(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 70341, 2559]*) (*NotebookOutlinePosition[ 71468, 2596]*) (* CellTagsIndexPosition[ 71424, 2592]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ "Introduction to the ", StyleBox["Mathematica", FontSlant->"Italic"], " Notebook Interface" }], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " is divided into two parts: the front end or interface that\nhandles \ interaction with the user and the kernel that does computations. A\nNotebook \ front end is used on Macintosh, DOS, and NeXT platforms. In this\nsection, \ you'll learn to do the following: start a ", StyleBox["Mathematica", FontSlant->"Italic"], " kernel,\nmanipulate cells, use menu commands, and get front end help. In \ the next\nsection, you'll learn how to perfom basic numeric and algebraic\n\ calculations, manipulate lists, and generate graphics.\nTo begin, double \ click above the half arrow below and to the right." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[{ "Starting ", StyleBox["Mathematica", FontSlant->"Italic"], "" }], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "To start ", StyleBox["Mathematica ", FontSlant->"Italic"], "on the Macintosh, double-click the ", StyleBox["Mathematica ", FontSlant->"Italic"], "icon in\nthe ", StyleBox["Mathematica", FontSlant->"Italic"], " folder. An empty Notebook, ", StyleBox["Untitled-1", FontFamily->"Chicago"], ", will be displayed on\nthe screen. \nThe empty Notebook looks similar to \ other Macintosh documents. A title bar\nextends across the top of the \ Notebook. There are small boxes on either end\nof the bar with the title in \ the middle. Clicking the left box closes the\nNotebook and clicking the right \ box resizes the window. Click the right box\nto make the Notebook larger. \ Horizontal and vertical scroll bars are at the\nright and bottom of the \ Notebook. Also at the bottom are the memory usage\nmeter and the percentage \ display size pop-up menu. Just below the title bar\nis a horizontal line that \ extends from the left side to the scroll bar on\nthe right side. This is the \ cell insertion point.\n\nWhen you start typing a ", StyleBox["cell", FontSlant->"Italic"], " will be created, which is indicated by a cell\nbracket along the right \ edge of the Notebook window. A cell can contain\neither explanatory text, \ kernel input or output, or graphics, but not a\ncombination of these. The \ cell that is created is an input cell by default." }], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Starting a Kernel"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "To start the kernel type a character or characters into the input cell.\n\ Type 2+2"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["2+2"], "Input", AspectRatioFixed->True], Cell[TextData[{ "Then press Enter (or Shift-Return or Command-Return). Since ", StyleBox["Mathematica", FontSlant->"Italic"], " is\na large program, it may take a while for the kernel to be loaded into\ \nmemory. Subsequent evaluations will be returned much more quickly. If the\n\ input is not accepted, a message is returned in a message cell and the\n\ cursor is placed in the input cell where a problem was encountered. If the\n\ input is accepted, output is returned in an output cell and the cell\n\ insertion line is left below the output. You can now type more input at the\n\ cell insertion line or move the pointer back to the first input cell, edit\n\ the input, and reenter that input. " }], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Cells"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Opening and Closing of Cells"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "The output is returned in an output cell that is grouped with the\n\ corresponding input cell, the grouping bracket encompasses the input and\n\ output cell brackets. Move the mouse to place the pointer on the group\n\ bracket and double-click; the group closes and the first cell in the group,\n\ in this case your input cell, is left displayed. Double-click on the group\n\ bracket to open the group."], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Refering to Previous Outputs"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Each input cell will have an associated ", StyleBox["In[n]:= ", FontSlant->"Italic"], "and each output cell will\nhave an associated ", StyleBox["Out[n]= ", FontSlant->"Italic"], ". Note the difference in the equal signs; ", StyleBox[":=", FontFamily->"Courier", FontWeight->"Bold"], " is\nshorthand for ", StyleBox["SetDelayed", FontFamily->"Courier", FontWeight->"Bold"], " and ", StyleBox["=", FontFamily->"Courier", FontWeight->"Bold"], " is shorthand for ", StyleBox["Set", FontFamily->"Courier", FontWeight->"Bold"], ". The difference will be\nexplained later. For now this means that earlier \ output can be referred to\nusing ", StyleBox["Out[n]", FontFamily->"Courier", FontWeight->"Bold"], ", or in shorthand notation ", StyleBox["%n", FontFamily->"Courier", FontWeight->"Bold"], ". The most recent output can be\nreferred to by ", StyleBox["%", FontFamily->"Courier", FontWeight->"Bold"], ".", StyleBox["\n", FontFamily->"Courier", FontWeight->"Bold"], " " }], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Changing Cells Styles"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "The Style/Cell Style menu, or the Ruler, which can be turned on or off in \ the Style menu, may be used to select other cell styles, in addition\nto \ input style and output style. Move the pointer above, between, or below\n\ cells; it becomes a horizontal I-bar. Click the mouse button while the\n\ pointer is a horizontal I-bar to create a cell insertion line. Move the\n\ pointer to the Style menu, click-hold while moving down the pop-up menu.