(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. 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 Mathematica- compatible 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[ 26740, 824]*) (*NotebookOutlinePosition[ 27977, 863]*) (* CellTagsIndexPosition[ 27933, 859]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ " Mathematics 117\n Curves and Surfaces Module\n ", StyleBox["CTJ Dodson, ", FontSize->12], StyleBox["Department of Mathematics, UMIST", "Text", FontSize->12] }], "Subtitle"], Cell["\<\ Assignment 2 Space Curves \ \>", "Subtitle", AspectRatioFixed->True], Cell[TextData[StyleBox["Date: Name: \ ", FontSize->18, FontColor->RGBColor[0, 0, 1]]], "Subsubtitle", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Instructions", FontSize->24, FontWeight->"Plain", FontColor->RGBColor[1, 0, 0], FontVariations->{"CompatibilityType"->0}], " ", StyleBox[" ", FontColor->RGBColor[1, 0, 0]], StyleBox["Double-Click on the check mark at the right to open or close a \ cell", FontSize->12, FontWeight->"Plain", FontColor->RGBColor[1, 0, 0], FontVariations->{"CompatibilityType"->0}] }], "Section"], Cell[TextData[{ "Save to a floppy disk a copy of the introductory Notebook ", StyleBox["TRYMMA.nb", FontWeight->"Bold"], " from the web server at \n", StyleBox["http://www.ma.umist.ac.uk/kd/mmaprogs/\n", FontWeight->"Bold"], StyleBox["Save also from this location onto this floppy disk the files: ", FontVariations->{"CompatibilityType"->0}], StyleBox[" \n117A2.nb, CSPROGS.m, PLTPROGS.m, CURVES.m, SURFS.m \n\ ", FontWeight->"Bold"], StyleBox["Note that these filenames and the Mathematica program are all \ case-sensitive.", FontVariations->{"CompatibilityType"->0}], StyleBox["\n\n", FontWeight->"Bold"], StyleBox["Try ", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], StyleBox["Mathematica:", FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[1, 0, 0]], StyleBox[" ", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], StyleBox["\n", FontWeight->"Bold"], "To use the introductory Notebook ", StyleBox["TRYMMA.nb", FontWeight->"Bold"], " , launch ", StyleBox["Mathematica", FontSlant->"Italic"], " by double clicking its icon, then from the menu bar. Select `File' \ then `Open' and select the Notebook from where you saved it. ", StyleBox[" ", FontWeight->"Bold", FontSlant->"Italic"], "That procedure should open the Notebook and allow you to type into it. \ Once the Notebook is open, you can activate any of the input lines already \ there by typing together the keys . Try altering some of the \ obvious mathematical entries. When using ", StyleBox["Mathematica", FontSlant->"Italic"], " functions, the resulting expressions may be quite complicated; you can \ ask ", StyleBox["Mathematica", FontSlant->"Italic"], " to attempt to simplify them by using ", StyleBox[" \n", FontWeight->"Bold"], "//", StyleBox["FullSimplify ", FontWeight->"Bold"], "at the end of the line of input; it may still leave you some trivial \ cancellations to do if it believes that a function of a complex number is \ involved. \n\n", StyleBox["Help and Printing Settings", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], "\nNote that the whole ", StyleBox["Mathematica", FontSlant->"Italic"], " manual is on-line via `Help' and its index will guide you to a \ particular operation, then give you an example that you can cut and paste. \ Greek letters and mathematical symbols are available via the menu under \ `File' `Palettes'. To help format for printing, under `File' Printing \ Settings' set to A4 paper, under menu `Format' switch on Show Rule, Show \ Page Breaks, and set to Word Wrapping at Paper Width. Note that A4 paper is \ approximately 8.25 inches wide (", StyleBox["Mathematica", FontSlant->"Italic"], " is an American package!) and your left hand margin will have been \ allowed for. A 10pt or 12pt font is usually convenient for general text; you \ can cut and paste from this Notebook to edit in a new Notebook, selecting \ examples of section headings etc.