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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 408815, 15756]*) (*NotebookOutlinePosition[ 454261, 17330]*) (* CellTagsIndexPosition[ 454091, 17321]*) (*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[{ "2D and 3D Measurements\n", StyleBox[ "Authors: Bill Davis, Horacio Porta 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[{ "VC.06 Sources, Sinks, Swirls and Singularities\n", StyleBox["Samples", FontSize->16, FontSlant->"Italic"] }], "Title"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " Initializations" }], "Special2"], Cell[BoxData[{ \(Off[General::spell]; \nOff[General::spell1]; \n \(Off[ParametricPlot3D::ppcom]; \)\), \(\(Off[ParametricPlot::ppcom]; \)\), \(\(Off[Plot::plnr]; \)\), \(\(SetOptions[Limit, Analytic \[Rule] True]; \)\)}], "Input", InitializationCell->True], Cell[BoxData[ \(\(CMView = {2.7, 1.6, 1.2}; \)\)], "Input", InitializationCell->True], Cell[CellGroupData[{ Cell["Vector Drawer", "Subsubsection"], 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[BoxData[ \(\(BeginPackage["\", "\"]; \)\)], "Input",\ InitializationCell->True], Cell[BoxData[{ \(Axes3D::"\" = "\"; \n \(Perpend::"\" = "\"; \)\), \(\(Arrow::"\" = "\"; \)\), \(\(Vector::"\" = "\"; \)\), \(\(ArrowHead::"\" = "\"; \)\), \(\(Tail::"\" = "\"; \)\), \(\(Norm::"\" = "\"; \)\), \(\(Normalize::"\" = "\"; \)\), \(\(Aperture::"\" = "\"; \)\), \(\(TipSize::"\" = "\"; \)\), \(\(TipRatio::"\" = "\"; \)\), \(\(HeadRatio::"\" = "\"; \)\), \(\(HeadSize::"\" = "\"; \)\), \(\(ScaleFactor::"\" = "\"; \)\), \(\(ZeroVectorPointSize::"\" = "\"; \)\), \(\(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}\)\); \)\), \(\(Begin["\<`Private`\>"]; \)\), \(\(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["\", {u + v, 0, 0}], {Blue, Line[{{0, \(-\(u\/3\)\), 0}, {0, u, 0}}]}, Text["\", {0, u + v, 0}], {Blue, Line[{{0, 0, \(-\(u\/3\)\)}, {0, 0, u}}]}, Text["\", {0, 0, u + v}]}]; \)\), \(\(Axes3D[u_] := Axes3D[u, u\/8]; \)\), \(\(Norm[a_?VectorQ] := N[\@\(a . a\)]; \)\), \(\(Normalize[a_?VectorQ] := If[a . a == 0, a, a\/Norm[a]]; \)\), \(Perpend[a_?VectorQ] := Normalize[{a\[LeftDoubleBracket]2\[RightDoubleBracket], \(-a\[LeftDoubleBracket]1\[RightDoubleBracket]\)}] /; Length[a] == 2\), \(Perpend[a_?VectorQ] := Normalize[ If[a\[LeftDoubleBracket]1\[RightDoubleBracket] == 0, {1, 0, 0}, {0, a\[LeftDoubleBracket]3\[RightDoubleBracket], \(-a\[LeftDoubleBracket]2\[RightDoubleBracket]\)}]] /; 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]}, \((Polygon[Append[#1, tip]]&)\)/@ Partition[ \((tip - hr\ diff + #1&)\)/@ \((\((ap\ hr\ Norm[diff]\ Base3D)\) . {temp, Normalize[temp\[Cross]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]; \n\nEnd[]; \n \(EndPackage[]; \)\)}], "Input", InitializationCell->True] }, Closed]], Cell[BoxData[ \(\(Graphics`Colors`GosiaGreen = RGBColor[0, \ 0.392187, \ 0]; \)\)], "Input", InitializationCell->True], Cell[CellGroupData[{ Cell["Spherical Setup", "Subsubsection"], Cell[BoxData[{ \(Clear[polar]; \n polar[{x_, y_, z_}] := {\@\(x\^2 + y\^2 + z\^2\), ArcTan[y\/x], ArcCos[z\/\@\(x\^2 + y\^2 + z\^2\)]}; \nClear[X, length]; \n length[X_] := \@\(X . X\); \n\(rr = 3; \)\), \(\(R = 5; \)\), \(\(scale = 0.8; \)\), \(\(red = RGBColor[1, 0, 0]; \)\), \(\(blue = RGBColor[0, 0, 1]; \)\), \(\(color1 = RGBColor[0.8, 0.2, 0.8]; \)\), \(\(color2 = RGBColor[0.2, 0.8, 0.2]; \)\), \(\(black = RGBColor[0, 0, 0]; \)\), \(Clear[axes]; \n axes := Graphics3D[{Line[{X, {1.5\ rr, 0, 0}}], Line[{X, {0, 1.5\ rr, 0}}], Line[{X, {0, 0, 2\ rr}}]}]; \n Clear[xyplane]; \n xyplane := Graphics3D[ Polygon[{{\(-2\)\ rr, \(-2\)\ rr, 0}, {\(-2\)\ rr, 2\ rr, 0}, {2\ rr, 2\ rr, 0}, {2\ rr, \(-2\)\ rr, 0}}]]; \n\(X = {0, 0, 0}; \)\), \(\(origin = Graphics3D[{black, Point[X]}]; \)\), \(\(Z = {2, 3, 5}; \)\), \(Clear[point]; \n point := Graphics3D[{red, PointSize[0.03], Point[scale\ Z]}]; \n \(Z0 = {0, 0, length[0.8\ Z]}; \)\), \(Clear[zpoint]; \nzpoint = Graphics3D[{black, Point[Z0]}]; \n \(XY = scale\ Z\ {1, 1, 0}; \)\), \(Clear[xypoint]; \nxypoint := Graphics3D[{black, Point[XY]}]; \n \(X0 = {length[scale\ Z\ {1, 1, 0}], 0, 0}; \)\), \(\(xpoint = Graphics3D[{black, Point[X0]}]; 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\)], "Input", InitializationCell->True] }, Closed]] }, Closed]], Cell["Basic Problems", "Subsubsection"], Cell[CellGroupData[{ Cell["B.2) Sources, sinks, and the divergence of a vector field", "Subsection", CellTags->"VC.06.B2"], Cell[TextData[{ "B.2.a) The meaning of the sign of\n ", Cell[BoxData[ \(divField[x, y] = D[m[x, y], x] + \ D[n[x, y], y]\)]] }], "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ "Given a vector field \n ", Cell[BoxData[ \(Field[x, y] = {m[x, y], n[x, y]}\)]], ",\nyou calculate \n ", Cell[BoxData[ \(divField[x, y] = D[m[x, y], x] + \ D[n[x, y], y]\)]], ".\nHow does the sign of \n ", Cell[BoxData[ \(divField[x, y] = D[m[x, y], x] + D[n[x, y], y]\)]], " \ntell you whether ", Cell[BoxData[ \({x, y}\)]], " is a source of new fluid or a sink (drain) for old fluid? " }], "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "If ", Cell[BoxData[ \(divField[x\_0, y\_0] > 0\)]], ", then the point ", Cell[BoxData[ \({x\_0, y\_0}\)]], " is a source of new fluid.\nIf ", Cell[BoxData[ \(divField[x\_0, y\_0] < 0\)]], ", then the point ", Cell[BoxData[ \({x\_0, y\_0}\)]], " is a sink for old fluid.\nHere's why: \nTake a small circle ", Cell[BoxData[ \(C\)]], " centered at ", Cell[BoxData[ \({x\_0, y\_0}\)]], ". Calculate the flow of ", Cell[BoxData[ \(Field[x, y]\)]], " across ", Cell[BoxData[ \(C\)]], " by calculating \n ", Cell[BoxData[ \(\[ContourIntegral]\_\(\ C\)\ \(-n[x, y]\) \[DifferentialD]x\ + \ m[x, y] \[DifferentialD]y\)]], "\n ", Cell[BoxData[ \(\( = \[Integral]\_\[Null]\%\[Null]\(\[Integral]\_\(\ R\)\ divField[x, y] \[DifferentialD]x \[DifferentialD]y\)\)\)]], ".\nHere's the kicker: \nIf \n ", Cell[BoxData[ \(divField[x\_0, y\_0] > 0\)]], ", \nthen it is positive for all ", Cell[BoxData[ \({x, y}\)]], "'s close to ", Cell[BoxData[ \({x\_0, y\_0}\)]], ". So if ", Cell[BoxData[ \(C\)]], " is so small that\n ", Cell[BoxData[ \(divField[x\_0, y\_0] > 0\)]], "\nat all ", Cell[BoxData[ \({x, y}\)]], "'s inside ", Cell[BoxData[ \(C\)]], ", then you see that\n ", Cell[BoxData[ \(\[ContourIntegral]\_\(\ C\)\ \(-n[x, y]\) \[DifferentialD]x\ + \ m[x, y] \[DifferentialD]y\)]], "\n ", Cell[BoxData[ \(\( = \[Integral]\_\[Null]\%\[Null]\(\[Integral]\_\(\ R\)\ divField[x, y] \[DifferentialD]x \[DifferentialD]y\) > 0\)\)]], ".\nThis means that if ", Cell[BoxData[ \(divField[x\_0, y\_0] > 0\)]], ", then the net flow of ", Cell[BoxData[ \(Field[x, y]\)]], " across small circles centered at ", Cell[BoxData[ \({x\_0, y\_0}\)]], " is from inside to outside. \nThe upshot: \nIf ", Cell[BoxData[ \(divField[x\_0, y\_0] > 0\)]], ", then the point ", Cell[BoxData[ \({x\_0, y\_0}\)]], " is a source of new fluid.