(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of 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[ 22079, 629]*) (*NotebookOutlinePosition[ 23261, 669]*) (* CellTagsIndexPosition[ 23150, 662]*) (*WindowFrame->Normal*) Notebook[{ Cell[BoxData[ StyleBox[\(Y . \(Tazawa\ : \ IntroDiffGeom\)\), FontFamily->"Times"]], "Input", TextAlignment->Right, TextJustification->0, FontSize->10], Cell[TextData[{ "4. 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D[x[u, vv], vv]] /. vv -> v gggg[x_][u_, v_] := Simplify[Sqrt[ee[x][u, v]*gg[x][u, v] - ff[x][u, v]^2]]\ \>", "Input", AspectRatioFixed->True], Cell[BoxData[ \(numarclgth[f_, a_, b_, t0_, t1_] := NIntegrate[ Sqrt[\n\t\t \(ee[x]\)[a[t], b[t]]*\((\((D[a[tt], tt] /. tt -> t)\)^2)\) + \n\t\t 2 \( ff[x]\)[a[t], b[t]]*\((D[a[tt], tt] /. tt -> t)\)* \((D[b[ss], ss] /. ss -> t)\) + \n\t\t \(gg[x]\)[a[t], b[t]]*\((\((D[b[ss], ss] /. ss -> t)\)^2)\)], {t, t0, t1}]\)], "Input"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ StyleBox["tangentplane00", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["[", FontWeight->"Bold"], RowBox[{ StyleBox["f_", FontWeight->"Bold"], StyleBox[",", FontWeight->"Bold"], StyleBox["u_", FontWeight->"Bold"], StyleBox[",", FontWeight->"Bold"], StyleBox["v_", FontWeight->"Bold"], ",", "d_"}], StyleBox["]", FontWeight->"Bold"]}], StyleBox[":=", FontWeight->"Bold"], StyleBox["\n", FontWeight->"Bold"], StyleBox[" ", FontWeight->"Bold"], StyleBox[ \(Graphics3D[\n \ \ {\ RGBColor[1, 0, 0], \n\ \ \ \ \ \ PointSize[ .015], \n \ \ \ \ \ \ Point[f[u, v]], \n\ \ RGBColor[0, 0, 1], \n \ \ \ \ \ \ Line[{f[u, v], f[u, v] + d*n[f, u, v]}], \n\ RGBColor[0, 0, 1], \n\ \ \ \ Thickness[ .004], \ \n\ \ \ \ \ \ Line[{f[u, v] + 1.2*d*puu11[f, u, v] + 1.2*d*pvv11[f, u, v], \n \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ f[u, v] + 1.2*d*puu11[f, u, v] - 1.2*d*pvv11[f, u, v]}], \n \ \ \ \ \ \ Line[{f[u, v] - 1.2*d*puu11[f, u, v] + 1.2*d*pvv11[f, u, v], \n \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ f[u, v] + 1.2*d*puu11[f, u, v] + 1.2*d*pvv11[f, u, v]}], \n \ \ \ \ \ \ Line[{f[u, v] - 1.2*d*puu11[f, u, v] + 1.2*d*pvv11[f, u, v], \n \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ f[u, v] - 1.2*d*puu11[f, u, v] - 1.2*d*pvv11[f, u, v]}], \n \ \ \ \ \ \ Line[{f[u, v] - 1.2*d*puu11[f, u, v] - 1.2*d*pvv11[f, u, v], \n \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ f[u, v] + 1.2*d*puu11[f, u, v] - 1.2*d*pvv11[f, u, v]}]\n \ \ \ \ \ \ \ \ \ \ \ }]\), FontWeight->"Bold"]}], StyleBox[";", FontWeight->"Bold"]}]], "Input"], Cell[CellGroupData[{ Cell[TextData[StyleBox["Christofel's Symbols", FontColor->RGBColor[1, 0, 0]]], "Subsection"], Cell[BoxData[ RowBox[{ RowBox[{\(xxuu[u_, v_]\), "=", RowBox[{ RowBox[{"(", RowBox[{ StyleBox[\(D[x[ww, v], ww]\), InitializationCell->True, AspectRatioFixed->True], "/.", \(ww -> u\)}], ")"}], "//", "Simplify"}]}], ";", "\n", RowBox[{\(xxvv[u_, v_]\), "=", RowBox[{ RowBox[{"(", RowBox[{ StyleBox[\(D[x[u, vv], vv]\), InitializationCell->True, AspectRatioFixed->True], "/.", \(vv -> v\)}], ")"}], "//", "Simplify"}]}], ";", "\n", "\n", \(ee[u_, v_] = xxuu[u, v] . xxuu[u, v] // Simplify\), ";", "\n", \(ff[u_, v_] = xxuu[u, v] . xxvv[u, v] // Simplify\), ";", "\n", \(gg[u_, v_] = xxvv[u, v] . xxvv[u, v] // Simplify\), ";", "\n", "\n", RowBox[{\(eeuu[u_, v_]\), "=", RowBox[{ StyleBox[\(D[ee[ww, v], ww]\), InitializationCell->True, AspectRatioFixed->True], "/.", \(ww -> u\)}]}], ";", "\n", RowBox[{\(ffuu[u_, v_]\), "=", RowBox[{ StyleBox[\(D[ff[ww, v], ww]\), InitializationCell->True, AspectRatioFixed->True], "/.", \(ww -> u\)}]}], ";", "\n", RowBox[{\(gguu[u_, v_]\), "=", RowBox[{ StyleBox[\(D[gg[ww, v], ww]\), InitializationCell->True, AspectRatioFixed->True], "/.", \(ww -> u\)}]}], ";", "\n", "\n", RowBox[{\(eevv[u_, v_]\), "=", RowBox[{ StyleBox[\(D[ee[u, vv], vv]\), InitializationCell->True, AspectRatioFixed->True], "/.", \(vv -> v\)}]}], ";", "\n", RowBox[{\(ffvv[u_, v_]\), "=", RowBox[{ StyleBox[\(D[ff[u, vv], vv]\), InitializationCell->True, AspectRatioFixed->True], "/.", \(vv -> v\)}]}], ";", "\n", RowBox[{\(ggvv[u_, v_]\), "=", RowBox[{ StyleBox[\(D[gg[u, vv], vv]\), InitializationCell->True, AspectRatioFixed->True], "/.", \(vv -> v\)}]}], ";"}]], "Input"], Cell[BoxData[ \(\(christfl[1, 1, 1]\)[u_, v_] = \((1/\((2*\((ee[u, v]*gg[u, v] - ff[u, v]^2)\))\))\)*\n\t\t \((gg[u, v]*eeuu[u, v] - 2 ff[u, v]*ffuu[u, v] + ff[u, v]*eevv[u, v]) \); \n\n\(christfl[2, 1, 1]\)[u_, v_] = \((1/\((2*\((ee[u, v]*gg[u, v] - ff[u, v]^2)\))\))\)*\n\t\t \((2*ee[u, v]*ffuu[u, v] - ee[u, v]*eevv[u, v] - ff[u, v]*eeuu[u, v]) \); \n\n\(christfl[1, 1, 2]\)[u_, v_] = \((1/\((2*\((ee[u, v]*gg[u, v] - ff[u, v]^2)\))\))\)*\n\ \t \((gg[u, v]*eevv[u, v] - ff[u, v]*gguu[u, v])\); \n\n \(christfl[2, 1, 2]\)[u_, v_] = \((1/\((2*\((ee[u, v]*gg[u, v] - ff[u, v]^2)\))\))\)*\n\ \t \((\(-ff[u, v]\)*eevv[u, v] + ee[u, v]*gguu[u, v])\); \n\n \(christfl[1, 2, 