(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' 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[ 28972, 976]*) (*NotebookOutlinePosition[ 59371, 1950]*) (* CellTagsIndexPosition[ 59327, 1946]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Notebook 1", "Title"], Cell[CellGroupData[{ Cell["N1.0", "Section"], Cell[TextData[{ "This notebook explains how to use ", StyleBox["Mathematica", FontSlant->"Italic"], " to study curves in the plane. It permits easy access to a wide variety \ of curves for further study and experimentation. In the past, plane curves \ were drawn by ingenious but laborious methods that one can find, for example, \ in the book [Frost]. Computer methods allow us not only to reproduce any \ given curve quickly and accurately, but also to plot families of curves \ effortlessly.\n\nAs well as plotting curves, one can also use ", StyleBox["Mathematica ", FontSlant->"Italic"], "to study them analytically. In this first notebook, we incorporate \ programs for computing the curvature and length of plane curves. Although \ one can compute the curvature of these curves symbolically, it is usually \ more informative to graph the curvature. The study of more complicated plane \ curves is relegated to Notebooks 2 and 3.\n\nTo illustrate a single curve \ effectively, it will often be necessary to add additional information to the \ overall plot, such as straight lines, arrows, and text. With this in mind, \ the first section explains how to define, combine and display individual \ graphics objects. The relevant commands form the basis of all the programs \ that follow, even if they are applied in a more sophisticated way in later \ notebooks." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["N1.1", "Section"], Cell[TextData[{ "Ordered sets are represented using curly brackets in ", StyleBox["Mathematica,", FontSlant->"Italic"], " and can also be generated with the commands ", StyleBox["Table ", FontFamily->"Verdana"], "or ", StyleBox["Array", FontFamily->"Verdana"], ". Vectors are expressed in this way, and can be added and multiplied by \ scalars:" }], "Text"], Cell["\<\ v= {x,y,z} Table[a[i],{i,3}] w= Array[a,3] 2v+w\ \>", "Input"], Cell["\<\ The printed versions of the notebooks display no output, but all the outputs \ can be inspected by evaluating cells in the electronic versions.\ \>", "Text"], Cell["\<\ 2v Sin[v] 1+v\ \>", "Input"], Cell[TextData[{ "Other operators on vectors are also possible, and reflect the \ \"listability\" of many ", StyleBox["Mathematica", FontSlant->"Italic"], " functions. The next two products produce different outputs:" }], "Text"], Cell["\<\ v w v.w\ \>", "Input"], Cell[TextData[{ "We can use the dot product to define the norm of a vector. We shall \ accomplish this with a command denoted ", StyleBox["L ", FontFamily->"Verdana"], "(for \"length\") to avoid confusion with the built-in ", StyleBox["Mathematica", FontSlant->"Italic"], " function ", StyleBox["Norm. ", FontFamily->"Verdana"], "The definition of ", StyleBox["L", FontFamily->"Verdana"], " also incorporates a simplification command:" }], "Text"], Cell["L[v_]:= Sqrt[Simplify[v.v]]", "Definition"], Cell[TextData[{ "This definition illustrates several important features of ", StyleBox["Mathematica", FontSlant->"Italic"], ". Square brackets are generally used to surround function arguments. The \ underscore is an important part of the syntax and ", StyleBox["v_ ", FontFamily->"Verdana"], "represents a generic vector, although its value on the right