(************** Content-type: application/mathematica ************** 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[ 650995, 14711]*) (*NotebookOutlinePosition[ 652119, 14748]*) (* CellTagsIndexPosition[ 652075, 14744]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ StyleBox["1. ", FontFamily->"New century schoolbook", FontSize->18], StyleBox["Introduction", FontSize->18] }], "Section", Evaluatable->False, AspectRatioFixed->True, FontSize->12], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " is a high level scientific and mathematical programming environment. The \ program consists of two components: the ", StyleBox["front-end ", FontWeight->"Bold"], "or ", StyleBox["notebook", FontWeight->"Bold"], " lets you interact with Mathematica in an editor-like format. You can \ type your input expressions, comments, format the text and see the output of \ the execution inside the notebook. The core engine of ", StyleBox["Mathematica", FontSlant->"Italic"], " that does all the computation is the ", StyleBox["kernel", FontWeight->"Bold"], ". ", StyleBox["Mathematica", FontSlant->"Italic"], " uses the concept of cells to hold or display all the data. A cell can be \ of type input, output, text or any of the various other types that ", StyleBox["Mathematica", FontSlant->"Italic"], " offers (see the menu Format \[LongRightArrow] Style). By default, \ whatever you type in a new cell is of type input. To enter comments or to \ describe your code (like this paragraph), you should change the cell style to \ Text using the above menu. You are encouraged to document your programs so \ that they can be easily read and used by others or yourself at a later time. \ Also, the best method to master a new program like ", StyleBox["Mathematica", FontSlant->"Italic"], " is to use it and explore all the menu items and help topics. " }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "After entering an expression, press ", StyleBox["[Shift-Enter] ", FontWeight->"Bold"], "to evaluate the cell. The very first evaluation may take a little longer \ time because the notebook needs to start the kernel. ", StyleBox["Mathematica", FontSlant->"Italic"], " labels each input cell and the corresponding output cells by unique \ numbers to keep track of all the evaluations made. After the evaluation, the \ input cells\nare designated by ", StyleBox["In", FontFamily->"Courier", FontWeight->"Bold"], " and the output cells are designated by ", StyleBox["Out", FontFamily->"Courier", FontWeight->"Bold"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(2 + 2\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(4\)], "Output"] }, Open ]], Cell["\<\ Since each evaluation is assigned a number (\"1\" in the example \ above), it can be referenced at a later time. Both the input and output of \ an expression are stored and can be used in other expressions. For example:\ \ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(a = %1\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(4\)], "Output"] }, Open ]], Cell["and", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(2 + In[1]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(6\)], "Output"] }, Open ]], Cell["\<\ Mathematica, by default, conducts all arithematic calculations with \ rational numbers when possible. For example:\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(3\/10 + 1\/2\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(4\/5\)], "Output"] }, Open ]], Cell["\<\ There are two ways of generating output as floating-point numbers: \ (1) Add decimal points in the input line\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(3. \/10 + 1\/2\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(0.8`\)], "Output"] }, Open ]], Cell["or (2) by using the built-in function \"N\". ", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(N[%4]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(0.8`\)], "Output"] }, Open ]], Cell["\<\ A quick way of referencing the last output is by using the % \ command. This has the same effect as using \"Out\" for the last evaluated \ output.