\nSimilarly, if ", Cell[BoxData[ \(divField[x\_0, y\_0] < 0\)]], ", then the net flow of ", Cell[BoxData[ \(Field[x, y]\)]], " across small circles centered at ", Cell[BoxData[ \({x\_0, y\_0}\)]], " is from outside to inside. \nSo:\nIf ", Cell[BoxData[ \(divField[x\_0, y\_0] < 0\)]], ", then the point ", Cell[BoxData[ \({x\_0, y\_0}\)]], " is a sink for old fluid.\nCheck this out:\nGo with ", Cell[BoxData[ \(Field[x, y] = {3 x\ y, y}\)]], " and look at ", Cell[BoxData[ \(divField[x, y]\)]], ":" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[x, y, Field, m, n, divField];\)\), "\n", \(\({m[x_, y_], n[x_, y_]} = {3\ x\ y, y};\)\), "\n", \(\(Field[x_, y_] = {m[x, y], n[x, y]};\)\), "\n", \(divField[x_, y_] = D[m[x, y], x] + D[n[x, y], y]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(1 + 3\ y\)], "Output"] }, Open ]], Cell[TextData[{ "See whether the point ", Cell[BoxData[ \({0, 2}\)]], " is a source or a sink:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(divField[0, 2]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(7\)], "Output"] }, Open ]], Cell[TextData[{ "Positive. This tells you that the point ", Cell[BoxData[ \({0, 2}\)]], " is a source.\nTake a look at this vector field on a small circle centered \ at ", Cell[BoxData[ \({0, 2}\)]], " to see whether this calculation agrees with reality:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(point = {0, 2};\)\), "\n", \(\(radius = 0.1;\)\), "\n", \(\(pointplot = Graphics[{PointSize[0.04], Point[point]}];\)\), "\n", \(\(Clear[x, y, t];\)\), "\n", \(\({x[t_], y[t_]} = point + radius\ {Cos[t], Sin[t]};\)\), "\n", \(\(tlow = 0;\)\), "\n", \(\(thigh = 2\ \[Pi];\)\), "\n", \(\(smallcircle = ParametricPlot[{x[t], y[t]}, {t, tlow, thigh}, PlotStyle \[Rule] {{Red, Thickness[0.01]}}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(scalefactor = 0.05;\)\), "\n", \(\(fieldplot = Table[Arrow[Field[x[t], y[t]], Tail \[Rule] {x[t], y[t]}, \ ScaleFactor \[Rule] scalefactor], {t, tlow, thigh, \(thigh - tlow\)\/16}];\)\), "\n", \(\(Show[pointplot, smallcircle, fieldplot, Axes \[Rule] True, AxesLabel \[Rule] 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Bob. The plot shows a lot more flow from inside to \ outside than from outside to inside. Just as you would expect on a small circle centered at a source. \ \>", "SmallText"] }, Closed]], Cell["B.2.b) Sources and sinks", "Subsubsection"], Cell["Here's a vector field:", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[Field, m, n, x, y];\)\), "\n", \(\({m[x_, y_], n[x_, y_]} = {Sin[x]\ Cos[y], Sin[y]\ Cos[x]};\)\), "\n", \(Field[x_, y_] = {m[x, y], n[x, y]}\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \({Cos[y]\ Sin[x], Cos[x]\ Sin[y]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Give a sample plot of some of the sources and sinks in this vector \ field.\ \>", "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "Here's ", Cell[BoxData[ \(divField[x, y]\)]], ":" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[divField];\)\), "\n", \(divField[x_, y_] = D[m[x, y], x] + D[n[x, y], y]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(2\ Cos[x]\ Cos[y]\)], "Output"] }, Open ]], Cell[TextData[{ "A point ", Cell[BoxData[ \({x, y}\)]], 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.54286 Mdot .5 g .025 w .5 .56667 Mdot .5 .59048 Mdot .5 .61429 Mdot .5 .6381 Mdot .5 .6619 Mdot .5 .68571 Mdot .5 .70952 Mdot .5 .73333 Mdot .5 .75714 Mdot .5 .78095 Mdot .52381 .01905 Mdot .52381 .04286 Mdot .52381 .06667 Mdot .52381 .09048 Mdot .52381 .11429 Mdot .52381 .1381 Mdot .52381 .1619 Mdot .52381 .18571 Mdot .52381 .20952 Mdot .52381 .23333 Mdot 1 0 0 r .015 w .52381 .25714 Mdot .52381 .28095 Mdot .52381 .30476 Mdot .52381 .32857 Mdot .52381 .35238 Mdot .52381 .37619 Mdot .52381 .4 Mdot .52381 .42381 Mdot .52381 .44762 Mdot .52381 .47143 Mdot .52381 .49524 Mdot .52381 .51905 Mdot .52381 .54286 Mdot .5 g .025 w .52381 .56667 Mdot .52381 .59048 Mdot .52381 .61429 Mdot .52381 .6381 Mdot .52381 .6619 Mdot .52381 .68571 Mdot .52381 .70952 Mdot .52381 .73333 Mdot .52381 .75714 Mdot .52381 .78095 Mdot .54762 .01905 Mdot .54762 .04286 Mdot .54762 .06667 Mdot .54762 .09048 Mdot .54762 .11429 Mdot .54762 .1381 Mdot .54762 .1619 Mdot .54762 .18571 Mdot .54762 .20952 Mdot .54762 .23333 Mdot 1 0 0 r .015 w .54762 .25714 Mdot .54762 .28095 Mdot .54762 .30476 Mdot .54762 .32857 Mdot .54762 .35238 Mdot .54762 .37619 Mdot .54762 .4 Mdot .54762 .42381 Mdot .54762 .44762 Mdot .54762 .47143 Mdot .54762 .49524 Mdot .54762 .51905 Mdot .54762 .54286 Mdot .5 g .025 w .54762 .56667 Mdot .54762 .59048 Mdot .54762 .61429 Mdot .54762 .6381 Mdot .54762 .6619 Mdot .54762 .68571 Mdot .54762 .70952 Mdot .54762 .73333 Mdot .54762 .75714 Mdot .54762 .78095 Mdot .57143 .01905 Mdot .57143 .04286 Mdot .57143 .06667 Mdot .57143 .09048 Mdot .57143 .11429 Mdot .57143 .1381 Mdot .57143 .1619 Mdot .57143 .18571 Mdot .57143 .20952 Mdot .57143 .23333 Mdot 1 0 0 r .015 w .57143 .25714 Mdot .57143 .28095 Mdot .57143 .30476 Mdot .57143 .32857 Mdot .57143 .35238 Mdot .57143 .37619 Mdot .57143 .4 Mdot .57143 .42381 Mdot .57143 .44762 Mdot .57143 .47143 Mdot .57143 .49524 Mdot .57143 .51905 Mdot .57143 .54286 Mdot .5 g .025 w .57143 .56667 Mdot .57143 .59048 Mdot .57143 .61429 Mdot .57143 .6381 Mdot .57143 .6619 Mdot .57143 .68571 Mdot .57143 .70952 Mdot .57143 .73333 Mdot .57143 .75714 Mdot .57143 .78095 Mdot .59524 .01905 Mdot .59524 .04286 Mdot .59524 .06667 Mdot .59524 .09048 Mdot .59524 .11429 Mdot .59524 .1381 Mdot .59524 .1619 Mdot .59524 .18571 Mdot .59524 .20952 Mdot .59524 .23333 Mdot 1 0 0 r .015 w .59524 .25714 Mdot .59524 .28095 Mdot .59524 .30476 Mdot .59524 .32857 Mdot .59524 .35238 Mdot .59524 .37619 Mdot .59524 .4 Mdot .59524 .42381 Mdot .59524 .44762 Mdot .59524 .47143 Mdot .59524 .49524 Mdot .59524 .51905 Mdot .59524 .54286 Mdot .5 g .025 w .59524 .56667 Mdot .59524 .59048 Mdot .59524 .61429 Mdot .59524 .6381 Mdot .59524 .6619 Mdot .59524 .68571 Mdot .59524 .70952 Mdot .59524 .73333 Mdot .59524 .75714 Mdot .59524 .78095 Mdot .61905 .01905 Mdot .61905 .04286 Mdot .61905 .06667 Mdot .61905 .09048 Mdot .61905 .11429 Mdot .61905 .1381 Mdot .61905 .1619 Mdot .61905 .18571 Mdot .61905 .20952 Mdot .61905 .23333 Mdot 1 0 0 r .015 w .61905 .25714 Mdot .61905 .28095 Mdot .61905 .30476 Mdot .61905 .32857 Mdot .61905 .35238 Mdot .61905 .37619 Mdot .61905 .4 Mdot .61905 .42381 Mdot .61905 .44762 Mdot .61905 .47143 Mdot .61905 .49524 Mdot .61905 .51905 Mdot .61905 .54286 Mdot .5 g .025 w .61905 .56667 Mdot .61905 .59048 Mdot .61905 .61429 Mdot .61905 .6381 Mdot .61905 .6619 Mdot .61905 .68571 Mdot .61905 .70952 Mdot .61905 .73333 Mdot .61905 .75714 Mdot .61905 .78095 Mdot .64286 .01905 Mdot .64286 .04286 Mdot .64286 .06667 Mdot .64286 .09048 Mdot .64286 .11429 Mdot .64286 .1381 Mdot .64286 .1619 Mdot .64286 .18571 Mdot .64286 .20952 Mdot .64286 .23333 Mdot 1 0 0 r .015 w .64286 .25714 Mdot .64286 .28095 Mdot .64286 .30476 Mdot .64286 .32857 Mdot .64286 .35238 Mdot .64286 .37619 Mdot .64286 .4 Mdot .64286 .42381 Mdot .64286 .44762 Mdot .64286 .47143 Mdot .64286 .49524 Mdot .64286 .51905 Mdot .64286 .54286 Mdot .5 g .025 w .64286 .56667 Mdot .64286 .59048 Mdot .64286 .61429 Mdot .64286 .6381 Mdot .64286 .6619 Mdot .64286 .68571 Mdot .64286 .70952 Mdot .64286 .73333 Mdot .64286 .75714 Mdot .64286 .78095 Mdot 1 0 0 r .015 w .66667 .01905 Mdot .66667 .04286 Mdot .66667 .06667 Mdot .66667 .09048 Mdot .66667 .11429 Mdot .66667 .1381 Mdot .66667 .1619 Mdot .66667 .18571 Mdot .66667 .20952 Mdot .66667 .23333 Mdot .5 g .025 w .66667 .25714 Mdot .66667 .28095 Mdot .66667 .30476 Mdot .66667 .32857 Mdot .66667 .35238 Mdot .66667 .37619 Mdot .66667 .4 Mdot .66667 .42381 Mdot .66667 .44762 Mdot .66667 .47143 Mdot .66667 .49524 Mdot .66667 .51905 Mdot .66667 .54286 Mdot 1 0 0 r .015 w .66667 .56667 Mdot .66667 .59048 Mdot .66667 .61429 Mdot .66667 .6381 Mdot .66667 .6619 Mdot .66667 .68571 Mdot .66667 .70952 Mdot .66667 .73333 Mdot .66667 .75714 Mdot .66667 .78095 Mdot .69048 .01905 Mdot .69048 .04286 Mdot .69048 .06667 Mdot .69048 .09048 Mdot .69048 .11429 Mdot .69048 .1381 Mdot .69048 .1619 Mdot .69048 .18571 Mdot .69048 .20952 Mdot .69048 .23333 Mdot .5 g .025 w .69048 .25714 Mdot .69048 .28095 Mdot .69048 .30476 Mdot .69048 .32857 Mdot .69048 .35238 Mdot .69048 .37619 Mdot .69048 .4 Mdot .69048 .42381 Mdot .69048 .44762 Mdot .69048 .47143 Mdot .69048 .49524 Mdot .69048 .51905 Mdot .69048 .54286 Mdot 1 0 0 r .015 w .69048 .56667 Mdot .69048 .59048 Mdot .69048 .61429 Mdot .69048 .6381 Mdot .69048 .6619 Mdot .69048 .68571 Mdot .69048 .70952 Mdot .69048 .73333 Mdot .69048 .75714 Mdot .69048 .78095 Mdot .71429 .01905 Mdot .71429 .04286 Mdot .71429 .06667 Mdot .71429 .09048 Mdot .71429 .11429 Mdot .71429 .1381 Mdot .71429 .1619 Mdot .71429 .18571 Mdot .71429 .20952 Mdot .71429 .23333 Mdot .5 g .025 w .71429 .25714 Mdot .71429 .28095 Mdot .71429 .30476 Mdot .71429 .32857 Mdot .71429 .35238 Mdot .71429 .37619 Mdot .71429 .4 Mdot .71429 .42381 Mdot .71429 .44762 Mdot .71429 .47143 Mdot .71429 .49524 Mdot .71429 .51905 Mdot .71429 .54286 Mdot 1 0 0 r .015 w .71429 .56667 Mdot .71429 .59048 Mdot .71429 .61429 Mdot .71429 .6381 Mdot .71429 .6619 Mdot .71429 .68571 Mdot .71429 .70952 Mdot .71429 .73333 Mdot .71429 .75714 Mdot .71429 .78095 Mdot .7381 .01905 Mdot .7381 .04286 Mdot .7381 .06667 Mdot .7381 .09048 Mdot .7381 .11429 Mdot .7381 .1381 Mdot .7381 .1619 Mdot .7381 .18571 Mdot .7381 .20952 Mdot .7381 .23333 Mdot .5 g .025 w .7381 .25714 Mdot .7381 .28095 Mdot .7381 .30476 Mdot .7381 .32857 Mdot .7381 .35238 Mdot .7381 .37619 Mdot .7381 .4 Mdot .7381 .42381 Mdot .7381 .44762 Mdot .7381 .47143 Mdot .7381 .49524 Mdot .7381 .51905 Mdot .7381 .54286 Mdot 1 0 0 r .015 w .7381 .56667 Mdot .7381 .59048 Mdot .7381 .61429 Mdot .7381 .6381 Mdot .7381 .6619 Mdot .7381 .68571 Mdot .7381 .70952 Mdot .7381 .73333 Mdot .7381 .75714 Mdot .7381 .78095 Mdot .7619 .01905 Mdot .7619 .04286 Mdot .7619 .06667 Mdot .7619 .09048 Mdot .7619 .11429 Mdot .7619 .1381 Mdot .7619 .1619 Mdot .7619 .18571 Mdot .7619 .20952 Mdot .7619 .23333 Mdot .5 g .025 w .7619 .25714 Mdot .7619 .28095 Mdot .7619 .30476 Mdot .7619 .32857 Mdot .7619 .35238 Mdot .7619 .37619 Mdot .7619 .4 Mdot .7619 .42381 Mdot .7619 .44762 Mdot .7619 .47143 Mdot .7619 .49524 Mdot .7619 .51905 Mdot .7619 .54286 Mdot 1 0 0 r .015 w .7619 .56667 Mdot .7619 .59048 Mdot .7619 .61429 Mdot .7619 .6381 Mdot .7619 .6619 Mdot .7619 .68571 Mdot .7619 .70952 Mdot .7619 .73333 Mdot .7619 .75714 Mdot .7619 .78095 Mdot .78571 .01905 Mdot .78571 .04286 Mdot .78571 .06667 Mdot .78571 .09048 Mdot .78571 .11429 Mdot .78571 .1381 Mdot .78571 .1619 Mdot .78571 .18571 Mdot .78571 .20952 Mdot .78571 .23333 Mdot .5 g .025 w .78571 .25714 Mdot .78571 .28095 Mdot .78571 .30476 Mdot .78571 .32857 Mdot .78571 .35238 Mdot .78571 .37619 Mdot .78571 .4 Mdot .78571 .42381 Mdot .78571 .44762 Mdot .78571 .47143 Mdot .78571 .49524 Mdot .78571 .51905 Mdot .78571 .54286 Mdot 1 0 0 r .015 w .78571 .56667 Mdot .78571 .59048 Mdot .78571 .61429 Mdot .78571 .6381 Mdot .78571 .6619 Mdot .78571 .68571 Mdot .78571 .70952 Mdot .78571 .73333 Mdot .78571 .75714 Mdot .78571 .78095 Mdot .80952 .01905 Mdot .80952 .04286 Mdot .80952 .06667 Mdot .80952 .09048 Mdot .80952 .11429 Mdot .80952 .1381 Mdot .80952 .1619 Mdot .80952 .18571 Mdot .80952 .20952 Mdot .80952 .23333 Mdot .5 g .025 w .80952 .25714 Mdot .80952 .28095 Mdot .80952 .30476 Mdot .80952 .32857 Mdot .80952 .35238 Mdot .80952 .37619 Mdot .80952 .4 Mdot .80952 .42381 Mdot .80952 .44762 Mdot .80952 .47143 Mdot .80952 .49524 Mdot .80952 .51905 Mdot .80952 .54286 Mdot 1 0 0 r .015 w .80952 .56667 Mdot .80952 .59048 Mdot .80952 .61429 Mdot .80952 .6381 Mdot .80952 .6619 Mdot .80952 .68571 Mdot .80952 .70952 Mdot .80952 .73333 Mdot .80952 .75714 Mdot .80952 .78095 Mdot .83333 .01905 Mdot .83333 .04286 Mdot .83333 .06667 Mdot .83333 .09048 Mdot .83333 .11429 Mdot .83333 .1381 Mdot .83333 .1619 Mdot .83333 .18571 Mdot .83333 .20952 Mdot .83333 .23333 Mdot .5 g .025 w .83333 .25714 Mdot .83333 .28095 Mdot .83333 .30476 Mdot .83333 .32857 Mdot .83333 .35238 Mdot .83333 .37619 Mdot .83333 .4 Mdot .83333 .42381 Mdot .83333 .44762 Mdot .83333 .47143 Mdot .83333 .49524 Mdot .83333 .51905 Mdot .83333 .54286 Mdot 1 0 0 r .015 w .83333 .56667 Mdot .83333 .59048 Mdot .83333 .61429 Mdot .83333 .6381 Mdot .83333 .6619 Mdot .83333 .68571 Mdot .83333 .70952 Mdot .83333 .73333 Mdot .83333 .75714 Mdot .83333 .78095 Mdot .85714 .01905 Mdot .85714 .04286 Mdot .85714 .06667 Mdot .85714 .09048 Mdot .85714 .11429 Mdot .85714 .1381 Mdot .85714 .1619 Mdot .85714 .18571 Mdot .85714 .20952 Mdot .85714 .23333 Mdot .5 g .025 w .85714 .25714 Mdot .85714 .28095 Mdot .85714 .30476 Mdot .85714 .32857 Mdot .85714 .35238 Mdot .85714 .37619 Mdot .85714 .4 Mdot .85714 .42381 Mdot .85714 .44762 Mdot .85714 .47143 Mdot .85714 .49524 Mdot .85714 .51905 Mdot .85714 .54286 Mdot 1 0 0 r .015 w .85714 .56667 Mdot .85714 .59048 Mdot .85714 .61429 Mdot .85714 .6381 Mdot .85714 .6619 Mdot .85714 .68571 Mdot .85714 .70952 Mdot .85714 .73333 Mdot .85714 .75714 Mdot .85714 .78095 Mdot .88095 .01905 Mdot .88095 .04286 Mdot .88095 .06667 Mdot .88095 .09048 Mdot .88095 .11429 Mdot .88095 .1381 Mdot .88095 .1619 Mdot .88095 .18571 Mdot .88095 .20952 Mdot .88095 .23333 Mdot .5 g .025 w .88095 .25714 Mdot .88095 .28095 Mdot .88095 .30476 Mdot .88095 .32857 Mdot .88095 .35238 Mdot .88095 .37619 Mdot .88095 .4 Mdot .88095 .42381 Mdot .88095 .44762 Mdot .88095 .47143 Mdot .88095 .49524 Mdot .88095 .51905 Mdot .88095 .54286 Mdot 1 0 0 r .015 w .88095 .56667 Mdot .88095 .59048 Mdot .88095 .61429 Mdot .88095 .6381 Mdot .88095 .6619 Mdot .88095 .68571 Mdot .88095 .70952 Mdot .88095 .73333 Mdot .88095 .75714 Mdot .88095 .78095 Mdot .90476 .01905 Mdot .90476 .04286 Mdot .90476 .06667 Mdot .90476 .09048 Mdot .90476 .11429 Mdot .90476 .1381 Mdot .90476 .1619 Mdot .90476 .18571 Mdot .90476 .20952 Mdot .90476 .23333 Mdot .5 g .025 w .90476 .25714 Mdot .90476 .28095 Mdot .90476 .30476 Mdot .90476 .32857 Mdot .90476 .35238 Mdot .90476 .37619 Mdot .90476 .4 Mdot .90476 .42381 Mdot .90476 .44762 Mdot .90476 .47143 Mdot .90476 .49524 Mdot .90476 .51905 Mdot .90476 .54286 Mdot 1 0 0 r .015 w .90476 .56667 Mdot .90476 .59048 Mdot .90476 .61429 Mdot .90476 .6381 Mdot .90476 .6619 Mdot .90476 .68571 Mdot .90476 .70952 Mdot .90476 .73333 Mdot .90476 .75714 Mdot .90476 .78095 Mdot .92857 .01905 Mdot .92857 .04286 Mdot .92857 .06667 Mdot .92857 .09048 Mdot .92857 .11429 Mdot .92857 .1381 Mdot .92857 .1619 Mdot .92857 .18571 Mdot .92857 .20952 Mdot .92857 .23333 Mdot .5 g .025 w .92857 .25714 Mdot .92857 .28095 Mdot .92857 .30476 Mdot .92857 .32857 Mdot .92857 .35238 Mdot .92857 .37619 Mdot .92857 .4 Mdot .92857 .42381 Mdot .92857 .44762 Mdot .92857 .47143 Mdot .92857 .49524 Mdot .92857 .51905 Mdot .92857 .54286 Mdot 1 0 0 r .015 w .92857 .56667 Mdot .92857 .59048 Mdot .92857 .61429 Mdot .92857 .6381 Mdot .92857 .6619 Mdot .92857 .68571 Mdot .92857 .70952 Mdot .92857 .73333 Mdot .92857 .75714 Mdot .92857 .78095 Mdot .5 g .025 w .95238 .01905 Mdot .95238 .04286 Mdot .95238 .06667 Mdot .95238 .09048 Mdot .95238 .11429 Mdot .95238 .1381 Mdot .95238 .1619 Mdot .95238 .18571 Mdot .95238 .20952 Mdot .95238 .23333 Mdot 1 0 0 r .015 w .95238 .25714 Mdot .95238 .28095 Mdot .95238 .30476 Mdot .95238 .32857 Mdot .95238 .35238 Mdot .95238 .37619 Mdot .95238 .4 Mdot .95238 .42381 Mdot .95238 .44762 Mdot .95238 .47143 Mdot .95238 .49524 Mdot .95238 .51905 Mdot .95238 .54286 Mdot .5 g .025 w .95238 .56667 Mdot .95238 .59048 Mdot .95238 .61429 Mdot .95238 .6381 Mdot .95238 .6619 Mdot .95238 .68571 Mdot .95238 .70952 Mdot .95238 .73333 Mdot .95238 .75714 Mdot .95238 .78095 Mdot .97619 .01905 Mdot .97619 .04286 Mdot .97619 .06667 Mdot .97619 .09048 Mdot .97619 .11429 Mdot .97619 .1381 Mdot .97619 .1619 Mdot .97619 .18571 Mdot .97619 .20952 Mdot .97619 .23333 Mdot 1 0 0 r .015 w .97619 .25714 Mdot .97619 .28095 Mdot .97619 .30476 Mdot .97619 .32857 Mdot .97619 .35238 Mdot .97619 .37619 Mdot .97619 .4 Mdot .97619 .42381 Mdot .97619 .44762 Mdot .97619 .47143 Mdot .97619 .49524 Mdot .97619 .51905 Mdot .97619 .54286 Mdot .5 g .025 w .97619 .56667 Mdot .97619 .59048 Mdot .97619 .61429 Mdot .97619 .6381 Mdot .97619 .6619 Mdot .97619 .68571 Mdot .97619 .70952 Mdot .97619 .73333 Mdot .97619 .75714 Mdot .97619 .78095 Mdot % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 230.375}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]], Cell["The larger points are sinks; the smaller points are sources.", "Special2"], Cell["\<\ Alternate squares of sources and sinks. Think of the sources as little individual springs feeding the flow. Think of the sinks as tiny little holes through which fluid seeps out as the \ flow goes by.\ \>", "SmallText"] }, Closed]], Cell[TextData[{ "B.2.c.i) All sources inside ", Cell[BoxData[ \(C\)]] }], "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ "If every point inside a closed curve (like a deformed circle) ", Cell[BoxData[ \(C\)]], " is a source of a given vector field, and if the vector field has no \ singularities inside ", Cell[BoxData[ \(C\)]], ", then how do you know that the net flow of the given vector field across \ ", Cell[BoxData[ \(C\)]], " is automatically from inside to outside?" }], "Text"], Cell["Answer:", "Subsubsection"], Cell[TextData[{ "If every point inside a closed curve ", Cell[BoxData[ \(C\)]], " is a source of a given vector field, then \n\[Rule] new fluid is oozing \ out of each point inside ", Cell[BoxData[ \(C\)]], ", and \n\[Rule] there there is no place within ", Cell[BoxData[ \(C\)]], " to absorb excess outside-to-inside flow.\nThe result:\nIf every point \ inside a closed curve ", Cell[BoxData[ \(C\)]], " is a source of a given vector field, then the flow of this vector field \ across ", Cell[BoxData[ \(C\)]], " is automatically from inside to outside.\nFor example, look at:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[Field, m, n, x, y, divField];\)\), "\n", \(\({m[x_, y_], n[x_, y_]} = {x\^3 - y, y\^3 + x};\)\), "\n", \(\(Field[x_, y_] = {m[x, y], n[x, y]};\)\), "\n", \(divField[x_, y_] = D[m[x, y], x] + D[n[x, y], y]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(3\ x\^2 + 3\ y\^2\)], "Output"] }, Open ]], Cell[TextData[{ "Unless ", Cell[BoxData[ \({x, y} = {0, 0}\)]], ", ", Cell[BoxData[ \(divField[x, y] > 0\)]], ". \nThis tells you that all points ", Cell[BoxData[ \({x, y}\)]], " except ", Cell[BoxData[ \({0, 0}\)]], " are sources for ", Cell[BoxData[ \(Field[x, y]\)]], " and the lone exception is not a sink.\nThis vector field has no \ singularities. \nSo:\nOn the basis of this information you can say with \ confidence and authority that the flow of this vector field across any closed \ curve is from inside to outside." }], "SmallText"] }, Closed]], Cell[TextData[{ "B.2.c.ii) All sinks inside ", Cell[BoxData[ \(C\)]] }], "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ "If every point inside a closed curve (like a deformed circle) ", Cell[BoxData[ \(C\)]], " is a sink of a given vector field, and if the vector field has no \ singularities inside ", Cell[BoxData[ \(C\)]], ", then how do you know that the net flow of the given vector field across \ ", Cell[BoxData[ \(C\)]], " is automatically from outside to inside?" }], "Text"], Cell["Answer:", "Subsubsection"], Cell[TextData[{ "If every point inside a closed curve ", Cell[BoxData[ \(C\)]], " is a sink of a given vector field, then \n\[Rule] old fluid is soaking \ into each point inside ", Cell[BoxData[ \(C\)]], ", and \n\[Rule] there there is no place within ", Cell[BoxData[ \(C\)]], " to generate excess inside-to-outside flow.\nThe result:\nIf every point \ inside a closed curve ", Cell[BoxData[ \(C\)]], " is a sink of a given vector field, then the flow of this vector field \ across ", Cell[BoxData[ \(C\)]], " is automatically from outside to inside.\nFor example, look at:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[Field, m, n, x, y, divField];\)\), "\n", \(\({m[x_, y_], n[x_, y_]} = {y - x\^3, x - y\^7};\)\), "\n", \(\(Field[x_, y_] = {m[x, y], n[x, y]};\)\), "\n", \(divField[x_, y_] = D[m[x, y], x] + D[n[x, y], y]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(-3\)\ x\^2 - 7\ y\^6\)], "Output"] }, Open ]], Cell[TextData[{ "Unless ", Cell[BoxData[ \({x, y} = {0, 0}\)]], ", ", Cell[BoxData[ \(divField[x, y] < 0\)]], ". \nThis tells you that all points ", Cell[BoxData[ \({x, y}\)]], " except ", Cell[BoxData[ \({0, 0}\)]], " are sinks for ", Cell[BoxData[ \(Field[x, y]\)]], " and the lone exception is not a source.