\n\ Move to the first selection, which is ", StyleBox["Cell Style", FontWeight->"Bold"], ", to see the many types of\ncell styles that can be assigned. There is a \ check mark by Input. Continue\ndown to ", StyleBox["Show Ruler", FontWeight->"Bold"], " and release the mouse button. The ruler is displayed\nbelow the title \ bar. Notice the pop-up menu in the lower left of the ruler;\nit should be at \ ", StyleBox["Input", FontFamily->"Chicago"], ". Select ", StyleBox["Text", FontWeight->"Bold"], " from the Cell Style menu or from the\npop-up menu on the ruler and a text \ cell will be created when you start\ntyping. An existing cell's style can be \ changed by highlighting the cell's\nbracket and selecting a new style as \ before." }], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]]}, Open]], Cell[CellGroupData[{Cell[TextData["Menus"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "For more information about menu selections not mentioned here, read either\ \nthe ", StyleBox["User's Guide", FontSlant->"Italic"], " for the Macintosh or the Front End Help, a Notebook\nlocated in the same \ folder as ", StyleBox["Mathematica", FontSlant->"Italic"], ". To open the Front End Help\nNotebook, choose the ", StyleBox["Open", FontWeight->"Bold"], " command from the File menu, then double-click the\nname of the Notebook \ in the dialog box that appears. Other ways to get\nfront end help are \ described in a later section." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["File Menu"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "The selections in the File menu, such as ", StyleBox["Open", FontWeight->"Bold"], ", ", StyleBox["Close", FontWeight->"Bold"], ", ", StyleBox["Save", FontWeight->"Bold"], ", and ", StyleBox["Quit", FontWeight->"Bold"], ", are\nsimilar to their respective selections in other Macintosh \ applications. As\nyou would in other applications, save often to protect your \ work." }], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Edit Menu"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Under ", StyleBox["Nesting", FontWeight->"Bold"], " you will find ", StyleBox["Balance", FontWeight->"Bold"], ", ", StyleBox["Parenthesize", FontWeight->"Bold"], ", ", StyleBox["Bracketize", FontWeight->"Bold"], ", ", StyleBox["List\nBracketize", FontWeight->"Bold"], ", and others commands.\n\n", StyleBox["Balance", FontWeight->"Bold"], " helps you check the structure of your input. The other commands\nsurround \ selected text by the respective bracket type." }], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Graph Menu"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "The selections in the Graph menu allow you to convert graphics to different\n\ graphics formats (PICT, Bitmap PICT, PostScript), align or animate selected\n\ graphics, or set animation parameters or other aspects the selected\n\ graphics."], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Find Menu"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "The selections in the Find menu can be used to highlight specified text or\n\ cells."], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Action Menu"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "The important set of selections for our purpose is under ", StyleBox["Prepare Input", FontWeight->"Bold"], ".\n" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Copy Input", FontWeight->"Bold"], "" }], "Special1", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Copy Input from Above", FontWeight->"Bold"], " and", StyleBox[" Copy Output from Above", FontWeight->"Bold"], " will copy the input or\noutput above, no matter how far above, to the \ insertion point. Note: In the menu, both of these look like Command-L, but \ in fact the second is Command-Shift-L. The macintosh puts only capitals in \ the memu commands. This works also for Command-F, which is ", StyleBox["Find", FontWeight->"Bold"], ", and Command-shift-F, which is ", StyleBox["Find in Function Browser.", FontWeight->"Bold"], "\n" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Complete Selection", FontWeight->"Bold"], "" }], "Special1", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Complete Selection", FontWeight->"Bold"], " will complete a function name you have started typing or\ngive a list of \ all possible completions. Type ", StyleBox["Pl", FontFamily->"Courier", FontWeight->"Bold"], " and press Command-k. A\nlist will appear with all possible completions \ for ", StyleBox["Pl", FontFamily->"Courier", FontWeight->"Bold"], ".\n" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Make Template", FontWeight->"Bold"], "" }], "Special1", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Make Template", FontWeight->"Bold"], " completes a selection then gives a template for the partial\nfunction \ name or function name that you have typed.