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(b = %\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(0.8`\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["Mathematica has several predefined constants for your use \ including: \[Pi], and e as well as Euler's gamma constant, the golden \ ratio, and Catalan's constant: (By the way, ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" has very good type-setting abilities. You can enter almost any \ mathematical symbol or character using the palettes or key combinations.)", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(N[{\[Pi], E, EulerGamma, GoldenRatio, Catalan}]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({3.141592653589793`, 2.718281828459045`, 0.5772156649015329`, 1.618033988749895`, 0.915965594177219`}\)], "Output"] }, Open ]], Cell["\<\ Irrational numbers can be evaluated to arbitrary accuracy by using \ \"N\". The second argument represents the number of digits required:\ \>", \ "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(N[\[Pi], 10]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(3.141592653589793`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(N[\[Pi], 200]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(3.1415926535897932384626433832795028841971693993751058209749445923078164\ 062862089986280348253421170679821480865132823066470938446095505822317253594081\ 284811174502841027019385211055596446229489549303819644288109757`200\)], \ "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["2. Variables", FontSize->18]], "Section", Evaluatable->False, AspectRatioFixed->True, FontSize->12], Cell["\<\ There are two different ways of assigning values to variables. \ The first is a \"one-time\" assignment using the equal sign.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(a = 10\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(10\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(2\ a - 2\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(18\)], "Output"] }, Open ]], Cell["Notice what happens in the following series of evaluations:", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(a = 10\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(10\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(b = a - 2\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(8\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(a = 5\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(5\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(b\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(8\)], "Output"] }, Open ]], Cell["\<\ After changing the value of \"a\" from 10 to 5, the value of \"b\" \ remained equal to 8. This is because \"b\" was evaluated when \"a=10\" and \ was unaffected by the reassignment of \"a\". \ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell["Now consider the following:", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(a = 10\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(10\)], "Output"] }, Open ]], Cell[BoxData[ \(b := a - 2\)], "Input", AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(b\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(8\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(a = 5\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(5\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(b\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(3\)], "Output"] }, Open ]], Cell["\<\ By assigning \"b\" using : = rather than just =, the value of \"b\" \ is re-evaluated every time \"b\" is referenced. This represents the second \ way variables are defined.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(a = c\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(c\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(b\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(-2\) + c\)], "Output"] }, Open ]], Cell["\<\ As can be seen, Mathematica can conduct algebraic manipulations as \ well as arithmetic.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ Note that variable names are case sensitive in that \"a\" is \ different from \"A\".\ \>", "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["3. Functions", FontSize->18]], "Section", Evaluatable->False, AspectRatioFixed->True, FontSize->12], Cell["\<\ The arguments of a function in Mathematica are contained in square \ brackets. Examples of functions are:\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \({f[x], g[x, y], My3DFunction[x, y, z]}\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({f[x], g[x, y], My3DFunction[x, y, z]}\)], "Output"] }, Open ]], Cell["\<\ Mathematica has many built-in functions. Some examples are (not \ nearly a complete list):\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\({Sin[x], Cos[x], Tan[x], Csc[x], Sec[x], Cot[x], ArcSin[x], Exp[x], Sinh[x], Cosh[x], Tanh[x], ArcSinh[x], Log[x], BesselI[n, x], BesselJ[n, x], EllipticE[x], Gamma[x], Hypergeometric0F1[n, x], LegendreP[n, x], Erf[x], Erfc[x], Zeta[x]};\)\)], "Input", AspectRatioFixed->True], Cell[TextData[{ "To get helpful information about a function (or variable) use the ", StyleBox["?", FontWeight->"Bold"], " or ", StyleBox["??", FontWeight->"Bold"], " commands:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(?? Zeta\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \("Zeta[s] gives the Riemann zeta function. Zeta[s, a] gives the \ generalized Riemann zeta function."