hand side \ crucially depends on whether the symbol ", StyleBox["=", FontFamily->"Verdana"], " or ", StyleBox[":= ", FontFamily->"Verdana"], "is used in the definition. The latter delays evaluation in order to use \ current values, which is often more appropriate. Had the colon been omitted, \ the norm of any future vector would always be output as ", Cell[BoxData[ \(\@\(x\^2 + y\^2 + z\^2\)\)]], ", this being the norm of the particular vector ", StyleBox["v", FontFamily->"Verdana"], " defined above." }], "Text"], Cell[TextData[{ "The faint gray shading used in the previous cell will always accompany \ important definitions. Cells containing these form part of the \ initialization data that the user is invited to evaluate when the notebook is \ opened. Here is another example: the rotation operator ", StyleBox["J ", FontFamily->"Verdana"], " in the plane is defined by" }], "Text"], Cell["J[{x_,y_}]:= {-y,x}", "Definition"], Cell[TextData[{ "The form of this definition ensures that it can only be applied to vectors \ with two components. We can compute ", StyleBox["J ", FontFamily->"Verdana"], "squared as follows, the second and third lines demonstrating alternative \ ways of applying a function:" }], "Text"], Cell["\<\ J[J[{x,y}]] {x,y}//J//J J@J@{x,y}\ \>", "Input"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " plotting commands will be introduced in subsequent sections, but we make \ a start by introducing some of the so-called graphics primitives." }], "Text"], Cell["\<\ z1:= Graphics[Line[{{0,0},{4,2}}]] z2:= Graphics[Point[{1,1}]] z3:= Graphics[Circle[{1,1},.5]] Show[z1,z2,z3]\ \>", "Input"], Cell[TextData[{ "Throughout the notebooks in this book, we shall often use ", StyleBox["z ", FontFamily->"Verdana"], "together with an index such as ", StyleBox["1,2,3,... ", FontFamily->"Verdana"], " as a call symbol for a graphics object. This will enable us to list such \ objects separately, making it easier to modify and combine plots. For just a \ couple of objects, it will be easier to write ", StyleBox["z1 ", FontFamily->"Verdana"], "and ", StyleBox["z2", FontFamily->"Verdana"], ", but when there are more it is more convenient to use automatic indexing \ of the form ", StyleBox["z[1],z[2],...", FontFamily->"Verdana"], " The symbol ", StyleBox["z ", FontFamily->"Verdana"], "might occasionally be used on its own as an algebraic symbol, such as the \ third coordinate in space, but context will make the meaning clear." }], "Text"], Cell[TextData[{ "The diagram can be improved by the use of graphics directives to emphasize \ each individual object, and graphics options with the command ", StyleBox["Show ", FontFamily->"Verdana"], "to improve overall features of the plot. In the present situation, one \ can equally well use ", StyleBox["= ", FontFamily->"Verdana"], "in place of ", StyleBox[":= ", FontFamily->"Verdana"], "though semicolons are then needed to prevent echoing. " }], "Text"], Cell["\<\ z[1]:= {Thickness[.01], Line[{{0,0},{4,2}}]} z[2]:= {PointSize[.02], Point[{1,1}]} z[3]:= {Hue[0], Circle[{1,1}, .5, {0,\[Pi]}]} z[4]:= Text[\"P\", {1,1.1}] Show[Graphics[Array[z,4]], Axes->True, AspectRatio->Automatic]\ \>", "Input"], Cell[TextData[{ "The ", StyleBox["Array", FontFamily->"Verdana"], " command saves having to apply ", StyleBox["Graphics", FontFamily->"Verdana"], " to each individual object. Setting the ", StyleBox["AspectRatio", FontFamily->"Verdana"], " option corrects the scaling that is essential when displaying circles. " }], "Text"], Cell["\<\ Graphics primitives will be used repeatedly to embellish the book's figures \ with points, lines, circles and text, and we shall carefully duplicate the \ programs used to draw each figure. However, these typically incorporate \ niceties that help improve the overall visual effect, and add nothing of \ substance; for example, a way of improving text can be found in the program \ for Figure 1.1 below. The code for more complicated figures is contained in \ cells, the first in N1.2.1, that are not visible in the printed book.