(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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[ 1193284, 37326]*) (*NotebookOutlinePosition[ 1238419, 38890]*) (* CellTagsIndexPosition[ 1238277, 38882]*) (*WindowFrame->Normal*) Notebook[{ Cell[GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHg"], "Graphics", ShowCellBracket->False, CellMargins->{{25, 24}, {5, 7}}, ImageSize->{81, 22}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "Differential Equations&", StyleBox["Mathematica", FontSlant->"Italic"], "\n", StyleBox["Authors: Bill Davis and Jerry Uhl ", "Subtitle"], StyleBox["\[Copyright]1999", "Subtitle", FontSize->12], StyleBox["\nProducer: Bruce Carpenter\n", "Subtitle"], StyleBox["Publisher: ", "Subtitle", FontSize->12], StyleBox[ButtonBox["Math Everywhere, Inc.", ButtonData:>{ URL[ "http://www.matheverywhere.com"], None}, ButtonStyle->"MEIHyperlink"], FontSize->12], StyleBox[" Distributor: ", "Subtitle", FontSize->12], StyleBox[ButtonBox["Wolfram Research, Inc.", ButtonData:>{ URL[ "http://www.wolfram.com"], None}, ButtonStyle->"MEIHyperlink", ButtonNote->"Makers of Mathematica!"], FontSize->12] }], "PrefaceTitle", CellMargins->{{Inherited, Inherited}, {Inherited, 0}}], Cell[TextData[{ "DE.09 The Heat Equation \n ", Cell[BoxData[ \(\[PartialD]\_{x, 2}temp[x, t] = \[PartialD]\_t temp[x, t]\)]], "\nand the Wave Equation\n ", Cell[BoxData[ \(\(\(\ \)\(\[PartialD]\_{x, 2}position[x, t] = \[PartialD]\_{t, 2}position[x, t]\)\)\)]], "\n", StyleBox[" Sampler", FontSize->16, FontSlant->"Italic"] }], "Title"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " Initializations" }], "Special2"], Cell["\<\ Needs[\"Graphics`Colors`\"]; Needs[\"Graphics`FilledPlot`\"]; If[MemberQ[{\"Global`\",\"Graphics`Common`GraphicsCommon`\",\"Vector3D`\"}, Context[ScaleFactor]],Remove[\"*`ScaleFactor\"]]; <<\"Graphics`Legend`\"; <<\"Graphics`Common`GraphicsCommon`\"; Remove[\"*`ScaleFactor\"]; Graphics`Colors`GosiaGreen=RGBColor[0, 0.392187, 0]; Off[General::spell]; Off[General::spell1]; Off[ParametricPlot3D::ppcom]; Off[ParametricPlot::ppcom]; Off[Plot::plnr]; Needs[\"Calculus`Master`\"]; Off[Unique::\"usym\"]; SetOptions[Limit,Analytic->True];\ \>", "Input", InitializationCell->True], Cell["\<\ CMView = {2.7, 1.6, 1.2}; Clear[FastFourierfit,Fourierfitters,F,Fvalues,n,k, jump,num,numtab,coeffs,t,L]; jump[n_] := jump[n] = N[1/(2 n)]; Fvalues[F_,L_,n_] := N[Table[F[L t],{t,0,1 - jump[n],jump[n]}]]; numtab[n_] := numtab[n] = Table[k,{k,1,n}]; Fourierfitters[L_,n_,t_] := Table[E^(2 Pi I k t/L), {k,-n + 1,n - 1}]; coeffs[n_,list_] := Join[Reverse[Part[Fourier[list],numtab[n]]], \tPart[InverseFourier[list],Drop[numtab[n],1]]]/ \t\tN[Sqrt[Length[list]]] FastFourierfit[F_,L_,n_,t_] := Chop[Fourierfitters[L,n,t].coeffs[n,Fvalues[F,L,n]]]; L:=Expand[a D[#1,{#2,2}] + b D[#1,#2] + c #1]&\ \>", "Input", InitializationCell->True], Cell["Off[Graphics::gpt]", "Input", InitializationCell->True], Cell[CellGroupData[{ Cell["New Vector Drawers", "Section"], Cell[BoxData[ \(\( (*\ \( : Date : \ \ Copyright\ 1999\), \ Math\ Everywhere, \ \(Inc . \)\ \n*) \n (*\ \ \( : Drawbacks : \ \ Since\ the\ vectors\ are\ drawn\ out\ of\ \n\ \ \ \ \ context\), \ the\ shapes\ of\ the\ arrowheads\ are\ only\ \n\ \ \ \ \ correct\ with\ AspectRatio\ set\ to\ \(Automatic . \)\n*) \)\)], "Input"], Cell["BeginPackage[\"Vector3D`\",\"Graphics`Colors`\"];", "Input", InitializationCell->True], Cell["\<\ Axes3D::usage = \"Axes3D[a, b] creates a Graphics3D object of \ cartesian axes with x, y, and z running from -a/3 to a, and with axes labels \ b units beyond the tips of the axes. Axes3D[a] is Axes3D[a, a/8].\"; Perpend::usage = \"Perpend[a] returns a unit vector which is perpendicular to \ the vector a.\"; Arrow::usage = \"Arrow[a] produces a 2 or 3 dimensional vector from the \ origin to a. Arrow[a, Tail \[Rule] tail] gives a vector from tail to a.\"; \t Vector::usage = \"Vector[a, b] gives a 2D or 3D vector from a to b.\"; ArrowHead::usage = \"ArrowHead[a, vec] produces an arrowhead with its tip \ placed at the point a, pointing in the direction of the vector vec.\"; Tail::usage = \"Tail \[Rule] point puts the tail of the vector at point.\"; Norm::usage = \"Norm[vec] returns the Euclidean length of vec.\"; Normalize::usage = \"Normalize[vec] returns a unit vector in the direction of \ vec, provided vec is nonzero.\"; Aperture::usage = \"Aperture is the ratio of the radius of the base to the \ length of the head of a vector.\"; TipSize::usage = \"TipSize specifies an absolute size for the inner tip \ length of the head of a vector.\"; TipRatio::usage = \"TipRatio is the ratio of the inner length of the \ arrowhead to the length of the shaft of a vector.\"; HeadRatio::usage = \"HeadRatio is the ratio of the outer length of the \ arrowhead to the length of the shaft of a vector.\"; HeadSize::usage = \"HeadSize specifies an absolute size for the head of a \ vector.\"; ScaleFactor::usage = \"ScaleFactor specifies the amount to scale a vector in \ length and may be either a number or a pure function.\"; ZeroVectorPointSize::usage = \"ZeroVectorPointSize is the size of the point \ used to represent a zero vector.\"; Options[Vector] = Options[Arrow] = Options[ArrowHead] = {HeadRatio \[Rule] 0.18, TipRatio \[Rule] 0.14, Aperture \[Rule] 0.3, \ ScaleFactor \[Rule] 1, HeadType \[Rule] Polygon, ShaftQ \[Rule] True, EdgesQ \ \[Rule] True, VectorColor \[Rule] RGBColor[0,0,1], ShaftWidth \[Rule] 0.005, \ ZeroVectorPointSize \[Rule] 0.01}; \t\t Begin[\"`Private`\"]; SetOptions[ParametricPlot,AspectRatio \[Rule] Automatic]; SetOptions[Plot,AspectRatio \[Rule] Automatic]; SetOptions[Graphics,AspectRatio \[Rule] Automatic]; Axes3D[u_,v_] := Graphics3D[{ {Blue,Line[{{-u/3,0,0},{u,0,0}}]}, Text[\"x\",{u + v,0,0}], {Blue,Line[{{0,-u/3,0},{0,u,0}}]}, Text[\"y\",{0, u + v,0}], {Blue,Line[{{0,0,-u/3},{0,0,u}}]}, Text[\"z\",{0,0,u + v}]}]; Axes3D[u_] := Axes3D[u,u/8]; \t\t\t\t Norm[a_?VectorQ] := N[Sqrt[a.a]]; Normalize[a_?VectorQ] := If[a.a == 0, a, a/Norm[a]]; Perpend[a_?VectorQ] := Normalize[{a[[2]],-a[[1]]}]/;Length[a]==2 Perpend[a_?VectorQ] := \ Normalize[If[a[[1]]==0,{1,0,0},{0,a[[3]],-a[[2]]}]]/;Length[a]==3 Base3D = Table[{Cos[2 k Pi/8.], Sin[2 k Pi/8.]},{k,0,8}]; Vector[from_?VectorQ, to_?VectorQ, opts___] := Module[ {sq, sw, ht, hs, hr, tr, ap, sf, vc, zvps, diff = to - from, tip}, {sq, sw, ht, hs, hr, ts, tr, ap, sf, vc, zvps} = {ShaftQ, ShaftWidth, \ HeadType, HeadSize, HeadRatio, TipSize, TipRatio, Aperture, ScaleFactor, \ VectorColor, ZeroVectorPointSize} /. {opts} /. Options[Vector]; If[from==to, Return[Graphics[{PointSize[zvps],vc,Point[from]}]]]; diff = If[NumberQ[N[sf]], sf diff, sf[diff]]; tip = from + diff; If[NumberQ[hs], hr = hs/Norm[diff]; tr = 0.75 hr]; If[NumberQ[ts], tr = ts/Norm[diff]]; Graphics[{If[sq,{vc,Thickness[sw],Line[{from,tip - tr diff}]},{}], With[{trans = ap hr Norm[diff] Perpend[diff]}, {vc, Thickness[sw], ht[{tip, (tip - hr diff + trans), (tip - tr diff), (tip - hr diff - trans),tip}]}]}] ] /; Length[from]==Length[to]==2 Vector[from_?VectorQ, to_?VectorQ, opts___] := Module[ {sq, eq, sw, hs, hr, tr,ap, sf, vc, zvps, diff = to - from, tip}, {sq, eq, sw, hs, hr, ap, sf, vc, zvps} = {ShaftQ, EdgesQ, ShaftWidth, \ HeadSize, HeadRatio, Aperture, ScaleFactor, VectorColor, ZeroVectorPointSize} \ /. {opts} /. Options[Vector]; If[from==to, Return[Graphics3D[{PointSize[zvps],vc,Point[from]}]]]; diff = If[NumberQ[N[sf]], sf diff, sf[diff]]; tip = from + diff; If[NumberQ[hs], hr = hs/Norm[diff]]; tr = hr/2; Graphics3D[{If[sq,{vc,Thickness[sw],Line[{from,tip-tr diff}]},{}], If[eq, \ {}, EdgeForm[]], {SurfaceColor[vc], With[{temp = Perpend[diff]}, Polygon[Append[#,tip]]& /@ Partition[(tip - hr diff + #)&/@((ap hr \ Norm[diff] Base3D) . {temp,Normalize[Cross[temp,diff]]}),2,1]]}}] ] /; Length[from]==Length[to]==3 Arrow[a_, Tail\[Rule]b_, opts___] := Vector[b,a+b,opts] Arrow[a_, opts___] := Vector[Table[0,{Length[a]}],a,opts] ArrowHead[a_, b_, opts___] := Vector[a-b,a,ShaftQ\[Rule]False,opts]; End[]; EndPackage[]; Null\ \>", "Input", InitializationCell->True], Cell[CellGroupData[{ Cell["VectorField and Phase Lines", "Subsubsection"], Cell[TextData[{ "(* :Title: VectorAndPhase.m *)\n\n(* :Author: Bill Davis *)\n\n(* \ :Dates: about March, 1996 *)\n\n(* :Drawbacks: I wish someone serious \ would come and turn these into\n Really carefully done \ packages. *)\n\nBeginPackage[\"PhaseStuff`\",\"Vector3D`\",\n\t\t\ \"Utilities`FilterOptions`\"];\n\nPhaseLine::usage = \ \"PhaseLine[fcn,{var,a,b},tint,pos]\n\tmakes a vertical phase line for the \ function, fcn, of the variable, var,\n\tfor the variable going from a to b, \ with \n\ta color given by tint, at the x-coordinate pos.\";\n\t\n\ PhaseLine3D::usage = \"PhaseLIne3D[fcn,{var,a,b},tint,loc]\n\tmakes a \ vertical phase line for the function, fcn, of the variable, var,\n\tfor the \ variable going from a to b, with \n\ta color given by tint, and base point at \ loc = {x,y}\";\n\t\nVectorField::usage = \ \"VectorField[{function,{var1,a,b,space1},{var2,c,d,space2}]\n\twill make a \ vector field of normalized vectors, colored by\n\tmagnitude, for the function \ of the variables on the\n\trange {r1,r2} with spacing between points given by \ \n\t{space1, space2}. If a single function of a single variable is given,\n\t\ the vector field {1,f} will be plotted.\";\n\t\nStikPikXZ::usage = \ \"StikPikXZ[picture,a:_0] takes a 2 dimensional\n\t", StyleBox["Mathematica", FontSlant->"Italic"], " picture and embeds it on the XZ plane at y = a. \n\tSome optional \ characteristics of the graphs are lost in the \n\tprocess. StikPikYZ and \ StikPikXZ are similar.\";\n\t\nStikPikYZ::usage = \"StikPikYZ[picture,a:_0] \ takes a 2 dimensional\n\t", StyleBox["Mathematica", FontSlant->"Italic"], " picture and embeds it on the XZ plane at y = a. \n\tSome optional \ characteristics of the graphs are lost in the \n\tprocess. StikPikYZ and \ StikPikXZ are similar.\";\n\t\nStikPikXY::usage = \"StikPikXY[picture,a:_0] \ takes a 2 dimensional\n\t", StyleBox["Mathematica", FontSlant->"Italic"], " picture and embeds it on the XZ plane at y = a. \n\tSome optional \ characteristics of the graphs are lost in the \n\tprocess. StikPikYZ and \ StikPikXZ are similar.\";\n\t\nBegin[\"`Private`\"];\t\n\n\ PhaseLine[fcn_,{var_,a_,b_},tint_,pos_:0] := \n\t\ Module[{len,len1,k,zset,zero,zero1,signs,stick,dots,pointers},\n\tzset = \ Solve[fcn==0,var];\n\tzero = If[Length[zset]==0,{},var/.zset];\n\tzero = \ Sort[zero];\n\tzero = Select[zero,((# >= a)&&(# <= b))&];\n\tzero1 = \ Sort[Union[zero,{a,b}]];\n\tlen = Length[zero];\n\tlen1 = Length[zero1];\n\t\ signs = Table[Sign[fcn/.var->((zero1[[k+1]] + zero1[[k]])/2)],\n\t\t\ {k,1,len1-1}];\n\tstick = {Thickness[0.007],Line[{{pos,a},{pos,b}}]};\n\tdots \ = Point[{pos,#}]&/@zero;\n\tdots = Join[{PointSize[0.05],tint},dots];\n\t\ heads = Table[MakeHead[zero1[[k]],zero1[[k+1]],pos,\n\t\t\ signs[[k]],b-a,tint],{k,1,len1-1}];\n\t\ Return[{Graphics[Join[{tint},stick,dots]],heads}]\n\t\t]\n\t\t\n\ MakeHead[a_,b_,c_,s_,slen_,tint_] := Module[{u,v,w},\n\tu = Min[2(b-a),3/4 \ slen];\n\tReturn[ArrowHead[{c,a(.5-s/5)+b(.5+s/5)},\n\t\t{0,s \ u},VectorColor->tint]]]\n\t\t\n\ PhaseLine3D[fcn_,{var_,a_,b_},tint_,pos_:{0,0}] := \n\t\ Module[{len,len1,k,zset,zero,zero1,signs,stick,dots,pointers},\n\tzset = \ Solve[fcn==0,var];\n\tzero = If[Length[zset]==0,{},var/.zset];\n\tzero = \ Sort[zero];\n\tzero = Select[zero,((# >= a)&&(# <= b))&];\n\tzero1 = \ Sort[Union[zero,{a,b}]];\n\tlen = Length[zero];\n\tlen1 = Length[zero1];\n\t\ signs = Table[Sign[fcn/.var->((zero1[[k+1]] + zero1[[k]])/2)],\n\t\t\ {k,1,len1-1}];\n\tstick = {Thickness[0.007],Line[{{pos[[1]],pos[[2]],a},\n\t\t\ {pos[[1]],pos[[2]],b}}]};\n\tdots = Point[{pos[[1]],pos[[2]],#}]&/@zero;\n\t\ dots = Join[{PointSize[0.03],tint},dots];\n\theads = \ Table[MakeHead3D[zero1[[k]],zero1[[k+1]],pos,\n\t\t\ signs[[k]],b-a,tint],{k,1,len1-1}];\n\t\ Return[{Graphics3D[Join[{tint},stick,dots]],heads}]\n\t\t]\n\t\t\n\ MakeHead3D[a_,b_,c_,s_,slen_,tint_] := Module[{u,v,w},\n\tu = Min[2(b-a),1/2 \ slen];\n\tReturn[ArrowHead[{c[[1]],c[[2]],a(.5-s/5)+b(.5+s/5)},\n\t\t{0,0,s \ u},VectorColor->tint]]]\n\t\t\n\t\t\n\ VectorField[function_List,{var1_,a_,b_,sp1_:1},{var2_,c_,d_,sp2_:1}] := \n\t\ Module[{i,j,U,V,UU,VV,ulow,uhigh,vlow,vhigh,maxval,minval,\n\t\t\ lengths,directions,locations,mlen},\n\tmlen = Min[sp1,sp2];\n\tnormf[uu_,vv_] \ = Sqrt[function[[1]]^2+function[[2]]^2]/.\n\t\t{var1->uu,var2->vv};\n\t\ lengths = Table[N[normf[U,V]],{U,a,b,sp1},{V,c,d,sp2}];\n\tmaxval = \ Max[lengths]; minval = Min[lengths];\n\tIf[minval == maxval,maxval += 1];\n\t\ locations = Table[{uu,vv},{uu,a,b,sp1},\n\t\t{vv,c,d,sp2}];\n\tdirections = \ Table[If[N[normf[U,V]]==0,{0,0},\n\t\tmlen (function/.{var1->U,var2->V})/(1.3 \ N[normf[U,V]])],\n\t\t\t{U,a,b,sp1},{V,c,d,sp2}];\n\t{UU,VV} = \ Dimensions[directions][[{1,2}]];\n\tReturn[Table[Arrow[directions[[i,j]],\n\t\ \t\tTail->locations[[i,j]],Aperture->1,\n\t\t\t\ HeadRatio->.4,ShaftWidth->0.008, VectorColor->\n\t\t\t\ RGBColor[(lengths[[i,j]] - minval)/(maxval-minval),0,\n\t\t\t\t(maxval \ -lengths[[i,j]])/(maxval-minval)]],{i,UU},{j,VV}]]\n\t\t]\n\t\t\n\ VectorField[function_,{var1_,a_,b_,sp1_:1},{var2_,c_,d_,sp2_:1}] :=\n\t\ VectorField[{1,function},{var1,a,b,sp1},{var2,c,d,sp2}]\n\t\n\t\n\ Needs[\"Utilities`FilterOptions`\"]\n\nFixOptions[a_] := Module[{picture = a, \ opt3},\nIf[Options[picture] != {}, \n\topt3 = \ FilterOptions[Graphics3D,#]&/@Options[picture];\n\t(* Fix things that might \ carry 2D characteristics. *)\n\topt3 = opt3//. {(AxesLabel -> _) ->(AxesLabel \ -> None),\n\t\t\t(PlotRange->_) -> (PlotRange->Automatic),\n\t\t\t(PlotRegion \ -> _) -> (PlotRegion -> Automatic),\n\t\t\t(Ticks -> _) -> (Ticks -> \ Automatic)};\n\tpicture = Apply[List,picture];\n\tpicture[[-1]] = opt3;\n\t\ picture = Apply[Graphics3D,picture]];\n\tReturn[picture]\n\t]\n\n\ StikPikXZ[picture_,a_:0] := Module[{PIK3 = picture},\n\tPIK3 = \ PIK3//.{Graphics->Graphics3D,\n\t\t{u_?NumberQ,v_?NumberQ}:>{u,a,v}};\n\tPIK3 \ = FixOptions[PIK3];\n\tReturn[PIK3]\n\t\t]\n\n\nStikPikYZ[picture_,a_:0] := \ Module[{PIK3 = picture},\n\tPIK3 = PIK3//.{Graphics->Graphics3D,\n\t\t\ {u_?NumberQ,v_?NumberQ}:>{a,u,v}};\n\tPIK3 = FixOptions[PIK3];\n\t\ Return[PIK3]\n\t\t]\n\n\nStikPikXY[picture_,a_:0] := Module[{PIK3 = picture},\ \n\tPIK3 = PIK3//.{Graphics->Graphics3D,\n\t\t\ {u_?NumberQ,v_?NumberQ}:>{u,v,a}};\n\tPIK3 = FixOptions[PIK3];\n\t\ Return[PIK3]\n\t\t]\n\t\t\nEnd[];\n\nEndPackage[];\nNull" }], "Input", InitializationCell->True] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["ThreeAxes[u,v]", "Subsubsection"], Cell["\<\ ThreeAxes[u_,v_] := Graphics3D[{ {Blue,Line[{{-u,0,0},{u,0,0}}]}, Text[\"x\",{u + v,0,0}], {Blue,Line[{{0,-u,0},{0,u,0}}]}, Text[\"y\",{0, u + v,0}], {Blue,Line[{{0,0,0},{0,0,u}}]}, Text[\"z\",{0,0,u + v}]}]; ThreeAxes[u_] := ThreeAxes[u,u/8]; ThreeAxes::usage = \"ThreeAxes[a,b] makes a standard cartesian axis graphics \ object \twith x, y, and z running from -a to a, and with axis \tlabels b units beyond the tips of the axes. 