(* 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[ 41374, 1333] NotebookOptionsPosition[ 37429, 1200] NotebookOutlinePosition[ 37816, 1217] CellTagsIndexPosition[ 37773, 1214] WindowFrame->Normal ContainsDynamic->True *) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"2", "-", "2"}]], "Input", CellChangeTimes->{{3.429288127222005*^9, 3.429288127459132*^9}}], Cell[BoxData["0"], "Output", CellChangeTimes->{3.429288129030663*^9}] }, Open ]], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.4292881329733763`*^9}], Cell[CellGroupData[{ Cell["Illustrative application of the Frenet-Serret formulae", "Subsection", CellChangeTimes->{{3.429288151563652*^9, 3.4292881690269814`*^9}}], Cell[BoxData[ RowBox[{"Needs", "[", "\"\\"", "]"}]], "Input"], Cell[BoxData[ RowBox[{"Clear", "[", RowBox[{"x", ",", "v", ",", "a"}], "]"}]], "Input", CellChangeTimes->{{3.4292858809315147`*^9, 3.429285892869905*^9}, { 3.429285940270233*^9, 3.429285957086857*^9}}], Cell["The equations ", "Text", CellChangeTimes->{{3.42928859124419*^9, 3.429288599553906*^9}}], Cell[BoxData[{ RowBox[{"x", "=", RowBox[{"r", " ", RowBox[{"Cos", "[", "t", "]"}]}]}], "\[IndentingNewLine]", RowBox[{"y", "=", RowBox[{"r", " ", RowBox[{"Sin", "[", "t", "]"}]}]}], "\[IndentingNewLine]", RowBox[{"z", "=", RowBox[{"h", " ", "t"}]}]}], "DisplayFormula", CellChangeTimes->{{3.4292886132198143`*^9, 3.429288644122818*^9}}], Cell[TextData[{ "provide parametric description (relative to a right-handed Cartesian frame) \ of a ", StyleBox["right handed helix", FontWeight->"Bold"], ", that spirals about the z-axis." }], "Text", CellChangeTimes->{{3.429288657336053*^9, 3.429288738276713*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"x", "=", RowBox[{"{", RowBox[{ RowBox[{"r", " ", RowBox[{"Cos", "[", "t", "]"}]}], ",", RowBox[{"r", " ", RowBox[{"Sin", "[", "t", "]"}]}], ",", RowBox[{"h", " ", "t"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"v", "=", RowBox[{ SubscriptBox["\[PartialD]", "t"], " ", "x"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"a", "=", RowBox[{ SubscriptBox["\[PartialD]", RowBox[{"t", ",", "t"}]], "x"}]}], ";"}]}], "Input", CellChangeTimes->{{3.4292859621763*^9, 3.4292860037957973`*^9}, { 3.429286076189753*^9, 3.429286076789706*^9}}], Cell[CellGroupData[{ Cell[BoxData[{"x", "\[IndentingNewLine]", "v", "\[IndentingNewLine]", "a"}], \ "Input", CellChangeTimes->{{3.429286016113864*^9, 3.4292860906245613`*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"r", " ", RowBox[{"Cos", "[", "t", "]"}]}], ",", RowBox[{"r", " ", RowBox[{"Sin", "[", "t", "]"}]}], ",", RowBox[{"h", " ", "t"}]}], "}"}]], "Output", CellChangeTimes->{3.429286092310295*^9, 3.4293687200660963`*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"-", "r"}], " ", RowBox[{"Sin", "[", "t", "]"}]}], ",", RowBox[{"r", " ", RowBox[{"Cos", "[", "t", "]"}]}], ",", "h"}], "}"}]], "Output", CellChangeTimes->{3.429286092310295*^9, 3.429368720073435*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"-", "r"}], " ", RowBox[{"Cos", "[", "t", "]"}]}], ",", RowBox[{ RowBox[{"-", "r"}], " ", RowBox[{"Sin", "[", "t", "]"}]}], ",", "0"}], "}"}]], "Output", CellChangeTimes->{3.429286092310295*^9, 3.429368720079547*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Manipulate", "[", RowBox[{ RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"{", " ", RowBox[{ RowBox[{"Cos", "[", "t", "]"}], ",", " ", RowBox[{"Sin", "[", "t", "]"}], ",", RowBox[{"h", " ", "t"}]}], "}"}], ",", " ", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"10", "\[Pi]"}]}], "}"}], ",", " ", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "10"}], "}"}]}], "}"}]}], ",", " ", RowBox[{"Ticks", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"5", ",", "10"}], "}"}]}], "}"}]}]}], "]"}], ",", RowBox[{"{", RowBox[{"h", ",", "0", ",", "0.3"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.429288244487733*^9, 3.429288329017247*^9}, { 3.429288386168602*^9, 3.429288438044736*^9}, {3.429288493170596*^9, 3.429288530616143*^9}}], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`h$$ = 0.2325, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{ Hold[$CellContext`h$$], 0, 0.3}}, Typeset`size$$ = {183., {266., 274.