
    HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"},
    AutoItalicWords->{},
    LanguageCategory->"Formula",
    FormatType->StandardForm,
    ShowStringCharacters->True,
    NumberMarks->True,
    LinebreakAdjustments->{0.85, 2, 10, 0, 1},
    CounterIncrements->"Input",
    FontWeight->"Bold"],
  
  Cell[StyleData["Input", "Printout"],
    ShowCellBracket->False,
    CellMargins->{{42, 4}, {3, 8}},
    LinebreakAdjustments->{0.85, 2, 10, 1, 1},
    FontSize->10]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["Output"],
    CellMargins->{{57, 4}, {5, 2}},
    CellEditDuplicate->True,
    CellGroupingRules->"OutputGrouping",
    PageBreakWithin->False,
    GroupPageBreakWithin->False,
    GeneratedCell->True,
    CellAutoOverwrite->True,
    CellLabelMargins->{{21, Inherited}, {Inherited, Inherited}},
    DefaultFormatType->DefaultOutputFormatType,
    HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"},
    AutoItalicWords->{},
    LanguageCategory->"Formula",
    FormatType->StandardForm,
    CounterIncrements->"Output"],
  
  Cell[StyleData["Output", "Printout"],
    ShowCellBracket->False,
    CellMargins->{{42, 4}, {4, 2}},
    FontSize->10]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["Message"],
    CellMargins->{{56, 4}, {3, 8}},
    CellGroupingRules->"OutputGrouping",
    PageBreakWithin->False,
    GroupPageBreakWithin->False,
    GeneratedCell->True,
    CellAutoOverwrite->True,
    ShowCellLabel->False,
    CellLabelMargins->{{21, Inherited}, {Inherited, Inherited}},
    DefaultFormatType->DefaultOutputFormatType,
    HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"},
    AutoItalicWords->{},
    FormatType->InputForm,
    CounterIncrements->"Message",
    StyleMenuListing->None,
    FontColor->RGBColor[0, 0.2, 1]],
  
  Cell[StyleData["Message", "Printout"],
    ShowCellBracket->False,
    CellMargins->{{42, 4}, {4, 2}},
    FontSize->10,
    FontColor->GrayLevel[0]]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["Print"],
    CellMargins->{{56, 4}, {3, 8}},
    CellGroupingRules->"OutputGrouping",
    PageBreakWithin->False,
    GroupPageBreakWithin->False,
    GeneratedCell->True,
    CellAutoOverwrite->True,
    ShowCellLabel->False,
    CellLabelMargins->{{21, Inherited}, {Inherited, Inherited}},
    DefaultFormatType->DefaultOutputFormatType,
    HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"},
    AutoItalicWords->{},
    FormatType->InputForm,
    CounterIncrements->"Print",
    StyleMenuListing->None],
  
  Cell[StyleData["Print", "Printout"],
    ShowCellBracket->False,
    CellMargins->{{42, 4}, {4, 2}},
    FontSize->10]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["Graphics"],
    CellMargins->{{56, Inherited}, {Inherited, Inherited}},
    CellGroupingRules->"GraphicsGrouping",
    CellHorizontalScrolling->True,
    PageBreakWithin->False,
    GeneratedCell->True,
    CellAutoOverwrite->True,
    ShowCellLabel->False,
    DefaultFormatType->DefaultOutputFormatType,
    FormatType->InputForm,
    CounterIncrements->"Graphics",
    StyleMenuListing->None],
  
  Cell[StyleData["Graphics", "Printout"],
    CellMargins->{{40, 4}, {4, 2}},
    ImageSize->{250, 250},
    FontSize->9]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["CellLabel"],
    StyleMenuListing->None,
    FontFamily->"Helvetica",
    FontSize->9,
    FontColor->RGBColor[0, 0.2, 1]],
  