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coefficients of the ", Cell[BoxData[ \(Sin[\(\((k\ \[Pi])\)\ x\)\/L]\)]], " terms:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[A, k];\)\), "\n", \(\(A[k_]\ := \ Coefficient[riggedsinefit[x], Sin[\((k\ Pi/L)\)\ x]];\)\n\), "\n", \(Table[A[k], {k, 1, n}]\)}], "Input"], Cell[BoxData[ \({1.225903745671842`, 0, \(-0.14555519988957555`\), 0, 0.06000000000000004`, 0, \(-0.03778848551047502`\), 0, 0.030752568928107404`, 0}\)], "Output"] }, Open ]], Cell[TextData[{ "The reason you run ", Cell[BoxData[ \(k\)]], " from ", Cell[BoxData[ \(1\)]], " to ", Cell[BoxData[ \(n\)]], " is to pick up all the coefficients." }], "Special5", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Now you're done because you can write down ", Cell[BoxData[ \(temp[x, t]\)]], ".\nIt's just:" }], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[temp, x, t];\)\), "\n", \(temp[x_, t_] = 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]], Cell[CellGroupData[{ Cell[TextData[{ "This is a plot of ", Cell[BoxData[ \(startertemp[x]\)]], " and ", Cell[BoxData[ \(temp[x, 0]\)]], " for ", Cell[BoxData[ \(0 \[LessEqual] x \[LessEqual] L\)]], ".\nWhy is it totally natural to expect the this plot to come out they way \ it did? " }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["Physical Answer:", "Special1", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "To begin with, \n ", Cell[BoxData[ \(temp[x, t]\)]], " \nestimates the temperature of the wire at position ", Cell[BoxData[ \(x\)]], " at ", Cell[BoxData[ \(t\)]], " time units after the experiment starts.\nBut \n ", Cell[BoxData[ \(startertemp[x]\)]], " \nmeasures the the temperature of the wire at position ", Cell[BoxData[ \(x\)]], " at the start (at time ", Cell[BoxData[ \(t = 0\)]], "). So it is natural that\n ", Cell[BoxData[ \(temp[x, 0]\)]], " should estimate ", Cell[BoxData[ \(starter[x]\)]], " \npretty well." }], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell["Math Answer: ", "Special1", Evaluatable->False, AspectRatioFixed->True], Cell["Look at:", "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(temp[x, t]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(1.225903745671842`\ \[ExponentialE]\^\(-\(\(\[Pi]\^2\ t\)\/25\)\)\ Sin[\ \(\[Pi]\ x\)\/5] - 0.14555519988957555`\ \[ExponentialE]\^\(-\(\(9\ \[Pi]\^2\ t\)\/25\)\)\ \ Sin[\(3\ \[Pi]\ x\)\/5] + 0.06000000000000004`\ \[ExponentialE]\^\(\(-\[Pi]\^2\)\ t\)\ Sin[\[Pi]\ \ x] - 0.03778848551047502`\ \[ExponentialE]\^\(-\(\(49\ \[Pi]\^2\ t\)\/25\)\)\ \ Sin[\(7\ \[Pi]\ x\)\/5] + 0.030752568928107404`\ \[ExponentialE]\^\(-\(\(81\ \[Pi]\^2\ \ t\)\/25\)\)\ Sin[\(9\ \[Pi]\ x\)\/5]\)], "Output"] }, Open ]], Cell[TextData[{ "When you put ", Cell[BoxData[ \(t = 0\)]], ", all the exponentials become ", Cell[BoxData[ \(1\)]], ":" }], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(temp[x, 0]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(1.225903745671842`\ Sin[\(\[Pi]\ x\)\/5] - 0.14555519988957555`\ Sin[\(3\ \[Pi]\ x\)\/5] + 0.06000000000000004`\ Sin[\[Pi]\ x] - 0.03778848551047502`\ Sin[\(7\ \[Pi]\ x\)\/5] + 0.030752568928107404`\ Sin[\(9\ \[Pi]\ x\)\/5]\)], "Output"] }, Open ]], Cell[TextData[{ "This is the same as the rigged sine fit of ", Cell[BoxData[ \(starter[x]\)]], ":" }], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(riggedsinefit[x]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(1.225903745671842`\ Sin[\(\[Pi]\ x\)\/5] - 0.14555519988957555`\ Sin[\(3\ \[Pi]\ x\)\/5] + 0.06000000000000004`\ Sin[\[Pi]\ x] - 0.03778848551047502`\ Sin[\(7\ \[Pi]\ x\)\/5] + 0.030752568928107404`\ Sin[\(9\ \[Pi]\ x\)\/5]\)], "Output"] }, Open ]], 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}], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(riggedsinefit[x]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(1.225903745671842`\ Sin[\(\[Pi]\ x\)\/5] - 0.14555519988957555`\ Sin[\(3\ \[Pi]\ x\)\/5] + 0.06000000000000004`\ Sin[\[Pi]\ x] - 0.03778848551047502`\ Sin[\(7\ \[Pi]\ x\)\/5] + 0.030752568928107404`\ Sin[\(9\ \[Pi]\ x\)\/5]\)], "Output"] }, Open ]], Cell[TextData[{ "And you get ", Cell[BoxData[ \(temp[x, t]\)]], " by inserting exponentials into each term of the rigged sine fit:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(temp[x, t]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(1.225903745671842`\ \[ExponentialE]\^\(-\(\(\[Pi]\^2\ t\)\/25\)\)\ Sin[\ \(\[Pi]\ x\)\/5] - 0.14555519988957555`\ \[ExponentialE]\^\(-\(\(9\ \[Pi]\^2\ t\)\/25\)\)\ \ Sin[\(3\ \[Pi]\ x\)\/5] + 0.06000000000000004`\ \[ExponentialE]\^\(\(-\[Pi]\^2\)\ t\)\ Sin[\[Pi]\ \ x] - 0.03778848551047502`\ \[ExponentialE]\^\(-\(\(49\ \[Pi]\^2\ t\)\/25\)\)\ \ Sin[\(7\ \[Pi]\ x\)\/5] + 0.030752568928107404`\ \[ExponentialE]\^\(-\(\(81\ \[Pi]\^2\ \ t\)\/25\)\)\ Sin[\(9\ \[Pi]\ x\)\/5]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Why do you insert those exponentials?", "Text", Evaluatable->False, AspectRatioFixed->True], Cell["Answer", "Special1", Evaluatable->False, AspectRatioFixed->True], Cell["This is the central question.", "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Engineering studies have shown that after the appropriate unit adjustments \ are made, the function temp[x,t] satisfies the partial diffeerential \ equation known as the heat equation\n ", Cell[BoxData[ \(\[PartialD]\_{x, 2}temp[x, t] = \[PartialD]\_t temp[x, t]\)]] }], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ This is the same as D[temp[x,t], {x,2}] = D[temp[x,t], t]; in other words the second derivative of temp[x,t[ with respect to x equals the first \ derivative of temp[x,t] with respect to t. \ \>", "Special5", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "with \n\[Rule] ", Cell[BoxData[ \(temp[x, 0] = startertemp[x]\)]], "\n and\n\[Rule] ", Cell[BoxData[ \(\(temp[0, t] = 0\ \ \)\)]], "and ", Cell[BoxData[ \(temp[L, t] = 0\)]], " for all ", Cell[BoxData[ \(t\)]], "'s.\n" }], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ Lots of folks call this a by the name \"Boundary Value Problem.\"\ \ \>", "Special5", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "The key is the boundary conditions \n ", Cell[BoxData[ \(\(temp[0, t] = 0\ \ \)\)]], "and ", Cell[BoxData[ \(temp[L, t] = 0\)]], " for all ", Cell[BoxData[ \(t\)]], "'s.\nThese match up well with the fact that\n ", Cell[BoxData[ \(Sin[\(k\ \[Pi]\ x\)\/L] = 0\)]], " for x = 0 and x = L\nfor all positive integers k. This suggests that for \ each fixed time t, you can fit temp[x,t] with a rigged Sine fit like this:" }], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[approxtemp, t, x, u, L];\)\), "\n", \(\(n = 6;\)\), "\n", \(approxtemp[x_, t_] = \[Sum]\+\(k = 1\)\%n u[t, k]\ Sin[\(\((k\ \[Pi])\)\ x\)\/L]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(Sin[\(\[Pi]\ x\)\/L]\ u[t, 1] + Sin[\(2\ \[Pi]\ x\)\/L]\ u[t, 2] + Sin[\(3\ \[Pi]\ x\)\/L]\ u[t, 3] + Sin[\(4\ \[Pi]\ x\)\/L]\ u[t, 4] + Sin[\(5\ \[Pi]\ x\)\/L]\ u[t, 5] + Sin[\(6\ \[Pi]\ x\)\/L]\ u[t, 6]\)], "Output"] }, Open ]], Cell["\<\ There is nothing magic taking n = 6. You can go with a bigger n or a smaller n.\ \>", "Special5", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "The Fourier fit coefficients ", Cell[BoxData[ \(u[t, k]\)]], " depend on t as well as k because you expect a different rigged sine fit \ at different times t.\nThe heat equation says\n D[temp[x,t],{x,2}] = \ D[temp[x,t],t].