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`h$7515$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`h$$ = 0}, "ControllerVariables" :> { Hold[$CellContext`h$$, $CellContext`h$7515$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> ParametricPlot3D[{ Cos[$CellContext`t], Sin[$CellContext`t], $CellContext`h$$ $CellContext`t}, \ {$CellContext`t, 0, 10 Pi}, PlotRange -> {{-1, 1}, {-1, 1}, {0, 10}}, Ticks -> {{-1, 1}, {-1, 1}, {5, 10}}], "Specifications" :> {{$CellContext`h$$, 0, 0.3}}, "Options" :> {}, "DefaultOptions" :> {}], ImageSizeCache->{306.5, {315.5, 322.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellChangeTimes->{ 3.429288335432065*^9, {3.4292884239541063`*^9, 3.4292884427920027`*^9}, 3.429288533106257*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"T", "=", RowBox[{ FractionBox["v", SqrtBox[ RowBox[{"DotProduct", "[", RowBox[{"v", ",", "v"}], "]"}]]], "//", "Simplify"}]}]], "Input", CellChangeTimes->{{3.429286107880726*^9, 3.429286148085093*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"-", FractionBox[ RowBox[{"r", " ", RowBox[{"Sin", "[", "t", "]"}]}], SqrtBox[ RowBox[{ SuperscriptBox["h", "2"], "+", SuperscriptBox["r", "2"]}]]]}], ",", FractionBox[ RowBox[{"r", " ", RowBox[{"Cos", "[", "t", "]"}]}], SqrtBox[ RowBox[{ SuperscriptBox["h", "2"], "+", SuperscriptBox["r", "2"]}]]], ",", FractionBox["h", SqrtBox[ RowBox[{ SuperscriptBox["h", "2"], "+", SuperscriptBox["r", "2"]}]]]}], "}"}]], "Output", CellChangeTimes->{{3.429286138620625*^9, 3.4292861495284967`*^9}, 3.429368748359021*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"DotProduct", "[", RowBox[{"T", ",", "T"}], "]"}], "//", "Simplify"}]], "Input", CellChangeTimes->{{3.4292861640113487`*^9, 3.429286185372662*^9}}], Cell[BoxData["1"], "Output", CellChangeTimes->{{3.429286178819468*^9, 3.429286186531797*^9}, 3.429368755037292*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ FractionBox[ RowBox[{"CrossProduct", "[", RowBox[{"v", ",", RowBox[{"CrossProduct", "[", RowBox[{"a", ",", "v"}], "]"}]}], "]"}], SqrtBox[ RowBox[{ RowBox[{"DotProduct", "[", RowBox[{"v", ",", "v"}], "]"}], RowBox[{"DotProduct", "[", RowBox[{ RowBox[{"CrossProduct", "[", RowBox[{"a", ",", "v"}], "]"}], ",", RowBox[{"CrossProduct", "[", RowBox[{"a", ",", "v"}], "]"}]}], "]"}]}]]], "//", "Simplify"}]], "Input", CellChangeTimes->{{3.429286196568755*^9, 3.429286399239201*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"-", FractionBox[ RowBox[{"r", " ", RowBox[{"(", RowBox[{ SuperscriptBox["h", "2"], "+", SuperscriptBox["r", "2"]}], ")"}], " ", RowBox[{"Cos", "[", "t", "]"}]}], SqrtBox[ RowBox[{ SuperscriptBox["r", "2"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ SuperscriptBox["h", "2"], "+", SuperscriptBox["r", "2"]}], ")"}], "2"]}]]]}], ",", RowBox[{"-", FractionBox[ RowBox[{"r", " ", RowBox[{"(", RowBox[{ SuperscriptBox["h", "2"], "+", SuperscriptBox["r", "2"]}], ")"}], " ", RowBox[{"Sin", "[", "t", "]"}]}], SqrtBox[ RowBox[{ SuperscriptBox["r", "2"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ SuperscriptBox["h", "2"], "+", SuperscriptBox["r", "2"]}], ")"}], "2"]}]]]}], ",", "0"}], "}"}]], "Output", CellChangeTimes->{3.429286371405546*^9, 3.4293687599750977`*^9}] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"U", "=", RowBox[{"{", RowBox[{ RowBox[{"-", " ", RowBox[{"Cos", "[", "t", "]"}]}], ",", RowBox[{"-", " ", RowBox[{"Sin", "[", "t", "]"}]}], ",", "0"}], "}"}]}], ";"}]], "Input", CellChangeTimes->{{3.429286405762727*^9, 3.429286414280835*^9}, { 3.429286449993086*^9, 3.4292864706355143`*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"DotProduct", "[", RowBox[{"T", ",", "U"}], "]"}], "//", "Simplify"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"DotProduct", "[", RowBox[{"U", ",", "U"}], "]"}], "//", "Simplify"}]}], "Input", CellChangeTimes->{{3.429286489341373*^9, 3.429286500746674*^9}}], Cell[BoxData["0"], "Output", CellChangeTimes->{3.4292865026114597`*^9, 