(************** Content-type: application/mathematica ************** Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 18395, 600]*) (*NotebookOutlinePosition[ 19093, 624]*) (* CellTagsIndexPosition[ 19049, 620]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Constant Coefficient Phase Portrait Types", "Title", CellFrame->True, TextAlignment->Center, Background->RGBColor[0, 1, 0]], Cell["\<\ This notebook is by Steven Amgott. Please send any questions or \ comments to samgott1@swarthmore.edu. Feel free to use and distribute this \ notebook, but keep this author information in any copy you use or distribute.\ \ \>", "SmallText"], Cell[TextData[{ "This notebook used the add-on package ", StyleBox["VisualDSolve", FontColor->RGBColor[0, 0, 1]], " to create the plots. ", StyleBox["VisualDSolve", FontColor->RGBColor[0, 0, 1]], " is not part of the standard ", StyleBox["Mathematica", FontSlant->"Italic"], " media set. It is a commercial third-party package available (together \ with an accompanying book) from Telos at" }], "Text"], Cell[TextData[{ " ", StyleBox["http://www.telospub.com/catalog/MATHEMATICA/VisualDSolve.html", FontColor->RGBColor[0, 0, 0.996109]], "." }], "Text", TextAlignment->Center, TextJustification->0], Cell["or by download from Wolfram Research at", "Text"], Cell[TextData[{ StyleBox[" ", TextAlignment->Center, TextJustification->0, FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->0}], StyleBox["http://store.wolfram.com/view/book/D0706.str.", TextAlignment->Center, TextJustification->0, FontFamily->"Times New Roman", FontColor->RGBColor[0, 0, 0.996109], FontVariations->{"CompatibilityType"->0}] }], "Text", TextAlignment->Center, TextJustification->0], Cell["\<\ Without this package you will not be able to create the plots in \ this notebook.\ \>", "Text"], Cell[TextData[{ "In any input cell containing ", StyleBox["xxx", FontColor->RGBColor[1, 0, 1]], " , you must replace it with your input before evaluating the cell. In \ general, anything in ", StyleBox["magenta", FontColor->RGBColor[1, 0, 1]], " is something you can, and possibly should, change." }], "Text"], Cell["\<\ The following cell is an initialization cell which will be \ automatically evaluated, provided you answer \"Yes\" to the initialization \ prompt. If you do not, you will need to evaluate it manually before \ evaluating any of the cells in this notebook which create the plots.\ \>", \ "Text"], Cell[BoxData[ \(Needs["\"]\)], "Input", InitializationCell->True], Cell[TextData[{ "You can check to see if the package loaded by evaluating the next cell. \ The output should include ", StyleBox["VisualDSolve`", FontColor->RGBColor[1, 0, 0]], ". If it does not, either the package is not on your computer or it did \ not load correctly, and you will be unable to produce the plots." }], "Text"], Cell[BoxData[ \($Packages\)], "Input"], Cell[CellGroupData[{ Cell["Proper Node", "Section"], Cell[TextData[{ "This is the case where there are two distinct real eigenvalues, either \ both positive or both negative. The general solution is of the form ", Cell[BoxData[ \(TraditionalForm\`c\_1\)]], Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\(\(\[Lambda]\_1\) t\)\)]], Cell[BoxData[ \(TraditionalForm\`\[Xi]\_1\)]], "+ ", Cell[BoxData[ \(TraditionalForm\`c\_2\)]], Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\(\(\[Lambda]\_2\) t\)\)]], Cell[BoxData[ \(TraditionalForm\`\[Xi]\_2\)]], ". We will assume ", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\_1\)]], ">", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\_2\)]], "." }], "Text"], Cell["\<\ Without trajectories (phase plane). If the eigenvalues are \ positive, the arrows move us away from the origin, and the node is unstable. \ If negative, the arrows move us toward the origin and the node is \ stable.