(*********************************************************************** 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[ 21797, 879]*) (*NotebookOutlinePosition[ 22670, 909]*) (* CellTagsIndexPosition[ 22626, 905]*) (*WindowFrame->Normal*) Notebook[{ Cell["\<\ 98.03.26. You can make a copy of this Notebook. Contact Davis Cope. Phone: 1-7765 e-mail: cope@plains.nodak.edu\ \>", "Subsubtitle"], Cell["\<\ This screen presentation may differ from your default Notebook presentation \ because the following menu items have been set: (1) MENU: FORMAT/ RULER, TOOLBAR, PAGEBREAKS. (2) MENU: FORMAT/ MAGNIFICATION (at 150%). (3) MENU: CELL / CELL GROUPING / MANUAL GROUPING. \ \>", "Section", CellMargins->{{18, Inherited}, {Inherited, Inherited}}], Cell["\<\ MATHEMATICA files are called \"Notebooks\". *When you entered MATHEMATICA, it automatically opened a Notebook for you. *You can save this Notebook at the end of your session by using \"FILE / SAVE AS\". This allows you to both name the Notebook and save it to your personal disk or directory. NOTE: MATHEMATICA 3.0 Notebooks must always end with the extension \".nb\" . *To save the Notebook without renaming it, just use \"FILE / SAVE\". *You may want to practice saving and recalling your Notebook at this point. \ \>", "Section", CellMargins->{{18, Inherited}, {Inherited, Inherited}}], Cell["\<\ Each Notebook is organized by \"cells\". You can change the format in a cell (such as this one) by (1) SELECTING the cell (click on the edge marker); (2) working with FORMAT in the menu bar above. \ \>", "Section", CellMargins->{{18, Inherited}, {Inherited, Inherited}}], Cell["\<\ The default format for each cell is \"Input\" format, explained below. This \ cell uses \"Section\" format, obtained from FORMAT / STYLE / SECTION\". You may want to practice changing the format of cells. Notice you can check the type of format for a cell by selecting it, then going to FORMAT / STYLE. You can also change alignment using FORMAT / TEXT ALIGNMENT to get \"alignment at left\" or \"centered alignment\". \ \>", "Section", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, TextAlignment->Left], Cell["(1) Grouping cells. ", "Section"], Cell["\<\ (2) To save presentation space, cells can be grouped together.\ \>", "Section"], Cell["\<\ (3) This is done by \"selecting\" the (consecutive) cells (click and move the cursor along their edges), then using CELL / CELL GROUPING / GROUP CELLS.\ \>", "Section"], Cell["\<\ (4) When the cells are grouped, you can close (or open) the group by CELL / CELL GROUPING / OPEN/CLOSE GROUP, and related commands.\ \>", "Section"], Cell["\<\ (5) Practice grouping some cells (such as these 5) and opening and closing the group. \ \>", "Section"], Cell["\<\ (Now, down to business.) (This is a \"title\" cell.) A MATHEMATICA Notebook of my very own. 98.03.26.\ \>", "Title", Editable->False, TextAlignment->Center], Cell["\<\ CONTENTS. 1. Getting started with some arithmetic. 2. Lists. 3. Some 2-dimensional graphing. Options. 4. Some algebra. 5. Some calculus. 6. Some linear algebra. 7. Some (3-dimensional) graphing. 8. Some programming.\ \>", "Subtitle", Editable->False, TextAlignment->Left, TextJustification->0], Cell[CellGroupData[{ Cell["\<\ (This is a \"subtitle\" cell.) 1. Getting started with some arithmetic.