\n" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["3D ViewPoint Selector", FontWeight->"Bold"], "" }], "Special1", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["3D ViewPoint Selector", FontWeight->"Bold"], " is a dialog box that helps to select a value for the\n", StyleBox["ViewPoint", FontFamily->"Courier", FontWeight->"Bold"], " option. This command will be discussed with graphics." }], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]]}, Open]], Cell[CellGroupData[{Cell[TextData["Getting More Front End Help"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "General information can be obtained by choosing the About ", StyleBox["Mathematica", FontSlant->"Italic"], "\ncommand from the Info menu and clicking the Help button when the Info \ panel\nappears. This provides an overview of the entire system.\n\nMenu \ information can be obtained about any menu command by holding down the\n\ Command key and pressing the Question Mark (or Slash) key. This turns the\n\ cursor into a question mark. Using the question mark cursor, choose the\nmenu \ command you want to know about." }], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData[{ "Introduction to the ", StyleBox["Mathematica", FontSlant->"Italic"], " Kernel" }], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ " In this section, you'll learn how to perfom basic numeric and algebraic\n\ calculations, manipulate lists, and generate graphics."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Numeric Calculations"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Arithmetic Calculations"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "You can do arithmetic calculations with ", StyleBox["Mathematica", FontSlant->"Italic"], " just as you would on a\ncalculator." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["This sums two numbers."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["2 + 3.45"], "Input", AspectRatioFixed->True], Cell[TextData[ "A space denotes multiplication, as does an asterisk or a number enclosed in\n\ parentheses."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["2 2*2(2)2"], "Input", AspectRatioFixed->True], Cell[TextData[{ "A sla", StyleBox["sh (", FontFamily->"Palatino"], StyleBox["/", FontFamily->"Courier", FontWeight->"Bold"], StyleBox[") ", FontFamily->"Palatino"], "denotes division." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["5/3"], "Input", AspectRatioFixed->True], Cell[TextData[{ "A caret (", StyleBox["^", FontFamily->"Courier", FontWeight->"Bold"], ") stands for exponentiation." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["2^5"], "Input", AspectRatioFixed->True], Cell[TextData[{ "Parentheses are reserved for ordering and grouping operations in\n", StyleBox["Mathematica", FontSlant->"Italic"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["2+4(2+9.5)^2"], "Input", AspectRatioFixed->True], Cell[TextData[{ "The left column shows a shorthand notation of arithmetic operations and\n\ equivalent functions.\n\n", StyleBox["x", FontSlant->"Italic"], StyleBox["^", FontFamily->"Courier", FontWeight->"Bold"], StyleBox["y", FontSlant->"Italic"], "\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t", StyleBox["Power[", FontFamily->"Courier", FontWeight->"Bold"], StyleBox["x", FontSlant->"Italic"], ", ", StyleBox["y", FontSlant->"Italic"], StyleBox["]", FontFamily->"Courier", FontWeight->"Bold"], "\n", StyleBox["x", FontSlant->"Italic"], StyleBox["", FontFamily->"Courier", FontWeight->"Bold"], StyleBox["-y", FontSlant->"Italic"], " ", StyleBox["Minus[", FontFamily->"Courier", FontWeight->"Bold"], StyleBox["x", FontSlant->"Italic"], ", ", StyleBox["y", FontSlant->"Italic"], StyleBox["]", FontFamily->"Courier", FontWeight->"Bold"], "\n", StyleBox["x", FontSlant->"Italic"], StyleBox["+", FontFamily->"Courier", FontWeight->"Bold"], StyleBox["y", FontSlant->"Italic"], " ", StyleBox["Plus[", FontFamily->"Courier", FontWeight->"Bold"], StyleBox["x", FontSlant->"Italic"], ", ", StyleBox["y", FontSlant->"Italic"], StyleBox["]", FontFamily->"Courier", FontWeight->"Bold"], "\n", StyleBox["x", FontSlant->"Italic"], StyleBox["/", FontFamily->"Courier", FontWeight->"Bold"], StyleBox["y", FontSlant->"Italic"], " ", StyleBox["Divide[", FontFamily->"Courier", FontWeight->"Bold"], StyleBox["x", FontSlant->"Italic"], ", ", StyleBox["y", FontSlant->"Italic"], StyleBox["]", FontFamily->"Courier", FontWeight->"Bold"], "\n", StyleBox["x y", FontSlant->"Italic"], " or ", StyleBox["x", FontSlant->"Italic"], StyleBox["*", FontFamily->"Courier", FontWeight->"Bold"], StyleBox["y", FontSlant->"Italic"], " ", StyleBox["Times", FontFamily->"Courier", FontWeight->"Bold"], "\:ffff", StyleBox["", FontSlant->"Italic"], StyleBox["", FontFamily->"Courier", FontWeight->"Bold"] }], "Text", CellFrame->True, CellMargins->{{8, Inherited}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Exact and Approximate Numbers"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "With ", StyleBox["Mathematica", FontSlant->"Italic"], " you can get exact results for large calculuations." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["2^1000"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " can give you an approximate result if you use the function ", StyleBox["N", FontFamily->"Courier", FontWeight->"Bold"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["2^1000//N"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " returns exact results if you input exact numbers. Here it\nreturns a \ rational number." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["5/7 + 2/3"], "Input", AspectRatioFixed->True], Cell[TextData[{ "An approximate result can be obtained by applying ", StyleBox["N", FontFamily->"Courier", FontWeight->"Bold"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["5/7 + 2/3 //N"], "Input", AspectRatioFixed->True], Cell[TextData[{ "Approximate results can also be obtained from numbers with decimal points.\ \n", StyleBox["Mathematica", FontSlant->"Italic"], " assumes numbers with decimal points to six decimal places." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["5./7 + 2/3"], "Input", AspectRatioFixed->True], Cell[TextData["1. + 5/7"], "Input", AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Mathematical Functions"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "The following functions are given without explanation. Information about \ a\nfunction can be obtained by preceding a function name with a question mark\ \n(", StyleBox["e.g.", FontSlant->"Italic"], " ", StyleBox["?Random", FontFamily->"Courier", FontWeight->"Bold"], ").\n The names of all built-in ", StyleBox["Mathematica", FontSlant->"Italic"], " functions begins with a capital\nletter.\n The arguments of all ", StyleBox["Mathematica", FontSlant->"Italic"], " functions are enclosed in square\nbrackets. " }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "N[]\nSqrt[x]\nExp[x]\nLog[x]\nLog[b,x]\nSin[x], Cos[x], Tan[x]\nArcSin[x], \ ArcCos[x], ArcTan[x]\nAbs[x]\nRound[x]\nMod[n, m]\nRandom[]\nMax[x, y, ...], \ Min[x, y, ...]\nFactorInteger[n]\nRealDigits[r]\nIntegerDigits[n]"], "Input", CellFrame->True, AspectRatioFixed->True], Cell[TextData["Log[.2345]"], "Input", AspectRatioFixed->True], Cell[TextData["ArcTan[1]"], "Input", AspectRatioFixed->True], Cell[TextData["This gives information about Random."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["?Random"], "Input", AspectRatioFixed->True], Cell[TextData[ "Random[ ] gives a uniformly distributed pseudorandom Real in the\n range 0 \ to 1. Random[type, range] gives a pseudorandom number of\n the specified \ type, lying in the specified range. Possible types\n are: Integer, Real and \ Complex. The default range is 0 to 1. You\n can give the range {min, max} \ explicitly; a range specification\n of max is equivalent to {0, max}."], "Info", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[{ "Here we use the information about ", StyleBox["Random", FontFamily->"Courier", FontWeight->"Bold"], " to obtain a pseudorandom integer\nin the range {1,6}, or to simulate the \ roll of a cubical die." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Random[Integer,{1,6}]"], "Input", AspectRatioFixed->True], Cell[TextData["This returns an exact result."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Sqrt[3]"], "Input", AspectRatioFixed->True], Cell[TextData["This returns an approximate result."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["N[Sqrt[3]]"], "Input", AspectRatioFixed->True], Cell[TextData[{ "The following constants are built into ", StyleBox["Mathematica", FontSlant->"Italic"], ", so the names start with\ncapital letters." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Pi\nE\nDegree\nI\nInfinity"], "Input", CellFrame->True, AspectRatioFixed->True], Cell[TextData["Here are some examples:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Sin[Pi]"], "Input", AspectRatioFixed->True], Cell[TextData["Log[E]"], "Input", AspectRatioFixed->True], Cell[TextData["E^(I Pi)"], "Input", AspectRatioFixed->True], Cell[TextData["E^-Infinity"], "Input", AspectRatioFixed->True], Cell[TextData["Log[2,256]"], "Input", AspectRatioFixed->True], Cell[TextData["Sqrt[-4]"], "Input", AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Arbitrary-Precision Calculations"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "You can specify the number significant figures to keep in a particular\n\ calculation. This feature allows you to do arbitrary-precision\n\ calculations."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "This input returns pi to a fixed number of significant digits."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["N[Pi]"], "Input", AspectRatioFixed->True], Cell[TextData["This input gives pi to 40 digits of precision."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["N[Pi,40]"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " can do many kinds of exact computations with integers. ", StyleBox["FactorInteger", FontFamily->"Courier", FontSize->10, FontWeight->"Bold"], StyleBox[" ", FontWeight->"Bold"], "gives the factors of an integer." }], "Text", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData["FactorInteger[70612139395722186]"], "Input", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, AspectRatioFixed->True]}, Open]]}, Open]], Cell[CellGroupData[{Cell[TextData["Algebraic Calculations"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "When you evaluate an algebraic expression, ", StyleBox["Mathematica", FontSlant->"Italic"], " will rearrange and\ndisplay the expression in an approximation to \ standard mathematical notation. Notice that the exponents go\nfrom small to \ large." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["5x^3 - 3x^2 + x"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " will carry out algebraic simplifications." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["x + 2x^3 - 4x + 5x^3"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Expand", FontFamily->"Courier", FontWeight->"Bold"], " multiplies out products and powers." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Expand[(x+y)^3]"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Factor", FontFamily->"Courier", FontWeight->"Bold"], " carries out the inverse of ", StyleBox["Expand", FontFamily->"Courier", FontWeight->"Bold"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Factor[%]"], "Input", AspectRatioFixed->True], Cell[TextData["Factor[x^105 -1]"], "Input", AspectRatioFixed->True], Cell[TextData["Expand[(1+2x+5y)^7]"], "Input", AspectRatioFixed->True], Cell[TextData[{ "With the option setting ", StyleBox["Trig -> True", FontFamily->"Courier", FontSize->10, FontWeight->"Bold"], " ", StyleBox["Mathematica ", FontSlant->"Italic"], " will do algebraic operations using trigonometric identities." }], "Text", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Expand[ Cos[x]^3 Sin[x]^2, Trig -> True]\n"], "Input", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData[ "Mathematica also does linear algebra on symbolic matrices."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Inverse[{{a, b}, {c, d}}]"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " automatically applies rules for transforming functions." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Sqrt[1+x]^2"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " has no rules for transforming this expression." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Sqrt[1+Cos[x]]"], "Input", AspectRatioFixed->True], Cell[TextData[{ "A general principle of ", StyleBox["Mathematica", FontSlant->"Italic"], " is to take any expression and apply\ntransformation rules until the \ result no longer changes." }], "Text", CellFrame->True, Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Applying Transformation Rules"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "When transforming an expression like ", StyleBox["x", FontSlant->"Italic"], " + ", StyleBox["x", FontSlant->"Italic"], " to 2", StyleBox["x", FontSlant->"Italic"], " , ", StyleBox["Mathematica", FontSlant->"Italic"], " treats ", StyleBox["x", FontSlant->"Italic"], " in\na purely symbolic manner. One way to replace", StyleBox[" x", FontSlant->"Italic"], " with a definite value is to\napply a transformation rule." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "This is how you apply a transformation rule. You can think of this as \"in\ \nthe expression 2", StyleBox["x", FontSlant->"Italic"], " replace all ", StyleBox["x", FontSlant->"Italic"], " with 3.\"" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["2 x /. x -> 3"], "Input", AspectRatioFixed->True], Cell[TextData[{ "You can replace ", StyleBox["x", FontSlant->"Italic"], " with an expression." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["1 + 2x + 4x^2 /. x -> 2 - y"], "Input", AspectRatioFixed->True], Cell[TextData["You can apply lists of rules."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["(x + y)(x - y) /. {x -> 3, y -> 2}"], "Input", AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Transforming Algebraic Expressions"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "There are a number of ways to transform algebraic expressions. We have\n\ looked briefly at ", StyleBox["Expand", FontFamily->"Courier", FontWeight->"Bold"], " and ", StyleBox["Factor", FontFamily->"Courier", FontWeight->"Bold"], ". This section introduces ", StyleBox["Simplify", FontFamily->"Courier", FontWeight->"Bold"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Because there are many simplified versions of an algebraic\nexpression,", StyleBox["Mathematica", FontSlant->"Italic"], " looks for the shortest equivalent form of the\nexpression." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "In some cases ", StyleBox["Expand", FontFamily->"Courier", FontWeight->"Bold"], " can give a shorter result than ", StyleBox["Factor", FontFamily->"Courier", FontWeight->"Bold"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Here is an earlier example."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Expand[(x+y)^3]"], "Input", AspectRatioFixed->True], Cell[TextData["Factor[%]"], "Input", AspectRatioFixed->True], Cell[TextData[ "This is what we expect, however, if we choose a different expression."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Factor[(x^6-1)]"], "Input", AspectRatioFixed->True], Cell[TextData["Expand[%]"], "Input", AspectRatioFixed->True], Cell[TextData[{ "Now if we use ", StyleBox["Simplify", FontFamily->"Courier", FontWeight->"Bold"], " on both expressions we will get the shorter results." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Simplify[(x^6-1)]"], "Input", AspectRatioFixed->True], Cell[TextData["Simplify[(x+y)^6]"], "Input", AspectRatioFixed->True], Cell[TextData[{ "Below are some functions which transform algebraic expressions. Use ", StyleBox["?", FontFamily->"Courier", FontWeight->"Bold"], StyleBox["name", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Italic"], "\nto get information on these functions and apply them to either the\n\ following expressions or to your own expressions.\n" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Expand[]\nExpandAll[]\nTogether[]\nApart[]\nFactor[]\nCancel[]\nSimplify[]\n\ Collect[]\nFactorTerms\nPowerExpand"], "Input", CellFrame->True, AspectRatioFixed->True], Cell[TextData[{ "\n", StyleBox["e", FontSlant->"Italic"], " = (", StyleBox["x", FontSlant->"Italic"], "-1)^2 (2+", StyleBox["x", FontSlant->"Italic"], ") / ((1+", StyleBox["x", FontSlant->"Italic"], ") (", StyleBox["x", FontSlant->"Italic"], "-3)^2)\n\n", StyleBox["v", FontSlant->"Italic"], " = Expand[(4 + 3", StyleBox["x", FontSlant->"Italic"], ")^2 (", StyleBox["x", FontSlant->"Italic"], " - 2", StyleBox["y", FontSlant->"Italic"], ")^2]" }], "Input", AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Getting Parts of Algebraic Expressions"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Here is an algebraic expression."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["e = Expand[(1 + 3 x + 4 y^2)^3]"], "Input", AspectRatioFixed->True], Cell[TextData[{ "This gives the coefficient of ", StyleBox["x", FontSlant->"Italic"], " in ", StyleBox["e", FontSlant->"Italic"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Coefficient[e,x]"], "Input", AspectRatioFixed->True], Cell[TextData[{ "This gives the highest power of ", StyleBox["y", FontSlant->"Italic"], " in ", StyleBox["e", FontSlant->"Italic"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Exponent[e,y]"], "Input", AspectRatioFixed->True], Cell[TextData[{ "This gives the 5th term in ", StyleBox["e", FontSlant->"Italic"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Part[e,5]"], "Input", AspectRatioFixed->True], Cell[TextData[{ "You can use ", StyleBox["Denominator", FontFamily->"Courier", FontWeight->"Bold"], " and ", StyleBox["Numerator", FontFamily->"Courier", FontWeight->"Bold"], " as well to get parts of expressions." }], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]]}, Open]], Cell[CellGroupData[{Cell[TextData["General Information"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Using Previous Results"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Here is a result."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["17+4"], "Input", AspectRatioFixed->True], Cell[TextData[{ "The precent sign (", StyleBox["%", FontFamily->"Courier", FontWeight->"Bold"], ") gives the previous result. This input multiplies the\nprevious result by \ 3." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["% 3"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Out[", FontFamily->"Courier", FontWeight->"Bold"], StyleBox["n", FontSlant->"Italic"], StyleBox["]", FontFamily->"Courier", FontWeight->"Bold"], " or ", StyleBox["%", FontFamily->"Courier", FontWeight->"Bold"], StyleBox["n", FontSlant->"Italic"], " gives output ", StyleBox["n", FontSlant->"Italic"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Assigning values to variables"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "This sets ", StyleBox["x", FontSlant->"Italic"], " to be 5" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["x = 5"], "Input", AspectRatioFixed->True], Cell[TextData[{ "The ", StyleBox["x", FontSlant->"Italic"], " may now be used in calculations and will have the value 5." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["x/7"], "Input", AspectRatioFixed->True], Cell[TextData[{ "This redefines the value for ", StyleBox["x", FontSlant->"Italic"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["x = 2 + 9"], "Input", AspectRatioFixed->True], Cell[TextData[{ "Asking for", StyleBox[" x", FontSlant->"Italic"], " gives the current value of ", StyleBox["x", FontSlant->"Italic"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["x"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Clear[", FontFamily->"Courier", FontWeight->"Bold"], StyleBox["x", FontSlant->"Italic"], StyleBox["]", FontWeight->"Bold"], " or ", StyleBox["x", FontSlant->"Italic"], " = . clears the value for ", StyleBox["x", FontSlant->"Italic"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["x = ."], "Input", AspectRatioFixed->True], Cell[TextData[{ "The result of the input ", StyleBox["x", FontSlant->"Italic"], " will now be ", StyleBox["x", FontSlant->"Italic"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["x"], "Input", AspectRatioFixed->True], Cell[TextData[{ "A common source of problems in using ", StyleBox["Mathematica", FontSlant->"Italic"], " is trying to use variables\nthat already have values assigned. Therefore, \ clear variables directly\nafter you are finished with them or before you use \ them." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Because ", StyleBox["Mathematica", FontSlant->"Italic"], " functions begin with capital letters, you may wish to\nbegin your \ function names with lower case letters to prevent confusion." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "There are some things to consider when using variables in formulas.