(************** 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). 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[ 34269, 1252]*) (*NotebookOutlinePosition[ 64621, 2225]*) (* CellTagsIndexPosition[ 64577, 2221]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Notebook 2", "Title", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell["N2.0", "Section", PageWidth->PaperWidth], Cell["\<\ This notebook is primarily concerned with the plotting of special curves and \ their curvature. Often, we shall plot two or more functions simultaneously \ in order to compare members of the same family of curves, or to visualize a \ given curve sharing a common coordinate with its curvature.\ \>", "Text", PageWidth->PaperWidth], Cell[TextData[{ "Based on the experience from Chapter 1, we shall proceed to set default \ options that will apply to the command ", StyleBox["ParametricPlot ", FontFamily->"Verdana"], "throughout Notebook 2. In particular, setting ", StyleBox["AspectRatio->Automatic", FontFamily->"Verdana"], " will generally ensure ", "that x,y", " have the same scale, whereas setting ", StyleBox["Compiled->False", FontFamily->"Verdana"], " will assist evaluation of the function to be plotted. " }], "Text", PageWidth->PaperWidth], Cell["\<\ SetOptions[ParametricPlot, Compiled->False, AspectRatio->Automatic, PlotStyle->{{RGBColor[1,0,0], AbsoluteThickness[2]}} ]\ \>", "Definition", PageWidth->PaperWidth], Cell[TextData[{ "The ", StyleBox["PlotStyle", FontFamily->"Verdana"], " option allows us to use thick red as the default, and double brackets \ bind together the color and thickness commands so that they apply equally to \ multiple plots. The output summarizes all the new defaults, many of which \ are unaffected, but can be suppressed by adding a semicolon after the final \ closing bracket." }], "Text", PageWidth->PaperWidth], Cell[TextData[{ "We next take the opportunity to reload some definitions made in Notebook \ 1, in order to make the present notebook self-contained. There is a similar \ subsection ", StyleBox["Nn.0.1 ", FontFamily->"Verdana"], "in Notebook n for n>2, in which we have placed relevant programs from \ previous notebooks that have no need of further comment. The contents of \ these subsections are not displayed in print, but can be inspected and used \ in the electronic version, and many of the encapsulated programs are \ automatically run when the initialization cells are evaluated." }], "Text", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell["N2.0.1", "Subsection", PageWidth->PaperWidth, FontColor->GrayLevel[0]], Cell[TextData[{ "The command ", StyleBox["L", FontFamily->"Verdana"], " computes the norm or \"length\" of a vector. The pairing of ", StyleBox["Unprotect", FontFamily->"Verdana"], " and ", StyleBox["Protect ", FontFamily->"Verdana"], "ensures that the symbol ", StyleBox["L", FontFamily->"Verdana"], " cannot be used in a different sense, though we shall only adopt this \ device for single upper case letters that might accidentally be used in \ another context." }], "Text", PageWidth->PaperWidth], Cell["\<\ Unprotect[L] L[v_]:= Sqrt[Simplify[v.v]] Protect[L]\ \>", "Definition", PageWidth->PaperWidth], Cell[TextData[{ "We also redefine the complex structure ", StyleBox["J ", FontFamily->"Verdana"], "with the same device,", " even though there is less danger that ", StyleBox["J", FontFamily->"Verdana"], " will be used in a different sense." }], "Text", PageWidth->PaperWidth], Cell["\<\ Unprotect[J] J[{x_,y_}]:= {-y,x} Protect[J]\ \>", "Definition", PageWidth->PaperWidth], Cell[TextData[{ "The curvature ", StyleBox["\[Kappa]2", FontFamily->"Verdana"], " of a plane curve was defined in ", StyleBox["N1.4 ", FontFamily->"Verdana"], "by the single line" }], "Text", PageWidth->PaperWidth], Cell["\<\ \[Kappa]2[\[Alpha]_][t_]:= \ \[Alpha]''[tt].J[\[Alpha]'[tt]]/L[\[Alpha]'[tt]]^3/.tt->t\ \>", "Definition", PageWidth->PaperWidth] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["N2.1", "Section", PageWidth->PaperWidth], Cell["\<\ The parametric equation of a cycloid depending on two parameters is \ \>", "Text", PageWidth->PaperWidth], Cell["cycloid[a_,b_][t_]:= {a t - b Sin[t], a - b Cos[t]}", "Definition", PageWidth->PaperWidth], Cell[TextData[{ "The acceleration vector (when applied to a fixed point) moves in a circle \ of radius ", StyleBox["b", FontFamily->"Verdana"], "." }], "Text", PageWidth->PaperWidth], Cell["\<\ cycloid[a,b]'[t] cycloid[a,b]''[t]\ \>", "Input", PageWidth->PaperWidth], Cell[TextData[{ "The simplest cycloid has ", StyleBox["a=b=1:", FontFamily->"Verdana"] }], "Text", PageWidth->PaperWidth], Cell["\<\ \[Alpha]:= cycloid[1,1] ParametricPlot[\[Alpha][t], {t,-3\[Pi],3\[Pi]}, PlotRange->{{-3\[Pi],3\[Pi]},{-2,4}}, Ticks->None]\ \>", "NumberedFigure", PageWidth->PaperWidth], Cell[TextData[{ "With \[Alpha] continuing to represent this particular cycloid, we can \ easily compute and plot its curvature, which we abbreviate with the symbol ", StyleBox["k:", FontFamily->"Verdana"] }], "Text", PageWidth->PaperWidth], Cell["\<\ k[t_]:= FullSimplify[\[Kappa]2[\[Alpha]][t]] k[t] Plot[k[t], {t,-\[Pi],3\[Pi]}, AspectRatio->1, PlotRange->{{-1,7},{-7,1}}, PlotStyle->{Hue[.7],Thickness[.01]}]\ \>", "NumberedFigure", PageWidth->PaperWidth], Cell[TextData[{ "Observe that the curvature is approximately constant except very close to \ the cusps. The ", StyleBox["AspectRatio", FontFamily->"Verdana"], " and ", StyleBox["PlotRange", FontFamily->"Verdana"], " options can be set by trial and error. Given that we have chosen the \ horizontal and vertical intervals to be equal, ", StyleBox["AspectRatio->1 ", FontFamily->"Verdana"], "is equivalent to ", StyleBox["AspectRatio->Automatic", FontFamily->"Verdana"], ".\n \nThe next program numerically computes the value ", StyleBox["g[x]", FontFamily->"Verdana"], " of the parameter ", StyleBox["t,", FontFamily->"Verdana"], " as a function of the cycloid's first coordinate ", StyleBox["x.", FontFamily->"Verdana"], " Thus", StyleBox[" g", FontFamily->"Verdana"], " is the inverse of the function", StyleBox[" f ", FontFamily->"Verdana"], "given by ", StyleBox["f[t] = t - Sin[t]/2", FontFamily->"Verdana"], ", and its graph is the reflection of that of ", StyleBox["f ", FontFamily->"Verdana"], "in the line ", StyleBox["y=x", FontFamily->"Verdana"], "." }], "Text", PageWidth->PaperWidth], Cell["\<\ \[Alpha]:=cycloid[1, 1/2] f[t_]:= \[Alpha][t][[1]] f[t] g[x_]:= t /. FindRoot[f[t]\[Equal]x, {t,1}] Plot[{f[s], s, g[s]}, {s,0,1}, PlotStyle->{Hue[0],Hue[.5],Hue[.7]}, AspectRatio->1, PlotRange->{{0,1},{0,1}}]\ \>", "Input", PageWidth->PaperWidth], Cell["\<\ Using definitions from the previous cell, we can plot the curvature in such a \ way that the curvature of a point on the cycloid is represented by the \ vertical coordinate of the point on the blue curve with the same horizontal \ coordinate.