2]\)[u_, v_] = \((1/\((2*\((ee[u, v]*gg[u, v] - ff[u, v]^2)\))\))\)*\n\ \t \((2*gg[u, v]*ffvv[u, v] - gg[u, v]*gguu[u, v] - ff[u, v]*ggvv[u, v]) \); \n\n\(christfl[2, 2, 2]\)[u_, v_] = \((1/\((2*\((ee[u, v]*gg[u, v] - ff[u, v]^2)\))\))\)*\n\ \t \((\(-2\)*ff[u, v]*ffvv[u, v] + ff[u, v]*gguu[u, v] + ee[u, v]*ggvv[u, v])\); \)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"(*", " ", StyleBox["mywireframe77", AspectRatioFixed->True, FontColor->RGBColor[1, 0, 0]], StyleBox[" ", AspectRatioFixed->True, FontColor->RGBColor[1, 0, 0]], StyleBox["*)", AspectRatioFixed->True, FontColor->RGBColor[1, 0, 0]]}]], "Input"], Cell[TextData[{ StyleBox["mywireframe77", AspectRatioFixed->True, FontColor->RGBColor[1, 0, 0]], StyleBox[ "[f_,u0_,u1_,v0_,v1_,m_,n_,opts___]:={\nDo[uuu[i]=u0+i*(u1-u0)/m,{i,0,m}];\n\ Do[vvv[j]=v0+j*(v1-v0)/n,{j,0,n}];\nDo[ggg[i,j]=f[uuu[i],vvv[j]]//N, \ {i,0,m},{j,0,n}];\n\ normalvector[u_,v_]=Cross[D[f[uu,v],uu]/.uu->u,D[f[u,vv],vv]/.vv->v];\n\ Do[norml[i,j]=Evaluate[normalvector[uuu[i],vvv[j]]],{i,0,m},{j,0,n}];\n\n\ Do[jaco[i,j]=Max[0.001,Evaluate[Sqrt[norml[i,j].norml[i,j]]//N]],\n \ {i,0,m},{j,0,n}];\n\nDo[lightcolor[i,j]=\nRGBColor[\n ", AspectRatioFixed->True], StyleBox[ " (1+(1/(jaco[i,j]*Sqrt[3]))*norml[i,j].{1,-1,1})/2 //N,\n \ (1+(1/(jaco[i,j]*Sqrt[2]))*norml[i,j].{ 1,0,1})/3 //N,\n \ (1+(1/(jaco[i,j]*Sqrt[3]))*norml[i,j].{-1,1,1})/2 //N],", AspectRatioFixed->True, FontColor->RGBColor[0, 0, 1]], StyleBox[ "\n {i,0,m},{j,0,n}];\n\n\ uucurve[i_,j_]=Graphics3D[{Thickness[0.001],lightcolor[i,j],\n \ Line[{ggg[i,j],ggg[i+1,j]}]}];\n\ vvcurve[i_,j_]=Graphics3D[{Thickness[0.001],lightcolor[i,j],\n \ Line[{ggg[i,j],ggg[i,j+1]}]}];\n\n\ hhh[i_,j_]=ggg[i,j]-0.001*norml[i,j]/jaco[i,j];\nuucurve11[i_,j_]=Graphics3D[\ \n {Thickness[0.002],RGBColor[1,1,1],Dashing[{0.005,0.01}],\n \ Line[{hhh[i,j],hhh[i+1,j]}]}];\nvvcurve11[i_,j_]=Graphics3D[\n \ {Thickness[0.002],RGBColor[1,1,1],Dashing[{0.005,0.01}],\n \ Line[{hhh[i,j],hhh[i,j+1]}]}];\n \nwireframe[f]=Show[Join[\n \ Table[uucurve[i,j],{i,0,m-1},{j,0,n}],\n \ Table[vvcurve[i,j],{i,0,m},{j,0,n-1}],\n \ Table[uucurve11[i,j],{i,0,m-1},{j,0,n}],\n \ Table[vvcurve11[i,j],{i,0,m},{j,0,n-1}]],Boxed->False,\n ", AspectRatioFixed->True], "DisplayFunction->", StyleBox["Identity];}", AspectRatioFixed->True] }], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Example ", "Subsection", FontColor->RGBColor[0, 0, 1], CellTags->"geodesic 8"], Cell[CellGroupData[{ Cell["Surface and initial