(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of 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[ 12294, 417]*) (*NotebookOutlinePosition[ 25152, 888]*) (* CellTagsIndexPosition[ 25108, 884]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[StyleBox["Packages", FontSlant->"Italic"]], "Title", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["why needed?", "Subsection", FontWeight->"Bold"], Cell[TextData[{ "Several hundred basic commands and mathematical functions are built into \ the ", StyleBox["Mathematica", FontSlant->"Italic"], " kernel (which is written in C, by the way), but this can't possibly cover \ all the thousands of special mathematical functions used in various fields. \ Rather, the more specialized functions (e.g., statistical functions) can be \ written in terms of the built-in ", StyleBox["Mathematica", FontSlant->"Italic"], " functions. ", StyleBox["Mathematica", FontSlant->"Italic"], " comes with about 200 extra files containing such specialized definitions; \ such files are called ", StyleBox["Add-Ons", FontWeight->"Bold"], " or ", StyleBox["Standard Packages", FontWeight->"Bold"], ", and they can easily be loaded and used." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Loading a package", "Section", FontWeight->"Bold"], Cell[TextData[{ "The shortest way to read the definitions from a package is to use \ double-left-arrow (<<) notation, naming the \"context\" of the package, such \ as ", StyleBox["Graphics`PlotField`", FontFamily->"Courier", FontWeight->"Bold"], ". The context name for a standard package always consists of two parts \ (category and topic) both ending with a back-quote." }], "Text"], Cell["<< Graphics`PlotField`", "Input"], Cell[TextData[{ "The reason for the backquotes? It's part of an overall scheme for naming \ things in ", StyleBox["Mathematica", FontSlant->"Italic"], " (including symbols, expressions, input, output, and files) which helps, \ among other things, to avoid conflicts between files with the same name, or \ definitions from different sources which use the same symbol name. If you \ are really interested, read about \"Contexts\"." }], "Text"], Cell[TextData[{ "More traditional-looking commands to load the contents of a package are ", StyleBox["Gets", FontWeight->"Bold"], " and ", StyleBox["Needs", FontWeight->"Bold"], ". With these, put the context name in double-quotes." }], "Text"], Cell["Get[ \"Graphics`PlotField`\" ];", "Input"], Cell[TextData[{ "The command ", StyleBox["Needs", FontWeight->"Bold"], " differs from ", StyleBox["Get, ", FontWeight->"Bold"], "in that ", StyleBox["Needs", FontWeight->"Bold"], " will ", StyleBox["not", FontSlant->"Italic"], " re-read a package which was read earlier." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Packages most likely to be needed", "Section", FontWeight->"Bold"], Cell[TextData[{ "In a differential equations class it is likely that you will want to \ produce vector field graphics -- a bunch of little arrows showing slopes over \ a grid on the ", StyleBox["(t,y", FontSlant->"Italic"], ") plane. It is also common to use some simple statistical functions \ (Mean, Variance) which, surprisingly, are not part of the kernel but must be \ defined by loading a package." }], "Text"], Cell[CellGroupData[{ Cell["Vector fields: Graphics`PlotField`", "Subsection"], Cell[TextData[{ "Here is a simple example of how to use ", StyleBox["PlotVectorField", FontWeight->"Bold"], " (first loading the necessary package)." }], "Text"], Cell["\<\ Get[\"Graphics`PlotField`\"]; PlotVectorField[ {-y/2,x}, {x,-2,2}, {y,-2,2} ];\ \>", "Input", FontWeight->"Bold"], Cell[TextData[{ "The first parameter is the vector ", StyleBox["{", FontSlant->"Italic"], Cell[BoxData[ StyleBox[\(f\_x\), FontWeight->"Bold"]], FontSlant->"Italic"], StyleBox["(x,y), ", FontSlant->"Italic"], Cell[BoxData[ StyleBox[\(f\_y\), FontWeight->"Bold"]], FontSlant->"Italic"], StyleBox["(x,y)}", FontSlant->"Italic"], " representing the arrow at gridpoint ", StyleBox["(x,y)", FontSlant->"Italic"], ". 