\ \>", "Text", PageWidth->PaperWidth], Cell["\<\ Off[FindRoot::cvnwt] {a,b}= {-\[Pi]/2, 5\[Pi]/2}; y[1]:= ParametricPlot[\[Alpha][t],{t,a,b}, DisplayFunction->Identity] y[2]:= Plot[\[Kappa]2[\[Alpha]][g[x]], {x,f[a],f[b]}, PlotStyle->Hue[.7], PlotPoints->100, DisplayFunction->Identity] Show[y[1], y[2], PlotRange->{{f[a],f[b]},{-2,4}}, AspectRatio->Automatic, DisplayFunction->$DisplayFunction]\ \>", "NumberedFigure", PageWidth->PaperWidth], Cell["Clear[\[Alpha],f,g,k,a,b,y]", "Input", PageWidth->PaperWidth], Cell[TextData[{ "It is best to clear the variables that were introduced in the course of \ the previous two cells in a separate third cell. In the future, we shall \ often avoid this problem by arranging each new program within the command ", StyleBox["Module", FontFamily->"Verdana"], ". " }], "Text", PageWidth->PaperWidth], Cell[TextData[{ "The prolate cycloid is animated by the inclusion of the ", StyleBox["Do", FontFamily->"Verdana"], " command in the following miniprogram. In the definition of", StyleBox[" z[1], ", FontFamily->"Verdana"], "it is important to color or shade the disks in ", StyleBox["decreasing", FontSlant->"Italic"], " size so as not to mask previous assignments. " }], "Text", PageWidth->PaperWidth], Cell["\<\ Module[{\[Alpha],c,C,z}, \[Alpha]= cycloid[1,3]; C= {c,1}; z[1]= {{GrayLevel[.9], Disk[C,3]}, {GrayLevel[.5], Disk[C,1]}}; z[2]= {PointSize[.02], Point[C]}; z[3]= {Circle[C,1], Circle[C,3]}; z[4]= {Line[{C,\[Alpha][c]}], Line[{{-5,0},{20,0}}]}; Do[ParametricPlot[\[Alpha][t], {t,0,4\[Pi]}, Axes->None, Prolog->Array[z,4]], {c,3,9,3}] ]\ \>", "NumberedFigure", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell["N2.1.1", "Subsection", PageWidth->PaperWidth], Cell["The curtate cycloid is similarly given by", "Text", PageWidth->PaperWidth], Cell["\<\ Module[{\[Alpha],z}, \[Alpha]= cycloid[2,1]; z[1]= {{GrayLevel[.6],Disk[{6,2},2]}, {GrayLevel[0.9],Disk[{6,2},1]}}; z[2]= {PointSize[.02],Point[{6,2}]}; z[3]= {Circle[{6,2},1],Circle[{6,2},2]}; z[4]= {Line[{{6,2},\[Alpha][3]}], Line[{{-5,0},{20,0}}]}; ParametricPlot[\[Alpha][t],{t,-\[Pi]/2,5\[Pi]/2}, Prolog->Array[z,4], PlotRange->{Automatic,{-1,5}}, Axes->None] ]; \ \>", "NumberedFigure", PageWidth->PaperWidth], Cell["\<\ The next animation illustrates properties of the normal line to a cycloid.\ \>", "Text", PageWidth->PaperWidth], Cell["\<\ Module[{\[Alpha],z,zz}, \[Alpha]= cycloid[1,1]; z[1,c_]= {AbsolutePointSize[5], Point[{c,1}], Point[\[Alpha][c]]}; z[2,c_]= {Hue[.6], Circle[{c,1},1]}; z[3,c_]= Table[Line[{\[Alpha][c],{c,n}}], {n,0,2}]; z[4,c_]= Line[{{c,0},{c,2}}]; zz[c_]:= ParametricPlot[\[Alpha][t], {t,-\[Pi], \[Pi]}, \ Epilog->Table[z[i,c],{i,4}], Ticks->None]; zzz:= Do[zz[c], {c,-2,1,3}]; Show[GraphicsArray[{{zz[-2]},{zz[-.5]},{zz[1]},{zz[2]}}]] ];\ \>", "NumberedFigure", PageWidth->PaperWidth] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["N2.2", "Section", PageWidth->PaperWidth], Cell[TextData[{ "In the following definition, the parameter ", StyleBox["a ", FontFamily->"Verdana"], "may seem redundant since it can in theory be added later, but this way it \ is easier to handle lemniscates of different sizes. The same remark applies \ to many other curves, and such a parameter was already used in ", StyleBox["N1.3 to ", FontFamily->"Verdana"], "define ", StyleBox["circle", FontFamily->"Verdana"], " and ", StyleBox["parabola", FontFamily->"Verdana"], "." }], "Text", PageWidth->PaperWidth], Cell["lemniscate[a_][t_]:= a{Cos[t], Sin[t]Cos[t]}/(1+Sin[t]^2)", "Definition", PageWidth->PaperWidth], Cell["lemniscate[a][t] - a lemniscate[1][t]", "Input", PageWidth->PaperWidth], Cell["\<\ Inclusion of a parameter in the definition is convenient, since the following \ syntax fails:\ \>", "Text", PageWidth->PaperWidth], Cell["\<\ lem:= a lemniscate[1] lem[t]\ \>", "Input", PageWidth->PaperWidth], Cell["\<\ The next calculation verifies the relationship between implicit and \ parametric forms of the lemniscate.\ \>", "Text", PageWidth->PaperWidth], Cell["\<\ t1= (x+f)^2+y^2 t2= t1/.f->-f t3= t1 t2 - f^4\ \>", "Input", PageWidth->PaperWidth], Cell["\<\ t4= Expand[t3]/.{y->x Sin[t], f->a Sqrt[2]} Simplify[Solve[t4\[Equal]0, x]]//PowerExpand\ \>", "Input", PageWidth->PaperWidth], Cell[TextData[{ "Next, we plot multiple lemniscates simultaneously by inserting a ", StyleBox["Table", FontFamily->"Verdana"], " command inside ", StyleBox["ParametricPlot", FontFamily->"Verdana"], ". In this case, our default option", StyleBox[" Compiled->False", FontFamily->"Verdana"], " will not prevent compilation problems that result from the multiple \ plotting, and it is essential to add ", StyleBox["Evaluate", FontFamily->"Verdana"], ". The function ", StyleBox["f ", FontFamily->"Verdana"], "has been defined to make it easier to ensure that each lemniscate passes \ through the previous one's foci. " }], "Text", PageWidth->PaperWidth], Cell["\<\ Module[{f,y,z}, f[a_]= (1/Sqrt[2])^a; y[a_]= lemniscate[f[a]]; z[a_]= {PointSize[.02],Point[{-f[a],0}],Point[{f[a],0}]}; ParametricPlot[Table[y[a][t],{a,0,3}]//Evaluate, {t,0,2\[Pi]}, Epilog->Array[z,4], Ticks->None, PlotRange->{{-1.2,1.2},{-.5,.5}}, ImageSize->400] ]\ \>", "NumberedFigure", PageWidth->PaperWidth], Cell["The curvature of a lemniscate is described by", "Text", PageWidth->PaperWidth], Cell["\<\ Module[{\[Beta],k}, \[Beta]:= lemniscate[1]; k[t_]= FullSimplify[\[Kappa]2[\[Beta]][t]]; Print[k[t]]; Plot[k[t], {t,0,2\[Pi]}, AspectRatio->Automatic, PlotStyle->Hue[.7]] ]\ \>", "Input", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell["N2.2.1", "Subsection", PageWidth->PaperWidth], Cell[TextData[{ "We can also adopt the same technique as for Figure 2.2 to plot a \ lemniscate and its curvature with common ", StyleBox["x", FontFamily->"Verdana"], "-coordinate." }], "Text", PageWidth->PaperWidth], Cell["\<\ Module[{\[Beta],f,g,y}, \[Beta]= lemniscate[1]; f[t_]= \[Beta][t][[1]]; g[x_]= ArcCos[(-1+Sqrt[1+8x^2])/(2x)]; y[1]= ParametricPlot[\[Beta][t],{t,0,2\[Pi]}, DisplayFunction->Identity]; y[2]= Plot[\[Kappa]2[\[Beta]][g[x]],{x,-1,1}, PlotStyle->{Hue[.7],Thickness[.01]}, DisplayFunction->Identity]; Show[y[1],y[2], AspectRatio->1, PlotRange->{{-1.2,1.2},{-3,3}}, DisplayFunction->$DisplayFunction]; f[g[x]]//Simplify ]\ \>", "NumberedFigure", PageWidth->PaperWidth] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["N2.3", "Section", PageWidth->PaperWidth], Cell[TextData[{ "The definition of the cardioid employs the same parameter-handling device \ explained at the start of ", StyleBox["N2.2.", FontFamily->"Verdana"] }], "Text", PageWidth->PaperWidth], Cell["cardioid[a_][t_]:= 