\nPlug \n ", Cell[BoxData[ \(approxtemp[x, t] = \[Sum]\+\(k = 1\)\%n u[t, k]\ Sin[\(\((k\ \[Pi])\)\ x\)\/L]\)]], "\ninto the heat equation and see that\n ", Cell[BoxData[ \(\[Sum]\+\(k = 1\)\%n u[t, k]\ \((\(k\ \[Pi]\)\/L)\)\^2\ \((\(-Sin[\(\((k\ \[Pi])\)\ x\)\/L]\))\) = \[Sum]\+\(k = 1\)\%6\[PartialD]\_t u[t, k]\ Sin[\(\((k\ \[Pi])\)\ x\)\/L]\)]], ".\nYou can make this happen by setting\n ", Cell[BoxData[ \(D[u[t, k], t] = \(-\((\(k\ \[Pi]\)\/L)\)\^2\)\ u[t, k]\)]], " .\nThis is a big break in your favor because this is the exponential \ differential equation and you know that this makes\n ", Cell[BoxData[ \(u[t, k] = A[k]\ E\^\(\(-\((\(k\ \[Pi]\)\/L)\)\^2\)\ t\)\)]], "\nwhere the real constant A[k] has yet to be determined." }], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell["You get a different constant for each k.", "Special5", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Substitute these u[t,k]'s into approxtemp[x,t] to get\n ", Cell[BoxData[ \(approxtemp[x, t] = \[Sum]\+\(k = 1\)\%n A[k]\ E\^\(\(-\((\(k\ \[Pi]\)\/L)\)\^2\)\ t\)\ Sin[\(\((k\ \[Pi])\)\ x\)\/L]\)]] }], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Look at what happens for t = 0:\n ", Cell[BoxData[ \(approxtemp[x, 0] = \[Sum]\+\(k = 1\)\%n A[k]\ \ Sin[\(\((k\ \[Pi])\)\ x\)\/L]\)]], "\n" }], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ This is to fit the starting temperature, so you lift the A[k]'s \ from the rigged sine fit of startertemp[x]. And now you see why you want to insert the exponentials.\ \>", "SmallText", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell["B.2.b) A new one", "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Start with a heated wire L units long with the temperature allowed to vary \ from position to position on the wire. Think of the wire as the interval 0 \ \[LessEqual] x \[LessEqual] L. ", Evaluatable->False, AspectRatioFixed->True, FontColor->RGBColor[0, 0, 0.493645]], StyleBox[ " At the start of the experiment, you instantly cool the ends at x = 0 and \ x = L and maintain these ends at temperature 0,", Evaluatable->False, AspectRatioFixed->True, FontColor->RGBColor[0, 0, 0.489738]], StyleBox[ " and you take pains to guarantee that the rest of the wire is perfectly \ insulated.\nAt the start of this particular experiment, the temperature of \ the wire at position x (for ", Evaluatable->False, AspectRatioFixed->True, FontColor->RGBColor[0, 0, 0.48191]], StyleBox["0 \[LessEqual] x \[LessEqual] ", Evaluatable->False, AspectRatioFixed->True, FontColor->RGBColor[0, 0, 0.493645]], StyleBox["L = 3) is given by the following function startertemp[x]:", Evaluatable->False, AspectRatioFixed->True, FontColor->RGBColor[0, 0, 0.48191]] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[{ \(\(L = 3;\)\), "\n", \(\(Clear[startertemp, x];\)\), 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The tip of the pointer tells you the starting temperature (at time t = 0) at \ the tail of the pointer. You now know how and understand how come up with a function temp[x,t] that \ estimates the temperature of the wire at position x at time t after the \ experiment begins. Do it. Make a good movie.\ \>", "Text"], Cell["Answer:", "Special1", Evaluatable->False, AspectRatioFixed->True], Cell["This is a copy, paste and edit job on B.2.a.i and a.ii)", "Special5", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "The function \n ", Cell[BoxData[ \(startertemp[x]\)]], "\n is ripe for a rigged Fourier Sine fit on ", Cell[BoxData[ \(\([0, L]\)\)]], "because \n ", Cell[BoxData[ \(startertemp[x] = 0\)]], " for ", Cell[BoxData[ \(x = 0\)]], " and ", Cell[BoxData[ \(x = L\)]], ":" }], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \({startertemp[0], startertemp[L]}\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({0, 0}\)], "Output"] }, Open ]], Cell["\<\ Rig startertemp[x] for a pure sine fit on [0,L] and get a good sine \ fit of startertemp[t] on [0,L]:\ \>", "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ If none of this makes sense to you, then look at B.1) immediately \ above.\ \>", 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Now look at the rigged sine fit of startertep[x]: \ \>", "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(riggedsinefit[x]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(0.379447765640153`\ Sin[\(\[Pi]\ x\)\/3] + 0.39396735644537356`\ Sin[\(2\ \[Pi]\ x\)\/3] + 0.297018655198057`\ Sin[\[Pi]\ x] + 0.06624330871697656`\ Sin[\(4\ \[Pi]\ x\)\/3] - 0.09499619818770802`\ Sin[\(5\ \[Pi]\ x\)\/3] - 0.06122865926876519`\ Sin[2\ \[Pi]\ x] - 0.028094336683998297`\ Sin[\(7\ \[Pi]\ x\)\/3] - 0.015807309025632543`\ Sin[\(8\ \[Pi]\ x\)\/3] - 0.006565563434718679`\ Sin[3\ \[Pi]\ x]\)], "Output"] }, Open ]], Cell["Pick off the coefficients of the Sin[(k Pi/L) x] terms:", "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[A, k];\)\), "\n", \(\(A[k_] := Coefficient[riggedsinefit[x], Sin[\(\((k\ \[Pi])\)\ x\)\/L]];\)\), "\n", \(coeffs = Table[A[k], {k, 1, n}]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \({0.379447765640153`, 0.39396735644537356`, 0.297018655198057`, 0.06624330871697656`, \(-0.09499619818770802`\), \ \(-0.06122865926876519`\), \(-0.028094336683998297`\), \ \(-0.015807309025632543`\), \(-0.006565563434718679`\), 0}\)], "Output"] }, Open ]], Cell["\<\ The reason you run k from 1 to n is to pick up all the \ coefficients.\ \>", "Special5", Evaluatable->False, AspectRatioFixed->True], Cell["Now you're done because you can write down temp[x,t]:", "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[temp, x, t];\)\), "\n", \(temp[x_, t_] = \[Sum]\+\(k = 1\)\%\(Length[coeffs]\)A[ k]\ E\^\(\(-\((\(k\ \[Pi]\)\/L)\)\^2\)\ t\)\ Sin[\(\((k\ \[Pi])\)\ \ x\)\/L]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(0.379447765640153`\ \[ExponentialE]\^\(-\(\(\[Pi]\^2\ t\)\/9\)\)\ \ Sin[\(\[Pi]\ x\)\/3] + 0.39396735644537356`\ \[ExponentialE]\^\(-\(\(4\ \[Pi]\^2\ t\)\/9\)\)\ \ Sin[\(2\ \[Pi]\ x\)\/3] + 0.297018655198057`\ \[ExponentialE]\^\(\(-\[Pi]\^2\)\ t\)\ Sin[\[Pi]\ \ x] + 0.06624330871697656`\ \[ExponentialE]\^\(-\(\(16\ \[Pi]\^2\ t\)\/9\)\)\ \ Sin[\(4\ \[Pi]\ x\)\/3] - 0.09499619818770802`\ \[ExponentialE]\^\(-\(\(25\ \[Pi]\^2\ t\)\/9\)\)\ \ Sin[\(5\ \[Pi]\ x\)\/3] - 0.06122865926876519`\ \[ExponentialE]\^\(\(-4\)\ \[Pi]\^2\ t\)\ Sin[ 2\ \[Pi]\ x] - 0.028094336683998297`\ \[ExponentialE]\^\(-\(\(49\ \[Pi]\^2\ t\)\/9\)\)\ \ Sin[\(7\ \[Pi]\ x\)\/3] - 0.015807309025632543`\ \[ExponentialE]\^\(-\(\(64\ \[Pi]\^2\ t\)\/9\)\)\ \ Sin[\(8\ \[Pi]\ x\)\/3] - 0.006565563434718679`\ \[ExponentialE]\^\(\(-9\)\ \[Pi]\^2\ t\)\ Sin[ 3\ \[Pi]\ x]\)], "Output"] }, Open ]], Cell["Here's the movie:", "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[tempplotter];\)\), "\n", \(\(tempplotter[t_] := Plot[temp[x, t], {x, 0, L}, PlotStyle \[Rule] {{Thickness[0.01], Blue}}, PlotRange \[Rule] {0, 1.2}, AxesLabel \[Rule] {"\", "\< temperature\>"}, 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Think of the wire as \ the interval 0 \[LessEqual] x \[LessEqual] L. ", Evaluatable->False, AspectRatioFixed->True, FontColor->RGBColor[0, 0, 0.493645]], StyleBox[" At the start of the experiment, you instantly cool the ends at \ x = 0 and x = L and maintain these ends at temperature 0,", Evaluatable->False, AspectRatioFixed->True, FontColor->RGBColor[0, 0, 0.489738]], StyleBox[" and you take pains to guarantee that the rest of the wire is \ perfectly insulated.