(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 23841, 838]*) (*NotebookOutlinePosition[ 36699, 1309]*) (* CellTagsIndexPosition[ 36655, 1305]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[StyleBox["Basic Mathematical Operations", FontSlant->"Italic"]], "Title", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Preliminaries", "Section"], Cell[CellGroupData[{ Cell["Input and Output Cells", "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Several different commands can be typed into a single input cell (marked \ by the skinny bracket along the right edge of a cell); either one per line, \ or several (separated by semicolons) on a line, or with long commands \ extending to several lines. 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When a session \ is saved to a file, all the In/Out labels are dropped.\n\nAny command \ followed by a semicolon will be evaluated, but the results not displayed. \ This is useful for commands with lengthy output that you don't really want to \ see." }], "Text"], Cell[BoxData[{ \(poly3\ \ = \ \ Expand[\ \((2 x - 5 y)\)^3\ ]\), \(\(poly99\ = \ Expand[\ \((2 x - 5 y)\)^99]; \)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Previous results; % and naming", "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "In building up results from previous calculations, there are three ways to \ refer to an earlier result.\n\n ", StyleBox["1", FontWeight->"Bold"], ". use the output number of the earlier result, e.g. ", StyleBox["Out[17]", FontFamily->"Courier", FontWeight->"Bold"], " or simply ", StyleBox["%17", FontFamily->"Courier", FontWeight->"Bold"] }], "Text"], Cell["%3", "Input"], Cell[TextData[{ StyleBox["\n 2", FontWeight->"Bold"], ". the most recent output can be referred to as ", StyleBox["%", FontFamily->"Courier"], ", the ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(2\^nd\), "-", RowBox[{"most", " ", "recent", " ", "output", " ", "as", " ", StyleBox["%%", FontFamily->"Courier", FontWeight->"Bold"]}]}], ",", " ", \(etc.\)}], TraditionalForm]]] }], "Text"], Cell["\<\ 6.4 * 10^6 m\t\t\t(* radius of earth *) 4 * Pi * %^2\t\t\t(* surface area of earth *) 2 Pi %%\t\t\t\t\t(* diameter of earth *)\ \>", "Input"], Cell[TextData[{ StyleBox["\n 3", FontWeight->"Bold"], ". name any result which you know will be used repeatedly later" }], "Text"], Cell["\<\ a=2^5 ; b=a-3 ; c=a/b ; (* defines a,b,c *) b Sqrt[c]\ \>", "Input"], Cell[TextData[{ "When entering multiple commands per cell, it is more reliable to name \ results (", StyleBox["3.", FontWeight->"Bold"], ") than to use ", StyleBox["%%", FontFamily->"Courier"], " notation (", StyleBox["2.", FontWeight->"Bold"], "), because of subtleties in how output results are counted in different \ kinds of cells." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["About Function Names", "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Names of ", StyleBox["all", FontWeight->"Bold", FontSlant->"Italic"], " built-in ", StyleBox["Mathematica", FontSlant->"Italic"], " functions are capitalized, e.g., ", StyleBox["Log", FontWeight->"Bold"], ", ", StyleBox["ArcTan", FontWeight->"Bold"], ", ", StyleBox["Factor", FontWeight->"Bold"], ", ", StyleBox["ParametricPlot", FontWeight->"Bold"], ". After you get used to this fact, many function names become \ predictable. For others, the quickest way to find the name of a function is \ to use a question mark to seek information, using * as a wildcard symbol. \ Using ? with a single function name gives basic information about that \ function." }], "Text"], Cell[BoxData[ \(\(?