  Cell[StyleData["CellLabel", "Printout"],
    FontSize->7,
    FontSlant->"Oblique",
    FontColor->GrayLevel[0]]
  }, Closed]]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell["Unique Styles", "Subsection"],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["Abstract"],
    CellFrame->False,
    CellMargins->{{45, 75}, {Inherited, 30}},
    Hyphenation->True,
    LineSpacing->{1, 1}],
  
  Cell[StyleData["Abstract", "Printout"],
    CellMargins->{{36, 67}, {Inherited, 50}},
    FontSize->10]
  }, Open  ]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["TextTop"],
    CellFrame->{{0, 0}, {0, 2}},
    CellMargins->{{10, 4}, {2, 80}},
    CellHorizontalScrolling->True,
    CellFrameMargins->4,
    ShowSpecialCharacters->Automatic,
    ParagraphSpacing->{0, 8},
    CounterIncrements->"Text"],
  
  Cell[StyleData["TextTop", "Printout"],
    CellMargins->{{2, 4}, {2, 80}},
    FontSize->10]
  }, Open  ]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["ItemizedText"],
    CellMargins->{{20, 4}, {0, 8}},
    ShowSpecialCharacters->Automatic,
    Hyphenation->True,
    ParagraphSpacing->{0, 8},
    ParagraphIndent->-15,
    CounterIncrements->"Text"],
  
  Cell[StyleData["ItemizedText", "Printout"],
    FontSize->10]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["ItemizedTextNote"],
    CellMargins->{{35, 4}, {0, 4}},
    ShowSpecialCharacters->Automatic,
    Hyphenation->True,
    ParagraphSpacing->{0, 4},
    CounterIncrements->"Text"],
  
  Cell[StyleData["ItemizedTextNote", "Printout"],
    FontSize->10]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["IndentedText"],
    CellMargins->{{20, 4}, {0, 6}},
    ShowSpecialCharacters->Automatic,
    Hyphenation->True,
    ParagraphSpacing->{0, 8},
    CounterIncrements->"Text"],
  
  Cell[StyleData["IndentedText", "Printout"],
    FontSize->10]
  }, Open  ]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["Note"],
    CellFrame->True,
    CellMargins->{{10, 4}, {0, 8}},
    CellHorizontalScrolling->True,
    ShowSpecialCharacters->Automatic,
    ParagraphSpacing->{0, 8},
    CounterIncrements->"Text",
    FontFamily->"Helvetica",
    FontSize->10],
  
  Cell[StyleData["Note", "Printout"],
    CellMargins->{{2, 4}, {0, 8}},
    FontSize->8]
  }, Open  ]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["1ColumnBox"],
    CellFrame->1,
    CellMargins->{{30, 30}, {0, 8}},
    CellHorizontalScrolling->True,
    LineIndent->0,
    Background->GrayLevel[0.900008],
    FrameBoxOptions->{BoxMargins->{{1, 1}, {1.5, 1.5}}},
    GridBoxOptions->{ColumnSpacings->1}],
  
  Cell[StyleData["1ColumnBox", "Printout"],
    CellMargins->{{30, 30}, {0, 8}},
    FontSize->10,
    Background->GrayLevel[0.900008]]
  }, Open  ]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["2ColumnBox"],
    CellFrame->True,
    CellMargins->{{10, 4}, {0, 8}},
    CellHorizontalScrolling->True,
    LineIndent->0,
    Background->GrayLevel[0.8],
    FrameBoxOptions->{BoxMargins->{{1, 1}, {1.5, 1.5}}},
    GridBoxOptions->{ColumnWidths->{0.39, 0.59}}],
  
  Cell[StyleData["2ColumnBox", "Printout"],
    CellMargins->{{2, 4}, {0, 8}},
    FontSize->10,
    Background->GrayLevel[0.900008]]
  }, Open  ]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["2ColumnSmallBox"],
    CellFrame->True,
    CellMargins->{{10, 4}, {0, 8}},
    CellHorizontalScrolling->True,
    LineIndent->0,
    Background->GrayLevel[0.8],
    FrameBoxOptions->{BoxMargins->{{1, 1}, {1.5, 1.5}}},
    GridBoxOptions->{ColumnSpacings->1.5,
    ColumnWidths->0.35,
    ColumnAlignments->{Right, Left}}],
  