conditions", "Subsection", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(\(x[u_, v_] = {u, v, 2*\((0.25*\ u + 0.27\ u\^2 - 0.31\ u\^3 - 0.31\ u\^4 - 0.046\ v - 0.040\ u\ v + 0.28\ u\^2\ v + 0.012 u\^3\ v + 0.26\ v\^2 - 0.38 u\ v\^2 + 0.43\ u\^2\ v\^2 - 0.13\ v\^3 - 0.29\ u\ v\^3 - 0.24\ v\^4)\)}; \)\)], "Input"], Cell[BoxData[ \(a[t_] = t; b[t_] = t; \)], "Input"], Cell[BoxData[ \(Timing[numarclgth[x, a, b, 0, 1]]\)], "Input"], Cell[BoxData[ \({u[0], v[0]} = {0, 0}; del00 = 0.16; \)], "Input"], Cell[BoxData[ \(\(vect00[a_, b_] = N[del00*\((a*puu[x, u[0], v[0]] + b*pvv[x, u[0], v[0]])\)/ Sqrt[\((a*puu[x, u[0], v[0]] + b*pvv[x, u[0], v[0]])\) . \((a*puu[x, u[0], v[0]] + b*pvv[x, u[0], v[0]])\)]]; \)\)], "Input"], Cell[BoxData[ \(\(vect[0] = vect00[1, 1]; \)\)], "Input"], Cell[BoxData[ \(\(cc[0] = {u[0], v[0]}; \)\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["ODE for Geodesics", FontColor->RGBColor[1, 0, 0]]], "Subsection"], Cell[BoxData[ \(pnt0 = {u[0], v[0]}; tangt0 = vect[0]/Sqrt[vect[0] . vect[0]]; \)], "Input"], Cell[BoxData[ \(Clear[aa, bb, aa11, bb11]\)], "Input"], Cell[BoxData[ RowBox[{"Timing", "[", " ", RowBox[{ RowBox[{ StyleBox["sol", AspectRatioFixed->True, FontColor->RGBColor[0, 0, 1]], StyleBox["=", AspectRatioFixed->True], RowBox[{ StyleBox[ RowBox[{ StyleBox["N", AspectRatioFixed->True, FontColor->RGBColor[1, 0, 0]], StyleBox["DSolve", AspectRatioFixed->True]}]], StyleBox["[", AspectRatioFixed->True], StyleBox[ \({\n\t\t\t\t\(aa'\)[t] == aa11[t], \n\t\(bb'\)[t] == bb11[t], \n\(aa11'\)[t] + \(christfl[1, 1, 1]\)[aa[t], bb[t]]*\((aa11[t]^2)\)\n \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \t + 2 \(christfl[1, 1, 2]\)[aa[t], bb[t]]*aa11[t]*bb11[t] + \n \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \(christfl[1, 2, 2]\)[aa[t], bb[t]]*\((bb11[t]^2)\) == 0, \n\(bb11'\)[t] + \(christfl[2, 1, 1]\)[aa[t], bb[t]]*\((aa11[t]^2)\)\n \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \t + 2 \(christfl[2, 1, 2]\)[aa[t], bb[t]]*aa11[t]*bb11[t] + \n \t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \(christfl[2, 2, 2]\)[aa[t], bb[t]]*\((bb11[t]^2)\) == 0, \n\t\naa[0] == pnt0[\([1]\)], bb[0] == pnt0[\([2]\)], \n \ \ \ \ \ \ \ \ aa11[0] == tangt0[\([1]\)], bb11[0] == tangt0[\([2]\)]\t\t\n\t\t\t}, {aa[t], bb[t], aa11[t], bb11[t]}, {t, 0, 8*del00}\), AspectRatioFixed->True], StyleBox["]", AspectRatioFixed->True]}]}], ";"}], StyleBox["]", AspectRatioFixed->True]}]], "Input"], Cell[BoxData[ \(aa[t_] = aa[t] /. sol[\([1]\)]; \nbb[t_] = bb[t] /. sol[\([1]\)]; \)], "Input"], Cell[BoxData[ \(\(c[t_] = {aa[t], bb[t]}; \)\)], "Input"], Cell[BoxData[ \(\(geod[t_] = x[aa[t], bb[t]]; \)\)], "Input"], Cell[BoxData[ \(\(geodfig0000 = Graphics3D[{RGBColor[0, 0.5, 0.5], Thickness[0.003], Line[Table[\n\t\t\t\tgeod[t], {t, 0, 8*del00, del00/8}]]}, Boxed -> False]; \)\)], "Input"] }, Closed]], Cell[BoxData[ \(Show[geodfig0000]\)], "Input"], Cell[BoxData[ \(Timing[numarclgth[x, aa, bb, 0, 8*del00]]\)], "Input"], Cell[BoxData[ \(unitnormalc[ t_] := {{0, \(-1\)}, {1, 0}} . \((D[c[tt], tt] /. tt -> t)\)/ Sqrt[\((D[c[tt], tt] /. tt -> t)\) . \((D[c[tt], tt] /. tt -> t)\)] \)], "Input"], Cell[BoxData[ \(unitnormalc[0.7]\)], "Input"], Cell[BoxData[ \(funct[t_] = \(-t\) \((t - 8*del00)\) \((Random[Real, {\(-1\), 1}] + Random[Real, {\(-1\), 1}]*t + Random[Real, {\(-1\), 1}]*t^2 + Random[Real, {\(-1\), 1}]*t^3 + Random[Real, {\(-1\), 1}]*t^4)\)\)], "Input"], Cell[BoxData[ \(funct[t_] = \(-\((\(-1.28000000000000003`\) + t)\)\)\ t\ \((\(0.544392402205752645`\[InvisibleSpace]\) - 0.290441081374759093`\ t + 0.222763043556791107`\ t\^2 + 0.498062437379438005`\ t\^3 - 0.857339818325579017`\ t\^4)\)\)], "Input"], Cell[BoxData[ \(Plot[funct[t], {t, 0, 8*del00}]\)], "Input"], Cell[BoxData[ \(\(varcc[k_]\)[t_] := c[t] + k*funct[t]*unitnormalc[t]; \n \(varcc11[k_]\)[t_] := \(\(varcc[k]\)[t]\)[\([1]\)]; \n \(varcc22[k_]\)[t_] := \(\(varcc[k]\)[t]\)[\([2]\)]; \)], "Input"], Cell[BoxData[ \(\(\(varxx[k_]\)[t_] = x[\(varcc11[k]\)[t], \(varcc22[k]\)[t]]; \)\)], "Input"], Cell[BoxData[ \(\(ccfig0000 := ParametricPlot[c[t] // Evaluate, {t, 0, 8*del00}, PlotStyle -> RGBColor[0, 0.5, 0.5], AspectRatio -> Automatic, Ticks -> None, AxesLabel -> {u, v}, PlotRange -> {{\(-0.1\), 1}, {\(-0.1\), 1}}, DisplayFunction -> Identity]; \)\)], "Input"], Cell[BoxData[ \(\(ccfig00[k_] := ParametricPlot[\(varcc[k]\)[t] // Evaluate, {t, 0, 8*del00}, AspectRatio -> Automatic, Ticks -> None, AxesLabel -> {u, v}, PlotRange -> {{\(-0.1\), 1}, {\(-0.1\), 1}}, DisplayFunction -> Identity]; \)\)], "Input"], Cell[BoxData[ \(\(ccfig[k_] := Show[ccfig0000, ccfig00[k], DisplayFunction -> Identity]; \)\)], "Input"], Cell[BoxData[ \(\(Show[ccfig[0.4], DisplayFunction -> $DisplayFunction]; \)\)], "Input"], Cell[BoxData[ \(\(geodfig00[k_] := ParametricPlot3D[\(varxx[k]\)[t] // Evaluate, {t, 0, 8*del00}, AspectRatio -> Automatic, Boxed -> False, Axes -> None, DisplayFunction -> Identity]; 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