(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' 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). 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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[ 307895, 7212]*) (*NotebookOutlinePosition[ 309073, 7250]*) (* CellTagsIndexPosition[ 308895, 7242]*) (*WindowFrame->Normal*) Notebook[{ Cell["\<\ K+K problem 1.20 gives the trajectory of some object in terms of \ its polar coordinates:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Theta]\ = \ \((1/2)\)\ \[Alpha]\ \ t^2\)], "Input"], Cell[BoxData[ \(\(t\^2\ \[Alpha]\)\/2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(r = \ \((1/\[Pi])\)\ \[Theta]\)], "Input"], Cell[BoxData[ \(\(t\^2\ \[Alpha]\)\/\(2\ \[Pi]\)\)], "Output"] }, Open ]], Cell["In rectilinear coordinates, the position is given by:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(x[t_]\ = \ r\ Cos[\[Theta]]\)], "Input"], Cell[BoxData[ \(\(t\^2\ \[Alpha]\ Cos[\(t\^2\ \[Alpha]\)\/2]\)\/\(2\ \[Pi]\)\)], \ "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(y[t_]\ = \ r*Sin[\[Theta]]\)], "Input"], Cell[BoxData[ \(\(t\^2\ \[Alpha]\ Sin[\(t\^2\ \[Alpha]\)\/2]\)\/\(2\ \[Pi]\)\)], \ "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(x[House]\)], "Input"], Cell[BoxData[ \(\(House\^2\ \[Alpha]\ Cos[\(House\^2\ \[Alpha]\)\/2]\)\/\(2\ \ \[Pi]\)\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " will plot this trajectory for us once we specify a concrete numerical \ value for the parameter alpha. Note the use of the AspectRatio option to \ make x and y distances the same on the page,\nand the use of one of the many \ PlotSyle options to color the curve. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(pplot\ = \ ParametricPlot[{x[t], y[t]} /. {\[Alpha] \[Rule] 1}, {t, 0, 5}, AspectRatio \[Rule] 1, PlotRange \[Rule] {{\(-4\), 4}, {\(-4\), 4}}, PlotStyle \[Rule] RGBColor[1, 0, 1]]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.125 0.5 0.125 [ [0 .4875 -6 -9 ] [0 .4875 6 0 ] [.125 .4875 -6 -9 ] [.125 .4875 6 0 ] [.25 .4875 -6 -9 ] [.25 .4875 6 0 ] [.375 .4875 -6 -9 ] [.375 .4875 6 0 ] [.625 .4875 -3 -9 ] [.625 .4875 3 0 ] [.75 .4875 -3 -9 ] [.75 .4875 3 0 ] [.875 .4875 -3 -9 ] [.875 .4875 3 0 ] [1 .4875 -3 -9 ] [1 .4875 3 0 ] [.4875 0 -12 -4.5 ] [.4875 0 0 4.5 ] [.4875 .125 -12 -4.5 ] [.4875 .125 0 4.5 ] [.4875 .25 -12 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For our \ convenience we first define a function which produces the point we are at as \ a function of time.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(pt[t_]\ = \ {x[t], y[t]} /. {\[Alpha] \[Rule] 1}\)], "Input"], Cell[BoxData[ \({\(t\^2\ Cos[t\^2\/2]\)\/\(2\ \[Pi]\), \(t\^2\ Sin[t\^2\/2]\)\/\(2\ \ \[Pi]\)}\)], "Output"] }, Open ]], Cell["For example, where are we at time t=2.5?", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(pt[2.5]\)], "Input"], Cell[BoxData[ \({\(-0.9945814664450304`\), 0.016504260397179542`}\)], "Output"] }, Open ]], Cell["\<\ Next lets pick a few random times and plot those points using \ ListPlot[]. Note the PlotStyle options used to make red circles of a given \ size.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?ListPlot\)\)], "Input"], Cell[BoxData[ RowBox[{"\<\"ListPlot[{y1, y2, ... }] plots points corresponding to a \ list of values. 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{575, 0}} -> {-4.17239, -4.12931, 0.0287256, \ 0.0287256}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[TextData[{ "OK, but part of the problem was to sketch not only the points, but the \ velocity and accelerations vectors at those points. Surely ", StyleBox["Mathematica", FontSlant->"Italic"], " will plot vectors for us, right? To find the right function, first make \ a stab by asking for all the known functions which have the string \"Vect\" \ in their name:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?*Vect*\)\)], "Input"], Cell[BoxData[ RowBox[{"\<\"VectorQ[expr] gives True if expr is a list or a \ one-dimensional SparseArray object, none of whose elements are themselves \ lists, and gives False otherwise. VectorQ[expr, test] gives True only if test \ yields True when applied to each of the elements in expr.