\nAt the start of this partricular experiment, the \ temperature of the wire at position x (for ", Evaluatable->False, AspectRatioFixed->True, FontColor->RGBColor[0, 0, 0.48191]], StyleBox["0 \[LessEqual] x \[LessEqual] L", Evaluatable->False, AspectRatioFixed->True, FontColor->RGBColor[0, 0, 0.493645]], StyleBox[" = 4) is given by the following startertemp data:", Evaluatable->False, AspectRatioFixed->True, FontColor->RGBColor[0, 0, 0.48191]] }], "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[{ \(\(startertempdata = {{0.0, 0.0}, {0.2, 0.25}, {0.4, 0.35}, {0.6, 0.48}, {0.8, 0.69}, 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l>H00`000000oo3V0?o`iP040?o`iP030000003ol>H0oo3V01@0oo3V00<000000?o`iP3ol>H09`3o l>H00`000000oo3V0?o`iP2j0?o`iP00o`3ol>HQ0?o`iP00o`3ol>HQ0?o`iP00\ \>"], ImageRangeCache->{{{0, 287}, {107, 0}} -> {-1.23883, -0.229926, 0.0197129, \ 0.0197129}}] }, Open ]], Cell["\<\ Think of the interval [0,L] = [0,4] as the wire. The tip of the pointer tells you the starting temperature (at time t = 0) at \ the tail of the pointer. The function startertemp[x] is ripe for a rigged Fourier Sine fit on [0,L] \ because startertemp[x] = 0 for x = 0 and x = L:\ \>", "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[ \({startertemp[0], startertemp[L]}\)], "Input"], Cell[BoxData[ \({\(-1.3877787807814457`*^-17\), 0.`}\)], "Output"] }, Open ]], Cell["\<\ Rig startertemp[x] for a pure sine fit on [0,L] and get a good sine \ fit of startertemp[t] on [0,L]:\ \>", "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[rigged];\)\), "\n", \(\(rigged[x_] := startertemp[x] /; 0 \[LessEqual] x \[LessEqual] L;\)\), "\n", \(\(rigged[x_] := \(-startertemp[2\ L - x]\) /; L < x \[LessEqual] 2\ L;\)\), "\n", \(\(n = 4;\)\), "\n", \(\(Clear[riggedsinefit];\)\), "\n", \(riggedsinefit[x_] = Chop[ComplexExpand[FastFourierfit[rigged, 2\ L, n, x]]]\)}], "Input"], Cell[BoxData[ \(1.2803300858899105`\ Sin[\(\[Pi]\ x\)\/4] - 0.21966991411008926`\ Sin[\(3\ \[Pi]\ x\)\/4]\)], "Output"] }, Open ]], 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Now look at the rigged sine fit of startertep[x]: \ \>", "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[ \(riggedsinefit[x]\)], "Input"], Cell[BoxData[ \(1.2768204387968707`\ Sin[\(\[Pi]\ x\)\/4] + 0.00039477370695176087`\ Sin[\(\[Pi]\ x\)\/2] - 0.11257724452067726`\ Sin[\(3\ \[Pi]\ x\)\/4] + 0.00022451398828978678`\ Sin[\[Pi]\ x] + 0.06979036790187179`\ Sin[\(5\ \[Pi]\ x\)\/4] + 0.007327655332728059`\ Sin[\(3\ \[Pi]\ x\)\/2] - 0.019982895983741138`\ Sin[\(7\ \[Pi]\ x\)\/4] + 0.0017633557568774599`\ Sin[2\ \[Pi]\ x] + 0.030991810984396035`\ Sin[\(9\ \[Pi]\ x\)\/4] + 0.0009999999999999705`\ Sin[\(5\ \[Pi]\ x\)\/2] - 0.00003124796748381065`\ Sin[\(11\ \[Pi]\ x\)\/4] + 0.002489898284882791`\ Sin[3\ \[Pi]\ x] + 0.02802867338636237`\ Sin[\(13\ \[Pi]\ x\)\/4] - 0.005709621343978297`\ Sin[\(7\ \[Pi]\ x\)\/2] + 0.02779036790187168`\ Sin[\(15\ \[Pi]\ x\)\/4] - 0.0028531695488853825`\ Sin[4\ \[Pi]\ x] - 0.0196746919732254`\ Sin[\(17\ \[Pi]\ x\)\/4] + 0.0002232602817980748`\ Sin[\(9\ \[Pi]\ x\)\/2] - 0.009242380333694023`\ Sin[\(19\ \[Pi]\ x\)\/4]\)], "Output"] }, Open ]], Cell["Pick off the coefficients of the Sin[(k Pi/L) x] terms:", "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[A, k];\)\), "\n", \(\(A[k_] := Coefficient[riggedsinefit[x], Sin[\(\((k\ \[Pi])\)\ x\)\/L]];\)\), "\n", \(coeffs = Table[A[k], {k, 1, n}]\)}], "Input"], Cell[BoxData[ \({1.2768204387968707`, 0.00039477370695176087`, \(-0.11257724452067726`\), 0.00022451398828978678`, 0.06979036790187179`, 0.007327655332728059`, \(-0.019982895983741138`\), 0.0017633557568774599`, 0.030991810984396035`, 0.0009999999999999705`, \(-0.00003124796748381065`\), 0.002489898284882791`, 0.02802867338636237`, \(-0.005709621343978297`\), 0.02779036790187168`, \(-0.0028531695488853825`\), \ \(-0.0196746919732254`\), 0.0002232602817980748`, \(-0.009242380333694023`\), 0}\)], "Output"] }, Open ]], Cell["\<\ The reason you run k from 1 to n is to pick up all the \ coefficients.\ \>", "Special5", Evaluatable->False], Cell["Now you're done because you can write down temp[x,t]:", "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[temp, x, t];\)\), "\n", \(temp[x_, t_] = \[Sum]\+\(k = 1\)\%n A[ k]\ E\^\(\(-\((\(k\ \[Pi]\)\/L)\)\^2\)\ t\)\ Sin[\(\((k\ \[Pi])\)\ \ x\)\/L]\)}], "Input"], Cell[BoxData[ \(1.2768204387968707`\ \[ExponentialE]\^\(-\(\(\[Pi]\^2\ t\)\/16\)\)\ \ Sin[\(\[Pi]\ x\)\/4] + 0.00039477370695176087`\ \[ExponentialE]\^\(-\(\(\[Pi]\^2\ t\)\/4\)\)\ \ Sin[\(\[Pi]\ x\)\/2] - 0.11257724452067726`\ \[ExponentialE]\^\(-\(\(9\ \[Pi]\^2\ t\)\/16\)\)\ \ Sin[\(3\ \[Pi]\ x\)\/4] + 0.00022451398828978678`\ \[ExponentialE]\^\(\(-\[Pi]\^2\)\ t\)\ Sin[\ \[Pi]\ x] + 0.06979036790187179`\ \[ExponentialE]\^\(-\(\(25\ \[Pi]\^2\ t\)\/16\)\)\ \ Sin[\(5\ \[Pi]\ x\)\/4] + 0.007327655332728059`\ \[ExponentialE]\^\(-\(\(9\ \[Pi]\^2\ t\)\/4\)\)\ \ Sin[\(3\ \[Pi]\ x\)\/2] - 0.019982895983741138`\ \[ExponentialE]\^\(-\(\(49\ \[Pi]\^2\ \ t\)\/16\)\)\ Sin[\(7\ \[Pi]\ x\)\/4] + 0.0017633557568774599`\ \[ExponentialE]\^\(\(-4\)\ \[Pi]\^2\ t\)\ Sin[ 2\ \[Pi]\ x] + 0.030991810984396035`\ \[ExponentialE]\^\(-\(\(81\ \[Pi]\^2\ \ t\)\/16\)\)\ Sin[\(9\ \[Pi]\ x\)\/4] + 0.0009999999999999705`\ \[ExponentialE]\^\(-\(\(25\ \[Pi]\^2\ \ t\)\/4\)\)\ Sin[\(5\ \[Pi]\ x\)\/2] - 0.00003124796748381065`\ \[ExponentialE]\^\(-\(\(121\ \[Pi]\^2\ t\)\/16\ \)\)\ Sin[\(11\ \[Pi]\ x\)\/4] + 0.002489898284882791`\ \[ExponentialE]\^\(\(-9\)\ \[Pi]\^2\ t\)\ Sin[ 3\ \[Pi]\ x] + 0.02802867338636237`\ \[ExponentialE]\^\(-\(\(169\ \[Pi]\^2\ \ t\)\/16\)\)\ Sin[\(13\ \[Pi]\ x\)\/4] - 0.005709621343978297`\ \[ExponentialE]\^\(-\(\(49\ \[Pi]\^2\ t\)\/4\)\)\ \ Sin[\(7\ \[Pi]\ x\)\/2] + 0.02779036790187168`\ \[ExponentialE]\^\(-\(\(225\ \[Pi]\^2\ \ t\)\/16\)\)\ Sin[\(15\ \[Pi]\ x\)\/4] - 0.0028531695488853825`\ \[ExponentialE]\^\(\(-16\)\ \[Pi]\^2\ t\)\ Sin[ 4\ \[Pi]\ x] - 0.0196746919732254`\ \[ExponentialE]\^\(-\(\(289\ \[Pi]\^2\ t\)\/16\)\)\ \ Sin[\(17\ \[Pi]\ x\)\/4] 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Now look at the rigged sine fit of starterposition[x]: \ \>", "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[ \(riggedsinefit[x]\)], "Input"], Cell[BoxData[ \(1.2771000543971815`\ Sin[\(\[Pi]\ x\)\/4] + 0.00041250033557908096`\ Sin[\(\[Pi]\ x\)\/2] - 0.11259077840142107`\ Sin[\(3\ \[Pi]\ x\)\/4] + 0.0002571712857632962`\ Sin[\[Pi]\ x] + 0.07033170515572852`\ Sin[\(5\ \[Pi]\ x\)\/4] + 0.007206538682408817`\ Sin[\(3\ \[Pi]\ x\)\/2] - 0.019159302070297552`\ Sin[\(7\ \[Pi]\ x\)\/4] + 0.0018419771952156096`\ Sin[2\ \[Pi]\ x] + 0.03191383427323069`\ Sin[\(9\ \[Pi]\ x\)\/4] + 0.00018856473338042627`\ Sin[\(5\ \[Pi]\ x\)\/2] + 0.00454808225658343`\ Sin[\(11\ \[Pi]\ x\)\/4] + 0.0015606320719520955`\ Sin[3\ \[Pi]\ x] + 0.01820073766121567`\ Sin[\(13\ \[Pi]\ x\)\/4] - 0.004322409735334761`\ Sin[\(7\ \[Pi]\ x\)\/2] + 0.01713835753061883`\ Sin[\(15\ \[Pi]\ x\)\/4] - 0.0018008901135731594`\ Sin[4\ \[Pi]\ x] - 0.007609972171873005`\ Sin[\(17\ \[Pi]\ x\)\/4]\)], "Output"] }, Open ]], Cell[TextData[{ "Pick off the coefficients of the ", Cell[BoxData[ \(Sin[\(\((k\ \[Pi])\)\ x\)\/L]\)]], " terms:" }], "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[ \(L\)], "Input"], Cell[BoxData[ \(4\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[A, k];\)\), "\n", \(\(A[k_] := Coefficient[riggedsinefit[x], Sin[\(\((k\ \[Pi])\)\ x\)\/L]];\)\), "\n", \(coeffs = Table[A[k], {k, 1, n}]\)}], "Input"], Cell[BoxData[ \({1.2771000543971815`, 0.00041250033557908096`, \(-0.11259077840142107`\), 0.0002571712857632962`, 0.07033170515572852`, 0.007206538682408817`, \(-0.019159302070297552`\), 0.0018419771952156096`, 0.03191383427323069`, 0.00018856473338042627`, 0.00454808225658343`, 0.0015606320719520955`, 0.01820073766121567`, \(-0.004322409735334761`\), 0.01713835753061883`, \(-0.0018008901135731594`\), \ \(-0.007609972171873005`\), 0}\)], "Output"] }, Open ]], Cell["Now you're done because you can write down position[x,t]:", "SmallText", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[position, x, t];\)\), "\n", \(position[x_, t_] = \[Sum]\+\(k = 1\)\%\(Length[coeffs]\)A[ k]\ Cos[\(\((k\ \[Pi])\)\ t\)\/L]\ Sin[\(\((k\ \[Pi])\)\ \ x\)\/L]\)}], "Input"], Cell[BoxData[ \(1.2771000543971815`\ Cos[\(\[Pi]\ t\)\/4]\ Sin[\(\[Pi]\ x\)\/4] + 0.00041250033557908096`\ Cos[\(\[Pi]\ t\)\/2]\ Sin[\(\[Pi]\ x\)\/2] - 0.11259077840142107`\ Cos[\(3\ \[Pi]\ t\)\/4]\ Sin[\(3\ \[Pi]\ x\)\/4] \ + 0.0002571712857632962`\ Cos[\[Pi]\ t]\ Sin[\[Pi]\ x] + 0.07033170515572852`\ Cos[\(5\ \[Pi]\ t\)\/4]\ Sin[\(5\ \[Pi]\ x\)\/4] \ + 0.007206538682408817`\ Cos[\(3\ \[Pi]\ t\)\/2]\ Sin[\(3\ \[Pi]\ x\)\/2] - 0.019159302070297552`\ Cos[\(7\ \[Pi]\ t\)\/4]\ Sin[\(7\ \[Pi]\ x\)\/4] \ + 0.0018419771952156096`\ Cos[2\ \[Pi]\ t]\ Sin[2\ \[Pi]\ x] + 0.03191383427323069`\ Cos[\(9\ \[Pi]\ t\)\/4]\ Sin[\(9\ \[Pi]\ x\)\/4] \ + 0.00018856473338042627`\ Cos[\(5\ \[Pi]\ t\)\/2]\ Sin[\(5\ \[Pi]\ x\)\/2] + 0.00454808225658343`\ Cos[\(11\ \[Pi]\ t\)\/4]\ Sin[\(11\ \[Pi]\ \ x\)\/4] + 0.0015606320719520955`\ Cos[3\ \[Pi]\ t]\ Sin[3\ \[Pi]\ x] + 0.01820073766121567`\ Cos[\(13\ \[Pi]\ t\)\/4]\ Sin[\(13\ \[Pi]\ \ x\)\/4] - 0.004322409735334761`\ Cos[\(7\ \[Pi]\ t\)\/2]\ Sin[\(7\ \[Pi]\ x\)\/2] \ + 0.01713835753061883`\ Cos[\(15\ \[Pi]\ t\)\/4]\ Sin[\(15\ \[Pi]\ x\)\/4] - 0.0018008901135731594`\ Cos[4\ \[Pi]\ t]\ Sin[4\ \[Pi]\ x] - 0.007609972171873005`\ Cos[\(17\ \[Pi]\ t\)\/4]\ Sin[\(17\ \[Pi]\ \ x\)\/4]\)], "Output"] }, Open ]], Cell["\<\ That's all there is to it. 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Show off your work with a plot.", Evaluatable->False, AspectRatioFixed->True, FontColor->RGBColor[0, 0, 0.493645]]], "Text", Evaluatable->False], Cell["G.1.a.iv)", "Subsubsection", Evaluatable->False], Cell["Look at these:", "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{\(Clear[f, t]\), ";", "\n", \(L\ = \ 2\ Pi\), ";", " ", "\n", RowBox[{\(f[t_]\), " ", StyleBox["=", FontColor->RGBColor[1, 0, 0]], StyleBox[" ", FontColor->RGBColor[1, 0, 0]], StyleBox[\(Sin[12\ t]\), FontColor->RGBColor[1, 0, 0]]}], StyleBox[" ", FontColor->RGBColor[1, 0, 0]], StyleBox[";", FontColor->RGBColor[1, 0, 0]], "\n", RowBox[{"Chop", "[", RowBox[{"ComplexExpand", "[", RowBox[{"FastFourierfit", "[", RowBox[{"f", ",", "L", ",", StyleBox["6", FontColor->RGBColor[1, 0, 0]], ",", "t"}], "]"}], "]"}], "]"}]}], "\n", RowBox[{"Chop", "[", RowBox[{"ComplexExpand", "[", RowBox[{"FastFourierfit", "[", RowBox[{"f", ",", "L", ",", StyleBox["14", FontColor->RGBColor[1, 0, 0]], ",", "t"}], "]"}], 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LineSpacing->{1, 0}], Cell[StyleData["Subsubtitle", "Printout"], CellMargins->{{14, Inherited}, {2, 2}}, FontColor->GrayLevel[0]], Cell[StyleData["Subsubtitle", "TwoColumn"], CellMargins->{{14, Inherited}, {2, 2}}, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Section"], CellDingbat->"\[GraySquare]", ShowCellBracket->True, ShowGroupOpenCloseIcon->True, CellMargins->{{22, Inherited}, {Inherited, 20}}, CellGroupingRules->{"SectionGrouping", 30}, PageBreakBelow->False, DefaultInlineFormatType->DefaultInputInlineFormatType, LineSpacing->{1.5, 1}, CounterIncrements->"Section", CounterAssignments->{{"Subsection", 0}, {"Subsubsection", 0}}, FontFamily->"Times", FontSize->16], Cell[StyleData["Section", "Presentation"], CellMargins->{{24, Inherited}, {Inherited, 20}}, LineSpacing->{1.5, 0}, FontSize->18], Cell[StyleData["Section", "Printout"], CellMargins->{{14, Inherited}, {2, 10}}], Cell[StyleData["Section", "TwoColumn"], CellMargins->{{14, Inherited}, {2, 10}}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsection"], CellDingbat->"", ShowCellBracket->True, ShowGroupOpenCloseIcon->True, CellMargins->{{19, Inherited}, {Inherited, 18}}, CellGroupingRules->{"SectionGrouping", 40}, PageBreakBelow->False, DefaultInlineFormatType->DefaultInputInlineFormatType, LineSpacing->{1.5, 1}, CounterIncrements->"Subsection", CounterAssignments->{{"Subsubsection", 0}}, FontFamily->"Times", FontSize->16, FontWeight->"Bold", FontColor->RGBColor[0, 0.392187, 0]], Cell[StyleData["Subsection", "Presentation"], CellMargins->{{24, Inherited}, {Inherited, 15}}], Cell[StyleData["Subsection", "Printout"], CellMargins->{{14, Inherited}, {2, 5}}, FontSize->12, FontColor->GrayLevel[0]], Cell[StyleData["Subsection", "TwoColumn"], CellFrame->{{0, 0}, {0, 1}}, CellMargins->{{14, Inherited}, {2, 10}}, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsubsection"], CellDingbat->"\[EmptySquare]", ShowClosedCellArea->True, CellMargins->{{18, Inherited}, {Inherited, 12}}, CellGroupingRules->{"SectionGrouping", 50}, PageBreakBelow->False, DefaultInlineFormatType->DefaultInputInlineFormatType, LineSpacing->{1.5, 1}, CounterIncrements->"Subsubsection", FontFamily->"Times", FontSize->14, FontWeight->"Bold", FontColor->RGBColor[0.689998, 0.0899977, 0.119997]], Cell[StyleData["Subsubsection", "Presentation"], CellMargins->{{24, Inherited}, {Inherited, 12}}, LineSpacing->{1, 0}], Cell[StyleData["Subsubsection", "Printout"], CellMargins->{{14, Inherited}, {2, 3}}, FontSize->12, FontColor->GrayLevel[0]], Cell[StyleData["Subsubsection", "TwoColumn"], CellMargins->{{14, Inherited}, {2, 3}}, FontColor->GrayLevel[0]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Body Text", "Section"], Cell[CellGroupData[{ Cell[StyleData["PrefaceText"], CellMargins->{{15, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, LineSpacing->{1, 1}, LimitsPositioningTokens->{}, StyleMenuListing->None, FontFamily->"Times", FontSize->10, FontWeight->"Plain"], Cell[StyleData["PrefaceText", "Presentation"], CellMargins->{{24, Inherited}, {Inherited, Inherited}}, LineSpacing->{2, 0}, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["PrefaceText", "Printout"], CellMargins->{{0, Inherited}, {2, 2}}, LineSpacing->{1, 1}, FontSize->12, FontColor->GrayLevel[0], Background->None], Cell[StyleData["PrefaceText", "TwoColumn"], CellMargins->{{0, Inherited}, {2, 2}}, LineSpacing->{1, 1}, FontSize->12, FontColor->GrayLevel[0], Background->None] }, Closed]], Cell[StyleData["PrefaceHyperlink"], CellMargins->{{15, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, LineSpacing->{1, 1}, LimitsPositioningTokens->{}, StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Times", FontSize->10, FontWeight->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonNote->ButtonData}], Cell[CellGroupData[{ Cell[StyleData["Text"], ShowClosedCellArea->True, CellMargins->{{15, 10}, {Inherited, Inherited}}, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, LineSpacing->{1.5, 1}, LimitsPositioningTokens->{}, FontFamily->"Times", FontSize->16, FontColor->RGBColor[0, 0, 0.500008]], Cell[StyleData["Text", "Presentation"], CellMargins->{{24, Inherited}, {Inherited, Inherited}}, LineSpacing->{2, 0}, FontSize->16, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["Text", "Printout"], CellMargins->{{14, Inherited}, {3, 1}}, PageBreakWithin->True, GroupPageBreakWithin->True, LineSpacing->{1, 