(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 479469, 12802] NotebookOptionsPosition[ 444319, 11940] NotebookOutlinePosition[ 459981, 12299] CellTagsIndexPosition[ 459209, 12277] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell["Math 421 \[FilledSmallCircle] Fall 2010", "Subsubtitle", CellChangeTimes->{{3.4910584115*^9, 3.4910584118125*^9}}, TextAlignment->Center], Cell[CellGroupData[{ Cell["Visualizing complex functions", "Subtitle", ShowCellTags->False, TextAlignment->Center, TextJustification->0], Cell["25 September 2010", "Subsubtitle", CellChangeTimes->{{3.48893028171875*^9, 3.48893030315625*^9}, { 3.489583451984375*^9, 3.489583455796875*^9}, 3.49028881771875*^9, { 3.4905515529375*^9, 3.490551553125*^9}, 3.490875430171875*^9, { 3.492293113328125*^9, 3.492293120375*^9}, 3.49346163509375*^9, { 3.49410698178125*^9, 3.494106982265625*^9}, 3.494416320359375*^9}, TextAlignment->Center, TextJustification->0], Cell["\<\ Copyright \[Copyright] 2004\[Dash]2010 by Murray Eisenberg. All rights \ reserved.\ \>", "SmallText", ShowCellTags->False, CellChangeTimes->{{3.4910584066875*^9, 3.49105843275*^9}}, TextAlignment->Center, TextJustification->0], Cell[CellGroupData[{ Cell["Introduction", "Section", CellChangeTimes->{{3.493499537953125*^9, 3.493499540421875*^9}}], Cell[TextData[{ "This notebook shows how to use David Park's ", StyleBox["Presentations", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], " add-on application to visualize functions ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", ":", "\[DoubleStruckCapitalC]"}], "\[Rule]", "\[DoubleStruckCapitalC]"}], TraditionalForm]]], ". The ", ButtonBox["appendix", BaseStyle->"Hyperlink", ButtonData:>"appendix 1"], " describes how to use, instead, the more limited capabilities provided by \ the built-in ", StyleBox["Mathematica", FontSlant->"Italic"], " function ", StyleBox["ParametricPlot", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "." }], "Text", ShowCellTags->False, CellChangeTimes->{{3.491058451234375*^9, 3.4910584870625*^9}, 3.4910585549375*^9, {3.49279786209375*^9, 3.492797896046875*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["Prerequisites", "Section", CellChangeTimes->{{3.466616366328125*^9, 3.46661637828125*^9}, { 3.466616604453125*^9, 3.466616607390625*^9}}], Cell[CellGroupData[{ Cell[TextData[StyleBox["Mathematica", FontSlant->"Italic"]], "Subsection", CellChangeTimes->{{3.466616614109375*^9, 3.466616621015625*^9}}], Cell[TextData[{ "Most of this notebook requires David Park's ", StyleBox["Mathematica", FontSlant->"Italic"], " add-on application ", StyleBox["Presentations", FontSlant->"Italic"], "." }], "Text", CellChangeTimes->{{3.46661644090625*^9, 3.466616483484375*^9}, { 3.466616542578125*^9, 3.466616591375*^9}, {3.4666167129375*^9, 3.466616714296875*^9}, {3.490983043921875*^9, 3.490983044875*^9}, { 3.490984276765625*^9, 3.4909843116875*^9}, 3.4910591349375*^9}], Cell[TextData[{ StyleBox["Presentations", FontSlant->"Italic"], " should be loaded by evaluating the expression:\n\t", StyleBox["<<", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["Presentations", FontFamily->"Courier", FontWeight->"Plain"], StyleBox["`", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "\nThat initialization is done below, in the ", ButtonBox["Initialization section", BaseStyle->"Hyperlink", ButtonData->"initialization"], ", below." }], "Text", CellChangeTimes->{{3.46661644090625*^9, 3.466616483484375*^9}, { 3.466616542578125*^9, 3.466616591375*^9}, {3.4666167129375*^9, 3.46661679353125*^9}, {3.4666168744375*^9, 3.466616901171875*^9}, { 3.4909830825*^9, 3.4909830910625*^9}, {3.490984498375*^9, 3.49098451746875*^9}, {3.491058732328125*^9, 3.491058740734375*^9}, 3.491058892734375*^9, {3.491058930921875*^9, 3.491059018609375*^9}, { 3.491059057390625*^9, 3.491059075953125*^9}, {3.49105918140625*^9, 3.49105918140625*^9}, {3.491059247640625*^9, 3.49105924765625*^9}, { 3.491059412859375*^9, 3.491059447375*^9}, {3.491059659734375*^9, 3.491059692453125*^9}, {3.49105976584375*^9, 3.491059767484375*^9}}, ParagraphSpacing->{0.5, 0.