(*********************************************************************** 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[ 203236, 6398]*) (*NotebookOutlinePosition[ 204002, 6425]*) (* CellTagsIndexPosition[ 203958, 6421]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ StyleBox["Students and ", FontSize->24], StyleBox["Mathematica", FontSize->24, FontSlant->"Italic"], StyleBox[":\nSwirl Discoveries. ", FontSize->24], StyleBox["A 2D Linear Differential Equations Investigation", "Subtitle", FontColor->GrayLevel[0]] }], "Title", TextAlignment->Left], Cell[TextData[{ "The coefficient matrix of 2-dimensional linear differential equation \ systems has complex eigenvalues and eigenvectors if and only if there are \ lines of attraction. The proof arose from and makes use of a graphical field \ plot representation of such systems. The use of ", StyleBox["Mathematica", FontSlant->"Italic"], " for both algebraic calculation and generation of these field plots \ allowed the proof to arise quite naturally." }], "Subsubtitle"], Cell["By Nathan Linger", "Text"], Cell[TextData[{ "\nThis paper gives a proof of a result I discovered in a Differential \ Equations class at the University of Illinois at Urbana-Champaign with \ Professor Jerry Uhl. The result is that the coefficient matrix of \ 2-dimensional linear differential equation systems has complex eigenvalues \ and eigenvectors if and only if there are lines of attraction. The proof \ arose from and makes use of a graphical field plot representation of such \ systems. The use of ", StyleBox["Mathematica", FontSlant->"Italic"], " for both algebraic calculation and generation of these field plots \ allowed the proof to arise quite naturally. " }], "Text"], Cell[CellGroupData[{ Cell["Introduction", "Section", TextAlignment->Left], Cell["\<\ My differential equations class was studying 2 dimensional linear \ differential equations of the form \ \>", "Text"], Cell[TextData[Cell[BoxData[ FormBox[GridBox[{ {\(x' = a\ x + b\ y\)}, {\(y' = c\ x + d\ y\)} }], TraditionalForm]]]], "Text", TextAlignment->Center], Cell["To visualize such a system we plotted the vector field", "Text"], Cell[TextData[Cell[BoxData[ \(TraditionalForm\`F[x, y] = {x', y'}\)]]], "Text", TextAlignment->Center], Cell["\<\ Comparisons between the resulting plots of vector fields reveals \ properties of the system. 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ool0oooo00h0oooo0P00000@0?ooo`80000000<0oooo0000003oool0303oool5000004X0oooo000E 0?ooo`030000003oool0oooo00h0oooo00<000000?ooo`3oool03P3oool00`000000oooo0?ooo`0? 0?ooo`030000003oool0000000l0oooo1000000=0?ooo`D00000B03oool001H0oooo00<000000?oo o`3oool03P3oool00`000000oooo0?ooo`0>0?ooo`800000403oool3000000h0oooo1@00001I0?oo o`005`3oool00`000000oooo0?ooo`0>0?ooo`030000003oool0oooo00l0oooo0P00000>0?ooo`@0 0000J`3oool001L0oooo00<000000?ooo`3oool03`3oool2000000l0oooo0`00000?0?ooo`<00000 JP3oool001P0oooo00<000000?ooo`3oool03P3oool200000100oooo0P00001l0?ooo`006@3oool2 000000l0oooo0P00000A0?ooo`030000003oool0oooo07T0oooo000H0?ooo`<00000403oool00`00 0000oooo0?ooo`2;0?ooo`006@3oool300000100oooo00<000000?ooo`3oool0RP3oool001/0oooo 