(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.1' 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). 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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[ 147097, 4133]*) (*NotebookOutlinePosition[ 147813, 4158]*) (* CellTagsIndexPosition[ 147769, 4154]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " Lesson II" }], "Title"], Cell["Physics 80301 Fall 2005", "Subtitle"], Cell[CellGroupData[{ Cell["Solving Algebraic Equations", "Section"], Cell[CellGroupData[{ Cell["Single Nonlinear Equation", "Subsection"], Cell[TextData[{ "Let us consider a nonlinear equation such as ", Cell[BoxData[ \(TraditionalForm\`\@\(4 - x\^2\)\)]], "= tan(x) that occurs in the quantum mechanical square-well problem. 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As we see below, the equation is solved to parts in \ ", Cell[BoxData[ \(TraditionalForm\`10\^16\)]], ".\n" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\@\(4 - x\^2\) - Tan[x] /. \ %\)], "Input", CellLabel->"In[3]:="], Cell[BoxData[ \(\(-2.220446049250313`*^-16\)\)], "Output", CellLabel->"Out[3]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Several Polynomial Equations", "Subsection"], Cell[TextData[{ "As an example consider the pair of equations ", Cell[BoxData[ \(TraditionalForm\`x\^2\)]], "+", Cell[BoxData[ \(TraditionalForm\`y\^2\)]], "=4 and x+y=1. These equations can be solved as follows:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Solve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ FormBox[\(x\^2\), "TraditionalForm"], "+", FormBox[\(y\^2\), "TraditionalForm"]}], "==", "4"}], ",", " ", \(x + y == 1\)}], "}"}], ",", \({x, y}\)}], "]"}]], "Input", CellLabel->"In[4]:="], Cell[BoxData[ \({{x \[Rule] 1\/2\ \((1 - \@7)\), y \[Rule] 1\/2\ \((1 + \@7)\)}, {x \[Rule] 1\/2\ \((1 + \@7)\), y \[Rule] 1\/2\ \((1 - \@7)\)}}\)], "Output", CellLabel->"Out[4]="] }, Open ]], Cell["\<\ Note the use of == above to designate equality. To check the solution, we substitute back into the equations\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ FormBox[\(x\^2\), "TraditionalForm"], "+", FormBox[\(y\^2\), "TraditionalForm"]}], ",", \(x + y\)}], "}"}], "/.", " ", "%"}], "]"}]], "Input", CellLabel->"In[5]:="], Cell[BoxData[ \({{4, 1}, {4, 1}}\)], "Output", CellLabel->"Out[5]="] }, Open ]], Cell["\<\ To get a numerical solution, use NSolve[] instead of Solve[]:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"NSolve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ FormBox[\(x\^2\), "TraditionalForm"], "+", FormBox[\(y\^2\), "TraditionalForm"]}], "==", "4"}], ",", " ", \(x + y == 1\)}], "}"}], ",", \({x, y}\)}], "]"}]], "Input", CellLabel->"In[6]:="], Cell[BoxData[ \({{x \[Rule] \(-0.8228756555322949`\), y \[Rule] 1.822875655532295`}, {x \[Rule] 1.8228756555322954`, y \[Rule] \(-0.8228756555322954`\)}}\)], "Output", CellLabel->"Out[6]="] }, Open ]], Cell["Check the solutions:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ FormBox[\(x\^2\), "TraditionalForm"], "+", FormBox[\(y\^2\), "TraditionalForm"]}], ",", \(x + y\)}], "}"}], "/.", " ", "%"}], "]"}]], "Input", CellLabel->"In[7]:="], Cell[BoxData[ \({{3.9999999999999982`, 1.`}, {4.`, 1.`}}\)], "Output", CellLabel->"Out[7]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Several Linear Equations", "Subsection"], Cell["\<\ Use matrix algebra! 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