2a(1+Cos[t]){Cos[t], Sin[t]}", "Definition", PageWidth->PaperWidth], Cell[TextData[{ "In the following multiple plot, colors have been added by inserting a ", StyleBox["Table", FontFamily->"Verdana"], " with a different ", StyleBox["Hue", FontFamily->"Verdana"], " for each curve. The coloring is determined by ", StyleBox["n,r,p", FontFamily->"Verdana"], ". The number of plots is ", StyleBox["n, ", FontFamily->"Verdana"], "while ", StyleBox["r ", FontFamily->"Verdana"], "is the number of times the color scheme is repeated, and ", StyleBox["p ", FontFamily->"Verdana"], "the fraction of the spectrum used. The values chosen give a smooth \ 3-dimensional maze effect. We prefer to append the ", StyleBox["Evaluate", FontFamily->"Verdana"], " command on the right as it is helpful to think of it as an \"afterthought\ \" rather than part of the mathematics." }], "Text", PageWidth->PaperWidth], Cell["\<\ Module[{n,r,p,cc}, n= 80; r= 5; p= 1; cc[t_]:= Table[cardioid[a][t], {a,n}]; ParametricPlot[cc[t]//Evaluate, {t,0,2\[Pi]}, PlotStyle->Table[Hue[a p r/n],{a, n/r}], Axes->None] ]\ \>", "NumberedFigure", PageWidth->PaperWidth], Cell["The curvature of the cardioid is given by", "Text", PageWidth->PaperWidth], Cell["\<\ \[Gamma]:=cardioid[1] Simplify[\[Kappa]2[\[Gamma]][t]]\ \>", "Input", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell["N2.3.1", "Subsection", PageWidth->PaperWidth], Cell[TextData[{ "A cardioid and its curvature as a function of the x-coordinate. In this \ case, the inverse function has been solved by hand, and the value of the \ curvature is again represented by the y-coordinate of the point vertically \ above or below the point on the cardioid. Two curvature plots are needed to \ cater for the respective branches of the cardioid between ", StyleBox["x=-1/2", FontFamily->"Verdana"], " and", StyleBox[" x=0, whilst t", FontFamily->"Verdana"], "he situation resembles that of a circle with constant curvarture for ", StyleBox["x>2. ", FontFamily->"Verdana"], "Notice how scale on the y-axis has been placed manually, to keep the \ numbers away from the curve." }], "Text", PageWidth->PaperWidth], Cell["\<\ Module[{\[Gamma],g1,g2,y}, th:= Thickness[.015]; \[Gamma]= cardioid[1]; g1[x_]= ArcCos[(-1-Sqrt[1+2x])/2]; g2[x_]= ArcCos[(-1+Sqrt[1+2x])/2]; y[1]= ParametricPlot[\[Gamma][t],{t,0,2\[Pi]}, DisplayFunction->Identity]; y[2]= Plot[\[Kappa]2[\[Gamma]][g1[x]],{x,-.5,0}, PlotStyle->{Hue[.6], th}, \ DisplayFunction->Identity]; y[3]= Plot[\[Kappa]2[\[Gamma]][g2[x]],{x,-.5,4}, PlotStyle->{Hue[.8], th}, \ DisplayFunction->Identity]; Show[Array[y,3], AspectRatio->Automatic, PlotRange->{{-2,4.5},{-3,7}}, Ticks->{None,{{4,4,{0,.05}},{6,6,{0,.05}}}}, DisplayFunction->$DisplayFunction] ];\ \>", "NumberedFigure", PageWidth->PaperWidth] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["N2.4", "Section", PageWidth->PaperWidth], Cell["\<\ We list some of the standard conic definitions that form part of Notebook 2's \ initialization data:\ \>", "Text", PageWidth->PaperWidth], Cell["\<\ parabola[a_][t_]:= a{2t, t^2} ellipse[a_,b_][t_]:= {a Cos[t], b Sin[t]} hyperbola[a_,b_][t_]:= {a Cosh[t], b Sinh[t]}\ \>", "Definition", PageWidth->PaperWidth], Cell["\<\ The catenary is essentially the graph of the hyperbolic cosine, though with \ the first coordinate expressed as a function of the second. It can be \ rotated in one of two ways:\ \>", "Text", PageWidth->PaperWidth], Cell["\<\ catenary[a_][t_]:= {a Cosh[t/a], t} revcatenary[a_][t_]:= J[catenary[a][-t]]\ \>", "Definition", PageWidth->PaperWidth], Cell["\<\ revcatenary[a][t] Reverse[catenary[a][t]]\ \>", "Input", PageWidth->PaperWidth], Cell["\<\ We now load a package to draw the arrows in Figure 2.11, relating to the \ catenary.\ \>", "Text", PageWidth->PaperWidth], Cell["<PaperWidth], Cell[TextData[{ "The following command is used for placing text in Figure 2.11. Care has \ to be taken to choose the font size carefully, as it does not scale if the \ image is resized, though the size of each graphics window can be adjusted \ manually or using the command ", StyleBox["ImageSize. ", FontFamily->"Verdana"], "For our purposes, 18pt is a good compromise though we shall modify the \ command", StyleBox[" tx ", FontFamily->"Verdana"], "in later notebooks." }], "Text", PageWidth->PaperWidth], Cell["tx[A_,v_]:= Text[FontForm[A, {\"Helvetica\",18}], v]", "Definition", PageWidth->PaperWidth], Cell["\<\ Module[{r,ar,y}, r= revcatenary[2]; ar[t_,v_]:= Arrow[r[t], r[t]+v]; y[1]= {Thickness[.007], Hue[.7], ar[0,-2r'[0]], ar[2,2r'[2]], ar[2, 2{r'[2][[1]],0}]}; y[2]= {PointSize[.02], Point[r[0]], Point[r[2]]}; y[3]= Circle[r[2], 1.3, {0,Apply[ArcTan,r'[2]]}]; y[4]= {tx[\"\[Theta]\",{2.8,3.5}], tx[\"T\",{-2.3,2.2}], tx[\"U\",{4.3,5.7}], tx[\"p\",{.3,2.4}], tx[\"q\",{1.6,3.2}]}; ParametricPlot[r[t], {t,-2,4}, PlotRange->{{-3,5},{1,7}}, Ticks->None, Epilog->Array[y,4], ImageSize->400] ]\ \>", "NumberedFigure", PageWidth->PaperWidth], Cell["\<\ Here is a comparison of the curvatures of the parabola and catenary:\ \>", "Text", PageWidth->PaperWidth], Cell["\<\ \[Kappa]2[parabola[a]][0] \[Kappa]2[catenary[a]][0]//PowerExpand\ \>", "Input", PageWidth->PaperWidth], Cell[TextData[{ "This enables us to plot the catenary and parabola with equal curvature at \ their vertices. The parabola has", StyleBox[" Hue[.65] ", FontFamily->"Verdana"], "and is therefore blue!" }], "Text", PageWidth->PaperWidth], Cell["\<\ ps[h_]:= {Thickness[.015], Hue[h]} ParametricPlot[Evaluate[{parabola[1][t]+{0,2}, revcatenary[2][t]}], {t,-50,50}, PlotRange->{{-4,4},{0,8}}, PlotStyle->{ps[.65],ps[.05]}]\ \>", "NumberedFigure", PageWidth->PaperWidth], Cell["\<\ f:= catenary[a] ff:= PowerExpand[L[f'[#]]] & Integrate[ff[tt],{tt,0,t}] f[t/.Solve[s\[Equal]%, t]]\ \>", "Input", PageWidth->PaperWidth] }, Closed]], Cell[CellGroupData[{ Cell["N2.5", "Section", PageWidth->PaperWidth], Cell[TextData[{ "In the next definition, the variable ", StyleBox["t ", FontFamily->"Verdana"], "can assume any real value:" }], "Text", PageWidth->PaperWidth], Cell["cissoid[a_][t_]:=2a{t^2, t^3}/(1+t^2)", "Definition", PageWidth->PaperWidth], Cell["\<\ Clear[a] \[Gamma]:= cissoid[a] \[Gamma][t]\ \>", "Input", PageWidth->PaperWidth], Cell["\<\ \[Gamma]'[t] %//Simplify\ \>", "Input", PageWidth->PaperWidth], Cell["ParametricPlot[cissoid[1][t], {t,-2,2}];", "Input", PageWidth->PaperWidth], Cell["\<\ y[1]:= {Thickness[.01], Line[{{-1,0},{3,0}}]} y[2]:= {PointSize[.025], Point[{-1,0}], Point[{0,0}], Point[{1,0}], Point[{2,0}]} y[3]:= {tx[\"A\",{-1.2,.2}], tx[\"Q\",{-.2,.2}], tx[\"P\",{.9,.2}], tx[\"R\",{1.8,.2}], tx[\"\[Xi]\",{.3,1.5}], tx[\"\[Eta]\",{2,1.1}]} ParametricPlot[{Reverse[parabola[1][t]], Reverse[parabola[1][t]]+{2,0}}, {t,-1,1}, PlotRange->{{-1.5,2.5},{-2,2}}, Epilog->Array[y,3], Axes->None, ImageSize->300]\ \>", "NumberedFigure", PageWidth->PaperWidth], Cell["\<\ The following computation verifies the implicit form of the cissoid.