00<000000?ooo`3oool0V`3oool00;T0oooo002i0?ooo`00^@3oool00;T0oooo0000\ \>"], ImageRangeCache->{{{0, 184.375}, {176.125, 0}} -> {-1.27691, -1.21977, \ 0.0138512, 0.0138512}}] }, Open ]], Cell[TextData[{ "All trajectories in this system end up rotating outwards forever. We \ learned that systems of this sort have coefficient matrices with non-real \ eigenvalues. These systems were called ", StyleBox["pure swirlers", FontWeight->"Bold"], "." }], "Text"] }, Open ]], Cell["\<\ Questions arose about other features of the field plot \ representations. In the case of a pure swirler, how can one determine from \ the coefficient matrix whether its trajectories will swirl clockwise or \ counter-clockwise. This is the question that I began investigating.\ \>", \ "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Definition of Swirl", "Section", TextAlignment->Left], Cell["\<\ It is useful to interpret x and y as the coordinates of a \ particle's position. Then solutions of the system are possible particle \ trajectories. We can already determine a particle's velocity from its \ position. Another useful quantity is its acceleration. This can easily be \ derived by taking derivatives of the original equations to obtain\ \>", "Text"], Cell[TextData[Cell[BoxData[ FormBox[GridBox[{ {\(x'' = a\ x' + b\ y'\)}, {\(y'' = c\ x' + d\ y'\)} }], TraditionalForm]]]], "Text", TextAlignment->Center], Cell["\<\ So acceleration is obtained by applying the same coefficient matrix \ to the velocity.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"velocity", "=", RowBox[{ RowBox[{"(", GridBox[{ {"a", "b"}, {"c", "d"} }], ")"}], ".", \({x, y}\)}]}], "\[IndentingNewLine]", RowBox[{"acceleration", "=", RowBox[{ RowBox[{"(", GridBox[{ {"a", "b"}, {"c", "d"} }], ")"}], ".", "velocity"}]}]}], "Input"], Cell[BoxData[ \({a\ x + b\ y, c\ x + d\ y}\)], "Output"], Cell[BoxData[ \({a\ \((a\ x + b\ y)\) + b\ \((c\ x + d\ y)\), c\ \((a\ x + b\ y)\) + d\ \((c\ x + d\ y)\)}\)], "Output"] }, Open ]], Cell["\<\ What we want to define is which way a particle is \"turning\" at a \ given point in its trajectory. Is it being pulled to the left or right of \ its current path? This is the same question as whether or not its \ acceleration vector is on the left or right of its velocity vector. The way \ to determine this is with a cross product.\ \>", "Text"], Cell[TextData[{ "For a cross product, we need 3 dimensions instead of just two. So we \ append a z-coordinate of ", StyleBox["0", "Input"], " to each vector." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(velocity = Append[velocity, 0]\), "\n", \(acceleration = Append[acceleration, 0]\)}], "Input"], Cell[BoxData[ \({a\ x + b\ y, c\ x + d\ y, 0}\)], "Output"], Cell[BoxData[ \({a\ \((a\ x + b\ y)\) + b\ \((c\ x + d\ y)\), c\ \((a\ x + b\ y)\) + d\ \((c\ x + d\ y)\), 0}\)], "Output"] }, Open ]], Cell[TextData[{ "Now we are ready to define ", StyleBox["Swirl", FontWeight->"Bold"], " as the z-coordinate of the cross product of the velocity and acceleration \ of a point at position {x,y}." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"CartesianSwirl", "[", RowBox[{"(", GridBox[{ {"a_", "b_"}, {"c_", "d_"} }], ")"}], "]"}], "[", \(x_, y_\), "]"}], "=", \(\(Cross[velocity, acceleration]\)\[LeftDoubleBracket]3\[RightDoubleBracket]\)}]], \ "Input"], Cell[BoxData[ \(\(-b\)\ c\^2\ x\^2 + a\ c\ d\ x\^2 + a\ b\ c\ x\ y - a\^2\ d\ x\ y - b\ c\ d\ x\ y + a\ d\^2\ x\ y + b\^2\ c\ y\^2 - a\ b\ d\ y\^2\)], "Output"] }, Open ]], Cell["It is also useful to think of Swirl in polar coordinates.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"PolarSwirl", "[", RowBox[{"(", GridBox[{ {"a_", "b_"}, {"c_", "d_"} }], ")"}], "]"}], "[", \(r_, \[Theta]_\), "]"}], "=", RowBox[{ RowBox[{ RowBox[{ RowBox[{"CartesianSwirl", "[", RowBox[{"(", GridBox[{ {"a", "b"}, {"c", "d"} }], ")"}], "]"}], "[", \(r\ Cos[\[Theta]], r\ Sin[\[Theta]]\), "]"}], "//", "TrigReduce"}], "//", "Simplify"}]}]], "Input"], Cell[BoxData[ \(\(-\(1\/2\)\)\ \((b\ c - a\ d)\)\ r\^2\ \((\(-b\) + c + \((b + c)\)\ Cos[2\ \[Theta]] + \((\(-a\) + d)\)\ Sin[ 2\ \[Theta]])\)\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Interpretation of Swirl", "Section", TextAlignment->Left], Cell["\<\ Swirl is simply a measurement of which way the imaginary particle \ at that point is turning. If it is veering to the left (turning \ counter-clockwise), Swirl is positive. If it is veering to the right \ (turning clockwise), Swirl is negative. If it is not veering to either the \ left or the right (it is going in a perfectly straight line), Swirl is \ zero.\ \>", "Text"], Cell["\<\ So the system is a pure swirler iff Swirl is never zero. 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Y03oool006P0oooo1@0000050?ooo`H00000103oool400000:00oooo001f0?ooo`@000001P3oool4 000009`0oooo00230?ooo`<00000VP3oool00?l0oooo8@3oool00?l0oooo8@3oool00?l0oooo8@3o ool00?l0oooo8@3oool00?l0oooo8@3oool00001\ \>"], ImageRangeCache->{{{0, 287}, {117.875, 0}} -> {-0.05546, -0.0206425, \ 0.00735513, 0.00735513}, {{7.5, 137}, {115.062, 2.75}} -> {-1.42422, \ -1.16241, 0.0197124, 0.0197124}, {{149.938, 279.438}, {115.062, 2.75}} -> \ {-8.64188, -25.7799, 0.0565887, 0.437177}}] }, Open ]], Cell["\<\ Here we have a \"partitioned\" swirler which turns \ counter-clockwise at some places and clockwise at others. The polar swirl \ plot show this: Swirl is positive (counter-clockwise) for some ranges of \ angles and negative (clockwise) for others.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["A Connection to Eigenvectors", "Section", TextAlignment->Left], Cell[TextData[{ "Let's investigate the points where Swirl is zero. Here are the \ constraints on ", StyleBox["x", "Input"], " and ", StyleBox["y", "Input"], " that make Swirl zero:" }], "Text", TextAlignment->Left], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"CartesianSwirl", "[", RowBox[{"(", GridBox[{ {"a", "b"}, {"c", "d"} }], ")"}], "]"}], "[", \(x, y\), "]"}], "==", "0"}], ",", \({x, y}\)}], "]"}]], "Input", TextAlignment->Left], Cell[BoxData[ \(Solve::"svars" \(\(:\)\(\ \)\) "Equations may not give solutions for all \"solve\" variables."