\)], "Print"], Cell[BoxData[ InterpretationBox[GridBox[{ {\(Attributes[Zeta] = {Listable, NumericFunction, Protected}\)}, {" "}, {GridBox[{ {\(Options[Zeta] = {IncludeSingularTerm \[Rule] False}\)} }, GridBaseline->{Baseline, {1, 1}}, ColumnWidths->0.999, ColumnAlignments->{Left}]} }, GridBaseline->{Baseline, {1, 1}}, ColumnAlignments->{Left}], Definition[ Zeta], Editable->False]], "Print"] }, Open ]], Cell["Some examples of function evaluation are given below:", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(Log[10. ]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(2.302585092994046`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Sin[\[Pi]\/4]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(1\/\@2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(N[%]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(0.7071067811865476`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Exp[y\ Log[x]]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(x\^y\)], "Output"] }, Open ]], Cell["\<\ An important \"function\" is the derivative \"D\" which has \ different levels of application. Let's look at the definition of this \ function:\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(?? D\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \("D[f, x] gives the partial derivative of f with respect to x. D[f, {x, \ n}] gives the nth partial derivative of f with respect to x. D[f, x1, x2, ... \ ] gives a mixed derivative."\)], "Print"], Cell[BoxData[ InterpretationBox[GridBox[{ {\(Attributes[D] = {Protected, ReadProtected}\)}, {" "}, {GridBox[{ {\(Options[D] = NonConstants \[Rule] {}\)} }, GridBaseline->{Baseline, {1, 1}}, ColumnWidths->0.999, ColumnAlignments->{Left}]} }, GridBaseline->{Baseline, {1, 1}}, ColumnAlignments->{Left}], Definition[ D], Editable->False]], "Print"] }, Open ]], Cell["For example try", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(\[PartialD]\_x f[x]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "x", "]"}]], "Output"] }, Open ]], Cell["\<\ This is the generic way of representing derivatives and will be \ used in differential equation solving latter. Also consider these symbolic \ derivatives:\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(\[PartialD]\_{x, 3}f[x]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ RowBox[{ SuperscriptBox["f", TagBox[\((3)\), Derivative], MultilineFunction->None], "[", "x", "]"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[PartialD]\_x f[x, y]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ RowBox[{ SuperscriptBox["f", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, y\), "]"}]], "Output"] }, Open ]], Cell["More examples:", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(\[PartialD]\_x Sin[x]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(Cos[x]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[PartialD]\_z z\^x\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(x\ z\^\(\(-1\) + x\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[PartialD]\_x z\^x\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(z\^x\ Log[z]\)], "Output"] }, Open ]], Cell["\<\ You can create your own functions using the : = assignment as \ follows.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(fun1[x_] := Sin[x]\ Exp[x - 1]\)], "Input", AspectRatioFixed->True], Cell["\<\ The expression \"x_\" is called a replacement pattern. It says, \ \"replace all occurences of 'x' in the following expression by the value of \ the argument.\" A multi-dimensional function can be created as follows:\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(fun2[x_, y_] := Sin[x + y]\ Cos[x\ y]\ Exp[x - 1]\)], "Input", AspectRatioFixed->True], Cell["\<\ These functions can be evaluated the same as built-in \ functions.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(fun1[\[Pi]\/2]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\[ExponentialE]\^\(\(-1\) + \[Pi]\/2\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[PartialD]\_x fun1[x]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\[ExponentialE]\^\(\(-1\) + x\)\ Cos[ x] + \[ExponentialE]\^\(\(-1\) + x\)\ Sin[x]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(fun2[fun1[x], x\^2]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\[ExponentialE]\^\(\(-1\) + \[ExponentialE]\^\(\(-1\) + x\)\ Sin[x]\)\ \ Cos[\[ExponentialE]\^\(\(-1\) + x\)\ x\^2\ Sin[x]]\ Sin[ x\^2 + \[ExponentialE]\^\(\(-1\) + x\)\ Sin[x]]\)], "Output"] }, Open ]], Cell["Some other useful functions you should explore are:", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\({Ceiling[x], Floor[x], Round[x], Chop[x], Rationalize[x]};\)\)], "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["4. Lists", FontSize->18]], "Section", Evaluatable->False, AspectRatioFixed->True, FontSize->12], Cell["\<\ A list is a collection of objects whose symbols are enclosed in \ braces { }, and separated by commas. The objects are not restricted to \ numbers. For example, here are some lists:\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \({3, 2, 5, \(-1\), 0, 0}\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({3, 2, 5, \(-1\), 0, 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \({3.4, x, y\^2, c, Sin[x], func[x], f[a], \[Pi]}\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({3.4`, x, y\^2, c, Sin[x], func[x], f[c], \[Pi]}\)], "Output"] }, Open ]], Cell["\<\ A list can be created by a number of different means. The most \ common method is to use the \"Table\" command.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(?? Table\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \("Table[expr, {imax}] generates a list of imax copies of expr. \ Table[expr, {i, imax}] generates a list of the values of expr when i runs \ from 1 to imax. Table[expr, {i, imin, imax}] starts with i = imin. \ Table[expr, {i, imin, imax, di}] uses steps di. Table[expr, {i, imin, imax}, \ {j, jmin, jmax}, ... ] gives a nested list. The list associated with i is \ outermost."\)], "Print"], Cell[BoxData[ InterpretationBox[GridBox[{ {\(Attributes[Table] = {HoldAll, Protected}\)} }, GridBaseline->{Baseline, {1, 1}}, ColumnAlignments->{Left}], Definition[ Table], Editable->False]], "Print"] }, Open ]], Cell["Here are a few examples:", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(list1 = Table[n, {n, 4}]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({1, 2, 3, 4}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(list2 = Table[x\^n, {n, 2, 5}]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({x\^2, x\^3, x\^4, x\^5}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(list3 = Table[1\/n, {n, 3, 24, 3}]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({1\/3, 1\/6, 1\/9, 1\/12, 1\/15, 1\/18, 1\/21, 1\/24}\)], "Output"] }, Open ]], Cell["\<\ There are several ways to manipulate lists once they are created. \ The following are functions of lists that do just that. \ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(lista = {2, 8, 0, 1, \(-2\), 0, 0, 12, 3, 8, 5}\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({2, 8, 0, 1, \(-2\), 0, 0, 12, 3, 8, 5}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Sort[lista]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({\(-2\), 0, 0, 0, 1, 2, 3, 5, 8, 8, 12}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Reverse[lista]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({5, 8, 3, 12, 0, 0, \(-2\), 1, 0, 8, 2}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RotateLeft[%, 2]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({3, 12, 0, 0, \(-2\), 1, 0, 8, 2, 5, 8}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RotateRight[%, 4]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({8, 2, 5, 8, 3, 12, 0, 0, \(-2\), 1, 0}\)], "Output"] }, Open ]], Cell["\<\ Any set of componants can be dropped or added to a list as follows:\ \ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(Drop[%, 3]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({8, 3, 12, 0, 0, \(-2\), 1, 0}\)], "Output"] }, Open ]], Cell["The first 3 elements of the list were dropped.", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(Drop[%, \(-2\)]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({8, 3, 12, 0, 0, \(-2\)}\)], "Output"] }, Open ]], Cell["The last 2 elements were dropped.", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(Append[%, 100]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({8, 3, 12, 0, 0, \(-2\), 100}\)], "Output"] }, Open ]], Cell["Try playing around with some of these functions:", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\({Rest[list1], Take[list1, 2], Append[list1, x], AppendTo[list1, x], Prepend[list1, x], PrependTo[list1, x], Length[list1], Insert[list1, x, 3], Complement[list1, list2], list1 \[Intersection] list2, list1 \[Union] list2, Join[list1, list2], etc};\)\)], "Input", AspectRatioFixed->True], Cell["\<\ An n-dimensional list can also be thought of as an n-dimensional \ vector. The rules of vector addition and multiplication (as well as \ calculus) can be applied. For example:\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[a, b]\)], "Input", AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(a = {a1, a2, a3}\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({a1, a2, a3}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(b = {b1, b2, b3}\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({b1, b2, b3}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(a + b\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({a1 + b1, a2 + b2, a3 + b3}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(a . b\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(a1\ b1 + a2\ b2 + a3\ b3\)], "Output"] }, Open ]], Cell["\<\ The last evaluation is an example of how to calculate the dot \ product.