(************** 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[ 22201, 842]*) (*NotebookOutlinePosition[ 51873, 1794]*) (* CellTagsIndexPosition[ 51829, 1790]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Notebook 27", "Title"], Cell[CellGroupData[{ Cell["N27.0", "Section"], Cell[TextData[{ "This notebook has three aims. One is to construct the two figures in \ Chapter 27 consisting of quadrilaterals together with angles and text. \ These, and similar figures in Chapters 12 and 25, were originally produced by \ the author using a ", Cell[BoxData[ StyleBox[ RowBox[{"T", AdjustmentBox["E", BoxMargins->{{-0.075, -0.085}, {0, 0}}, BoxBaselineShift->0.5], "X"}]]]], "-based program. The editors' exercise for the Third Edition was therefore \ to convert these pictures into ", StyleBox["Mathematica", FontSlant->"Italic"], ", if only to bolster the content of the notebooks." }], "Text"], Cell[TextData[{ "In ", StyleBox["N27.1,", FontFamily->"Verdana"], " the curves are all parts of parabolas of the type ", StyleBox["y=", FontFamily->"Verdana"], Cell[BoxData[ \(TraditionalForm\`ax\^2\)], FontFamily->"Verdana"], StyleBox["+bx+c.", FontFamily->"Verdana"], " In ", StyleBox["N27.2, ", FontFamily->"Verdana"], "we generalize this by allowing parabolas ", StyleBox["x=", FontFamily->"Verdana"], Cell[BoxData[ \(TraditionalForm\`ay\^2\)], FontFamily->"Verdana"], StyleBox["+by+c,", FontFamily->"Verdana"], " obtained by rotating the former. For this purpose we load" }], "Text"], Cell["<"Verdana"] }], "Text"], Cell["tx[A_, v_]:= Text[FontForm[A,{\"Helvetica\",18}], v]", "Definition"], Cell["\<\ The second aim is to construct the figures in Section 27.3, based on the \ building blocks\ \>", "Text"], Cell["\<\ torus[a_,b_,c_][u_,v_]:= {(a+b Cos[v])Cos[u], (a+b Cos[v])Sin[u], c Sin[v]} crosscap[u_,v_]:= {(Sin[u]Sin[2v])/2, Sin[2u]Cos[v]^2, Cos[2u]Cos[v]^2} torus[a_,b_]:= torus[a,b,b] ellipsoid[a_,c_]:= torus[0,a,c] sphere[a_]:= ellipsoid[a,a]\ \>", "Definition", FontFamily->"Verdana"], Cell[TextData[{ "The first two definitions were introduced in ", StyleBox["N7.6 ", FontFamily->"Verdana"], "and ", StyleBox["N11.5 ", FontFamily->"Verdana"], "respectively. Another approach will rely on the package" }], "Text"], Cell["<"Verdana"], " with the help of the definitions of ", StyleBox["L, metric, gcurv", FontFamily->"Verdana"], " reloaded in ", StyleBox["N27.0.1.", FontFamily->"Verdana"] }], "Text"], Cell[CellGroupData[{ Cell["N27.0.1", "Subsection"], Cell[TextData[StyleBox["metric[x_][u_,v_]:= \nModule[{xu,xv,U,V,si},\n si= \ Simplify[#]/.{U->u, V->v} &;\n xu= D[x[U,V],U]; \n xv= D[x[U,V],V]; \n \ ee= xu.xu//si;\n ff= xu.xv//si;\n gg= xv.xv//si;\n {{ee,ff},{ff,gg}}\n\ ]", ScreenRectangle->{{0, 1280}, {0, 717}}, AutoGeneratedPackage->None, ScreenStyleEnvironment->"Working", WindowSize->{1016, 668}, WindowMargins->{{0, Automatic}, {Automatic, 0}}]], "Miniprogram", FontVariations->{"CompatibilityType"->0}], Cell["\<\ gcurv[x_][u_,v_]:= Module[{xu,xv,xuu,xuv,xvv,U,V,A,B}, xu= D[x[U,V],U]; xv= D[x[U,V],V]; xuu= D[x[U,V],U,U]; xuv= D[x[U,V],U,V]; xvv= D[x[U,V],V,V]; B= Det[{xuu,xu,xv}]Det[{xvv,xu,xv}] - Det[{xuv,xu,xv}]^2; A= (xu.xu)(xv.xv)-(xu.xv)^2; Simplify[B/A^2]/.