2}, FontColor->GrayLevel[0], Background->None], Cell[StyleData["Text", "TwoColumn"], CellMargins->{{14, Inherited}, {3, 1}}, PageBreakWithin->True, GroupPageBreakWithin->True, LineSpacing->{1, 2}, FontSize->14, FontColor->GrayLevel[0], Background->None] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["SmallText"], ShowClosedCellArea->True, CellMargins->{{15, Inherited}, {Inherited, Inherited}}, DefaultInlineFormatType->DefaultInputInlineFormatType, LineSpacing->{1.5, 1}, LimitsPositioningTokens->{}, FontFamily->"Times", FontSize->16], Cell[StyleData["SmallText", "Presentation"], CellMargins->{{24, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}, FontSize->16], Cell[StyleData["SmallText", "Printout"], CellMargins->{{14, Inherited}, {2, 2}}, PageBreakWithin->True, GroupPageBreakWithin->True], Cell[StyleData["SmallText", "TwoColumn"], CellMargins->{{14, Inherited}, {2, 2}}, PageBreakWithin->True, GroupPageBreakWithin->True, FontSize->14] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Input/Output", "Section"], 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"], Cell[CellGroupData[{ Cell[StyleData["Input"], CellFrame->{{3, 0}, {0, 0}}, CellMargins->{{45, Inherited}, {Inherited, Inherited}}, Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, GroupPageBreakWithin->False, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultInputFormatType, AutoItalicWords->{}, FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, CounterIncrements->"Input", FontWeight->"Bold"], Cell[StyleData["Input", "Presentation"], CellFrame->{{3, 0}, {0, 0}}, CellMargins->{{45, Inherited}, {20, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Input", "Printout"], CellFrame->{{3, 0}, {0, 0}}, CellMargins->{{30, Inherited}, {2, 2}}, PageBreakWithin->True, GroupPageBreakWithin->True, FontSize->9], Cell[StyleData["Input", "TwoColumn"], CellFrame->{{3, 0}, {0, 0}}, CellMargins->{{30, Inherited}, {2, 2}}, PageBreakWithin->True, GroupPageBreakWithin->True, FontSize->12] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Output"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, CellEditDuplicate->True, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, LineSpacing->{1.5, 0}, FormatType->StandardForm, FontFamily->"Courier", FontSize->14, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["Output", "Presentation"], CellMargins->{{45, Inherited}, {20, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Output", "Printout"], CellMargins->{{30, Inherited}, {2, 2}}, PageBreakWithin->True, GroupPageBreakWithin->True, LineSpacing->{1, 0}, FontSize->10, FontColor->GrayLevel[0]], Cell[StyleData["Output", "TwoColumn"], CellMargins->{{30, Inherited}, {2, 2}}, PageBreakWithin->True, GroupPageBreakWithin->True, LineSpacing->{1, 0}, FontSize->12, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Message"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, FormatType->InputForm, StyleMenuListing->None, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["Message", "Presentation"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Message", "Printout"], CellMargins->{{30, Inherited}, {2, 2}}, FontSize->9, FontColor->GrayLevel[0]], Cell[StyleData["Message", "TwoColumn"], CellMargins->{{30, Inherited}, {2, 2}}, FontSize->9, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Print"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, FormatType->InputForm, StyleMenuListing->None], Cell[StyleData["Print", "Presentation"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Print", "Printout"], CellMargins->{{30, Inherited}, {2, 2}}, FontSize->9], Cell[StyleData["Print", "TwoColumn"], CellMargins->{{30, Inherited}, {2, 2}}, FontSize->16] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Info"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, FormatType->InputForm, StyleMenuListing->None], Cell[StyleData["Info", "Presentation"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Info", "Printout"], CellMargins->{{30, Inherited}, {Inherited, Inherited}}, FontSize->10], Cell[StyleData["Info", "TwoColumn"], CellMargins->{{30, Inherited}, {Inherited, Inherited}}, FontSize->12] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Graphics"], CellMargins->{{15, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"GraphicsGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, FormatType->InputForm, ImageMargins->{{35, Inherited}, {Inherited, 0}}, AnimationDisplayTime->0.2, StyleMenuListing->None, FontSize->14], Cell[StyleData["Graphics", "Presentation"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Graphics", "Printout"], CellMargins->{{30, Inherited}, {0, 0}}, CellFrameMargins->False, ImageSize->{Inherited, 150}, ImageMargins->{{45, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, FontSize->9], Cell[StyleData["Graphics", "TwoColumn"], CellMargins->{{20, Inherited}, {0, 0}}, CellFrameMargins->False, ImageSize->{Inherited, 150}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["CellLabel"], StyleMenuListing->None, FontFamily->"Times", FontSize->9, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["CellLabel", "Presentation"], FontSize->14], Cell[StyleData["CellLabel", "Printout"], FontColor->GrayLevel[1]], Cell[StyleData["CellLabel", "TwoColumn"], FontColor->GrayLevel[1]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Special Headings", "Section"], Cell[CellGroupData[{ Cell[StyleData["PrefaceTitle"], CellFrame->{{1, 1}, {1, 5}}, ShowCellBracket->False, CellMargins->{{24, 24}, {0, 10}}, CellGroupingRules->{"SectionGrouping", 30}, PageBreakBelow->False, CellFrameMargins->{{15, Inherited}, {Inherited, Inherited}}, DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Center, LineSpacing->{1.4, 1}, FontFamily->"Times", FontSize->24, FontWeight->"Bold", FontColor->GrayLevel[1], Background->RGBColor[0, 0.392187, 0]], Cell[StyleData["PrefaceTitle", "Presentation"], CellMargins->{{24, Inherited}, {60, Inherited}}, TextAlignment->Center, FontSize->38, FontColor->GrayLevel[1], Background->RGBColor[0.596078, 0.65098, 0.0196078]], Cell[StyleData["PrefaceTitle", "Printout"], CellMargins->{{0, Inherited}, {0, Inherited}}, FontSize->16, FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[StyleData["PrefaceTitle", "TwoColumn"], CellFrame->{{1, 1}, {0, 5}}, CellMargins->{{0, Inherited}, {0, Inherited}}, FontSize->16, FontColor->GrayLevel[0], Background->GrayLevel[1]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Special Body Text and Index", "Section"], Cell[CellGroupData[{ Cell[StyleData["Accident"], CellFrame->3, ShowCellBracket->False, CellMargins->{{24, 24}, {0, 10}}, CellFrameMargins->{{15, Inherited}, {Inherited, Inherited}}, TextAlignment->Center, LineSpacing->{1.4, 1}, FontFamily->"Times", FontSize->24, FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], Cell[StyleData["Accident", "Presentation"], CellMargins->{{24, Inherited}, {60, Inherited}}, TextAlignment->Center, FontSize->36], Cell[StyleData["Accident", "Printout"], CellFrame->2, CellMargins->{{0, Inherited}, {0, Inherited}}, FontSize->16, FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[StyleData["Accident", "TwoColumn"], CellFrame->2, CellMargins->{{0, Inherited}, {0, Inherited}}, FontSize->16, FontColor->GrayLevel[0], Background->GrayLevel[1]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["ContentsText"], CellMargins->{{50, 10}, {5, 