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ Lists can also have other lists as elements. Consider for example:\ \ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(m = {{1, 2, 4}, {2, 0, 1}, {\(-1\), \(-1\), 3}}\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({{1, 2, 4}, {2, 0, 1}, {\(-1\), \(-1\), 3}}\)], "Output"] }, Open ]], Cell["\<\ This is a three-dimensional list containing three-dimensional lists \ as elements. In Mathematica, this is how matrices are represented. Use the \ function \"MatrixForm\" to display 'm' in a more natural form. \ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[m]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "2", "4"}, {"2", "0", "1"}, {\(-1\), \(-1\), "3"} }], "\[NoBreak]", ")"}], (MatrixForm[ #]&)]], "Output"] }, Open ]], Cell["\<\ Note: MatrixForm is only for displaying a matrix not to do \ computations. Do not use it in expressions. \ \>", "Text"], Cell["\<\ You can conduct matrix operations like addition, matrix product, \ determinant, inverse, and eigen analysis as follows:\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(p = {{3, 0, 2}, {0, 1, 2}, {2, 2, 0}}\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({{3, 0, 2}, {0, 1, 2}, {2, 2, 0}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[p]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"3", "0", "2"}, {"0", "1", "2"}, {"2", "2", "0"} }], "\[NoBreak]", ")"}], (MatrixForm[ #]&)]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[m + p]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"4", "2", "6"}, {"2", "1", "3"}, {"1", "1", "3"} }], "\[NoBreak]", ")"}], (MatrixForm[ #]&)]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[m . p]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"11", "10", "6"}, {"8", "2", "4"}, {"3", "5", \(-4\)} }], "\[NoBreak]", ")"}], (MatrixForm[ #]&)]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Det[p]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(-16\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[Inverse[p]]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(1\/4\), \(-\(1\/4\)\), \(1\/8\)}, {\(-\(1\/4\)\), \(1\/4\), \(3\/8\)}, {\(1\/8\), \(3\/8\), \(-\(3\/16\)\)} }], "\[NoBreak]", ")"}], (MatrixForm[ #]&)]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Chop[N[Eigenvalues[p]]]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({4.297070544363273`, \(-2.0838723594356066`\), 1.7868018150723326`}\)], "Output"] }, Open ]], Cell["\<\ Chop was used here to eliminate erroneous, small imaginary \ components in the evaluation.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ Elements of a list can be easily referenced using double square \ brackets as follows: \ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(a\[LeftDoubleBracket]2\[RightDoubleBracket]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(a2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(p\[LeftDoubleBracket]2, 3\[RightDoubleBracket]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(2\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["5. Numerical Evaluation", FontSize->18]], "Section", Evaluatable->False, AspectRatioFixed->True, FontSize->12], Cell["\<\ There are a variety of tools available to calculate systems of \ equations, differential equations, and integrals, the most versatile being \ the \"FindRoot\" functions. \"FindRoot\" is a Newton-Raphson based tool for \ calculating equation systems.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(?? FindRoot\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \("FindRoot[lhs==rhs, {x, x0}] searches for a numerical solution to the \ equation lhs==rhs, starting with x=x0."