Mathematica notebook_1

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Unformatted text preview: (************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematicacompatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ (*NotebookOutlinePosition[ (* CellTagsIndexPosition[ (*WindowFrame->Normal*) 1425881, 1426776, 1426684, 38606]*) 38637]*) 38631]*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ "Second Steps with ", StyleBox["Mathematica", FontSlant->"Italic"] }], "Title", FontColor->RGBColor[0, 0, 1]], Cell[CellGroupData[{ Cell["1. Lists and Tables, Vectors and Matrices", "Section"], Cell[TextData[{ "One of the most fundamental notions in ", StyleBox["Mathematica", FontSlant->"Italic"], " is that of a ", StyleBox["list. ", FontSlant->"Italic"], "A list is an ", StyleBox["ordered ", FontVariations->{"Underline"->True}], "collection of objects. " }], "Text"], Cell[BoxData[ \(A = {2, \(-1\), 3, 0, 4, \(-2\), 5}; B = {3, 2, 1, 0};\)], "Input"], Cell[TextData[{ StyleBox["\[FilledSmallCircle] ", FontColor->RGBColor[0, 0, 1]], "The ", Cell[BoxData[ \(TraditionalForm\`n\^th\)]], "element of a list ", Cell[BoxData[ \(TraditionalForm\`L\)]], " is ", Cell[BoxData[ \(TraditionalForm\`L[\([n]\)]\)]], " or ", Cell[BoxData[ FormBox[ StyleBox[\(Part[L, n]\), FontFamily->"Courier New"], TraditionalForm]]], "and the number of entries in ", Cell[BoxData[ \(TraditionalForm\`L\)]], " is ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["Length", FontFamily->"Courier New"], "[", "L", "]"}], TraditionalForm]]], "." }], "Text"], Cell[BoxData[ \(A[\([4]\)]\)], "Input"], Cell[BoxData[ \(Length[B]\)], "Input"], Cell[TextData[{ StyleBox["\[FilledSmallCircle] ", FontColor->RGBColor[0, 0, 1]], "In the following example, ", Cell[BoxData[ \(TraditionalForm\`c\)]], " is a list whose entries are themselves lists." }], "Text"], Cell[BoxData[ \(\(c = {{2, \(-1\)}, {1, 3, 5}, {0, 2, 1, \(-3\)}};\)\)], "Input"], Cell[BoxData[ \(c[\([2]\)]\)], "Input"], Cell[TextData[{ StyleBox["\[FilledSmallCircle] ", FontColor->RGBColor[0, 0, 1]], "The ", Cell[BoxData[ \(TraditionalForm\`j\^th\)]], " entry of ", Cell[BoxData[ \(TraditionalForm\`c[\([i]\)]\)]], " is ", Cell[BoxData[ \(TraditionalForm\`c[\([i, j]\)]\)]] }], "Text"], Cell[BoxData[ \(c[\([2, 3]\)]\)], "Input"], Cell[BoxData[ \(c[\([3, 1]\)]\)], "Input"], Cell[CellGroupData[{ Cell["1.1 An application to solutions of equations", "Subsection", FontColor->RGBColor[1, 0, 1]], Cell[TextData[{ "The ", StyleBox["Solve", FontFamily->"Courier New"], " command returns a list of rules corresponding to solutions of an \ equation." }], "Text"], Cell[BoxData[ \(solutions = NSolve[x\^3 - 2 x + 3 \[Equal] 0, x]\)], "Input"], Cell[BoxData[ \(FullForm[solutions]\)], "Input"], Cell[TextData[{ Cell[BoxData[ FormBox[ StyleBox[\(solutions[\([1, 1]\)]\), FontFamily->"Courier New"], TraditionalForm]]], " is a rule:" }], "Text"], Cell[BoxData[ \(solutions[\([1, 1]\)]\)], "Input"], Cell["Thus the real solution of the equation is", "Text"], Cell[BoxData[ \(realsol = solutions[\([1, 1, 2]\)]\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["1.2 Tables", "Subsection", FontColor->RGBColor[1, 0, 1]], Cell[TextData[{ StyleBox["\[FilledSmallCircle] ", FontColor->RGBColor[0, 0, 1]], "A convenient way to construct lists is with a table.Given a function ", Cell[BoxData[ \(TraditionalForm\`f\)]], ", the command ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["Table", FontFamily->"Courier New"], "[", \(f[k], {k, \ n, m}\), "]"}], TraditionalForm]]], " generates a list ", Cell[BoxData[ \(TraditionalForm\`{f(n), f(n + 1), ... , f(m)}\)]], ". " }], "Text"], Cell[BoxData[ \(a = Table[\((k^2 - 2 k)\)/\((k + 3)\), {k, 0, 5}]\)], "Input"], Cell[BoxData[ \(a[\([3]\)]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[Random[Integer, {\(-2\), 3}], {i, 1, 3}]\)], "Input"], Cell[BoxData[ \({0, 1, 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Table[Random[Integer, {\(-2\), 3}], {i, 1, 3}]\)], "Input"], Cell[BoxData[ \({\(-2\), 1, 3}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[" 1.2.1 How to label a table", "Subsubsection", FontColor->RGBColor[1, 0, 0]], Cell[TextData[{ StyleBox["\[FilledSmallCircle] ", FontColor->RGBColor[0, 0, 1]], "The command ", StyleBox["TableForm", FontFamily->"Courier New"], " prints a list in a rectangular array. A single level list is printed as a \ column, and a prints a two\[Hyphen]level list is displayed as a \ two\[Hyphen]dimensional table that can have rows and columns labeled." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(TableForm[{22, 30, 11, 19, 25}]\)], "Input"], Cell[BoxData[ InterpretationBox[GridBox[{ {"22"}, {"30"}, {"11"}, {"19"}, {"25"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], TableForm[ {22, 30, 11, 19, 25}]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(TableForm[{{a, b, c}, {d, e, f}}, TableHeadings -> {{"\<I\>", "\<II\>"}, {"\<first\>", "\<second\>", \ "\<third\>"}}]\)], "Input"], Cell[BoxData[ TagBox[GridBox[{ >"}, {"\<\"\"\>", "\<\"first\"\>", "\<\"second\"\>", "\<\"third\"\ {"\<\"I\"\>", "a", "b", "c"}, {"\<\"II\"\>", "d", "e", "f"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$, TableHeadings -> {{"I", "II"}, {"first", "second", "third"}}]]]], "Output"] }, Open ]], Cell[BoxData[""], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(TableForm[{{a, b, c}, {d, e, f}}, TableHeadings -> {{"\<I\>", "\<II\>"}, {"\<first\>", "\<second\>", \ "\<third\>"}}, \[IndentingNewLine]TableAlignments -> Center]\)], "Input"], Cell[BoxData[ TagBox[GridBox[{ {"\<\"\"\>", "\<\"first\"\>", "\<\"second\"\>", "\<\"third\"\ >"}, {"\<\"I\"\>", "a", "b", "c"}, {"\<\"II\"\>", "d", "e", "f"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Center}], Function[ BoxForm`e$, TableForm[ BoxForm`e$, TableHeadings -> {{"I", "II"}, {"first", "second", "third"}}, TableAlignments -> Center]]]], "Output"] }, Open ]], Cell[TextData[{ StyleBox["\[FilledSmallCircle] ", FontColor->RGBColor[0, 0, 1]], "An application." }], "Text"], Cell[TextData[{ "Find and classify the critical points of ", Cell[BoxData[ \(TraditionalForm\`f(x) = 2 x\^5 - 4\ x\^4 - 2\ x\^2 + 7 }], "Text"], Cell[BoxData[ \(f[x_] := 2 x\^5 - 4\ x\^4 - 2\ x\^2 + 7 x\)], "Input"], x\)]] Cell[CellGroupData[{ Cell[BoxData[ \(crits = NSolve[\(f'\)[x] \[Equal] 0, x]\)], "Input"], Cell[BoxData[ \({{x \[Rule] \(-0.38119418236749675`\) 0.6497116306651218`\ \[ImaginaryI]}, {x \[Rule] \ \(-0.38119418236749675`\) + 0.6497116306651218`\ \[ImaginaryI]}, {x \ [Rule] 0.779203508480791`}, {x \[Rule] 1.5831848562542026`}}\)], "Output", GeneratedCell->False, CellAutoOverwrite->False] }, Open ]], Cell["\<\ Since we're only interested in real values, we drop the first two \ solutions:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(crits = Drop[crits, 2]\)], "Input"], Cell[BoxData[ \({{x \[Rule] 0.779203508480791`}, {x \[Rule] 1.5831848562542026`}}\)], "Output"] }, Open ]], Cell[BoxData[ \(\(?Drop\)\)], "Input"], Cell[TextData[{ StyleBox["\[FilledSmallCircle] ", FontColor->RGBColor[0, 0, 1]], " Read the following input command from the inside-out: the heart of it is \ the expression ", Cell[BoxData[ FormBox[ StyleBox[\({x, f[x], \(f''\)[x]} /. crits\), FontFamily->"Courier New"], TraditionalForm]]], " which constructs a list of 2 ordered triples. \n", StyleBox["TableForm[...,TableHeadings\[Rule]{Automatic, {\"x\",\"y\",\"f''[x]\ \"}}]", FontFamily->"Courier New"], " \nthen arranges this nested list as a ", Cell[BoxData[ \(TraditionalForm\`2\[Times]3\)]], " table with numbered rows and columns labeled ", Cell[BoxData[ \(TraditionalForm\`x\)]], ", ", Cell[BoxData[ \(TraditionalForm\`y\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\(f''\)[x]\)]], ", respectively." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(TableForm[{x, f[x], \(f''\)[x]} /. crits, TableHeadings \[Rule] {Automatic, {"\<x\>", "\<y\>", \ "\<f''[x]\>"}}]\)], "Input"], Cell[BoxData[ TagBox[GridBox[{ {"\<\"\"\>", "\<\"x\"\>", "\<\"y\"\>", "\<\"f''[x]\"\>"}, {"1", "0.779203508480791`", "3.340037144210418`", \(-14.219600059505218`\)}, {"2", "1.5831848562542026`", "0.8321353587106088`", "34.41771960652616`"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$, TableHeadings -> {Automatic, {"x", "y", "f''[x]"}}]]]], "Output"] }, Open ]], Cell["\<\ We conclude that the first point is a local maximum, while point #2 is a \ local minimum. 