(*********************************************************************** 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["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[{ "To run this ", StyleBox["Mathematica", FontSlant->"Italic"], " Notebook, you need ", StyleBox["FIRST", FontWeight->"Bold"], " to input the following lines to use Gray's ", StyleBox["Mathematica", FontSlant->"Italic"], " code, after first saving them from the web to floppy A. Put 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. \nYou can ask about and call \ any curve from the collection into your ", StyleBox["Mathematica", FontSlant->"Italic"], " NoteBook, eg" }], "Text", PageWidth->PaperWidth], Cell[BoxData[ \(\(?ellipse\)\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(\(ellipse[2, 1]\)\ [s]\)], "Input", PageWidth->PaperWidth], Cell[TextData[{ "So, ", StyleBox["ellipse[2,1]", FontWeight->"Bold"], " is a function of real parameter s. We can plot the graph with \ command" }], "Text", PageWidth->PaperWidth], Cell[BoxData[ \(\(ParametricPlot[\(ellipse[2, 1]\)[s] // Evaluate, \ {s, 0, 2 \[Pi]}, \ AspectRatio -> Automatic];\)\)], "Input", PageWidth->PaperWidth], Cell["\<\ Note that here we have set the parameter s to run over the interval \ [0,2\[Pi]].\ \>", "Text", PageWidth->PaperWidth], Cell[TextData[{ "The ", StyleBox["derivative", FontVariations->{"CompatibilityType"->0}], StyleBox[" ", FontWeight->"Bold"], " at s of the function ellipse[2,1] is the tangent or ", StyleBox["velocity", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], StyleBox[" vector", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " to this ellipse, " }], "Text", PageWidth->PaperWidth], Cell[BoxData[ \(\(\(\ \)\(\(ellipse[2, 1]'\)\ [s]\)\(\ \ \)\)\)], "Input", PageWidth->PaperWidth], Cell[TextData[{ "Vectors are ordered triplets in curly brackets; their ", StyleBox["dot product", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], StyleBox[" ", FontColor->RGBColor[1, 0, 0]], " is given by " }], "Text", PageWidth->PaperWidth], Cell[BoxData[ \({l, m, n} . {u, v, w}\)], "Input", PageWidth->PaperWidth], Cell[TextData[{ "and their ", StyleBox["cross product", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " is given (using \[Cross] , 3rd item on 2nd row of palette) by" }], "Text"], Cell[BoxData[ \({l, m, n}\[Cross]{u, v, w}\)], "Input"], Cell[TextData[{ " You can use also ", StyleBox["Dot[{l,m,n},{u,v,w}]", FontWeight->"Bold"], " and ", StyleBox["Cross[{l,m,n},{u,v,w}]", FontWeight->"Bold"], " for these products.\n The ", StyleBox[" ", FontWeight->"Bold"], StyleBox["norm", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], StyleBox[" ", FontWeight->"Bold"], " or ", StyleBox[" ", FontWeight->"Bold"], StyleBox["magnitude", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " of a vector is the square root of its self dot product, so the norm of \ the velocity vector to our ellipse is given by" }], "Text", PageWidth->PaperWidth], Cell[BoxData[ \(\@\(\(ellipse[2, 1]'\)[s] . \(ellipse[2, 1]'\)[s]\)\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(\(ellipse[2, 1]'\)[s] . \(ellipse[2, 1]'\)[s]\)], "Input", PageWidth->PaperWidth], Cell[TextData[{ "The ", StyleBox["distance", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " or ", StyleBox["length of the line", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " between two points in ", Cell[BoxData[ \(TraditionalForm\`E\^3\)]], " is