}], Cell[TextData[{ "You should already know about the ", StyleBox["Presentations", FontSlant->"Italic"], " functions ", StyleBox["ComplexPoint", FontFamily->"Courier"], StyleBox[", ", FontFamily->"Times"], StyleBox["ComplexCurve", FontFamily->"Courier"], StyleBox[", ", FontFamily->"Times"], StyleBox["ComplexLine", FontFamily->"Courier"], StyleBox[", ", FontFamily->"Times"], StyleBox["ComplexCircle", FontFamily->"Courier"], StyleBox[", etc., that form complex objects as well as the ", FontFamily->"Times"], StyleBox["Presentations", FontFamily->"Times", FontSlant->"Italic"], StyleBox[" function ", FontFamily->"Times"], StyleBox["Draw2D", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], StyleBox[". See notebooks ", FontFamily->"Times"], StyleBox["nthRoots.nb", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], StyleBox[" and ", FontFamily->"Times"], StyleBox["DrawingComplexObjects.nb", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], StyleBox[".", FontFamily->"Times"] }], "Text", CellChangeTimes->{{3.4910597709375*^9, 3.4910599115*^9}, { 3.49247129434375*^9, 3.492471300390625*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["Mathematics", "Subsection", CellChangeTimes->{{3.46661662978125*^9, 3.466616631109375*^9}}], Cell[TextData[{ "You should already know the algebra of complex numbers and polar \ representation of complex numbers as well as the idea of a function ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", ":", "\[DoubleStruckCapitalC]"}], "\[Rule]", "\[DoubleStruckCapitalC]"}], TraditionalForm]], FormatType->"TraditionalForm"], ", that is, a complex-valued function of a complex variable." }], "Text", CellChangeTimes->{{3.466616634859375*^9, 3.46661670978125*^9}, { 3.49105958428125*^9, 3.4910595870625*^9}, {3.49105962053125*^9, 3.491059636984375*^9}, {3.4910599269375*^9, 3.4910599291875*^9}, { 3.4924712228125*^9, 3.492471275453125*^9}}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Initialization", "Section", ShowGroupOpener->True, CellChangeTimes->{{3.4906339376875*^9, 3.49063394653125*^9}, { 3.490638317984375*^9, 3.49063833975*^9}, {3.490735161125*^9, 3.490735165125*^9}, 3.490984367984375*^9}, CellTags->"initialization"], Cell[TextData[{ "When you opened this notebook, it should have prompted you whether you want \ to evaluate Initialization Cells. You should have answered \ \[OpenCurlyDoubleQuote]yes.\[CloseCurlyDoubleQuote]\nIf you did not, then ", StyleBox["evaluate the following Input cell now", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], ". 