3.429368772987898*^9}], Cell[BoxData["1"], "Output", CellChangeTimes->{3.4292865026114597`*^9, 3.42936877304105*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"CrossProduct", "[", RowBox[{"T", ",", " ", "U"}], "]"}], "//", "Simplify"}]], "Input", CellChangeTimes->{{3.429286774442807*^9, 3.429286796718355*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ FractionBox[ RowBox[{"h", " ", RowBox[{"Sin", "[", "t", "]"}]}], SqrtBox[ RowBox[{ SuperscriptBox["h", "2"], "+", SuperscriptBox["r", "2"]}]]], ",", RowBox[{"-", FractionBox[ RowBox[{"h", " ", RowBox[{"Cos", "[", "t", "]"}]}], SqrtBox[ RowBox[{ SuperscriptBox["h", "2"], "+", SuperscriptBox["r", "2"]}]]]}], ",", FractionBox["r", SqrtBox[ RowBox[{ SuperscriptBox["h", "2"], "+", SuperscriptBox["r", "2"]}]]]}], "}"}]], "Output", CellChangeTimes->{{3.429286790017188*^9, 3.4292867979877167`*^9}, 3.4293687770270576`*^9}] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"V", "=", RowBox[{"{", RowBox[{ FractionBox[ RowBox[{"h", " ", RowBox[{"Sin", "[", "t", "]"}]}], SqrtBox[ RowBox[{" ", RowBox[{"(", RowBox[{ SuperscriptBox["h", "2"], "+", SuperscriptBox["r", "2"]}], ")"}]}]]], ",", RowBox[{"-", FractionBox[ RowBox[{"h", " ", RowBox[{"Cos", "[", "t", "]"}]}], SqrtBox[ RowBox[{" ", RowBox[{"(", RowBox[{ SuperscriptBox["h", "2"], "+", SuperscriptBox["r", "2"]}], ")"}]}]]]}], ",", FractionBox["r", SqrtBox[ RowBox[{" ", RowBox[{"(", RowBox[{ SuperscriptBox["h", "2"], "+", SuperscriptBox["r", "2"]}], ")"}]}]]]}], "}"}]}], ";"}]], "Input", CellChangeTimes->{{3.429286620831539*^9, 3.429286664306408*^9}, 3.4292868142237*^9}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"DotProduct", "[", RowBox[{"T", ",", "V"}], "]"}], "//", "Simplify"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"DotProduct", "[", RowBox[{"U", ",", "V"}], "]"}], "//", "Simplify"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"DotProduct", "[", RowBox[{"V", ",", "V"}], "]"}], "//", "Simplify"}]}], "Input", CellChangeTimes->{{3.429286696825004*^9, 3.4292867115826197`*^9}}], Cell[BoxData["0"], "Output", CellChangeTimes->{3.42928671283978*^9, 3.429286844842534*^9, 3.4293687876220713`*^9}], Cell[BoxData["0"], "Output", CellChangeTimes->{3.42928671283978*^9, 3.429286844842534*^9, 3.429368787630105*^9}], Cell[BoxData["1"], "Output", CellChangeTimes->{3.42928671283978*^9, 3.429286844842534*^9, 3.429368787635173*^9}] }, Open ]], Cell["\<\ Working from the Frenet-Serret formula\ \>", "Text", CellChangeTimes->{{3.429369032730166*^9, 3.429369062885006*^9}, 3.429450370034169*^9}], Cell[BoxData[ RowBox[{ RowBox[{ SubscriptBox["\[PartialD]", "t"], " ", "U"}], "=", RowBox[{ SqrtBox[ RowBox[{"DotProduct", "[", RowBox[{ RowBox[{ SubscriptBox["\[PartialD]", "t"], " ", "x"}], ",", RowBox[{ SubscriptBox["\[PartialD]", "t"], " ", "x"}]}], "]"}]], RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", "\[Kappa]"}], " ", "T"}], "+", RowBox[{"\[Tau]", " ", "V"}]}], ")"}]}]}]], "Input", CellChangeTimes->{{3.4293691099956293`*^9, 3.429369203568008*^9}}], Cell[TextData[{ "\[LongDash]we have (by the orthonormality of ", StyleBox["T", "Input"], " and ", StyleBox["V", "Input"], ")" }], "Text", CellChangeTimes->{{3.429369224119564*^9, 3.4293692282188168`*^9}, { 3.429369418026162*^9, 3.429369428033033*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"\[Kappa]", " ", "=", RowBox[{ RowBox[{"-", " ", RowBox[{"DotProduct", "[", RowBox[{ RowBox[{ SubscriptBox["\[PartialD]", "t"], " ", "U"}], ",", "T"}], "]"}]}], "//", "Simplify"}]}], "\[IndentingNewLine]", RowBox[{"\[Tau]", " ", "=", RowBox[{ RowBox[{"+", " ", RowBox[{"DotProduct", "[", RowBox[{ RowBox[{ SubscriptBox["\[PartialD]", "t"], " ", "U"}], ",", "V"}], "]"}]}], "//", "Simplify"}]}]}], "Input", CellChangeTimes->{{3.429369388207595*^9, 3.4293693967307777`*^9}}], Cell[BoxData[ FractionBox["r", SqrtBox[ RowBox[{ SuperscriptBox["h", "2"], "+", SuperscriptBox["r", "2"]}]]]], "Output", CellChangeTimes->{{3.429369381571607*^9, 3.429369398526658*^9}}], Cell[BoxData[ FractionBox["h", SqrtBox[ RowBox[{ SuperscriptBox["h", "2"], "+", SuperscriptBox["r", "2"]}]]]], "Output", CellChangeTimes->{{3.429369381571607*^9, 