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Find the absolute extrema of ", Cell[BoxData[ \(TraditionalForm\`f(x) = 2 x\^5 - 4\ x\^4 - 2\ x\^2 + 7 " on the interval ", Cell[BoxData[ \(TraditionalForm\`\([0, 1.75]\)\)]], "." }], "Text"], Cell[BoxData[ \(f[x_] := 2 x\^5 - 4\ x\^4 - 2\ x\^2 + 7 x\)], "Input"], x\)]], Cell[CellGroupData[{ Cell[BoxData[ \(Clear[crits]; crits = NSolve[\(f'\)[x] \[Equal] 0, x]\)], "Input"], Cell[BoxData[ \({{x \[Rule] \(-0.38119418236749675`\) 0.6497116306651218`\ \[ImaginaryI]}, {x \[Rule] \ \(-0.38119418236749675`\) + 0.6497116306651218`\ \[ImaginaryI]}, {x \ [Rule] 0.779203508480791`}, {x \[Rule] 1.5831848562542026`}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(crits = Drop[crits, 2]\)], "Input"], Cell[BoxData[ \({{x \[Rule] 0.779203508480791`}, {x \[Rule] 1.5831848562542026`}}\)], "Output"] }, Open ]], Cell["Now, we must also consider the endpoints of the interval:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(crits = Union[%, {{x \[Rule] 0}, {x -> 1.75}}]\)], "Input"], Cell[BoxData[ \({{x \[Rule] 0}, {x \[Rule] 0.779203508480791`}, {x \[Rule] 1.5831848562542026`}, {x \[Rule] 1.75`}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(TableForm[{x, f[x]} /. crits, TableHeadings \[Rule] {Automatic, {"\<x\>", "\<f[x]\>"}}]\)], "Input"], Cell[BoxData[ TagBox[GridBox[{ {"\<\"\"\>", "\<\"x\"\>", "\<\"f[x]\"\>"}, {"1", "0", "0"}, {"2", "0.779203508480791`", "3.340037144210418`"}, {"3", "1.5831848562542026`", "0.8321353587106088`"}, {"4", "1.75`", "1.435546875`"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$, TableHeadings -> {Automatic, {"x", "f[x]"}}]]]], "Output"] }, Open ]], Cell[TextData[{ "We conclude that the minimum value of ", Cell[BoxData[ \(TraditionalForm\`f\)]], " on ", Cell[BoxData[ \(TraditionalForm\`\([0, 1.75]\)\)]], " is ", Cell[BoxData[ \(TraditionalForm\`0 = f(0)\)]], " and the max is ", Cell[BoxData[ \(TraditionalForm\`3.340037144210418` = f(0.779203508480791`)\)]], "." }], "Text"], Cell[BoxData[ \(\[IndentingNewLine]\)], "Input"], Cell[TextData[{ StyleBox["\[FilledSmallCircle] ", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ FormBox[ RowBox[{ StyleBox["Intersection", FontFamily->"Courier New"], "[", \(a\_1, a\_2, a\_n\), "]"}], TraditionalForm]]], " returs the intersection of the lists ", Cell[BoxData[ \(TraditionalForm\`a\_1, a\_2, ... , a\_n\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?Intersection\)\)], "Input"], Cell[BoxData[ RowBox[{"\<\"Intersection[list1, list2, ... ] gives a sorted list of the \ elements common to all the listi.\"\>", " ", ButtonBox[ StyleBox["More\[Ellipsis]", "SR"], ButtonData:>"Intersection", Active->True, ButtonStyle->"RefGuideLink"]}]], "Print", CellTags->"Info3291009626-4518664"] }, Open ]], Cell[TextData[{ StyleBox["\[FilledSmallCircle] ", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ ... , FormBox[ RowBox[{ StyleBox["Append", FontFamily->"Courier New"], "[", \(a, x\), "]"}], TraditionalForm]]], " inserts the object ", Cell[BoxData[ \(TraditionalForm\`x\)]], " at the end of the list ", Cell[BoxData[ \(TraditionalForm\`a\)]], ", and ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["Prepend", FontFamily->"Courier New"], "[", \(a, x\), "]"}], TraditionalForm]]], " puts ", Cell[BoxData[ \(TraditionalForm\`x\)]], " at the beginning of ", Cell[BoxData[ \(TraditionalForm\`a\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Append[a, 0]\)], "Input"], Cell[BoxData[ \({1, 2, 1, 3, 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Prepend[a, 0]\)], "Input"], Cell[BoxData[ \({0, 1, 2, 1, 3}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Append[a, b]\)], "Input"], Cell[BoxData[ \({1, 2, 1, 3, {2, \(-1\), 3, 5, 5}}\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["\[FilledSmallCircle] ", FontColor->RGBColor[0, 0, 1]], " Given a nested list ", Cell[BoxData[ \(TraditionalForm\`a\)]], ", the command ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["Flatten", FontFamily->"Courier New"], StyleBox["[", FontFamily->"Times New Roman"], StyleBox["a", FontFamily->"Times New Roman"], StyleBox["]", FontFamily->"Times New Roman"]}], TraditionalForm]]], " removes internal grouping. " }], "Text"], Cell[BoxData[ \(Clear[s]; \ s = Table[\((i - j)\)/\((i + j)\), {i, 1, 3}, {j, 1, 4}]\)], "Input"], Cell[BoxData[ \(Flatten[s]\)], "Input"], Cell[TextData[{ StyleBox["\[FilledSmallCircle] ", FontColor->RGBColor[0, 0, 1]], " ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["Flatten", FontFamily->"Courier New"], StyleBox["[", FontFamily->"Times New Roman"], StyleBox[\(a, \ n\), FontFamily->"Times New Roman"], StyleBox["]", FontFamily->"Times New Roman"]}], TraditionalForm]]], " removes internal grouping to the level ", Cell[BoxData[ \(TraditionalForm\`n\)]], ". " }], "Text"], Cell[TextData[{ StyleBox["\[FilledSmallCircle]", FontColor->RGBColor[0, 1, 0]], StyleBox[" ", FontColor->RGBColor[1, 0, 0]], " As an example, ", "suppose that we wish to plot a collection of points that depend on two \ parameters." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Clear[s]; s = Table[\((i - j + k)\)/\((i + j + k)\), {i, 1, 3}, {j, 1, 4}, {k, 1, 2}]\)], "Input"], Cell[BoxData[ \({{{1\/3, 1\/2}, {0, 1\/5}, {\(-\(1\/5\)\), 0}, {\(-\(1\/3\)\), \(-\(1\/7\)\)}}, {{1\/2, 3\/5}, {1\/5, 1\/3}, {0, 1\/7}, {\(-\(1\/7\)\), 0}}, {{3\/5, 2\/3}, {1\/3, 3\/7}, {1\/7, 1\/4}, {0, 1\/9}}}\)], "Output"] }, Open ]], Cell[TextData[{ "We cannot apply ", StyleBox["ListPlot", FontFamily->"Courier New"], " directly to this nested list of points" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(ListPlot[s, PlotStyle \[Rule] {RGBColor[1, 0, 0], PointSize[ .02]}];\)\)], "Input"], Cell[BoxData[ RowBox[{\(ListPlot::"lpn"\), \(\(:\)\(\ \)\), "\<\"\\!\\({\\(\\({\\ (\\({1\ \\/3, 1\\/2}\\)\\), \\(\\({0, 1\\/5}\\)\\), \ \\(\\({\\(\\(-\\(\\(1\\/5\\)\\)\\)\\), 0}\\)\\), \\(\\({\\(\\(-\\(\\ (1\\/3\\)\ \\)\\)\\), \\(\\(-\\(\\(1\\/7\\)\\)\\)\\)}\\)\\)}\\)\\), \\(\\({\\(\ \[LeftSkeleton] 1 \[RightSkeleton]\\)}\\)\\), \\(\\({\\(\[LeftSkeleton] 1 \ \[RightSkeleton]\\)}\\)\\)}\\) is not a list of numbers or pairs of numbers. \ \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", ButtonStyle>\\\"RefGuideLinkText\ \\\", ButtonFrame->None, ButtonData:>\\\"ListPlot::lpn\\\"]\\)\"\>"}]], \ "Message"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(t = Flatten[s, 1]\)], "Input"], Cell[BoxData[ \({{1\/3, 1\/2}, {0, 1\/5}, {\(-\(1\/5\)\), 0}, {\(-\(1\/3\)\), \(-\(1\/7\)\)}, {1\/2, 3\/5}, {1\/5, 1\/3}, {0, 1\/7}, {\(-\(1\/7\)\), 0}, {3\/5, 2\/3}, {1\/3, 3\/7}, {1\/7, 1\/4}, {0, 1\/9}}\)], "Output"] }, Open ]], Cell[BoxData[ \(\(ListPlot[t, PlotStyle \[Rule] {RGBColor[1, 0, 0], PointSize[ .02]}];\)\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["1.6 Vector and matrix operations", "Subsection", FontColor->RGBColor[1, 0, 1]], Cell[TextData[{ StyleBox["\[FilledSmallCircle] ", FontColor->RGBColor[0, 0, 1]], "Sums and scalar multiples of matrices and vectors are performed just like \ you'd guess." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(v = {2, 3, 4}; w = {\(-1\), 0, 2}; \ 2 Cell[BoxData[ \({7, 6, 2}\)], "Output"] }, Open ]], Cell[BoxData[{ \(Clear[a, b, c]; a = Table[i\ j, {i, 1, 3}, {j, 1, 4}]; b = Table[2 i/\ j, {i, 1, 3}, {j, 1, 4}]; \ c = Table[ 2 \((i\ j)\) - 3 \((2 i/\ j)\), {i, 1, 3}, {j, 1, 4}];\), "\[IndentingNewLine]", \(\((2 a - 3 b)\) - c // MatrixForm\)}], "Input"], Cell[TextData[{ StyleBox["\[FilledSmallCircle] ", FontColor->RGBColor[0, 0, 1]], "In ", StyleBox["Mathematica", FontSlant->"Italic"], ", the product of two matrices is indicated by a period. So, for \n \ matrices ", Cell[BoxData[ FormBox[ RowBox[{"m", "=", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(-1\), "2", "0"}, {"3", "1", \(-4\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, v - 3 w\)], "Input"], MatrixForm[ BoxForm`e$]]]}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{"n", "=", RowBox[{"(", "\[NoBreak]", GridBox[{ {"4", "1", \(-5\)}, {\(-3\), "1", "5"}, {"4", "1", \(-3\)} }], "\[NoBreak]", ")"}]}], TraditionalForm]]], ", the product ", Cell[BoxData[ \(TraditionalForm\`m\ n\)]], " is obtained with the command ", Cell[BoxData[ FormBox[ StyleBox[\(m . n\), FontFamily->"Courier New"], TraditionalForm]]], "." }], "Text"], Cell[BoxData[{\(Clear[m, n, v];\), "\[IndentingNewLine]", RowBox[{" ", RowBox[{ FormBox[ RowBox[{ StyleBox["m", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "=", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(-1\), "2", "0"}, {"3", "1", \(-4\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]}], "TraditionalForm"], ";", FormBox[ RowBox[{ StyleBox["n", FontSlant->"Plain"], "=", RowBox[{"(", "\[NoBreak]", GridBox[{ {"4", "1", \(-5\)}, {\(-3\), "1", "5"}, {"4", "1", \(-3\)} }], "\[NoBreak]", ")"}]}], "TraditionalForm"], ";", \(MatrixForm[m . n]\)}]}]}], "Input"], Cell[TextData[{ StyleBox["\[FilledSmallCircle] ", FontColor->RGBColor[0, 0, 1]], "The product of an ", Cell[BoxData[ \(TraditionalForm\`m\[Times]n\)]], " matrix ", Cell[BoxData[ \(TraditionalForm\`\(\(A\)\(\ \)\)\)]], "and a vector ", Cell[BoxData[ \(TraditionalForm\`v \[Element] \[DoubleStruckCapitalR]\^n\)]], " is also indicated by a period. " }], "Text"], Cell[BoxData[ \(\(\(A = {{3, 2, 2}, {2, \(-1\), 1}}\)\(;\)\(v = {2, \(-3\), 4}\)\(;\)\(\ \)\)\)], "Input"], Cell[BoxData[ \(A . v\)], "Input"], Cell[BoxData[ \(% // MatrixForm\)], "Input"], Cell[BoxData[ \(n . v // MatrixForm\)], "Input"], Cell[TextData[{ StyleBox["Caution! ", FontColor->RGBColor[1, 0, 0]], "Notice that, in the computation of ", Cell[BoxData[ FormBox[ StyleBox[\(m . v\), FontFamily->"Courier New"], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ StyleBox[\(n . v\), FontFamily->"Courier New"], TraditionalForm]]], ", ", StyleBox["Mathematica", FontSlant->"Italic"], " interprets ", Cell[BoxData[ FormBox[ StyleBox[\(v = {2, \(-3\), 4}\), FontFamily->"Courier New"], TraditionalForm]]], " as a ", StyleBox["column", FontSlant->"Italic"], " vector, that is, an ", Cell[BoxData[ \(TraditionalForm\`3\[Times]1\)]], " matrix. Sometimes