\ \>", "Text"], Cell[TextData[{ "The next package is needed for Figure 1.1 and subsequent pictures. The \ command ", StyleBox["Arrow", FontFamily->"Verdana"], " is then used much in the same way as", StyleBox[" Line", FontFamily->"Verdana"], " above, but is only valid in 2 dimensions." }], "Text"], Cell["<"Verdana"], " option in ", StyleBox["Show ", FontFamily->"Verdana"], "produces larger labels on the picture. The vector-valued function ", StyleBox["\[Gamma][t] ", FontFamily->"Verdana"], "represents a point of the unit circle, and is used to place arrows and \ text." }], "Text"], Cell["\<\ \[Gamma][t_]:= {Cos[t], Sin[t]} z[1]:= {Hue[0], Thickness[.01], Arrow[{0,0}, \[Gamma][\[Pi]/6]]} z[2]:= {Hue[.7], Thickness[.01], Arrow[{0,0}, 1.5\[Gamma][\[Pi]/3]]} z[3]:= {Hue[0], Thickness[.01], Arrow[{0,0}, \[Gamma][2\[Pi]/3]]} z[4]:= Circle[{0,0}, 0.25, {\[Pi]/6,\[Pi]/3}] z[5]:= {Text[\"\[Theta]\", .3\[Gamma][\[Pi]/4]], Text[\"q\", \ 1.1\[Gamma][.53]], Text[\"p\",1.6\[Gamma][1.05]], Text[\"Jq\", 1.1\[Gamma][2]]} ga:= Graphics[Array[z,5]] Show[ga, Axes->True, Ticks->None, AspectRatio->Automatic, TextStyle->Section] Clear[\[Gamma]]\ \>", "NumberedFigure"], Cell[TextData[{ "The last line saves us getting into difficulty when we try to redefine ", StyleBox["\[Gamma]", FontFamily->"Verdana"], ". The symbols ", StyleBox["z", FontFamily->"Verdana"], " and ", StyleBox["ga", FontFamily->"Verdana"], " are less problematic as they will not be used for functions." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["N1.2", "Section"], Cell["\<\ There are equivalent ways of computing derivatives of vector-valued \ functions, and these can be used to determine the velocity, speed and \ acceleration of a parametrized curve.\ \>", "Text"], Cell["\<\ \[Alpha][t_]:= {t, t^2, t^3} D[\[Alpha][t],t] \[Alpha]'[u] L[\[Alpha]'[t]]\ \>", "Input"], Cell["\<\ D[\[Alpha][t], {t,2}] \[Alpha]''[t]\ \>", "Input"], Cell[TextData[{ "The chain rule and resulting Leibniz formula are illustrated next in \ various formats. The first line clears the previous definition of ", StyleBox["\[Alpha]", FontFamily->"Verdana"], "." }], "Text"], Cell["\<\ Clear[\[Alpha]] \[Beta][t_]:= \[Alpha][f[t]] Table[D[\[Beta][t],{t,k}], {k,4}] ColumnForm[%] Do[Print[D[\[Beta][t],{t,k}]], {k,5,6}] \ \>", "Input"], Cell[TextData[{ "Whilst ordinary functions of one variable are plotted with ", StyleBox["Plot", FontFamily->"Verdana"], ", the main command in ", StyleBox["Mathematica", FontSlant->"Italic"], " for plotting plane curves is", StyleBox[" ParametricPlo", FontFamily->"Verdana"], "t." }], "Text"], Cell["\<\ \[Alpha][t_]:={Cos[5t], Cos[7t]} ParametricPlot[\[Alpha][t]//Evaluate, {t,1,\[Pi]}, AspectRatio->Automatic]\ \>", "Input"], Cell[CellGroupData[{ Cell["N1.2.1", "Subsection"], Cell["\<\ f[t_]:={Cos[1.1t], Cos[.7t]} z[0]:= ParametricPlot[f[t]//Evaluate,{t,0,9}, PlotStyle\[Rule]Thickness[.01], DisplayFunction->Identity] a:= 3.6 