{U->u,V->v} ]\ \>", "Miniprogram", PageWidth->PaperWidth, FontVariations->{"CompatibilityType"->0}], Cell["\<\ mcurv[x_][u_,v_]:= Module[{xu,xv,xuu,xuv,xvv,U,V}, xu= D[x[U,V],U]; xv= D[x[U,V],V]; xuu= D[x[U,V],U,U]; xuv= D[x[U,V],U,V]; xvv= D[x[U,V],V,V]; B= Det[{xuu,xu,xv}](xv.xv)-2Det[{xuv,xu,xv}](xu.xv) +Det[{xvv,xu,xv}](xu.xu); A= (xu.xu)(xv.xv)-(xu.xv)^2; Simplify[B/(2A^(3/2))]/.{U->u,V->v} ]\ \>", "Miniprogram", PageWidth->PaperWidth, FontVariations->{"CompatibilityType"->0}], Cell["Unprotect[L]; L[v_]:= Sqrt[Simplify[v.v]]; Protect[L];", "Definition", PageWidth->PaperWidth], Cell["\<\ $DisplayFunction = Identity; Unprotect[S]; S[X__]:= Show[X, DisplayFunction->(Display[$Display,#1] & )] Protect[S]; \ \>", "Definition"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["N27.1", "Section"], Cell["\<\ Here is a program that constructs a parabola passing through three assigned \ points.\ \>", "Text"], Cell["\<\ pts2parabola[u_,v_,w_]:= Module[{f,eq,ta,so}, f[t_]= a t^2 + b t + c; eq[x_]:= f[x[[1]]]==x[[2]]; ta= eq/@{u,v,w}; so= Solve[ta, {a,b,c}][[1]]; {#, f[#]/.so} & ]\ \>", "Miniprogram"], Cell[TextData[{ "The output is a parametrization, with parameter ", StyleBox["t", FontFamily->"Verdana"], " equal to ", "the x-coordinate:" }], "Text"], Cell["\<\ u:= {0,0} v:= {2,1} w:= {1,-3} \[Alpha]:= pts2parabola[u,v,w] \[Alpha][t]\ \>", "Input"], Cell[TextData[{ "We formulate the plotting command into the following ", StyleBox["Module. ", FontFamily->"Verdana"], "This will draw the parabola from the point ", StyleBox["u", FontFamily->"Verdana"], " to the point ", StyleBox["v, and ", FontFamily->"Verdana"], "the third argument of ", StyleBox["edge", FontFamily->"Verdana"], " will often be used merely as a control point to modify the chosen curve \ through the first two points." }], "Text"], Cell["\<\ edge[u_,v_,w_]:= Module[{\[Alpha]}, \[Alpha]= pts2parabola[u,v,w]; ParametricPlot[\[Alpha][t], {t, u[[1]], v[[1]]}, Axes->False, PlotStyle->{Thickness[.01],Hue[0]}] ]\ \>", "Miniprogram"], Cell[TextData[{ "We check that the parabola", StyleBox[" does", FontSlant->"Italic"], " pass through the assigned points:" }], "Text"], Cell["\<\ z0:= {PointSize[.03], Point[u], Point[v], Point[w]}//Graphics z1:= edge[u,v,w] S[z0, z1, Axes->True, PlotRange->All]\ \>", "Input"], Cell[TextData[{ "Figure 27.1 also consists of short lines tangent at both ends to the \ parabola through ", StyleBox["u,v,w", FontFamily->"Verdana"], ", and their plotting is accomplished by the following miniprogram. Note \ that the parametrization of the parabola cannot be differentiated directly; \ this explains the dummy variable in the definition of \[Beta]." }], "Text"], Cell["\<\ tangents[u_,v_,w_]:= Module[{\[Alpha],\[Beta],\[Lambda]}, \[Alpha]:= pts2parabola[u,v,w]; \[Beta][t_]:= D[\[Alpha][tt], tt]/.tt->t; pm= If[u[[1]]", "Miniprogram"], Cell["\<\ We break up the data for Figure 27.1 into separate units. The next cell \ lists coordinates of the four vertices, and the subsequent one lists the \ triples of points needed to determine the curves.