\ \>", "Text"], Cell[BoxData[{ \(Clear[x, y, t]\), \(\(PhasePlot[{\(x'\)[t] == \(-2\)\ x[t] - y[t], \(y'\)[t] == \(-y[t]\)}, {x[t], y[t]}, \n \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {t, 0, 10}, {x, \(-2\), 2}, {\ y, \(-2\), 2}, \n \t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ VectorField -> True, FieldLogScale \[Rule] 10]; \)\ \)}], "Input"], Cell[TextData[{ "With some trajectories. Notice that the trajectories become tangent to ", Cell[BoxData[ \(TraditionalForm\`\[Xi]\_1\)]], "as we approach the origin, except the one along ", Cell[BoxData[ \(TraditionalForm\`\[Xi]\_2\)]], "." }], "Text"], Cell[BoxData[{ \(Clear[x, y, t]\), \(\(PhasePlot[{\(x'\)[t] == \(-2\)\ x[t] - y[t], \(y'\)[t] == \(-y[t]\)}, {x[t], y[t]}, \n \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {t, 0, 10}, {x, \(-2\), 2}, {\ y, \(-2\), 2}, \n \t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ VectorField -> True, FieldLogScale \[Rule] 10, \n \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ InitialValues\ -> Flatten[Table[{ .5\ i, .5 j}, {i, \(-3\), 3}, {j, \(-3\), 3}], 1]]; \)\ \)}], "Input"], Cell[CellGroupData[{ Cell["Eigenvalues and eigenvectors for this system", "Subsection"], Cell[BoxData[{\(Clear[a]\), RowBox[{ RowBox[{"a", "=", RowBox[{"(", GridBox[{ {\(-2\), \(-1\)}, {"0", \(-1\)} }], ")"}]}], ";"}]}], "Input"], Cell[BoxData[{ \(Clear[\[Lambda], v]\), \(\[Lambda] = Eigenvalues[a]\), \(v = Eigenvectors[a]\)}], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Saddle", "Section"], Cell[TextData[{ "This is the case where there are two distinct real eigenvalues of opposite \ sign. The general solution is of the form ", Cell[BoxData[ \(TraditionalForm\`c\_1\)]], Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\(\(\[Lambda]\_1\) t\)\)]], Cell[BoxData[ \(TraditionalForm\`\[Xi]\_1\)]], "+ ", Cell[BoxData[ \(TraditionalForm\`c\_2\)]], Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\(\(\[Lambda]\_2\) t\)\)]], Cell[BoxData[ \(TraditionalForm\`\[Xi]\_2\)]], ". We will assume ", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\_1\)]], ">0>", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\_2\)]], "." }], "Text"], Cell["Without trajectories.", "Text"], Cell[BoxData[{ \(Clear[x, y, t]\), \(\(PhasePlot[{\(x'\)[t] == x[t], \(y'\)[t] == x[t] - y[t]}, {x[t], y[t]}, \n\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {t, \(-10\), 10}, {x, \(-2\), 2}, {\ y, \(-2\), 2}, \n \t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ VectorField -> True, FieldLogScale \[Rule] 10]; \)\ \)}], "Input"], Cell[TextData[{ "With some trajectories. Notice that there are two trajectories along the \ eigenvectors that act as asymptotes for all other trajectories. For positive \ t, the trajectories approach ", Cell[BoxData[ \(TraditionalForm\`\[Xi]\_1\)]], "and for negative t they approach ", Cell[BoxData[ \(TraditionalForm\`\[Xi]\_2\)]], "." }], "Text"], Cell[BoxData[{ \(Clear[x, y, t]\), \(\(PhasePlot[{\(x'\)[t] == x[t], \(y'\)[t] == x[t] - y[t]}, {x[t], y[t]}, \n\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {t, \(-10\), 10}, {x, \(-2\), 2}, {\ y, \(-2\), 2}, \n \t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ VectorField -> True, FieldLogScale \[Rule] 10, \n \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ InitialValues\ -> Flatten[Table[{0, .5\ i, .5 j}, {i, \(-3\), 3}, {j, \(-3\), 3}], 1]]; \)\ \)}], "Input"], Cell[CellGroupData[{ Cell["Eigenvalues and eigenvectors for this system.", "Subsection"], Cell[BoxData[{\(Clear[a]\), RowBox[{ RowBox[{"a", "=", RowBox[{"(", GridBox[{ {"1", "0"}, {"1", \(-1\)} }], ")"}]}], ";"}]}], "Input"], Cell[BoxData[{ \(Clear[\[Lambda], v]\), \(\[Lambda] = Eigenvalues[a]\), \(v = Eigenvectors[a]\)}], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Star (Proper) Node", "Section"], Cell[TextData[{ "This is the case where there is only one real eigenvalue, and the \ eigenspace is two-dimensional. The general solution is of the form ", Cell[BoxData[ \(TraditionalForm\`c\_1\)]], Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\[Lambda]t\)]], Cell[BoxData[ \(TraditionalForm\`\[Xi]\_1\)]], "+ ", Cell[BoxData[ \(TraditionalForm\`c\_2\)]], Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\[Lambda]t\)]], Cell[BoxData[ \(TraditionalForm\`\[Xi]\_2\)]], "." }], "Text"], Cell["\<\ Without trajectories (phase plane). If the eigenvalue is positive, \ the arrows move us away from the origin, and the node is unstable. If \ negative, the arrows move us toward the origin and the node is stable..\ \>", "Text"], Cell["Without trajectories.", "Text"], Cell[BoxData[{ \(Clear[x, y, t]\), \(\(PhasePlot[{\(x'\)[t] == x[t], \(y'\)[t] == y[t]}, {x[t], y[t]}, \n \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {t, \(-10\), 10}, {x, \(-5\), 5}, {\ y, \(-5\), 5}, \n \t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ VectorField -> True, FieldLogScale \[Rule] 10]; \)\ \)}], "Input"], Cell["\<\ With some trajectories. Notice that any straight half-line through \ the origin is a trajectory.\ \>", "Text"], Cell[BoxData[{ \(Clear[x, y, t]\), \(\(PhasePlot[{\(x'\)[t] == x[t], \(y'\)[t] == y[t]}, {x[t], y[t]}, \n \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {t, \(-10\), 10}, {x, \(-5\), 5}, {\ y, \(-5\), 5}, \n \t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ VectorField -> True, FieldLogScale \[Rule] 10, \n \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ InitialValues\ -> Flatten[Table[{0, \ i, j}, {i, \(-2\), 2}, {j, \(-2\), 2}], 1]]; \)\ \)}], "Input"], Cell[CellGroupData[{ Cell["Eigenvalues and eigenvectors for this system.", "Subsection"], Cell[BoxData[{\(Clear[a]\), RowBox[{ RowBox[{"a", "=", RowBox[{"(", GridBox[{ {"1", "0"}, {"0", "1"} }], ")"}]}], ";"}]}], "Input"], Cell[BoxData[{ \(Clear[\[Lambda], v]\), \(\[Lambda] = Eigenvalues[a]\), \(v = Eigenvectors[a]\)}], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Improper Node", "Section"], Cell[TextData[{ "This is the case where there are one real eigenvalue, and the eigenspace \ is one-dimensional. The general solution is of the form ", Cell[BoxData[ \(TraditionalForm\`c\_1\)]], Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\[Lambda]t\)]], Cell[BoxData[ \(TraditionalForm\`\[Xi]\)]], "+ ", Cell[BoxData[ \(TraditionalForm\`c\_2\)]], " ", Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\[Lambda]t\)]], "(", Cell[BoxData[ \(TraditionalForm\`\[Xi]\)]], " t + \[Eta])." }], "Text"], Cell["\<\ Without trajectories (phase plane). If the eigenvalue is positive, \ the arrows move us away from the origin, and the node is unstable. If \ negative, the arrows move us toward the origin and the node is stable.\ \>", "Text"], Cell["Without trajectories.", "Text"], Cell[BoxData[{ \(Clear[x, y, t]\), \(\(PhasePlot[{\(x'\)[t] == \(-x[t]\) + 2 y[t], \(y'\)[t] == \(-y[t]\)}, {x[t], y[t]}, \n \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {t, 0, 10}, {x, \(-2\), 2}, {\ y, \(-2\), 2}, \n \t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ VectorField -> True, FieldLogScale \[Rule] 10]; \)\ \)}], "Input"], Cell[TextData[{ "With some trajectories. Notice that all trajectories become tangent to ", Cell[BoxData[ \(TraditionalForm\`\[Xi]\)]], " as we approach the origin. This is because the dominant term as t\[Rule]\ \[Infinity] or -\[Infinity] is ", Cell[BoxData[ \(TraditionalForm\`c\_2\)]], " ", Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\[Lambda]t\)]], Cell[BoxData[ \(TraditionalForm\`\[Xi]\)]], " t unless ", Cell[BoxData[ \(TraditionalForm\`c\_2\)]], "= 