\ *Factor*\)\)], "Input"], Cell[BoxData[ \(\(?\ Factor*\)\)], "Input"], Cell[BoxData[ \(\(?FactorInteger\)\)], "Input"], Cell[TextData[{ "For functions which can't be guessed in this way, or for ", StyleBox["complete", FontSlant->"Italic"], " information about a function, use the Help Browser. 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Each \ parameter is an ", StyleBox["expression", FontSlant->"Italic"], ", which can be anything from a simple variable to a complicated formula.\n\ \nMany functions yield a different kind of result depending on whether the \ parameter is an exact value (integer or rational number), a floating point \ number (i.e. has a decimal point), or a symbol." }], "Text"], Cell[BoxData[ \(Sqrt[\ z\ + \ 3\ - \ \((phi\ + \ 2 theta)\)/Pi]\)], "Input"], Cell[BoxData[ \(\(Exp[4 n]\ /\ \((\(3!\) E) \)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (*\ factorial\ \(notation!\)\ *) \)\)], "Input"], Cell[BoxData[ \(\(FactorInteger[\ Mod[154, 100]] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (*\ factorize\ 54\ *) \)\)], "Input"], Cell[BoxData[ \({\ \ \ \ Log[8], \ \ Log[8.], \ \ Log[2, 8], \ \ \ Log[n, 8], \ \ Log[n, 8.]\ \ \ \ }\)], "Input"], Cell["(Curly brackets, {}, delimit a list of items.)", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Algebraic Manipulation", "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Here's where Mathematica goes beyond what a clever calculator can do. 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For simple derivatives one can use the \ \"prime\" notation analogous to y'(x). \n\nWe start by removing any possible \ previous definitions of ", StyleBox["t", FontWeight->"Bold"], ", ", StyleBox["x", FontWeight->"Bold"], ", or ", StyleBox["y", FontWeight->"Bold"], "; we want them to be undefined symbols." }], "Text"], Cell[BoxData[{ \(Clear[t, x, y]; \n\(Sin'\)[x]\), \(\(ArcTan'\)[t]\ - \ \(\(Sin'\)'\)[t]\ + \ \(\(\(y'\)'\)'\)[x]\)}], "Input"], Cell["\<\ \"D\" notation is also provided, for more complex expressions.\ \>", "Text"], Cell[BoxData[ \(D[\ \ \ 4 x^3\ - \ t\ x\ + \ Sqrt[x] - 17, \ \ \ \ x]\)], "Input"], Cell[TextData[{ "We wish to differentiate the function defined below, twice with respect to \ variable ", Cell[BoxData[ \(TraditionalForm\`x\)]], ", and once with respect to ", Cell[BoxData[ \(TraditionalForm\`t\)]], ":" }], "Text"], Cell[BoxData[ \(func\ \ = \ \ t\ \ Exp[x]\ - \ t^2*\ x^2\ + \ x^7\ ArcTan[t]; \n\n D[\ \ func\ , \ \ {x, 2}, t]\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Integration", "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Integration is more difficult. A great deal of work has been done to \ enable ", StyleBox["Mathematica", FontSlant->"Italic"], " to evaluate all indefinite and definite integrals found in standard \ tables compiled around the world, starting with the standard Abramowitz & \ Stegun. This still leaves many integrals which cannot be expressed in \ closed form, of course." }], "Text"], Cell[CellGroupData[{ Cell["Indefinite Integrals", "Subsection"], Cell[TextData[{ "Indefinite integrals are entered in the form ", StyleBox["Integrate[integrand, variable]", FontWeight->"Bold"], ":" }], "Text"], Cell[BoxData[ \(\(Integrate[\ \ 1/\((x^4 - a^4)\), \ x\ ] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (*\ can\ do\ *) \)\)], "Input"], Cell[BoxData[ \(\(Integrate[\ \ \ \ x^x, \ x\ ] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (*\ cannot\ do\ *) \)\)], "Input"], Cell[TextData[{ "In the 2nd example above, ", StyleBox["Mathematica", FontSlant->"Italic"], " cannot