(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. 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[ 368635, 12863]*) (*NotebookOutlinePosition[ 369623, 12896]*) (* CellTagsIndexPosition[ 369579, 12892]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[StyleBox["DATA ANALYSIS", FontFamily->"Times New Roman", FontSize->14, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]]], "Title", TextAlignment->Center, FontVariations->{"Underline"->True}], Cell[TextData[{ StyleBox["\n", FontFamily->"Times New Roman", FontWeight->"Bold", FontSlant->"Plain", FontColor->RGBColor[0, 0, 1]], StyleBox["Introduction", FontFamily->"Times New Roman", FontSize->14, FontWeight->"Bold", FontSlant->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}] }], "Subtitle", TextAlignment->Center], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["In this ", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["Notebook", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" you will learn some of the commands needed for data analysis in \ PHYS 375 using ", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["Mathematica", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[". Topics to be covered include:\n\t1. assignments and \ transformation rules,\n\t2. the lists,\n\t3. creating data and/or reading a \ data file from disk,\n\t4. plotting of data values,\n\t5. statistical \ analysis of the data,\n\t6. leat-squares fits to data,\n\t7. nonlinear fits \ to data.\n\t8. propagation of errors\nIt is a good idea to load all the \ packages that you expect to use in a ", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["Mathematica", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" session in the beginning of your notebooks.", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]] }], "Section", PageBreakBelow->False], Cell[BoxData[{ \(\(Off[General::spell, General::spelll];\)\), "\[IndentingNewLine]", \(Needs["\"]\), "\[IndentingNewLine]", \(Needs["\"]\), "\n", \(Needs["\"]\), "\[IndentingNewLine]", \(Needs["\"]\), "\[IndentingNewLine]", \(Needs["\"]\)}], "Input", PageBreakAbove->False, PageBreakBelow->False] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["\n", FontFamily->"Times New Roman", FontSize->24, FontWeight->"Bold", FontSlant->"Plain", FontColor->RGBColor[0, 0, 1]], StyleBox["Assignments and Transformation Rules", FontFamily->"Times New Roman", FontWeight->"Bold", FontSlant->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}] }], "Subtitle", TextAlignment->Center, FontSize->14, FontWeight->"Plain"], Cell[TextData[{ StyleBox["You can assign symbolic or numerical values to variables. To make \ an assignment, enter ", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["variable = value", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[". Names of variables can consist of any number of alphanumeric \ characters. ", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["Mathematica", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" version 3.0 and higher allows the use of Greek letters and \ subscripts as names of variables. By convention, the variables you define \ should begin with lowercase letters because names of built-in ", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["Mathematica", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" variables start with uppercase letters.", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]] }], "Section", Evaluatable->False, PageBreakBelow->False, AspectRatioFixed->True, FontWeight->"Plain"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Assignments in a ", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["Mathematica", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" session remain in effect until the values are changed or the \ variables are removed with the command ", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["Clear", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox["[variable]. Forgetting that certain variables were assigned \ values earlier in a session and proceeding to use them is a common mistake \ and will lead to erroneous results. Some examples are given below.", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]] }], "Section", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Plain"], Cell[CellGroupData[{ Cell[BoxData[ \(t = \((a + b)\)*\((a - b)\) // Expand\)], "Input", CellLabel->"In[7]:="], Cell[BoxData[ \(a\^2 - b\^2\)], "Output", CellLabel->"Out[7]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(t + b\^2\)], "Input", CellLabel->"In[8]:="], Cell[BoxData[ \(a\^2\)], "Output", CellLabel->"Out[8]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(t = c\^2\)], "Input", CellLabel->"In[9]:="], Cell[BoxData[ \(c\^2\)], "Output", CellLabel->"Out[9]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(t + b\^2\)], "Input", CellLabel->"In[10]:="], Cell[BoxData[ \(b\^2 + c\^2\)], "Output", CellLabel->"Out[10]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Let's solve numerically the differential equation (don't worry \ about the syntax for now):\n\t\t\t", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`\(\(d\^2\) x\)\/dt\^2\)], FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox[" + ", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`dx\/dt\)], FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox[" + x = 0\nwith x=1 and dx/dt=0 at t=0:", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]] }], "Section", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Plain"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"NDSolve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["x", "\[DoublePrime]", MultilineFunction->None], "[", "t", "]"}], "+", RowBox[{ SuperscriptBox["x", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], " ", "+", " ", \(x[t]\)}], "\[Equal]", " ", "0"}], ",", " ", RowBox[{ RowBox[{ SuperscriptBox["x", "\[Prime]", MultilineFunction->None], "[", "0", "]"}], " ", "\[Equal]", " ", "0"}], ",", " ", \(x[0]\ \[Equal] \ 1\)}], "}"}], ",", "x", ",", \({t, 0, 5}\)}], "]"}]], "Input", CellLabel->"In[11]:="], Cell[BoxData[ \(General::"ivar" \(\(:\)\(\ \)\) "\!\(c\^2\) is not a valid variable."\)], "Message", CellLabel->"From In[11]:="], Cell[BoxData[ \(NDSolve::"dsvar" \(\(:\)\(\ \)\) "\!\(c\^2\) cannot be used as a variable."