0. If ", Cell[BoxData[ \(TraditionalForm\`c\_2\)]], " = 0, the remaining term is still along \[Xi]." }], "Text"], Cell[BoxData[{ \(Clear[x, y, t]\), \(\(PhasePlot[{\(x'\)[t] == \(-x[t]\) + 2 y[t], \(y'\)[t] == \(-y[t]\)}, {x[t], y[t]}, \n \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {t, 0, 10}, {x, \(-2\), 2}, {\ y, \(-2\), 2}, \n \t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ VectorField -> True, FieldLogScale \[Rule] 10, \n \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ InitialValues\ -> Flatten[Table[{ .5\ i, .5 j}, {i, \(-3\), 3}, {j, \(-3\), 3}], 1]]; \)\ \)}], "Input"], Cell[CellGroupData[{ Cell["Eigenvalues and eigenvectors for this system.", "Subsection"], Cell[BoxData[{\(Clear[a]\), RowBox[{ RowBox[{"a", "=", RowBox[{"(", GridBox[{ {\(-1\), "2"}, {"0", \(-1\)} }], ")"}]}], ";"}]}], "Input"], Cell[BoxData[{ \(Clear[\[Lambda], v]\), \(\[Lambda] = Eigenvalues[a]\), \(v = Eigenvectors[a]\)}], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Focus or Spiral Node", "Section"], Cell["\<\ Here we have complex eigenvalues whose real parts are not \ zero.\ \>", "Text"], Cell["Without trajectories.", "Text"], Cell[BoxData[{ \(Clear[x, y, t]\), \(\(PhasePlot[{\(x'\)[t] == \(-x[t]\) - 4\ y[t], \(y'\)[t] == x[t] - y[t]}, {x[t], y[t]}, \n \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {t, 0, 10}, {x, \(-3\), 3}, {\ y, \(-3\), 3}, \n \t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ VectorField -> True, FieldLogScale \[Rule] 10]; \)\ \)}], "Input"], Cell["\<\ With some trajectories. The page 465 of Boyce and DiPrima \ contains a derivation which shows that the trajectories are spirals.\ \>", "Text"], Cell[BoxData[{ \(Clear[x, y, t]\), \(\(PhasePlot[{\(x'\)[t] == \(-x[t]\) - 4\ y[t], \(y'\)[t] == x[t] - y[t]}, {x[t], y[t]}, \n \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {t, 0, 10}, {x, \(-3\), 3}, {\ y, \(-3\), 3}, \n \t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ VectorField -> True, FieldLogScale \[Rule] 10, \n \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ InitialValues\ -> Flatten[Table[{\ i, j}, {i, \(-2\), 2}, {j, \(-2\), 2}], 1]]; \)\ \)}], "Input"], Cell[CellGroupData[{ Cell["Eigenvalues and eigenvectors for this system.", "Subsection"], Cell[BoxData[{\(Clear[a]\), RowBox[{ RowBox[{"a", "=", RowBox[{"(", GridBox[{ {\(-1\), \(-4\)}, {"1", \(-1\)} }], ")"}]}], ";"}]}], "Input"], Cell[BoxData[{ \(Clear[\[Lambda], v]\), \(\[Lambda] = Eigenvalues[a]\), \(v = Eigenvectors[a]\)}], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Center", "Section"], Cell["\<\ This time we have complex eigenvalues whose real parts are \ zero.\ \>", "Text"], Cell["Without trajectories.", "Text"], Cell[BoxData[{ \(Clear[x, y, t]\), \(\(PhasePlot[{\(x'\)[t] == x[t] + 2\ y[t], \(y'\)[t] == \(-5\)\ x[t] - y[t]}, {x[t], y[t]}, \n \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {t, 0, 10}, {x, \(-5\), 5}, {\ y, \(-5\), 5}, \n \t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ VectorField -> True, FieldLogScale \[Rule] 10]; \)\ \)}], "Input"], Cell["With some trajectories. Notice they are all ellipses.", "Text"], Cell[BoxData[{ \(Clear[x, y, t]\), \(\(PhasePlot[{\(x'\)[t] == x[t] + 2\ y[t], \(y'\)[t] == \(-5\)\ x[t] - y[t]}, {x[t], y[t]}, \n \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {t, 0, 10}, {x, \(-5\), 5}, {\ y, \(-5\), 5}, \n \t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ VectorField -> True, FieldLogScale \[Rule] 10, \n \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ InitialValues\ -> Flatten[Table[{\ i, j}, {i, \(-2\), 2}, {j, \(-2\), 2}], 1]]; \)\ \)}], "Input"], Cell[CellGroupData[{ Cell["Eigenvalues and eigenvectors for this system.", "Subsection"], Cell[BoxData[{\(Clear[a]\), RowBox[{ RowBox[{"a", "=", RowBox[{"(", GridBox[{ {"1", "2"}, {\(-5\), \(-1\)} }], ")"}]}], ";"}]}], "Input"], Cell[BoxData[{ \(Clear[\[Lambda], v]\), \(\[Lambda] = Eigenvalues[a]\), \(v = Eigenvectors[a]\)}], "Input"] }, Closed]] }, Closed]] }, Open ]] }, FrontEndVersion->"4.1 for Microsoft Windows", ScreenRectangle->{{0, 1024}, {0, 722}}, AutoGeneratedPackage->None, WindowToolbars->"EditBar", WindowSize->{520, 509}, WindowMargins->{{9, Automatic}, {Automatic, 9}} ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. *******************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[1727, 52, 135, 3, 186, "Title"], Cell[1865, 57, 252, 5, 60, "SmallText"], Cell[2120, 64, 430, 12, 71, "Text"], Cell[2553, 78, 207, 7, 33, "Text"], Cell[2763, 87, 55, 0, 33, "Text"], Cell[2821, 89, 462, 14, 33, "Text"], Cell[3286, 105, 105, 3, 33, "Text"], Cell[3394, 110, 328, 9, 52, "Text"], Cell[3725, 121, 303, 6, 71, "Text"], Cell[4031, 129, 87, 2, 30, "Input", InitializationCell->True], Cell[4121, 133, 340, 7, 71, "Text"], Cell[4464, 142, 42, 1, 30, "Input"], Cell[CellGroupData[{ Cell[4531, 147, 30, 0, 59, "Section"], Cell[4564, 149, 722, 23, 47, "Text"], Cell[5289, 174, 241, 5, 62, "Text"], Cell[5533, 181, 383, 7, 91, "Input"], Cell[5919, 190, 275, 8, 46, "Text"], Cell[6197, 200, 553, 11, 123, "Input"], Cell[CellGroupData[{ Cell[6775, 215, 66, 0, 46, "Subsection"], Cell[6844, 217, 202, 6, 58, "Input"], Cell[7049, 225, 121, 3, 59, "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[7219, 234, 25, 0, 39, "Section"], Cell[7247, 236, 702, 23, 47, "Text"], Cell[7952, 261, 37, 0, 30, "Text"], Cell[7992, 263, 369, 6, 91, "Input"], Cell[8364, 271, 373, 10, 62, "Text"], Cell[8740, 283, 548, 10, 123, "Input"], Cell[CellGroupData[{ Cell[9313, 297, 67, 0, 46, "Subsection"], Cell[9383, 299, 196, 6, 58, "Input"], Cell[9582, 307, 121, 3, 59, "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[9752, 316, 37, 0, 39, "Section"], Cell[9792, 318, 543, 17, 47, "Text"], Cell[10338, 337, 241, 5, 62, "Text"], Cell[10582, 344, 37, 0, 30, "Text"], Cell[10622, 346, 360, 6, 75, "Input"], Cell[10985, 354, 121, 3, 30, "Text"], Cell[11109, 359, 525, 10, 107, "Input"], Cell[CellGroupData[{ Cell[11659, 373, 67, 0, 46, "Subsection"], Cell[11729, 375, 193, 6, 58, "Input"], Cell[11925, 383, 121, 3, 59, "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[12095, 392, 32, 0, 39, "Section"], Cell[12130, 394, 559, 19, 52, "Text"], Cell[12692, 415, 240, 5, 71, "Text"], Cell[12935, 422, 37, 0, 33, "Text"], Cell[12975, 424, 383, 7, 110, "Input"], Cell[13361, 433, 638, 20, 71, "Text"], Cell[14002, 455, 553, 11, 150, "Input"], Cell[CellGroupData[{ Cell[14580, 470, 67, 0, 47, "Subsection"], Cell[14650, 472, 199, 6, 58, "Input"], Cell[14852, 480, 121, 3, 59, "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[15022, 489, 39, 0, 39, "Section"], Cell[15064, 491, 89, 3, 30, "Text"], Cell[15156, 496, 37, 0, 30, "Text"], Cell[15196, 498, 385, 7, 91, "Input"], Cell[15584, 507, 158, 4, 46, "Text"], Cell[15745, 513, 547, 11, 123, "Input"], Cell[CellGroupData[{ Cell[16317, 528, 67, 0, 46, "Subsection"], Cell[16387, 530, 202, 6, 58, "Input"], Cell[16592, 538, 121, 3, 59, "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[16762, 547, 25, 0, 39, "Section"], Cell[16790, 549, 90, 3, 30, "Text"], Cell[16883, 554, 37, 0, 30, "Text"], Cell[16923, 556, 388, 7, 91, "Input"], Cell[17314, 565, 70, 0, 30, "Text"], Cell[17387, 567, 550, 11, 123, "Input"], Cell[CellGroupData[{ Cell[17962, 582, 67, 0, 46, "Subsection"], Cell[18032, 584, 199, 6, 58, "Input"], Cell[18234, 592, 121, 3, 59, "Input"] }, Closed]] }, Closed]] }, Open ]] } ] *) (******************************************************************* End of Mathematica Notebook file. *******************************************************************)