3.429369398547474*^9}}] }, Open ]], Cell["\<\ \[LongDash]both fo which turn out, in this instance, to be constant. \ \>", "Text", CellChangeTimes->{{3.429288848692474*^9, 3.42928888361404*^9}, { 3.429369466837785*^9, 3.429369498986883*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.429289149350322*^9}], Cell[TextData[{ "Look now to the case of a ", StyleBox["left-handed helix", FontWeight->"Bold"], ":" }], "Text", CellChangeTimes->{{3.429289158447528*^9, 3.42928922871653*^9}}], Cell[BoxData[ RowBox[{"Clear", "[", RowBox[{ "x", ",", "v", ",", "a", ",", "T", ",", "U", ",", "V", ",", " ", "\[Kappa]", ",", " ", "\[Tau]"}], "]"}]], "Input", CellChangeTimes->{{3.4292892367261543`*^9, 3.42928924743069*^9}, { 3.429369531251359*^9, 3.4293695353352213`*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"x", "=", RowBox[{"{", RowBox[{ RowBox[{"r", " ", RowBox[{"Cos", "[", "t", "]"}]}], ",", RowBox[{ RowBox[{"-", "r"}], " ", RowBox[{"Sin", "[", "t", "]"}]}], ",", RowBox[{"h", " ", "t"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"v", "=", RowBox[{ SubscriptBox["\[PartialD]", "t"], " ", "x"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"a", "=", RowBox[{ SubscriptBox["\[PartialD]", RowBox[{"t", ",", "t"}]], "x"}]}], ";"}]}], "Input", CellChangeTimes->{{3.4292859621763*^9, 3.4292860037957973`*^9}, { 3.429286076189753*^9, 3.429286076789706*^9}, 3.429289276403315*^9}], Cell[CellGroupData[{ Cell[BoxData[{"x", "\[IndentingNewLine]", "v", "\[IndentingNewLine]", "a"}], \ "Input", CellChangeTimes->{{3.429286016113864*^9, 3.4292860906245613`*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"r", " ", RowBox[{"Cos", "[", "t", "]"}]}], ",", RowBox[{ RowBox[{"-", "r"}], " ", RowBox[{"Sin", "[", "t", "]"}]}], ",", RowBox[{"h", " ", "t"}]}], "}"}]], "Output", CellChangeTimes->{3.429289299438941*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"-", "r"}], " ", RowBox[{"Sin", "[", "t", "]"}]}], ",", RowBox[{ RowBox[{"-", "r"}], " ", RowBox[{"Cos", "[", "t", "]"}]}], ",", "h"}], "}"}]], "Output", CellChangeTimes->{3.429289299505999*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"-", "r"}], " ", RowBox[{"Cos", "[", "t", "]"}]}], ",", RowBox[{"r", " ", RowBox[{"Sin", "[", "t", "]"}]}], ",", "0"}], "}"}]], "Output", CellChangeTimes->{3.42928929955446*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Manipulate", "[", RowBox[{ RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"{", " ", RowBox[{ RowBox[{"Cos", "[", "t", "]"}], ",", " ", RowBox[{"-", RowBox[{"Sin", "[", "t", "]"}]}], ",", RowBox[{"h", " ", "t"}]}], "}"}], ",", " ", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"10", "\[Pi]"}]}], "}"}], ",", " ", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "10"}], "}"}]}], "}"}]}], ",", " ", RowBox[{"Ticks", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"5", ",", "10"}], "}"}]}], "}"}]}]}], "]"}], ",", RowBox[{"{", RowBox[{"h", ",", "0", ",", "0.3"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.429288244487733*^9, 3.429288329017247*^9}, { 3.429288386168602*^9, 3.429288438044736*^9}, {3.429288493170596*^9, 3.429288530616143*^9}, 3.429289340089995*^9}], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`h$$ = 0.1575, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{ Hold[$CellContext`h$$], 0, 0.3}}, Typeset`size$$ = {183., {266., 274.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`h$7792$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`h$$ = 0}, "ControllerVariables" :> { Hold[$CellContext`h$$, $CellContext`h$7792$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> ParametricPlot3D[{ Cos[$CellContext`t], - Sin[$CellContext`t], $CellContext`h$$ $CellContext`t}, \ {$CellContext`t, 0, 10 Pi}, PlotRange -> {{-1, 1}, {-1, 1}, {0, 10}}, Ticks -> {{-1, 1}, {-1, 1}, {5, 10}}], "Specifications" :> {{$CellContext`h$$, 0, 0.3}}, "Options" :> {}, "DefaultOptions" :> {}], ImageSizeCache->{306.5, {315.5, 322.