\n\n ", StyleBox["x y", FontSlant->"Italic"], " means ", StyleBox["x", FontSlant->"Italic"], " times ", StyleBox["y", FontSlant->"Italic"], ".\n\n ", StyleBox["xy", FontSlant->"Italic"], " (with no space) is the variable ", StyleBox["xy", FontSlant->"Italic"], ". Note; Watch out for this.\n\n 5", StyleBox["x", FontSlant->"Italic"], " is five times ", StyleBox["x", FontSlant->"Italic"], ".\n\n ", StyleBox["x", FontSlant->"Italic"], "^2", StyleBox["y", FontSlant->"Italic"], " is (", StyleBox["x", FontSlant->"Italic"], "^2) ", StyleBox["y", FontSlant->"Italic"], " not ", StyleBox["x", FontSlant->"Italic"], "^(2", StyleBox["y", FontSlant->"Italic"], ").\n\nWhen in doubt use an asterisk or parentheses to group expressions." }], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]]}, Open]], Cell[CellGroupData[{Cell[TextData["Calculus"], "Subsection", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "You can use ", StyleBox["Mathematica", FontSlant->"Italic"], " to do calculus. Here's a simple integral." }], "Text", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Integrate[x^n, x]"], "Input", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData["Here's a more complicated example."], "Text", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Integrate[x/(x^3-1), x]"], "Input", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData["Now let's try differentiating again."], "Text", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData["D[%, x]"], "Input", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData[{ "This gives the expression in a different algebraic form. We can get back \ our original form using ", StyleBox["Simplify", FontFamily->"Courier", FontSize->10, FontWeight->"Bold"], "." }], "Text", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Simplify[%]"], "Input", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " can also give exact solutions to many definite integrals." }], "Text", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Integrate[ Sin[x]^10 Cos[x]^5 / x^10, {x, 0, Infinity}]"], "Input", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData["\nHere's another example."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Integrate[ Log[x] (1 + x^2)^(-2), {x, 0, 1}]"], "Input", AspectRatioFixed->True], Cell[TextData[{ "Most integrals found in any book of tables can be done by ", StyleBox["Mathematica", FontSlant->"Italic"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Integrate[ Log[x]^6/(1 + x^2), {x, 0, 1}]"], "Input", AspectRatioFixed->True], Cell[TextData[{ "Many integrals do not have a \[OpenCurlyDoubleQuote]closed form solution\ \[CloseCurlyDoubleQuote]. If you give ", StyleBox["Mathematica", FontSlant->"Italic"], " such a definite integral it will be returned unevaluated. You can still \ use ", StyleBox["N", FontSize->10, FontWeight->"Bold"], " to get a numerical answer." }], "Text", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Integrate[ Sin[ Sin[x]], {x, 0, Pi}]"], "Input", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData["N[ % ]"], "Input", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " can also solve differential equations. Here is a pair of simultaneous \ differential equations. The solution you get involves two undetermined \ coefficients." }], "Text", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "DSolve[{x'[t] == -y[t], y'[t] == x[t]}, \n \ {x[t], y[t]}, t]"], "Input", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData[{ "It is a mathematical fact that most differential equations do not have an \ explicit symbolic solution. In these cases you can get a numerical \ approximation to the solution using ", StyleBox["NDSolve", FontFamily->"Courier", FontSize->10, FontWeight->"Bold"], ". After the solution is computed it is plotted." }], "Text", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "NDSolve[{y''[x] + Sin[x]^2 y'[x] + y[x] == Cos[x]^2,\n y[0] == 1, y'[0] \ == 0}, y, {x, 0, 20}]"], "Input", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData["Plot[ Evaluate[ y[x] /. %], {x, 0, 20} ]"], "Input", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Making Lists of Objects"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Lists are a way to make collections of objects in ", StyleBox["Mathematica", FontSlant->"Italic"], ". Lists are\nimportant structures in ", StyleBox["Mathematica", FontSlant->"Italic"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Here is a list of three numbers."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["{3,5,2}"], "Input", AspectRatioFixed->True], Cell[TextData[ "You can perform arithmetic operations on the whole list."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["{3,5,2} - 2"], "Input", AspectRatioFixed->True], Cell[TextData[ "This takes the difference between corresponding elements in the lists. The\n\ lists must be of equal lengths."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["{3,5,2} - {6,4,7}"], "Input", AspectRatioFixed->True], Cell[TextData["You can apply mathematical functions to whole lists."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Exp[%]"], "Input", AspectRatioFixed->True], Cell[TextData["You can assign the list to a variable."