the ", StyleBox["norm of the difference vector", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " between them. The ", StyleBox["length of a curve", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " is the integral of the norm of the velocity vector and is difficult to \ find analytically in terms of elementary functions, but we can always compute \ it numerically to any desired accuracy, for our ellipse it is, correct to 20 \ digits:" }], "Text", PageWidth->PaperWidth], Cell[BoxData[ \(N[\[Integral]\_0\%\(2 \[Pi]\)\(\@\(\(ellipse[2, 1]'\)[ s] . \(ellipse[2, 1]'\)[s]\)\) \[DifferentialD]s, 20]\)], "Input", PageWidth->PaperWidth], Cell["\<\ If we change curve to the ellipse[1,1] it is actually a unit circle; here \ we can find the length analytically:\ \>", "Text", PageWidth->PaperWidth], Cell[BoxData[ \(\[Integral]\_0\%\(2 \[Pi]\)\(\@\(\(ellipse[1, 1]'\)[ s] . \(ellipse[1, 1]'\)[s]\)\) \[DifferentialD]s\)], "Input", PageWidth->PaperWidth], Cell[TextData[{ StyleBox["Plane Curves", FontSize->14, FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], "\nWe use in this assignment special ", StyleBox["Mathematica", FontSlant->"Italic"], " functions from CSPROGS..m ", StyleBox[" \ntangent ", FontWeight->"Bold"], StyleBox["which finds the (", FontVariations->{"CompatibilityType"->0}], StyleBox["unit", FontWeight->"Bold", FontVariations->{"CompatibilityType"->0}], StyleBox[") tangent vector to a curve---it divides the tangent vector by \ its norm", FontVariations->{"CompatibilityType"->0}], StyleBox["\n kappa2 ", FontWeight->"Bold"], StyleBox["which finds the signed curvature of a ", FontVariations->{"CompatibilityType"->0}], StyleBox["plane", FontWeight->"Bold", FontVariations->{"CompatibilityType"->0}], StyleBox[" curve (ie one with only 2 components)", FontVariations->{"CompatibilityType"->0}], StyleBox["\n", FontWeight->"Bold"], StyleBox["try these on the ellipse ", FontVariations->{"CompatibilityType"->0}], StyleBox[" ", FontWeight->"Bold"], "by activating the following commands in turn---note that these functions \ are applied to the curve function ", StyleBox["ellipse[2,1]", FontWeight->"Bold"], ", which then gives a function of a function of the parameter:" }], "Text", PageWidth->PaperWidth], Cell[BoxData[ \(\(tangent[ellipse[2, 1]]\)\ \ [s]\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(\(kappa2[ellipse[2, 1]]\)\ \ [s]\)], "Input", PageWidth->PaperWidth], Cell[TextData[{ StyleBox["Space Curves", FontSize->14, FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], "\nFor a space curve, which must have ", StyleBox["three", FontWeight->"Bold"], " components, we have also the functions", StyleBox["\ntangent ", FontWeight->"Bold"], StyleBox["which finds the (", FontVariations->{"CompatibilityType"->0}], StyleBox["unit", FontWeight->"Bold", FontVariations->{"CompatibilityType"->0}], StyleBox[") tangent vector to a curve---it divides the tangent vector by \ its norm", FontVariations->{"CompatibilityType"->0}], StyleBox["\nkappa ", FontWeight->"Bold"], StyleBox["which finds the curvature of a ", FontVariations->{"CompatibilityType"->0}], StyleBox["space", FontWeight->"Bold", FontVariations->{"CompatibilityType"->0}], StyleBox[" curve\n", FontVariations->{"CompatibilityType"->0}], StyleBox["tau ", FontWeight->"Bold"], StyleBox["which finds the torsion of a ", FontVariations->{"CompatibilityType"->0}], StyleBox["space", FontWeight->"Bold", FontVariations->{"CompatibilityType"->0}], StyleBox[" curve", FontVariations->{"CompatibilityType"->0}], "\nWe can illustrate by re-defining our ellipse to lie in the z=0 plane of \ ", Cell[BoxData[ \(TraditionalForm\`E\^3\)]], " as the space curve ", StyleBox["ellz0", FontWeight->"Bold"], ", say:" }], "Text", PageWidth->PaperWidth], Cell[BoxData[ \(ellz0[s_] := \ {2\ Cos[s], Sin[s], 0}\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(\(ellz0'\)[s]\)], "Input"], Cell[BoxData[ \(\(tangent[ellz0]\)\ [s]\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(\(kappa[ellz0]\)\ [s]\)], "Input", PageWidth->PaperWidth], Cell[BoxData[ \(\(tau[ellz0]\)\ [s]\)], "Input", PageWidth->PaperWidth] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Question A (Necessary, but not for submission)", FontColor->RGBColor[1, 0, 0]]], "Subtitle", PageWidth->PaperWidth], Cell[TextData[{ "Consider an ellipse with eccentricity e<1 and focii at (\[PlusMinus]", Cell[BoxData[ \(TraditionalForm\`\@\(a\^2 - b\^2\)\)]], ",0) , so we have a>b \nand implicit equation ", Cell[BoxData[ \(TraditionalForm\`x\^2\/a\^2\)]], "+", Cell[BoxData[ \(TraditionalForm\`y\^2\/b\^2\)]], "= 1 where ", Cell[BoxData[ \(TraditionalForm\`\(\(\ \ \ \)\(b\^\(\(2\)\(\ \)\)\)\)\)]], "=", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(a\^2\)\)\)]], "(1 - ", Cell[BoxData[ \(TraditionalForm\`e\^2\)]], ") = ", Cell[BoxData[ \(TraditionalForm\`a\^2\)]], " -", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(c\^2\)\)\)]], ".\n", "For this question we use the parametric form of this equation for standard \ angular parameter 0\[LessEqual]s<2\[Pi] to study the ellipse for the \ case a=5, b=3 " }], "Text", PageWidth->PaperWidth], Cell[TextData[{ StyleBox["(1)", FontColor->RGBColor[1, 0, 0]], " Use ", StyleBox["ParametricPlot", FontWeight->"Bold"], " to plot the ellipse for the case a=5, b=3 by cutting and pasting:\n \ ", StyleBox[" p=ParametricPlot[Evaluate[ellipse[5,3][s]], {s,0,2\[Pi]}, \ AspectRatio->Automatic];", FontWeight->"Bold"], "\n", StyleBox["(2)", FontColor->RGBColor[1, 0, 0]], " Find c. What are the maximum and minimum distances from the ellipse to \ the point (c,0) ? \n", StyleBox["(3)", FontColor->RGBColor[1, 0, 0]], " Find the two points where the ellipse intersects the line y=x.\n", StyleBox["(4)", FontColor->RGBColor[1, 0, 0]], " Find expressions for the velocity vector, the acceleration vector and the \ signed curvature of this ellipse. Plot the magnitudes of these three \ quantities as a function of angle ", Cell[BoxData[ \(TraditionalForm\`0 \[LessEqual] s\)]], "<2\[Pi] . Show that the function ", StyleBox[" tangen", FontWeight->"Bold"], "t gives the ", StyleBox["unit ", FontWeight->"Bold"], " tangent vector. \n", StyleBox["(5)", FontColor->RGBColor[1, 0, 0]], " Find the length of perimeter of the ellipse correct to 17 digits. \n", StyleBox["(6)", FontColor->RGBColor[1, 0, 0]], " What is the equation of an ellipse with ", Cell[BoxData[ \(TraditionalForm\`a = 5, \ \(\(b\)\(=\)\(3\)\(\ \)\)\)]], "but with its focii shifted to (0,0) and (-2c,0)? \n Plot this \ and call it q. Show it with the previous ellipse on the same graph. \n \ ", StyleBox["Hint: Create a composite plot to show two previous plots p and q \ with the command ", FontColor->RGBColor[1, 0, 0]], StyleBox["Show[p,q]", FontWeight->"Bold"], StyleBox[". \n", FontColor->RGBColor[1, 0, 0]] }], "Text", PageWidth->PaperWidth] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Question