\nThis vector field has no \ singularities.\nSo:\nOn the basis of this information you can say with \ confidence and authority that the flow of this vector field across any closed \ curve is from outside to inside.\nThe region inside ", Cell[BoxData[ \(C\)]], " is like a big vacuum sucking up fluid." }], "SmallText"] }, Closed]], Cell[TextData[{ "B.2.c.iii) No sources or sinks inside ", Cell[BoxData[ \(C\)]] }], "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ "If there are no sinks and there are no sources of a given vector field \ inside a closed curve (like a deformed circle) ", Cell[BoxData[ \(C\)]], ", and if the vector field has no singularities inside ", Cell[BoxData[ \(C\)]], ", then how do you know that the net flow of the given vector field across \ ", Cell[BoxData[ \(C\)]], " is ", Cell[BoxData[ \(0\)]], "?" }], "Text"], Cell["Answer:", "Subsubsection"], Cell[TextData[{ "If there are no sinks and there are no sources of a given vector field \ inside ", Cell[BoxData[ \(C\)]], ", and there are no singularities inside ", Cell[BoxData[ \(C\)]], ", then no new fluid is injected and no old fluid is sucked up inside ", Cell[BoxData[ \(C\)]], ".\nSo:\n\[Rule] What flows from inside to outside must be replaced by \ equal outside to inside flow.\n\[Rule] What flows from outside to inside must \ be replaced by equal inside to outside flow.\nThe result:\nIf there are no \ sinks and there are no sources of a given vector field inside ", Cell[BoxData[ \(C\)]], ", and there are no singularities inside ", Cell[BoxData[ \(C\)]], ", then the net flow of the vector field across ", Cell[BoxData[ \(C\)]], " is ", Cell[BoxData[ \(0\)]], ".\nFor example, look at:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[Field, m, n, x, y, divField];\)\), "\n", \(\({m[x_, y_], n[x_, y_]} = {Cos[x]\ Cosh[y], Sin[x]\ Sinh[y]};\)\), "\n", \(\(Field[x_, y_] = {m[x, y], n[x, y]};\)\), "\n", \(divField[x_, y_] = D[m[x, y], x] + D[n[x, y], y]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[TextData[{ "This tells you that this vector field has no sources or sinks.\nThis \ vector field has no singularities, so:\nOn the basis if this information you \ can say with confidence and authority that the net flow of this vector field \ across any closed curve is ", Cell[BoxData[ \(0\)]], "." }], "SmallText"] }, Closed]], Cell["B.2.d) Divergence", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ "Why do most folks call ", Cell[BoxData[ \(divField[x, y]\)]], " the divergence of a vector field ", Cell[BoxData[ \(Field[x, y]\)]], "?" }], "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "The name fits. \nIf ", Cell[BoxData[ \(divField[x, y] > 0\)]], ", then new fluid is oozing out of the point \n", Cell[BoxData[ \({x, y}\)]], " and diverges elsewhere.\nIf ", Cell[BoxData[ \(divField[x, y] < 0\)]], ", then old fluid is sucked into the point \n", Cell[BoxData[ \({x, y}\)]], " and converges onto this point.\nIf ", Cell[BoxData[ \(divField[x, y] = 0\)]], ", then no new fluid is introduced, and no old fluid is sucked off as the \ flow passes by ", Cell[BoxData[ \({x, y}\)]], ". 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Was this an accident?\ \>", "Text"], Cell["Answer:", "Special1"], Cell["You got it right.", "SmallText"], Cell["In mathematics, there are no accidents.", "Accident"], Cell["Take a look at the vector field.", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(Field[x, y]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({\(6\ x\)\/\(x\^2 + y\^2\), \(6\ y\)\/\(x\^2 + y\^2\)}\)], "Output"] }, Open ]], Cell[TextData[{ "Note the nasty singularity at ", Cell[BoxData[ \({0, 0}\)]], "." }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(Field[0, 0]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(Power::"infy" \(\(:\)\(\ \)\) "Infinite expression \!\(1\/0\) encountered."\)], "Message"], Cell[BoxData[ \(Power::"infy" \(\(:\)\(\ \)\) "Infinite expression \!\(1\/0\) encountered."\)], "Message"], Cell[BoxData[ \({Indeterminate, Indeterminate}\)], "Output"] }, Open ]], Cell[TextData[{ "Look at ", Cell[BoxData[ \(divField[x, y]\)]], ":" }], "SmallText"], Cell[CellGroupData[{ 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%%IncludeFont: Courier /Courier findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 0.000 0.000 rmoveto 1.000 setlinewidth grestore % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 239.938}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Calculate the flow of \n ", Cell[BoxData[ RowBox[{\(Field[x, y]\), "=", RowBox[{"{", RowBox[{ StyleBox[ \(\(2\ \((x\ - \ 2)\)\)\/\(\((x\ - \ 2)\)\^2\ + \ \((y\ - \ 1)\)\^2\)\), FontSize->16], ",", StyleBox[ \(\(2\ \((y\ - \ 1)\)\)\/\(\((x\ - \ 2)\)\^2\ + \ \((y\ - \ 1)\)\^2\)\), FontSize->16]}], "}"}]}]]], "\nacross ", Cell[BoxData[ \(C\)]], " without going to all the bother of parameterizing ", Cell[BoxData[ \(C\)]], "." }], "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "This is definitely a case in which a little knowledge can save a lot of \ work. Only a bean-counting dweeb would find pleasure in parameterizing that \ silly polygonal curve ", Cell[BoxData[ \(C\)]], ".\n" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[Field, m, n, x, y];\)\), "\n", \(\(m[x_, y_] = \(2\ \((x - 2)\)\)\/\(\((x - 2)\)\^2 + \((y - 1)\)\^2\);\)\), \ "\n", \(\(n[x_, y_] = \(2\ \((y - 1)\)\)\/\(\((x - 2)\)\^2 + \((y - 1)\)\^2\);\)\), \ "\n", \(Field[x_, y_] = {m[x, y], n[x, y]}\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \({\(2\ \((\(-2\) + x)\)\)\/\(\((\(-2\) + x)\)\^2 + \((\(-1\) + \ y)\)\^2\), \(2\ \((\(-1\) + y)\)\)\/\(\((\(-2\) + x)\)\^2 + \((\(-1\) + \ y)\)\^2\)}\)], "Output"] }, Open ]], Cell[TextData[{ "Note the singularity at ", Cell[BoxData[ \({2, 1}\)]], "." }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(Field[2, 1]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(Power::"infy" \(\(:\)\(\ \)\) "Infinite expression \!\(1\/0\) encountered."\)], "Message"], Cell[BoxData[ \(Power::"infy" \(\(:\)\(\ \)\) "Infinite expression \!\(1\/0\) encountered."\)], "Message"], Cell[BoxData[ \({Indeterminate, Indeterminate}\)], "Output"] }, Open ]], Cell[TextData[{ "Look at ", Cell[BoxData[ \(divField[x, y]\)]], ":" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(Together[D[m[x, y], x] + D[n[x, y], y]]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[TextData[{ "Good.