\ \>", "Text", PageWidth->PaperWidth], Cell["\<\ rr:= x^2+y^2 Solve[4a^2 - (2a/x-1)^2 rr \[Equal] (2a y/x)^2 - rr, y] Simplify[y^2/.%[[1]]]\ \>", "Input", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell["N2.5.1", "Subsection", PageWidth->PaperWidth], Cell["Here is the figure illustrating the cissoid's definition:", "Text", PageWidth->PaperWidth], Cell["\<\ Module[{\[Alpha],f}, \[Alpha]= cissoid[1]; y[1]= {Hue[.7],Circle[{1,0},1]}; y[2]= {Line[{{2,3.0},{2,-3.0}}], Line[{{0,0},1.45 \[Alpha][1.5]}], Line[{{2,-3.3},{2,3.3}}], Line[{{0,0},{2,0}}], Line[{{2,0},0.43\[Alpha][1.5]}]}; y[3]= {tx[\"A\",{-.3,0}], tx[\"Q\",{.3,1.1}], tx[\"P\",{1.1,2.2}], tx[\"R\",{2.4,3.1}], tx[\"S\",{2.4,0}]}; y[4]= {AbsolutePointSize[5], Point[{0,0}], Point[{2,0}], Point[0.43\[Alpha][1.5]], Point[\[Alpha][1.5]], Point[1.45 \ \[Alpha][1.5]]}; ParametricPlot[\[Alpha][t], {t,-2,2}, Axes->None, PlotRange->{{-1,3},{-3.5,3.5}}, Epilog->Array[y,4], ImageSize->250] ]\ \>", "NumberedFigure", PageWidth->PaperWidth], Cell[TextData[{ "In the simplification of the cissoid's curvature,", StyleBox[" PowerExpand", FontFamily->"Verdana"], " incorrectly has", StyleBox[" t", FontFamily->"Verdana"], " rather than", StyleBox[" |t|", FontFamily->"Verdana"], ". The graph is correct." }], "Text", PageWidth->PaperWidth], Cell["\<\ k:= \[Kappa]2[cissoid[1]] Simplify[k[t]] PowerExpand[%] Plot[k[t], {t,-1,1}, PlotStyle->{Thickness[.01],Hue[.7]}, AspectRatio->1]\ \>", "NumberedFigure", PageWidth->PaperWidth] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["N2.6", "Section", PageWidth->PaperWidth], Cell[TextData[{ "By contrast to the cissoid in ", StyleBox["N2.5,", FontFamily->"Verdana"], " the variable", StyleBox[" t ", FontFamily->"Verdana"], "is now restricted to the open interval", StyleBox[" (0,\[Pi])", FontFamily->"Verdana"], ":" }], "Text", PageWidth->PaperWidth], Cell["tractrix[a_][t_]:= a{Sin[t], Cos[t]+Log[Tan[t/2]]}", "Definition", PageWidth->PaperWidth], Cell["\<\ \[Tau]:= tractrix[1] ParametricPlot[\[Tau][t], {t,0,\[Pi]}, PlotRange->{{-1,2},{-2,2}}];\ \>", "Input", PageWidth->PaperWidth], Cell["\<\ The curvature of the tractrix should be everywhere negative, and the graph is \ correct.\ \>", "Text", PageWidth->PaperWidth], Cell["\<\ k:=\[Kappa]2[\[Tau]] FullSimplify[k[t]] PowerExpand[%] Plot[-Abs[k[t]], {t,0,\[Pi]}, PlotStyle->Hue[.7]]\ \>", "Input", PageWidth->PaperWidth], Cell["\<\ The next command draws a tangent line of length s to a curve at an assigned \ point, and is used in Figure 2.13.\ \>", "Text", PageWidth->PaperWidth], Cell["\<\ tangentline[\[Alpha]_][s_,t_]:= Line[{\[Alpha][t], \[Alpha][t] + s \ \[Alpha]'[t]/L[\[Alpha]'[t]]}]\ \>", "Definition", PageWidth->PaperWidth], Cell["\<\ Module[{\[Tau],a,h,p,q,y}, \[Tau]= tractrix[1]; a=.6\[Pi]; h=.05\[Pi]; p[i_]= Point[\[Tau][a+i h]]; q[i_]= tangentline[\[Tau]][1,a+i h]; y[1]= {Hue[.7], Array[q,6]}; y[2]= {Hue[.7], PointSize[.02], Array[p,6]}; ParametricPlot[\[Tau][t], {t,.5,\[Pi]}, Epilog->Array[y,2], PlotRange->{{-.5,1.5},{-.5,2}}, Ticks->None] ]\ \>", "NumberedFigure", PageWidth->PaperWidth], Cell["\<\ The next computation helps one to find a valid parametrization of the \ tractrix.\ \>", "Text"], Cell["\<\ DSolve[y'[x]\[Equal]-Sqrt[a^2-x^2]/x, y[x], x]\ \>", "Input", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell["N2.6.1", "Subsection", PageWidth->PaperWidth], Cell["\<\ Module[{c,\[Tau]}, c= .8\[Pi]; \[Tau]= tractrix[1]; y[1]= {PointSize[.03], Point[\[Tau][c]]}; y[2]= {tangentline[\[Tau]][1,c], Line[{\[Tau][c],{0,\[Tau][c][[2]]}}]}; y[3]= {tx[\"(a,0)\",{1,-.1}], tx[\"(x,y)\",\[Tau][c]+{.1,.07}], tx[\"x\",\[Tau][c+.15]/2], tx[\"a\",\[Tau][c]/2+{-.1,.6}]}; ParametricPlot[\[Tau][t],{t,\[Pi]/2,\[Pi]}, Ticks->None, PlotRange->{{-.2,1.2},{-.2,1.5}}, Epilog->Array[y,3], ImageSize->300] ];\ \>", "NumberedFigure", PageWidth->PaperWidth], Cell["\<\ Module[{\[Gamma],f,g,y}, \[Tau]= tractrix[1]; g[x_]= ArcSin[x]; y[1]=ParametricPlot[\[Tau][t],{t,0,\[Pi]}, DisplayFunction->Identity]; y[2]=Plot[\[Kappa]2[\[Tau]][g[x]],{x,0,1}, PlotStyle->{Hue[.5],Thickness[.015]}, DisplayFunction->Identity]; Show[Array[y,2], AspectRatio->1.5, PlotRange->{{-.4,1.5},{-2.2,2.2}}, Ticks->{None,Automatic}, DisplayFunction->$DisplayFunction, ImageSize->250] ];\ \>", "NumberedFigure", PageWidth->PaperWidth], Cell["D[a(Cos[t]+Log[Tan[t/2]]), t]", "Input", PageWidth->PaperWidth] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["N2.7", "Section", PageWidth->PaperWidth], Cell["Here are the definitions needed to plot clothoids:", "Text", PageWidth->PaperWidth], Cell["\<\ clothoidprime[n_,a_][t_]:= a{Sin[t^(n+1)/(n+1)], Cos[t^(n+1)/(n+1)]} clothoid[n_,a_][t_]:= Integrate[clothoidprime[n,a][tt], {tt,0,t}]\ \>", "Definition", PageWidth->PaperWidth], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " has internal commands ", StyleBox["FresnelS", FontFamily->"Verdana"], " and", StyleBox[" FresnelC", FontFamily->"Verdana"], " that define the Fresnel integrals. In fact, ", StyleBox["clothoid[1,a]", FontFamily->"Verdana"], " will be converted automatically into an equivalent expression using these \ two functions." }], "Text", PageWidth->PaperWidth], Cell["\<\ s:= Sqrt[\[Pi]] f[t_]:= a s{FresnelS[t/s], FresnelC[t/s]} f'[t] Clear[s,f]\ \>", "Input", PageWidth->PaperWidth], Cell["ParametricPlot[clothoid[1,1][t], {t,-10,10}, PlotPoints->10]", "Input", PageWidth->PaperWidth], Cell["Integrate[clothoidprime[1,1][tt], {tt,0,Infinity}]", "Input", PageWidth->PaperWidth], Cell["\<\ The ouput gives the coordinates around which the top right end of the first \ clothoid curls. Here is another example:\ \>", "Text", PageWidth->PaperWidth], Cell["\<\ ParametricPlot[Evaluate[clothoid[2,3][t]], {t,-4,4}, PlotPoints->2]\ \>", "Input", PageWidth->PaperWidth], Cell["Here is a verification of a clothoid's curvature:", "Text", PageWidth->PaperWidth], Cell["\<\ \[Kappa]2[clothoid[1,a]][t] PowerExpand[Simplify[%]]\ \>", "Input", PageWidth->PaperWidth], Cell[TextData[{ "The command", StyleBox[" clothoid", FontFamily->"Verdana"], " can be extremely slow because ", StyleBox["Mathematica", FontSlant->"Italic"], " uses numerical integration. In the cell below, we use a different \ definition of the clothoid curve which calls ", StyleBox["NDSolve", FontFamily->"Verdana"], ", so that the curve is plotted much more quickly." }], "Text", PageWidth->PaperWidth], Cell["\<\ clothoidnds[min_,max_][n_,a_][s_]:= Module[{so,ss}, so=NDSolve[{ x'[ss]\[Equal]clothoidprime[n,a][ss][[1]], y'[ss]\[Equal]clothoidprime[n,a][ss][[2]], x[0]\[Equal]0,y[0]\[Equal]0}, {x,y}, {ss, min, max}]; First[{x[s],y[s]} /.so] ]\ \>", "Miniprogram"], Cell[TextData[{ "Since