z[1]:={Hue[0], Thickness[.01], Arrow[f[a],f[a]+f'[a]]} z[2]:= {Hue[0], Thickness[.01], Arrow[f[a],f[a]+J[f'[a]]]} z[3]:= {Hue[.7],Thickness[.01], Arrow[f[a],f[a]+f''[a]]} z[4]:= {Text[\"\[Alpha]'(t)\", f[a]+1.2f'[a]], Text[\"J\[Alpha]'(t)\", \ f[a]+1.1J[f'[a]]], Text[\"\[Alpha]'\[NegativeThinSpace]'(t)\", f[a]+1.2f''[a]]} ga:= Graphics[Array[z,4]] Show[z[0], ga, AspectRatio->Automatic, Axes->False, TextStyle->Section, ImageSize->400, DisplayFunction->$DisplayFunction]\ \>", "NumberedFigure"], Cell[TextData[{ "In this program, it was convenient to first define the parameter value ", StyleBox["a ", FontFamily->"Verdana"], "at which the action takes place. This can then be changed and the whole \ plot quickly recompiled. In fact, the program above can easily be modified \ into a function of ", StyleBox["a", FontFamily->"Verdana"], " that displays the arrows for a given value of a, and this can be used to \ carry out an animation. This approach will be taken in Figure 1.10." }], "Text"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["N1.3", "Section"], Cell["We begin by defining some simple curves:", "Text"], Cell["\<\ circ[a_][t_]:= {a Cos[t], a Sin[t]} parabola[a_][t_]:= {2a t, a t^2} ellipse[a_,b_][t_]:= {a Cos[t], b Sin[t]} \ \>", "Definition"], Cell[TextData[{ "Each is a function, depending on one or two parameters ", StyleBox["a,b", FontFamily->"Verdana"], " that determine the curve's size or shape. The abbreviation ", StyleBox["circ ", FontFamily->"Verdana"], "is used to avoid a clash with ", StyleBox["Mathematica", FontSlant->"Italic"], "'s built-in command", StyleBox[" Circle", FontFamily->"Verdana"], "." }], "Text"], Cell[TextData[{ "It is more efficient to compute the derivative of the arc length function \ before computing the arc length itself. The command ", StyleBox["arclength[\[Alpha]] ", FontFamily->"Verdana"], "is an indefinite integral of ", StyleBox["arclengthprime[\[Alpha]]", FontFamily->"Verdana"], "." }], "Text"], Cell["\<\ arclengthprime[\[Alpha]_][t_]:= L[\[Alpha]'[t]] arclength[\[Alpha]_][t_]:= Integrate[L[\[Alpha]'[tt]], tt]/.tt->t\ \>", "Definition"], Cell[TextData[{ "We retain the notation of the book's first edition, whereby the symbol ", StyleBox["tt", FontFamily->"Verdana"], " is reserved for a dummy variable. Here it is the variable of integration \ that, when complete, is replaced by ", StyleBox["t", FontFamily->"Verdana"], "; without its use, the computation of something like ", Cell[BoxData[ \(TraditionalForm\`\(\(arclength[\[Alpha]]\)\([\)\(t\^2\)\)\)]], StyleBox["] would fail", FontFamily->"Verdana"], ". The command ", StyleBox["Module", FontFamily->"Verdana"], " can be used to bind dummy variables, but for short definitions the above \ formalism is clearer. In practice, one also needs to add various \ simplification commands to obtain useful results." }], "Text"], Cell["\<\ Clear[a] arclength[circ[a]][t] %//Simplify//PowerExpand\ \>", "Input"], Cell["\<\ arclength[parabola[a]][t]; %//Simplify//PowerExpand\ \>", "Input"], Cell[TextData[{ "The command", StyleBox[" PowerExpand", FontFamily->"Verdana"], " converts", StyleBox[" ", FontFamily->"Verdana"], Cell[BoxData[ \(TraditionalForm\`\((ab)\)\^s\)]], " into ", Cell[BoxData[ \(TraditionalForm\`a\^s\)]], StyleBox[" times ", FontFamily->"Verdana"], Cell[BoxData[ \(TraditionalForm\`b\^s\)]], "; sometimes this is correct mathematically, but sometimes it is not. For \ example, the square root of ", Cell[BoxData[ \(TraditionalForm\`x\^2\)]], " may not equal ", StyleBox["x", FontFamily->"Verdana"], ", so ", StyleBox["PowerExpand", FontFamily->"Verdana"], " must be