\nIf there were no singularities inside ", Cell[BoxData[ \(C\)]], ", then this information would be enough to tell you that the flow of this \ field across ", Cell[BoxData[ \(C\)]], " is ", Cell[BoxData[ \(0\)]], ". But the nasty singularity (blow-up) at ", Cell[BoxData[ \({2, 1}\)]], " inside the curve ", Cell[BoxData[ \(C\)]], " might be a sink or source. \nThis won't hold you back because you can \ pull off a pretty neat stunt:\nEncapsulate the singularity by centering a \ little circle ", Cell[BoxData[ \(C\_1\)]], " around the singularity at ", Cell[BoxData[ \({2, 1}\)]], ", taking care that the little circle lies completely within ", Cell[BoxData[ \(C\)]], ". \nHere's one:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(singularity = {2, 1};\)\), "\n", \(\(Clear[x1, y1, t];\)\), "\n", \(\({x1[t_], y1[t_]} = singularity + 0.5\ {Cos[t], Sin[t]};\)\), "\n", \(\(C1plot = ParametricPlot[{x1[t], y1[t]}, {t, 0, 2\ \[Pi]}, PlotStyle \[Rule] {{Thickness[0.01], Red}}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(singularityplot = Graphics[{Blue, PointSize[0.03], Point[singularity]}];\)\), "\n", \(\(extralabel = Graphics[Text["\", {1.3, 0.6}]];\)\n\), "\n", \(\(Show[Cplot, C1plot, 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two curves, you can \ be certain that that the flow of this vector field across the little circle ", Cell[BoxData[ \(C\_1\)]], " is the same as the flow of this vector field across ", Cell[BoxData[ \(C\)]], ". 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go with just a couple flicks of your fingers:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"flowacrossC1", "=", RowBox[{"NIntegrate", "[", RowBox[{ RowBox[{ RowBox[{\(-n[x1[t], y1[t]]\), " ", RowBox[{ SuperscriptBox["x1", "\[Prime]", MultilineFunction->None], "[", "t", "]"}]}], "+", RowBox[{\(m[x1[t], y1[t]]\), " ", RowBox[{ SuperscriptBox["y1", "\[Prime]", MultilineFunction->None], "[", "t", "]"}]}]}], ",", \({t, 0, 2\ \[Pi]}\)}], "]"}]}], ";"}], "\n", RowBox[{ RowBox[{"flowacrossC2", "=", RowBox[{"NIntegrate", "[", RowBox[{ RowBox[{ RowBox[{\(-n[x2[t], y2[t]]\), " ", RowBox[{ SuperscriptBox["x2", "\[Prime]", MultilineFunction->None], "[", "t", "]"}]}], "+", RowBox[{\(m[x2[t], y2[t]]\), " ", RowBox[{ SuperscriptBox["y2", "\[Prime]", MultilineFunction->None], "[", "t", "]"}]}]}], ",", \({t, 0, 2\ \[Pi]}\)}], "]"}]}], ";", "\n", "\n", \(flowacrossC = flowacrossC1 + flowacrossC2\)}]}], "Input", AspectRatioFixed->True], Cell[BoxData[ 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At least one of the singularities must have \ been a big time sink." }], "SmallText"] }, Closed]], Cell["Tutorial Problem", "Subsubsection"], Cell[CellGroupData[{ Cell["T.4) Summary of main ideas.", "Subsection", CellTags->"VC.06.T4"], Cell[TextData[{ "Vector Calculus&", StyleBox["Mathematica", FontSlant->"Italic"], " offers this summary to you \nfor your good use and enjoyment.\nIt comes \ from the home office to you.\n'99" }], "Special2"], Cell["T.4.a.i) Sources versus sinks", "Subsubsection"], Cell[TextData[{ "If \n ", Cell[BoxData[ \(Field[x, y] = {m[x, y], n[x, y]}\)]], "\nis a vector field, then a point ", Cell[BoxData[ \({x, y}\)]], " is:\n\[Rule] a source of new fluid if\n ", Cell[BoxData[ \(divField[x, y] = D[m[x, y], x] + \ D[n[x, y], y] > 0\)]], "\n\[Rule] a sink for old fluid if\n ", Cell[BoxData[ \(divField[x, y] = D[m[x, y], x] + \ D[n[x, y], y] < 0\)]], "\n\[Rule] neither a source nor a sink if\n ", Cell[BoxData[ \(divField[x, y] = \(D[m[x, y], x] + \ D[n[x, y], y] = 0\)\)]], "." }], "Text"], Cell["T.4.a.ii) Counterclockwise versus clockwise swirl", "Subsubsection"], Cell[TextData[{ "If \n ", Cell[BoxData[ \(Field[x, y] = {m[x, y], n[x, y]}\)]], "\nis a vector field, then a the swirl about a point ", Cell[BoxData[ \({x, y}\)]], " is:\n\[Rule] counterclockwise provided\n ", Cell[BoxData[ \(rotField[x, y] = D[n[x, y], x] - 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" }], "Text"] }, Closed]], Cell["Give It a Try Problem", "Subsubsection"], Cell[CellGroupData[{ Cell["\<\ G.5) 2D electric fields, dipole fields, and Gauss's law in \ physics*\ \>", "Subsection", CellTags->"VC.06.G5"], Cell["G.5.a)", "Subsubsection"], Cell[TextData[{ "When you put a charge proportional to a number ", Cell[BoxData[ \(q\)]], " at a point ", Cell[BoxData[ \({a, b}\)]], ", the resulting two-dimensional electrical field is:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[x, y, a, b, q, Electfield];\)\), "\n", \(Electfield[x_, y_, a_, b_, q_] = \(q\ {x - a, y - b}\)\/\(\((x - a)\)\^2 + \((y - b)\)\^2\)\)}], \ "Input", AspectRatioFixed->True], Cell[BoxData[ \({\(q\ \((\(-a\) + x)\)\)\/\(\((\(-a\) + x)\)\^2 + \((\(-b\) + \ y)\)\^2\), \(q\ \((\(-b\) + y)\)\)\/\(\((\(-a\) + x)\)\^2 + \((\(-b\) + \ y)\)\^2\)}\)], "Output"] }, Open ]], Cell[TextData[{ "Here's a plot in the case ", Cell[BoxData[ \({a, b} = {1.25, 0.25}\)]], " and ", Cell[BoxData[ \(q = 2\)]], ":" }], 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-3.40807, 0.0255689, \ 0.0255689}}] }, Open ]], Cell["Describe what you see.", "Text"], Cell["G.5.b)", "Subsubsection"], Cell[TextData[{ "Here are the divergence and the rotation of the electric field resulting \ from a point charge of strength ", Cell[BoxData[ \(q\)]], " placed at ", Cell[BoxData[ \({a, b}\)]], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[x, y, a, b, q, Electfield, m, n, divElectfield, rotElectfield];\)\), "\n", \(Electfield[x_, y_, a_, b_, q_] = \(q\ {x - a, y - b}\)\/\(\((x - a)\)\^2 + \((y - b)\)\^2\)\)}], \ "Input", AspectRatioFixed->True], Cell[BoxData[ \({\(q\ \((\(-a\) + x)\)\)\/\(\((\(-a\) + x)\)\^2 + \((\(-b\) + \ y)\)\^2\), \(q\ \((\(-b\) + y)\)\)\/\(\((\(-a\) + x)\)\^2 + \((\(-b\) + \ y)\)\^2\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\({m[x_, y_], n[x_, y_]} = Electfield[x, y, a, b, q];\)\), "\n", \(divElectfield[x_, y_] = Simplify[D[m[x, y], x] + D[n[x, y], y]]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(rotElectfield[x_, y_] = Simplify[D[n[x, y], x] - D[m[x, y], y]]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[TextData[{ "Does this electric field have sources or sinks at points other than ", Cell[BoxData[ \({a, b}\)]], "?\nHow do you know?\nWhat is the net flow of this electric field across \ any closed curve without loops that does not wrap around the singularity at \ ", Cell[BoxData[ \({a, b}\)]], "?" }], "Text"], Cell["G.5.c)", "Subsubsection"], Cell[TextData[{ "Here is the measurement of the flow of the electric field across any \ circle of radius ", Cell[BoxData[ \(r\)]], " centered at ", Cell[BoxData[ \({a, b}\)]], ". " }], "Text"], Cell[TextData[{ "Electrical folks call the net flow measurement of an electric field \n\ across a curve ", Cell[BoxData[ \(C\)]], " by the name \"", Cell[BoxData[ \(flux\)]], " ", Cell[BoxData[ \(across\)]], " ", Cell[BoxData[ \(C\)]], ".