  Cell[StyleData["2ColumnSmallBox", "Printout"],
    CellMargins->{{2, 4}, {0, 8}},
    FontSize->10,
    Background->GrayLevel[0.900008]]
  }, Open  ]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["3ColumnBox"],
    CellFrame->True,
    CellMargins->{{10, 4}, {0, 8}},
    CellHorizontalScrolling->True,
    LineIndent->0,
    Background->GrayLevel[0.8],
    FrameBoxOptions->{BoxMargins->{{1, 1}, {1.5, 1.5}}},
    GridBoxOptions->{ColumnWidths->0.325}],
  
  Cell[StyleData["3ColumnBox", "Printout"],
    CellMargins->{{2, 4}, {0, 8}},
    FontSize->10,
    Background->GrayLevel[0.900008]]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["3ColumnSmallBox"],
    CellFrame->True,
    CellMargins->{{10, 4}, {0, 8}},
    CellHorizontalScrolling->True,
    LineIndent->0,
    Background->GrayLevel[0.8],
    FrameBoxOptions->{BoxMargins->{{1, 1}, {1.5, 1.5}}},
    GridBoxOptions->{ColumnSpacings->1.5,
    ColumnWidths->0.23,
    ColumnAlignments->{Right, Center, Left}}],
  
  Cell[StyleData["3ColumnSmallBox", "Printout"],
    CellMargins->{{2, 4}, {0, 8}},
    FontSize->10,
    Background->GrayLevel[0.900008]]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["4ColumnBox"],
    CellFrame->True,
    CellMargins->{{10, 4}, {0, 8}},
    CellHorizontalScrolling->True,
    LineIndent->0,
    Background->GrayLevel[0.8],
    FrameBoxOptions->{BoxMargins->{{1, 1}, {1.5, 1.5}}},
    GridBoxOptions->{ColumnWidths->{0.145, 0.345, 0.145, 0.345}}],
  
  Cell[StyleData["4ColumnBox", "Printout"],
    CellMargins->{{2, 4}, {0, 8}},
    FontSize->10,
    Background->GrayLevel[0.900008]]
  }, Open  ]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["5ColumnBox"],
    CellFrame->True,
    CellMargins->{{10, 4}, {0, 8}},
    CellHorizontalScrolling->True,
    LineIndent->0,
    Background->GrayLevel[0.8],
    FrameBoxOptions->{BoxMargins->{{1, 1}, {1.5, 1.5}}},
    GridBoxOptions->{ColumnWidths->0.195}],
  
  Cell[StyleData["5ColumnBox", "Printout"],
    CellMargins->{{2, 4}, {0, 8}},
    FontSize->10,
    Background->GrayLevel[0.900008]]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["6ColumnBox"],
    CellFrame->True,
    CellMargins->{{10, 4}, {0, 8}},
    CellHorizontalScrolling->True,
    LineIndent->0,
    Background->GrayLevel[0.8],
    FrameBoxOptions->{BoxMargins->{{1, 1}, {1.5, 1.5}}},
    GridBoxOptions->{ColumnWidths->{0.13, 0.23, 0.13, 0.13, 0.23, 0.13}}],
  
  Cell[StyleData["6ColumnBox", "Printout"],
    CellMargins->{{2, 4}, {0, 8}},
    FontSize->10,
    Background->GrayLevel[0.900008]]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["Picture"],
    CellMargins->{{10, Inherited}, {0, 8}},
    CellHorizontalScrolling->True],
  
  Cell[StyleData["Picture", "Printout"],
    CellMargins->{{2, Inherited}, {0, 8}}]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["Caption"],
    CellMargins->{{10, 50}, {0, 3}},
    PageBreakAbove->False,
    Hyphenation->True,
    FontFamily->"Helvetica",
    FontSize->9],
  