B (Submit a solution--your own work) ", FontColor->RGBColor[1, 0, 0]]], "Subtitle", PageWidth->PaperWidth], Cell[TextData[{ "Here you will use the above results from A to investigate the ellipses \ that approximate certain orbits around the sun. In such cases it is normal \ to use time ", Cell[BoxData[ \(TraditionalForm\`t\)]], " measured in Earth years as the parameter, so we need ", Cell[BoxData[ \(TraditionalForm\`0 \[LessEqual] t\)]], "PaperWidth], Cell["\<\ Period Semi-major axis \ Eccentricity T years a AU \ e ------------------------------------------------------\ ------- Mercury 0.241 0.387 \ 0.206 Venus 0.615 0.723 \ 0.007 Earth 1.000 1.000 \ 0.017 Mars 1.880 1.524 \ 0.093 Halley 76.2 18.000 \ 0.97 \ \>", "Text", PageWidth->PaperWidth], Cell[TextData[{ StyleBox["(1)", FontColor->RGBColor[1, 0, 0]], " State explicitly the values of b and c for the ellipses. Find the \ equations of each of these ellipses in terms of parameter time ", Cell[BoxData[ \(TraditionalForm\`0 \[LessEqual] t\)]], "RGBColor[1, 0, 0]], " Find in each case the minimum and maximum distance of the orbit from the \ sun, which is represented by the focus (0,0).. \n", StyleBox["(3)", FontColor->RGBColor[1, 0, 0]], " Plot all of the five ellipses", StyleBox[" on one graph", FontWeight->"Bold"], ", using ", StyleBox["ParametricPlot", FontWeight->"Bold"], " with the sun at the origin (0,0). \n", StyleBox["(4)", FontColor->RGBColor[1, 0, 0]], " Give a second plot showing the region contained within about 3 AU of the \ sun. Hint: find out how to use the option ", StyleBox["PlotRange", FontWeight->"Bold"], " to show part of a plot." }], "Text", PageWidth->PaperWidth] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Mathematics 117 \nCurves and Surfaces Assignment 1 Submission\n", FontColor->RGBColor[0, 0, 1]], StyleBox["Date:", FontSize->18, FontColor->RGBColor[0, 0, 1]], StyleBox[" ", FontColor->RGBColor[0, 0, 1]], StyleBox["Name:", FontSize->18, FontColor->RGBColor[0, 0, 1]], StyleBox[" ", FontColor->RGBColor[0, 0, 1]] }], "Title", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell[TextData[StyleBox["Answer to Question B ", FontSize->24, FontColor->RGBColor[1, 0, 0]]], "Section"], Cell[BoxData[ RowBox[{ StyleBox[\((1)\), FontColor->RGBColor[1, 0, 0]], " "}]], "Input"], Cell[BoxData[ RowBox[{ StyleBox[\((2)\), FontColor->RGBColor[1, 0, 0]], " "}]], "Input"], Cell[BoxData[ RowBox[{ StyleBox[\((3)\), FontColor->RGBColor[1, 0, 0]], " "}]], "Input"], Cell[BoxData[ RowBox[{ StyleBox[\((4)\), FontColor->RGBColor[1, 0, 0]], " "}]], "Input"] }, Open ]] }, Open ]] }, FrontEndVersion->"4.0 for Microsoft Windows", ScreenRectangle->{{0, 1152}, {0, 791}}, WindowToolbars->"RulerBar", WindowSize->{667, 695}, WindowMargins->{{73, Automatic}, {Automatic, 0}}, PrintingCopies->1, PrintingPageRange->{3, 3}, PageHeaders->{{Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"], Inherited, Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"]}, {Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"], Inherited, Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"]}}, PrintingOptions->{"PrintingMargins"->{{36, 36}, {36, 36}}, "PrintCellBrackets"->False, "PrintRegistrationMarks"->True, "PrintMultipleHorizontalPages"->False} ] (*********************************************************************** Cached data follows. 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