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellChangeTimes->{3.429289344600634*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"T", "=", RowBox[{ FractionBox["v", SqrtBox[ RowBox[{"DotProduct", "[", RowBox[{"v", ",", "v"}], "]"}]]], "//", "Simplify"}]}]], "Input", CellChangeTimes->{{3.429286107880726*^9, 3.429286148085093*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"-", FractionBox[ RowBox[{"r", " ", RowBox[{"Sin", "[", "t", "]"}]}], SqrtBox[ RowBox[{ SuperscriptBox["h", "2"], "+", SuperscriptBox["r", "2"]}]]]}], ",", RowBox[{"-", FractionBox[ RowBox[{"r", " ", RowBox[{"Cos", "[", "t", "]"}]}], SqrtBox[ RowBox[{ SuperscriptBox["h", "2"], "+", SuperscriptBox["r", "2"]}]]]}], ",", FractionBox["h", SqrtBox[ RowBox[{ SuperscriptBox["h", "2"], "+", SuperscriptBox["r", "2"]}]]]}], "}"}]], "Output", CellChangeTimes->{3.429289381790784*^9, 3.42936955829277*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"DotProduct", "[", RowBox[{"T", ",", "T"}], "]"}], "//", "Simplify"}]], "Input", CellChangeTimes->{{3.4292861640113487`*^9, 3.429286185372662*^9}}], Cell[BoxData["1"], "Output", CellChangeTimes->{3.429289397267787*^9, 3.429369561494233*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ FractionBox[ RowBox[{"CrossProduct", "[", RowBox[{"v", ",", RowBox[{"CrossProduct", "[", RowBox[{"a", ",", "v"}], "]"}]}], "]"}], SqrtBox[ RowBox[{ RowBox[{"DotProduct", "[", RowBox[{"v", ",", "v"}], "]"}], RowBox[{"DotProduct", "[", RowBox[{ RowBox[{"CrossProduct", "[", RowBox[{"a", ",", "v"}], "]"}], ",", RowBox[{"CrossProduct", "[", RowBox[{"a", ",", "v"}], "]"}]}], "]"}]}]]], "//", "Simplify"}]], "Input", CellChangeTimes->{{3.429286196568755*^9, 3.429286399239201*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"-", FractionBox[ RowBox[{"r", " ", RowBox[{"(", RowBox[{ SuperscriptBox["h", "2"], "+", SuperscriptBox["r", "2"]}], ")"}], " ", RowBox[{"Cos", "[", "t", "]"}]}], SqrtBox[ RowBox[{ SuperscriptBox["r", "2"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ SuperscriptBox["h", "2"], "+", SuperscriptBox["r", "2"]}], ")"}], "2"]}]]]}], ",", FractionBox[ RowBox[{"r", " ", RowBox[{"(", RowBox[{ SuperscriptBox["h", "2"], "+", SuperscriptBox["r", "2"]}], ")"}], " ", RowBox[{"Sin", "[", "t", "]"}]}], SqrtBox[ RowBox[{ SuperscriptBox["r", "2"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ SuperscriptBox["h", "2"], "+", SuperscriptBox["r", "2"]}], ")"}], "2"]}]]], ",", "0"}], "}"}]], "Output", CellChangeTimes->{3.429289410429083*^9, 3.429369564290433*^9}] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"U", "=", RowBox[{"{", RowBox[{ RowBox[{"-", " ", RowBox[{"Cos", "[", "t", "]"}]}], ",", RowBox[{"Sin", "[", "t", "]"}], ",", "0"}], "}"}]}], ";"}]], "Input", CellChangeTimes->{{3.429286405762727*^9, 3.429286414280835*^9}, { 3.429286449993086*^9, 3.4292864706355143`*^9}, {3.429289432201532*^9, 3.429289460414864*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"DotProduct", "[", RowBox[{"T", ",", "U"}], "]"}], "//", "Simplify"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"DotProduct", "[", RowBox[{"U", ",", "U"}], "]"}], "//", "Simplify"}]}], "Input", CellChangeTimes->{{3.429286489341373*^9, 3.429286500746674*^9}}], Cell[BoxData["0"], "Output", CellChangeTimes->{3.429289479314177*^9, 3.429369575008107*^9}], Cell[BoxData["1"], "Output", CellChangeTimes->{3.429289479314177*^9, 3.429369575054982*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"CrossProduct", "[", RowBox[{"T", ",", " ", "U"}], "]"}], "//", "Simplify"}]], "Input", CellChangeTimes->{{3.429286774442807*^9, 3.429286796718355*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"-", FractionBox[ RowBox[{"h", " ", RowBox[{"Sin", "[", "t", "]"}]}], SqrtBox[ RowBox[{ SuperscriptBox["h", "2"], "+", SuperscriptBox["r", "2"]}]]]}], ",", RowBox[{"-", FractionBox[ RowBox[{"h", " ", RowBox[{"Cos", "[", "t", "]"}]}], SqrtBox[ RowBox[{ SuperscriptBox["h", "2"], "+", SuperscriptBox["r", "2"]}]]]}], ",", RowBox[{"-", FractionBox["r", SqrtBox[ RowBox[{ SuperscriptBox["h", "2"], "+", SuperscriptBox["r", "2"]}]]]}]}], "}"}]], "Output", CellChangeTimes->{3.429289491373362*^9, 3.429369582998537*^9}] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"V", "=", RowBox[{"{", RowBox[{ RowBox[{"-", FractionBox[ RowBox[{"h", " ", RowBox[{"Sin", "[", "t", "]"}]}], SqrtBox[ RowBox[{" ", RowBox[{"(", RowBox[{ SuperscriptBox["h", "2"], "+", SuperscriptBox["r", "2"]}], ")"}]}]]]}], ",", RowBox[{"-", FractionBox[ RowBox[{"h", " ", RowBox[{"Cos", "[", "t", "]"}]}], SqrtBox[ RowBox[{" ", RowBox[{"(", RowBox[{ SuperscriptBox["h", "2"], "+", SuperscriptBox["r", "2"]}], ")"}]}]]]}], ",", RowBox[{"-", FractionBox["r", SqrtBox[ RowBox[{" ", RowBox[{"(", RowBox[{ SuperscriptBox["h", "2"], "+", SuperscriptBox["r", "2"]}], ")"}]}]]]}]}], "}"}]}], ";"}]], "Input",\ CellChangeTimes->{{3.429286620831539*^9, 3.429286664306408*^9}, 3.4292868142237*^9, {3.429289660798767*^9, 3.429289664258552*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"DotProduct", "[", RowBox[{"T", ",", "V"}], "]"}], "//", "Simplify"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"DotProduct", "[", RowBox[{"U", ",", "V"}], "]"}], "//", "Simplify"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"DotProduct", "[", RowBox[{"V", ",", "V"}], "]"}], "//", "Simplify"}]}], "Input", CellChangeTimes->{{3.429286696825004*^9, 3.4292867115826197`*^9}}], Cell[BoxData["0"], "Output", CellChangeTimes->{3.4292895173391314`*^9, 3.429289670467634*^9, 3.429369591120356*^9}], Cell[BoxData["0"], "Output", CellChangeTimes->{3.4292895173391314`*^9, 3.429289670467634*^9, 3.4293695911566353`*^9}], Cell[BoxData["1"], "Output", CellChangeTimes->{3.4292895173391314`*^9, 3.429289670467634*^9, 3.429369591227016*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"\[Kappa]", " ", "=", RowBox[{ RowBox[{"-", " ", RowBox[{"DotProduct", "[", RowBox[{ RowBox[{ SubscriptBox["\[PartialD]", "t"], " ", "U"}], ",", "T"}], "]"}]}], "//", "Simplify"}]}], "\[IndentingNewLine]", RowBox[{"\[Tau]", " ", "=", RowBox[{ RowBox[{"+", " ", RowBox[{"DotProduct", "[", RowBox[{ RowBox[{ SubscriptBox["\[PartialD]", "t"], " ", "U"}], ",", "V"}], "]"}]}], "//", "Simplify"}]}]}], "Input", CellChangeTimes->{{3.429369388207595*^9, 3.4293693967307777`*^9}}], Cell[BoxData[ FractionBox["r", SqrtBox[ RowBox[{ SuperscriptBox["h", "2"], "+", SuperscriptBox["r", "2"]}]]]], "Output", CellChangeTimes->{3.429369673358137*^9}], Cell[BoxData[ RowBox[{"-", FractionBox["h", SqrtBox[ RowBox[{ SuperscriptBox["h", "2"], "+", SuperscriptBox["r", "2"]}]]]}]], "Output", CellChangeTimes->{3.4293696733848543`*^9}] }, Open ]], Cell["Reversing the helicity has reversed the sign of the torsion.", "Text", CellChangeTimes->{{3.429369691273058*^9, 3.429369706861412*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Manipulate", "[", RowBox[{ RowBox[{"Graphics3D", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"{", RowBox[{"Thick", ",", RowBox[{"{", RowBox[{"Red", ",", RowBox[{"Line", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", FractionBox[ RowBox[{" ", RowBox[{"Sin", "[", "t", "]"}]}], SqrtBox[ RowBox[{ SuperscriptBox["h", "2"], "+", "1"}]]]}], ",", FractionBox[ RowBox[{" ", RowBox[{"Cos", "[", "t", "]"}]}], SqrtBox[ RowBox[{ SuperscriptBox["h", "2"], "+", "1"}]]], ",", FractionBox["h", SqrtBox[ RowBox[{ SuperscriptBox["h", "2"], "+", "1"}]]]}], "}"}]}], "}"}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"RGBColor", "[", RowBox[{"0", ",", ".5", ",", ".5"}], "]"}], ",", RowBox[{"Line", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", " ", RowBox[{"Cos", "[", "t", "]"}]}], ",", RowBox[{"-", " ", RowBox[{"Sin", "[", "t", "]"}]}], ",", "0"}], "}"}]}], "}"}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{"Blue", ",", RowBox[{"Line", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{ FractionBox[ RowBox[{"h", " ", RowBox[{"Sin", "[", "t", "]"}]}], SqrtBox[ RowBox[{" ", RowBox[{"(", RowBox[{ SuperscriptBox["h", "2"], "+", "1"}], ")"}]}]]], ",", RowBox[{"-", FractionBox[ RowBox[{"h", " ", RowBox[{"Cos", "[", "t", "]"}]}], SqrtBox[ RowBox[{" ", RowBox[{"(", RowBox[{ SuperscriptBox["h", "2"], "+", "1"}], ")"}]}]]]}], ",", FractionBox["1", SqrtBox[ RowBox[{" ", RowBox[{"(", RowBox[{ SuperscriptBox["h", "2"], "+", "1"}], ")"}]}]]]}], "}"}]}], "}"}], "]"}]}], "}"}]}], "}"}], ",", " ", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "1.5"}], ",", "1.5"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "1.5"}], ",", "1.5"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "1.5"}], ",", "1.5"}], "}"}]}], "}"}]}], ",", " ", RowBox[{"PlotLabel", "\[Rule]", "\"\<\!