\"" }], "Special2"], Cell[CellGroupData[{ Cell[BoxData[{\(Clear[x, y, a, b, q, Electfield, m, n];\), "\n", \(Electfield[x_, y_, a_, b_, q_] = \(q\ {x - a, y - b}\)\/\(\((x - a)\)\^2 + \((y - \ b)\)\^2\);\), "\n", \({m[x_, y_], n[x_, y_]} = Electfield[x, y, a, b, q];\), "\n", RowBox[{\({x[t_], y[t_]} = {a, b} + r\ {Cos[t], Sin[t]}\), ";", "\n", "\n", RowBox[{\(\[Integral]\_0\%\(2\ \[Pi]\)\), RowBox[{ RowBox[{"(", RowBox[{ RowBox[{\(-n[x[t], y[t]]\), " ", RowBox[{ SuperscriptBox["x", "\[Prime]", MultilineFunction->None], "[", "t", "]"}]}], "+", RowBox[{\(m[x[t], y[t]]\), " ", RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "t", "]"}]}]}], ")"}], \(\[DifferentialD]t\)}]}]}]}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(6.283185307179586`\ q\)], "Output"] }, Open ]], Cell[TextData[{ "Usually when the divergence of a vector field is ", Cell[BoxData[ \(0\)]], ", then its flow across any closed curve is ", Cell[BoxData[ \(0\)]], ". But this is not true here for closed curves wrapping around ", Cell[BoxData[ \({a, b}\)]], " because of the singularity (blow up) of the electric field at ", Cell[BoxData[ \({a, b}\)]], ".\nWhat is the flow of the electric field across any closed curve without \ loops that wraps around the singularity at ", Cell[BoxData[ \({a, b}\)]], "?" }], "Text"], Cell["G.5.d) Dipoles", "Subsubsection"], Cell["\<\ A dipole can be approximated by two charges of the same magnitude \ but opposite sign separated by a small distance. Dipoles are especially important in atomic theory. The great scientist \ Richard Feynman explained it this way:\ \>", "Text"], Cell["\<\ \"Although an atom or molecule remains neutral in an external electronic field, there is a tiny separation of its positive and negative charges and it becomes a microscopic dipole.\" .....................The Feynman Lectures\ \>", "Special2"], Cell[TextData[{ "Here is a plot of the dipole electrical field resulting from a positive \ charge at ", Cell[BoxData[ \({0.25, 0}\)]], " and the opposite charge of the same magnitude at ", Cell[BoxData[ \({\(-0.25\), 0}\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[x, y, a, b, q, Electfield, m, n];\)\), "\n", \(\(Electfield[x_, y_, a_, b_, q_] = \(q\ {x - a, y - b}\)\/\(\((x - a)\)\^2 + \((y - b)\)\^2\);\)\ \), "\n", \(\({aplus, bplus} = {0.25, 0};\)\), "\n", \(\({aminus, bminus} = {\(-0.25\), 0};\)\), "\n", \(\(q = 2;\)\), "\n", \(\(singularitysource = Graphics[{PointSize[0.04], Red, Point[{aplus, bplus}]}];\)\), "\n", \(\(singularitysink 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aplus, bplus, q] + Electfield[x, y, aminus, bminus, \(-q\)]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \({\(q\ \((\(-0.25`\) + x)\)\)\/\(\((\(-0.25`\) + x)\)\^2 + y\^2\) - \(q\ \ \((\(\(0.25`\)\(\[InvisibleSpace]\)\) + x)\)\)\/\(\((\(\(0.25`\)\(\ \[InvisibleSpace]\)\) + x)\)\^2 + y\^2\), \(q\ y\)\/\(\((\(-0.25`\) + x)\)\^2 \ + y\^2\) - \(q\ y\)\/\(\((\(\(0.25`\)\(\[InvisibleSpace]\)\) + x)\)\^2 + y\^2\ \)}\)], "Output"] }, Open ]], Cell[TextData[{ "Report on:\n\[Rule] ", Cell[BoxData[ \(divDipole[x, y]\)]], "\n\[Rule] ", Cell[BoxData[ \(rotDipole[x, y]\)]], "\n\[Rule] The flow of ", Cell[BoxData[ \(Dipole[x, y]\)]], " across the square with corners at ", Cell[BoxData[ \({\(-1\), \(-1\)}\)]], ", ", Cell[BoxData[ \({1, \(-1\)}\)]], ", ", Cell[BoxData[ \({1, 1}\)]], " and ", Cell[BoxData[ \({\(-1\), 1}\)]], ".\n\[Rule] The flow of ", Cell[BoxData[ \(Dipole[x, y]\)]], " across any closed curve without loops enclosing both of the singularites \ at ", Cell[BoxData[ \({aplus, bplus}\)]], " and ", Cell[BoxData[ \({aminus, bminus}\)]], "." }], "Text"], Cell["G.5.e.i) Lots of point charges", "Subsubsection"], Cell[TextData[{ "You can build intricate electric fields by taking ", Cell[BoxData[ \(n\)]], " different points \n ", Cell[BoxData[ \({a\_1, b\_1}, {a\_2, b\_2}, \[Ellipsis], {a\_n, b\_n}\)]], " \nand placing point charges of strength \n ", Cell[BoxData[ \(q\_1\)]], " at ", Cell[BoxData[ \({a\_1, b\_1}\)]], ",\n ", Cell[BoxData[ \(q\_2\)]], " at ", Cell[BoxData[ \({a\_2, b\_2}\)]], ",\n ", Cell[BoxData[ \(\[Ellipsis]\)]], " , and\n ", Cell[BoxData[ \(q\_n\)]], " at ", Cell[BoxData[ \({a\_n, b\_n}\)]], ".\nThe strengths ", Cell[BoxData[ \(q\_1\)]], ", ", Cell[BoxData[ \(q\_2\)]], ", ", Cell[BoxData[ \(\[Ellipsis]\)]], " , and ", Cell[BoxData[ \(q\_n\)]], " can be positive (singularity sources) or negative (singularity sinks).\n\ The combined electric field resulting from this placement of point charges is \ the sum of the individual electric fields resulting from the point charges of \ strength ", Cell[BoxData[ \(q\_i\)]], " at ", Cell[BoxData[ \({a\_i, b\_i}\)]], ". \nAs you can imagine, the formula for the resulting vector field can be \ complicated as all get-out. Nevertheless you can explain why this \ complicated electric field has no sources or sinks other than at the \ singularities at\n ", Cell[BoxData[ \({a\_1, b\_1}, {a\_2, b\_2}, \[Ellipsis], {a\_n, b\_n}\)]], ".\nGive your explanation." }], "Text"], Cell["G.5.e.ii) Gauss's law in physics", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ "Gauss's law in physics says that if you take ", Cell[BoxData[ \(n\)]], " different points \n ", Cell[BoxData[ \({a\_1, b\_1}, {a\_2, b\_2}, \[Ellipsis], {a\_n, b\_n}\)]], "\ninside a closed curve ", Cell[BoxData[ \(C\)]], " and you place electrical charges of strength \n ", Cell[BoxData[ \(q\_1\)]], " at ", Cell[BoxData[ \({a\_1, b\_1}\)]], ",\n ", Cell[BoxData[ \(q\_2\)]], " at ", Cell[BoxData[ \({a\_2, b\_2}\)]], ",\n ", Cell[BoxData[ \(\[Ellipsis]\)]], " , and\n ", Cell[BoxData[ \(q\_n\)]], " at ", Cell[BoxData[ \({a\_n, b\_n}\)]], ",\nthen the ", Cell[BoxData[ \(flux\)]], " (= flow) of the resulting electric field across ", Cell[BoxData[ \(C\)]], " is simply\n ", Cell[BoxData[ \(2 \[Pi] \((q\_1 + \ q\_2 + \ q\_3 + \ \[Ellipsis]\ + \ q\_n)\)\)]], ".