\"\>", " ", ButtonBox[ StyleBox["More\[Ellipsis]", "SR"], ButtonData:>"VectorQ", Active->True, ButtonStyle->"RefGuideLink"]}]], "Print", CellTags->"Info3400525590-7946949"] }, Open ]], Cell["\<\ Oops, nothing too useful. So going to the Help Browser (button in \ the upper right corner) and looking for PlotList in the Master Index, we see \ that there is a function called ListPlotVectorField which might do the trick. \ To get access to the function we need to load a package, which we do as \ follows:\ \>", "Text"], Cell[BoxData[ \(<< Graphics`PlotField`\)], "Input"], Cell["Now when we repeat the question above ...", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?*Vect*\)\)], "Input"], Cell[BoxData[GridBox[{ { StyleBox["System`", FontFamily->"Helvetica", FontSize->12, FontWeight->"Bold"]}, {GridBox[{ { ButtonBox[ StyleBox["VectorQ", "Hyperlink"], ButtonFunction:>( Internal`PutInformation[ #, LongForm -> False]&), ButtonEvaluator->Automatic, ButtonData:>{"Info3400525590-8557759", "System`VectorQ"}, ButtonFrame->"None", ButtonNote->"System`"], "", "", "", "", "", "", ""} }, RowMinHeight->1.05]}, { StyleBox["Graphics`PlotField`", FontFamily->"Helvetica", FontSize->12, FontWeight->"Bold"]}, {GridBox[{ { ButtonBox[ StyleBox["ListPlotVectorField", "Hyperlink"], ButtonFunction:>( Internal`PutInformation[ #, LongForm -> False]&), ButtonEvaluator->Automatic, ButtonData:>{"Info3400525590-8557759", "Graphics`PlotField`ListPlotVectorField"}, ButtonFrame->"None", ButtonNote->"Graphics`PlotField`"], ButtonBox[ StyleBox["PlotVectorField", "Hyperlink"], ButtonFunction:>( Internal`PutInformation[ #, LongForm -> False]&), ButtonEvaluator->Automatic, ButtonData:>{"Info3400525590-8557759", "Graphics`PlotField`PlotVectorField"}, ButtonFrame->"None", ButtonNote->"Graphics`PlotField`"], ""} }, RowMinHeight->1.05]} }, RowSpacings->{2, 3, 2, 3}, ColumnsEqual->True]], "Print", CellMargins->{{20, Inherited}, {Inherited, Inherited}}, ShowCellLabel->False, CellFrameMargins->{{Inherited, Inherited}, {14, 14}}, Background->RGBColor[0.964706, 0.929412, 0.839216], ButtonBoxOptions->{Active->True}, GridBoxOptions->{ColumnAlignments->{Left}}] }, Open ]], Cell[TextData[{ "... ", StyleBox["Mathematica", FontSlant->"Italic"], " knows a couple new functions.\nHow does ListPlotVectorField work?" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?ListPlotVectorField\)\)], "Input"], Cell[BoxData[ RowBox[{"\<\"ListPlotVectorField[{{vec11,vec12,..},...}] accepts a \ rectangular array of two-dimensional vectors (larger than 2x2) and displays \ them, with each vector positioned in the same location graphically as the \ matrix would be (ie, vector 1,1 in the upper left corner). \ ListPlotVectorField[{{pt,vec},{pt,vec},...}] displays a list of vectors, each \ based at the corresponding point.\"\>", " ", ButtonBox[ StyleBox["More\[Ellipsis]", "SR"], ButtonData:>"Graphics`PlotField`", Active->True, ButtonStyle->"AddOnsLink"]}]], "Print", CellTags->"Info3400525591-5567270"] }, Open ]], Cell["\<\ Oh, so we need to make a list of points and the vectors to draw at \ those points. Lets make a function which will produce these pairs of \ {pt,vec}:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(vec[ t_]\ = \ {pt[ t], {\(x'\)[t], \(y'\)[t]} /. \[Alpha] \[Rule] 1}\)], "Input"], Cell[BoxData[ \({{\(t\^2\ Cos[t\^2\/2]\)\/\(2\ \[Pi]\), \(t\^2\ Sin[t\^2\/2]\)\/\(2\ \ \[Pi]\)}, {\(t\ Cos[t\^2\/2]\)\/\[Pi] - \(t\^3\ Sin[t\^2\/2]\)\/\(2\ \[Pi]\), \ \(t\^3\ Cos[t\^2\/2]\)\/\(2\ \[Pi]\) + \(t\ Sin[t\^2\/2]\)\/\[Pi]}}\)], \ "Output"] }, Open ]], Cell["For example, at t=1 here is our position and velocity:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(vec[1.0]\)], "Input"], Cell[BoxData[ \({{0.13967160269610202`, 0.0763029443133532`}, {0.20304026107885081`, 0.29227749132280845`}}\)], "Output"] }, Open ]], Cell["\<\ Next lets draw a few of these points on the screen. 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