\ \>", "Text"], Cell["\<\ p[0]:= {0,0} p[1]:= {6,1} p[2]:= {8,6} p[3]:= {2,7}\ \>", "Input", InitializationCell->True], Cell["\<\ pp[0]:= {p[0], p[1], {3,1.5}} pp[1]:= {p[1], p[2], {6.5,2.8}} pp[2]:= {p[2], p[3], {4,6}} pp[3]:= {p[3], p[0], {.5,2.5}} \ \>", "Input", InitializationCell->True], Cell[TextData[{ "The graphics object ", StyleBox["z[i] ", FontFamily->"Verdana"], "now consists of the curve together with its two tangents:" }], "Text"], Cell["\<\ Clear[z] z[i_]:= {edge@@pp[i], tangents@@pp[i]}\ \>", "Definition"], Cell["S @ z[3]", "Input"], Cell[TextData[{ "Circular arcs indicating the exterior angles are formed from ", StyleBox["Mathematica'", FontSlant->"Italic"], "s command ", StyleBox["Circle", FontFamily->"Verdana"], ", introduced in ", StyleBox["N1.1.", FontFamily->"Verdana"], " The center of the circle", StyleBox[" y[i]", FontFamily->"Verdana"], " is defined automatically to be ", StyleBox["p[i]", FontFamily->"Verdana"], ". The extent of the arc is represented by the pair ", StyleBox["d[i]={a,b}, ", FontFamily->"Verdana"], "which defines an arc from", StyleBox[" a\[Pi]", FontFamily->"Verdana"], " to ", StyleBox["b\[Pi]", FontFamily->"Verdana"], " radians. In theory, the value of ", StyleBox["d[i]", FontFamily->"Verdana"], " could be defined automatically; in practice, all the intervals were added \ by hand after plotting the edges and inspecting the diagram! " }], "Text"], Cell["\<\ Module[{d,r}, r[i_]= 1; d[0]= {-.55,.22}; r[0]=.8; d[1]= {-.14,.42}; d[2]= {.25,1.1}; r[2]=.8; d[3]= {.81,1.3}; y[i_]:= Circle[p[i], r[i], d[i]\[Pi]] ]\ \>", "Input", InitializationCell->True], Cell[TextData[{ "The default radius of each circle is 1, but this is modified twice above \ by adding the required radius by hand after the definition of ", StyleBox["d[i].", FontFamily->"Verdana"], " For the final diagram to take account of such modifications, it is \ essential that the definition of ", StyleBox["y", FontFamily->"Verdana"], " be made with the symbol", StyleBox[" := ", FontFamily->"Verdana"], "in place of mere equality, as it is sensitive to when evaluation is made. \ The final ingredient is the text, which needs no comment:" }], "Text"], Cell[TextData[{ "txt:= {tx[", Cell[BoxData[ \(TraditionalForm\`\[Epsilon]\_0\)]], ", {1.1,-.6}], tx[", Cell[BoxData[ \(TraditionalForm\`\[Epsilon]\_1\)]], ", {7.4,1.3}], tx[", Cell[BoxData[ \(TraditionalForm\`\[Epsilon]\_2\)]], ", {7.4,7.2}], \n tx[", Cell[BoxData[ \(TraditionalForm\`\[Epsilon]\_3\)]], ", {.7, 6.4}]}//Graphics" }], "Input", InitializationCell->True], Cell[TextData[{ "We put everything together by letting ", StyleBox["z[i] ", FontFamily->"Verdana"], "with ", StyleBox["i=4,...,7", FontFamily->"Verdana"], " correspond to the circular arcs. The conditional at the end of line 3 \ below stops the definition being applied unless", StyleBox[" i>3, ", FontFamily->"Verdana"], "so that ", StyleBox["z[i] ", FontFamily->"Verdana"], "refers to the definition above for", StyleBox[" i=0,...,3.", FontFamily->"Verdana"] }], "Text"], Cell["\<\ Module[{hk,ay}, hk= {RGBColor[0,.6,0], Thickness[.01]}; z[i_]:= Graphics[Append[hk, y[i-4]]]/; i>3 ; z[8]= txt; ay:= Flatten[Table[z[i], {i,0,8}]]; S[ay, AspectRatio->Automatic] ]\ \>", "NumberedFigure"] }, Closed]], Cell[CellGroupData[{ Cell["N27.3", "Section"], Cell[TextData[{ "We can easily generalize the construction of ", StyleBox["N27.1 ", FontFamily->"Verdana"], "to allow quadratic curves that are not necessarily graphs over the \ horizontal axis. For our purposes, it will be sufficient to include \ parabolas ", StyleBox["x=", FontFamily->"Verdana"], Cell[BoxData[ \(TraditionalForm\`ay\^2\)], FontFamily->"Verdana"], StyleBox["+by+c,", FontFamily->"Verdana"], " that are obtained by rotating the previous ones by", StyleBox[" \[Pi]/2.", FontFamily->"Verdana"], " If we designate these new ones \"type 2,\" and the previous ones \"type \ 1,\" we can insert the index", StyleBox[" k=1,2", FontFamily->"Verdana"], " as a fourth argument of the new command ", StyleBox["pts2parabolas. ", FontFamily->"Verdana"], "The latter is now plural, as there are two choices; the ", StyleBox["Off", FontFamily->"Verdana"], " command is used to eliminate the resulting spelling query." }], "Text"], Cell["\<\ Off[General::spell1] pts2parabolas[u_,v_,w_,k_]:= Module[{\[Theta],R,f,eq,ta,so}, \[Theta]= If[k==1, 0, \[Pi]/2]; R[x_]= Rotate2D[x, \[Theta]]; f[t_]= a t^2 + b t + c; eq[x_]:= f[x[[1]]]==x[[2]]; ta= eq/@R/@{u,v,w}; so= Solve[ta, {a,b,c}][[1]]; Rotate2D[{#, f[#]/.so}, -\[Theta]] & ]\ \>", "Miniprogram"], Cell[TextData[{ "In the plural version of", StyleBox[" edge, ", FontFamily->"Verdana"], "the index", StyleBox[" k=1,2", FontFamily->"Verdana"], " is also inserted into the plotting range, so as to determine whether it \ is the x- or y-coordinate that determines the parameter range. To emphasize \ the difference between the two types of parabolas, they are also colored \ slightly differently. " }], "Text"], Cell["\<\ Off[General::spell1] edges[u_,v_,w_,k_]:= Module[{\[Alpha]}, \[Alpha]= pts2parabolas[u,v,w, k]; ParametricPlot[\[Alpha][t], {t, u[[k]], v[[k]]}, Axes->False, PlotStyle->{Thickness[.01],Hue[.4+k/5]}] ]\ \>", "Miniprogram"], Cell[TextData[{ "Here is the first example from ", StyleBox["N27.1.", FontFamily->"Verdana"] }], "Text"], Cell["\<\ u:= {0,0} v:= {2,1} w:= {1,-3} \[Beta]:= pts2parabolas[u,v,w,2] \[Beta][t]\ \>", "Input"], Cell[TextData[{ "It is essential to order the arguments of ", StyleBox["edges", FontFamily->"Verdana"], " correctly in order that the two types of parabolas extend through all \ three points:" }], "Text"], Cell["\<\ z0:= {PointSize[.03], Point[u], Point[v], Point[w]}//Graphics z1:= edges[u,v,w,1] z2:= edges[v,w,u,2] S[z0, z1, z2, Axes->True, PlotRange->All]\ \>", "Input"], Cell["\<\ With these modifications, the remaining code in the construction of Figure \ 27.2 mimics that of Figure 27.1, and we suppress the details.\ \>", "Text"], Cell[CellGroupData[{ Cell["N27.3.1", "Subsection"], Cell["Clear[u,v,w,p,z]", "Input"], Cell[TextData[{ "The remaining definitions follow, using notation analogous to that of ", StyleBox["N27.1.", FontFamily->"Verdana"] }], "Text"], Cell["\<\ q[0]:= {0,0} q[1]:= {3,1} q[2]:= {6,0} q[3]:= {-1,2.5} q[4]:= {2.5,3.5} q[5]:= {5,3} q[6]:= {3,7}\ \>", "Input", InitializationCell->True], Cell["\<\ qq[0]:= {q[0], q[2], q[1], 1} qq[1]:= {q[0], q[6], q[3], 2} qq[2]:= {q[2], q[6], q[5], 1} qq[3]:= {q[1], q[4], {2.5,1.5}, 2} qq[4]:= {q[3], q[4], {1,2.7}, 1} qq[5]:= {q[4], q[5], {3, 5}, 2}\ \>", "Input", InitializationCell->True], Cell["\<\ arcs= Module[{r,d}, r[i_]= .55; d[0]= {.2, .7}; r[0]= .9; d[1]= {0, 1}; d[2]= {.6, .82}; r[2]= 1.3; d[3]= {-.48, .41}; d[4]= {0, 2}; r[4]= .5; d[5]= {.64, 1.6}; d[6]= {1.17, 1.67}; r[6]= .9; y[i_]:= Circle[q[i], r[i], d[i]\[Pi]] ];\ \>", "Input", InitializationCell->True], Cell["z[i_]:= edges@@qq[i]", "Definition"], Cell[TextData[{ "z[13]:= {tx[\"2\[Pi]\", {1.7, 