\(\* StyleBox[\"T\", FontColor->RGBColor[1, 0, 0]]\), \!\(\* StyleBox[\"U\", FontColor->RGBColor[0, 1, 0]]\), \!\(\* StyleBox[\"V\", FontColor->RGBColor[0, 0, 1]]\)\>\""}], ",", RowBox[{"LabelStyle", "\[Rule]", RowBox[{"Directive", "[", "Bold", "]"}]}]}], "]"}], ",", RowBox[{"{", RowBox[{"h", ",", "0", ",", "0.3"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"2", "\[Pi]"}]}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.429289863841416*^9, 3.429289896896927*^9}, { 3.429289944389841*^9, 3.429289965969309*^9}, {3.429290009388576*^9, 3.4292900225373096`*^9}, {3.429290057229106*^9, 3.4292901650658607`*^9}, { 3.429290205760358*^9, 3.42929026362994*^9}, {3.429290329459352*^9, 3.429290356461672*^9}, {3.429290447562498*^9, 3.429290450041586*^9}, { 3.429290525349844*^9, 3.429290618351123*^9}, {3.429290693025201*^9, 3.4292907241846647`*^9}, {3.429290789878953*^9, 3.429290793561328*^9}, { 3.4292908680487223`*^9, 3.429290880941063*^9}, {3.429291031606594*^9, 3.4292910830498047`*^9}, {3.429291149958054*^9, 3.4292912086685257`*^9}}], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`h$$ = 0, $CellContext`t$$ = 0, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{ Hold[$CellContext`h$$], 0, 0.3}, { Hold[$CellContext`t$$], 0, 2 Pi}}, Typeset`size$$ = { 450., {248., 255.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`h$8509$$ = 0, $CellContext`t$8510$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`h$$ = 0, $CellContext`t$$ = 0}, "ControllerVariables" :> { Hold[$CellContext`h$$, $CellContext`h$8509$$, 0], Hold[$CellContext`t$$, $CellContext`t$8510$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> Graphics3D[{Thick, {Red, Line[{{0, 0, 0}, {-(Sin[$CellContext`t$$]/($CellContext`h$$^2 + 1)^ Rational[1, 2]), Cos[$CellContext`t$$]/($CellContext`h$$^2 + 1)^ Rational[1, 2], $CellContext`h$$/($CellContext`h$$^2 + 1)^ Rational[1, 2]}}]}, { RGBColor[0, 0.5, 0.5], Line[{{0, 0, 0}, {-Cos[$CellContext`t$$], -Sin[$CellContext`t$$], 0}}]}, { Blue, Line[{{0, 0, 0}, {$CellContext`h$$ ( Sin[$CellContext`t$$]/($CellContext`h$$^2 + 1)^ Rational[1, 2]), -($CellContext`h$$ ( Cos[$CellContext`t$$]/($CellContext`h$$^2 + 1)^ Rational[1, 2])), 1/($CellContext`h$$^2 + 1)^ Rational[1, 2]}}]}}, PlotRange -> {{-1.5, 1.5}, {-1.5, 1.5}, {-1.5, 1.5}}, PlotLabel -> "\!\(\*\nStyleBox[\"T\",\nFontColor->RGBColor[1, 0, 0]]\), \!\(\*\n\ StyleBox[\"U\",\nFontColor->RGBColor[0, 1, 0]]\), \!\(\*\nStyleBox[\"V\",\n\ FontColor->RGBColor[0, 0, 1]]\)", LabelStyle -> Directive[Bold]], "Specifications" :> {{$CellContext`h$$, 0, 0.3}, {$CellContext`t$$, 0, 2 Pi}}, "Options" :> {}, "DefaultOptions" :> {}], ImageSizeCache->{494.5, {314.5, 321.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellChangeTimes->{ 3.429290172331469*^9, 3.429290268947318*^9, {3.42929036202916*^9, 3.4292904089441957`*^9}, {3.4292904553746147`*^9, 3.4292904717941647`*^9}, 3.429290552934869*^9, {3.429290622949787*^9, 3.4292906560882263`*^9}, 3.4292907774174833`*^9, 3.4292908846160793`*^9, {3.4292910716588497`*^9, 3.429291084289733*^9}, {3.429291160067568*^9, 3.429291213174964*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Hyperlink", "[", RowBox[{ "\"\\"", ",", "\"\\""}], "]"}]], "Input", CellChangeTimes->{{3.429366320748993*^9, 3.429366394303315*^9}}], Cell[BoxData[ TagBox[ ButtonBox[ PaneSelectorBox[{False->"\<\"Curvature & Torsion\"\>", True-> StyleBox["\<\"Curvature & Torsion\"\>", "HyperlinkActive"]}, Dynamic[ CurrentValue["MouseOver"]], BaselinePosition->Baseline, FrameMargins->0, ImageSize->Automatic], BaseStyle->"Hyperlink", ButtonData->{ URL["http://demonstrations.wolfram.com/CurvatureAndTorsion/"], None}, ButtonNote->"http://demonstrations.wolfram.com/CurvatureAndTorsion/"], Annotation[#, "http://demonstrations.wolfram.com/CurvatureAndTorsion/", "Hyperlink"]& ]], "Output", CellChangeTimes->{3.4293686724603786`*^9}] }, Open ]] }, Open ]] }, WindowToolbars->"EditBar", WindowSize->{927, 599}, WindowMargins->{{71, Automatic}, {Automatic, 24}}, Magnification->1.25, FrontEndVersion->"6.0 for Mac OS X PowerPC (32-bit) (May 21, 2008)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[590, 23, 115, 2, 33, "Input"], Cell[708, 27, 70, 1, 33, "Output"] }, Open ]], Cell[793, 31, 89, 1, 53, "Input"], Cell[CellGroupData[{ Cell[907, 36, 144, 1, 42, "Subsection"], Cell[1054, 39, 80, 1, 33, "Input"], Cell[1137, 42, 208, 4, 33, "Input"], Cell[1348, 48, 95, 1, 32, "Text"], Cell[1446, 51, 359, 9, 64, "DisplayFormula"], Cell[1808, 62, 272, 7, 32, "Text"], Cell[2083, 71, 665, 21, 73, "Input"], Cell[CellGroupData[{ Cell[2773, 96, 155, 2, 72, "Input"], Cell[2931, 100, 285, 8, 33, "Output"], Cell[3219, 110, 279, 8, 33, "Output"], Cell[3501, 120, 299, 9, 33, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[3837, 134, 1343, 39, 72, "Input"], Cell[5183, 175, 1779, 37, 658, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[6999, 217, 249, 7, 59, "Input"], Cell[7251, 226, 673, 24, 68, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[7961, 255, 186, 4, 33, "Input"], Cell[8150, 261, 120, 2, 33, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[8307, 268, 594, 18, 62, "Input"], Cell[8904, 288, 1052, 36, 73, "Output"] }, Open ]], Cell[9971, 327, 363, 10, 33, "Input"], Cell[CellGroupData[{ Cell[10359, 341, 308, 8, 53, "Input"], Cell[10670, 351, 94, 1, 33, "Output"], Cell[10767, 354, 93, 1, 33, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[10897, 360, 191, 4, 33, "Input"], Cell[11091, 366, 675, 24, 68, "Output"] }, Open ]], Cell[11781, 393, 930, 32, 71, "Input"], Cell[CellGroupData[{ Cell[12736, 429, 432, 12, 72, "Input"], Cell[13171, 443, 118, 2, 33, "Output"], Cell[13292, 447, 116, 2, 33, "Output"], Cell[13411, 451, 116, 2, 33, "Output"] }, Open ]], Cell[13542, 456, 154, 4, 32, "Text"], Cell[13699, 462, 532, 17, 48, "Input"], Cell[14234, 481, 259, 8, 32, "Text"], Cell[CellGroupData[{ Cell[14518, 493, 567, 17, 53, "Input"], Cell[15088, 512, 202, 6, 64, "Output"], Cell[15293, 520, 202, 6, 67, "Output"] }, Open ]], Cell[15510, 529, 207, 4, 32, "Text"], Cell[15720, 535, 87, 1, 53, "Input"], Cell[15810, 538, 182, 6, 32, "Text"], Cell[15995, 546, 288, 6, 33, "Input"], Cell[16286, 554, 710, 22, 73, "Input"], Cell[CellGroupData[{ Cell[17021, 580, 155, 2, 72, "Input"], Cell[17179, 584, 281, 9, 33, "Output"], Cell[17463, 595, 277, 9, 33, "Output"], Cell[17743, 606, 256, 8, 33, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[18036, 619, 1391, 40, 72, "Input"], Cell[19430, 661, 1701, 35, 658, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[21168, 701, 249, 7, 59, "Input"], Cell[21420, 710, 668, 24, 68, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[22125, 739, 186, 4, 33, "Input"], Cell[22314, 745, 92, 1, 33, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[22443, 751, 594, 18, 62, "Input"], Cell[23040, 771, 1016, 35, 73, "Output"] }, Open ]], Cell[24071, 809, 385, 10, 33, "Input"], Cell[CellGroupData[{ Cell[24481, 823, 308, 8, 53, "Input"], Cell[24792, 833, 92, 1, 33, "Output"], Cell[24887, 836, 92, 1, 33, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[25016, 842, 191, 4, 33, "Input"], Cell[25210, 848, 693, 25, 68, "Output"] }, Open ]], Cell[25918, 876, 1036, 35, 71, "Input"], Cell[CellGroupData[{ Cell[26979, 915, 432, 12, 72, "Input"], Cell[27414, 929, 119, 2, 33, "Output"], Cell[27536, 933, 121, 2, 33, "Output"], Cell[27660, 937, 119, 2, 33, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[27816, 944, 567, 17, 53, "Input"], Cell[28386, 963, 178, 6, 64, "Output"], Cell[28567, 971, 202, 7, 67, "Output"] }, Open ]], Cell[28784, 981, 142, 1, 32, "Text"], Cell[CellGroupData[{ Cell[28951, 986, 4389, 117, 242, "Input"], Cell[33343, 1105, 3140, 63, 656, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[36520, 1173, 248, 6, 53, "Input"], Cell[36771, 1181, 630, 15, 33, "Output"] }, Open ]] }, Open ]] } ] *) (* End of internal cache information *)