(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.1' 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[ 210591, 4471]*) (*NotebookOutlinePosition[ 213313, 4553]*) (* CellTagsIndexPosition[ 213149, 4544]*) (*WindowFrame->Normal*) Notebook[{ Cell["Introduction to Mathematica", "Title", TextAlignment->Center, TextJustification->0], Cell["\<\ 16.21 Spring 2004 February 8, 2005\ \>", "Subtitle", TextAlignment->Center, TextJustification->0], Cell["\<\ Daniel Jamous Academic Computing Information Services & Technology jamous@mit.edu x2-1383\ \>", "Subsubtitle", TextAlignment->Center, TextJustification->0], Cell[TextData[StyleBox["Note: to open the cells below, double-click on the \ \"arrow\" icons. To close the cells, double-click again.", FontSize->12, FontWeight->"Bold"]], "Text"], Cell[CellGroupData[{ Cell["First Steps in Mathematica", "Section"], Cell[TextData[{ "When launching ", StyleBox["Mathematica", FontSlant->"Italic"], ", you'll see on the left a blank \"white\" window called a \"notebook.\" \ On the right, you'll see another window called the \"palette\" window. To do \ simple calculations, simply click on the notebook window and type what you \ want to do." }], "Text"], Cell[CellGroupData[{ Cell["first calculation with Mathematica", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(2 + 2\)], "Input"], Cell[BoxData[ \(4\)], "Output"] }, Open ]], Cell["\<\ To evaluate your expression, press \"Shift+Enter\" and not just \"Enter\". \ Let's see what happens if you just press \"Enter\":\ \>", "Text"], Cell[BoxData[ \(\(\(2 + 2\)\(\[IndentingNewLine]\) \)\)], "Input"], Cell[TextData[{ "Nothing happens. The cursor goes to the next line but the expression is \ not evaluated.\n\nNote: your first calculation will take longer than \ subsequent calculations because the ", StyleBox["Mathematica", FontSlant->"Italic"], " kernel has to start up (", StyleBox["Mathematica", FontSlant->"Italic"], " is made of two parts: a kernel and a front end. The kernel is the part \ that does the calculations. The front end is the part that handles notebooks \ and interaction with the user)." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["other numerical examples", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(3^50\)], "Input"], Cell[BoxData[ \(717897987691852588770249\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(34*765\)], "Input"], Cell[BoxData[ \(26010\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(1/0.786\)], "Input"], Cell[BoxData[ \(1.272264631043257`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Pi\)], "Input"], Cell[BoxData[ \(\[Pi]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Cos[Pi]\)], "Input"], Cell[BoxData[ \(\(-1\)\)], "Output"] }, Open ]], Cell[TextData[{ "Note: ", StyleBox["Mathematica", FontSlant->"Italic"], " built-in functions and symbols start with a capital letter. Function \ arguments are enclosed in brackets [ ]. So if we try to evaluate this:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(cos[Pi]\)], "Input"], Cell[BoxData[ RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \ error: new symbol name \\\"\\!\\(cos\\)\\\" is similar to existing symbol \ \\\"\\!\\(Cos\\)\\\". \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \ ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"General::spell1\\\"]\\)\"\>"}]], "Message"], Cell[BoxData[ \(cos[\[Pi]]\)], "Output"] }, Open ]], Cell["We get an error message.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Log[1]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Exp[1]\)], "Input"], Cell[BoxData[ \(\[ExponentialE]\)], "Output"] }, Open ]], Cell[TextData[{ "This is, in ", StyleBox["Mathematica", FontSlant->"Italic"], ", the symbol for ", StyleBox["e ", FontSlant->"Italic"], "the exponential constant. One can also use the letter E. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(E\)], "Input"], Cell[BoxData[ \(\[ExponentialE]\)], "Output"] }, Open ]], Cell["\<\ To evaluate the numerical value of a constant, one can use the function N.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(N[E]\)], "Input"], Cell[BoxData[ \(2.718281828459045`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(N[Pi]\)], "Input"], Cell[BoxData[ \(3.141592653589793`\)], "Output"] }, Open ]], Cell[TextData[{ "\n", StyleBox["Mathematica", FontSlant->"Italic"], " also handles complex numbers. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Sqrt[\(-1\)]\)], "Input"], Cell[BoxData[ \(\[ImaginaryI]\)], "Output"] }, Open ]], Cell["This is the symbol for the imaginary unit.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Re[Sqrt[\(-1\)]]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Im[Sqrt[\(-1\)]]\)], "Input"], Cell[BoxData[ \(1\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Im[3 - 5 I]\)], "Input"], Cell[BoxData[ \(\(-5\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Im[3 - 2 \[ImaginaryI]]\)], "Input"], Cell[BoxData[ \(\(-2\)\)], "Output"] }, Open ]], Cell[TextData[{ "\nHowever, ", StyleBox["Mathematica", FontSlant->"Italic"], " is much more powerful than just a calculator. Let's try some examples \ from the input palette (called the \"Basic Input\" palette) to the left. " }], "Text"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Using the input palette", "Section"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " can be used to evaluate algebraic expressions. Example:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Integral]\(1\/x\) \[DifferentialD]x\)], "Input"], Cell[BoxData[ \(Log[x]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[Integral]\_a\%b 2 x + 3 \[DifferentialD]x\)], "Input"], Cell[BoxData[ RowBox[{\(Integrate::"nodiffd"\), \(\(:\)\(\ \)\), "\<\"\\!\\(\\(\ \[Integral]\\_a\\%b\\) \\(\\(2x\\)\\)\\) cannot be interpreted. Integrals are \ entered in the form \\!\\(\[Integral]f\[DifferentialD]x\\), where \\!\\(\ \[DifferentialD]\\) is entered as \[EscapeKey]dd\[EscapeKey]. \ \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\ \\\", ButtonFrame->None, ButtonData:>\\\"Integrate::nodiffd\\\"]\\)\"\>"}]], \ "Message"], Cell[BoxData[ StyleBox[ RowBox[{ ErrorBox[\(\[Integral]\_a\%b 2 x\)], "+", \(3 \[DifferentialD]x\)}], ShowStringCharacters->True]], "Message"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[Integral]\_a\%b\((2 x + 3)\) \[DifferentialD]x\)], "Input"], Cell[BoxData[ \(\(-3\)\ a - a\^2 + 3\ b + b\^2\)], "Output"] }, Open ]], Cell["\<\ Note: use the \[TabKey] key to move from one placeholder to the next.\ \>", "Text"], Cell["\<\ Note: as the example above shows, you need to use parentheses. Other examples:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[PartialD]\_x\ x\^2\)], "Input"], Cell[BoxData[ \(2\ x\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[PartialD]\_x\ x - 3 x\^2\)], "Input"], Cell[BoxData[ \(1 - 3\ x\^2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[PartialD]\_x\ \((x - 3 x\^2)\)\)], "Input"], Cell[BoxData[ \(1 - 6\ x\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[Sum]\+\(i = 1\)\%n i\)], "Input"], Cell[BoxData[ \(1\/2\ n\ \((1 + n)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[Product]\+\(i = 1\)\%n i\)], "Input"], Cell[BoxData[ \(\(n!\)\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Use of other palettes", "Section"], Cell[TextData[{ "Now let's look at other things that ", StyleBox["Mathematica", FontSlant->"Italic"], " can do. A good way to do this is to see what \"palettes\" are available \ and try them (in File\[Rule]Palettes). We'll leave the OpenAuthor Tools, \ CreateSlideShow, International Characters, and Notebook Launcher palettes for \ now. Also we already saw the basic input palette (the one appearing by \ default when launching ", StyleBox["Mathematica", FontSlant->"Italic"], ").