numerical solution of a differential equation requires the \ specification of an interval, ", StyleBox["clothoidnds", FontFamily->"Verdana"], " will only be defined on the interval ", StyleBox["(min,max)", FontFamily->"Verdana"], ". We use this method to draw Figure 2.19, and then show what goes wrong \ when the solution is defined on too small an interval." }], "Text", PageWidth->PaperWidth], Cell["\<\ m:= 6 r:= 1.5 th:= Thickness[.01] cc[t_]:= {clothoidnds[-m,m][1,1][t], clothoidnds[-m,m][2,5/8][t]} zz:= ParametricPlot[cc[t]//Evaluate, {t,-6,6}, PlotRange->{{-r,r},{-r,r}}, PlotStyle->{{Hue[0],th},{Hue[.65],th}}, AspectRatio->Automatic, ImageSize->400] zz\ \>", "NumberedFigure", PageWidth->PaperWidth], Cell["\<\ m:= 4 zz\ \>", "Input", PageWidth->PaperWidth] }, Closed]], Cell[CellGroupData[{ Cell["N2.8", "Section", PageWidth->PaperWidth], Cell[TextData[{ "Special packages are used to produce the sketch for Figure 2.20. The \ first one requires no comment, though we shall have occasion to use ", StyleBox["Rotations", FontFamily->"Verdana"], " in a subsequent notebook." }], "Text", PageWidth->PaperWidth], Cell[TextData[{ "<PaperWidth], Cell["\<\ The following definition is copied from the text, but we must not forget to \ manufacture the \"boat.\" \ \>", "Text", PageWidth->PaperWidth], Cell["\<\ linearpursuit[a_,k_][t_]:= a k/(k^2-1) + k(a - t)^(1+1/k)/(2a^(1/k)(1+k)) - k a^(1/k)(a-t)^(1-1/k)/(2(k-1)) \ \>", "Definition", PageWidth->PaperWidth], Cell["\<\ boat[c_,d_,e_,f_]:= {{-c,d},{0,d-f},{1,d},{1,d+e},{0,d+e+f},{-1,d+e}}\ \>", "Definition", PageWidth->PaperWidth], Cell["\<\ Module[{b,z}, b= boat[1,1,4,1.5]/40; z[1]= {Dashing[{.02}], Thickness[.007], Line[{{1,-1},{1,2}}]}; z[2]= {Polygon[Map[{1,.4}+#&, b]], Polygon[Map[{.6,.2}+Rotate2D[#,.25\[Pi]]&,b]]}; z[3]= {tx[\"\[Beta]\",{1.1,.45}], tx[\"\[Alpha]\",{.5,.25}]}; Plot[linearpursuit[1,1.2][t], {t,0,1}, Axes->None, PlotStyle->Thickness[.007], PlotRange->{{-.2,1.3},{-.2,1.3}}, Epilog->Array[z,3], AspectRatio->1] ]\ \>", "NumberedFigure", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell["N2.8.1", "Subsection", PageWidth->PaperWidth], Cell["\<\ Clear[k] Module[{f,e}, f[k_,t_]= linearpursuit[1,k][t]; e= {Line[{{1,-0.2},{1,2.4}}], PointSize[.02], Table[Point[{1, f[k,1]}], {k,2,5}]}; Plot[Table[f[6-i,t],{i,4}]//Evaluate, {t,0,1}, PlotStyle->Table[{Hue[2i/3], Dashing[{i/150}]},{i,4}], Epilog->e, PlotRange->All, AxesLabel->{StyleForm[\"t\", SubSection], StyleForm[\"y(t)\", SubSection]}, PlotLegend->{\"k=5\",\"k=4\",\"k=3\",\"k=2\"}, LegendPosition->{-.5,0}, ImageSize->600] ];\ \>", "NumberedFigure", PageWidth->PaperWidth] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["N2.9", "Section", PageWidth->PaperWidth], Cell["\<\ A triple of cycloids is achieved by setting up a function whose values are \ plots whose shape and color depends on the given parameter. Another way of \ plotting multiple curves is given in Section 2.2.\ \>", "Text", PageWidth->PaperWidth], Cell["\<\ Module[{y}, y[a_]:=ParametricPlot[cycloid[1,a][t], {t,-\[Pi],3\[Pi]}, PlotStyle->{Hue[1.5/a],Thickness[.008]}, DisplayFunction->Identity]; Show[y[1], y[2], y[.5], DisplayFunction->$DisplayFunction] ]\ \>", "Input", PageWidth->PaperWidth], Cell["\<\ We finish with some further definitions of curves cited in the Exercises.\ \>", "Text", PageWidth->PaperWidth], Cell["\<\ deltoid[a_][t_]:= {2a Cos[t](1+Cos[t])-a, 2a Sin[t](1-Cos[t])} lissajous[n_,d_,a_,b_][t_]:= {a Sin[n t+d], b Sin[t]} limacon[a_,b_][t_]:= (2a Cos[t]+b){Cos[t], Sin[t]} agnesi[a][t_]:= 2a{Tan[t], Cos[t]^2}\ \>", "Definition", PageWidth->PaperWidth] }, Closed]] }, Open ]] }, FrontEndVersion->"5.2 for Microsoft Windows", ScreenRectangle->{{0, 1280}, {0, 717}}, AutoGeneratedPackage->None, PrintingStyleEnvironment->"Working", WindowToolbars->"RulerBar", WindowSize->{1267, 683}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, PrintingCopies->1, PrintingStartingPageNumber->60, PrintingPageRange->{Automatic, Automatic}, PageHeaders->{{Cell[ TextData[ { CounterBox[ "Page"]}], FontSlant -> Italic], Inherited, Cell[ "NOTEBOOK 2", FontSlant -> Italic]}, { Cell[ "NOTEBOOK 2", 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"->Left, "Magnification"->0.75}, CellLabelAutoDelete->True, 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", 0}, PageBreakBelow->False, CellFrameMargins->{{100, 4}, {8, 10}}, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-0.36, -0.1}, {0, -0}}, BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica"}, TextAlignment->Left, LineSpacing->{1, 13}, CounterIncrements->"Title", CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}}, FontFamily->"Verdana", FontSize->36, FontColor->GrayLevel[0]], Cell[StyleData["Title", "Printout"], PageWidth->PaperWidth, FontSize->40] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subtitle"], PageWidth->PaperWidth, CellDingbat->"\[FilledSquare]", CellMargins->{{54, 25}, {2, 30}}, CellGroupingRules->{"TitleGrouping", 10}, PageBreakBelow->False, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-0.36, -0.1}, {0, -0}}, BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica"}, TextAlignment->Left, CounterIncrements->"Subtitle", CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}}, FontSize->18, FontWeight->"Bold", FontSlant->"Plain", FontColor->GrayLevel[0]], Cell[StyleData["Subtitle", "Printout"], PageWidth->PaperWidth, FontSize->18, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsubtitle"], PageWidth->PaperWidth, CellMargins->{{54, 25}, {30, 10}}, CellGroupingRules->{"TitleGrouping", 20}, PageBreakBelow->False, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-0.36, -0.1}, {0, -0}}, BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica"}, TextAlignment->Center, CounterIncrements->"Subsubtitle", CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}}, FontSize->14, FontSlant->"Italic"], Cell[StyleData["Subsubtitle", "Printout"], PageWidth->PaperWidth, FontSize->12] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Section"], PageWidth->PaperWidth, CellDingbat->"\[FilledSquare]", CellMargins->{{54, 25}, {2, 30}}, CellGroupingRules->{"SectionGrouping", 30}, PageBreakBelow->False, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-0.36, -0.1}, {0, -0}}, BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica"}, TextAlignment->Left, LineSpacing->{1, 9}, CounterIncrements->"Section", CounterAssignments->{{"Subsection", 0}, {"Subsubsection", 0}}, FontFamily->"Verdana", FontSize->24, FontWeight->"Plain", FontColor->GrayLevel[0]], Cell[StyleData["Section", "Printout"], PageWidth->PaperWidth, FontSize->12, FontColor->GrayLevel[0]] }, Open ]], Cell[CellGroupData[{ Cell[StyleData["Subsection"], PageWidth->PaperWidth, CellDingbat->"\[FilledSmallSquare]", CellMargins->{{54, 