\)], \ "Message"], Cell[BoxData[ \({{x \[Rule] \(\(-\((\(-a\) + d)\)\)\ y - \@\(a\^2 + 4\ b\ c - 2\ a\ d + \ d\^2\)\ y\)\/\(2\ c\)}, {x \[Rule] \(\(-\((\(-a\) + d)\)\)\ y + \@\(a\^2 + 4\ \ b\ c - 2\ a\ d + d\^2\)\ y\)\/\(2\ c\)}}\)], "Output"] }, Open ]], Cell["This is a familiar looking result.", "Text", TextAlignment->Left], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Eigenvectors", "[", RowBox[{"(", GridBox[{ {"a", "b"}, {"c", "d"} }], ")"}], "]"}]], "Input", TextAlignment->Left], Cell[BoxData[ \({{\(-\(\(\(-a\) + d + \@\(a\^2 + 4\ b\ c - 2\ a\ d + d\^2\)\)\/\(2\ c\)\)\), 1}, {\(-\(\(\(-a\) + d - \@\(a\^2 + 4\ b\ c - 2\ a\ d + d\^2\)\)\/\(2\ c\)\)\), 1}}\)], "Output"] }, Open ]], Cell["Let's set y=1 in the original solution", "Text", TextAlignment->Left], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"CartesianSwirl", "[", RowBox[{"(", GridBox[{ {"a", "b"}, {"c", "d"} }], ")"}], "]"}], "[", \(x, 1\), "]"}], "==", "0"}], ",", \({x}\)}], "]"}]], "Input", TextAlignment->Left], Cell[BoxData[ \({{x \[Rule] \(-\(\(\(-a\) - \@\(4\ b\ c + \((a - d)\)\^2\) + d\)\/\(2\ c\)\)\)}, {x \[Rule] \(-\(\(\(-a\) + \@\(4\ b\ c \ + \((a - d)\)\^2\) + d\)\/\(2\ c\)\)\)}}\)], "Output"] }, Open ]], Cell[TextData[{ "The ", StyleBox["{x,y}", "Input"], " coordinates where swirl is ", StyleBox["0", "Input"], " are two lines through the origin having the same slope as their \ corresponding eigenvectors. So these eigenvector lines partition the plane \ into sections of equivalent direction (sign) of Swirl." }], "Text", TextAlignment->Left], Cell["\<\ This is why all trajectories coming from systems with non-real \ eigenvectors swirl in the same direction (clockwise or counterclockwise) at \ all points. If they didn't, i.e. there is one trajectory turning clockwise \ and another turning counterclockwise, then swirl changes from negative to \ positive. Since Swirl is a continuous function it must be zero at some \ point. This is impossible since that \"point\" is lies on a non-real \ eigenvector.\ \>", "Text", TextAlignment->Left] }, Open ]], Cell[CellGroupData[{ Cell["A Closed Form for Direction of Pure Swirlers", "Section", TextAlignment->Left], Cell["\<\ How can we decide which direction it will swirl (counter-clockwise \ or clockwise) using only the coefficient matrix? This problem is equivalent \ to finding the sign of swirl given only the coefficient matrix (if we know it \ will have the same sign at all points).\ \>", "Text"], Cell[TextData[{ "A good starting point is to find the extrema of PolarSwirl (when ", StyleBox["r", "Input"], " is fixed and ", StyleBox["t", "Input"], " varies). So solve for \[Theta] after setting the derivative of \ PolarSwirl (with respect to ", StyleBox["t", "Input"], ") to ", StyleBox["0", "Input"], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"D", "[", RowBox[{ RowBox[{ RowBox[{"PolarSwirl", "[", RowBox[{"(", GridBox[{ {"a", "b"}, {"c", "d"} }], ")"}], "]"}], "[", \(r, \[Theta]\), "]"}], ",", "\[Theta]"}], "]"}], "==", "0"}]], "Input"], Cell[BoxData[ \(\(-\(1\/2\)\)\ \((b\ c - a\ d)\)\ r\^2\ \((2\ \((\(-a\) + d)\)\ Cos[2\ \[Theta]] - 2\ \((b + c)\)\ Sin[2\ \[Theta]])\) == 0\)], "Output"], Cell["This can be simplified to the equivalent expression", "Text"], Cell[BoxData[ \(\(\(-a\) + d\)\/\(b + c\) = Tan[2\ \[Theta]]\)], "Text", TextAlignment->Center] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Tsols = Solve[\(\(-a\) + d\)\/\(b + c\) == Tan[2\ \[Theta]], \[Theta]]\)], "Input"], Cell[BoxData[ \(Solve::"ifun" \(\(:\)\(\ \)\) "Inverse functions are being used by \!