used with care. To compute the length of a curve on some \ interval requires a definite integral, though the method may fail if ", StyleBox["L[\[Alpha]'[t]] ", FontFamily->"Verdana"], " is too complicated. A numerical value for the length can always be \ obtained using ", StyleBox["Mathematica", FontSlant->"Italic"], "'s numerical integrator." }], "Text"], Cell["\<\ length[a_,b_][\[Alpha]_]:= Integrate[L[\[Alpha]'[tt]], {tt,a,b}, GenerateConditions->False] lengthn[a_,b_][\[Alpha]_]:= NIntegrate[L[\[Alpha]'[tt]], {tt,a,b}] \ \>", "Definition"], Cell[TextData[{ "The following example involves the semicubical parabola ", StyleBox["sc. ", FontFamily->"Verdana"], "The numerical computation tells us that a string stretched along the curve \ from ", StyleBox["sc[1]", FontFamily->"Verdana"], " to ", StyleBox["sc[2] ", FontFamily->"Verdana"], "has length 7.63371." }], "Text"], Cell["sc[t_]:= {t^2, t^3} ", "Definition"], Cell["\<\ length[a,b][sc] %//PowerExpand %//Simplify\ \>", "Input"], Cell["\<\ lengthn[1,2][sc] N[length[1,2][sc]]\ \>", "Input"], Cell[TextData[{ "The next figure uses the 3-dimensional plotting command ", StyleBox["ParametricPlot3D", FontFamily->"Verdana"], ", but we shall not use this extensively until Notebook 7." }], "Text"], Cell["\<\ f[t_]:={Cos[t], Sin[t], t/3} z1:= Graphics3D[{Hue[0], Thickness[.015], Line[Table[f[t], {t,0,4\[Pi],\[Pi]/3}]]}] ff[t_]:= Append[f[t], {Hue[.6], Thickness[.01]}] z2:= ParametricPlot3D[ff[t], {t,0,4\[Pi]}, DisplayFunction->Identity] Show[z1, z2, DisplayFunction->$DisplayFunction]\ \>", "NumberedFigure"], Cell[TextData[{ "The ", StyleBox["DisplayFunction", FontFamily->"Verdana"], " options ensure that only the final combined plot is displayed. The", StyleBox[" Append", FontFamily->"Verdana"], " command is used to define a new function", StyleBox[" ff", FontFamily->"Verdana"], " representing a thickened 3-dimensional curve." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["N1.4", "Section"], Cell[TextData[{ "The definition of the curvature of a plane curve translates easily into ", StyleBox["Mathematica", FontSlant->"Italic"], ". Once again, ", StyleBox["tt", FontFamily->"Verdana"], " is used as a dummy variable. Examples follow." }], "Text"], Cell["\<\ \[Kappa]2[\[Alpha]_][t_]:= \ \[Alpha]''[tt].J[\[Alpha]'[tt]]/L[\[Alpha]'[tt]]^3/.tt->t\ \>", "Definition"], Cell["\<\ \[Kappa]2[circ[a]][t] PowerExpand[Simplify[%]] \[Kappa]2[parabola[1]][t]; Simplify[%]\ \>", "Input"], Cell[CellGroupData[{ Cell["N1.4.1", "Subsection"], Cell["\<\ The next command combines a basic parametric plot with arrows and labels. \ \>", "Text"], Cell["\<\ para[f_, A_] := ParametricPlot[Evaluate[f[t]], {t, -2, 2}, PlotStyle->{AbsoluteThickness[2], RGBColor[0,.5,0]}, AspectRatio->Automatic, Ticks->False, Epilog -> {Arrow[f[1.2], f[1.4], HeadWidth->2], Arrow[f[-1.4], f[-1.2], HeadWidth->2]}, PlotLabel -> StyleForm[A, Subsection], DisplayFunction -> Identity]\ \>", "Definition"], Cell["\<\ Enclosing the program inside Module is the best way of preventing future \ definition clashes,and definitions are given with = followed by a \ semicolon.\ \>", "Text"], Cell["\<\ Module[{z}, z[1] = para[{#, #^2 - 1} & , {\" t, \\!\\(t\\^2\\) - 1\"}]; z[2] = para[{-#, 1 - #^2} & , {\"-t, 1- \\!\\(t\\^2\\)\"}]; Show[GraphicsArray[Array[z,2]], DisplayFunction->$DisplayFunction] ]\ \>", "NumberedFigure"], Cell["\<\ Module[{z}, z[1] = para[{-#, #^2 - 1} & , {\"-t, \\!\\(t\\^2\\) - 1\"}]; z[2] = para[{#, 1 - #^2} & , {\" t, 1 - \\!