\)], "Message", CellLabel->"From In[11]:="], Cell[BoxData[ RowBox[{"NDSolve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{\(x[c\^2]\), "+", RowBox[{ SuperscriptBox["x", "\[Prime]", MultilineFunction->None], "[", \(c\^2\), "]"}], "+", RowBox[{ SuperscriptBox["x", "\[Prime]\[Prime]", MultilineFunction->None], "[", \(c\^2\), "]"}]}], "==", "0"}], ",", RowBox[{ RowBox[{ SuperscriptBox["x", "\[Prime]", MultilineFunction->None], "[", "0", "]"}], "==", "0"}], ",", \(x[0] == 1\)}], "}"}], ",", "x", ",", \({c\^2, 0, 5}\)}], "]"}]], "Output", CellLabel->"Out[11]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["You get an error message because ", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["t", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" was assigned the value", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox[" ", FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ FormBox[ StyleBox[\(c\^2\), FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], TraditionalForm]], FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]], StyleBox[" in the previous example. To solve (or avoid) this problem, let's \ clear ", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["t ", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox["before using it.", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]] }], "Section", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Plain"], Cell[CellGroupData[{ Cell[BoxData[{\(Clear[t];\), "\[IndentingNewLine]", RowBox[{"NDSolve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["x", "\[DoublePrime]", MultilineFunction->None], "[", "t", "]"}], "+", RowBox[{ SuperscriptBox["x", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], " ", "+", " ", \(x[t]\)}], "\[Equal]", " ", "0"}], ",", " ", RowBox[{ RowBox[{ SuperscriptBox["x", "\[Prime]", MultilineFunction->None], "[", "0", "]"}], " ", "\[Equal]", " ", "0"}], ",", " ", \(x[0]\ \[Equal] \ 1\)}], "}"}], ",", "x", ",", \({t, 0, 5}\)}], "]"}]}], "Input", CellLabel->"In[12]:="], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{"x", "\[Rule]", TagBox[\(InterpolatingFunction[{{0.`, 5.`}}, "<>"]\), False, Editable->False]}], "}"}], "}"}]], "Output", CellLabel->"Out[13]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["A simple solution to avoid forgetting earlier assignments is to \ use transformation rules. A transformation rule is written as ", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["variable -> value", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" with no space between \"-\" and \">\". The transformation rule \ is applied to an expression with the replacement operator /. with no space \ between \"/\" and \".\". To make a replacement, write ", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["expr /. variable -> value", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[". For example, ", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["2x + 10 /. x -> 5", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" can be interpreted as ", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["2x + 10", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" given that ", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["x", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" goes to 5. Note that ", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["x", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" remains unassigned.", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]] }], "Section", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Plain"], Cell[CellGroupData[{ Cell[BoxData[ \(2 x\ + \ 10\ /. \ x \[Rule] \ 5\)], "Input", CellLabel->"In[14]:="], Cell[BoxData[ \(20\)], "Output", CellLabel->"Out[14]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(x\)], "Input", CellLabel->"In[15]:="], Cell[BoxData[ \(x\)], "Output", CellLabel->"Out[15]="] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["\n", FontFamily->"Times New Roman", FontSize->24, FontWeight->"Bold", FontSlant->"Plain", FontColor->RGBColor[0, 0, 1]], StyleBox["The Lists", FontFamily->"Times New Roman", FontWeight->"Bold", FontSlant->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}] }], "Subtitle", TextAlignment->Center, FontSize->14, FontWeight->"Plain"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Lists are the fundamental data structure in ", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["Mathematica", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[". Lists can represent vectors, rectangular arrays, matrices, \ tensors, and tables. A ", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["List", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" is an ordered set of objects separated by commas and enclosed in \ curly brackets. A list can contain anything, including other lists. Some \ examples are given below, where x is a vector; m is a matrix; and t is a \ tensor.", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]] }], "Section", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Plain"], Cell[BoxData[{ \(\(x = {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0};\)\), "\[IndentingNewLine]", \(\(m = {{a, b, c}, {d, e, f}};\)\), "\[IndentingNewLine]", \(\(t = {{{0.1, 0.2}, {0.3, 0.4}}, {{0.5, 0.6}, {0.7, 0.8}}, {{0.9, 1.0}, {1.1, 1.2}}, {{1.3, 1.4}, {1.5, 1.6}}};\)\)}], "Input", CellLabel->"In[16]:="] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["A list can always be made by explicitly entering the expression \ for it, as we have done above. However, we often need lists that represent a \ set of equally spaced values of some parameters, or the values of a function. \ Two commands are available in ", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["Mathematica", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" to create lists that meet these needs. The commands are ", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["Range ", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox["and ", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["Table", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[". The starting value, ending value, and step for the indexing are \ given in iterator format.", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]] }], "Section", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Plain"], Cell[CellGroupData[{ Cell[BoxData[{ \(Range[5]\), "\[IndentingNewLine]", \(Range[5, 10]\), "\[IndentingNewLine]", \(Range[1, 5, 0.5]\)}], "Input", CellLabel->"In[19]:="], Cell[BoxData[ \({1, 2, 3, 4, 5}\)], "Output", CellLabel->"Out[19]="], Cell[BoxData[ \({5, 6, 7, 8, 9, 10}\)], "Output", CellLabel->"Out[20]="], Cell[BoxData[ \({1, 1.5`, 2.`, 2.5`, 3.`, 3.5`, 4.`, 4.5`, 5.