though, ", StyleBox["Mathematica", FontSlant->"Italic"], " will interpret a list of numbers as a ", StyleBox["row", FontSlant->"Italic"], " vector, ie, a ", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(1\[Times]n\)\)\)]], " matrix. You must be careful to interpret matrices and lists of vectors \ appropriately. " }], "Text"], Cell[BoxData[ \(Clear[x, a, b, c]; x = {a, b, c};\)], "Input"], Cell[BoxData[ \(m . x // MatrixForm\)], "Input"], Cell[BoxData[ \(x . n\)], "Input"], Cell[TextData[{ StyleBox["\[FilledSmallCircle] ", FontColor->RGBColor[0, 0, 1]], "Notice that in the product ", Cell[BoxData[ \(TraditionalForm\`m . x\)]], ", ", StyleBox["Mathematica", FontSlant->"Italic"], " interprets ", Cell[BoxData[ \(TraditionalForm\`x\)]], " as a column matrix, while in the second product ", Cell[BoxData[ \(TraditionalForm\`x . n\)]], ", ", Cell[BoxData[ \(TraditionalForm\`x = {a, b, c}\)]], "is regarded as a row matrix. If " }], "Text"], Cell[BoxData[{ \(\(Clear[u, v, w, y];\)\), "\[IndentingNewLine]", \(\(y = {u, v, w};\)\)}], "Input"], Cell[TextData[{ StyleBox["\[FilledSmallCircle] ", FontColor->RGBColor[0, 0, 1]], "The ", StyleBox["dot product", FontSlant->"Italic"], " of vectors ", Cell[BoxData[ \(TraditionalForm\`x\)]], " and ", Cell[BoxData[ \(TraditionalForm\`y\)]], " is" }], "Text"], Cell[BoxData[ \(x . y\)], "Input"], Cell["\<\ In this case, the first vector is considered to be a row matrix and the \ second, a column matrix.\ \>", "Text"], Cell[TextData[{ StyleBox["\[FilledSmallCircle] ", FontColor->RGBColor[0, 0, 1]], "The Euclidian norm of a vectors ", Cell[BoxData[ \(TraditionalForm\`x\)]], " is ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ StyleBox["Norm", FontFamily->"Courier New"], "[", "x", "]"}], "=", \(\@\(x . x\)\)}], TraditionalForm]]], ", and the cross product of two vectors ", Cell[BoxData[ FormBox[ RowBox[{"x", "=", RowBox[{"(", GridBox[{ {\(x\_1\)}, {\(x\_2\)}, {\(x\_3\)} }], ")"}]}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{"y", "=", RowBox[{"(", GridBox[{ {\(y\_1\)}, {\(y\_2\)}, {\(y\_3\)} }], ")"}]}], TraditionalForm]]], " is ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["Cross", FontFamily->"Courier New"], "[", \(x, y\), "]"}], TraditionalForm]]], " or ", Cell[BoxData[ \(TraditionalForm\`x\[Cross]y\)]] }], "Text"], Cell[BoxData[ \(x = {3, 2, \(-1\)}; y = {2, 0, 1};\)], "Input"], Cell[BoxData[ \(x . y\)], "Input"], Cell[BoxData[ \(Norm[x]\)], "Input"], Cell[BoxData[ \(x . x\)], "Input"], Cell[BoxData[ \(z = x\[Cross]y\)], "Input"], Cell[TextData[{ StyleBox["\[FilledSmallCircle] ", FontColor->RGBColor[0, 0, 1]], "Notice that ", Cell[BoxData[ \(TraditionalForm\`x\[Cross]y\)]], " is orthogonal to both ", Cell[BoxData[ \(TraditionalForm\`x\)]], " and ", Cell[BoxData[ \(TraditionalForm\`y\)]], "." }], "Text"], Cell[BoxData[ \(z . x\)], "Input"], Cell[BoxData[ \(z . y\)], "Input"], Cell[TextData[{ StyleBox["Caution! ", FontColor->RGBColor[1, 0, 0]], "It's easy to confuse the symbols for \"Times\" ", Cell[BoxData[ \(TraditionalForm\`\[Times]\)]], " and \"Cross Product\" ", Cell[BoxData[ \(TraditionalForm\`\[Cross]\)]], " of the BasicInput Palette. ", Cell[BoxData[ \(TraditionalForm\`x\[Times]y = Times[x, y]\)]], " performs a coordinate-wise multiplication." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(w = x\[Times]y\)], "Input"], Cell[BoxData[ \({6, 0, \(-1\)}\)], "Output"] }, Open ]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["2. 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For example, consider the spiral with parameterization ", Cell[BoxData[ \(TraditionalForm\`r(t) = \[LeftAngleBracket]t\ \(cos(\[Pi]\ t)\), t\ \(sin(\[Pi]\ t)\)\[RightAngleBracket], \ 0 \[LessEqual] t \[LessEqual] 4\)]], ". Remember that in ", StyleBox["Mathematica", FontSlant->"Italic"], ", a vector or point is represented as a ", StyleBox["list", FontSlant->"Italic"], "." }], "Text"], Cell[BoxData[ \(\(ParametricPlot[{t\ Cos[\[Pi]\ t], t\ Sin[\[Pi]\ t]}, {t, 0, 4}];\)\)], "Input"], Cell[BoxData[ \(\(?ParametricPlot\)\)], "Input"], Cell[TextData[{ StyleBox["\[FilledSmallCircle] ", FontColor->RGBColor[0, 0, 1]], "Notice that the scales on ", Cell[BoxData[ \(TraditionalForm\`x\)]], " and ", Cell[BoxData[ \(TraditionalForm\`y\)]], " axes is slightly different. You can make ", StyleBox["Mathematica", FontSlant->"Italic"], " apply the same scale on each axis with the command ", Cell[BoxData[ FormBox[ StyleBox[\(AspectRatio \[Rule] Automatic\), FontFamily->"Courier New"], TraditionalForm]]], ". You can also specify ", StyleBox["PlotStyle", FontFamily->"Courier New"], " as well as other options" }], "Text"], Cell[BoxData[ \(\(ParametricPlot[{t\ Cos[\[Pi]\ t], t\ Sin[\[Pi]\ t]}, {t, 0, 4}, PlotStyle \[Rule] {Thickness[ .005], Hue[0]}, AspectRatio \[Rule] Automatic];\)\)], "Input"], Cell[BoxData[ \(\(ParametricPlot[{Cos[3\ t], Sin[5 2 \[Pi]}];\)\)], "Input"], t]}, {t, 0, Cell[TextData[{ StyleBox["\[FilledSmallCircle] ", FontColor->RGBColor[0, 0, 1]], "A ", StyleBox["rose", FontSlant->"Italic"], " has polar equation ", Cell[BoxData[ \(TraditionalForm\`r = cos(n\ \[Theta])\)]], " for some natural number ", Cell[BoxData[ \(TraditionalForm\`n \[GreaterEqual] 2\)]], ". Investigate this family of curves by plotting several examples. Since ", Cell[BoxData[ \(TraditionalForm\`x = r\ \(cos(\[Theta])\)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`y = r\ \(sin(\[Theta])\)\)]], ", we obtain parameterizations of the roses" }], "Text"], Cell[BoxData[ \(s[\[Theta]_, n_] := {Cos[n\ \[Theta]] Cos[\[Theta]], Cos[n\ \[Theta]] Sin[\[Theta]]}\)], "Input"], Cell[TextData[{ StyleBox["Warning!", FontColor->RGBColor[1, 0, 0]], StyleBox[" ", FontColor->RGBColor[0, 0, 1]], "Because of the order of involved, we must use the command ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["Evaluate", FontFamily->"Courier New"], "[", \(s[\[Theta], n]\), "]"}], TraditionalForm]]], " within ", StyleBox["ParametricPlot", FontFamily->"Courier New"], StyleBox[".", FontSlant->"Italic"] }], "Text"], Cell[BoxData[ \(\(Table[ ParametricPlot[ Evaluate[s[\[Theta], n]], {\[Theta], 0, 2 \[Pi]}, \[IndentingNewLine]AspectRatio \[Rule] Automatic, \[IndentingNewLine]PlotStyle \[Rule] {Thickness[ \ .005], Hue[\((n - 2)\)/5]}, \[IndentingNewLine]PlotLabel \[Rule] n "\<=n\>"], {n, 2, 6}];\)\)], "Input"], Cell[TextData[{ "It seems that for ", Cell[BoxData[ \(TraditionalForm\`n\)]], " odd, ", Cell[BoxData[ \(TraditionalForm\`s(\[Theta], n)\)]], " traces a rose with ", Cell[BoxData[ \(TraditionalForm\`n\)]], " petals, while ", Cell[BoxData[ \(TraditionalForm\`s(\[Theta], n)\)]], " traces a rose with ", Cell[BoxData[ \(TraditionalForm\`2 n\)]], " petals in case ", Cell[BoxData[ \(TraditionalForm\`n\)]], " is even. We can better understand what's going on using ", StyleBox["Mathematica's ", FontSlant->"Italic"], StyleBox["animation capability", FontVariations->{"CompatibilityType"->0}], StyleBox[".", FontSlant->"Italic"] }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["2.2 Animation", "Subsection", FontColor->RGBColor[1, 0, 1]], Cell[TextData[{ "You can run any sequence of graphics as a movie. To create an animated \ movie\n", StyleBox["1) ", FontColor->RGBColor[0, 0, 1]], "Create a collection of frames for the movie grouped together in a cell. \ One way to do this is with the Table command.\n", StyleBox["2) ", FontColor->RGBColor[0, 0, 1]], "Double-click on the cell-bracket that contains all of the graphics cells \ created in step 1. This collapses the cell, so only the first graphics cell \ is visible.\n", StyleBox["3) ", FontColor->RGBColor[0, 0, 1]], "Now, double-click inside the visible graphics cell to play the sequence. \ Playback controls are located in the lower left margin of the notebook." }], "Text"], Cell[TextData[{ "As an example, let's animate the motion of the particle tracing an \ 8-petaled rose ", Cell[BoxData[ \(TraditionalForm\`\((n = 4)\)\)]], " and another tracing a 3-petaled rose.