3.6}], tx[\[Pi], {3, .7}], tx[\[Pi], {5.4, \ 3}], \n tx[\[Pi], {-1.4, 2.5}], tx[", Cell[BoxData[ \(TraditionalForm\`\[Iota]\_1\)]], ", {.1, .5}], tx[", Cell[BoxData[ \(TraditionalForm\`\[Iota]\_2\)]], ", {5.5, .62}], \n tx[", Cell[BoxData[ \(TraditionalForm\`\[Iota]\_3\)]], ", {2.9, 6.5}], tx[", Cell[BoxData[ \(TraditionalForm\`R\_1\)]], ", {1.1, 1.7}], tx[", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(R\_2\)\)\)]], ", {3.8, 2.2}], \n tx[", Cell[BoxData[ \(TraditionalForm\`R\_3\)]], ", {2.5, 5}]}//Graphics;" }], "Input"], Cell["\<\ draw= Module[{hk,ar}, hk= {Hue[0], Thickness[.01]}; z[i_]:= Graphics[Append[hk, y[i-6]]]/; i>5 ; ar:= Table[z[i], {i,0,13}]; S[ar, AspectRatio->Automatic, ImageSize->400] ];\ \>", "NumberedFigure"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["N27.4", "Section"], Cell[TextData[{ "We shall need the abbreviation for the identity matrix given in ", StyleBox["N7.1.", FontFamily->"Verdana"] }], "Text"], Cell["id:= IdentityMatrix[3]", "Definition"], Cell["Here is a program to plot the four tori of Figure 23.3:", "Text"], Cell["\<\ holes[a_,d_,p_,R_:id]:= Module[{y,yy,r,s,z,ta}, y= torus[a,1,1]; yy[u_,v_]:= RotateRight[y[u,v]]; r= {d,0,0}; s= {0,0,d}; z[X_]:= ParametricPlot3D[R.X, {u,0,2\[Pi]}, {v,-\[Pi]/2,3\[Pi]/2}, PlotPoints->p]; ta= {z[y[u,v]+r], z[y[u,v]-r], z[yy[u,v]+s], z[yy[u,v]-s]}; Show[ta, Boxed->False, Axes->False, PlotRange->All] ]\ \>", "Miniprogram"], Cell[TextData[{ "The next two pictures can be seen by adding ", StyleBox["S@, ", FontFamily->"Verdana"], "but similar ones are combined for Figure 27.3." }], "Text"], Cell["holes[4,4,7]", "Input"], Cell[TextData[{ "By varying the value of ", StyleBox["PlotPoints,", FontFamily->"Verdana"], " we can pass from a polyhedral representation of the surface to one that \ is approximately smooth." }], "Text"], Cell["holes[5,6,20]", "Input"], Cell["\<\ S @ GraphicsArray[{holes[4,4,30], holes[3,3,5]}, GraphicsSpacing->-.2]\ \>", "NumberedFigure"], Cell[TextData[{ "The next program computes a rotation matrix associated to a vector ", StyleBox["x", FontFamily->"Verdana"], " (of norm less than or equal to 1). It is a variant of one defined in ", StyleBox["N23.1, ", FontFamily->"Verdana"], "but incorporates a conditional to avoid evaluating ", StyleBox["B", FontFamily->"Verdana"], " when ", StyleBox["x", FontFamily->"Verdana"], " is the null vector." }], "Text"], Cell["\<\ EAif[x_]:= Module[{l,A,B}, A[y_]:= Table[Cross[id[[i]],y], {i,3}]; l= L[x]; B:= Sin[l] A[x]/l + (1-Cos[l]) A[x].A[x]/l^2; id + If[l==0, 0, B] ]\ \>", "Miniprogram"], Cell[TextData[{ "We use it to rotate the polyhedral surface. To display the plot, \ replace", StyleBox[" Show", FontFamily->"Verdana"], " by ", StyleBox["S. ", FontFamily->"Verdana"], "Without introducing extra small triangular faces, the central part of the \ picture incorporates 14 vertices and 20 edges." }], "Text"], Cell["\<\ Module[{w,h}, w[t_]= EAif[.6\[Pi]{Cos[t], Sin[t], 0}]; h[t_]:= Show[holes[3,3,5, w[t]], PlotRange->{{-6,6},{-6,6},{-6,6}}]; Do[h[\[Pi] k/18], {k,36}] ]\ \>", "Animation"], Cell[TextData[{ "Here is another approach to topological sketching using an implicit plot \ of the set of zeros of the ", StyleBox["product", FontSlant->"Italic"], " of two torus functions: " }], "Text"], Cell["\<\ torusimplicit[a_,b_][x_,y_,z_]:= z^2 + (Sqrt[x^2+y^2]-a)^2 - b^2\ \>", "Definition", FontFamily->"Verdana"], Cell["\<\ f[x_,y_,z_]:= torusimplicit[8,3][x-8,y,z] torusimplicit[8,3][x+8,y,z] S @ ImplicitPlot3D[f[x,y,z], {x,-20,20}, {y,-12,12}, {z,-10,10}, PlotPoints->{{7},{3},{2}}, Boxed->False]\ \>", "Input", FontFamily->"Verdana"], Cell[CellGroupData[{ Cell["N27.4.1", "Subsection"], Cell[TextData[{ "This is similar to Figure 27.3, but uses the", StyleBox[" Append ", FontFamily->"Verdana"], "command to customize color, and the following command to suppress grid \ lines on the sphere." }], "Text"], Cell["dip[ins_][g_]:= $DisplayFunction[Insert[g, ins, {1,1}]]", "Definition"], Cell["\<\ Module[{f,g,h,z}, f[u_,v_]:= Append[7crosscap[u,v], RGBColor[1,.7,1]]; g[u_,v_]:= Append[sphere[9][u,v]-{0,0,8.5}, RGBColor[.7,.7,1]]; h[u_,v_]:= Append[torus[5,1.5][u,v]+{8,0,-8}, Hue[.5]]; z[1]:= ParametricPlot3D[f[u,v], {u,0,\[Pi]}, {v,-\[Pi]/2, \[Pi]/2}, PlotPoints->{30, 30}]; z[2]:= ParametricPlot3D[g[u,v], {u,0,2\[Pi]}, {v,-\[Pi]/2, \[Pi]/2}, DisplayFunction->dip[EdgeForm[]]]; z[3]= ParametricPlot3D[h[u,v], {u,0,2\[Pi]},{v,-\[Pi]/2, 3\[Pi]/2}, PlotPoints->{20,20}]; S[Array[z,3], Boxed->False, Axes->False, Lighting->False] ];\ \>", "NumberedFigure"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["N27.7", "Section"], Cell["\<\ The following computations support the exercises to verify the Gauss-Bonnet \ theorem. First we define the relevant integrand.\ \>", "Text"], Cell["infarea:= Sqrt[Simplify[ee gg - ff^2]]", "Definition", FontFamily->"Verdana"], Cell["\<\ tor:= torus[a,b] metric[tor][u,v]\ \>", "Input", FontFamily->"Verdana"], Cell["The Euler characteristic of a torus is zero:", "Text"], Cell["\<\ Integrate[PowerExpand[gcurv[tor][u,v]infarea], {u,0,2\[Pi]}, {v,0,2\[Pi]}]\ \>", "Input", FontFamily->"Verdana"], Cell["\<\ We can also compute the integral of the square of the mean curvature.\ \>", "Text"], Cell["\<\ Integrate[PowerExpand[mcurv[tor][u,v]^2], {u,0,2\[Pi]}, {v,0,2\[Pi]}]\ \>", "Input", FontFamily->"Verdana"], Cell["\<\ Finally, we verify the Gauss-Bonnet theorem for an ellipsoid:\ \>", "Text"], Cell["\<\ ell:= ellipsoid[a,b] metric[ell][u,v]; Integrate[gcurv[ell][u,v]infarea, {u,0,2\[Pi]}, {v,-\[Pi]/2,\[Pi]/2}] %//PowerExpand\ \>", "Input", FontFamily->"Verdana"] }, Closed]] }, Open ]] }, FrontEndVersion->"5.2 for Microsoft Windows", ScreenRectangle->{{0, 1280}, {0, 717}}, AutoGeneratedPackage->None, WindowToolbars->"EditBar", WindowSize->{1005, 651}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, PrintingCopies->1, PrintingStartingPageNumber->923, PrintingPageRange->{Automatic, Automatic}, PageHeaders->{{ Inherited, Inherited, Cell[ "NOTEBOOK 27", FontSlant -> Italic]}, { Cell[ "NOTEBOOK 27", FontSlant -> Italic], Inherited, Inherited}}, PrintingOptions->{"FirstPageFace"->Right}, CounterBoxOptions->{CounterFunction:>Identity}, 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. 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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", 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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}, 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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]], 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