(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 28683, 802] NotebookOptionsPosition[ 27871, 770] NotebookOutlinePosition[ 28230, 786] CellTagsIndexPosition[ 28187, 783] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[BoxData[ RowBox[{"(*", "\[IndentingNewLine]", StyleBox[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"#23", " ", StyleBox[ RowBox[{"r", "[", "t_", "]"}], FormatType->StandardForm]}], StyleBox[":=", FormatType->StandardForm], StyleBox[ RowBox[{"{", " ", RowBox[{ RowBox[{"3", " ", "+", " ", RowBox[{"Cos", "[", "t", "]"}]}], " ", ",", " ", RowBox[{"2", "+", " ", RowBox[{"Sin", "[", "t", "]"}]}], " ", ",", " ", "0"}], " ", "}"}], FormatType->StandardForm]}], StyleBox[" ", FormatType->StandardForm], StyleBox[";", FormatType->StandardForm], StyleBox[" ", FormatType->StandardForm], StyleBox[ RowBox[{"a", "=", " ", "0"}], FormatType->StandardForm, FontSize->14], StyleBox[" ", FormatType->StandardForm, FontSize->14], StyleBox[";", FormatType->StandardForm, FontSize->14], StyleBox[" ", FormatType->StandardForm, FontSize->14], StyleBox[ RowBox[{"b", "=", " ", RowBox[{"2", "Pi"}]}], FormatType->StandardForm, FontSize->14], StyleBox[" ", FormatType->StandardForm, FontSize->14], StyleBox[";", FormatType->StandardForm, FontSize->14]}], StyleBox["\[IndentingNewLine]", FormatType->StandardForm, FontSize->14], RowBox[{ RowBox[{ RowBox[{"#24", " ", StyleBox[ RowBox[{"r", "[", "t_", "]"}], FormatType->StandardForm]}], StyleBox[":=", FormatType->StandardForm], StyleBox[ RowBox[{"{", " ", RowBox[{ RowBox[{ RowBox[{"Sin", "[", "t", "]"}], "^", "3"}], " ", ",", " ", RowBox[{ RowBox[{"Cos", "[", "t", "]"}], "^", "3"}], " ", ",", " ", "0"}], " ", "}"}], FormatType->StandardForm]}], StyleBox[" ", FormatType->StandardForm], StyleBox[";", FormatType->StandardForm], StyleBox[" ", FormatType->StandardForm], StyleBox[ RowBox[{"a", "=", " ", "0"}], FormatType->StandardForm, FontSize->14], StyleBox[" ", FormatType->StandardForm, FontSize->14], StyleBox[";", FormatType->StandardForm, FontSize->14], StyleBox[" ", FormatType->StandardForm, FontSize->14], StyleBox[ RowBox[{"b", "=", " ", RowBox[{"Pi", "/", "2"}]}], FormatType->StandardForm, FontSize->14], StyleBox[";", FormatType->StandardForm, FontSize->14]}], StyleBox["\[IndentingNewLine]", FormatType->StandardForm, FontSize->14], RowBox[{ RowBox[{ RowBox[{"#27", " ", StyleBox[ RowBox[{"r", "[", "t_", "]"}], FormatType->StandardForm]}], StyleBox[":=", FormatType->StandardForm], StyleBox[ RowBox[{"{", " ", RowBox[{ RowBox[{ RowBox[{"Exp", "[", "t", "]"}], "*", RowBox[{"Cos", "[", "t", "]"}]}], " ", ",", " ", RowBox[{ RowBox[{"Exp", "[", "t", "]"}], "*", RowBox[{"Sin", "[", "t", "]"}]}], " ", ",", " ", "0"}], " ", "}"}], FormatType->StandardForm]}], StyleBox[" ", FormatType->StandardForm], StyleBox[";", FormatType->StandardForm], StyleBox[" ", FormatType->StandardForm], StyleBox[ RowBox[{"a", "=", " ", "0"}], FormatType->StandardForm, FontSize->14], StyleBox[" ", FormatType->StandardForm, FontSize->14], StyleBox[";", FormatType->StandardForm, FontSize->14], StyleBox[" ", FormatType->StandardForm, FontSize->14], StyleBox[ RowBox[{"b", "=", " ", RowBox[{"Pi", "/", "2"}]}], FormatType->StandardForm, FontSize->14], StyleBox[";", FormatType->StandardForm, FontSize->14]}], StyleBox["\[IndentingNewLine]", FormatType->StandardForm, FontSize->14], RowBox[{ RowBox[{ RowBox[{"#28", " ", StyleBox[ RowBox[{"r", "[", "t_", "]"}], FormatType->StandardForm]}], StyleBox[":=", FormatType->StandardForm], StyleBox[ RowBox[{"{", " ", RowBox[{ RowBox[{"Sin", "[", " ", RowBox[{"Exp", "[", "t", "]"}], "]"}], " ", ",", " ", RowBox[{"Cos", "[", " ", RowBox[{"Exp", "[", "t", "]"}], "]"}], ",", " ", RowBox[{ RowBox[{"Sqrt", "[", "3", "]"}], RowBox[{"Exp", "[", "t", "]"}]}]}], " ", "}"}], FormatType->StandardForm]}], StyleBox[" ", FormatType->StandardForm], StyleBox[";", FormatType->StandardForm], StyleBox[" ", FormatType->StandardForm], StyleBox[ RowBox[{"a", "=", " ", "0"}], FormatType->StandardForm, FontSize->14], StyleBox[" ", FormatType->StandardForm, FontSize->14], StyleBox[";", FormatType->StandardForm, FontSize->14], StyleBox[" ", FormatType->StandardForm, FontSize->14], StyleBox[ RowBox[{"b", "=", " ", RowBox[{"Pi", "/", "2"}]}], FormatType->StandardForm, FontSize->14], StyleBox[" ", FormatType->StandardForm, FontSize->14], StyleBox[";", FormatType->StandardForm, FontSize->14]}]}], FontColor->RGBColor[0, 1, 0]], StyleBox["\[IndentingNewLine]", FormatType->StandardForm, FontSize->14, FontColor->RGBColor[0, 1, 0]], "*)"}]], "Input", TextAlignment->Left], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"\"\< 13.3 Arc Length Parametrization \>\ \"", ";"}], "\n", RowBox[{ RowBox[{ RowBox[{"r", "[", "t_", "]"}], ":=", RowBox[{"{", RowBox[{ RowBox[{ SuperscriptBox["\[ExponentialE]", "t"], " ", RowBox[{"Cos", "[", "t", "]"}]}], ",", RowBox[{ SuperscriptBox["\[ExponentialE]", "t"], " ", RowBox[{"Sin", "[", "t", "]"}]}], ",", "0"}], "}"}]}], ";"}], "\n", RowBox[{ RowBox[{"a", "=", "0"}], ";", RowBox[{"b", "=", FractionBox["\[Pi]", "2"]}], ";"}], "\n", RowBox[{ RowBox[{"Tref", "=", "0"}], ";"}], "\n", RowBox[{ RowBox[{"Print", "[", RowBox[{ RowBox[{"Style", "[", RowBox[{"\"\\"", ",", RowBox[{"FontWeight", "\[Rule]", "\"\\""}], ",", RowBox[{"FontColor", "\[Rule]", RowBox[{"RGBColor", "[", RowBox[{"1", ",", "0", ",", "0"}], "]"}]}]}], "]"}], ",", RowBox[{"S", "[", "b", "]"}]}], "]"}], ";"}], "\n", RowBox[{ RowBox[{"Print", "[", RowBox[{ RowBox[{"Style", "[", RowBox[{"\"\\"", ",", RowBox[{"FontWeight", "\[Rule]", "\"\\""}], ",", RowBox[{"FontColor", "\[Rule]", RowBox[{"RGBColor", "[", RowBox[{"1", ",", "0", ",", "0"}], "]"}]}]}], "]"}], ",", RowBox[{"S", "[", "t", "]"}]}], "]"}], ";"}], "\n", RowBox[{ RowBox[{"Print", "[", RowBox[{ RowBox[{"Style", "[", RowBox[{"\"\