  Cell[StyleData["Caption", "Printout"],
    CellMargins->{{2, 50}, {2, 4}},
    FontSize->7]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["ExerciseTitle"],
    CellDingbat->"\[FilledDownTriangle]",
    CellMargins->{{23, Inherited}, {4, 18}},
    CellGroupingRules->{"SectionGrouping", 40},
    PageBreakBelow->False,
    CounterIncrements->"Subsection",
    CounterAssignments->{{"Subsubsection", 0}},
    FontFamily->"Times",
    FontSize->13,
    FontWeight->"Bold",
    FontColor->GrayLevel[0]],
  
  Cell[StyleData["ExerciseTitle", "Printout"],
    CellDingbat->"\[FilledDownTriangle]",
    CellMargins->{{9, 0}, {6, 20}},
    FontSize->11,
    FontColor->GrayLevel[0]]
  }, Open  ]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["ExerciseText"],
    CellMargins->{{32, 10}, {5, 5}},
    LineSpacing->{1, 3},
    ParagraphSpacing->{0, 8},
    CounterIncrements->"ExerciseText",
    FontFamily->"Times"],
  
  Cell[StyleData["ExerciseText", "Printout"],
    CellMargins->{{32, 0}, {4, 4}},
    ParagraphSpacing->{0, 6},
    FontSize->10]
  }, Open  ]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["Problem"],
    CellDingbat->"\[FilledDownTriangle]",
    CellMargins->{{23, Inherited}, {4, 18}},
    CellGroupingRules->{"SectionGrouping", 40},
    PageBreakBelow->False,
    CounterIncrements->"Subsection",
    CounterAssignments->{{"Subsubsection", 0}},
    FontFamily->"Times",
    FontSize->13,
    FontWeight->"Plain",
    FontColor->GrayLevel[0]],
  
  Cell[StyleData["Problem", "Printout"]]
  }, Open  ]]
  }, Open  ]],
  
  Cell[CellGroupData[{
  
  Cell["Tables", "Subsection"],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["2ColumnTable"],
    CellMargins->{{10, 4}, {0, 8}},
    CellHorizontalScrolling->True,
    GridBoxOptions->{ColumnWidths->{0.39, 0.59},
    ColumnAlignments->{Left}}],
  
  Cell[StyleData["2ColumnTable", "Printout"],
    CellMargins->{{2, 4}, {0, 8}},
    FontSize->10]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["3ColumnTable"],
    CellMargins->{{10, 4}, {0, 8}},
    CellHorizontalScrolling->True,
    StyleMenuListing->None,
    GridBoxOptions->{ColumnWidths->0.325,
    ColumnAlignments->{Left}}],
  
  Cell[StyleData["3ColumnTable", "Printout"],
    CellMargins->{{2, 4}, {0, 8}},
    FontSize->10]
  }, Closed]]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell["Formulas and Programming", "Subsection"],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["ChemicalFormula"],
    CellMargins->{{42, Inherited}, {Inherited, Inherited}},
    CellHorizontalScrolling->True,
    DefaultFormatType->DefaultInputFormatType,
    HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"},
    LanguageCategory->"Formula",
    AutoSpacing->False,
    ScriptLevel->1,
    ScriptBaselineShifts->{0.6, Automatic},
    SingleLetterItalics->False,
    ZeroWidthTimes->True],
  
  Cell[StyleData["ChemicalFormula", "Printout"],
    CellMargins->{{34, Inherited}, {Inherited, Inherited}},
    FontSize->10]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["DisplayFormula"],
    CellMargins->{{42, Inherited}, {Inherited, Inherited}},
    CellHorizontalScrolling->True,
    DefaultFormatType->TraditionalForm,
    HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"},
    LanguageCategory->"Formula",
    ScriptLevel->0,
    SingleLetterItalics->True,
    SpanMaxSize->Infinity,
    UnderoverscriptBoxOptions->{LimitsPositioning->True},
    GridBoxOptions->{ColumnWidths->Automatic}],
  