25}, {2, 12}}, CellGroupingRules->{"SectionGrouping", 40}, PageBreakBelow->False, CellFrameMargins->{{2, 12}, {0, 12}}, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-0.36, -0.1}, {0, -0}}, BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica"}, CounterIncrements->"Subsection", CounterAssignments->{{"Subsubsection", 0}}, FontFamily->"Verdana", FontSize->16, FontWeight->"Plain", FontSlant->"Plain", FontColor->GrayLevel[0.4]], Cell[StyleData["Subsection", "Printout"], PageWidth->PaperWidth, FontSize->10] }, Open ]], Cell[CellGroupData[{ Cell[StyleData["Subsubsection"], PageWidth->PaperWidth, CellMargins->{{54, 25}, {2, 12}}, CellGroupingRules->{"SectionGrouping", 50}, PageBreakBelow->False, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-0.36, -0.1}, {0, -0}}, BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica"}, CounterIncrements->"Subsubsection", FontSlant->"Italic", FontColor->RGBColor[0.771908, 0.399634, 0.262867]], Cell[StyleData["Subsubsection", "Printout"], PageWidth->PaperWidth, FontSize->9, FontColor->GrayLevel[0]] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["Styles for Body Text", "Section", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell[StyleData["Text"], PageWidth->PaperWidth, CellMargins->{{54, 25}, {6, 6}}, CellFrameMargins->{{4, 10}, {6, 0}}, CellFrameLabelMargins->4, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-0.36, -0.1}, {0, -0}}, BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica"}, TextJustification->1, Hyphenation->True, LineSpacing->{1.25, 0}, CounterIncrements->"Text", FontFamily->"Georgia", FontSize->12, FontColor->RGBColor[0, 0.2, 0]], Cell[StyleData["Text", "Printout"], PageWidth->PaperWidth, LineSpacing->{1.5, 0}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["SmallText"], PageWidth->PaperWidth, CellFrame->{{0, 0}, {0, 0.25}}, CellMargins->{{72, 44}, {6, 6}}, CellFrameMargins->2, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-0.36, -0.1}, {0, -0}}, BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica"}, TextJustification->1, Hyphenation->True, LineSpacing->{1, 2}, CounterIncrements->"SmallText", FontSize->10], Cell[StyleData["SmallText", "Printout"], PageWidth->PaperWidth, LineSpacing->{1, 3}, FontSize->9] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["Styles for Input/Output", "Section", PageWidth->PaperWidth], Cell["\<\ The cells in this section define styles used for input and output to the \ kernel. Be careful when modifying, renaming, or removing these styles, \ because the front end associates special meanings with these style names.\ \>", "Text", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell[StyleData["Input"], PageWidth->PaperWidth, CellFrame->True, CellMargins->{{72, 25}, {8, 12}}, Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, CellLabelMargins->{{33, Inherited}, {Inherited, Inherited}}, TextAlignment->Left, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, LanguageCategory->"Formula", ShowStringCharacters->True, NumberMarks->True, LinebreakAdjustments->{0.85, 2, 10, 0, 1}, CounterIncrements->"Input", FontFamily->"Verdana", FontSize->12, FontWeight->"Plain", Background->RGBColor[1, 1, 0.700008], FontVariations->{"CompatibilityType"->0}], Cell[StyleData["Input", "Printout"], PageWidth->PaperWidth, LinebreakAdjustments->{0.85, 2, 10, 1, 1}, FontSize->9.5] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Output"], PageWidth->PaperWidth, CellMargins->{{72, 25}, {12, 4}}, CellEditDuplicate->True, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, CellLabelMargins->{{33, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, TextAlignment->Left, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, LanguageCategory->"Formula", FormatType->InputForm, CounterIncrements->"Output", FontFamily->"Courier", FontSize->12, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["Output", "Printout"], PageWidth->PaperWidth, FontSize->9.5] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Message"], PageWidth->PaperWidth, CellDingbat->"\[LongDash]", CellMargins->{{72, 25}, {12, 4}}, CellGroupingRules->"OutputGrouping", PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{33, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Message", StyleMenuListing->None, FontFamily->"Helvetica", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontColor->RGBColor[1, 0, 0], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], Cell[StyleData["Message", "Printout"], PageWidth->PaperWidth, FontSize->8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Print"], PageWidth->PaperWidth, CellMargins->{{72, 25}, {12, 4}}, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{33, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, TextAlignment->Left, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Print", StyleMenuListing->None], Cell[StyleData["Print", "Printout"], PageWidth->PaperWidth, FontSize->9.5] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Graphics"], PageWidth->PaperWidth, CellMargins->{{72, 25}, {12, 4}}, CellGroupingRules->"GraphicsGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, FormatType->InputForm, CounterIncrements->"Graphics", ImageMargins->{{35, Inherited}, {Inherited, 0}}, StyleMenuListing->None, FontSize->10], Cell[StyleData["Graphics", "Printout"], PageWidth->PaperWidth, FontSize->8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["CellLabel"], PageWidth->PaperWidth, CellLabelAutoDelete->False, StyleMenuListing->None, FontSize->10, FontSlant->"Italic", FontColor->RGBColor[0.551492, 0.231144, 0.313466]], Cell[StyleData["CellLabel", "Printout"], PageWidth->PaperWidth, FontSize->8, FontColor->GrayLevel[0]] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["Hyperlink Styles", "Section", PageWidth->PaperWidth], Cell["\<\ The cells below define styles useful for making hypertext ButtonBoxes. The \ \"Hyperlink\" style is for links within the same Notebook, or between \ Notebooks.\ \>", "Text", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell[StyleData["Hyperlink"], PageWidth->PaperWidth, StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonNote->ButtonData}], Cell[StyleData["Hyperlink", "Printout"], PageWidth->PaperWidth, FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell["\<\ The following styles are for linking automatically to the on-line help \ system.