(*********************************************************************** 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). 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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["117A3.nb", FontWeight->"Bold"], " and enter your name and the date at the top of this page. Retain the \ original questions as given below. Complete the assignment by answering the \ questions in it, delete any unwanted working and graphics and then submit a \ printed version of your final completed NoteBook. 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. Include in \ the printout of your submission the questions but not the \ Instructions---you can hide them by double clicking their cell. \n\nBegin \ your answers after the end of the questions; you can copy the given ", StyleBox["Mathematica", FontSlant->"Italic"], " 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. \n", StyleBox["Use a new cell for each command.", FontWeight->"Bold"], " " }], "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 surfaces programs. You can use any editor to look at the \ parametric equations for any surface by opening the plain text file \ SURFS.m. You can call the surface into your ", StyleBox["Mathematica", FontSlant->"Italic"], " NoteBook, eg" }], "Text"], Cell[BoxData[ \(\(?torus\)\)], "Input"], Cell[BoxData[ \(\(?moebiusstrip\)\)], "Input"], Cell[BoxData[ \(\(moebiusstrip[1]\)[u, v]\)], "Input"], Cell[BoxData[ \(\(ParametricPlot3D[\(moebiusstrip[0.3]\)[u, v] // Evaluate, \ {u, 0, 2 Pi}, \ {v, \(-0.3\), 0.3}];\)\)], "Input"], Cell[TextData[{ "Find out what these ", StyleBox["Mathematica", FontSlant->"Italic"], " surface functions from Gray do: ", StyleBox[" \ngcurvature mcurvature\n", FontWeight->"Bold"], "by activating these enquiry commands in turn [You must already have input \ \nthe source files CSPROGS.m and SURFS.m above]:" }], "Text"], Cell[BoxData[ \(\(?gcurvature\)\)], "Input"], Cell[BoxData[ \(\(?mcurvature\)\)], "Input"], Cell[BoxData[ \(\(?kappa\)\)], "Input"], 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[" //Simplify ", FontWeight->"Bold"], "at the end of the line of input; it may still leave you some trivial \ cancellations \nto do if it believes that a function of a complex number is \ involved. " }], "Text"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Question A---Required", FontColor->RGBColor[1, 0, 0]]], "Subtitle"], Cell[TextData[{ StyleBox["(1)", FontColor->RGBColor[1, 0, 0]], " ", StyleBox["[25 Marks]", FontColor->RGBColor[1, 0, 0]], "\nState the parametric equation and use ", StyleBox["ParametricPlot3D", FontWeight->"Bold"], " to create graphs of the following surfaces: \n", StyleBox["(i)", FontColor->RGBColor[1, 0, 0]], " Cup defined by cup[u,v]:= ", Cell[BoxData[ \(TraditionalForm\`{u, v, u\^2\)]], "+", Cell[BoxData[ \(TraditionalForm\`v\^2\)]], "} over [-1,1]\[Times][-1,1]\n", StyleBox["(ii)", FontColor->RGBColor[1, 0, 0]], " Cap defined by cap[u,v]:={u, v, -", Cell[BoxData[ \(TraditionalForm\`u\^2\)]], "-", Cell[BoxData[ \(TraditionalForm\`\(\(v\^2\)\()\)\)\)]], "} over [-1,1]\[Times][-1,1]\n", StyleBox["(iii)", FontColor->RGBColor[1, 0, 0]], " Saddle defined by sad[u,v]:= ", Cell[BoxData[ \(TraditionalForm\`{u, \ v, \ u\^2\)]], "-", Cell[BoxData[ \(TraditionalForm\`v\^2\)]], "} over [-1,1]\[Times][-1,1]\n", StyleBox["(iv)", FontColor->RGBColor[1, 0, 0]], " Ellipsoid ", Cell[BoxData[ \(TraditionalForm\`\(\(\ \ \ \ \)\(x\^2\/9\)\)\)]], "+", Cell[BoxData[ \(TraditionalForm\`y\^2\/4\)]], "+", Cell[BoxData[ \(TraditionalForm\`z\^2\/1\)]], "=1\n", StyleBox["(v)", FontColor->RGBColor[1, 0, 0]], " Torus of tube radius 1 and wheel radius 3\n", StyleBox["(vi)", FontColor->RGBColor[1, 