\n" }], "Text"], Cell[CellGroupData[{ Cell["Algebraic Manipulation", "Subsection"], Cell[TextData[StyleBox["First, let's try the function \"Expand\". To see what \ it does, we can either open the \"Help Browser\" and type in \"Expand\" in \ the Go To box or enter Expand from the palette to the notebook, highlight \ Expand and press the F1 key:", "Text"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[\((a + b)\)\^5]\)], "Input"], Cell[BoxData[ \(a\^5 + 5\ a\^4\ b + 10\ a\^3\ b\^2 + 10\ a\^2\ b\^3 + 5\ a\ b\^4 + b\^5\)], "Output"] }, Open ]], Cell[TextData[StyleBox["You can do all sort of symbolic algebraic \ manipulations with this palette. Try some of the other functions.", "Text"]], \ "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Basic Calculations", "Subsection"], Cell[TextData[{ "This palette covers most of what ", StyleBox["Mathematica", FontSlant->"Italic"], " can do. Look at examples in each of the subcategories. This will allow us \ to introduce more ", StyleBox["Mathematica", FontSlant->"Italic"], " built-in functions" }], "Text"], Cell[CellGroupData[{ Cell["Arithmetic and Numbers", "Subsubsection"], Cell["\<\ We already saw many of the basic operations. Note the function N[ ] that will \ be used extensively in class:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(N[\[ExponentialE]]\)], "Input"], Cell[BoxData[ \(2.718281828459045`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(N[\[Pi], 10]\)], "Input"], Cell[BoxData[ \(3.1415926535897932385`10.000000000000004\)], "Output"] }, Open ]], Cell["\<\ The second argument specifies the number of digits after the decimal \ point.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(N[\[Pi], 50]\)], "Input"], Cell[BoxData[ \(3.14159265358979323846264338327950288419716939937510582097494459`50\)], \ "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Algebra", "Subsubsection"], Cell["Many interesting things here.", "Text"], Cell[CellGroupData[{ Cell[TextData[StyleBox["Solving Equations", FontSlant->"Italic"]], "Text", FontVariations->{"CompatibilityType"->0}], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(Solve[a\ x\^2 + b\ x\ + c \[Equal] 0, \ x]\)\(\n\) \)\)], "Input", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ \({{x \[Rule] \(\(-b\) - \@\(b\^2 - 4\ a\ c\)\)\/\(2\ a\)}, {x \[Rule] \(\ \(-b\) + \@\(b\^2 - 4\ a\ c\)\)\/\(2\ a\)}}\)], "Output"], Cell[BoxData[ \({{x \[Rule] \(\(-b\) - \@\(b\^2 - 4\ a\ c\)\)\/\(2\ a\)}, {x \[Rule] \(\ \(-b\) + \@\(b\^2 - 4\ a\ c\)\)\/\(2\ a\)}}\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[ax\^2 + bx\ + c \[Equal] 0, \ x]\)], "Input", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ \({{}}\)], "Output"] }, Open ]], Cell[TextData[StyleBox["No answer in the second case because there are no \ variables x in the equation (the spaces have been shrunk).\n\nNote: you can \ also use Solve to solve a system of equations:", FontSlant->"Plain", FontVariations->{"CompatibilityType"->0}]], "Text", FontSlant->"Italic"], Cell[CellGroupData[{ Cell[BoxData[ StyleBox[\(Solve[{4\ x + 5\ y \[Equal] 3, \(-2\)\ x\ + \ 6\ y \[Equal] \(-2\)}, {x, y}]\), FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}]], "Input", FontWeight->"Plain", FontSlant->"Italic"], Cell[BoxData[ \({{x \[Rule] 14\/17, y \[Rule] \(-\(1\/17\)\)}}\)], "Output"] }, Open ]], Cell[TextData[StyleBox["Note: expressions wrapped with curly brackets { } are \ called \"lists.\" We'll see other examples below.", FontSlant->"Plain", FontVariations->{"CompatibilityType"->0}]], "Text", FontSlant->"Italic"] }, Open ]], Cell[CellGroupData[{ Cell["Polynomial Manipulation", "Text", FontSlant->"Italic"], Cell["\<\ We already saw these functions in the \"Agebraic Manipulation\" palette. \ Another example:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Together[1/x + 1/\((1 - x)\)]\)], "Input"], Cell[BoxData[ \(\(-\(1\/\(\((\(-1\) + x)\)\ x\)\)\)\)], "Output"] }, Open ]] }, Open ]], Cell["Simplification", "Text", FontSlant->"Italic"], Cell[CellGroupData[{ Cell["Complex Numbers", "Text", FontSlant->"Italic"], Cell[CellGroupData[{ Cell[BoxData[ \(\((1 + 3 I)\)\^2\)], "Input"], Cell[BoxData[ \(\(-8\) + 6\ \[ImaginaryI]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Re[%]\)], "Input"], Cell[BoxData[ \(\(-8\)\)], "Output"] }, Open ]], Cell["Note: % refers to the last output or calculation.