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The following example represents the \ vector field associated with the differential equation\n\n", StyleBox["\t\t\t\tt y' + 2y = sin(t) / t\n\n", FontWeight->"Bold", FontSlant->"Italic"], "(equivalently, ", StyleBox["y'= ", FontSlant->"Italic"], StyleBox["(sin(t)-2ty)", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" / ", FontSlant->"Italic"], StyleBox["t^2 ", FontWeight->"Bold", FontSlant->"Italic"], ")\t" }], "Text"], Cell["\<\ vec = {t^2, Sin[t] - 2 t y}; PlotVectorField[ vec, {t,0.2, 5}, {y,-1.5, 1.5}, \t\tAxes->True, AxesLabel->{\"t\",\"y\"}, \t\tScaleFunction->(1&), ScaleFactor->0.15, \t\tPlotPoints->20, ColorFunction->(Hue[0.1]&)];\ \>", "Input"], Cell[TextData[{ "The ", StyleBox["ScaleFunction", FontWeight->"Bold"], " and ", StyleBox["ScaleFactor", FontWeight->"Bold"], " options make all the arrows of the same length, in this case, 0.15 (in \ terms of the graph). The denser the grid (here 20\[Times]20) the smaller the \ arrow length should be to avoid overlapping." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Statistics", "Subsection"], Cell[TextData[{ "There are 12 statistics-related packages! The ", StyleBox["DescriptiveStatistics", FontWeight->"Bold"], " package has all the basic functions like ", StyleBox["Mean", FontWeight->"Bold"], ", ", StyleBox["Median", FontWeight->"Bold"], ", ", StyleBox["Mode", FontWeight->"Bold"], ", ", StyleBox["Variance", FontWeight->"Bold"], ", ", StyleBox["GeometricMean", FontWeight->"Bold"], ", etc." }], "Text"], Cell["\<\ Get[\"Statistics`DescriptiveStatistics`\"]; ? Statistics`DescriptiveStatistics`*\ \>", "Input", FontWeight->"Bold"], Cell["\<\ data = {9, 7, 4, 4, 7, 8, 3, 6, 4, 10}; {Mean[data], StandardDeviation[data], Median[data],Mode[data]}\ \>", "Input"], Cell[TextData[{ "Another statistics package may be useful when students read in data in the \ form of a multiple columns of (text) numbers. The ", StyleBox["DataManipulation", FontWeight->"Bold"], " package includes a function ", StyleBox["Column", FontWeight->"Bold"], " which extracts one or more columns of the data. Extracting two columns \ produces a list of number-pairs (", StyleBox["x-y", FontSlant->"Italic"], ") which can be plotted with ", StyleBox["ListPlot", FontWeight->"Bold"], "." }], "Text"], Cell["\<\ data = Table[ N[{i,Sqrt[i], Sin[i/4], (i+9)/4},3], {i,0,9}]; data // TableForm\ \>", "Input", FontWeight->"Bold"], Cell["\<\ Get[\"Statistics`DataManipulation`\"]; col2 = Column[data, 2] ListPlot[ Column[data, {1,3}], (* col#3 vs. col#1 *) \t\tPlotStyle->{Hue[0.8],PointSize[0.03]}];\ \>", "Input", FontWeight->"Bold"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Common pitfall", "Section", FontWeight->"Bold"], Cell[TextData[{ "One of the most comon errors encountered when using a standard package is \ trying to use a function (like ", StyleBox["PlotVectorField", FontWeight->"Bold"], ") before its definition has been read in. Not only does the function not \ work, but the function name is now \"tainted\" in the sense that ", StyleBox["Mathematica", FontSlant->"Italic"], " assumes you have your own definition in mind for that function/variable \ name. Reading in the standard package at this point will NOT fix the \ problem, because the package's definition will only be considered a \ secondary, \"shadow\" definition. To fix this situation there are two \ choices:" }], "Text"], Cell[TextData[{ "\t1. Save the notebook, quit, restart. Do it right the next time. Or,\n\ \n\t2. 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Initialization cells are perfect for any \ definitions which need to be made for the notebook, but which students don't \ really have to understand." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["More information", "Section"], Cell[TextData[{ "For more complete information about what is available in the Standard \ Packages, see the Help Browser, or the printed book \"", StyleBox["Mathematica", FontSlant->"Italic"], " 3.0 Standard Add-on Packages\". 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