\n\n", StyleBox["Assignment", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], "\nFor the assignment,launch ", StyleBox["Mathematica,", FontSlant->"Italic"], " open your copy of the NoteBook ", StyleBox["117A1.nb", FontWeight->"Bold"], " and enter your name and the date at the top of this page and again at \ the beginning of the submission section. Complete the assignment by answering \ the necessary questions in it, delete any unwanted working and graphics and \ save this Notebook with questions and your complete solutions; these form an \ integral part of the notes for the course. \n\n", StyleBox["Submit a printed version of just the required solution, without \ the questions, as a complete NoteBook; this submission must be your own work \ and obviously copied sections will gain zero marks.", FontWeight->"Bold"], " Note that you can look at the formatting of the printed version by \ expanding the NoteBook window to full screen size; this will help avoid \ losing text off the edge in printing. Keep a copy of the electronic form and \ of the submitted hardcopy. You can hide sections not wanted in a printout by \ double clicking their cell. \n\n", StyleBox["Begin your answers after the end of the questions; you can copy \ the given ", FontColor->RGBColor[0.500008, 0, 0.996109]], StyleBox["Mathematica", FontSlant->"Italic", FontColor->RGBColor[0.500008, 0, 0.996109]], StyleBox[" inputs to the answer section using the Edit menu's `Copy' and \ `Paste'. A new cell is opened up when the cursor changes to a horizontal line \ instead of the vertical line for inside a cell. ", FontColor->RGBColor[0.500008, 0, 0.996109]], "\n ", StyleBox["Use a new cell for each mathematical statement", FontSize->14, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0.500008, 0, 0.996109]], StyleBox[" ", FontSize->14, FontColor->RGBColor[0.500008, 0, 0.996109]] }], "Text", PageWidth->PaperWidth], Cell[TextData[{ "\nYou can find the necessary files for this NoteBook at:\n", StyleBox["http://www.ma.umist.ac.uk/kd/mmaprogs/", FontWeight->"Bold"], "\nTo run this as a ", StyleBox["Mathematica", FontSlant->"Italic"], " Notebook, you need to input the following to use Gray's ", StyleBox["Mathematica", FontSlant->"Italic"], " code after first saving from the web to floppy disk the files: \n", StyleBox[" CSPROGS.m PLTPROGS.m CURVES.m SURFS.m ", FontWeight->"Bold"], "\nPut the correct pathname in as necessary." }], "Text", PageWidth->PaperWidth], Cell[BoxData[{ \(<< A:\\CSPROGS.m\), "\n", \(<< A:\\PLTPROGS.m\), "\n", \(<< A:\\CURVES.m\), "\[IndentingNewLine]", \(<< A:\\SURFS.m\)}], "Input", PageWidth->PaperWidth], Cell[TextData[{ "You need to re-enter these lines every time you restart ", StyleBox["Mathematica", FontSlant->"Italic"], " to use the programs. \nUse any editor to look at the equations for any \ curve by opening the plain text file CURVES.m. \nCall any curve from the \ collection into your ", StyleBox["Mathematica", FontSlant->"Italic"], " NoteBook, eg to plot a figure eight knot" }], "Text", PageWidth->PaperWidth], Cell[BoxData[ \(\(?eightknot\)\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(eightknot[t]\)], "Input"], Cell[BoxData[ \(\(ParametricPlot3D[ eightknot\ [t] // Evaluate, {t, 0, 2 \[Pi]}];\)\)], "Input", PageWidth->PaperWidth], Cell[TextData[{ "Find out what these ", StyleBox["Mathematica", FontSlant->"Italic"], " functions from Gray do: ", StyleBox[" \ntangent ", FontWeight->"Bold"], " ", StyleBox["normal", FontWeight->"Bold"], " ", StyleBox["binormal kappa tau \n", FontWeight->"Bold"], "by activating these enquiry commands in turn [You must already have