\)], "Print"], Cell[BoxData[ InterpretationBox[GridBox[{ {\(Attributes[FindRoot] = {HoldAll, Protected}\)}, {" "}, {GridBox[{ {\(Options[FindRoot] = {AccuracyGoal \[Rule] Automatic, Compiled \[Rule] True, DampingFactor \[Rule] 1, Jacobian \[Rule] Automatic, MaxIterations \[Rule] 15, WorkingPrecision \[Rule] 16}\)} }, GridBaseline->{Baseline, {1, 1}}, ColumnWidths->0.999, ColumnAlignments->{Left}]} }, GridBaseline->{Baseline, {1, 1}}, ColumnAlignments->{Left}], Definition[ FindRoot], Editable->False]], "Print"] }, Open ]], Cell["Consider the following examples:", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(FindRoot[x\ Log[x] == 3, {x, 2.0}]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({x \[Rule] 2.857390831837419`}\)], "Output"] }, Open ]], Cell["\<\ The expression ' x Log[x] == 3 ' is an example of an equation in \ Mathematica. Note that there are two successive equal signs (==) in the \ above equation. This is necessary to distinguish an equation from an \ assignment. Trying ' x Log[x] = 3 ' would have resulted in an error message. \ \ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell["Multidimensional equation solving is also possible.", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(FindRoot[{x\^2 + y\ x == 1.0, x + y == 2.0}, {x, 1.0}, {y, 1.0}]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({x \[Rule] 0.5`, y \[Rule] 1.5`}\)], "Output"] }, Open ]], Cell["\<\ Note that in both of the above examples, the variables to be solved \ for were given as lists with the initial guess values included. This is \ required for FindRoot. Also note the output is written in a list format with \ the symbol ' -> '. This is an example of a replacement rule. It can be used \ in conjunction with the \"ReplaceAll\" symbol as follows:\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(x\^2 /. x \[Rule] 3\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(9\)], "Output"] }, Open ]], Cell["\<\ The above expression reads, \"evaluate x^2 by making the \ replacement 'x->3'.\" Some of the tools in this section produce output in the \ form of replacement rules. \ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell["It is convenient to rewrite the above 1D problem as follows:", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(eq = x\ Log[x]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(x\ Log[x]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(answer = FindRoot[eq == 3.0, {x, 2.0}]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({x \[Rule] 2.857390831837419`}\)], "Output"] }, Open ]], Cell["\<\ Now we can use the replacement rule 'answer' to check our \ work.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(eq /. answer\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(3.0000000990578575`\)], "Output"] }, Open ]], Cell["\<\ Integration in Mathematica is conducted by similar means. The \ functions \"Integrate\" and \"NIntegrate\" can be utilized for symbolic and \ numerical integration as follows:\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(integrand = 1\/\(x\^2 + c\)\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(1\/\(c + x\^2\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[Integral]integrand \[DifferentialD]x\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(ArcTan[x\/\@c]\/\@c\)], "Output"] }, Open ]], Cell["\<\ Complicated integrals can be evaluated numerically with \ \"NIntegrate\".\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(?? NIntegrate\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \("NIntegrate[f, {x, xmin, xmax}] gives a numerical approximation to the \ integral of f with respect to x from xmin to xmax."\)], "Print"], Cell[BoxData[ InterpretationBox[GridBox[{ {\(Attributes[NIntegrate] = {HoldAll, Protected}\)}, {" "}, {GridBox[{ { RowBox[{\(Options[NIntegrate]\), "=", RowBox[{"{", RowBox[{ RowBox[{"AccuracyGoal", "\[Rule]", InterpretationBox["\[Infinity]", DirectedInfinity[ 1]]}], ",", \(Compiled \[Rule] True\), ",", \(GaussPoints \[Rule] Automatic\), ",", \(MaxPoints \[Rule] Automatic\), ",", \(MaxRecursion \[Rule] 6\), ",", \(Method \[Rule] Automatic\), ",", \(MinRecursion \[Rule] 0\), ",", \(PrecisionGoal \[Rule] Automatic\), ",", \(SingularityDepth \[Rule] 4\), ",", \(WorkingPrecision \[Rule] 16\)}], "}"}]}]} }, GridBaseline->{Baseline, {1, 1}}, ColumnWidths->0.999, ColumnAlignments->{Left}]} }, GridBaseline->{Baseline, {1, 1}}, ColumnAlignments->{Left}], Definition[ NIntegrate], Editable->False]], "Print"] }, Open ]], Cell["For example:", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(NIntegrate[Sin[x]\ Exp[\(-x\^2\)], {x, 0, \[Pi]}]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(0.4244375107724583`\)], "Output"] }, Open ]], Cell["\<\ Likewise, differential equations can be solved both symbolicly or \ numerically with the commands \"DSolve\" and \"NDSolve\". Consider the \ following differential equation:\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"diffeq", "=", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "+", \(\((t - 3)\)\ f[t]\)}], "==", \(Exp[t]\)}]}]], "Input", AspectRatioFixed->True], Cell[BoxData[ RowBox[{ RowBox[{\(\((\(-3\) + t)\)\ f[t]\), "+", RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "t", "]"}]}], "==", \(\[ExponentialE]\^t\)}]], "Output"] }, Open ]], Cell["\<\ This can be solved for the initial condition: f[0] = 1.0 as \ follows:\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(?? DSolve\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \("DSolve[eqn, y, x] solves a differential equation for the function y, \ with independent variable x. DSolve[{eqn1, eqn2, ... }, {y1, y2, ... }, x] \ solves a list of differential equations. DSolve[eqn, y, {x1, x2, ... }] \ solves a partial differential equation."