`}\)], "Output", CellLabel->"Out[21]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[x, y];\)\), "\[IndentingNewLine]", \(Table[x\^i + y\^j, {i, 3}, {j, 4}]\)}], "Input", CellLabel->"In[23]:="], Cell[BoxData[ \({{x + y, x + y\^2, x + y\^3, x + y\^4}, {x\^2 + y, x\^2 + y\^2, x\^2 + y\^3, x\^2 + y\^4}, {x\^3 + y, x\^3 + y\^2, x\^3 + y\^3, x\^3 + y\^4}}\)], "Output", CellLabel->"Out[24]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(N[Table[{n, Sin[n*Pi/180]}, {n, 0, 180, 15}]]\)], "Input", CellLabel->"In[25]:="], Cell[BoxData[ \({{0.`, 0.`}, {15.`, 0.2588190451025207`}, {30.`, 0.5`}, {45.`, 0.7071067811865475`}, {60.`, 0.8660254037844386`}, {75.`, 0.9659258262890682`}, {90.`, 1.`}, {105.`, 0.9659258262890682`}, {120.`, 0.8660254037844386`}, {135.`, 0.7071067811865475`}, {150.`, 0.5`}, {165.`, 0.2588190451025207`}, {180.`, 0.`}}\)], "Output", CellLabel->"Out[25]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Last result can be put in table form by using the command ", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["TableForm", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[". In ", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["Mathematica", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" a percent sign (%) is used to refer to the previous result (%% \ refers to the second previous result, and so on).", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]] }], "Section", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Plain"], Cell[CellGroupData[{ Cell[BoxData[ \(% // TableForm\)], "Input", CellLabel->"In[26]:="], Cell[BoxData[ TagBox[GridBox[{ {"0.`", "0.`"}, {"15.`", "0.2588190451025207`"}, {"30.`", "0.5`"}, {"45.`", "0.7071067811865475`"}, {"60.`", "0.8660254037844386`"}, {"75.`", "0.9659258262890682`"}, {"90.`", "1.`"}, {"105.`", "0.9659258262890682`"}, {"120.`", "0.8660254037844386`"}, {"135.`", "0.7071067811865475`"}, {"150.`", "0.5`"}, {"165.`", "0.2588190451025207`"}, {"180.`", "0.`"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], (TableForm[ #]&)]], "Output", CellLabel->"Out[26]//TableForm="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["The list and sublist elements can be obtained using the ", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["Mathematica", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" commands such as ", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["First", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox["[list], ", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["Last", FontFamily->"Times New Roman", 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The ", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["Mathematica", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" functions ", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["Fit", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" and ", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["Regress", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" enable you to do just that. The built\[Hyphen]in function ", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["Fit", "MR", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" finds a least\[Hyphen]squares fit to the data as a linear \ combination of the specified basis functions. The functions ", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["Regress", "MR", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" augments ", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["Fit", "MR", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" by giving a list of commonly required diagnostics such as the \ coefficient of determination ", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["RSquared", "MR", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[", the analysis of variance table ", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["ANOVATable", "MR", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[", and the mean squared error ", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["EstimatedVariance", "MR", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[". The output of regression functions can be controlled so that \ only needed information is produced. As an example, let's measure the \ temperature and pressure of an ideal gas kept at a constant volume. 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In a nonlinear fit, not only should you provide \ the fit function (or model), you should also provide initial search values \ for the parameters. The function ", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["Timing", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" is used to monitor how long it takes to do the fit. In some \ cases, it is useful to have this information. 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The acceleration can be \ written as\n", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["\ta = (", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`l\^\(\(\ \)\(2\)\)\/\(2 s\)\)], FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox[") (", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`1\/t\_2\^2\)], FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["- ", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`1\/t\_1\^2\)], FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox[").", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["\n", FontSize->12, FontColor->RGBColor[0, 0, 1]], StyleBox["We assume that\n\tl = 5.0 \[PlusMinus] 0.05 cm (1%)\n\ts = 100.0 \ \[PlusMinus] 0.2 cm (0.2%)\n\t", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`t\_1\)], FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["= 0.054 \[PlusMinus] 0.001 sec (2%)\n\t", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`t\_2\)], FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox[" = 0.031 \[PlusMinus] 0.001 sec (3%)\nIn the Notes on Error \ Analysis we calculate", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox[" ", FontSize->12, FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`\[Sigma]\_a\)], FontSize->12, FontColor->RGBColor[0, 0, 1]], StyleBox[" in steps (almost two pages long) and obtaine the result:\n\ta = \ (87.3 \[PlusMinus] 8) cm/", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`sec\^2\)], FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox[".\n", FontSize->12, FontColor->RGBColor[0, 0, 1]], StyleBox["Now let's find the result with the original formula for error \ propagation. 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