0, 0]], " Monkeysaddle over [-1,1]\[Times][-1,1]\n", StyleBox["(2)", FontColor->RGBColor[1, 0, 0]], " ", StyleBox["[25 Marks]", FontColor->RGBColor[1, 0, 0]], "\nFor", StyleBox[" one", FontWeight->"Bold"], " of the surfaces in Question 1 compute the Gaussian and mean curvatures \ and use ", StyleBox["Plot3D", FontWeight->"Bold"], " to show \n their graphs over the appropriate range of (u,v) \ values. \n", StyleBox["(3) ", FontColor->RGBColor[1, 0, 0]], " ", StyleBox["[25 Marks]", FontColor->RGBColor[1, 0, 0]], "\nFor ", StyleBox["one", FontWeight->"Bold"], " of the surfaces in Question 1, choose the point with parameters (0,0) \ and find through it the two curves:\n ", StyleBox["curve1", FontWeight->"Bold"], " corresponding to v=0 and ", StyleBox["curve2", FontWeight->"Bold"], " corresponding to u=0 \n Use ", StyleBox["ParametricPlot3D", FontWeight->"Bold"], " to plot the pair of curves on the same graph.\n ", StyleBox["(4) ", FontColor->RGBColor[1, 0, 0]], " ", StyleBox["[25 Marks]", FontColor->RGBColor[1, 0, 0]], "\n For the torus surface in Question 1, find a unit normal vector field \ n[u,v] and from it construct the shape operator \n S as a matrix with \ respect to the basis vectors ", Cell[BoxData[ \(TraditionalForm\`\[PartialD]\_u\ \(torus[3, 1]\)[u, v]\)]], " , ", Cell[BoxData[ \(TraditionalForm\`\[PartialD]\_v\ \(torus[3, 1]\)[u, v]\)]], " " }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Question B---Optional", FontColor->RGBColor[1, 0, 0]]], "Subtitle"], Cell[TextData[{ StyleBox[" (1) ", FontColor->RGBColor[1, 0, 0]], "On each surface in Question A, choose the point with parameters (0,0) \ and \n find through it the two curves:\n ", StyleBox["curve1", FontWeight->"Bold"], " corresponding to v=0 and ", StyleBox["curve2", FontWeight->"Bold"], " corresponding to u=0 \n Use ", StyleBox["ParametricPlot3D", FontWeight->"Bold"], " to plot each of the pairs of curves on the same graph.\n", StyleBox["(2)", FontColor->RGBColor[1, 0, 0]], " For each surface, compute the curvature of the curves and investigate \ these values at the the point of intersection. \n Do the sums and \ products of curvatures of the curves give information about the surface \ curvature at the point\n of intersection? " }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Question C---Optional", FontColor->RGBColor[1, 0, 0]]], "Subtitle"], Cell[TextData[{ StyleBox["(1)", FontColor->RGBColor[1, 0, 0]], " The ", StyleBox["sphere", FontWeight->"Bold"], " of radius ", Cell[BoxData[ \(TraditionalForm\`\(\(\ \ \)\(a\)\)\)]], " has ", StyleBox["constant positive Gaussian curvature", FontWeight->"Bold"], ", ", Cell[BoxData[ \(TraditionalForm\`K\)]], " = +", Cell[BoxData[ \(TraditionalForm\`1\/a\^2\)]], ".\n Investigate the ", StyleBox["pseudosphere", FontWeight->"Bold"], ", a surface of ", StyleBox["constant negative Gaussian curvature", FontWeight->"Bold"], ". \n ", StyleBox[" [You need to look at the formulae to find an appropriate range \ over which to plot. \n The log function can only take positive \ arguments!]", FontSlant->"Italic"], "\n\n", StyleBox["(2)", FontColor->RGBColor[1, 0, 0]], " A surface of ", StyleBox["zero mean curvature", FontWeight->"Bold"], " is called a ", StyleBox["minimal surface", FontWeight->"Bold"], " . \n Investigate the following minimal surfaces \n ", StyleBox[" helicoid catenoid scherk enneper ", FontWeight->"Bold"], " \n by plotting them and finding their Gaussian and mean \ curvatures. Note that soap films \n on wire frames are minimal \ surfaces---the helicoid is the surface you obtain using a frame \n \ made from two parallel wires that have been twisted to form the edges of the \ helicoid. 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