\)], "Print"], Cell[BoxData[ InterpretationBox[GridBox[{ {\(Attributes[DSolve] = {Protected, ReadProtected}\)}, {" "}, {GridBox[{ {\(Options[DSolve] = DSolveConstants \[Rule] C\)} }, GridBaseline->{Baseline, {1, 1}}, ColumnWidths->0.999, ColumnAlignments->{Left}]} }, GridBaseline->{Baseline, {1, 1}}, ColumnAlignments->{Left}], Definition[ DSolve], Editable->False]], "Print"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(DSolve[{diffeq, f[0] == 1.0}, f[t], t]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({{f[t] \[Rule] 0.5`\ \[ExponentialE]\^\(\(-2.`\) + 3.`\ t - 0.5`\ t\^2\)\ \ \((\(\(24.23592776908214`\)\(\[InvisibleSpace]\)\) + 2.5066282746310002`\ Erfi[ 0.7071067811865476`\ \((\(-2.`\) + t)\)])\)}}\)], "Output"] }, Open ]], Cell["\<\ Numerical solutions are calculated by similar means using \"NDSolve\ \".\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(?? NDSolve\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \("NDSolve[eqns, y, {x, xmin, xmax}] finds a numerical solution to the \ ordinary differential equations eqns for the function y with the independent \ variable x in the range xmin to xmax. NDSolve[eqns, y, {x, xmin, xmax}, {t, \ tmin, tmax}] finds a numerical solution to the partial differential equations \ eqns. NDSolve[eqns, {y1, y2, ... }, {x, xmin, xmax}] finds numerical \ solutions for the functions yi."\)], "Print"], Cell[BoxData[ InterpretationBox[GridBox[{ {\(Attributes[NDSolve] = {Protected}\)}, {" "}, {GridBox[{ { RowBox[{\(Options[NDSolve]\), "=", RowBox[{"{", RowBox[{\(AccuracyGoal \[Rule] Automatic\), ",", \(Compiled \[Rule] True\), ",", \(DifferenceOrder \[Rule] Automatic\), ",", \(InterpolationPrecision \[Rule] Automatic\), ",", \(MaxRelativeStepSize \[Rule] 1\), ",", \(MaxSteps \[Rule] Automatic\), ",", RowBox[{"MaxStepSize", "\[Rule]", InterpretationBox["\[Infinity]", DirectedInfinity[ 1]]}], ",", \(Method \[Rule] Automatic\), ",", \(PrecisionGoal \[Rule] Automatic\), ",", \(SolveDelayed \[Rule] False\), ",", \(StartingStepSize \[Rule] Automatic\), ",", \(StoppingTest \[Rule] None\), ",", \(WorkingPrecision \[Rule] 16\)}], "}"}]}]} }, GridBaseline->{Baseline, {1, 1}}, ColumnWidths->0.999, ColumnAlignments->{Left}]} }, GridBaseline->{Baseline, {1, 1}}, ColumnAlignments->{Left}], Definition[ NDSolve], Editable->False]], "Print"] }, Open ]], Cell["\<\ There is a major difference here compared to numerical integration \ where the output is an number as opposed to a function. The numerical \ solution (function) to a differential equation is represented by what's \ called an \"Interpolating function\". For example, consider the differential \ equation:\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"diffeq2", "=", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["y", "\[DoublePrime]", MultilineFunction->None], "[", "x", "]"}], "-", RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "x", "]"}], "+", \(x\ y[x]\)}], "==", "0"}]}]], "Input", AspectRatioFixed->True], Cell[BoxData[ RowBox[{ RowBox[{\(x\ y[x]\), "-", RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "x", "]"}], "+", RowBox[{ SuperscriptBox["y", "\[Prime]\[Prime]", MultilineFunction->None], "[", "x", "]"}]}], "==", "0"}]], "Output"] }, Open ]], Cell["\<\ with the initial conditions: y'[0] = -1 and y[0]=1. The numerical \ solution is calculated by\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"NDSolve", "[", RowBox[{ RowBox[{"{", RowBox[{"diffeq2", ",", RowBox[{ RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "0", "]"}], "==", \(-1\)}], ",", \(y[0] == 1\)}], "}"}], ",", "y", ",", \({x, 0, 5}\)}], "]"}]], "Input", AspectRatioFixed->True], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{"y", "\[Rule]", TagBox[\(InterpolatingFunction[{{0.`, 5.`}}, "<>"]\), False, Editable->False]}], "}"}], "}"}]], "Output"] }, Open ]], Cell["\<\ where we evaluated the solution from (0", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(y = y /. %\[LeftDoubleBracket]1, 1\[RightDoubleBracket]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ TagBox[\(InterpolatingFunction[{{0.`, 5.