\nExplain where Gauss's law comes from." }], "Text"], Cell["Tip:", "Special1"], Cell["Look at this:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[ElectricField, a, b, k, m, n, q, x, y];\)\), "\n", \(\({m\_k[x_, y_], n\_k[x_, y_]} = \(q\_k\ {x - a\_k, y - b\_k}\)\/\(\((x - a\_k)\)\^2 + \((y \ - b\_k)\)\^2\);\)\), "\n", \(ElectricField\_k[x_, y_] = {m\_k[x, y], n\_k[x, y]}\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \({\(\((x - a\_k)\)\ q\_k\)\/\(\((x - a\_k)\)\^2 + \((y - b\_k)\)\^2\), \ \(\((y - b\_k)\)\ q\_k\)\/\(\((x - a\_k)\)\^2 + \((y - b\_k)\)\^2\)}\)], \ "Output"] }, Open ]], Cell[TextData[{ "and look at the flow-across measurement \n ", Cell[BoxData[ \(\[ContourIntegral]\_\(\(\ \)\(C\_i\)\)\ \(-n[x, y]\) \[DifferentialD]x\ + \ m[x, y] \[DifferentialD]y\)]], "\nwhere ", Cell[BoxData[ \(C\_i\)]], " is a small circle centered at ", Cell[BoxData[ \({a\_i, b\_i}\)]], ": " }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{\(Clear[r, t]\), ";", "\n", \({x[t_], y[t_]} = {a\_k, b\_k} + r\_k\ {Cos[t], Sin[t]}\), ";", "\n", RowBox[{\(\[Integral]\_0\%\(2\ \[Pi]\)\), RowBox[{ RowBox[{"(", RowBox[{ RowBox[{\(-n\_k[x[t], y[t]]\), " ", RowBox[{ SuperscriptBox["x", "\[Prime]", MultilineFunction->None], "[", "t", "]"}]}], "+", RowBox[{\(m\_k[x[t], y[t]]\), " ", RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "t", "]"}]}]}], ")"}], \(\[DifferentialD]t\)}]}]}]], "Input", AspectRatioFixed->True], Cell[BoxData[ \(2\ \[Pi]\ q\_k\)], "Output"] }, Open ]] }, Closed]] }, Closed]], Cell["Literacy Problem", "Subsubsection"], Cell["L.12)", "Subsubsection"], Cell["Here are two curves: ", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[x1, y1, x2, y2, t];\)\), "\n", \(\(tlow = 0;\)\), "\n", \(\(thigh = 2\ \[Pi];\)\), "\n", \(\({x1[t_], y1[t_]} = 2\ {Cos[t], 0.6\ Sin[t] + 0.2\ Sin[4\ t]\^2};\)\), "\n", 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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}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonFrame->"None", ButtonNote->"Come visit us!"}], Cell[StyleData["MEIHyperlink", "Printout"], FontColor->GrayLevel[0], Background->GrayLevel[1], FontVariations->{"Underline"->False}], Cell[StyleData["MEIHyperlink", "TwoColumn"], FontColor->GrayLevel[0], Background->GrayLevel[1], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["BasicsHyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->None, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0.848096, 0.171878, 0.228321], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonFrame->"None", ButtonNote->"Go to Basics"}], Cell[StyleData["BasicsHyperlink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}], Cell[StyleData["BasicsHyperlink", "TwoColumn"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["BasicsIndexHyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->None, FontWeight->"Bold", FontColor->RGBColor[0.848096, 0.171878, 0.228321], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2], FrontEnd`SelectionMove[ FrontEnd`SelectedNotebook[ ], Next, CellGroup], FrontEndToken[ "SelectionCloseAllGroups"], FrontEndToken[ "OpenCloseGroup"]}]&), Active->True, ButtonFrame->"None", ButtonNote->"Link into Basics"}], Cell[StyleData["BasicsIndexHyperlink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}], Cell[StyleData["BasicsIndexHyperlink", "TwoColumn"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["TutorialsHyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->None, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0.0199588, 0.346716, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonFrame->"None", ButtonNote->"Go to Tutorials"}], Cell[StyleData["TutorialsHyperlink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}], Cell[StyleData["TutorialsHyperlink", "TwoColumn"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["TutorialsIndexHyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->None, FontWeight->"Bold", FontColor->RGBColor[0.0199588, 0.346716, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2], FrontEnd`SelectionMove[ FrontEnd`SelectedNotebook[ ], Next, CellGroup], FrontEndToken[ "SelectionCloseAllGroups"], FrontEndToken[ "OpenCloseGroup"]}]&), Active->True, ButtonFrame->"None", ButtonNote->"Link into Tutorials"}], Cell[StyleData["TutorialsIndexHyperlink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}], Cell[StyleData["TutorialsIndexHyperlink", "TwoColumn"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["GiveItaTryHyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->None, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0.497459, 0.196094, 0.543877], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonFrame->"None", ButtonNote->"Go to GiveItaTry"}], Cell[StyleData["GiveItaTryHyperlink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}], Cell[StyleData["GiveItaTryHyperlink", "TwoColumn"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["GiveItaTryIndexHyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->None, FontWeight->"Bold", FontColor->RGBColor[0.497459, 0.196094, 0.543877], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2], FrontEnd`SelectionMove[ FrontEnd`SelectedNotebook[ ], Next, CellGroup], FrontEndToken[ "SelectionCloseAllGroups"], FrontEndToken[ "OpenCloseGroup"]}]&), Active->True, ButtonFrame->"None", ButtonNote->"Link into GiveItaTry"}], Cell[StyleData["GiveItaTryIndexHyperlink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}], Cell[StyleData["GiveItaTryIndexHyperlink", "TwoColumn"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["LiteracyHyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->None, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[1, 0.433326, 0], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonFrame->"None", ButtonNote->"Go to Literacy"}], Cell[StyleData["LiteracyHyperlink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}], Cell[StyleData["LiteracyHyperlink", "TwoColumn"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["LiteracyIndexHyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->None, FontWeight->"Bold", FontColor->RGBColor[1, 0.433326, 0], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2], FrontEnd`SelectionMove[ FrontEnd`SelectedNotebook[ ], Next, CellGroup], FrontEndToken[ "SelectionCloseAllGroups"], FrontEndToken[ "OpenCloseGroup"]}]&), Active->True, ButtonFrame->"None", ButtonNote->"Link into Literacy"}], Cell[StyleData["LiteracyIndexHyperlink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}], Cell[StyleData["LiteracyIndexHyperlink", "TwoColumn"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["PreviewHyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->None, FontWeight->"Bold", FontSlant->"Italic", FontColor->GrayLevel[0.250004], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2], FrontEndToken[ "OpenCloseGroup"]}]&), Active->True, ButtonFrame->"None", ButtonNote->"Preview of Lesson"}], Cell[StyleData["PreviewHyperlink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}], Cell[StyleData["PreviewHyperlink", "TwoColumn"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["RefGuideLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "RefGuideLink", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["RefGuideLink", "Presentation"]], Cell[StyleData["RefGuideLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}], Cell[StyleData["RefGuideLink", "TwoColumn"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["GettingStartedLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "GettingStarted", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["GettingStartedLink", "Presentation"]], Cell[StyleData["GettingStartedLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}], Cell[StyleData["GettingStartedLink", 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