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["l = %"], "Input", AspectRatioFixed->True], Cell[TextData[{ "This variable ", StyleBox["l", FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" ", FontWeight->"Bold"], "can be used instead of the list in subsequent calculations." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Log[l]"], "Input", AspectRatioFixed->True]}, Open]]}, Open]], Cell[CellGroupData[{Cell[TextData["Solving Equations"], "Subsection", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "This is how you solve a quadratic equation in ", StyleBox["Mathematica", FontSlant->"Italic"], "." }], "Text", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Solve[x^2 + 2 a x + 1 == 0, x]"], "Input", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData["Here's a more complicated example."], "Text", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Solve[x^5 + 3x + 1 == 0, x]"], "Input", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData[{ "It is a fact of mathematics that there is no way to get an exact formula \ for the solutions of a quintic equation like this. You can nevertheless ask \ ", StyleBox["Mathematica ", FontSlant->"Italic"], " to give you numerical results. You get the five complex number roots to \ the equation." }], "Text", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData["N[%]"], "Input", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData[{ "When equations contain complicated functions there is in general no \ systematic procedure for finding all solutions, even numerically. The ", StyleBox["Mathematica", FontSlant->"Italic"], " function ", StyleBox["FindRoot", FontFamily->"Courier", FontSize->10, FontWeight->"Bold"], " searches for a numerical solution to an equation, starting at a specified \ point. " }], "Text", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "FindRoot[{Sin[x] == x - y, Cos[y] == x + y},\n {x, \ 1}, {y, 0}]"], "Input", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Mathematica ", FontSlant->"Italic"], " also has an efficient routine for finding the solution to linear \ equations. Here's a simple example." }], "Text", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData["LinearSolve[{{1.02, 5.9}, {2.87, 1.9}}, {1.9, 1.06}]"], "Input", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, AspectRatioFixed->True]}, Open]]}, Open]], Cell[CellGroupData[{Cell[TextData["Graphics"], "Section", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Here is a simple ", StyleBox["Mathematica", FontSlant->"Italic"], " plot. " }], "Text", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Plot[Sin[x], {x, 0, 2Pi}]"], "Input", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData[{ "There are many options you can specify for a plot. Using ", StyleBox["Show", FontFamily->"Courier", FontSize->10, FontWeight->"Bold"], " you can redraw the previous plot with specified options." }], "Text", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Show[% , Frame -> True, \n PlotLabel -> \"The Sine Function\"]"], "Input", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData["Now for some three-dimensional graphics."], "Text", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Plot3D[Sin[x] Sin[3 y] , {x, -2, 2}, {y,-2, 2}]"], "Input", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData["Here is a contour plot of the same function."], "Text", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Show[ ContourGraphics[ % ], ContourShading -> False,\n \ ContourSmoothing -> True]"], "Input", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData[ "This generates a three-dimensional parametric surface."], "Text", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "ParametricPlot3D[ {u Sin[t], u Cos[t], t/3},\n {t, 0, 12}, {u, \ -1, 1}, Ticks -> None]"], "Input", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " generates all graphics in PostScript, so that you can resize pictures, \ and make use of the resolution available on different types of printers. \ (Note, however, that to save disk space the graphics in this Notebook have \ been converted into bitmap images, which have lower resolution and do not \ look as good when resized or printed. The ability to convert images into \ bitmap form is useful when space is at a premium, and for animations, which \ are normally not printed.)" }], "Text", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Animated Graphics "], "Section", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "You can use sequences of graphics cells in a Notebook as frames in a \ \[OpenCurlyDoubleQuote]movie\[CloseCurlyDoubleQuote]."], "Text", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData["To view the movie:"], "Text", CellMargins->{{Inherited, 145}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "First execute the command below. This will take a couple of minutes as it \ produces a sequence of graphs. Then scroll until the first graphics cell is \ completely visible in the window. (The movie is shown in this cell.)"], "Text", CellDingbat->"\[FilledCircle]", CellMargins->{{54, 145}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Double-click the first picture in the sequence to start the movie. Click \ again anywhere to stop. 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