\n", StyleBox["\[FilledSmallCircle] ", FontColor->RGBColor[0, 0, 1]], "Define a function ", Cell[BoxData[ \(TraditionalForm\`\(\(f[x]\)\(\ \)\)\)]], "that traces the rose from time ", Cell[BoxData[ \(TraditionalForm\`t = \(-\[Pi]\)/8\)]], " to time ", Cell[BoxData[ \(TraditionalForm\`t = x, \ \(-\[Pi]\)/8 < x \[LessEqual] 15 \[Pi]/8\)]], ". In order to make the movie smooth, we specify the PlotRange to be fixed \ for all of the graphs." }], "Text"], Cell[BoxData[ \(\(\(Clear[f]; \ f[x_] := ParametricPlot[ Evaluate[s[\[Theta], 4]], {\[Theta], \(-\[Pi]\)/8, x}, \[IndentingNewLine]AspectRatio \[Rule] Automatic, \[IndentingNewLine]PlotStyle \[Rule] {Thickness[ .005], Hue[0]}, PlotRange \[Rule] {{\(-1.25\), 1.25}, {\(-1.25\), 1.25}}, \[IndentingNewLine]Ticks \[Rule] None]\)\(\[IndentingNewLine]\) \)\)], "Input"], Cell[BoxData[ \(\(f[15 \[Pi]/8];\)\)], "Input"], Cell[TextData[{ StyleBox["\[FilledSmallCircle] ", FontColor->RGBColor[0, 0, 1]], "Now, we construct a table ", Cell[BoxData[ \(TraditionalForm\`{f[\[Pi]/16\ \ k] : \(-1\) \[LessEqual] k \[LessEqual] 30}\)]], " " }], "Text"], Cell[BoxData[ \(\(Table[f[\[Pi]/16\ k], {k, \(-1\), 30}];\)\)], "Input"], Cell[TextData[{ "We can add an additional frame to the beginning of our animation using the \ command ", Cell[BoxData[ FormBox[ StyleBox["Join", FontFamily->"Courier New"], TraditionalForm]]] }], "Text"], Cell[BoxData[ \(\(Join[{f[\(-\[Pi]\)/8 + .025]}, Table[f[\[Pi]/16\ k], {k, \(-1\), 30}]];\)\)], "Input"], Cell[TextData[{ StyleBox["\[FilledSmallCircle] ", FontColor->RGBColor[0, 0, 1]], "For the 3-petaled rose, we use a piecewise-defined function that sketches \ the curve in red for time ", Cell[BoxData[ \(TraditionalForm\`\(-\[Pi]\)/6 < t \[LessEqual] 5 \[Pi]/6\)]], " and in blue for ", Cell[BoxData[ \(TraditionalForm\`5 \[Pi]/6 < t \[LessEqual] 11 \[Pi]/6\)]], " " }], "Text"], Cell[BoxData[{ \ .005], Hue[0]}, PlotRange \[Rule] {{\(-1.25\), 1.25}, {\(-1.25\), 1.25}}, \[IndentingNewLine]Ticks \[Rule] None] /; \(-\ [Pi]\)/ 6 \[LessEqual] x \[LessEqual] 5 \[Pi]/6;\), "\[IndentingNewLine]", \(\(f[x_] := ParametricPlot[ Evaluate[s[\[Theta], 3]], {\[Theta], 5 \[Pi]/6, x}, \[IndentingNewLine]AspectRatio \[Rule] Automatic, \[IndentingNewLine]PlotStyle \[Rule] {Thickness[ \ .005], Hue[ .6]}, PlotRange \[Rule] {{\(-1.25\), 1.25}, {\(-1.25\), 1.25}}, \[IndentingNewLine]Ticks \[Rule] None] /; 5 \[Pi]/6 < x \[LessEqual] 11 \[Pi]/6;\)\)}], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Join[{f[\(-\[Pi]\)/6 + .025]}, Table[f[\[Pi]/12\ k], {k, \(-1\), 22}]];\)\)], "Input"], Cell[CellGroupData[{ Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.4 0.5 0.4 [ [ 0 0 0 0 ] [ 1 1 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath \(Clear[f]; \ \ f[x_] := ParametricPlot[ Evaluate[s[\[Theta], 3]], {\[Theta], \(-\[Pi]\)/6, x}, \[IndentingNewLine]AspectRatio \[Rule] Automatic, \[IndentingNewLine]PlotStyle \[Rule] {Thickness[ 0 g .25 Mabswid [ ] 0 setdash 0 .5 m 1 .5 L s .5 0 m .5 1 L s 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath 1 0 0 r .005 w .5 .5 m .50105 .49939 L .50221 .49873 L .50329 .49812 L .50433 .49752 L .50544 .4969 L .50651 .49629 L .50765 .49566 L .50875 .49505 L .50981 .49446 L .51094 .49384 L .51202 .49324 L .51307 .49267 L .51418 .49207 L .51526 .49149 L .5164 .49087 L .5175 .49028 L .51856 .48972 L .51969 .48912 L .52078 .48855 L .52194 .48794 L .52305 .48736 L .52413 .4868 L .52528 .48621 L .52632 .48567 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 288}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHg<PAVmbKF5d0`40006`0001/2000`400? 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ooo`00o`3ooooo 0?ooog80oooo003o0?ooool0ooooLP3oool00001\ \>"], ImageRangeCache->{{{76.4375, 408.438}, {434.125, 60.0625}} -> {0.533506, \ 0.323377, 0.0050603, 0.0050603}, {{84.3125, 234.875}, {425.188, 255.563}} -> \ {-2.25329, -0.473778, 0.0112513, 0.0112513}, {{249.938, 400.5}, {425.188, \ 255.563}} -> {-4.58269, -0.473785, 0.0112513, 0.0112513}, {{84.3125, \ 234.875}, {238.563, 69}} -> {-1.52539, -2.40511, 0.00873471, 0.00873471}, \ {{249.938, 400.5}, {238.563, 69}} -> {-4.14411, -2.85433, 0.0103661, \ 0.0103661}}] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["2.4 Adding points, lines and text to graphics", "Subsection", FontColor->RGBColor[1, 0, 1]], Cell[TextData[{ "Points, lines and text can be incorporated into graphics through the use \ of the ", StyleBox["Graphics", FontFamily->"Courier New"], " environment. Specifically, ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["Show", FontFamily->"Courier New"], "[", RowBox[{ StyleBox["Graphics", FontFamily->"Courier New"], "[", StyleBox["list", FontSlant->"Italic"], "]"}], "]"}], TraditionalForm]]], " displays the graphics objects within ", StyleBox["list. ", FontSlant->"Italic"], "Among these objects are:\n\n", StyleBox["\[FilledSmallCircle] ", FontColor->RGBColor[0, 0, 1]], " ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["Point", FontFamily->"Courier New"], "[", \({a, b}\), "]"}], TraditionalForm]]], " draws a circular point at coordinates ", Cell[BoxData[ \(TraditionalForm\`\((a, b)\)\)]], ". \nThe size of the point is controlled with the command ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["PointSize", FontFamily->"Courier New"], "[", "r", "]"}], TraditionalForm]]], " where ", Cell[BoxData[ \(TraditionalForm\`r = diam\ of\ the\ point/width\ of\ entire\ graph\)]], ". \n(We've also seen the specification ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["PlotStyle", FontFamily->"Courier New"], "->", RowBox[{"{", RowBox[{ RowBox[{ StyleBox["PointSize", FontFamily->"Courier New"], "[", ".02", "]"}], ",", " ", RowBox[{ StyleBox["Hue", FontFamily->"Courier New"], "[", "0", "]"}]}], "}"}]}], TraditionalForm]]], " within ", StyleBox["ListPlot", FontFamily->"Courier New"], ".)\n\n", StyleBox["\[FilledSmallCircle] ", FontColor->RGBColor[0, 0, 1]], " If ", Cell[BoxData[ \(TraditionalForm\`{p\_1, p\_2, \(\(...\) \(p\_n\)\)}\)]], " is a list of points, ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["Line", FontFamily->"Courier New"], "[", \({p\_1, p\_2, \(\(...\) \(p\_n\)\)}\), "]"}], TraditionalForm]]], " draws a polygonal line joining the points. \nThe thickness of the line is \ controlled with the command ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["Thickness", FontFamily->"Courier New"], "[", "r", "]"}], TraditionalForm]]], " where ", Cell[BoxData[ \(TraditionalForm\`r = thickness/width\ of\ entire\ graph\)]], ". \n(We've encountered ", Cell[BoxData[ FormBox[ StyleBox[\(\(Thickness\)\(\ \)\), FontFamily->"Courier New"], TraditionalForm]]], "in the ", StyleBox["PlotStyle", FontFamily->"Courier New"], " environment.)\n\n", StyleBox["\[FilledSmallCircle] ", FontColor->RGBColor[0, 0, 1]], " ", StyleBox["Text[\"", FontFamily->"Courier New"], StyleBox["text", FontSlant->"Italic"], StyleBox["\",{a,b}]", FontFamily->"Courier New"], " inserts ", StyleBox["text", FontSlant->"Italic"], " at the point with coordinates ", Cell[BoxData[ \(TraditionalForm\`\((a, b)\)\)]], ". \nThe ", StyleBox["text", FontSlant->"Italic"], " is contained in a bounding rectangle with a local coordinate system such \ that ", StyleBox["text", FontSlant->"Italic"], " occupies the region ", Cell[BoxData[ \(TraditionalForm\`\([\(-1\), 1]\)\[Times]\([\(-1\), 1]\)\)]], ". Ordinarily, the box is oriented so that the local coordinates ", Cell[BoxData[ \(TraditionalForm\`\((0, 0)\)\)]], " are placed at the point ", Cell[BoxData[ \(TraditionalForm\`\((a, b)\)\)]], ". There is an optional offset: ", StyleBox["Text[\"", FontFamily->"Courier New"], StyleBox["text", FontSlant->"Italic"], StyleBox["\",{a,b},{c,d}]", FontFamily->"Courier New"], " orients the box so that the point with local coordinates ", Cell[BoxData[ \(TraditionalForm\`\((c, d)\)\)]], " is placed at ", Cell[BoxData[ \(TraditionalForm\`\((a, b)\)\)]], "." }], "Text"], Cell[TextData[{ StyleBox["\[FilledSmallCircle]", FontColor->RGBColor[0, 1, 0]], StyleBox[" ", FontColor->RGBColor[0, 0, 1]], " ", StyleBox["An example.", FontColor->RGBColor[0, 1, 0]], " Plot a curve traced by a particle with position function ", Cell[BoxData[ \(TraditionalForm\`r(t) = \[LeftAngleBracket]t + cos(t), \ 1 + sin(2 t)\[RightAngleBracket]\)]], " and display its position and velocity at time ", Cell[BoxData[ \(TraditionalForm\`t = \[Pi]/3\)]], "." }], "Text"], Cell[BoxData[ \(Clear[r]; r[t_] := {t + Cos[t], 1 + Sin[2 t]}\)], "Input"], Cell[BoxData[{ \(\(curve = ParametricPlot[Evaluate[r[t]], {t, 0, \[Pi]}, Ticks \[Rule] None, DisplayFunction \[Rule] Identity];\)\n\), "\ [IndentingNewLine]", \(\(Show[\[IndentingNewLine]curve, \[IndentingNewLine]Graphics[\ \[IndentingNewLine]{PointSize[ .02], \[IndentingNewLine]{Hue[0], Point[r[\[Pi]/3]], \[IndentingNewLine]Text["\<r[\[Pi]/3]\ >", r[\[Pi]/ 3], {\(-1\), \(-1\)}]}, \[IndentingNewLine] {Thickness[ \ .005], Hue[ .6], Line[{r[\[Pi]/3], r[\[Pi]/3] + \(r'\)[\[Pi]/ 3]}], \[IndentingNewLine]Text["\<r'[\ [Pi]/3]\>", r[\[Pi]/3] + 1/2\ \(r'\)[\[Pi]/ 3], {\(-1\), \(-1\)}]}\[IndentingNewLine]}], \ \[IndentingNewLine]DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"], Cell[TextData[{ "For a list of other objects that can be included in Graphics, go to the ", StyleBox["Help", FontVariations->{"Underline"->True}], " browser." }], "Text"], Cell[BoxData[ \(\(?Graphics\)\)], "Input"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox[" 2.4.1 A dirty trick.", FontColor->RGBColor[1, 0, 0]], " " }], "Subsubsection", FontColor->RGBColor[1, 0, 0]], Cell[TextData[{ "Sometimes we want to include a variable expression within a string of \ text. Say, for example, that we have a parameterized sequence of graphs (eg, \ the roses in the ", Cell[BoxData[ \(TraditionalForm\`1\^st\)]], "subsection above) and that we'd like to insert text that refers to the \ parameter. (Notice that the labels for the roses are written in a unusual \ way. It would have been nice if we had more control.)