\(Solve\), so some solutions may \ not be found."\)], "Message"], Cell[BoxData[ \({{\[Theta] \[Rule] 1\/2\ ArcTan[\(\(-a\) + d\)\/\(b + c\)]}}\)], "Output"] }, Open ]], Cell[TextData[{ "Since ", Cell[BoxData[ \(TraditionalForm\`Tan[2 \[Theta]]\)]], " has a period of ", Cell[BoxData[ \(TraditionalForm\`\[Pi]\/2\)]], ", any addition of ", Cell[BoxData[ \(TraditionalForm\`\[Pi]\/2\)]], " is also a solution." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Tsols\ = \ Append[Tsols, {\[Theta] -> \((\[Pi]\/2 + \[Theta] /. Tsols)\)\[LeftDoubleBracket]1\[RightDoubleBracket]}]\)], \ "Input"], Cell[BoxData[ \({{\[Theta] \[Rule] 1\/2\ ArcTan[\(\(-a\) + d\)\/\(b + c\)]}, {\[Theta] \[Rule] \ \[Pi]\/2 + 1\/2\ ArcTan[\(\(-a\) + d\)\/\(b + c\)]}}\)], "Output"] }, Open ]], Cell["\<\ Plugging these two values back into the function gives us the \ extrema of PolarSwirl. Since the signs of coefficients are unknown, the \ signs of the extrema are also unknown.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"MaxAndMin", " ", "=", " ", RowBox[{ RowBox[{"(", RowBox[{"FullSimplify", " ", "//@", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"PolarSwirl", "[", RowBox[{"(", GridBox[{ {"a", "b"}, {"c", "d"} }], ")"}], "]"}], "[", \(r, \[Theta]\), "]"}], "/.", "Tsols"}], ")"}]}], ")"}], "/.", " ", \(\@\(1 + a_\^2\/b_\^2\) -> \@\(b\^2 + a\^2\)\/b\)}]}]], "Input"], Cell[BoxData[ \({\(-\(1\/2\)\)\ \((\(-b\) + c + \@\(\((b + c)\)\^2 + \((a - d)\)\^2\))\)\ \((b\ c - a\ d)\)\ r\^2, \(-\(1\/2\)\)\ \((\(-b\) + c - \@\(\((b + c)\)\^2 + \((a - d)\)\^2\))\)\ \((b\ c - a\ d)\)\ r\^2}\)], "Output"] }, Open ]], Cell[TextData[{ "These two extrema values are of the form ", Cell[BoxData[ \(TraditionalForm\`X \[PlusMinus] Y\)]], " where" }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{"X", "=", RowBox[{\(1\/2\), \(r\^2\), \((c - b)\), RowBox[{"Det", "[", RowBox[{"(", GridBox[{ {"a", "b"}, {"c", "d"} }], ")"}], "]"}]}]}], ";"}]], "Input"], Cell["and", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"Y", "=", RowBox[{\(1\/2\), \(r\^2\), RowBox[{"Det", "[", RowBox[{"(", GridBox[{ {"a", "b"}, {"c", "d"} }], ")"}], "]"}], \(\@\(\((b + c)\)\^2 + \((a - d)\)\^2\)\)}]}], ";"}]], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Expand /@ MaxAndMin === Expand /@ {X + Y, X - Y}\)], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[TextData[{ "Now we have what we were looking for. The sign of Swirl in a pure swirler \ is the sign of X\[PlusMinus]Y, or just the sign of X. X is easiest to \ compute, so we'll use it. The ", Cell[BoxData[ \(TraditionalForm\`1\/2\)]], " and ", Cell[BoxData[ \(TraditionalForm\`r\^2\)]], " terms don't affect the sign at all, so we are left with the condition ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\((c - b)\), RowBox[{"Det", "[", GridBox[{ {"a", "b"}, {"c", "d"} }], "]"}]}], " ", ">", "0"}], TraditionalForm]]], " for counter-clockwise swirl and ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\((c - b)\), RowBox[{"Det", "[", GridBox[{ {"a", "b"}, {"c", "d"} }], "]"}]}], " ", "<", "0"}], TraditionalForm]]], " for clockwise swirl (in the case of a pure swirler)." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Evidence of Consistency", "Section", TextAlignment->Left], Cell["\<\ Toward the end of the previous section we found that the extrema of \ PolarSwirl are given by:\ \>", "Text"], Cell[BoxData[ \(\(MaxAndMin = \n\t{\(-\(1\/2\)\)\ \((\(-b\) + c + \@\(\((b + c)\)\^2 + \((a - d)\)\^2\))\)\ \((b\ c - a\ d)\)\ r\^2, 1\/2\ \((b - c + \@\(\((b + c)\)\^2 + \((a - d)\)\^2\))\)\ \((b\ c - a\ d)\)\ r\^2};\)\)], "Input"], Cell["\<\ Now all \"pure swirlers\" will have PolarSwirl extrema of the same \ sign, since Swirl never equals zero in \"pure swirler\" systems. We can test \ whether the extrema have the same sign by testing whether or not their \ product is positive.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(SameSignCondition\ = \n Factor[Expand[\n\t\tTimes\ @@ \ MaxAndMin\n\t\t]]\ > \ 0\)], "Input"], Cell[BoxData[ \(\(-\(1\/4\)\)\ \((\(-b\)\ c + a\ d)\)\^2\ \((a\^2 + 4\ b\ c - 2\ a\ d + d\^2)\)\ r\^4 > 0\)], "Output"] }, Open ]], Cell["We can divide out all positive values to obtain", "Text"], Cell[BoxData[ \(a\^2 + 4\ b\ c - 2\ a\ d + d\^2 < 0\)], "Text", TextAlignment->Center], Cell["\<\ This is the exact same condition for the eigenvectors of the \ coefficient matrix to be complex. A nice check of consistency.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Eigenvalues", "[", RowBox[{"(", GridBox[{ {"a", "b"}, {"c", "d"} }], ")"}], "]"}]], "Input"], Cell[BoxData[ \({1\/2\ \((a + d - \@\(a\^2 + 4\ b\ c - 2\ a\ d + d\^2\))\), 1\/2\ \((a + d + \@\(a\^2 + 4\ b\ c - 2\ a\ d + d\^2\))\)}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["References", "Section", TextAlignment->Left], Cell[TextData[{ StyleBox["Davis, B. and Jerry Uhl, J. .", "Text", FontFamily->"Times New Roman", FontSize->12], StyleBox["Differential Equations and Mathematica. ", "Text", FontFamily->"Times New Roman", FontSize->12, FontSlant->"Italic"], StyleBox["MathEverywhere. Columbus, Ohio 1998", "Text", FontFamily->"Times New Roman", FontSize->12] }], "SmallText"] }, Open ]], Cell["About the author", "Section"], Cell["\<\ Nathan Linger Computer Science / Mathematics Undergraduate University of Illinois at Urbana-Champaign rlinger@uiuc.edu\ \>", "Text"], Cell[CellGroupData[{ Cell["Electronic Subscriptions", "Section"], Cell[TextData[{ "Included in the distribution for each electronic subscription is the file \ ", StyleBox["swirl.nb", "Input", FontWeight->"Plain"], " containing ", StyleBox["Mathematica", FontSlant->"Italic"], " code for the material described in this article." }], "Text"] }, Open ]] }, FrontEndVersion->"4.0 for Microsoft Windows", ScreenRectangle->{{0, 1024}, {0, 695}}, AutoGeneratedPackage->None, WindowToolbars->"EditBar", CellGrouping->Manual, WindowSize->{863, 616}, WindowMargins->{{Automatic, 68}, {Automatic, 18}}, StyleDefinitions -> "Default.nb" ] (*********************************************************************** Cached data follows. 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