\\(t\\^2\\)\"}]; Show[GraphicsArray[Array[z,2]], DisplayFunction -> $DisplayFunction] ]\ \>", "NumberedFigure"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["N1.5", "Section"], Cell[TextData[{ "In this section, we display the relatively long program needed for Figure \ 1.6. The ", StyleBox["Module", FontFamily->"Verdana"], " command is one of ", StyleBox["Mathematica", FontSlant->"Italic"], "'s ways of setting local variables; it ensures that the various \ definitions retain their validity solely for the course of the program. \ Definitions are usually given with = followed by a semicolon at the end of \ the line, though the colon before = is essential in line 5, so as to delay \ evaluation." }], "Text"], Cell["\<\ Module[{f,g,a,p,z}, f[t_]= {1+Sin[1.3t], Sin[.7t]}; g[t_]= 2{Cos[1.3t], Cos[.7t]} ; a= 2.8; p[f_,h_]:= ParametricPlot[f[t]//Evaluate, {t,.5,8}, PlotStyle->{Hue[h], Thickness[.01]}, DisplayFunction->Identity]; z[1]= {Thickness[.007], Arrow[f[a], f[a]+f'[a]]}; z[2]= {Thickness[.007], Arrow[g[a], g[a]+g'[a]]}; z[3]= {Thickness[.007], Arrow[g[a], g[a]+f'[a]]}; z[4]= {Text[\"\[Alpha]'(t)\", f[a]+1.3f'[a]], Text[\"\[Beta]'(t)\", \ g[a]+1.15g'[a]]}; z[5]= {Hue[.7], Thickness[.007], Circle[g[a], .2, {-.75,3.3}]}; ga= Graphics[Array[z,5]]; Show[p[f,.6], p[g,0], ga, AspectRatio->Automatic, Axes->False, PlotRange->All, TextStyle->Section, DisplayFunction->$DisplayFunction, ImageSize->450] ]\ \>", "NumberedFigure"], Cell[TextData[{ "The whole thing can be replotted after changing the value of the parameter \ ", StyleBox["a", FontFamily->"Verdana"], ", though ", StyleBox["z[5] ", FontFamily->"Verdana"], "requires a more sophisticated definition if the arc of the circle \ indicating the angle is to transform correctly. " }], "Text"], Cell[CellGroupData[{ Cell["N1.5.1", "Subsection"], Cell[TextData[{ "In the following picture, the angle, lines and text are added using ", StyleBox["Epilog", FontFamily->"Verdana"], ". To hide the lines under the graph, one can us ", StyleBox["Prolog", FontFamily->"Verdana"], ". However, it is best to avoid these commands if the extra graphics data \ forms an essential part of the plot. For example, this method would have cut \ off some of the text in the previous figure. " }], "Text"], Cell["\<\ Module[{f,a,u,v,y,z}, f[t_]={t, t-t^2+.8t^3}; a=.7; u=f[a]; v=f'[a]; z[0]= ParametricPlot[f[t]//Evaluate,{t,0,2}, PlotStyle->Thickness[.01], AspectRatio->Automatic, PlotRange->{{-.5,2},{-.2,2}}, Ticks->None, DisplayFunction\[Rule]Identity]; z[1]= {Hue[.6], Thickness[.01], Arrow[u,u+v]}; z[2]= {Hue[.6], Thickness[.008], Line[{u,u+{2,0}}]}; z[3]= {Hue[.6], PointSize[.04], Point[u]}; z[4]= Circle[u, .2, {0, ArcTan[v[[2]]/v[[1]]]}]; z[5]= Text[\"\[Alpha](t)\", u, {1.2,-2}]; z[6]= Text[\"\[Alpha]'(t)\", u+1.15v, {0,0}]; z[7]= Text[\"\[Theta](t)\", u, {-3.2,-1.9}]; Show[z[0], Epilog->Array[z,7], ImageSize->400, DisplayFunction->$DisplayFunction, TextStyle->Section] ];\ \>", "NumberedFigure"], Cell[TextData[{ "Module[{f,P,F},\n f[t_]= {5Cos[t], 3Sin[t]};\n z[0]= \ ParametricPlot[f[t]//Evaluate,{t,0,2\[Pi]}, \n \ PlotStyle->{Thickness[.007],Hue[.65]},\n \ PlotRange->{{-6,6},{-4,4}},\n DisplayFunction->Identity];\n \ P=f[\[Pi]/4]; F={4,0};\n z[1]= {PointSize[.02],Point[P]};\n z[2]= \ {PointSize[.02],Point[F]};\n z[3]= {PointSize[.02],Point[-F]};\n z[4]= \ Text[\"P\", P, {-1,-1.5}];\n z[5]= Text[\"F", Cell[BoxData[ \(TraditionalForm\`\(\_+\)\)]], "\", F, {0,1.5}];\n z[6]= Text[\"F", Cell[BoxData[ \(TraditionalForm\`\(\_-\)\)]], "\", -F, {-.5,1.5}];\n z[7]= Line[{P,F}];\n z[8]= Line[{P,-F}];\n \ Show[z[0], Epilog->Array[z,8], Axes->True, \n Ticks->False, \ AspectRatio->Automatic,\n TextStyle->Section, \ DisplayFunction->$DisplayFunction,\n ImageSize->450];\n];" }], "NumberedFigure"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["N1.6", "Section"], Cell[TextData[{ "The following program illustrates the power of the ", StyleBox["Table", FontFamily->"Verdana"], " command." }], "Text"], Cell["\<\ cir[x_]:= {Hue[x/5], Circle[{x,0}, Abs[x]]} family:= Table[Graphics[cir[x]], {x,-50,50}] Show[family, AspectRatio->Automatic]\ \>", "Input"], Cell["The logarithmic spiral is defined by", "Text"], Cell["logspiral[a_,b_][t_]:= a Exp[b t]{Cos[t], Sin[t]} ", "Definition"], Cell[TextData[{ "The one plotted has ", StyleBox["b=0.2", FontFamily->"Verdana"], ", and its curvature can be computed as follows:" }], "Text"], Cell["\<\ \[Alpha]:= logspiral[1,.2] PowerExpand[Simplify[\[Kappa]2[\[Alpha]][t]]] \ \>", "Input"], Cell[TextData[{ "Open cells below show how to plot both the spiral and its curvature. The \ next few lines compute the curvature", StyleBox[" \[Kappa]2", FontFamily->"Verdana"], " of an ellipse, and then plot it:" }], "Text"], Cell["\<\ \[Epsilon]:= ellipse[3,4] \[Kappa]2[\[Epsilon]][t]//Simplify Plot[\[Kappa]2[\[Epsilon]][t]//Evaluate,{t,0,2\[Pi]}, \ PlotRange->{{0,2\[Pi]},{-.2,.75}}, AspectRatio->1/3, PlotStyle->{Thickness[.01], RGBColor[0,.5,0]}, ImageSize->450]\ \>", "NumberedFigure"], Cell[TextData[{ "The next program plots a spiral together with tangent and radial vector as \ a function of the variable", StyleBox[" a", FontFamily->"Verdana"], ". " }], "Text"], Cell["\<\ spiral[a_]:= Module[{f,ep}, f= logspiral[1,.2]; ep= {Arrow[{0,0}, f[a]], Arrow[f[a], f[a]+f'[a]]}; ParametricPlot[f[t]//Evaluate, {t,-4\[Pi],4\[Pi]}, PlotStyle->{Hue[.05],Thickness[.01]}, AspectRatio->Automatic, Epilog->ep, ImageSize->300] ] spiral[2]\ \>", "NumberedFigure"], Cell[TextData[{ "A program of this sort can be used to make multiple plots. Clicking on \ one of the resulting graphics windows will produce an animation. Effects \ such as timing can be customized using the menu ", StyleBox["Edit->Preferences->Graphics Options.", FontFamily->"Verdana"] }], "Text"], Cell["Do[spiral[a/3], {a,20}]", "Animation"], Cell["The plot for Figure 1.11 is more straightforward.", "Text"], Cell["\<\ Plot[\[Kappa]2[logspiral[1,.2]][t], {t,-4\[Pi],4\[Pi]}, ImageSize->400, PlotStyle->{Thickness[.01], RGBColor[0,.5,0]}]\ \>", "NumberedFigure"] }, Closed]], Cell[CellGroupData[{ Cell["N1.7", "Section"], Cell[TextData[{ "The semicubical parabola ", StyleBox["sc", FontFamily->"Verdana"], " was defined in ", StyleBox["N1.3,", FontFamily->"Verdana"], " and is now plotted." }], "Text"], Cell["\<\ ParametricPlot[sc[t]//Evaluate, {t,-1,1}, PlotStyle->{Thickness[.01],Hue[.7]}, PlotRange->{{0,1.2},{-1,1.1}}, AspectRatio->Automatic, Ticks->{{.5,1},Automatic}, ImageSize->250]\ \>", "NumberedFigure"] }, Closed]], Cell[CellGroupData[{ Cell["N1.8", "Section"], Cell["The following curves from Exercise 1 are plotted in N1.8.1.", "Text"], Cell["\<\ circleinvolute[a_][t_]:= circ[a][t] - t circ[a]'[t] catenary[a_][t_]:= {a Cosh[t/a], t} astroid[a_][t_]:= a{Cos[t]^3, Sin[t]^3} \ \>", "Definition"], Cell[TextData[{ "Here the ", StyleBox["Table", FontFamily->"Verdana"], " command is combined with", StyleBox[" ParametricPlot:", FontFamily->"Verdana"] }], "Text"], Cell["\<\ p[a_,b_,h_]:= ParametricPlot[logspiral[a,b][t]//Evaluate, {t,-4\[Pi],4\[Pi]}, PlotStyle->{Hue[h], Thickness[.01]}, AspectRatio->Automatic, DisplayFunction->Identity]\ \>", "Input"], Cell["\<\ Show[Table[p[n/10,.2,.1], {n,20,35}], Table[p[-n/10,.2,.7], {n,20,35}], PlotRange->{{-20,20}, {-20,20}}, DisplayFunction->$DisplayFunction]\ \>", "Input"], Cell[CellGroupData[{ Cell["N1.8.1", "Subsection"], Cell["\<\ Using the double subscipt __ in line 2 below allows us to insert not just a \ function, but also a plotting range.