\n\n", StyleBox["\[FilledSmallCircle]", FontColor->RGBColor[0, 1, 0]], StyleBox[" ", FontColor->RGBColor[0, 0, 1]], "To define a function ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["label1", FontSlant->"Italic"], "(", \(t, s\), ")"}], TraditionalForm]]], "that returns the string \"", Cell[BoxData[ \(TraditionalForm\`\[Theta] = t, \ r = s\)]], "\", we make use of the ", StyleBox["Mathematica", FontSlant->"Italic"], " functions ", Cell[BoxData[ FormBox[ StyleBox["ToString", FontFamily->"Courier New"], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ StyleBox["StringReplace", FontFamily->"Courier New"], TraditionalForm]]], "." }], "Text"], Cell[BoxData[ \(\(?ToString\)\)], "Input"], Cell[BoxData[ \(\(?StringReplace\)\)], "Input"], Cell[BoxData[ \(Clear[ label1]; \(\(label1[t_, s_]\)\(:=\)\(StringReplace["\<\[Theta]=a, r=b\>", {"\<a\>" \ [Rule] ToString[t, TraditionalForm], "\<b\>" \[Rule] ToString[s, TraditionalForm]}]\)\(\ \)\)\)], "Input"], Cell[BoxData[ \(label1[2 \[Pi]/4, 2 + 1]\)], "Input"], Cell[BoxData[ \(\(Show[\[IndentingNewLine]Table[\[IndentingNewLine]Graphics[{\ \[IndentingNewLine]Line[{{0, 0}, \((k\ \[Pi]/6 + 1)\)/2 {Cos[k\ \[Pi]/6], Sin[k\ \[Pi]/6]}}], \[IndentingNewLine]Text[ label1[k\ \[Pi]/6, \((k\ \[Pi]/6 + 1)\)/ 2], \[IndentingNewLine]\((k\ \[Pi]/6 + 1.5)\)/ 2 {Cos[k\ \[Pi]/6], Sin[k\ \[Pi]/6]}, \ {\(-Cos[k\ \[Pi]/6]\), Sin[k\ \[Pi]/ 6]}]\[IndentingNewLine]}], \[IndentingNewLine] {k, 0, (-1\), 3}], \[IndentingNewLine]PlotRange \[Rule] {{\(-1\), 3}, {\ 2}}];\)\)], "Input"], Cell[BoxData[""], "Input"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["2.5 Three dimensional graphics", "Subsection", FontColor->RGBColor[1, 0, 1]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " provides three-dimensional graphics command analogous to the \ two-dimensional functions we've already encountered. \n\n", StyleBox["\[FilledSmallCircle] ", FontColor->RGBColor[0, 0, 1]], " ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["ParametricPlot3D", FontFamily->"Courier New"], "[", RowBox[{ RowBox[{ StyleBox["Evaluate", FontFamily->"Courier New"], "[", \(r[t]\), "]"}], ",", \({t, a, b}\)}], "]"}], TraditionalForm]]], " plots the curve with parameterization ", Cell[BoxData[ \(TraditionalForm\`\((x, y, z)\) = r(t), \ a \[LessEqual] t \[LessEqual] b\)]], ". ", Cell[BoxData[ FormBox[ StyleBox["ParametricPlot3D", FontFamily->"Courier New"], TraditionalForm]]], " can also be used to plot surfaces. If \[ScriptCapitalS] is the surface \ with parameterization ", Cell[BoxData[ \(TraditionalForm\`\((x, y, z)\) = G(s, t), \ a \[LessEqual] s \[LessEqual] b, \ c \[LessEqual] t \[LessEqual] d\)]], ", then ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["ParametricPlot3D", FontFamily->"Courier New"], "[", RowBox[{ RowBox[{ StyleBox["Evaluate", FontFamily->"Courier New"], "[", \(G[s, t]\), "]"}], ",", \({s, a, b}\), ",", " ", \({t, c, d}\)}], "]"}], TraditionalForm]]], " creates a picture of \[ScriptCapitalS]." }], "Text"], Cell[BoxData[ \(Clear[r]; r[t_] := {\((4 - t)\) Cos[6 \[Pi]\ \ t], \((4 - t)\) Sin[6 \[Pi]\ \ t], \ \ t\^\(3/2\)}\)], "Input"], Cell[BoxData[ \(\(ParametricPlot3D[Evaluate[r[t]], {t, 0, 4}];\)\)], "Input"], Cell[TextData[{ StyleBox["\[FilledSmallCircle] ", FontColor->RGBColor[0, 0, 1]], "The results in this case are not very satisfying, but they do give an \ indication of what's going on. The 3D graphics functions are not as \ sophisticated as their two-dimensional counterparts. When evaluating ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["ParametricPlot3D", FontFamily->"Courier New"], "[", RowBox[{ RowBox[{ StyleBox["Evaluate", FontFamily->"Courier New"], "[", \(r[t]\), "]"}], ",", \({t, a, b}\)}], "]"}], TraditionalForm]]], ", ", StyleBox["Mathematica", FontSlant->"Italic"], " simply partitions the interval ", Cell[BoxData[ \(TraditionalForm\`\([a, b]\)\)]], "into 75 subintervals of equal length, ", Cell[BoxData[ \(TraditionalForm\`a = \(t\_0 < t\_1 < \[CenterEllipsis] < t\_75 = b\)\)]], "and constructs a polygon with vertices ", Cell[BoxData[ \(TraditionalForm\`r(t\_k), \ 0 \[LessEqual] k \[LessEqual] 75\)]], ". We can refine the partition by increasing the value of ", Cell[BoxData[ FormBox[ StyleBox["PlotPoints", FontFamily->"Courier New"], TraditionalForm]]], " within ", Cell[BoxData[ "." }], "Text"], FormBox[ StyleBox["ParametricPlot3D", FontFamily->"Courier New"], TraditionalForm]]], Cell[BoxData[ \(\(ParametricPlot3D[Evaluate[r[t]], {t, 0, 4}, PlotPoints \[Rule] 350];\)\)], "Input"], Cell[TextData[{ StyleBox["\[FilledSmallCircle] ", FontColor->RGBColor[0, 0, 1]], "There are a variety of options available for 3D graphics. Some are the \ same as the two dimensional ones, but some are different. Instead of \ AspectRatio, the relative scales for the ", Cell[BoxData[ \(TraditionalForm\`x\)]], ", ", Cell[BoxData[ \(TraditionalForm\`y\)]], ", and ", Cell[BoxData[ \(TraditionalForm\`z\)]], " axes are determined by specifying the ratios of the sides of the bounding \ box." }], "Text"], Cell[BoxData[ \(\(ParametricPlot3D[Evaluate[r[t]], {t, 0, 4}, PlotPoints \[Rule] 350, BoxRatios \[Rule] {1, 1, 2}];\)\)], "Input"], Cell[BoxData[ \(\(Show[%, Boxed \[Rule] False, \ Axes \[Rule] None];\)\)], "Input"], Cell[TextData[{ StyleBox["\[FilledSmallCircle] ", FontColor->RGBColor[0, 0, 1]], "One big difference is how ", StyleBox["PlotStyle", FontFamily->"Courier New"], " is specified within the ", Cell[BoxData[ FormBox[ StyleBox["ParametricPlot3D", FontFamily->"Courier New"], TraditionalForm]]], " environment. \n", StyleBox["Pa", FontFamily->"Courier New"], StyleBox["rametricPlot3D[", "MR"], Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{\(f\_x\), ",", \(f\_y\), ",", \(f\_z\), ",", StyleBox["style", FontSlant->"Italic"]}], "}"}], TraditionalForm]]], StyleBox[",", "MR"], " \[Ellipsis]", StyleBox["]", FontFamily->"Courier New"], " plots the curve with parameterization ", Cell[BoxData[ \(TraditionalForm\`\((x, y, z)\) = {f\_x, f\_y, f\_z}\)]], " according to the style specified\[NonBreakingSpace]by ", StyleBox["style", FontSlant->"Italic"], ". One advantage is that the style may vary for each of the subintervals ", Cell[BoxData[ \(TraditionalForm\`t\_\(k - 1\) \[LessEqual] t \[LessEqual] t\_k\)]], ", that is, ", Cell[BoxData[ FormBox[ StyleBox[\(style = f(t)\), FontSlant->"Italic"], TraditionalForm]]] }], "Text"], Cell[BoxData[ \(Clear[s]; s[t_] := Append[r[t], {Hue[t], Thickness[ .0075]}]\)], "Input"], Cell[BoxData[ \(s[t]\)], "Input"], Cell[BoxData[ \(\(ParametricPlot3D[Evaluate[s[t]], {t, 0, 4}, PlotPoints \[Rule] 350, Boxed -> False, Axes \[Rule] None];\)\)], "Input"], Cell[TextData[{ StyleBox["\[FilledSmallCircle] ", FontColor->RGBColor[0, 0, 1]], "Probably the single most important issue in plotting a \ three\[Hyphen]dimensional surface is specifying where you want to look at the \ surface from. The ", StyleBox["ViewPoint", "MR"], " option for 3D graphics allows you to specify the point ", Cell[BoxData[ \(TraditionalForm\`{x, y, z}\)], "InlineFormula"], " in space from which you view a surface. It's not necessary to know the \ details of how the coordinates for this point are defined since ", StyleBox["Mathematica", "TI"], " provides a way to choose view points interactively. \n\nTo adjust the ", StyleBox["ViewPoint", "MR"], " in a 3D graphics environment:\n", StyleBox["1) ", FontColor->RGBColor[1, 0, 0]], "Click at the location within the environment at the location you wasn't \ the option ", StyleBox["ViewPoint->\[CenterEllipsis]", "MR"], " to be pasted. \n\n", StyleBox["2)", FontColor->RGBColor[1, 0, 0]], " Select the ", StyleBox["3DViewPoint Selector", FontColor->RGBColor[0, 0, 1]], " from the ", StyleBox["Input", FontVariations->{"Underline"->True}], " menu on the toolbar. This opens a window that shows the bounding box for \ your graphics.\n\n", StyleBox["3)", FontColor->RGBColor[1, 0, 0]], " Place your cursor on the diagram of the box in the ViewPoint Selector \ window and, holding down the left mouse button, drag the box into the \ orientation you prefer for your graphics. \n\n", StyleBox["4) ", FontColor->RGBColor[1, 0, 0]], "Click on the \"Paste\" button in the ViewPoint Selector window. This \ inserts the appropriate ", StyleBox["ViewPoint", "MR"], " specification into your graphics command.\n\nHere are views of our \ spiral from directly above and from the side." }], "Text"], Cell[BoxData[ \(\(Show[%, ViewPoint -> {0.000, \ 0.000, \ 3}];\)\)], "Input"], Cell[BoxData[ \(\(Show[%, ViewPoint -> {2.964, \ 1.574, \ \(-0.431\)}];\)\)], "Input"], Cell[CellGroupData[{ Cell["2.5.1 Graphs of functions of two variables", "Subsubsection", FontColor->RGBColor[0, 0, 1]], Cell[TextData[{ "Functions of two or more variable are defined in ", StyleBox["Mathematica", FontSlant->"Italic"], " in the same way as single-variable functions. The function ", Cell[BoxData[ \(TraditionalForm\`f(x, y) = sin \@\( x\^2 + \((x + 3)\) y\^2\)\)]], "is defined in the following input cell:" }], "Text"], Cell[BoxData[ \(Clear[f]; f[x_, y_] := Sin[\@\(x\^2 + \((x + 3)\) y\^2\)]\)], "Input"], Cell[TextData[{ StyleBox["\[FilledSmallCircle] ", FontColor->RGBColor[0, 0, 1]], "Its graph is obtained with the command ", StyleBox["Plot3D", FontFamily->"Courier New"], "." }], "Text"], Cell[BoxData[ \(\(Plot3D[f[x, y], {x, \(-3\), 3}, {y, \(-4\), 4}];\)\)], "Input"], Cell[TextData[{ StyleBox["\[FilledSmallCircle] ", FontColor->RGBColor[0, 0, 1]], "As in the case of ", StyleBox["ParametricPlot3D", FontFamily->"Courier New"], ", ", StyleBox["Mathematica", FontSlant->"Italic"], " constructs the graph of ", Cell[BoxData[ \(TraditionalForm\`z = f(x, y), \ a \[LessEqual] x \[LessEqual] y, \ c \[LessEqual] y \[LessEqual] d\)]], ", by partitioning the intervals into 15 subintervals each. It then \ constructs a polyhedron with vertices the points ", Cell[BoxData[ \(TraditionalForm\`f(x\_j, y\_k), \ 0 \[LessEqual] j \[LessEqual] 15, \ 0 \[LessEqual] k \[LessEqual] 15\)]], ". This corresponds to the default setting ", StyleBox["PlotPoints->{15,15}", FontFamily->"Courier New"], ". You can improve the resolution by assigning another value. Other options \ are illustrated below as well." }], "Text"], Cell[BoxData[ \(\(Plot3D[f[x, y], {x, \(-3\), 3}, {y, \(-4\), 4}, PlotPoints \[Rule] {50, 50}, Boxed \[Rule] False, Axes \[Rule] None, BoxRatios \[Rule] {1, 1, 1/4}, Mesh \[Rule] False, \ ViewPoint -> {0.045, \ 0.002, \ 3.383}];\)\)], "Input"], Cell[TextData[{ StyleBox["\[FilledSmallCircle] ", FontColor->RGBColor[0, 0, 1]], "A contour plot of a function ", Cell[BoxData[ \(TraditionalForm\`z = f(x, y)\)]], " is a collection of graphs of level curves of the function ", Cell[BoxData[ \(TraditionalForm\`f(x, y) = c\)]], ", for various values ", Cell[BoxData[ \(TraditionalForm\`c\)]], ". This is another way to represent the distribution of values of a \ function." }], "Text"], Cell[BoxData[{ \(\(Clear[f];\)\), "\[IndentingNewLine]", \(f[x_, y_] := \(9\/2\) x\ y\^2 - 27\ x + 9 2 y\^3\)}], "Input"], y\^2 - \(81\/32\) x\^5 Cell[BoxData[ \(\(Plot3D[f[x, y], {x, \(-1\), 2.5}, {y, 1, 6.5}, PlotPoints \ [Rule] 50, BoxRatios -> {1, 1, 1.5}, PlotRange \[Rule] {\(-20\), 100}, AxesLabel -> {x, y, z}, ViewPoint -> {\(-1.323\), \ \(-2.803\), \ 1.356}];\)\)], "Input"], Cell[BoxData[ \(\(ContourPlot[f[x, y], {x, \(-1\), 2.5}, {y, 1, 6.5}, PlotPoints -> 50, Contours \[Rule] 100];\)\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ 2.5.2 Parameterized Surfaces, cylindrical and spherical coordinates \ \>", "Subsubsection", FontColor->RGBColor[0, 0, 1]], Cell[TextData[{ "Here, we some examples of using ", Cell[BoxData[ \(TraditionalForm\`ParametricPlot3D\)], FontFamily->"Courier New"], " to plot surfaces. " }], "Text"], Cell[BoxData[ \(Clear[s, t, G]; \ G[s_, t_] := {t\ Cos[t], t\^\(5/4\), t\ Sin[t]} + t {Cos[s], 0, Sin[s]}\)], "Input"], Cell[BoxData[ \(ParametricPlot3D[Evaluate[G[s, t]], {s, 0, 2 \[Pi]}, {t, 0, 18}, PlotPoints \[Rule] {30, 70}, Boxed \[Rule] False, Axes \[Rule] None, ViewPoint -> {3.010, \ \(-1.012\), \ 1.168}]\)], "Input"], Cell[TextData[{ StyleBox["\[FilledSmallCircle] ", FontColor->RGBColor[0, 1, 0]], "Graph the cone with polar equation ", Cell[BoxData[ \(TraditionalForm\`z = 4 - 2 r, \ 0 \[LessEqual] r \[LessEqual] 2\)]], ". " }], "Text"], Cell[BoxData[{ \(Clear[x, y, z]; \ x = r\ Cos[\[Theta]]; y = r\ Sin[\[Theta]]; \ z = 4 - 2 r;\), "\[IndentingNewLine]", \(\(ParametricPlot3D[{x, y, z}, {\[Theta], 0, 2 \[Pi]}, {r, 0, 2}, PlotPoints \[Rule] {50, 2}];\)\)}], "Input"], Cell[TextData[{ StyleBox["\[FilledSmallCircle] ", FontColor->RGBColor[0, 1, 0]], "Graph the cylinder with polar equation ", Cell[BoxData[ \(TraditionalForm\`r = 1, \ 0 \[LessEqual] z \[LessEqual] 2\)]], ". " }], "Text"], Cell[BoxData[{ \(Clear[x, y, z]; \ x = 1\ Cos[\[Theta]]; y = 1\ Sin[\[Theta]];\), "\[IndentingNewLine]", \(\(ParametricPlot3D[{x, y, z}, {\[Theta], 0, 2 \[Pi]}, {z, 0, 2}, PlotPoints \[Rule] {50, 2}];\)\)}], "Input"], Cell[TextData[{ StyleBox["\[FilledSmallCircle] ", FontColor->RGBColor[0, 1, 0]], " Graph the sphere ", Cell[BoxData[ \(TraditionalForm\`x\^2 + y\^2 + z\^2 = 4\)]], ". \n\n", StyleBox["\[FilledSmallCircle] ", FontColor->RGBColor[0, 0, 1]], " ", StyleBox["A Solution using spherical coordinates.", FontColor->RGBColor[0, 0, 1]], " The sphere has equation ", Cell[BoxData[ \(TraditionalForm\`\[Rho] = 2\)]], "; hence its parameterization is \n ", Cell[BoxData[ \(TraditionalForm\`x = 2 \( sin(\[CurlyPhi])\) \(cos(\ [Theta])\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`y = 2 \( sin(\[CurlyPhi])\) \(sin(\ [Theta])\)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`z = 2 \( cos(\[CurlyPhi])\), \ 0 \[LessEqual] \[Theta] \[LessEqual] 2 \[Pi], \ \(\(0\)\(\[LessEqual]\)\(\[CurlyPhi]\)\(\ [LessEqual]\)\(\ \[Pi]\)\(\ \)\)\)]] }], "Text"], Cell[BoxData[{ \(\(Clear[x, y, z, G, \[Theta], \[CurlyPhi]];\)\), "\[IndentingNewLine]", \(G[\[CurlyPhi]_, \[Theta]_] := {2\ Sin[\[CurlyPhi]]\ Cos[\ [Theta]], 2\ Sin[\[CurlyPhi]]\ Sin[\[Theta]], 2 Cos[\[CurlyPhi]]}\)}], "Input"], Cell[BoxData[ \(\(ParametricPlot3D[ Evaluate[G[\[CurlyPhi], \[Theta]]], {\[Theta], 0, 2 \[Pi]}, {\[CurlyPhi], 0, \[Pi]}, PlotPoints \[Rule] {70, 30}, Boxed \[Rule] False, Axes \[Rule] None];\)\)], "Input"], Cell[TextData[{ StyleBox["\[FilledSmallCircle] ", FontColor->RGBColor[0, 1, 0]], "Suppose that the temperature on the surface of the sphere \ \[ScriptCapitalS] above is given by ", Cell[BoxData[ \(TraditionalForm\`T(x, y, z) = 2 x\ \ y\ z + \((x + 2 y - z + 2)\)\^2 + \((2 x + y + z 1)\)\^2 \ + \((4 + x - y + 2 z)\)\^2\)]], ". \nWe want to construct a picture of the sphere representing the \ temperature distribution by the color (", StyleBox["style", FontFamily->"Times New Roman", FontSlant->"Italic"], "). \n\n", StyleBox["\[FilledSmallCircle] ", FontColor->RGBColor[0, 0, 1]], " ", StyleBox["A Solution.", FontColor->RGBColor[0, 0, 1]], " By plotting ", Cell[BoxData[ \(TraditionalForm\`\(\(T(G(\[CurlyPhi], \[Theta]))\)\(,\)\)\)]], " we see that the range of temperatures is about ", Cell[BoxData[ \(TraditionalForm\`20 \[LessEqual] T \[LessEqual] 80\)]], ", so we map the interval ", Cell[BoxData[ \(TraditionalForm\`\([20, 80]\)\)]], " onto ", Cell[BoxData[ \(TraditionalForm\`\([0, .6]\)\)]], " by ", Cell[BoxData[ \(TraditionalForm\`t = .6 - .01 \((T - 20)\)\)]], ". Our color function is then ", Cell[BoxData[ \(TraditionalForm\`f(\[CurlyPhi], \[Theta]) = Hue[ .6 - .01 \((T(G(\[CurlyPhi], \[Theta])) - 20)\)]\)]], " (This will make warmer regions red and cooler ones blue)" }], "Text"], Cell[BoxData[{ \(\(Clear[f];\)\), "\[IndentingNewLine]", \(\ f[\[CurlyPhi]_, \[Theta]_] := Hue[ .6 - .01 \((5\ \((9 + 4\ Cos[\[CurlyPhi]])\) + 8\ Cos[\[Theta]]\ \((2 + 3\ Cos[\[CurlyPhi]])\)\ Sin[\[CurlyPhi]]\ \[IndentingNewLine] 4\ \((1 + 6\ Cos[\[CurlyPhi]])\)\ Sin[\[Theta]]\ Sin[\ [CurlyPhi]] \ + 4\ \((3 + 2\ Cos[\[CurlyPhi]])\)\ Sin[2\ \[Theta]]\ Sin[\[CurlyPhi]]\ ^2 20)\)]\)}], "Input"], Cell[BoxData[ \(Clear[H]; H[\[CurlyPhi]_, \[Theta]_] := Append[G[\[CurlyPhi], \[Theta]], f[\[CurlyPhi], \[Theta]]]\)], "Input"], Cell[TextData[{ StyleBox["\[FilledSmallCircle]", FontColor->RGBColor[0, 0, 1]], StyleBox[" ", FontColor->RGBColor[0, 1, 0]], "In order to see the effects of our color function, we must turn off the \ external lighting in our ", StyleBox["3D graphics", 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.24212 .49233 0 g s .645 1 0 r .46888 .68134 m .47328 .67962 0 g s .624 1 0 r .49712 .69408 m .49473 .69488 0 g s .99 1 0 r .59975 .67034 m .60443 .67353 0 g s .99 1 0 r .5761 .68779 m .57299 .6856 L 0 g L .78186 .48533 L p F P L .76205 .55225 L p F P L .62107 .15648 L p F P L .61199 .13655 L p F P L .46123 .68528 L p F P L .43473 .66987 L p F P L .76836 .28238 L p F P L .79108 .35082 L p F P L .25373 .48411 L p F P L .23108 .46121 L p F P L .49473 .69488 L p F P L .47328 .67962 L p F P L .5761 .68779 L p F P .59975 .67034 L p F P s .791 1 0 r .80441 .39024 m .78989 .38274 L .79108 .35082 L p F P 0 g s .899 1 0 r .78989 .38274 m .80441 .39024 L .79996 .42241 L p F P 0 g s .132 1 0 r .37075 .64125 m .36499 .64667 L .33876 .6261 L p F P 0 g s .277 1 0 r .36499 .64667 m .37075 .64125 L .39755 .66015 L p F P 0 g s .89 1 0 r .56112 .67987 m .56547 .68162 L .54151 .69592 L p F P 0 g s .89 1 0 r .54151 .69592 m .53932 .69501 L .56112 .67987 L p F P 0 g s 1 .78 0 r .6639 .64206 m .66949 .64751 L .64134 .66536 L p F P 0 g s 1 .78 0 r .64134 .66536 m .63691 .66082 L .6639 .64206 L p F P 0 g s 1 .75 0 r .79318 .49361 m .78186 .48533 L .79241 .45421 L p F P 0 g s 1 .695 0 r .78186 .48533 m .79318 .49361 L .77904 .52363 L p F P 0 g s 0 1 .93 r .3138 .1936 m .33055 .18981 L .31969 .21529 L p F P 0 g s 0 1 .93 r .31969 .21529 m .30187 .2195 L .3138 .1936 L p F P 0 g s .67 1 0 r .46485 .68324 m .46888 .68134 L .49252 .69577 L p F P 0 g s .645 1 0 r .49473 .69488 m .49252 .69577 L .46888 .68134 L p F P 0 g s 0 1 .927 r .28544 .22409 m .30187 .2195 L .29251 .24709 L p F P 0 g s 0 1 .927 r .29251 .24709 m .27527 .25213 L .28544 .22409 L p F P 0 g s 0 1 .67 r .501 .1157 m .51815 .11558 L .51831 .13232 L p F P 0 g s 0 1 .67 r .51831 .13232 m .49908 .13246 L .501 .1157 L p F P 0 g s 0 1 .639 r .53754 .13251 m .51831 .13232 L .51815 .11558 L p F P 0 g s .935 1 0 r .56547 .68162 m .56943 .68353 L .5435 .6969 L p F P 0 g s .935 1 0 r .5435 .6969 m .54151 .69592 L .56547 .68162 L p F P 0 g s 0 1 .483 r .22318 .42931 m .23578 .42121 L .24327 .45299 L p F P 0 g s 0 1 .483 r .24327 .45299 m .23108 .46121 L .22318 .42931 L p F P 0 g s 0 1 .701 r .48395 .1161 m .501 .1157 L .49908 .13246 L p F P 0 g s 0 1 .701 r .49908 .13246 m .47995 .13293 L .48395 .1161 L p F P 0 g s 0 1 .639 r .51815 .11558 m .5353 .11574 L .53754 .13251 L p F P 0 g s 0 1 .607 r .55663 .13303 m .53754 .13251 L .5353 .11574 L p F P 0 g s .343 1 0 r .42994 .67303 m .42583 .67638 L .39755 .66015 L p F P 0 g s .549 1 0 r .42583 .67638 m .42994 .67303 L .45805 .68745 L p F P 0 g s .487 1 0 r .46123 .68528 m .45805 .68745 L .42994 .67303 L p F P 0 g s 0 1 .158 r .68801 .18688 m .70577 .19036 L .7168 .2159 L p F P 0 g s 0 1 .091 r .73441 .22017 m .7168 .2159 L .70577 .19036 L p F P 0 g s 0 1 .001 r .7168 .2159 m .73441 .22017 L .7439 .24782 L p F P 0 g s .277 1 0 r .39755 .66015 m .39297 .66466 L .36499 .64667 L p F P 0 g s .439 1 0 r .39297 .66466 m .39755 .66015 L .42583 .67638 L p F P 0 g s 0 1 .766 r .23207 .325 m .24683 .31825 L .245 .34977 L p F P 0 g s 0 1 .766 r .245 .34977 m .23011 .35689 L .23207 .325 L p F P 0 g s 0 1 .851 r .3901 .14014 m .4069 .13807 L .39637 .15821 L p F P 0 g s 0 1 .851 r .39637 .15821 m .37789 .16058 L .3901 .14014 L p F P 0 g s .083 1 0 r .76091 .25292 m .7439 .24782 L .73441 .22017 L p F P 0 g s .19 1 0 r .7439 .24782 m .76091 .25292 L .76836 .28238 L p F P 0 g s 0 1 .731 r .4671 .11679 m .48395 .1161 L .47995 .13293 L p F P 0 g s 0 1 .731 r .47995 .13293 m .46105 .13373 L .4671 .11679 L p F P 0 g s 1 .932 0 r .60443 .67353 m .60843 .67689 L .57874 .69009 L p F P 0 g s 1 .932 0 r .57874 .69009 m .5761 .68779 L .60443 .67353 L p F P 0 g s 0 1 .607 r .5353 .11574 m .55232 .11619 L .55663 .13303 L p F P 0 g s 0 1 .575 r .57549 .13388 m .55663 .13303 L .55232 .11619 L p F P 0 g s 1 .821 0 r .63691 .66082 m .64134 .66536 L .61171 .68041 L p F P 0 g s 1 .821 0 r .61171 .68041 m .60843 .67689 L .63691 .66082 L p F P 0 g s .699 1 0 r .46123 .68528 m .46485 .68324 L .4905 .69675 L p F P 0 g s .67 1 0 r .49252 .69577 m .4905 .69675 L .46485 .68324 L p F P 0 g s 0 1 .904 r .3437 .16633 m .36029 .16329 L .34848 .18638 L p F P 0 g s 0 1 .904 r .34848 .18638 m .33055 .18981 L .3437 .16633 L p F P 0 g s 0 1 .432 r .61199 .13655 m .62939 .13836 L .64019 .15855 L p F P 0 g s 0 1 .393 r .65853 .16097 m .64019 .15855 L .62939 .13836 L p F P 0 g s 1 .832 0 r .80429 .4625 m .79241 .45421 L .79996 .42241 L p F P 0 g s 1 .75 0 r .79241 .45421 m .80429 .4625 L .79318 .49361 L p F P 0 g s .981 1 0 r .56943 .68353 m .57299 .6856 L .54527 .69796 L p F P 0 g s .981 1 0 r .54527 .69796 m .5435 .6969 L .56943 .68353 L p F P 0 g s 0 1 .882 r .25961 .25757 m .27527 .25213 L .26791 .28152 L p F P 0 g s 0 1 .882 r .26791 .28152 m .2517 .28741 L .25961 .25757 L p F P 0 g s .665 1 0 r .80568 .35801 m .79108 .35082 L .78923 .31925 L p F P 0 g s .791 1 0 r .79108 .35082 m .80568 .35801 L .80441 .39024 L p F P 0 g s .09 1 0 r .28513 .58581 m .29255 .57804 L .31453 .60315 L p F P 0 g s .09 1 0 r .31453 .60315 m .30802 .61034 L .28513 .58581 L p F P 0 g s .183 1 0 r .30802 .61034 m .31453 .60315 L .33876 .6261 L p F P 0 g s 0 1 .286 r .65853 .16097 m .67599 .16373 L .68801 .18688 L p F P 0 g s 0 1 .236 r .70577 .19036 m .68801 .18688 L .67599 .16373 L p F P 0 g s .183 1 0 r .33876 .6261 m .3332 .63258 L .30802 .61034 L p F P 0 g s .287 1 0 r .3332 .63258 m .33876 .6261 L .36499 .64667 L p F P 0 g s .006 1 0 r .26477 .55923 m .27302 .55103 L .29255 .57804 L p F P 0 g s .006 1 0 r .29255 .57804 m .28513 .58581 L .26477 .55923 L p F P 0 g s 0 1 .759 r .45058 .11776 m .4671 .11679 L .46105 .13373 L p F P 0 g s 0 1 .759 r .46105 .13373 m .4425 .13486 L .45058 .11776 L p F P 0 g s 0 1 .575 r .55232 .11619 m .56912 .11692 L .57549 .13388 L p F P 0 g s 0 1 .545 r .59398 .13505 m .57549 .13388 L .56912 .11692 L p F P 0 g s 0 1 .536 r .21852 .39695 m .23137 .38907 L .23578 .42121 L p F P 0 g s 0 1 .536 r .23578 .42121 m .22318 .42931 L .21852 .39695 L p F P 0 g s .289 1 0 r .78431 .28834 m .76836 .28238 L .76091 .25292 L p F P 0 g s .414 1 0 r .76836 .28238 m .78431 .28834 L .78923 .31925 L p F P 0 g s .731 1 0 r .45805 .68745 m .46123 .68528 L .48869 .69779 L p F P 0 g s .699 1 0 r .4905 .69675 m .48869 .69779 L .46123 .68528 L p F P 0 g s 1 .495 0 r .72653 .61146 m .72026 .60422 L .74239 .5792 L p F P 0 g s 1 .535 0 r .72026 .60422 m .72653 .61146 L .70123 .63359 L p F P 0 g s 1 .535 0 r .70123 .63359 m .69588 .62707 L .72026 .60422 L p F P 0 g s .287 1 0 r .36499 .64667 m .3604 .65232 L .3332 .63258 L p F P 0 g s .405 1 0 r .3604 .65232 m .36499 .64667 L .39297 .66466 L p F P 0 g s 1 .59 0 r .69588 .62707 m .70123 .63359 L .67389 .65319 L p F P 0 g s 1 .59 0 r .67389 .65319 m .66949 .64751 L .69588 .62707 L p F P 0 g s 0 1 .072 r .24716 .53085 m .25615 .52237 L .27302 .55103 L p F P 0 g s 0 1 .072 r .27302 .55103 m .26477 .55923 L .24716 .53085 L p F P 0 g s 1 .973 0 r .57299 .6856 m .5761 .68779 L .54682 .69908 L p F P 0 g s 1 .973 0 r .54682 .69908 m .54527 .69796 L .57299 .6856 L p F P 0 g s .939 1 0 r .51685 .70724 m .51405 .69144 L .51708 .69139 L closepath p F P 0 g s .945 1 0 r .51685 .70724 m .51708 .69139 L .52012 .69145 L closepath p F P 0 g s .439 1 0 r .42583 .67638 m .42244 .67987 L .39297 .66466 L p F P 0 g s .615 1 0 r .42244 .67987 m .42583 .67638 L .45533 .68974 L p F P 0 g s .549 1 0 r .45805 .68745 m .45533 .68974 L .42583 .67638 L p F P 0 g s .935 1 0 r .51685 .70724 m .51103 .6916 L .51405 .69144 L closepath p F P 0 g s .951 1 0 r .51685 .70724 m .52012 .69145 L .52313 .69164 L closepath p F P 0 g s 1 .475 .74954 0 g s 1 .495 .74239 0 g s .933 1 .51685 0 g s .959 1 .51685 0 g s .932 1 .51685 0 g s .969 1 .51685 0 g s 1 .857 .60843 0 g s 1 .857 .58088 0 g s 1 .654 .66949 0 g s 1 .654 .64479 0 g s 1 .939 .81226 0 g s 1 .832 .79996 0 g s .933 1 .51685 0 g s .979 1 .51685 0 g 0 r .58701 m .74239 .5792 L .76205 .55225 L p F P 0 r .5792 m .74954 .58701 L .72653 .61146 L p F P 0 r .70724 m .50807 .69188 L .51103 .6916 L closepath p F P 0 r .70724 m .52313 .69164 L .52608 .69193 L closepath p F P 0 r .70724 m .50517 .69227 L .50807 .69188 L closepath p F P 0 r .70724 m .52608 .69193 L .52897 .69234 L closepath p F P 0 r .67689 m .61171 .68041 L .58088 .6925 L p F P 0 r .6925 m .57874 .69009 L .60843 .67689 L p F P 0 r .64751 m .67389 .65319 L .64479 .67006 L p F P 0 r .67006 m .64134 .66536 L .66949 .64751 L p F P 0 r .43058 m .79996 .42241 L .80441 .39024 L p F P 0 r .42241 m .81226 .43058 L .80429 .4625 L p F P 0 r .70724 m .50236 .69277 L .50517 .69227 L closepath p F P 0 r .70724 m .52897 .69234 L .53176 .69286 L closepath p F P s 1 .482 0 r .77002 .5605 m .76205 .55225 L .77904 .52363 L p F P 0 g s 1 .475 0 r .76205 .55225 m .77002 .5605 L .74954 .58701 L p F P 0 g s .405 1 0 r .39297 .66466 m .38937 .66934 L .3604 .65232 L p F P 0 g s .536 1 0 r .38937 .66934 m .39297 .66466 L .42244 .67987 L p F P 0 g s 0 1 .786 r .43449 .119 m .45058 .11776 L .4425 .13486 L p F P 0 g s 0 1 .786 r .4425 .13486 m .42441 .1363 L .43449 .119 L p F P 0 g s .935 1 0 r .51685 .70724 m .49968 .69338 L .50236 .69277 L closepath p F P 0 g s .991 1 0 r .51685 .70724 m .53176 .69286 L .53443 .69348 L closepath p F P 0 g s 0 1 .145 r .23249 .50096 m .24212 .49233 L .25615 .52237 L p F P 0 g s 0 1 .145 r .25615 .52237 m .24716 .53085 L .23249 .50096 L p F P 0 g s .766 1 0 r .45533 .68974 m .45805 .68745 L .48711 .69891 L p F P 0 g s .731 1 0 r .48869 .69779 m .48711 .69891 L .45805 .68745 L p F P 0 g s 0 1 .545 r .56912 .11692 m .58559 .11793 L .59398 .13505 L p F P 0 g s 0 1 .516 r .61199 .13655 m .59398 .13505 L .58559 .11793 L p F P 0 g s 0 1 .867 r .37411 .1425 m .3901 .14014 L .37789 .16058 L p F P 0 g s 0 1 .867 r .37789 .16058 m .36029 .16329 L .37411 .1425 L p F P 0 g s .939 1 0 r .51685 .70724 m .49712 .69408 L .49968 .69338 L closepath p F P 0 g s 0 1 .786 r .23725 .29372 m .2517 .28741 L .24683 .31825 L p F P 0 g s 0 1 .786 r .24683 .31825 m .23207 .325 L .23725 .29372 L p F P 0 g s 1 .721 0 r .64134 .66536 m .64479 .67006 L .61423 .68404 L p F P 0 g s 1 .721 0 r .61423 .68404 m .61171 .68041 L .64134 .66536 L p F P 0 g s 1 .927 0 r .5761 .68779 m .57874 .69009 L .54811 .70026 L p F P 0 g s 1 .927 0 r .54811 .70026 m .54682 .69908 L .5761 .68779 L p F P 0 g s 1 .997 0 r .51685 .70724 m .53443 .69348 L .53696 .6942 L closepath p F P 0 g s 0 1 .393 r .62939 .13836 m .64608 .14047 L .65853 .16097 L p F P 0 g s 0 1 .358 r .67599 .16373 m .65853 .16097 L .64608 .14047 L p F P 0 g s .944 1 0 r .51685 .70724 m .49473 .69488 L .49712 .69408 L closepath p F P 0 g s 1 .984 0 r .51685 .70724 m .53696 .6942 L .53932 .69501 L closepath p F P 0 g s 1 .516 0 r .78774 .53217 m .77904 .52363 L .79318 .49361 L p F P 0 g s 1 .482 0 r .77904 .52363 m .78774 .53217 L .77002 .5605 L p F P 0 g s 0 1 .586 r .21719 .36445 m .23011 .35689 L .23137 .38907 L p F P 0 g s 0 1 .586 r .23137 .38907 m .21852 .39695 L .21719 .36445 L p F P 0 g s 0 1 .907 r .29838 .19773 m .3138 .1936 L .30187 .2195 L p F P 0 g s 0 1 .907 r .30187 .2195 m .28544 .22409 L .29838 .19773 L p F P 0 g s .951 1 0 r .51685 .70724 m .49252 .69577 L .49473 .69488 L closepath p F P 0 g s .524 1 0 r .80371 .32606 m .78923 .31925 L .78431 .28834 L p F P 0 g s .665 1 0 r .78923 .31925 m .80371 .32606 L .80568 .35801 L p F P 0 g s .536 1 0 r .42244 .67987 m .4198 .68349 L .38937 .66934 L p F P 0 g s .683 1 0 r .4198 .68349 m .42244 .67987 L .45311 .69213 L p F P 0 g s .615 1 0 r .45533 .68974 m .45311 .69213 L .42244 .67987 L p F P 0 g s 1 .97 0 r .51685 .70724 m .53932 .69501 L .54151 .69592 L closepath p F P 0 g s 0 1 .217 r .22094 .46986 m .23108 .46121 L .24212 .49233 L p F P 0 g s 0 1 .217 r .24212 .49233 m .23249 .50096 L .22094 .46986 L p F P 0 g s .803 1 0 r .45311 .69213 m .45533 .68974 L .48577 .70008 L p F P 0 g s .766 1 0 r .48711 .69891 m .48577 .70008 L .45533 .68974 L p