\ \>", "Text", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell[StyleData["MainBookLink"], PageWidth->PaperWidth, StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "MainBook", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["MainBookLink", "Printout"], PageWidth->PaperWidth, FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["AddOnsLink"], PageWidth->PaperWidth, StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "AddOns", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["AddOnsLink", "Printout"], PageWidth->PaperWidth, FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["RefGuideLink"], PageWidth->PaperWidth, StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "RefGuide", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["RefGuideLink", "Printout"], PageWidth->PaperWidth, FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["GettingStartedLink"], PageWidth->PaperWidth, StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "GettingStarted", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["GettingStartedLink", "Printout"], PageWidth->PaperWidth, FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["OtherInformationLink"], PageWidth->PaperWidth, StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "OtherInformation", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["OtherInformationLink", "Printout"], PageWidth->PaperWidth, FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Palette Styles", "Section", PageWidth->PaperWidth], Cell["\<\ The cells below define styles that define standard ButtonFunctions, for use \ in palette buttons.\ \>", "Text", PageWidth->PaperWidth], Cell[StyleData["Paste"], PageWidth->PaperWidth, StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, After]}]&)}], Cell[StyleData["Evaluate"], PageWidth->PaperWidth, StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], SelectionEvaluate[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["EvaluateCell"], PageWidth->PaperWidth, StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionMove[ FrontEnd`InputNotebook[ ], All, Cell, 1], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["CopyEvaluate"], PageWidth->PaperWidth, StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[ ], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluate[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["CopyEvaluateCell"], PageWidth->PaperWidth, StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[ ], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Automatic Numbering", "Section", PageWidth->PaperWidth], Cell["\<\ The following styles are useful for numbered equations, figures, etc. They \ automatically give the cell a FrameLabel containing a reference to a \ particular counter, and also increment that counter.\ \>", "Text", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell[StyleData["NumberedEquation"], PageWidth->PaperWidth, CellMargins->{{72, 10}, {0, 10}}, CellFrameLabels->{{None, Cell[ TextData[ {"(", CounterBox[ "NumberedEquation"], ")"}]]}, {None, None}}, DefaultFormatType->DefaultInputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, CounterIncrements->"NumberedEquation", FormatTypeAutoConvert->False], Cell[StyleData["NumberedEquation", "Printout"], PageWidth->PaperWidth, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["NumberedFigure"], PageWidth->PaperWidth, CellFrame->True, CellMargins->{{72, 25}, {8, 12}}, Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, CellLabelMargins->{{33, Inherited}, {Inherited, Inherited}}, CellFrameLabels->{{None, None}, {Cell[ TextData[ {"Figure 2.", CounterBox[ "NumberedFigure"]}], FontSlant -> Italic, FontFamily -> "Georgia", FontColor -> RGBColor[ .5, .1, .1]], None}}, DefaultFormatType->DefaultInputFormatType, TextAlignment->Left, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, LanguageCategory->"Formula", ShowStringCharacters->True, NumberMarks->True, LinebreakAdjustments->{0.85, 2, 10, 0, 1}, CounterIncrements->"NumberedFigure", FormatTypeAutoConvert->False, FontFamily->"Verdana", FontSize->12, FontWeight->"Plain", Background->RGBColor[1, 0.8, 0.700008]], Cell[StyleData["NumberedFigure", "Printout"], PageWidth->PaperWidth, FontSize->10] }, Open ]], Cell[CellGroupData[{ Cell[StyleData["NumberedTable"], PageWidth->PaperWidth, CellMargins->{{72, 145}, {2, 10}}, CellFrameLabels->{{None, None}, {Cell[ TextData[ {"Table ", CounterBox[ "NumberedTable"]}]], None}}, TextAlignment->Center, CounterIncrements->"NumberedTable", FormatTypeAutoConvert->False], Cell[StyleData["NumberedTable", "Printout"], PageWidth->PaperWidth, FontSize->10] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["Formulas and Programming", "Section", PageWidth->PaperWidth], Cell[StyleData["Definition"], PageWidth->PaperWidth, CellFrame->True, CellMargins->{{72, 25}, {8, 12}}, Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, InitializationCell->True, CellLabelMargins->{{33, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultInputFormatType, TextAlignment->Left, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, LanguageCategory->"Formula", ShowStringCharacters->True, NumberMarks->True, LinebreakAdjustments->{0.85, 2, 10, 0, 1}, CounterIncrements->"Input", FormatTypeAutoConvert->False, FontFamily->"Verdana", FontSize->12, FontWeight->"Plain", Background->RGBColor[0.8, 1, 1]], Cell[StyleData["Miniprogram"], PageWidth->PaperWidth, CellFrame->True, CellDingbat->None, CellMargins->{{72, 25}, {8, 12}}, Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, InitializationCell->True, CellLabelMargins->{{33, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultInputFormatType, TextAlignment->Left, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, LanguageCategory->"Formula", ShowStringCharacters->True, NumberMarks->True, LinebreakAdjustments->{0.85, 2, 10, 0, 1}, CounterIncrements->"Input", FormatTypeAutoConvert->False, FontFamily->"Verdana", FontSize->12, FontWeight->"Plain", Background->RGBColor[0.8, 1, 0.700008]], Cell[StyleData["Animation"], PageWidth->PaperWidth, CellFrame->True, CellMargins->{{72, 25}, {8, 12}}, Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, InitializationCell->True, CellLabelMargins->{{33, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultInputFormatType, TextAlignment->Left, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, LanguageCategory->"Formula", ShowStringCharacters->True, NumberMarks->True, LinebreakAdjustments->{0.85, 2, 10, 0, 1}, CounterIncrements->"Input", FormatTypeAutoConvert->False, FontFamily->"Verdana", FontSize->12, FontWeight->"Plain", FontColor->GrayLevel[1], Background->RGBColor[0.4, 0.4, 1]], Cell[CellGroupData[{ Cell[StyleData["Package"], PageWidth->PaperWidth, CellFrame->True, CellMargins->{{72, 25}, {8, 12}}, Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, InitializationCell->True, CellLabelMargins->{{33, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultInputFormatType, TextAlignment->Left, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, LanguageCategory->"Formula", ShowStringCharacters->True, NumberMarks->True, LinebreakAdjustments->{0.85, 2, 10, 0, 