provide a closed form, so the integral is simply echoed back. \ Sometimes an expression is echoed back because of misspelling, though ", StyleBox["Mathematica", FontSlant->"Italic"], " often catches what it judges to be possible spelling mistakes." }], "Text"], Cell[BoxData[{ \(Integate[\ \ x^2\ - \ 3 x, \ x]\), \(IntegrXXXate[\ \ x^2\ - \ 3 x, \ x]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Definite Integrals", "Subsection"], Cell["\<\ Definite integrals are expressed by giving the variable of integration with \ its limits in a triplet bound by curly brackets:\ \>", "Text"], Cell[BoxData[ \(Integrate[\ \ 1/\((x^2 - a^2)\), \ {x, 3, 7}]\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Multiple Integrals", "Subsection"], Cell["\<\ Multiple integrals are expressed by listing the variables of integration, \ separated by commas. Each variable of integration can be given either with \ or without upper/lower bounds.\ \>", "Text"], Cell[BoxData[{ \(Integrate[\ \ x^2\ - \ 3\ x\ y\ + \ 7 y^2, \ \ \ x, \ y]\), \(Integrate[\ \ x^2\ - \ 3\ x\ y\ + \ 7\ y^2, \ \ \ x, \ {y, \ 2.1, \ 2, 6}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Numerical Integration", "Subsection"], Cell[TextData[{ "Even when an integral cannot be expressed analytically, it is usually \ possible to evaluate the definite integral numerically, to any desired degree \ of accuracy, using ", StyleBox["NIntegrate", FontWeight->"Bold"], ". The default ", StyleBox["working precision", FontSlant->"Italic"], " for numerical calculations on workstations is 16 decimal digits (machine \ precision), though the precision of the answer may be as many as 10 digits ", StyleBox["fewer", FontSlant->"Italic"], " than the working precision." }], "Text"], Cell[BoxData[{ \(NIntegrate[\ x^x, \ \ {x, 0, 5}]\), \(NIntegrate[\ x^x, \ \ {x, 0, 5}, \ WorkingPrecision -> 40]\)}], "Input"], Cell[TextData[{ StyleBox["NIntegrate", FontWeight->"Bold"], " uses an adaptive algorithm which recursively subdivides the integration \ region as needed. It is possible to fool ", StyleBox["NIntegrate", FontWeight->"Bold"], " with sufficiently pathological functions..." }], "Text"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Solving equations", "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Solve[equations,variables] ", FontWeight->"Bold"], " is used for an equation or set of equations in given variables, usually \ polynomial. (", StyleBox["Solve", FontWeight->"Bold"], " can handle certain non-polynomial equations, but usually with a warning.) \ ", StyleBox["Important", FontWeight->"Bold"], ": use ", StyleBox["double", FontSlant->"Italic"], " equal signs to indicate equality, as opposed to assigning a value -- just \ like the C convention; e.g.,\n\n", Cell[BoxData[ FormBox[ StyleBox[\(X = 3\), FontWeight->"Bold"], TraditionalForm]]], " assigns a value to variable ", Cell[BoxData[ \(TraditionalForm\`X\)]], ", whereas ", Cell[BoxData[ FormBox[ StyleBox[\(X == 3\), FontWeight->"Bold"], TraditionalForm]]], " is an equation." }], "Text", TextAlignment->Center], Cell[BoxData[{ \(Solve[\ \ x^3 + 7 x == 22, \ \ x] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (*\ simple\ equation\ *) \), \(Solve[\ \ x^3 + 7 x == 22.0, \ \ x] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (*\ \*"\""floating\ point \*"\"\<\> "*) \), \(Solve[\ {3 x + 2 y == 9, \ \ x - 5 y == 2}, \ \ {x, y}]\ \ \ \ (*\ simultaneous\ eqs\ *) \)}], "Input"], Cell[TextData[{ StyleBox["NOTE", FontWeight->"Bold"], ": the function ", StyleBox["Solve", FontWeight->"Bold"], " does ", StyleBox["not", FontSlant->"Italic"], " set values for the variables (", StyleBox["x", FontSlant->"Italic"], " and ", StyleBox["y", FontSlant->"Italic"], " above); instead, \"arrow notation\" is used to show what the solutions \ would be. To select one of the solutions, use \"slash dot\" syntax with the \ number of the solution you want (index number in double brackets):" }], "Text"], Cell[BoxData[{ RowBox[{\(soln\ \ = \ \ Solve[x^3 + 7 x == 22, x]\), " ", RowBox[{ StyleBox["(*", FontSlant->"Italic"], StyleBox[" ", FontSlant->"Italic"], StyleBox[\(list\ of\ 3\ solutions\), FontSlant->"Italic"], StyleBox[" ", FontSlant->"Italic"], StyleBox["*)", FontSlant->"Italic"]}]}], RowBox[{\(soln[\([2]\)]\), " ", RowBox[{ StyleBox["(*", FontSlant->"Italic"], StyleBox[" ", FontSlant->"Italic"], StyleBox[\(solution\ #2\), FontSlant->"Italic"], StyleBox[" ", FontSlant->"Italic"], StyleBox["*)", FontSlant->"Italic"]}], "\n"}], RowBox[{\(x\ \ /. \ \ soln[\([2]\)]\), " ", RowBox[{ StyleBox["(*", FontSlant->"Italic"], StyleBox[" ", FontSlant->"Italic"], StyleBox[\(the\ x - value\ of\ soln #2\), FontSlant->"Italic"], StyleBox[" ", FontSlant->"Italic"], StyleBox["*)", FontSlant->"Italic"]}], "\n"}], \(x1\ \ = \ \ x\ /. \ soln[\([1]\)]; \n x3\ \ = \ \ x\ /. \ soln[\([3]\)]; \n\ \ \ \ 9 x1\ + \ x3\)}], "Input"], Cell[TextData[{ "For many equations one must settle for a numerical solution rather than an \ analytic solution; one can use ", StyleBox["NSolve", FontWeight->"Bold"], " or ", StyleBox["FindRoot", FontWeight->"Bold"], ", discussed later. For 5th-degree and higher polynomials, you must \ usually settle for numerical values anyway; with ", StyleBox["NSolve", FontWeight->"Bold"], " you can easily specify the precision you want." }], "Text"], Cell[BoxData[ RowBox[{\(NSolve[\ \ x^5 - 3 x^2 == 1, \ x, \ 20]\), " ", StyleBox[\( (*\ 20\ digits\ of\ precision\ *) \), FontSlant->"Italic"]}]], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Differential equations", "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "The function DSolve is used to solve a differential equation, or a list of \ differential equations possibly including initial conditions or boundary \ conditions. Each equation has a ", StyleBox["double equal sign", FontWeight->"Bold"], ", ==, rather than a single equal sign." }], "Text"], Cell[BoxData[ \(DSolve[\ \ \(\(y'\)'\)[t] - 3 y[t] == 7, \ \ y[t], \ \ t]\)], "Input"], Cell["\<\ Notice that the two arbitrary scalar constants in the general \ solution above are C[1] and C[2]. Multiple (simultaneous) equations are \ given in curly brackets, {}; each additional equation (including initial or \ boundary conditions) reduces the number of undetermined constants.\ \>", "Text"], Cell[BoxData[ \(DSolve[{\ \ \(\(y'\)'\)[t] - 3 y[t] == 7, \ \ y[1] == 0}, \ \ \ \ y[t], \ \ t]\)], "Input"], Cell["\<\ In both examples above there have been exactly one solution. \ Sometimes there are more than one, sometimes none; each solution is presented \ in curly brackets with the arrow notation, and they are listed in an outer \ set of curly brackets.\ \>", "Text"], Cell[BoxData[{ \(DSolve[\ \ \((\(y'\)[t] - 3 y[t])\) \((3 \( y'\)[t] + 2 y[t])\) == 0, \ \ \ y[t], \ \ \ t]\n\), \(DSolve[\ \ {\ \(y'\)[t] == 1, \ \ y[0] == 0, \ \ y[1] == 0}, \ \ \ \ y[t], \ \ \ \ t]\)}], "Input"], Cell["\<\ In making further use of a solution, it is useful to indicate which \ of the solutions is wanted (even if there is only one) using double-bracket \ index notation. 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