F P 0 g s 1 .784 0 r .61171 .68041 m .61423 .68404 L .58251 .69498 L p F P 0 g s 1 .784 0 r .58251 .69498 m .58088 .6925 L .61171 .68041 L p F P 0 g s 0 1 .902 r .32823 .16969 m .3437 .16633 L .33055 .18981 L p F P 0 g s 0 1 .902 r .33055 .18981 m .3138 .1936 L .32823 .16969 L p F P 0 g s .958 1 0 r .51685 .70724 m .4905 .69675 L .49252 .69577 L closepath p F P 0 g s 1 .882 0 r .57874 .69009 m .58088 .6925 L .54916 .70148 L p F P 0 g s 1 .882 0 r .54916 .70148 m .54811 .70026 L .57874 .69009 L p F P 0 g s 1 .956 0 r .51685 .70724 m .54151 .69592 L .5435 .6969 L closepath p F P 0 g s 0 1 .812 r .41893 .12051 m .43449 .119 L .42441 .1363 L p F P 0 g s 0 1 .812 r .42441 .1363 m .4069 .13807 L .41893 .12051 L p F P 0 g s 0 1 .879 r .27054 .22905 m .28544 .22409 L .27527 .25213 L p F P 0 g s 0 1 .879 r .27527 .25213 m .25961 .25757 L .27054 .22905 L p F P 0 g s 0 1 .091 r .70577 .19036 m .72233 .1942 L .73441 .22017 L p F P 0 g s 0 1 .034 r .75062 .22482 m .73441 .22017 L .72233 .1942 L p F P 0 g s .083 1 0 r .73441 .22017 m .75062 .22482 L .76091 .25292 L p F P 0 g s .93 1 0 r .81695 .39818 m .80441 .39024 L .80568 .35801 L p F P 0 g s 1 .939 0 r .80441 .39024 m .81695 .39818 L .81226 .43058 L p F P 0 g s 0 1 .516 r .58559 .11793 m .60161 .11921 L .61199 .13655 L p F P 0 g s 0 1 .49 r .62939 .13836 m .61199 .13655 L .60161 .11921 L p F P 0 g s .967 1 0 r .51685 .70724 m .48869 .69779 L .4905 .69675 L closepath p F P 0 g s 1 .582 0 r .80249 .5023 m .79318 .49361 L .80429 .4625 L p F P 0 g s 1 .516 0 r .79318 .49361 m .80249 .5023 L .78774 .53217 L p F P 0 g s 0 1 .236 r .67599 .16373 m .69242 .16682 L .70577 .19036 L p F P 0 g s 0 1 .193 r .72233 .1942 m .70577 .19036 L .69242 .16682 L p F P 0 g s 1 .941 0 r .51685 .70724 m .5435 .6969 L .54527 .69796 L closepath p F P 0 g s .441 1 0 r .32906 .63929 m .3332 .63258 L .3604 .65232 L p F P 0 g s .441 1 0 r .3604 .65232 m .35705 .65814 L .32906 .63929 L p F P 0 g s .532 1 0 r .35705 .65814 m .3604 .65232 L .38937 .66934 L p F P 0 g s .359 1 0 r .30312 .6178 m .30802 .61034 L .3332 .63258 L p F P 0 g s .359 1 0 r .3332 .63258 m .32906 .63929 L .30312 .6178 L p F P 0 g s .154 1 0 r .77632 .25843 m .76091 .25292 L .75062 .22482 L p F P 0 g s .289 1 0 r .76091 .25292 m .77632 .25843 L .78431 .28834 L p F P 0 g s .532 1 0 r .38937 .66934 m .38678 .67416 L .35705 .65814 L p F P 0 g s .636 1 0 r .38678 .67416 m .38937 .66934 L .4198 .68349 L p F P 0 g s .283 1 0 r .27948 .59387 m .28513 .58581 L .30802 .61034 L p F P 0 g s .283 1 0 r .30802 .61034 m .30312 .6178 L .27948 .59387 L p F P 0 g s .977 1 0 r .51685 .70724 m .48711 .69891 L .48869 .69779 L closepath p F P 0 g s .683 1 0 r .45311 .69213 m .45141 .6946 L .4198 .68349 L p F P 0 g s .842 1 0 r .45141 .6946 m .45311 .69213 L .48469 .70129 L p F P 0 g s .803 1 0 r .48577 .70008 m .48469 .70129 L .45311 .69213 L p F P 0 g s 1 .457 0 r .67705 .65903 m .67389 .65319 L .70123 .63359 L p F P 0 g s 1 .538 0 r .67389 .65319 m .67705 .65903 L .64723 .6749 L p F P 0 g s 1 .538 0 r .64723 .6749 m .64479 .67006 L .67389 .65319 L p F P 0 g s 1 .926 0 r .51685 .70724 m .54527 .69796 L .54682 .69908 L closepath p F P 0 g s 0 1 .288 r .21266 .43784 m .22318 .42931 L .23108 .46121 L p F P 0 g s 0 1 .288 r .23108 .46121 m .22094 .46986 L .21266 .43784 L p F P 0 g s 1 .388 0 r .70514 .64032 m .70123 .63359 L .72653 .61146 L p F P 0 g s 1 .457 0 r .70123 .63359 m .70514 .64032 L .67705 .65903 L p F P 0 g s 1 .838 0 r .58088 .6925 m .58251 .69498 L .54994 .70274 L p F P 0 g s 1 .838 0 r .54994 .70274 m .54916 .70148 L .58088 .6925 L p F P 0 g s 1 .626 0 r .64479 .67006 m .64723 .6749 L .61597 .68777 L p F P 0 g s 1 .626 0 r .61597 .68777 m .61423 .68404 L .64479 .67006 L p F P 0 g s .636 1 0 r .4198 .68349 m .41795 .68721 L .38678 .67416 L p F P 0 g s .752 1 0 r .41795 .68721 m .4198 .68349 L .45141 .6946 L p F P 0 g s 0 1 .881 r .35907 .14516 m .37411 .1425 L .36029 .16329 L p F P 0 g s 0 1 .881 r .36029 .16329 m .3437 .16633 L .35907 .14516 L p F P 0 g s .988 1 0 r .51685 .70724 m .48577 .70008 L .48711 .69891 L closepath p F P 0 g s 0 1 .635 r .21925 .33216 m .23207 .325 L .23011 .35689 L p F P 0 g s 0 1 .635 r .23011 .35689 m .21719 .36445 L .21925 .33216 L p F P 0 g s 1 .339 0 r .73118 .61894 m .72653 .61146 L .74954 .58701 L p F P 0 g s 1 .388 0 r .72653 .61146 m .73118 .61894 L .70514 .64032 L p F P 0 g s .211 1 0 r .25843 .56775 m .26477 .55923 L .28513 .58581 L p F P 0 g s .211 1 0 r .28513 .58581 m .27948 .59387 L .25843 .56775 L p F P 0 g s 1 .911 0 r .51685 .70724 m .54682 .69908 L .54811 .70026 L closepath p F P 0 g s 0 1 .806 r .24568 .26339 m .25961 .25757 L .2517 .28741 L p F P 0 g s 0 1 .806 r .2517 .28741 m .23725 .29372 L .24568 .26339 L p F P 0 g s 1 .714 0 r .61423 .68404 m .61597 .68777 L .5836 .69752 L p F P 0 g s 1 .714 0 r .5836 .69752 m .58251 .69498 L .61423 .68404 L p F P 0 g s 1 .678 0 r .81411 .4712 m .80429 .4625 L .81226 .43058 L p F P 0 g s 1 .582 0 r .80429 .4625 m .81411 .4712 L .80249 .5023 L p F P 0 g s 0 1 .358 r .64608 .14047 m .66193 .14289 L .67599 .16373 L p F P 0 g s 0 1 .329 r .69242 .16682 m .67599 .16373 L .66193 .14289 L p F P 0 g s 0 1 .836 r .40402 .12228 m .41893 .12051 L .4069 .13807 L p F P 0 g s 0 1 .836 r .4069 .13807 m .3901 .14014 L .40402 .12228 L p F P 0 g s 1 1 0 r .51685 .70724 m .48469 .70129 L .48577 .70008 L closepath p F P 0 g s 1 .896 0 r .51685 .70724 m .54811 .70026 L .54916 .70148 L closepath p F P 0 g s 1 .316 0 r .75491 .59511 m .74954 .58701 L .77002 .5605 L p F P 0 g s 1 .339 0 r .74954 .58701 m .75491 .59511 L .73118 .61894 L p F P 0 g s .752 1 0 r .45141 .6946 m .45024 .69713 L .41795 .68721 L p F P 0 g s .883 1 0 r .45024 .69713 m .45141 .6946 L .48387 .70255 L p F P 0 g s .842 1 0 r .48469 .70129 m .48387 .70255 L .45141 .6946 L p F P 0 g s 0 1 .49 r .60161 .11921 m .61707 .12076 L .62939 .13836 L p F P 0 g s 0 1 .468 r .64608 .14047 m .62939 .13836 L .61707 .12076 L p F P 0 g s .78 1 0 r .81829 .36563 m .80568 .35801 L .80371 .32606 L p F P 0 g s .93 1 0 r .80568 .35801 m .81829 .36563 L .81695 .39818 L p F P 0 g s 1 .796 0 r .58251 .69498 m .5836 .69752 L .55045 .70403 L p F P 0 g s 1 .796 0 r .55045 .70403 m .54994 .70274 L .58251 .69498 L p F P 0 g s .372 1 0 r .79847 .29471 m .78431 .28834 L .77632 .25843 L p F P 0 g s .524 1 0 r .78431 .28834 m .79847 .29471 L .80371 .32606 L p F P 0 g s .138 1 0 r .24018 .53968 m .24716 .53085 L .26477 .55923 L p F P 0 g s .138 1 0 r .26477 .55923 m .25843 .56775 L .24018 .53968 L p F P 0 g s 0 1 .695 r .50315 .10181 m .51797 .10171 L .51815 .11558 L p F P 0 g s 0 1 .695 r .51815 .11558 m .501 .1157 L .50315 .10181 L p F P 0 g s 0 1 .67 r .5353 .11574 m .51815 .11558 L .51797 .10171 L p F P 0 g s 0 1 .72 r .48843 .10215 m .50315 .10181 L .501 .1157 L p F P 0 g s 0 1 .72 r .501 .1157 m .48395 .1161 L .48843 .10215 L p F P 0 g s 0 1 .67 r .51797 .10171 m .53278 .10184 L .5353 .11574 L p F P 0 g s 0 1 .647 r .55232 .11619 m .5353 .11574 L .53278 .10184 L p F P 0 g s 1 .987 0 r .51685 .70724 m .48387 .70255 L .48469 .70129 L closepath p F P 0 g s 1 .882 0 r .51685 .70724 m .54916 .70148 L .54994 .70274 L closepath p F P 0 g s 0 1 .36 r .20778 .40524 m .21852 .39695 L .22318 .42931 L p F P 0 g s 0 1 .36 r .22318 .42931 m .21266 .43784 L .20778 .40524 L p F P 0 g s 0 1 .746 r .47389 .10272 m .48843 .10215 L .48395 .1161 L p F P 0 g s 0 1 .746 r .48395 .1161 m .4671 .11679 L .47389 .10272 L p F P 0 g s 0 1 .647 r .53278 .10184 m .54749 .10222 L .55232 .11619 L p F P 0 g s 0 1 .625 r .56912 .11692 m .55232 .11619 L .54749 .10222 L p F P 0 g s 1 .324 0 r .77606 .56906 m .77002 .5605 L .78774 .53217 L p F P 0 g s 1 .316 0 r .77002 .5605 m .77606 .56906 L .75491 .59511 L p F P 0 g s .735 1 0 r .38524 .67909 m .38678 .67416 L .41795 .68721 L p F P 0 g s .735 1 0 r .41795 .68721 m .41689 .69099 L .38524 .67909 L p F P 0 g s .823 1 0 r .41689 .69099 m .41795 .68721 L .45024 .69713 L p F P 0 g s .659 1 0 r .35498 .6641 m .35705 .65814 L .38678 .67416 L p F P 0 g s .659 1 0 r .38678 .67416 m .38524 .67909 L .35498 .6641 L p F P 0 g s .883 1 0 r .48387 .70255 m .48332 .70384 L .45024 .69713 L p F P 0 g s 1 .974 0 r .51685 .70724 m .48332 .70384 L .48387 .70255 L closepath p F P 0 g s 1 .867 0 r .51685 .70724 m .54994 .70274 L .55045 .70403 L closepath p F P 0 g s .823 1 0 r .45024 .69713 m .44961 .6997 L .41689 .69099 L p F P 0 g s .924 1 0 r .44961 .6997 m .45024 .69713 L .48332 .70384 L p F P 0 g s 1 .539 0 r .6169 .69157 m .61597 .68777 L .64723 .6749 L p F P 0 g s 1 .649 0 r .61597 .68777 m .6169 .69157 L .58414 .7001 L p F P 0 g s 1 .649 0 r .58414 .7001 m .5836 .69752 L .61597 .68777 L p F P 0 g s .063 1 0 r .22497 .50996 m .23249 .50096 L .24716 .53085 L p F P 0 g s .063 1 0 r .24716 .53085 m .24018 .53968 L .22497 .50996 L p F P 0 g s .591 1 0 r .32643 .64616 m .32906 .63929 L .35705 .65814 L p F P 0 g s .591 1 0 r .35705 .65814 m .35498 .6641 L .32643 .64616 L p F P 0 g s 0 1 .899 r .31402 .17335 m .32823 .16969 L .3138 .1936 L p F P 0 g s 0 1 .899 r .3138 .1936 m .29838 .19773 L .31402 .17335 L p F P 0 g s 1 .755 0 r .5836 .69752 m .58414 .7001 L .55068 .70534 L p F P 0 g s 1 .755 0 r .55068 .70534 m .55045 .70403 L .5836 .69752 L p F P 0 g s 1 .433 0 r .6486 .67984 m