`}}, "<>"]\), False, Editable->False]], "Output"] }, Open ]], Cell["\<\ Now, the function \"y\" can be treated as any function as long as \ the argument is restricted within 0 and 5. For example\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(y[1]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(-0.7913286108173958`\)\)], "Output"] }, Open ]], Cell["\<\ Raw data can also be converted into an interpolating function by \ using the command \"Interpolation\".\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(?? Interpolation\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \("Interpolation[data] constructs an InterpolatingFunction object which \ represents an approximate function that interpolates the data. The data can \ have the forms {{x1, f1}, {x2, f2}, ... } or {f1, f2, ... }, where in the \ second case, the xi are taken to have values 1, 2, ... ."\)], "Print"], Cell[BoxData[ InterpretationBox[GridBox[{ {\(Attributes[Interpolation] = {Protected}\)}, {" "}, {GridBox[{ {\(Options[Interpolation] = {InterpolationOrder \[Rule] 3}\)} }, GridBaseline->{Baseline, {1, 1}}, ColumnWidths->0.999, ColumnAlignments->{Left}]} }, GridBaseline->{Baseline, {1, 1}}, ColumnAlignments->{Left}], Definition[ Interpolation], Editable->False]], "Print"] }, Open ]], Cell["Consider the list of data points {x,y}:", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(data = {{0, 0}, {1, 0.632}, {2, 0.917}, {3, 1.655}, {4, 2.828}, {5, 4.158}, {6, 5.465}, {7, 6.690}, {8, 7.828}, {9, 8.910}, {10, 9.955}}\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({{0, 0}, {1, 0.632`}, {2, 0.917`}, {3, 1.655`}, {4, 2.828`}, {5, 4.158`}, {6, 5.465`}, {7, 6.69`}, {8, 7.828`}, {9, 8.91`}, {10, 9.955`}}\)], "Output"] }, Open ]], Cell["The interpolating function can be generated by", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(interp = Interpolation[data]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ TagBox[\(InterpolatingFunction[{{0.`, 10.`}}, "<>"]\), False, Editable->False]], "Output"] }, Open ]], Cell["\<\ Now the function \"interp\" can be used:\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(interp[6.5]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(6.0880625`\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["6. Plotting Applications", FontSize->18]], "Section", Evaluatable->False, AspectRatioFixed->True, FontSize->12], Cell["\<\ Mathematica has several different plotting routines for \ visualization of functions and data. The most simple application is the \ \"Plot\" command.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(?? Plot\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \("Plot[f, {x, xmin, xmax}] generates a plot of f as a function of x from \ xmin to xmax. Plot[{f1, f2, ... }, {x, xmin, xmax}] plots several functions \ fi."\)], "Print"], Cell[BoxData[ InterpretationBox[GridBox[{ {\(Attributes[Plot] = {HoldAll, Protected}\)}, {" "}, {GridBox[{ {\(Options[Plot] = {AspectRatio \[Rule] 1\/GoldenRatio, Axes \[Rule] Automatic, AxesLabel \[Rule] None, AxesOrigin \[Rule] Automatic, AxesStyle \[Rule] Automatic, Background \[Rule] Automatic, ColorOutput \[Rule] Automatic, Compiled \[Rule] True, DefaultColor \[Rule] Automatic, Epilog \[Rule] {}, Frame \[Rule] False, FrameLabel \[Rule] None, FrameStyle \[Rule] Automatic, FrameTicks \[Rule] Automatic, GridLines \[Rule] None, ImageSize \[Rule] Automatic, MaxBend \[Rule] 10.`, PlotDivision \[Rule] 30.`, PlotLabel \[Rule] None, PlotPoints \[Rule] 25, PlotRange \[Rule] Automatic, PlotRegion \[Rule] Automatic, PlotStyle \[Rule] Automatic, Prolog \[Rule] {}, RotateLabel \[Rule] True, Ticks \[Rule] Automatic, DefaultFont \[RuleDelayed] $DefaultFont, DisplayFunction \[RuleDelayed] $DisplayFunction, FormatType \[RuleDelayed] $FormatType, TextStyle \[RuleDelayed] $TextStyle}\)} }, GridBaseline->{Baseline, {1, 1}}, ColumnWidths->0.999, ColumnAlignments->{Left}]} }, GridBaseline->{Baseline, {1, 1}}, ColumnAlignments->{Left}], Definition[ Plot], Editable->False]], "Print"] }, Open ]], Cell["\<\ This command plots 2-dimensional data or functions. Consider the \ following examples:\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[Sin[x], {x, 0, 2\ \[Pi]}]\)], "Input", AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.151576 0.309016 0.294303 [ [.17539 .29652 -3 -9 ] [.17539 .29652 3 0 ] [.32696 .29652 -3 -9 ] [.32696 .29652 3 0 ] [.47854 .29652 -3 -9 ] [.47854 .29652 3 0 ] [.63011 .29652 -3 -9 ] [.63011 .29652 3 0 ] [.78169 .29652 -3 -9 ] [.78169 .29652 3 0 ] [.93327 .29652 -3 -9 ] [.93327 .29652 3 0 ] [.01131 .01471 -12 -4.5 ] [.01131 .01471 0 4.5 ] [.01131 .16187 -24 -4.5 ] [.01131 .16187 0 4.5 ] [.01131 .45617 -18 -4.5 ] [.01131 .45617 0 4.5 ] 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