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Sqrt[\(-1\)]\)], "Input"], Cell[BoxData[ \(\[ImaginaryI]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Conjugate[\(-8\) + 6\ \[ImaginaryI]]\)], "Input"], Cell[BoxData[ \(\(-8\) - 6\ \[ImaginaryI]\)], "Output"] }, Open ]] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Lists and Matrices", "Subsubsection"], Cell[CellGroupData[{ Cell["Creating Lists and Matrices", "Text", FontSlant->"Italic"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"(", GridBox[{ {"1", "0"}, {"0", "1"} }], ")"}]], "Input"], Cell[BoxData[ \({{1, 0}, {0, 1}}\)], "Output"] }, Open ]], Cell[TextData[{ "A matrix is represented as a \"list\" in ", StyleBox["Mathematica", FontSlant->"Italic"], ". You can nest lists with each other." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"(", GridBox[{ {"1"}, {"0"} }], ")"}]], "Input"], Cell[BoxData[ \({{1}, {0}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"(", GridBox[{ {"1", "0"} }], ")"}]], "Input"], Cell[BoxData[ \({{1, 0}}\)], "Output"] }, Open ]], Cell["See the difference between a row or a column vector.", "Text"], Cell["\<\ You can use the function \"Table\" to build up vectors, matrices, and \ tensors.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[i, 5]\)], "Input"], Cell[BoxData[ RowBox[{\(Table::"itform"\), \(\(:\)\(\ \)\), "\<\"Argument \\!\\(5\\) at \ position \\!\\(2\\) does not have the correct form for an iterator. \ \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\ \\\", ButtonFrame->None, ButtonData:>\\\"General::itform\\\"]\\)\"\>"}]], \ "Message"], Cell[BoxData[ \(Table[i, 5]\)], "Output"] }, Open ]], Cell["Note that the second argument has to be entered as a list.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[i, {5}]\)], "Input"], Cell[BoxData[ \({i, i, i, i, i}\)], "Output"] }, Open ]], Cell["This creates a 1x5 row vector of i.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[i, {i, 5}]\)], "Input"], Cell[BoxData[ \({1, 2, 3, 4, 5}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Table[i, {i, 11, 15}]\)], "Input"], Cell[BoxData[ \({11, 12, 13, 14, 15}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Table[i, {i, 1, 20, 2}]\)], "Input"], Cell[BoxData[ \({1, 3, 5, 7, 9, 11, 13, 15, 17, 19}\)], "Output"] }, Open ]], Cell["\<\ Note the general syntax of a list {i, imin, imax, di} where i goes from imin \ to imax with increments di. By default imin and di are 1. Table is handy to \ create matrices.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[KroneckerDelta[i, j], {i, 2}, {j, 2}]\)], "Input"], Cell[BoxData[ \({{1, 0}, {0, 1}}\)], "Output"] }, Open ]], Cell["This is the 2x2 identity matrix.", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Matrix Operations", "Text", FontSlant->"Italic"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"(", GridBox[{ {"1", "2"}, {"3", "4"} }], ")"}], ".", RowBox[{"(", GridBox[{ {\(-1\), "2"}, {\(-4\), "5"} }], ")"}]}]], "Input"], Cell[BoxData[ \({{\(-9\), 12}, {\(-19\), 26}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"(", GridBox[{ {"1", "2"}, {"3", "4"} }], ")"}], ".", RowBox[{"(", GridBox[{ {\(-1\), "8"}, {"1", "2"}, {"4", "6"} }], ")"}]}]], "Input"], Cell[BoxData[ RowBox[{\(Dot::"dotsh"\), \(\(:\)\(\ \)\), "\<\"Tensors \\!\\({\\(\\({1, \ 2}\\)\\), \\(\\({3, 4}\\)\\)}\\) and \\!\\({\\(\\({\\(\\(-1\\)\\), 8}\\)\\), \ \\(\\({1, 2}\\)\\), \\(\\({4, 6}\\)\\)}\\) 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