5}}, FontFamily->"Times", FontSize->16], Cell[StyleData["ContentsText", "Presentation"]], Cell[StyleData["ContentsText", "Printout"], FontColor->GrayLevel[0], Background->None], Cell[StyleData["ContentsText", "TwoColumn"], FontColor->GrayLevel[0], Background->None] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Special1"], CellDingbat->"\[EmptySquare]", ShowClosedCellArea->True, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, FontFamily->"Times", FontSize->14, FontWeight->"Bold", FontColor->RGBColor[0.689998, 0.0899977, 0.119997]], Cell[StyleData["Special1", "Presentation"], FontSize->16], Cell[StyleData["Special1", "Printout"], FontSize->12, FontColor->GrayLevel[0]], Cell[StyleData["Special1", "TwoColumn"], FontSize->12, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Special2"], CellMargins->{{6, 0}, {0, 0}}, CellGroupingRules->{"SectionGrouping", 40}, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Center, FontFamily->"Courier", FontSize->10, FontColor->GrayLevel[0.333333]], Cell[StyleData["Special2", "Presentation"], FontSize->12], Cell[StyleData["Special2", "Printout"], FontSize->10, FontColor->GrayLevel[0]], Cell[StyleData["Special2", "TwoColumn"], FontSize->10, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Special3"], CellDingbat->"\[GraySquare]", ShowClosedCellArea->True, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Left, FontFamily->"Courier", FontSize->10, FontColor->GrayLevel[0.333333]], Cell[StyleData["Special3", "Presentation"]], Cell[StyleData["Special3", "Printout"], FontColor->GrayLevel[0]], Cell[StyleData["Special3", "TwoColumn"], FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Special4"], ShowClosedCellArea->True, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Center, FontFamily->"Courier", FontSize->10, FontColor->GrayLevel[0.333333]], Cell[StyleData["Special4", "Presentation"], FontSize->12], Cell[StyleData["Special4", "Printout"], FontColor->GrayLevel[0]], Cell[StyleData["Special4", "TwoColumn"], FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Special5"], ShowClosedCellArea->True, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Center, FontFamily->"Courier", FontSize->10, FontColor->GrayLevel[0.333333]], Cell[StyleData["Special5", "Presentation"], FontSize->12], Cell[StyleData["Special5", "Printout"]], Cell[StyleData["Special5", "TwoColumn"]] }, Closed]], Cell[StyleData["IndexEntry"], ShowCellBracket->False, CellMargins->{{15, 5}, {0, 5}}, PageBreakBelow->False, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, ParagraphIndent->-40, StyleMenuListing->None, FontSize->16], Cell[StyleData["IndexSubEntry"], ShowCellBracket->False, CellMargins->{{45, 5}, {0, 0}}, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, ParagraphIndent->-40, StyleMenuListing->None, FontSize->16] }, Closed]], Cell[CellGroupData[{ Cell["Styles for License Agreement", "Section"], Cell[CellGroupData[{ Cell[StyleData["LicenseHeading"], ShowCellBracket->True, ShowGroupOpenCloseIcon->True, CellMargins->{{24, 24}, {-1, 2}}, CellGroupingRules->{"SectionGrouping", 40}, PageBreakBelow->False, CounterIncrements->"Subsection", CounterAssignments->{{"Subsubsection", 0}}, StyleMenuListing->None, FontFamily->"Helvetica", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0.4, 0.300008, 0.6]], Cell[StyleData["LicenseHeading", "Presentation"], FontSize->12], Cell[StyleData["LicenseHeading", "Printout"], FontSize->10, FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[StyleData["LicenseHeading", "TwoColumn"], FontSize->10, FontColor->GrayLevel[0], Background->GrayLevel[1]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["LicenseText"], CellFrame->True, ShowCellBracket->False, CellMargins->{{24, 24}, {5, -1}}, StyleMenuListing->None, FontFamily->"Helvetica", FontSize->12, Background->RGBColor[1, 0.537743, 0.509071]], Cell[StyleData["LicenseText", "Presentation"], FontSize->18], Cell[StyleData["LicenseText", "Printout"], FontSize->10, FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[StyleData["LicenseText", "TwoColumn"], FontSize->10, FontColor->GrayLevel[0], Background->GrayLevel[1]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Automatic Numbering", "Section"], 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"], Cell[CellGroupData[{ Cell[StyleData["NumberedEquation"], CounterIncrements->"NumberedEquation"], Cell[StyleData["NumberedEquation", "Presentation"]], Cell[StyleData["NumberedEquation", "Printout"]], Cell[StyleData["NumberedEquation", "TwoColumn"]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["NumberedFigure"], CellFrameLabels->{{None, None}, {Cell[ TextData[ {"Figure ", CounterBox[ "NumberedFigure"]}]], None}}, CounterIncrements->"NumberedFigure", FormatTypeAutoConvert->False, FontFamily->"Times"], Cell[StyleData["NumberedFigure", "Presentation"], CellMargins->{{24, Inherited}, {20, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["NumberedFigure", "Printout"], CellMargins->{{14, Inherited}, {Inherited, Inherited}}, FontSize->10], Cell[StyleData["NumberedFigure", "TwoColumn"], CellMargins->{{14, Inherited}, {Inherited, Inherited}}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["NumberedTable"], CellMargins->{{6, Inherited}, {Inherited, Inherited}}, CellFrameLabels->{{None, None}, {Cell[ TextData[ {"Table ", CounterBox[ "NumberedTable"]}]], None}}, CounterIncrements->"NumberedTable", FormatTypeAutoConvert->False, FontFamily->"Times"], Cell[StyleData["NumberedTable", "Presentation"], CellMargins->{{24, Inherited}, {20, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["NumberedTable", "Printout"], CellMargins->{{14, Inherited}, {Inherited, Inherited}}, FontSize->10], Cell[StyleData["NumberedTable", "TwoColumn"], CellMargins->{{14, Inherited}, {Inherited, Inherited}}, FontSize->10] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Headers and Footers", "Section"], Cell[StyleData["Header"], CellMargins->{{0, 0}, {4, 1}}, StyleMenuListing->None, FontSize->10], Cell[StyleData["Footer"], CellMargins->{{0, 0}, {0, 4}}, StyleMenuListing->None, FontSize->9], Cell[StyleData["PageNumber"], CellMargins->{{0, 0}, {4, 1}}, StyleMenuListing->None, FontFamily->"Times", FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell["Palette Styles", "Section"], Cell["\<\ The cells below define styles that define standard \ ButtonFunctions, for use in palette buttons.\ \>", "Text"], Cell[StyleData["Paste"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, After]}]&)}], Cell[StyleData["Evaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], SelectionEvaluate[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["EvaluateCell"], 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"], 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"], 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["Hyperlink Styles", "Section"], 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"], Cell[CellGroupData[{ Cell[StyleData["Hyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontSize->14, FontColor->RGBColor[0, 0.392187, 0], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["Hyperlink", "Printout"], FontColor->GrayLevel[0], Background->GrayLevel[1], FontVariations->{"Underline"->False}], Cell[StyleData["Hyperlink", "TwoColumn"], FontColor->GrayLevel[0], Background->GrayLevel[1], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["MEIHyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->None, FontSize->14, FontWeight->"Bold", FontColor->RGBColor[0.650004, 0.680003, 0.0800031], Background->RGBColor[0, 0.392187, 0], FontVariations->{"Underline"->True}, 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