(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' 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[ 16546, 605]*) (*NotebookOutlinePosition[ 46610, 1698]*) (* CellTagsIndexPosition[ 46566, 1694]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ StyleBox["Math 421", FontSlant->"Italic"], " ", StyleBox["\[FilledSmallCircle]", FontSlant->"Plain"], " September 2004" }], "Subsubtitle", TextAlignment->Center, TextJustification->0], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " Introduction\[LongDash]Part 3" }], "Subtitle"], Cell[TextData[StyleBox["Your name(s):\n\n", FontWeight->"Bold"]], "ExampleSection"], Cell[CellGroupData[{ Cell["What else can Mathematica do?", "Section"], Cell[TextData[{ "Already you have seen some examples of numerical, symbolic, and graphical \ work with ", StyleBox["Mathematica", FontSlant->"Italic"], ". Here are a few more. Try to predict the output before evaluating each \ input cell" }], "Text"], Cell[CellGroupData[{ Cell["Simplifying expressions", "Subsection"], Cell["\<\ Mathematica has only a limited number of rules automatically applied to \ expressions you evaluate. For example:\ \>", "Text"], Cell[BoxData[ \(Log[2\ Exp[5]]\)], "Input"], Cell[TextData[{ "You can undoubtedly see another, possibly simpler form for this. (Of \ course simplicity is in the eye of the beholder.) Often you can something \ more closely resembling the \"simple\" form you expected by using the ", StyleBox["Simplify", FontFamily->"Courier"], " function:" }], "Text"], Cell[BoxData[ \(Simplify[Log[2\ Exp[5]]]\)], "Input"], Cell["Here's another one:", "Text"], Cell[BoxData[ \(Sqrt[\((x^2\ - \ 9)\)/\((x - 3)\)]\)], "Input"], Cell[TextData[{ "To simplify the output, you could use any of the following methods:\n \ \[FilledSmallCircle] Edit the original input cell so as to apply ", StyleBox["Simplify", FontFamily->"Courier"], ".\n \[FilledSmallCircle] Copy the input cell to a new one and apply ", StyleBox["Simplify", FontFamily->"Courier"], " in the new one.\n \[FilledSmallCircle] In the original or a copied input \ cell, assign the value to a name, for example: ", StyleBox["quantity = Sqrt[(x^2 - 9)/(x-3)]", FontFamily->"Courier"], ". Then evaluate ", StyleBox["Simplify[quantity]", FontFamily->"Courier"], ".\n \[FilledSmallCircle] Apply ", StyleBox["Simplify", FontFamily->"Courier"], " to the ", StyleBox["preceding-result", FontWeight->"Bold", FontSlant->"Italic"], " variable ", StyleBox["%", FontFamily->"Courier"], ", which holds the output of the most recently evaluated input cell:" }], "Text"], Cell[BoxData[ \(Simplify[%]\)], "Input"], Cell[TextData[{ "The preceding-result variable (", StyleBox["%", FontFamily->"Courier"], ") can be used in any expression and not just for ", StyleBox["Simplify", FontFamily->"Courier"], ". For example:" }], "Text"], Cell[BoxData[ \(25\/3\ 9\/100\)], "Input"], Cell[BoxData[ \(%\ + \ 1\/4\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Working with lists", "Subsection"], Cell[TextData[{ "Several objects may be combined into a ", StyleBox["list", FontWeight->"Bold"], ", which is denoted with braces. For example:" }], "Text"], Cell[BoxData[ \({15\/30, \ \[Pi], \ "\", \(3\ + \ 4 I\)\/\(5\ + \ 6 \ I\), \ Expand[\((a + \ b\ I)\)\^3]}\)], "Input"], Cell[BoxData[{ \(\(z\ = \ 3\ + \ 4 I;\)\), "\[IndentingNewLine]", \({Re[z], \ Im[z]}\)}], "Input"], Cell[TextData[{ "A quick way to form a list of the positive integers 1, 2, \[Ellipsis]", Cell[BoxData[ \(TraditionalForm\`n\)]], " for