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FontColor->RGBColor[0, 0, 1]], StyleBox[ "\[Copyright] Bertil Nilsson\nHalmstad University\ne-mail: \ bertiln@itn.hh.se", FontSize->12, FontColor->RGBColor[0, 0, 1]] }], "Title", CellFrame->True, ShowCellBracket->False, Evaluatable->False, CellHorizontalScrolling->False, TextAlignment->Center, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->24, FontWeight->"Bold", FontSlant->"Italic", Background->RGBColor[0.250004, 0.500008, 0.500008]], Cell[TextData[StyleBox["Please hit \[RightTriangle] to enter chapter", FontSlant->"Italic"]], "Text", ShowCellBracket->False, TextAlignment->Right, FontFamily->"Times New Roman", FontColor->RGBColor[1, 0, 0]], Cell[CellGroupData[{ Cell["Introduction", "Section", CellDingbat->None, ShowGroupOpenCloseIcon->True, Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontColor->RGBColor[0, 0, 1]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " is a computer system for 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It does not \ provide a detailed discussion of ", StyleBox["Mathematica", FontSlant->"Italic"], ", but it will give you some examples of the capabilities and expose you \ to the NoteBook structure. In order to use ", StyleBox["Mathematica", FontSlant->"Italic"], " effectively you must sooner or later take a close look at \"The Book\": \ ", StyleBox["Mathematica", FontSlant->"Italic"], ", a System for Doing Mathematics by Computer, by Stephen Wolfram. He is \ also the main designer of ", StyleBox["Mathematica", FontSlant->"Italic"], " and head of Wolfram Research, the makers of ", StyleBox["Mathematica.", FontSlant->"Italic"], " It is is god habit to visit their homepage ", ButtonBox["www.wolfram.com", ButtonData:>{ URL[ "http://www.wolfram.com/"], None}, Active->True, ButtonStyle->"Hyperlink", ButtonNote->"www.wolfram.com"] }], "Text"], Cell["\<\ It is understood that you have basic skills in using your computer, such as \ clicking on icons, selecting with the mouse, using menus and menue bars, drag \ and drop, dialog boxes and so on. The NoteBook outfit and functionality is \ the same on all computers to which it applies and is stored in a format that \ makes it easy to copy between different computer systems.\ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ "On a Macintosh or under MicroSoft Windows you \"launch\" ", StyleBox["Mathematica", FontSlant->"Italic"], " by clicking on the ", StyleBox["Mathematica", FontSlant->"Italic"], " icon or on a previosly saved NoteBook. In UNIX you must start ", StyleBox["Mathematica", FontSlant->"Italic"], " by issuing the command mathematica in a cmdtool window. Hopefully will \ this be worked around in next release." }], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ "The best way to learn both ", StyleBox["Mathematica", FontSlant->"Italic"], " and the NoteBook functionality is to play with the examples presented \ and botanize in the menue bar at the top. Don't forget Help! In some \ sections of this NoteBook it is assumed that you have executed the previous \ commands. So, it is best to work through the examples in the sequence \ presented." }], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell["\<\ In order to send an input cell to the kernel for evaluation, you must select \ the cell with the mouse and then hold the shift key and hit the return key \ or use Action-Evaluate selection on the menue bar. A plain return (enter) \ will simply cause a line feed in the cell.\ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell["Scrolling", "Subsection", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell["\<\ Using the mouse, click the scroll buttons on the right side of the window. \ Your view of this file will scroll in the direction indicated by the arrow \ on the scroll button you choose. Holding the mouse button down makes the \ scrolling continue until you release the button. Notice that this scrolling \ causes the scroll bar to move vertically up and down. The mouse can be used \ for directly drag the scroll bar to any desired position.\ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell["Cell Structure", "Subsection", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell["\<\ NoteBooks are hierarchically structured into cells in a manner similar to a \ table of contents in a book. However, if one of the topics catches your eye, \ you do not thumb to the page corresponding to that topic; preferably, you \ double click on the cell grouping line that corresponds to that heading. This \ will open the group and reveal individual cells that contain the text, \ formulae, and graphics associated with that heading. A closed group of cells \ is indicated by a hook on the bottom of the cell grouping line. An open \ grouping will not have a hook. This section, Introduction, is an open group \ of cells and this cell is a cell within it. The grouping line is on the far \ right, and it does not have a hook on the bottom. To close a group, \ double-click on the grouping line to the right. You should notice how the \ pointer changes shape when in position to click on the cell bracket or when \ you are moving around over the cells. All theese shapes tell you something \ about the background. To open a group, double-click again on the hooked line. \ You can create a new cell by placing the cell insertion point at the bottom \ or top edge of the existing cell and typing. The mouse pointer will change to \ a horizontal I-beam when the mouse is properly positioned. Using the options \ under \"Edit\", \"Cell\" and \"Style\" you can easily re-group, move, copy, \ delete and set different attributes to the cells or the text within them.\ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ " In order to send an input cell to the kernel for evaluation you must \ select the cell with the mouse and then press shift-return \[ShiftKey] \ \[ReturnIndicator] or enter \[EnterKey] or use Action-Evaluate selection on \ the menue bar. A plain return (enter) ", StyleBox["\[ReturnIndicator]", FontFamily->"Courier"], " will simply cause a line feed in the cell. However, this option opens for \ the possibility to arrange text and sending multiple commands to the kernel. \ You will also see that the kernel keeps track of the conversation using In[] \ and Out[] labels in the cells. You can later on refer to the cells using \ Out[nr] or %nr. On the first contact with the kernel it may take a while for \ loading it into memory, so please be patient. The following commands will be \ taken care of at once." }], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " uses three types of forms. InputForm, StandardForm and TraditionalForm. \ Input and output may take any of theese three forms." }], "Text"], Cell[CellGroupData[{ Cell["Limit[(1+a/x)^x, x -> Infinity]", "Input"], Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^a\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Limit[\((1 + a\/x)\)\^x, x \[Rule] \[Infinity]]\)], "Input"], Cell[BoxData[ \(E\^a\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(TraditionalForm \`lim\+\(x \[Rule] \[Infinity]\)\[ThinSpace]\((1 + a\/x)\)\^x\)], "Input"], Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^a\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["Information in ", FontWeight->"Bold"], StyleBox["Mathematica", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[ " is written and collected in cells.\n\nCells can be grouped to form \ chapters and subchapters.\n\nCells, or groups of cells, forms a NoteBook.", FontWeight->"Bold"] }], "Text", Background->RGBColor[1, 1, 0]], Cell[TextData[{ StyleBox[ "Send cell to kernel: \[EnterKey] or \[ShiftKey]\[ReturnIndicator] \n\n\ Newline in cell: ", FontWeight->"Bold"], StyleBox["\[ReturnIndicator]", FontFamily->"Courier", FontWeight->"Bold"] }], "Text", Background->RGBColor[1, 1, 0]], Cell[TextData[StyleBox[ "Kernel evaluates input cells only. So, please arrange all mathematical input \ as input cells, see Format->Style.", FontWeight->"Bold"]], "Text", Background->RGBColor[1, 1, 0]], Cell[TextData[{ "All ", StyleBox["other", FontSize->14, FontColor->RGBColor[0, 1, 0]], StyleBox[" ", FontSize->14], StyleBox["cells", FontSize->18, FontColor->RGBColor[0, 1, 1]], " ", StyleBox["are", FontSize->24, FontColor->RGBColor[1, 0, 1]], " ", StyleBox["used", FontSize->24, FontSlant->"Italic"], " for document generation purposes, see Format." }], "Text", Background->GrayLevel[0.666667]], Cell["\<\ From now on, it is up to you to open the group in which you are \ interested.\ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Delimiters", FontColor->RGBColor[0, 0, 1]]], "Section", CellDingbat->None, ShowGroupOpenCloseIcon->True, Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontColor->RGBColor[0, 0, 1]], Cell[CellGroupData[{ Cell["Parentheses ()", "Subsection", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell["\<\ Parentheses are used for grouping and control of evaluation. Without \ parentheses, multiplication and division have a higher precedence than \ addition and subtraction.\ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell["Square Brackets []", "Subsection", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[TextData[{ "Square brackets are used for specifying arguments to functions. For \ example, the following statement calculates the square root of the sine of ", Cell[BoxData[ \(TraditionalForm\`\(-\(\[Pi]\/2\)\)\)]], "." }], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[BoxData[ \(\@Sin[\(-\(\[Pi]\/2\)\)]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True], Cell["\<\ Some functions take a fixed number of arguments while others can be called \ with zero or more arguments.\ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell["Curly Braces {}", "Subsection", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell["\<\ Curly braces are used for specifying lists, vectors, and matrices. A list or \ vector is several expressions separated by commas and enclosed in curly \ braces. A list may be nested to arbitrary depth and acts similar to the data \ structures found in C++, LISP or APL2. The command shown below sets the \ variable m equal to a 3 by 3 matrix, represented as a vector of vectors.\ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell["m = {{1,2,3},{4,5,6},{7,8,9}}", "Input", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell["You can also use a template from the palette.", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{"m", "=", RowBox[{"(", GridBox[{ {"1", "2", "3"}, {"4", "5", "6"}, {"7", "8", "9"} }], ")"}]}], InputForm]], "Input"]], "Input", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Add row: \[ControlKey]\[ReturnIndicator]\nAdd column: \[ControlKey],\nNext \ placeholder ", "Text"], Cell[BoxData[ \(TraditionalForm\`\[Placeholder]\ \ or\ \ \[FilledSquare]\)], "Text"], StyleBox[ ": Tab\nDelete: Select delete\nMove out of matrix or table: \[ControlKey] \ \[SpaceIndicator] ", "Text"] }], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontWeight->"Bold", Background->RGBColor[1, 1, 0]] }, Closed]], Cell[CellGroupData[{ Cell["Double Square Brackets [[]]", "Subsection", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell["\<\ Double square brackets are used for indexing, i.e., for referencing an \ object or set of objects in a list. Suppose we have a vector v shown below.\ \ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell["v = {b,c,d}", "Input", PageWidth->Infinity, AspectRatioFixed->True], Cell["\<\ The notation v[[i]] returns the element i in the vector or list called v. \ See Part.\ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell["v[[2]]", "Input", PageWidth->Infinity, AspectRatioFixed->True], Cell["\<\ Using the matrix m, we can manipulate rows or elements. Recalling the \ assignment for m from the above section, we can add row 1 to row 2.\ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell["m[[2]] = m[[1]] + m[[2]]", "Input", PageWidth->Infinity, AspectRatioFixed->True], Cell["m", "Input", PageWidth->Infinity, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell["Comments (* *)", "Subsection", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->14, FontWeight->"Bold"], Cell["\<\ Text between (* and *) delimiters is not evaluated and may be inserted \ everywhere between two tokens. It is taken to be a comment. \ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData["6 (* This is a legal comment. *) \[Pi]"], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Other fancy characters...", "Subsection", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[BoxData[ FormBox[ StyleBox[GridBox[{ {"^", RowBox[{ RowBox[{ RowBox[{ StyleBox[ RowBox[{"M", StyleBox["eans", "Text"]}]], StyleBox[" ", "Text"], StyleBox[ \(exponentiation : \ 2^4\ is\ two\ to\ the\ power\ of\ four\), "Text"]}], StyleBox["=", "Text"], StyleBox["16.", "Text"]}], StyleBox[" ", "Text"], StyleBox["\n", "Text"], StyleBox[ \(Written\ as\ \(2\^4\) in\ Mathematica\ \ version\ 3.0\), "Text"]}]}, {"*", \(Multiplication : \ 4*5\ is\ the\ same\ as\ 4\ 5, \ i . e . \ space\ can\ be\ used\ to\ \ indicate\ multiplication . \nHowever, \ spaces\ often\ means\ nothing\ at\ all\ \ when\ used\ between\ \(tokens . \)\)}, {"/", "Division"}, {"%", \(Represents\ the\ result\ of\ the\ last\ calculation, \n%%\ the\ one\ before\ that\ and\ % n\ a\ \ specified\ \(cell . \)\ \)}, {"!", "Factorial"}, {"=", \(Means\ \(assignment : \ putte\)\ = \ 5\ defines\ a\ variable\ named\ putte\ to\ hold\ the\ value\ 5. \)}, {":=", \(Means\ delayed\ assignment . \ That\ is\ assign\ at\ \(reference . \)\)}, {"==", \(Is\ equality\ \(test : \ a == 5\ returns\ True\ if\ a\ is\ 5\ otherwise\ \(False . \)\)\)}, {"!=", \(Not\ \(equal . \)\)}, {\(\( < \n > \n <= \) ... \), \(Can\ not\ be\ \(missunderstood . \)\ \)} }, ColumnAlignments->{Left}, GridFrame->True, RowLines->True, ColumnLines->False], Background->RGBColor[0, 1, 1]], TextForm]], "Text"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Naming Conventions", "Section", CellDingbat->None, ShowGroupOpenCloseIcon->True, Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontColor->RGBColor[0, 0, 1]], Cell[TextData[{ "All build-in functions in ", StyleBox["Mathematica", FontSlant->"Italic"], " are given names according to definite rules. The names are usually \ complete English words, fully spelled out, starting with a capital letter. ", StyleBox["Mathematica", FontSlant->"Italic"], " rarely uses abbreviations, but for a few very common functions, ", StyleBox["Mathematica", FontSlant->"Italic"], " uses the traditional abbreviations: ", StyleBox["Sin, Table, Cos, Abs, Pi, E, I, Integrate", FontSlant->"Italic"], ", etc. Some of them are derived from more than one word: ", StyleBox["NestList, ContourPlot, ListDensityPlot,", FontSlant->"Italic"], " etc. Although this convention does lead to longer function names, it \ avoids any ambiguity or confusion. It is good manner to use lower case \ letters in user defined variables and functions." }], "Text", CellDingbat->None, ShowGroupOpenCloseIcon->True, Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ "When the standard notation for a mathematical function involves both \ subscripts and superscripts, the subscripts are given before the superscripts \ in the ", StyleBox["Mathematica", FontSlant->"Italic"], " form. Thus, for example, the associated Legendre polynomials are denoted \ ", StyleBox["LegendreP[n, m, x ]", FontSlant->"Italic"], "." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["On-line Help", "Section", CellDingbat->None, ShowGroupOpenCloseIcon->True, Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontColor->RGBColor[0, 0, 1]], Cell[TextData[{ "Help on the menue bar is the key to all info you need: Demos, Getting \ started, Introduktion to ", StyleBox["Mathematica... ", FontSlant->"Italic"], "Everything is there, including Wolframs 1400 pages Reference Book!" }], "Text", ShowCellBracket->False, Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[TextData[{ "Besides Help on the menue bar the ? operator can be used to obtain online \ help. In the spirit of UNIX the symbol * used with the ? operator acts as a \ wild card character, i.e., it can match any alphanumeric character or \ sequence of characters. If more than one command matches the request, ", StyleBox["Mathematica", FontSlant->"Italic"], " lists the names of all the commands. Given ?Plot* as input, Mathematica \ lists the commands that begin with the word Plot." }], "Text", ShowCellBracket->False, Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell["?Plot*", "Input", PageWidth->Infinity, ShowCellBracket->False, AspectRatioFixed->True], Cell[TextData[{ "If only one command matches the request, ", StyleBox["Mathematica", FontSlant->"Italic"], " will print the usage statement associated with the command. The usage \ statement typically consists of a template showing how to invoke the command \ and a brief description. Here is the information for Plot." }], "Text", ShowCellBracket->False, Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell["?Plot", "Input", PageWidth->Infinity, ShowCellBracket->False, AspectRatioFixed->True], Cell["\<\ The usage statement tells us that this command plots a function over a \ specified domain. For example, evaluate the command below. \ \>", "Text", ShowCellBracket->False, Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[BoxData[ \(Plot[Sin[x], {x, \(-1\), 5}]\)], "Input", PageWidth->Infinity, ShowCellBracket->False, AspectRatioFixed->True], Cell["\<\ Use ?? to obtain more information. This often provides you with more details \ than you need.\ \>", "Text", ShowCellBracket->False, Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell["??Plot", "Input", PageWidth->Infinity, ShowCellBracket->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell["Numerical Capabilities", "Section", CellDingbat->None, ShowGroupOpenCloseIcon->True, Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontColor->RGBColor[0, 0, 1]], Cell[TextData[{ "You can do arithmetic with ", StyleBox["Mathematica", FontSlant->"Italic"], " just as you would on a calculator. You type the input 5+7, and ", StyleBox["Mathematica", FontSlant->"Italic"], " prints the result 12. It's already there, just try it!" }], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[BoxData[ \(5 + 7\)], "Input", PageWidth->Infinity, AspectRatioFixed->True], Cell[TextData[{ "Unlike a calculator, however, ", StyleBox["Mathematica", FontSlant->"Italic"], " can give you exact results. Here is the exact result for 3 to the power \ 100. " }], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[BoxData[ \(3\^100\)], "Input", PageWidth->Infinity, AspectRatioFixed->True], Cell["\<\ Commands followed by a semicolon are performed, but the output is not \ printed. The semicolon also serves as a delimiter for regular expressions. In \ this case the output of the first two expressions is suppressed. A tailing ; \ will suppress the last output to.\ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[ \(3\^100; 2\ \[Pi]; 5 + 7\)], "Input", PageWidth->Infinity, AspectRatioFixed->True, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold"], Cell[TextData[{ "You can use the ", StyleBox["Mathematica", FontSlant->"Italic"], " function N to get approximate numerical results. The % stands for the \ last resul", StyleBox["t", FontColor->RGBColor[1, 1, 0]], ". The answer is given in scientific notation." }], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell["N[%]", "Input", PageWidth->Infinity, AspectRatioFixed->True], Cell["\<\ You can find numerical results to any degree of precision. This calculates \ the square root of 10 to 40 decimal places.\ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[BoxData[ \(N[\@10, 40]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " can also handle complex numbers. Here is ", Cell[BoxData[ \(TraditionalForm\`\((3\ + \ i4)\)\^10\)]], ". In ", StyleBox["Mathematica", FontSlant->"Italic"], ", I or \[ImaginaryI] stands for the imaginary unit, i.e. ", Cell[BoxData[ \(TraditionalForm\`\[ImaginaryI]\^2 == \(-1\)\)]], "." }], "Text", Evaluatable->False, CellHorizontalScrolling->False, PageBreakAbove->True, AspectRatioFixed->True], Cell[BoxData[ \(\((3 + 4 \[ImaginaryI])\)\^10\)], "Input", PageWidth->Infinity, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " can evaluate all standard mathematical functions. Here is the value of \ the Bessel function Jo(10.5)." }], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[BoxData[ \(BesselJ[0, 10.5]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True], Cell["\<\ Many build in functions has a N as prefix, such as NIntegrate and NSolve. \ This implies numeric version of the symbolic one. Here is the numerical \ value of the integral of sin(sin(x)) from 0 to pi.\ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[BoxData[ \(NIntegrate[Sin[Sin[x]], {x, 0, \[Pi]}]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True], Cell["\<\ For solving polynomial equations, or system of, you have Solve or NSolve. \ \ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False], Cell[BoxData[ \(Solve[x\^3 + 2\ x\^2 - x + 5 == 0, x]\)], "Input", PageWidth->Infinity], Cell["\<\ The solution is stated in terms of a list of replacement rules. With the \ operator /., one can apply the rule or list of rules to an expression. This \ is a simple application of the very powerful replacement techniques using \ pattern match, see ?Replace*. \ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False], Cell[TextData[ "You may read the sequence exp/.old\[ShortRightArrow]new as \"replace old \ with new in exp\". "], "Text", Evaluatable->False, CellHorizontalScrolling->False, FontWeight->"Bold", Background->RGBColor[1, 1, 0]], Cell["\<\ This is the standard \"Mathematica way\" to insert numerical values in \ symbolic expressions in order to keep the symbolic and numeric spaces apart. \ Adopting this behavior gives you a clean environment and minimizes the number \ of nasty surprises which will often show up if you \"polute\" the work space \ with many globally defined numeric variables. \ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False], Cell[BoxData[ \(x + 5\ y /. {x \[Rule] 1, y \[Rule] 2\/7}\)], "Input", PageWidth->Infinity], Cell["For general nonlinear equations there is FindRoot.", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[ \(FindRoot[Cos[x] == x, {x, 0.5}]\)], "Input", PageWidth->Infinity], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " can do many kinds of exact computations with integers. FactorInteger \ gives the factors of an integer." }], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[BoxData[ \(FactorInteger[20654065386]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True], Cell[TextData[{ "Remember that all built-in ", StyleBox["Mathematica", FontSlant->"Italic"], " functions start with a capital letter." }], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell["Algebraic and Symbolic Capabilities ", "Section", CellDingbat->None, ShowGroupOpenCloseIcon->True, Evaluatable->False, CellHorizontalScrolling->False, PageBreakAbove->True, AspectRatioFixed->True, FontFamily->"Times New Roman", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " can work not only with numbers, but also with algebraic formulas. " }], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[ \(\((x + y)\)\^2 + 9\ \((2 + x)\)\ \((x + y)\)\)], "Input", PageWidth->Infinity, AspectRatioFixed->True, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold"], Cell["This does the algebra to expand products and powers.", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[ \(Expand[%]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold"], Cell["Here is the third power of the expression.", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[ \(%\^3\)], "Input", PageWidth->Infinity, AspectRatioFixed->True, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold"], Cell[" Expanding this gives a somewhat more complicated result.", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[ \(Expand[%]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold"], Cell["\<\ Factoring the previous expression puts it in a much simpler form.\ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[ \(Factor[%]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " can do many kinds of algebraic computations. This gives an exact \ formula for the second derivative and integral of ", Cell[BoxData[ \(TraditionalForm\`x\/\(1 - x\^3\)\)]], "." }], "Text", Evaluatable->False, CellHorizontalScrolling->False, PageBreakAbove->True, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[ \(\[PartialD]\_{x, 2}\(x\/\(1 - x\^3\)\)\)], "Input", PageWidth->Infinity, AspectRatioFixed->True, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold"], Cell[BoxData[ \(\[Integral]\(x\/\(1 - x\^3\)\) \[DifferentialD]x\)], "Input", PageWidth->Infinity, AspectRatioFixed->True, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold"], Cell["Multivariate calculus causes no problem...", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[ \(\[Integral]\(\[Integral]\(x\^2\ Sin[y]\) \[DifferentialD]y \[DifferentialD]x\)\)], "Input", PageWidth->Infinity], Cell[BoxData[ \(\[Integral]\_0 \%\(\[Pi]\/3\)\(\[Integral]\_0\%x\( x\^2\ Sin[y]\) \[DifferentialD]y \[DifferentialD]x\)\)], "Input", PageWidth->Infinity], Cell[TextData[{ "Note Log[] is the natural logaritm ln. ", StyleBox["Mathematica", FontSlant->"Italic"], " is of course familiar with integrating by parts, product rule and chain \ rule." }], "Text", Evaluatable->False, CellHorizontalScrolling->False, PageBreakAbove->True, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[ \(\[Integral]\(Log[x]\^2\) \[DifferentialD]x\)], "Input", PageWidth->Infinity], Cell[BoxData[ \(\[PartialD]\_x\((u[x]\ v[x]\ w[x])\)\)], "Input", PageWidth->Infinity], Cell[BoxData[ \(\[PartialD]\_x u[v[x]]\)], "Input", PageWidth->Infinity], Cell[TextData[{ "Find limiting value of an expression when ", StyleBox["x", FontSlant->"Italic"], " approaches a certain value." }], "Text", Evaluatable->False, CellHorizontalScrolling->False, PageBreakAbove->True, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[{ \(Limit[Sin[x]\/x, x \[Rule] 0]\), \(Limit[Tan[x], x \[Rule] \[Pi]\/2, Direction \[Rule] 1]\), \(Limit[\(1 - Cos[x\^2]\)\/x\^4, x \[Rule] 0]\)}], "Input", PageWidth->Infinity], Cell["\<\ We can ask for the exact roots of a polynomial equation, (or system of!).\ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, PageBreakAbove->True, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[ \(Solve[x\^3 + 3\ x\^2 + 5\ x == 7, x]\)], "Input", PageWidth->Infinity], Cell["Differential equations.", "Text", Evaluatable->False, CellHorizontalScrolling->False, PageBreakAbove->True, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[ RowBox[{"DSolve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["y", "\[DoublePrime]", MultilineFunction->None], "[", "x", "]"}], "+", RowBox[{"2", " ", RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}], "-", \(5\ y[x]\)}], "==", \(\[ExponentialE]\^x\ Sin[x]\)}], "}"}], ",", \(y[x]\), ",", "x"}], "]"}]], "Input", PageWidth->Infinity], Cell["\<\ This ought to be real so try the useful simplification. Sometimes you need to \ use FullSimplify\ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[ \(Simplify[%]\)], "Input", PageWidth->Infinity], Cell["\<\ ...or system of ODE. You can also add initial or boundary conditions if you \ wish.\ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, PageBreakAbove->True, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[ RowBox[{"DSolve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["u", "\[Prime]", MultilineFunction->None], "[", "x", "]"}], "==", \(v[x]\)}], ",", RowBox[{ RowBox[{ SuperscriptBox["v", "\[Prime]", MultilineFunction->None], "[", "x", "]"}], "==", \(u[x]\)}], ",", \(u[0] == 1\), ",", \(v[1] == 2\)}], "}"}], ",", "\n", "\t ", \({u[x], v[x]}\), ",", "x"}], "]"}]], "Input", PageWidth->Infinity, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold"], Cell["Partial fractions.", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[ \(x\/\(\((x + 2)\)\^3\ \((x - 5)\)\)\)], "Input", PageWidth->Infinity], Cell[BoxData[ \(Apart[%]\)], "Input", PageWidth->Infinity], Cell[BoxData[ \(Together[%]\)], "Input", PageWidth->Infinity], Cell[TextData[{ "You can also find approximate formulae. This computes the power series \ expansion (Taylor) of ", Cell[BoxData[ \(TraditionalForm\`\(\[ExponentialE]\^\(-x\)\) sinx\)]], " about the point ", Cell[BoxData[ \(TraditionalForm\`x = 0\)]], " up to order ", Cell[BoxData[ \(TraditionalForm\`x\^6\)]], "." }], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[ \(Series[\[ExponentialE]\^\(-x\)\ Sin[2\ x], {x, 0, 6}]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold"] }, Closed]], Cell[CellGroupData[{ Cell["Making Definitions", "Section", CellDingbat->None, ShowGroupOpenCloseIcon->True, Evaluatable->False, CellHorizontalScrolling->False, PageBreakAbove->True, AspectRatioFixed->True, FontFamily->"Times New Roman", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], Cell["\<\ This defines a value for the variable v. The reverse operation is v=., \ Clear[v] or Remove[v].\ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[ \(v = 1 + x\)], "Input", PageWidth->Infinity, AspectRatioFixed->True, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold"], Cell["\<\ Now the value you have defined for v is used whenever v appears.\ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[ \(5 + 2\ v + 3\ v\^2\)], "Input", PageWidth->Infinity, AspectRatioFixed->True, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold"], Cell["\<\ This defines a function f with a formal parameter x. Note the underscore in \ the head indicating a formal parameter and the delayed assignment, i.e. \ evaluate at application time. \ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[ \(f[x_] := x\^2\)], "Input", PageWidth->Infinity, AspectRatioFixed->True, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold"], Cell["\<\ The occurrences of f in an expression like this are transformed according \ to the rules you have given.\ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[ \(f[3] + f[a + b]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold"], Cell["\<\ You can define functions of multiple variables. This defines the function \ f1, which depends on two variables.\ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[ \(f1[x_, y_] := 2\ x\^2 + x\ y + \@y\)], "Input", PageWidth->Infinity, AspectRatioFixed->True, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " will evaluate as much of the expression as possible. Unknowns are left \ as unknowns. Here are two examples." }], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[ \(f1[a, 4]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold"], Cell[BoxData[ \(f1[3, 4]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold"], Cell["Here is the recursive rule for the factorial function.", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[ \(fac[n_] := n\ fac[n - 1]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold"], Cell["\<\ This gives a rule for the end condition of the factorial function.\ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[ \(fac[1] = 1\)], "Input", PageWidth->Infinity, AspectRatioFixed->True, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold"], Cell["Here are the two rules you have defined for fac.", "Text", Evaluatable->False, CellHorizontalScrolling->False, PageBreakAbove->True, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell["?fac", "Input", PageWidth->Infinity, AspectRatioFixed->True, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " can now apply these rules to find values for factorials." }], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[ \(fac[20]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold"] }, Closed]], Cell[CellGroupData[{ Cell["Two-Dimensional Plots", "Section", CellDingbat->None, ShowGroupOpenCloseIcon->True, Evaluatable->False, CellHorizontalScrolling->False, PageBreakAbove->True, AspectRatioFixed->True, FontFamily->"Times New Roman", FontColor->RGBColor[0, 0, 1]], Cell["\<\ The graphics primitives are versatile. You can plot one or more functions, \ produce contour and density plots and draw arbitrary figures and combine \ them all.\ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[BoxData[ \(p1 = Plot[{Sin[x\^3], Sin[x\^4]}, {x, \(-2\), 2}, \n \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ PlotStyle \[Rule] {Hue[0], Hue[0.7]}]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True], Cell[BoxData[ \(p2 = Plot[Cos[x], {x, 1, 4}, \n\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ PlotStyle \[Rule] Thickness[0.03]]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True], Cell[BoxData[ \(Show[p1, p2]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True], Cell[BoxData[ \(Show[GraphicsArray[{p1, p2, %}]]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True], Cell["A bunch of dots.", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[ \(ListPlot[Table[{x, x + Sin[4\ x]}, {x, 0, \[Pi], \[Pi]\/20}], \n \t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ PlotStyle \[Rule] PointSize[0.02]]\)], "Input", PageWidth->Infinity], Cell["Option PlotJoined connects the points.", "Text", Evaluatable->False, CellHorizontalScrolling->False], Cell[BoxData[ \(ListPlot[{3, 1, \(-2\), 3, 1, 0, 4, \(-1\)}, \n \t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ PlotJoined \[Rule] True]\)], "Input", PageWidth->Infinity], Cell["Parametric plots.", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[ \(ParametricPlot[{4\ Cos[\(-\(\(11\ t\)\/4\)\)] + 7\ Cos[t], \n \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4\ Sin[\(-\(\(11\ t\)\/4\)\)] + 7\ Sin[t]}, {t, 0, 8\ \[Pi]}, \n \t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Axes \[Rule] None, PlotStyle \[Rule] Hue[0.6]]\)], "Input", PageWidth->Infinity], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " chooses appropriate scales for plots, even when there are \ singularities." }], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[ \(Plot[1\/Sin[x], {x, 0, 10}, PlotStyle \[Rule] Hue[0]]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold"], Cell["Contour plot, density plot, polar plots, implit functions...", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[BoxData[ \(ContourPlot[x\ Sin[x\ y], {x, 0, 1}, {y, 0, 2\ \[Pi]}, \n \t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ColorFunction \[Rule] Hue]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell["Three-Dimensional Plots", "Section", CellDingbat->None, ShowGroupOpenCloseIcon->True, Evaluatable->False, CellHorizontalScrolling->False, PageBreakAbove->True, AspectRatioFixed->True, FontFamily->"Times New Roman", FontColor->RGBColor[0, 0, 1]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " can make fancy three-dimensional pictures." }], "Text", Evaluatable->False, CellHorizontalScrolling->False, PageBreakAbove->True, AspectRatioFixed->True], Cell[BoxData[ \(Plot3D[x\^3 + y\^3, {x, \(-2\), 2}, {y, \(-3\), 3}]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold"], Cell[BoxData[ \(Plot3D[Sin[x\ Sin[x\ y]], {x, 0, 4}, {y, 0, 3}]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True], Cell["\<\ The plot does not come out very well with the default settings, so here we \ go again with a finer grid of sample points, and with shading determined by \ simulated illumination.\ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, PageBreakAbove->True, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[ \(Plot3D[Sin[x\ Sin[x\ y]], {x, 0, 4}, {y, 0, 3}, \n \t\ \ \ \ \ \ \ \ \ \ \ \ \ \ PlotPoints \[Rule] 50, Lighting \[Rule] True]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold"], Cell["\<\ This displays the surface as seen from a different view point. You can also \ use the automatic 3D viewpoint selector command in the Prepare Input submenu \ of the Action menu.\ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, PageBreakAbove->True, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[ \(Show[%, ViewPoint \[Rule] {1, 0, 1}]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold"], Cell["\<\ As in 2D you can do parametric plot, cylindrical plot, spherical... You can \ also plot general primitives as cylinders, cones, spheres...\ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, PageBreakAbove->True, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell["Animations", "Section", CellDingbat->None, ShowGroupOpenCloseIcon->True, Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(\(bilder = Table[Plot3D[Sin[\(x\ y\)\/t], {x, 0, \[Pi]\/2}, {y, 0, \[Pi]}, \n \t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ PlotPoints \[Rule] 9, Ticks \[Rule] None], {t, 1, 6, 3\/4}]; \)\)], "Input", PageWidth->Infinity], Cell[BoxData[ \(Show[GraphicsArray[bilder], GraphicsSpacing \[Rule] 0.2]\)], "Input", PageWidth->Infinity], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " allows a sequence of graphics frames to be animated. In the example \ below, the graphics frames have already been generated by commands that are \ not shown." }], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell["\<\ An animation is begun by selecting a group of graphics cell and then holding \ down the Command key and hitting the y key. Before starting the animation, \ make sure you can see all of the first two paragraphs which follow the box \ below. Now, start the animation in the following cell by clicking on the \ line with the hook at the bottom and then typing Command-y. Then, read the \ paragraphs below.\ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell["\<\ Notice the control buttons which appeared in the lower right corner of this \ window after the start of animation. As read from left to right, these \ buttons control the direction of the animation (backwards, loop, or forward), \ pausing, and the speed of the animation (decrease or increase). Click once \ on the pause button to freeze animation. Click on the pause button again \ restart the animation.\ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[TextData[{ "An animation is stopped by typing Command-y again or by clicking the \ mouse button while the mouse pointer is within the ", StyleBox["Mathematica", FontSlant->"Italic"], " window (but not pointing to one of the animation control buttons). If \ you stop the animation with the mouse button, you will most likely have \ un-selected the graphics cells. To reselect these cells, click only ONCE on \ the grouping line with the hook on the bottom that is closest to the desired \ graphics. (If you do click twice on this grouping line, you will open the \ cell and see each \.aaframe\.ba of the animation [feel free to do this, but \ it isn't very interesting].)" }], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell["\<\ An animated sequence may go very slowly when it is first run. This is \ because each frame has to be loaded into memory from the disk. Just be \ patient \[CapitalEth] once all the frames are in memory the speed of the \ animation should increase. \ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell["\<\ Again, you have not been shown the commands that generated the frames for \ this animation. That will come later. All that is important here is that \ you learn how to animate a collection of frames.\ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell["Lists", "Section", CellDingbat->None, ShowGroupOpenCloseIcon->True, Evaluatable->False, CellHorizontalScrolling->False, PageBreakAbove->True, AspectRatioFixed->True, FontFamily->"Times New Roman", FontColor->RGBColor[0, 0, 1]], Cell[TextData[{ " Lists provide a mechanism for representing arrays, both regular and \ irregular, vectors, matrices, and for grouping together objects such as data, \ variables or expressions, separate them by commas and enclose in curly \ braces. A list may be nested to any depth and rank. Lists is a very powerful \ data structure and the most efficient and cleanest way to approach a problem \ is by using lists. \"Thinking lists\" can be traced deep down in the soal of \ ", StyleBox["Mathematica", FontSlant->"Italic"], " and the importence is reflected by the fact that ", StyleBox["Mathematica", FontSlant->"Italic"], " uses them frequently both as input and output from functions." }], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[BoxData[ \({4, 6.7, a, 5\ x + 7, x + Sin[x], 3 + 5\ I}\)], "Input", PageWidth->Infinity], Cell["\<\ Table is a typical function for generating regular arrays with certain \ structure.\ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False], Cell[BoxData[ \(list = Table[\(n!\), {n, 1, 20}]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True], Cell[TextData[{ "This takes the logarithm of each entry in the list, and evaluates the \ result numerically. Functions like Log have the property of being \"listable\ \", so that they apply separately to each element in a list. All funtions \ for which it makes sense have the attribute \"listable\". This opens for \ powerful pure functional programming avoiding much of Do, For and other \ repetitive statements. We should try to express ourself in forms of lists in \ order to use ", StyleBox["Mathematica", FontSlant->"Italic"], " effectively. " }], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[BoxData[ \(N[Log[%]]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True], Cell["Here is a plot of the entries in the list. ", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[BoxData[ \(ListPlot[%]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True], Cell["\<\ The importence of lists is reflected in the number of functions that \ operates directly on them. Functions as Sort, Reverse, Rotate, Drop, Take, \ First, Last, Select, Insert, Append and Join needs no further explanation.\ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " uses lists to represent vectors. Here is the dot product of two \ three-dimensional vectors." }], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[BoxData[ \({x, y, z} . {a, b, c}\)], "Input", PageWidth->Infinity, AspectRatioFixed->True], Cell["\<\ You can also do purely symbolic operations with lists. Permutations gives \ all possible permutations of a list, i.e. all possible orderings of the \ elements in a list.\ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[BoxData[ \(Permutations[{a, b, c}]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True], Cell["Flatten ``un-nests'' lists.", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[BoxData[ \(Flatten[%]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True], Cell["\<\ The list given as the permutations of {a,b,c} is actually a list of lists. \ In this sense, the list is two dimensional, a matrix. \ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[BoxData[ \(list1 = Permutations[{a, b, c}]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True], Cell["\<\ The number of elements in each dimension of list1 can be determined with \ the Dimensions command. The first dimension of list1 has 6 elements, and \ the second dimension has 3.\ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[BoxData[ \(Dimensions[list1]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True], Cell["\<\ As noted in the Delimiters section, element locations within a list are \ specified with double brackets. To get the second element of list1 we write, \ (or Part[list1,2])\ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[BoxData[ \(list1\[LeftDoubleBracket]2\[RightDoubleBracket]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True], Cell["\<\ Notice that the second element is a list with three element in itself. To \ obtain the first element from this sub-list, we can add another subscript. \ \ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[BoxData[ \(list1\[LeftDoubleBracket]2, 1\[RightDoubleBracket]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True], Cell[TextData[{ "This generates a matrix whose (i , j)th element is 1/(i+j+1). ", StyleBox["Mathematica", FontSlant->"Italic"], " represents the matrix as a list of lists." }], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[BoxData[ \(m = Table[1\/\(i + j + 1\), {i, 3}, {j, 3}]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True], Cell["Here is the inverse of the matrix.", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[BoxData[ \(Inverse[m]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True], Cell["\<\ Multiplying the inverse by the original matrix gives an identity matrix.\ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[BoxData[ \(% . m\)], "Input", PageWidth->Infinity, AspectRatioFixed->True], Cell["This gives a new matrix, with a modified diagonal.", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[BoxData[ \(m - x\ IdentityMatrix[3]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True], Cell["\<\ The determinant of the new matrix gives the characteristic polynomial for \ the original matrix.\ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[BoxData[ \(Det[%]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True], Cell["\<\ This finds (numerically) the roots of the characteristic polynomial using \ the Solve function. These roots correspond to the eigenvalues of m.\ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, PageBreakAbove->True, AspectRatioFixed->True], Cell[BoxData[ \(N[Solve[% == 0, x]]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True], Cell["\<\ Using the function Eigensystem, you can find the numerical eigenvalues and \ vectors.\ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[ \(Eigensystem[N[m]]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold"], Cell["System of linear equations.", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[ \(LinearSolve[m, {1, 2, 3}]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold"], Cell["\<\ Lists may be nested in raged form to any depth, shape and rank. The data \ format is therefor as rich and powerful as thoose in LISP and APL. Lists can \ also be treated as sets and operated upon with set functions as Union, \ Complement and Intersection.\ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[{ \({3, 4, 2, 1} \[Union] {4, 3, 5}\), \({3, 4, 2, 1} \[Intersection] {4, 3, 5} \[Intersection] {6, 7, 4, 5, 3}\)}], "Input", PageWidth->Infinity, AspectRatioFixed->True, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold"], Cell[TextData[{ "As mentioned above we should try to apply functions to lists. ", StyleBox["Mathematica", FontSlant->"Italic"], " has a very powerful mechanism for doing this. The first functions you \ need to know about are Map and Apply." }], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[{ \(fkn/@{a, b, c, d}\), \(fkn@@{a, b, c, d}\), \(Plus@@{a, b, c, d}\)}], "Input", PageWidth->Infinity, AspectRatioFixed->True, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold"], Cell["\<\ In order to be familiar with Map, Apply and the other operators it is wise to \ experiment for a while. You may save some time...\ \>", "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[BoxData[ \(Timing[Plus@@Range[10000]]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True], Cell[BoxData[ \(Timing[\[Sum]\+\(i = 1\)\%10000 i]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell["Procedural Programming", "Section", CellDingbat->None, ShowGroupOpenCloseIcon->True, Evaluatable->False, CellHorizontalScrolling->False, PageBreakAbove->True, AspectRatioFixed->True, FontFamily->"Times New Roman", FontColor->RGBColor[0, 0, 1]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " supports the usual structural statements like Do, For, While, \ If-Then-Else and Switch. Se Help and ??. User defined functions are derived \ from the primitive ones and often packed in Block and Module if they \ contains several statements. See Help. Here is a simple ", StyleBox["Mathematica", FontSlant->"Italic"], " function that generates and expands products. Note the _ after the \ formal argument. " }], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[BoxData[ \(exprod[n_] := Expand[\[Product]\+\(i = 1\)\%n\((x + i)\)]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True], Cell[TextData[{ "The result of this function (or program) is the expansion of \ (x+1)(x+2)(x+3)...(x+n), where n is specified by the user. The following \ statement runs the function with ", Cell[BoxData[ \(TraditionalForm\`n = 4\)]], "." }], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[BoxData[ \(exprod[4]\)], "Input", PageWidth->Infinity, AspectRatioFixed->True] }, Closed]] }, FrontEndVersion->"Microsoft Windows 3.0", ScreenRectangle->{{0, 1024}, {0, 712}}, WindowToolbars->"EditBar", CellGrouping->Manual, WindowSize->{695, 611}, WindowMargins->{{3, Automatic}, {Automatic, 5}}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, -1}}, ShowCellLabel->False, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, StyleDefinitions -> Notebook[{ Cell[CellGroupData[{ Cell["Style Definitions", "Subtitle"], Cell["\<\ Modify the definitions below to change the default appearance of all cells in \ a given style. 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