\ \>", "Text"], Cell["\<\ Module[{p}, p[X__]:= ParametricPlot[X, Compiled\[Rule]False, AspectRatio\[Rule]Automatic, PlotStyle\[Rule]{Thickness[.02], Hue[.9]}, Ticks\[Rule]None, DisplayFunction\[Rule]Identity]; z[1]= p[circleinvolute[1][t],{t,0,2\[Pi]}]; z[2]= p[catenary[1][t],{t,0,2}]; z[3]= p[astroid[1][t],{t,0,2\[Pi]}]; z[4]= p[{t,t^2},{t,0,1}]; Show[GraphicsArray[Array[z,4], GraphicsSpacing->.3]] ];\ \>", "NumberedFigure"], Cell["\<\ \[Gamma][a_]:= ParametricPlot[{t, Abs[t]^a Sin[1/t]}, {t,-0.2,0.2}, PlotPoints\[Rule]10, PlotStyle\[Rule]Hue[.7], PlotRange\[Rule]All, Ticks\[Rule]None] \[Gamma][1.5];\ \>", "NumberedFigure"] }, Closed]] }, Closed]] }, Open ]] }, FrontEndVersion->"5.2 for Microsoft Windows", ScreenRectangle->{{0, 1280}, {0, 717}}, AutoGeneratedPackage->None, ScreenStyleEnvironment->"Working", PrintingStyleEnvironment->"Working", WindowToolbars->{"RulerBar", "EditBar"}, WindowSize->{1262, 683}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, PrintingCopies->1, PrintingStartingPageNumber->29, PrintingPageRange->{Automatic, Automatic}, PageHeaders->{{Cell[ TextData[ { CounterBox[ "Page"]}], FontSlant -> Italic], Inherited, Cell[ "NOTEBOOK 1", FontSlant -> Italic]}, { Cell[ "NOTEBOOK 1", FontSlant -> Italic], Inherited, Cell[ TextData[ { CounterBox[ "Page"]}], FontSlant -> Italic]}}, PageFooters->{{Inherited, Cell[ TextData[ { CounterBox[ "Page"]}], FontSlant -> Italic], Inherited}, { Inherited, Cell[ TextData[ { CounterBox[ "Page"]}], FontSlant -> Italic], Inherited}}, PageHeaderLines->{False, False}, PrintingOptions->{"PrintingMargins"->{{155, 20}, {150, 300}}, "PaperSize"->{720, 1200}, "PageSize"->{720, 1200}, "PageHeaderMargins"->{275, 275}, "PageFooterMargins"->{300, 300}, "FirstPageHeader"->False, "RestPagesFooter"->False, "FacingPages"->True, "FirstPageFace"->Right, "Magnification"->0.75}, FontFamily->"Verdana", StyleDefinitions -> Notebook[{ Cell[CellGroupData[{ Cell["Style Definitions", "Subtitle", PageWidth->PaperWidth], Cell["\<\ Modify the definitions below to change the default appearance of all cells in \ a given style. Make modifications to any definition using commands in the Format menu.\ \>", "Text", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell["Style Environment Names", "Section", PageWidth->PaperWidth], Cell[StyleData[All, "Working"], PageWidth->PaperWidth, CellBracketOptions->{"Color"->RGBColor[0.771908, 0.399634, 0.262867]}, ScriptMinSize->9], Cell[StyleData[All, "Printout"], PageWidth->PaperWidth, PrivateFontOptions->{"FontType"->"Outline"}] }, Open ]], Cell[CellGroupData[{ Cell["Notebook Options", "Section", PageWidth->PaperWidth], Cell["\<\ The options defined for the style below will be used at the Notebook level.\ \>", "Text", PageWidth->PaperWidth], Cell[StyleData["Notebook"], PageWidth->PaperWidth, CellFrameLabelMargins->6, StyleMenuListing->None] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Headings", "Section", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell[StyleData["Title"], PageWidth->PaperWidth, CellMargins->{{36, 20}, {0, 10}}, CellGroupingRules->{"TitleGrouping", 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