\ \>", "Subtitle", Editable->False, TextAlignment->Center, TextJustification->0], Cell["\<\ MATHEMATICA works with INPUT and OUTPUT cells. INPUT format is the default cell type. INPUT format is for calculations and instructions. Use SHIFT+ENTER to enter INPUT. (Use ENTER for new lines within a cell.) \ \>", "Section"], Cell["Some arithmetic, etc. ", "Section", CellMargins->{{17.625, Inherited}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell["\<\ Numbers are either EXACT (no decimal point) or FLOATING POINT (with decimal point). 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Observing convergence to a limit.", "Section"], Cell[BoxData[ \(a = Log[2]\)], "Input"], Cell[BoxData[ \(n = 100\)], "Input"], Cell[BoxData[ \(\((1 + a/n)\)^n\)], "Input"], Cell[BoxData[ \(a = Log[2]; \nn = Range[100, 1000, 100]; \nsequence = N[1 + a/n, 25]^n; \nTableForm[sequence]\)], "Input"] }, Open ]], Cell["\<\ Here are ALL the ways of BRACKETING in MATHEMATICA. For calculations: parentheses. ( ) For function input and indexing variables: square brackets. [ ] For lists: curly brackets. { } For part of a list: double square brackets. [[ ]] \ \>", "Section"] }, Closed]], Cell[CellGroupData[{ Cell["3. Some (2-dimensional) graphing. Options.", "Subtitle", Editable->False, TextAlignment->Center, TextJustification->0], Cell["\<\ We examine \"Plot\" in some detail; related commands are \"ListPlot\" and \"ParametricPlot\".\ \>", "Subsection"], Cell[BoxData[ \(Plot[BesselJ[0, x], {x, 0, 10}]\)], "Input"], Cell["\<\ The plot is the basic graph. To make it look pretty, use the \"Options\" for the command \"Plot\". You can find the \"Options\" (for any command) this way: \ \>", "Subsection"], Cell[BoxData[ \(TableForm[Options[Plot]]\)], "Input"], Cell["\<\ Here is some quick information on some options. Go to HELP for detailed information.\ \>", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?AspectRatio\)\)], "Input"], Cell[BoxData[ \(\(?Frame\)\)], "Input"], Cell[BoxData[ \(\(?PlotRange\)\)], "Input"] }, Open ]], Cell["\<\ We now plot two graphs, one of the Bessel function and one of a known asymptotic approximation for \ large x.\ \>", "Subsection", TextAlignment->Left, TextJustification->0], Cell[BoxData[ \(mrgraph1 = Plot[BesselJ[0, x], {x, 0, 10}, \n\t\tAspectRatio -> Automatic, \n\t\t Frame -> True, \n\t\tPlotRange -> {\(-1\), 1}]\)], "Input"], Cell[BoxData[ \(mrgraph2 = Plot[Sqrt[2/\((Pi*x)\)]*Cos[x - Pi/4], {x, 0, 10}, \n\t\t AspectRatio -> Automatic, \n\t\tFrame -> True, \n\t\t PlotRange -> {\(-1\), 1}]\)], "Input"], Cell["\<\ This command shows both graphs together. (And the asymptotic approximation is quite good!)\ \>", "Subsection"], Cell[BoxData[ \(Show[mrgraph1, mrgraph2]\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["4. Some algebra.", "Subtitle", Editable->False, TextAlignment->Center, TextJustification->0], Cell[CellGroupData[{ Cell["\<\ For symbolic calculations, make sure your variables are CLEAR!\ \>", "Section"], Cell[BoxData[ \(Clear[x, x1, x2, x3, x4, y]\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ In symbolic calculations, nothing is done unless you command it. So, you have to command \"expand\", \"factor\", etc. \ \>", "Section"], Cell[BoxData[ \(z = \((x - y)\)*\((x + y)\)\)], "Input"], Cell[BoxData[ \(z1 = Expand[z]\)], "Input"], Cell["But the original expression \"z\" is unchanged.", "Subsubsection"], Cell[BoxData[ \({z, z1}\)], "Input"], Cell[CellGroupData[{ Cell["We can always recover the factors in an expanded expression.", "Subsubsection"], Cell[BoxData[ \(Factor[z1]\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Some more experimentation.", "Subsubsection"], Cell[BoxData[ \(a = \(+3\)\ x\^5\ y + x\^2\ y\^2 - 2\ x\^3\ y\^2 - 6\ x\^4\ y\^2 - 2\ x\ y\^3 - x\^2\ y\^3 - 6\ x\^3\ y\^3 - x\ y\^4 + 12\ x\^2\ y\^4 + 6\ y\^5 + 3\ x\ y\^5 - 6\ y\^6 + x\^4 + 3\ x\^5 - 2\ x\^3\ y - 5\ x\^4\ y\)], "Input"], Cell[BoxData[ \(a1 = Factor[a]\)], "Input"], Cell[BoxData[ \(Expand[a1]\)], "Input"] }, Closed]], Cell["\<\ In symbolic calculations, nothing is done except to put variables in standard order!\ \>", "Section"] }, Closed]], Cell[CellGroupData[{ Cell["Substitution is done this way: ", "Section"], Cell["We use \"a\" from the previous cell:", "Subsubsection"], Cell[BoxData[ \(a = x\^4 + 3\ x\^5 - 2\ x\^3\ y - 5\ x\^4\ y + 3\ x\^5\ y + x\^2\ y\^2 - 2\ x\^3\ y\^2 - 6\ x\^4\ y\^2 - 2\ x\ y\^3 - x\^2\ y\^3 - 6\ x\^3\ y\^3 - x\ y\^4 + 12\ x\^2\ y\^4 + 6\ y\^5 + 3\ x\ y\^5 - 6\ y\^6\)], "Input"], Cell[BoxData[ \(b = a /. y -> 2\)], "Input"], Cell[BoxData[ \(Factor[b]\)], "Input"], Cell[BoxData[ \(Clear[z, w]; \nb = a /. {x -> z^2, y -> 2*\((w - z)\)^2}\)], "Input"], Cell[BoxData[ \(b1 = Expand[b]\)], "Input"], Cell[BoxData[ \(b2 = Factor[b1]\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ We can the length and the components of an algebraic expression.\ \>", "Section"], Cell[BoxData[ \(Length[b1]\)], "Input"], Cell[BoxData[ \(Part[b1, 12]\)], "Input"], Cell[BoxData[ \(Length[b2]\)], "Input"], Cell[BoxData[ \(Part[b2, 3]\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Solving equations.", "Section"], Cell[BoxData[ \(Clear[a, b, c, d, x, y, z]\)], "Input"], Cell[CellGroupData[{ Cell["Example.", "Subsection"], Cell[BoxData[ \(eqnList = {2*x + 3*y + 4*z == 15, \n \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3*x + 4*y - 5*z == 2, \n\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4*x - 5*y + 7*z == 23}\)], "Input"], Cell[BoxData[ \(solution = Solve[eqnList, {x, y, z}]\)], "Input"], Cell[BoxData[ \(solution = Flatten[solution]\)], "Input"], Cell[BoxData[ \({s1, s2, s3} = {x, y, z} /. solution\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Example upgraded.", "Subsection"], Cell[BoxData[ \(eqnList = {2*x + 3*y + d*z == a, \n \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3*x + d*y - 5*z == b, \n\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ d*x - 5*y + 7*z == c}\)], "Input"], Cell[BoxData[ \(solution = Solve[eqnList, {x, y, z}]\)], "Input"], Cell[BoxData[ \({s1, s2, s3} = {x, y, z} /. 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Advantages: (1) by using symbolic calculations, we can directly observe that the \ procedure is working correctly; (2) we just combine the statements (tinkering with a few parameters) to \ generate a construction for the Lagrange interpolation polynomials at an \ arbitrary number of points. \ \>", "Subsubsection"], Cell[BoxData[ \(Clear[a, x]\)], "Input"], Cell[BoxData[ \(aList = Array[a, 3, 0]\)], "Input"], Cell[BoxData[ \(xList = x - aList\)], "Input"], Cell[BoxData[ \(Apply[Times, xList]\)], "Input"], Cell[BoxData[ \(xList = Apply[Times, xList]/xList\)], "Input"], Cell[BoxData[ \(cList = Table[xList[\([k]\)] /. x -> a[k - 1], {k, 3}]\)], "Input"], Cell[BoxData[ \(LagrangeList = xList/cList\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Put it all together for a general solution. 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