(************** Content-type: application/mathematica ************** 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). 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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[ 3218196, 82674]*) (*NotebookOutlinePosition[ 3218826, 82696]*) (* CellTagsIndexPosition[ 3218782, 82692]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["\<\ 14.4 Motion in Space Tangents, Normals, and Binormals\ \>", "Subsection"], Cell[BoxData[ \(Norm[v_]\ := \ \@\(v . v\)\)], "Input"], Cell[CellGroupData[{ Cell["Example: Twisted Cubic Curve", "Subsubsection"], Cell[BoxData[ \(\(r[t_] = {t, t^2, t^3};\)\)], "Input"], Cell[CellGroupData[{ Cell["v[t_] = r'[t]", "Input", AspectRatioFixed->True], Cell[BoxData[ \({1, 2\ t, 3\ t\^2}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["a[t_] = v'[t]", "Input", AspectRatioFixed->True], Cell[BoxData[ \({0, 2, 6\ t}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["speed[t_] = Norm[v[t]] //Simplify", "Input", AspectRatioFixed->True], Cell[BoxData[ \(\@\(1 + 4\ t\^2 + 9\ t\^4\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["T[t_] = r'[t]/Norm[r'[t]] //Simplify", "Input", AspectRatioFixed->True], Cell[BoxData[ \({1\/\@\(1 + 4\ t\^2 + 9\ t\^4\), \(2\ t\)\/\@\(1 + 4\ t\^2 + 9\ t\^4\), \ \(3\ t\^2\)\/\@\(1 + 4\ t\^2 + 9\ t\^4\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["n[t_] = T'[t]/Norm[T'[t]] //Simplify", "Input", AspectRatioFixed->True], Cell[BoxData[ \({\(-\(\(t\ \((2 + 9\ t\^2)\)\)\/\(\((1 + 4\ t\^2 + 9\ t\^4)\)\^\(3/2\)\ \ \@\(\(1 + 9\ t\^2 + 9\ t\^4\)\/\((1 + 4\ t\^2 + 9\ t\^4)\)\^2\)\)\)\), \(1 - \ 9\ t\^4\)\/\(\((1 + 4\ t\^2 + 9\ t\^4)\)\^\(3/2\)\ \@\(\(1 + 9\ t\^2 + 9\ \ t\^4\)\/\((1 + 4\ t\^2 + 9\ t\^4)\)\^2\)\), \(3\ \((t + 2\ t\^3)\)\)\/\(\((1 \ + 4\ t\^2 + 9\ t\^4)\)\^\(3/2\)\ \@\(\(1 + 9\ t\^2 + 9\ t\^4\)\/\((1 + 4\ \ t\^2 + 9\ t\^4)\)\^2\)\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(B[t_]\ = \ Cross[T[t], n[t]] // Simplify\)], "Input"], Cell[BoxData[ \({\(3\ t\^2\)\/\(\((1 + 4\ t\^2 + 9\ t\^4)\)\ \@\(\(1 + 9\ t\^2 + 9\ \ t\^4\)\/\((1 + 4\ t\^2 + 9\ t\^4)\)\^2\)\), \(-\(\(3\ t\)\/\(\((1 + 4\ t\^2 + 9\ t\^4)\)\ \@\(\(1 + 9\ t\^2 + 9\ t\^4\)\/\((1 + 4\ t\^2 + \ 9\ t\^4)\)\^2\)\)\)\), 1\/\(\((1 + 4\ t\^2 + 9\ t\^4)\)\ \@\(\(1 + 9\ t\^2 + 9\ t\^4\)\/\((1 + \ 4\ t\^2 + 9\ t\^4)\)\^2\)\)}\)], "Output"] }, Open ]], Cell[TextData[{ "The length of the curve from ", Cell[BoxData[ \(TraditionalForm\`t\ = 0\)]], " to ", Cell[BoxData[ \(TraditionalForm\`t = 1\)]], "is:" }], "Text"], Cell[CellGroupData[{ Cell["NIntegrate[speed[t], {t,0,1}]", "Input", AspectRatioFixed->True], Cell[BoxData[ \(1.8630229825122526`\)], "Output"] }, Open ]], Cell["\<\ curve = ParametricPlot3D[r[t], {t,-1.2,1.2}, PlotRange->{{-2,2},{-1,2.5},{-3,3}}, ViewPoint->{3,0.5,1.5}];\ \>", "Input", AspectRatioFixed->True], Cell["\<\ tangents = \ Table[Graphics3D[{RGBColor[0,0,1],Line[{r[t],r[t]+T[t]}]}], \ {t,-1.2,1.2,0.1}];\ \>", "Input", AspectRatioFixed->True], Cell["\<\ normals = \ Table[Graphics3D[{RGBColor[0,1,0],Line[{r[t],r[t]+n[t]}]}], \ {t,-1.2,1.2,0.1}];\ \>", "Input", AspectRatioFixed->True], Cell["\<\ binormals = \ Table[Graphics3D[{RGBColor[1,0,0],Line[{r[t],r[t]+B[t]}]}], \ {t,-1.2,1.2,0.1}];\ \>", "Input", AspectRatioFixed->True], Cell[TextData[{ "Select the graphics cell below and type ", StyleBox["Alt-y", FontSlant->"Italic"], " to animate it. 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