1}, CounterIncrements->"Input", FormatTypeAutoConvert->False, FontFamily->"Verdana", FontSize->12, FontWeight->"Plain", Background->RGBColor[0.900008, 0.800015, 1]], Cell[StyleData["Program", "Printout"], PageWidth->PaperWidth, FontSize->9.5] }, Open ]] }, Open ]] }, Open ]] }] ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. *******************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[1776, 53, 52, 1, 68, "Title"], Cell[CellGroupData[{ Cell[1853, 58, 48, 1, 71, "Section"], Cell[1904, 61, 343, 6, 88, "Text"], Cell[2250, 69, 552, 15, 107, "Text"], Cell[2805, 86, 184, 6, 100, "Definition"], Cell[2992, 94, 442, 10, 107, "Text"], Cell[3437, 106, 637, 12, 145, "Text"], Cell[CellGroupData[{ Cell[4099, 122, 80, 2, 34, "Subsection"], Cell[4182, 126, 537, 17, 88, "Text"], Cell[4722, 145, 109, 5, 84, "Definition"], Cell[4834, 152, 298, 10, 50, "Text"], Cell[5135, 164, 101, 5, 84, "Definition"], Cell[5239, 171, 234, 9, 31, "Text"], Cell[5476, 182, 141, 4, 52, "Definition"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[5666, 192, 48, 1, 45, "Section"], Cell[5717, 195, 117, 3, 31, "Text"], Cell[5837, 200, 98, 1, 52, "Definition"], Cell[5938, 203, 194, 7, 50, "Text"], Cell[6135, 212, 84, 4, 68, "Input"], Cell[6222, 218, 131, 5, 31, "Text"], Cell[6356, 225, 185, 5, 107, "NumberedFigure"], Cell[6544, 232, 249, 6, 50, "Text"], Cell[6796, 240, 223, 6, 123, "NumberedFigure"], Cell[7022, 248, 1198, 41, 183, "Text"], Cell[8223, 291, 262, 8, 132, "Input"], Cell[8488, 301, 291, 6, 69, "Text"], Cell[8782, 309, 414, 9, 171, "NumberedFigure"], Cell[9199, 320, 69, 1, 52, "Input"], Cell[9271, 323, 339, 8, 69, "Text"], Cell[9613, 333, 430, 12, 69, "Text"], Cell[10046, 347, 422, 12, 219, "NumberedFigure"], Cell[CellGroupData[{ Cell[10493, 363, 53, 1, 34, "Subsection"], Cell[10549, 366, 82, 1, 31, "Text"], Cell[10634, 369, 532, 13, 235, "NumberedFigure"], Cell[11169, 384, 123, 3, 31, "Text"], Cell[11295, 389, 570, 14, 235, "NumberedFigure"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[11914, 409, 48, 1, 45, "Section"], Cell[11965, 412, 551, 17, 88, "Text"], Cell[12519, 431, 104, 1, 52, "Definition"], Cell[12626, 434, 79, 1, 52, "Input"], Cell[12708, 437, 142, 4, 50, "Text"], Cell[12853, 443, 78, 4, 68, "Input"], Cell[12934, 449, 154, 4, 50, "Text"], Cell[13091, 455, 95, 5, 84, "Input"], Cell[13189, 462, 138, 4, 68, "Input"], Cell[13330, 468, 695, 20, 107, "Text"], Cell[14028, 490, 357, 11, 203, "NumberedFigure"], Cell[14388, 503, 86, 1, 31, "Text"], Cell[14477, 506, 241, 9, 148, "Input"], Cell[CellGroupData[{ Cell[14743, 519, 53, 1, 34, "Subsection"], Cell[14799, 522, 227, 7, 50, "Text"], Cell[15029, 531, 524, 15, 267, "NumberedFigure"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[15602, 552, 48, 1, 45, "Section"], Cell[15653, 555, 207, 6, 50, "Text"], Cell[15863, 563, 94, 1, 52, "Definition"], Cell[15960, 566, 890, 26, 145, "Text"], Cell[16853, 594, 259, 10, 187, "NumberedFigure"], Cell[17115, 606, 82, 1, 31, "Text"], Cell[17200, 609, 104, 4, 68, "Input"], Cell[CellGroupData[{ Cell[17329, 617, 53, 1, 34, "Subsection"], Cell[17385, 620, 766, 17, 145, "Text"], Cell[18154, 639, 696, 19, 299, "NumberedFigure"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[18899, 664, 48, 1, 45, "Section"], Cell[18950, 667, 149, 4, 50, "Text"], Cell[19102, 673, 172, 5, 84, "Definition"], Cell[19277, 680, 227, 5, 69, "Text"], Cell[19507, 687, 131, 4, 68, "Definition"], Cell[19641, 693, 91, 4, 68, "Input"], Cell[19735, 699, 133, 4, 50, "Text"], Cell[19871, 705, 61, 1, 52, "Package"], Cell[19935, 708, 529, 13, 107, "Text"], Cell[20467, 723, 99, 1, 52, "Definition"], Cell[20569, 726, 587, 14, 251, "NumberedFigure"], Cell[21159, 742, 117, 3, 31, "Text"], Cell[21279, 747, 114, 4, 68, "Input"], Cell[21396, 753, 247, 7, 50, "Text"], Cell[21646, 762, 240, 6, 123, "NumberedFigure"], Cell[21889, 770, 148, 6, 100, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[22074, 781, 48, 1, 45, "Section"], Cell[22125, 784, 171, 6, 31, "Text"], Cell[22299, 792, 84, 1, 52, "Definition"], Cell[22386, 795, 92, 5, 84, "Input"], Cell[22481, 802, 74, 4, 68, "Input"], Cell[22558, 808, 82, 1, 52, "Input"], Cell[22643, 811, 505, 12, 219, "NumberedFigure"], Cell[23151, 825, 117, 3, 31, "Text"], Cell[23271, 830, 140, 5, 84, "Input"], Cell[CellGroupData[{ Cell[23436, 839, 53, 1, 34, "Subsection"], Cell[23492, 842, 98, 1, 31, "Text"], Cell[23593, 845, 703, 17, 283, "NumberedFigure"], Cell[24299, 864, 323, 12, 50, "Text"], Cell[24625, 878, 191, 7, 139, "NumberedFigure"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[24865, 891, 48, 1, 45, "Section"], Cell[24916, 894, 304, 12, 50, "Text"], Cell[25223, 908, 97, 1, 52, "Definition"], Cell[25323, 911, 138, 4, 68, "Input"], Cell[25464, 917, 137, 4, 50, "Text"], Cell[25604, 923, 154, 6, 100, "Input"], Cell[25761, 931, 161, 4, 50, "Text"], Cell[25925, 937, 154, 4, 52, "Definition"], Cell[26082, 943, 407, 13, 235, "NumberedFigure"], Cell[26492, 958, 105, 3, 31, "Text"], Cell[26600, 963, 96, 3, 52, "Input"], Cell[CellGroupData[{ Cell[26721, 970, 53, 1, 34, "Subsection"], Cell[26777, 973, 525, 13, 235, "NumberedFigure"], Cell[27305, 988, 501, 14, 251, "NumberedFigure"], Cell[27809, 1004, 71, 1, 52, "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[27929, 1011, 48, 1, 45, "Section"], Cell[27980, 1014, 91, 1, 31, "Text"], Cell[28074, 1017, 193, 5, 84, "Definition"], Cell[28270, 1024, 450, 15, 69, "Text"], Cell[28723, 1041, 124, 6, 100, "Input"], Cell[28850, 1049, 102, 1, 52, "Input"], Cell[28955, 1052, 92, 1, 52, "Input"], Cell[29050, 1055, 168, 4, 50, "Text"], Cell[29221, 1061, 117, 3, 52, "Input"], Cell[29341, 1066, 90, 1, 31, "Text"], Cell[29434, 1069, 102, 4, 68, "Input"], Cell[29539, 1075, 436, 13, 69, "Text"], Cell[29978, 1090, 285, 9, 164, "Miniprogram"], Cell[30266, 1101, 439, 11, 88, "Text"], Cell[30708, 1114, 329, 12, 219, "NumberedFigure"], Cell[31040, 1128, 58, 4, 68, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[31135, 1137, 48, 1, 45, "Section"], Cell[31186, 1140, 281, 7, 69, "Text"], Cell[31470, 1149, 151, 6, 70, "Package"], Cell[31624, 1157, 153, 4, 50, "Text"], Cell[31780, 1163, 170, 5, 84, "Definition"], Cell[31953, 1170, 128, 4, 68, "Definition"], Cell[32084, 1176, 495, 13, 235, "NumberedFigure"], Cell[CellGroupData[{ Cell[32604, 1193, 53, 1, 34, "Subsection"], Cell[32660, 1196, 570, 15, 267, "NumberedFigure"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[33279, 1217, 48, 1, 45, "Section"], Cell[33330, 1220, 253, 5, 69, "Text"], Cell[33586, 1227, 268, 8, 132, "Input"], Cell[33857, 1237, 122, 3, 31, "Text"], Cell[33982, 1242, 259, 6, 100, "Definition"] }, Closed]] }, Open ]] } ] *) (******************************************************************* End of Mathematica Notebook file. *******************************************************************)