  Cell[StyleData["DisplayFormula", "Printout"],
    CellMargins->{{34, Inherited}, {Inherited, Inherited}},
    FontSize->10]
  }, Open  ]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["Program"],
    CellMargins->{{10, 4}, {0, 8}},
    CellHorizontalScrolling->True,
    Hyphenation->False,
    LanguageCategory->"Formula",
    FontFamily->"Courier"],
  
  Cell[StyleData["Program", "Printout"],
    CellMargins->{{2, Inherited}, {Inherited, Inherited}},
    FontSize->9.5]
  }, Closed]]
  }, Open  ]],
  
  Cell[CellGroupData[{
  
  Cell["Hyperlink Styles", "Subsection"],
  
  Cell["\<\
The cells below define styles useful for making hypertext ButtonBoxes.  The \
\"Hyperlink\" style is for links within the same Notebook, or between \
Notebooks.\
\>", "Text"],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["Hyperlink"],
    StyleMenuListing->None,
    ButtonStyleMenuListing->Automatic,
    FontColor->RGBColor[0, 0, 1],
    FontVariations->{"Underline"->True},
    ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ {
        FrontEnd`NotebookLocate[ #2]}]&),
    Active->True,
    ButtonNote->ButtonData}],
  
  Cell[StyleData["Hyperlink", "Printout"],
    FontSize->10,
    FontColor->GrayLevel[0],
    FontVariations->{"Underline"->False}]
  }, Closed]],
  
  Cell["\<\
The following styles are for linking automatically to the on-line help \
system.\
\>", "Text"],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["MainBookLink"],
    StyleMenuListing->None,
    ButtonStyleMenuListing->Automatic,
    FontColor->RGBColor[0, 0, 1],
    FontVariations->{"Underline"->True},
    ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ {
        FrontEnd`HelpBrowserLookup[ "MainBook", #]}]&),
    Active->True,
    ButtonFrame->"None"}],
  
  Cell[StyleData["MainBookLink", "Printout"],
    FontSize->10,
    FontColor->GrayLevel[0],
    FontVariations->{"Underline"->False}]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["AddOnsLink"],
    StyleMenuListing->None,
    ButtonStyleMenuListing->Automatic,
    FontFamily->"Courier",
    FontColor->RGBColor[0, 0, 1],
    FontVariations->{"Underline"->True},
    ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ {
        FrontEnd`HelpBrowserLookup[ "AddOns", #]}]&),
    Active->True,
    ButtonFrame->"None"}],
  
  Cell[StyleData["AddOnsLink", "Printout"],
    FontSize->10,
    FontColor->GrayLevel[0],
    FontVariations->{"Underline"->False}]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["RefGuideLink"],
    StyleMenuListing->None,
    ButtonStyleMenuListing->Automatic,
    FontFamily->"Courier",
    FontColor->RGBColor[0, 0, 1],
    FontVariations->{"Underline"->True},
    ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ {
        FrontEnd`HelpBrowserLookup[ "RefGuide", #]}]&),
    Active->True,
    ButtonFrame->"None"}],
  
  Cell[StyleData["RefGuideLink", "Printout"],
    FontSize->10,
    FontColor->GrayLevel[0],
    FontVariations->{"Underline"->False}]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["GettingStartedLink"],
    StyleMenuListing->None,
    ButtonStyleMenuListing->Automatic,
    FontColor->RGBColor[0, 0, 1],
    FontVariations->{"Underline"->True},
    ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ {
        FrontEnd`HelpBrowserLookup[ "GettingStarted", #]}]&),
    Active->True,
    ButtonFrame->"None"}],
  
  Cell[StyleData["GettingStartedLink", "Printout"],
    FontSize->10,
    FontColor->GrayLevel[0],
    FontVariations->{"Underline"->False}]
  }, Closed]],
  
  Cell[CellGroupData[{
  
  Cell[StyleData["OtherInformationLink"],
    StyleMenuListing->None,
    ButtonStyleMenuListing->Automatic,
    FontColor->RGBColor[0, 0, 1],
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