input \ the source files CSPROGS.m and CURVES.m above]:" }], "Text", PageWidth->PaperWidth], Cell[BoxData[ \(\(?tangent\)\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(\(?normal\)\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(\(?binormal\)\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(\(?tau\)\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(\(?kappa\)\)], "Input", PageWidth->PaperWidth], Cell[TextData[{ "When using such functions the resulting expressions may be quite \ complicated; you can ask ", StyleBox["Mathematica", FontSlant->"Italic"], " to attempt to simplify them by using ", StyleBox[" FullSimplify ", FontWeight->"Bold"] }], "Text", PageWidth->PaperWidth], Cell[BoxData[ \(\(?FullSimplify\)\)], "Input", PageWidth->PaperWidth], Cell[TextData[{ "These should convince you of the value of computers to do routine \ calculus! \nYou can use the above functions in the assignment.\n\nAs with any \ ", 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PageWidth->PaperWidth] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Question A", FontColor->RGBColor[1, 0, 0]]], "Subtitle"], Cell["In this question we study the curve ", "Text"], Cell[BoxData[ \(a[t_] := {\((4/5)\) Cos[t], 1 - Sin[t], \(-\((3/5)\)\) Cos[t]}\)], "Input"], Cell[TextData[{ StyleBox["(1)", FontColor->RGBColor[1, 0, 0]], " Plot the curve for t \[Epsilon] [0,2\[Pi]] the curve ", StyleBox["\n(2)", FontColor->RGBColor[1, 0, 0]], " Find the velocity vector, acceleration, speed, and the Frenet frame \ {T,N,B}. \nShow that this curve satisfies the Frenet-Serret equations and \ find the curvature and torsion. \n", StyleBox["(3)", FontColor->RGBColor[1, 0, 0]], " Explain why you expect that this curve lies in a plane and show that it \ does lie in a plane.\n", StyleBox["Hint: then the vector from a[0] to a[t]=(x(t),y(t),z(t)) \ for all t\[Epsilon] [0,2\[Pi]] will be \nperpendicular to B(t), so a \ general point on the plane is given by: ", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], "\n", StyleBox["Dot[{x,y,z}-a[0],binormal[a][t]]==0", FontWeight->"Bold"], "\nFind this plane in the form ux+vy+wz=0 for some constants u,v,w.\n", StyleBox["(4)", FontColor->RGBColor[1, 0, 0]], " Plot the arc length function of ", StyleBox["a", FontWeight->"Bold"], " against t for t \[Epsilon] [0,2\[Pi]] . What is the total length of ", StyleBox["a", FontWeight->"Bold"], " over this interval? " }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Question B", FontColor->RGBColor[1, 0, 0]]], "Subtitle"], Cell["\<\ In this question we study Viviani's curve [it is the intersection of a \ cylinder of radius 1 and a sphere of radius 2]: \ \>", "Text"], Cell[BoxData[ \(\(viviani[1]\)\ [t]\)], "Input"], Cell[TextData[{ StyleBox["(1)", FontColor->RGBColor[1, 0, 0]], " Plot the curve for t \[Epsilon] [0,4\[Pi]] . Plot also the speed v[t] \ , and the norm a[t] of the acceleration vector on the \n same graph \ and find ", StyleBox["by inspection", FontSlant->"Italic"], " the values and positions of their maxima and minima. \n", StyleBox["(2)", FontColor->RGBColor[1, 0, 0]], " Find Frenet frame {T,N,B} of this curve. \n", StyleBox["(3)", FontColor->RGBColor[1, 0, 0]], " Show that the curve satisfies the Frenet-Serret equations. \n", StyleBox["To see the Frenet-Serret frame on Viviani's curve use a web \ browser to open up the animated gif file at\n ", FontColor->RGBColor[0, 0, 1]], StyleBox[" http://www.ma.umist.ac.uk/kd/mmaprogs/viviani.gif", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], "\n", StyleBox["(4)", FontColor->RGBColor[1, 0, 0]], " Find the curvature and torsion and plot their graphs. \n", StyleBox["(5)", FontColor->RGBColor[1, 0, 0]], " Plot the arc length function against t for t \[Epsilon] [0,4\[Pi]] . 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