(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 4.2' 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[ 6388, 221]*) (*NotebookOutlinePosition[ 7078, 245]*) (* CellTagsIndexPosition[ 7034, 241]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ Cell[BoxData[{ StyleBox[\(MATH\ 257\ Calculus\ III\t\t\t\tWeek\ of\ May\ 19, \ 2003\), FontSize->14], "\n", StyleBox[\(Lab\ 8 : \ Curvature\), "Title"], "\n", StyleBox[\(Name\ 1\), "Section"], "\n", StyleBox[\(Name\ 2\), "Section"], "\n", StyleBox[\(\(Section\)\(:\)\), "Section"]}], "Input"], "\n" }], "Text", FontColor->GrayLevel[1], Background->RGBColor[0, 0, 1]], Cell["\<\ This lab is quick and easy. Do it and then go down to the 7th \ floor of Old Chem and get some food!\ \>", "Text", FontFamily->"Helvetica", FontSize->14], Cell[CellGroupData[{ Cell[TextData[StyleBox["Doing Curvature on Mathematica", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]]], "Subtitle"], Cell[TextData[{ "As we've seen in class, computing the curvature of a space curve can, and \ usually is, a royal pain. On ", StyleBox["Mathematica", FontSlant->"Italic"], " it's much easier, as long as the program doesn't barf on you.\n\nFor \ example, let's find a general formula for the curvature of the twisted cubic. \ First we gotta define all the functions we'll need..." }], "Text", FontSize->14], Cell[TextData[{ "To do all this we'll definitely need to define a function that returns the \ ", StyleBox["magnitude", FontWeight->"Bold"], " of a general vector:" }], "Text", FontSize->14], Cell[BoxData[ \(mag[ v_] := \@\(v[\([1]\)]\^2 + v[\([2]\)]\^2 + v[\([3]\)]\^2\)\)], "Input"], Cell[TextData[{ "Then we're ready to define the ", StyleBox["unit tangent vector", FontWeight->"Bold"], ":" }], "Text", FontSize->14], Cell[BoxData[ \(T[t_] := \(r'\)[t]\/mag[\(r'\)[t]]\)], "Input"], Cell[TextData[{ "Then we can go right ahead and define the ", StyleBox["curvature function", FontWeight->"Bold"], ":" }], "Text", FontSize->14], Cell[BoxData[ \(kappa[t_] := mag[\(T'\)[t]]\/mag[\(r'\)[t]]\)], "Input"], Cell[TextData[{ "All right, then! Let's see what the general formula for the curvature of \ the twisted cubic looks like! We need to define ", StyleBox["r[t]", "Input"], " to be the twisted cubic vector function, and then just ask what ", StyleBox["kappa[t] ", "Input"], "is!" }], "Text", FontSize->14], Cell[BoxData[{ \(r[t_] = {t, t\^2, t\^3}\), "\[IndentingNewLine]", \(kappa[t]\)}], "Input"], Cell[TextData[{ "Doesn't that look ", StyleBox["wonderful", FontWeight->"Bold"], "? OK, we probably should simplify it:" }], "Text", FontSize->14], Cell[BoxData[ \(Simplify[%]\)], "Input"], Cell["\<\ And there it is; our very own function for the curvature of the \ twisted cubic at any point we want. Why do we care? Well, we could graph \ this sucker\ \>", "Text", FontSize->14], Cell[BoxData[ \(Plot[kappa[t], \ {t, \ \(-3\), 3}]\)], "Input"], Cell["\<\ So it looks like the twisted cubic has the greatest curvature when \ t=0. Neat!\ \>", "Text", FontSize->14], Cell[TextData[{ "\n", StyleBox["Question 1:", FontWeight->"Bold"], " Find a general formula for the curvature of the ", StyleBox["helix", FontWeight->"Bold"], " vector function < 3Sin[t], 3Cos[t], t/2 >. (You do NOT need to redefine \ the functions ", StyleBox["T[t]", "Input"], " and ", StyleBox["kappa[t]", "Input"], ". Just redefine ", StyleBox["r[t]", "Input"], " and you should be all set to go. Ain't ", StyleBox["Mathematica", FontSlant->"Italic"], " kewl?) Make sure you ", StyleBox["simplify", FontWeight->"Bold"], "." }], "Text", FontSize->14, FontColor->RGBColor[1, 0, 0]], Cell[BoxData[ \(\[IndentingNewLine]\)], "Input"], Cell[TextData[{ StyleBox["Question 2:", FontWeight->"Bold"], " ", "Find a general formula for the curvature of the vector function", " < Cos[t]-Cos[3t]Sin[t], 2Sin[t]-Sin[3t], 0 >. Yes, this curve lives in \ the x-y plane, but we can still find the curvature. Do the following: plot \ this space curve, find the formula for it's curvature, and make a plot of the \ curvature formula. (You should make 0 \[LessEqual] t \[LessEqual] 2\[Pi].)" }], "Text", FontSize->14, FontColor->RGBColor[1, 0, 0]], Cell[BoxData[ \(\[IndentingNewLine]\)], "Input"], Cell[TextData[StyleBox["Yep, that's it. You're done. Send it to me via the \ Digital Dropbox and then go to the Kade Center (7th Floor of Old Chem, in the \ center of the floor. It's the atrium area) and get some food and cheer on \ Tom Warburg, who won the Calculus Contest!)", FontWeight->"Plain"]], "Text", FontSize->14, FontWeight->"Bold"] }, Open ]] }, FrontEndVersion->"4.2 for Macintosh", ScreenRectangle->{{0, 754}, {0, 578}}, AutoGeneratedPackage->None, WindowToolbars->"EditBar", WindowSize->{663, 488}, WindowMargins->{{19, Automatic}, {Automatic, 1}} ] (******************************************************************* Cached data follows. 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