some positive integer ", Cell[BoxData[ \(TraditionalForm\`n\)]], " is to use the ", StyleBox["Range", FontFamily->"Courier"], " function:" }], "Text"], Cell[BoxData[ \(Range[10]\)], "Input"], Cell[TextData[{ "Then to form the list ", Cell[BoxData[ \(TraditionalForm\`{1\^2, \ 2\^2, \ 3\^2, \ \[Ellipsis], \ 10\^2}\)]], " of squares of those numbers, you may use the ", StyleBox["Sum", FontFamily->"Courier"], " function with a \"counter\" ", StyleBox["j", FontFamily->"Courier"], ". " }], "Text"], Cell[BoxData[ \(\(\(Sum[j\^2, \ {j, \ 1, \ 10}]\)\(\ \)\)\)], "Input"], Cell["Here's another, \"functional\", way that involves no counter:", "Text"], Cell[BoxData[ \(Apply[Plus, \ Range[10]\^2]\)], "Input"], Cell[TextData[{ "Individual entries in a list can be extracted by ", StyleBox["indexing", FontWeight->"Bold"], ", as follows:" }], "Text"], Cell[BoxData[{ \(w\ = \ 5\ + \ 6 I; \ theParts\ = \ {Re[w], \ Im[w]}\), "\n", \(theParts[\([2]\)]\)}], "Input"], Cell[TextData[{ "For indexing, you may prefer to type ", StyleBox["\[EscapeKey]", Magnification->1.5], StyleBox["[[", FontFamily->"Courier"], StyleBox["\[EscapeKey]", Magnification->1.5], " and ", StyleBox["\[EscapeKey]", Magnification->1.5], StyleBox["]]", FontFamily->"Courier"], StyleBox["\[EscapeKey]", Magnification->1.5], " to get more distinctive double brackets:" }], "Text"], Cell[BoxData[ \(theParts\[LeftDoubleBracket]2\[RightDoubleBracket]\)], "Input"], Cell[TextData[{ StyleBox["Try it now:", Background->GrayLevel[0.900008]], " Type and evaluate an expression that creates the list {1 + \ \[ImaginaryI], 2 + 2\[ImaginaryI], 3 + 3\[ImaginaryI], \[Ellipsis], 6 + 6\ \[ImaginaryI]} and assigns it to the name ", StyleBox["nums", FontFamily->"Courier"], ". Then use indexing to extract the third entry in ", StyleBox["nums", FontFamily->"Courier"], ". Finally, type another expression that uses indexing to extract both the \ first and third entry in ", StyleBox["nums", FontFamily->"Courier"], " together in the form of a list with two entries." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Numerical values", "Subsection"], Cell[TextData[{ "As you have already seen, ", StyleBox["Mathematica", FontSlant->"Italic"], " can work with exact rational and irrational quantities. When you want to \ convert such an exact quantity, say one involving ", Cell[BoxData[ \(TraditionalForm\`\@2\)]], ", into its numerical equivalent, use the function ", StyleBox["N", FontFamily->"Courier"], "." }], "Text"], Cell[BoxData[ \({\@2, \ N[\@2]}\)], "Input"], Cell[TextData[{ "By default, ", StyleBox["Mathematica", FontSlant->"Italic"], " displays only 6 significant digits, but you can ask for more if you start \ with exact numbers:" }], "Text"], Cell[BoxData[ \(N[\@2, \ 40]\)], "Input"], Cell[TextData[{ "If, though, your input is a floating-point (approximate) number including \ a decimal point, then ", StyleBox["Mathematica", FontSlant->"Italic"], " will only give meaningful results to machine precision:" }], "Text"], Cell[BoxData[ \(N[\@2.0]\)], "Input"], Cell["Predict the result of evaluating the following cell:", "Text"], Cell[BoxData[ \(N[\@2. , \ 12]\)], "Input"], Cell[TextData[{ "Are your surprised at the result? Did you expect to see 20 digits? \ Actually, ", StyleBox["Mathematica", FontSlant->"Italic"], " ", StyleBox["did", FontVariations->{"Underline"->True}], " approximate ", Cell[BoxData[ \(TraditionalForm\`\@2. \)]], " to 20 digits, but it printed it with its default 6 digits precision. To \ see all 20 digits, wrap the expression inside the NumberForm function:" }], "Text"], Cell[BoxData[ \(NumberForm[N[\@2. , \ 20], \ 20]\)], "Input"], Cell[TextData[{ "High precision arithmetic is possible with ", StyleBox["Mathematica. ", FontSlant->"Italic"], StyleBox["Try it yourself now:", Background->GrayLevel[0.900008]], " Create a cell below with an expression that will give 100 digits of ", Cell[BoxData[ \(TraditionalForm\`\[Pi]\)]], ", then evaluate that cell to see the result." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Solving equations", "Subsection"], Cell[TextData[{ StyleBox["\[WarningSign]", FontColor->RGBColor[1, 0, 0], Magnification->2], " Equations are formed with ", StyleBox["two", FontSlant->"Italic"], " equal signs!" }], "Text"], Cell[BoxData[ \(2\ + \ 2\ == \ 4\)], "Input"], Cell[BoxData[ \(2\ + \ 3\ \[Equal] \ 4\)], "Input"], Cell[BoxData[ \(Clear[z]\)], "Input"], Cell[BoxData[ \(z \((z\ + \ z)\)\ == \ 2\ z\^2\)], "Input"], Cell[TextData[{ StyleBox["To solve an equation", FontWeight->"Bold", FontSlant->"Italic"], ", use the ", StyleBox["Solve", FontFamily->"Courier"], " function:" }], "Text"], Cell[BoxData[ \(Solve[3 z\ - \ 7\ == \ 8\/5\ + \ 2 I, \ z]\)], "Input"], Cell[TextData[{ "Surprised at the output? It does not directly give you the solution, but \ rather a ", StyleBox["replacement rule", FontWeight->"Bold", FontSlant->"Italic"], " for the solve variable. (Moreover, the rule is in the form of a \ brace-enclosed list; more about that later.)" }], "Text"], Cell[TextData[{ StyleBox["To convert a replacement rule into an actual value,", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" use the form ", FontWeight->"Bold"], StyleBox["expr /. rule", FontFamily->"Courier", FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"] }], "Text", CellFrame->True, TextAlignment->Center, TextJustification->0, Background->GrayLevel[0.833326]], Cell[BoxData[ \(z\ /. %\)], "Input"], Cell[TextData[{ "Better yet, anticipate having to do this and assign the replacement rule \ result of the ", StyleBox["Solve", FontFamily->"Courier"], " to a name: " }], "Text"], Cell[BoxData[ \(mysoln\ = \ Solve[3 z\ - \ 7\ == \ 8\/5\ + \ 2 I, \ z]\)], "Input"], Cell[BoxData[ \(z\ /. \ mysoln\)], "Input"], Cell[TextData[{ "There is still one nasty thing about the answer\[LongDash]it's a list, \ albeit with a single entry. And that's because the result of ", StyleBox["Solve", FontFamily->"Courier"], " was a one-entry list, and that entry is itself a list. To get rid of the \ outer list braces, you can index with ", StyleBox["[[1]]", FontFamily->"Courier"], " or use in this case the equivalent function ", StyleBox["First", FontFamily->"Courier"], ":" }], "Text"], Cell[BoxData[{ \(mysoln\ = \ First[Solve[3 z\ - \ 7\ == \ 8\/5\ + \ 2 I, \ z]]\), "\n", \(z /. \ mysoln\)}], "Input"], Cell["\<\ Of course equations may have more than one solution\[LongDash]especially when \ the solutions may be non-real complex numbers:\ \>", "Text"], Cell[BoxData[ \(Solve[z\^2\ + \ 1\ \[Equal] \ 0, \ z]\)], "Input"], Cell[TextData[{ "Something like the following is perfectly legal in ", StyleBox["Mathematica", FontSlant->"Italic"], ":" }], "Text"], Cell[BoxData[ \(z\ = \ z\ /. \ Solve[z\^2\ + \ 1\ \[Equal] \ 0, \ z]\)], "Input"], Cell[TextData[{ "But then z now has a fixed value\[LongDash]a list of the two complex \ square-roots of ", Cell[BoxData[ \(TraditionalForm\`\(-1\)\)]], "\[LongDash]and so if you want to use ", StyleBox["z", FontFamily->"Courier"], " again as a variable, you'll have to clear it first. Watch what happens \ if you don't:" }], "Text"], Cell[BoxData[ \(Solve[z\/\(1\ + \ 2 I\)\ \ \[Equal] \ 3, \ z]\)], "Input"], Cell[TextData[{ "So now clear the value assigned to ", StyleBox["z", FontFamily->"Courier"], " and try again:" }], "Text"], Cell[BoxData[{ \(Clear[z]\), "\[IndentingNewLine]", \(Solve[z\/\(1\ + \ 2 I\)\ \ \[Equal] \ 3, \ z]\)}], "Input"], Cell["\<\ To solve several equations simultaneously, put them into a list:\ \>", "Text"], Cell[BoxData[ \(Solve[{2\ x\ - \ 3\ y\ == \ 7, \ 5\ x\ + \ 6\ y\ == \ 4}, \ {x, \ y}]\)], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Creating notebooks", "Section"], Cell[TextData[{ StyleBox["To create a new notebook", FontWeight->"Bold", FontSlant->"Italic"], ", from the ", StyleBox["File", FontFamily->"Helvetica"], " menu you would select ", StyleBox["New", FontFamily->"Helvetica"], ". The new, blank notebook that opens will be untitled; you will give it a \ name when you first save the notebook." }], "Text"], Cell[TextData[{ StyleBox["Try it now:", Background->GrayLevel[0.900008]], " Create a new notebook. In it type some input cell and evaluate that \ cell. At the top of the notebook, create a text cell and type your name(s) \ there. " }], "Text"], Cell[TextData[{ "If you create a new notebook, it will doubtless look plainer than this \ one. This one has had a distinctive ", StyleBox["style sheet", FontWeight->"Bold"], " applied to format the entire notebook in order to make it more attractive \ (I hope) and to more readily distinguish input and output cells from others." }], "Text"], Cell[TextData[{ "You may have several notebooks open at once in a ", StyleBox["Mathematica", FontSlant->"Italic"], " session. To select the one in which you want to work, just click \ anywhere in its window." }], "Text"] }, Closed]], Cell[TextData[{ "Before proceeding, print a copy of this notebook\[LongDash]one copy for \ each of you if you are working with a partner\[LongDash]to keep for \ reference. (Do ", StyleBox["not", FontSlant->"Italic"], " hand it in!)" }], "BoxedText"], Cell[CellGroupData[{ Cell["What's next?", "Section"], Cell[TextData[{ "Congratulations! You have completed this brief introduction to ", StyleBox["Mathematica.", FontSlant->"Italic"], " \nNow you're ready to start using ", StyleBox["Mathematica", FontSlant->"Italic"], " to learn topics in this course.\nGo to the ", ButtonBox["Math 421 web site Files page", ButtonData:>{ URL[ "http://www.math.umass.edu/Courses/Math_421/Files/files.html"], None}, ButtonStyle->"Hyperlink"], " again and open the notebook ", StyleBox["nthRoots.nb", FontFamily->"Courier"], ". (You did click on the underlined, blue hyperlink in the preceding \ sentence, didn't you?) Work through that notebook.\nIf you're out of time, \ \"clean up\" as indicated below." }], "Text"] }, Open ]], Cell["Cleaning up", "Section"], Cell[TextData[{ "When you are finished at your ", StyleBox["Mathematica", FontSlant->"Italic"], " session for today, please:\n \[FilledSmallCircle] Save whatever notebooks \ you want.\n \[FilledSmallCircle] Close all open notebooks.\n \ \[FilledSmallCircle] Exit (quit) ", StyleBox["Mathematica", FontSlant->"Italic"], " from the ", StyleBox["File", FontFamily->"Helvetica"], " menu.\n \[FilledSmallCircle] Log out." }], "BoxedText"], Cell["Copyright \[Copyright] 2004 by Murray Eisenberg. 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