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A couple of other examples are treated more briefly below." }], "Text", ShowCellTags->False, CellChangeTimes->{{3.49105857428125*^9, 3.491058577375*^9}}], Cell["\<\ Be sure to evaluate the following input cell before proceeding:\ \>", "Text", ShowCellTags->False], Cell[BoxData[ RowBox[{ RowBox[{"f", "[", "z_", "]"}], ":=", SuperscriptBox["z", "2"]}]], "Input", ShowCellTags->False, CellChangeTimes->{{3.491058565625*^9, 3.49105856721875*^9}}, CellTags->"define f"] }, Closed]], Cell[CellGroupData[{ Cell["Plotting real and imaginary parts of a function", "Section", ShowGroupOpener->True, ShowCellTags->False, CellChangeTimes->{{3.49247137328125*^9, 3.492471380671875*^9}, { 3.492607134515625*^9, 3.49260714234375*^9}}], Cell[TextData[{ "The objective is to visualize the composite functions ", Cell[BoxData[ FormBox[ RowBox[{"Re", " ", "\[SmallCircle]", "f"}], TraditionalForm]], FormatType->"TraditionalForm"], " and ", Cell[BoxData[ FormBox[ RowBox[{"Im", " ", "\[SmallCircle]", "f"}], TraditionalForm]], FormatType->"TraditionalForm"], "\[LongDash]the ", StyleBox["real part", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " and the ", StyleBox["imaginary part", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], ", respectively, of the complex function ", Cell[BoxData[ FormBox["f", TraditionalForm]], FormatType->"TraditionalForm"], ". Each is a function from the complex plane ", Cell[BoxData[ FormBox["\[DoubleStruckCapitalC]", TraditionalForm]], FormatType->"TraditionalForm"], " to the real line ", Cell[BoxData[ FormBox["\[DoubleStruckCapitalR]", TraditionalForm]], FormatType->"TraditionalForm"], ". If we regard ", Cell[BoxData[ FormBox["\[DoubleStruckCapitalC]", TraditionalForm]], FormatType->"TraditionalForm"], " as the Cartesian plane ", Cell[BoxData[ FormBox[ SuperscriptBox["\[DoubleStruckCapitalR]", "2"], TraditionalForm]], FormatType->"TraditionalForm"], ", then the graph of each is a surface in ", Cell[BoxData[ FormBox[ SuperscriptBox["\[DoubleStruckCapitalR]", "3"], TraditionalForm]], FormatType->"TraditionalForm"], "." }], "Text", CellChangeTimes->{{3.49260716709375*^9, 3.492607182828125*^9}, { 3.492616773515625*^9, 3.49261679475*^9}, {3.492616831796875*^9, 3.492617046109375*^9}}], Cell[CellGroupData[{ Cell[TextData[{ "Using Cartesian coordinates to plot real and imaginary parts of a function \ with built-in ", StyleBox["Mathematica", FontSlant->"Italic"], " functions" }], "Subsection", CellChangeTimes->{{3.492607242109375*^9, 3.492607262796875*^9}, { 3.4926075283125*^9, 3.49260754809375*^9}, {3.492616467921875*^9, 3.49261647075*^9}}], Cell[TextData[{ "In this section we use only built-in ", StyleBox["Mathematica", FontSlant->"Italic"], " functions, and so we have to resort directly to Cartesian coordinates. " }], "Text", CellChangeTimes->{{3.492607586359375*^9, 3.49260761921875*^9}}], Cell[CellGroupData[{ Cell["\<\ Finding the real and imaginary parts of a function in terms of Cartesian \ coordinates\ \>", "Subsubsection", CellChangeTimes->{{3.492607300546875*^9, 3.492607317453125*^9}, 3.4944412561875*^9}], Cell[TextData[{ "Here is the formula for the function ", Cell[BoxData[ FormBox["f", TraditionalForm]]], " in terms of the real part ", Cell[BoxData[ FormBox["x", TraditionalForm]]], " and the imaginary part ", Cell[BoxData[ FormBox["y", TraditionalForm]]], " of an input ", Cell[BoxData[ FormBox[ RowBox[{"z", " ", "=", " ", RowBox[{"x", " ", "+", " ", RowBox[{"\[ImaginaryI]", " ", "y"}]}]}], TraditionalForm]]], "\[Ellipsis]" }], "Text", ShowCellTags->False], Cell[BoxData[ RowBox[{"f", "[", RowBox[{"x", "+", RowBox[{"\[ImaginaryI]", " ", "y"}]}], "]"}]], "Input", ShowCellTags->False, CellChangeTimes->{{3.491059938453125*^9, 3.491059939578125*^9}}], Cell["\<\ \[Ellipsis]and here is that formula in Cartesian form after the squaring is \ done:\ \>", "Text", ShowCellTags->False, CellChangeTimes->{{3.4910599746875*^9, 3.491059978796875*^9}, { 3.492471390078125*^9, 3.49247139215625*^9}, 3.492520122515625*^9}], Cell[BoxData[ RowBox[{"Expand", "[", RowBox[{"f", "[", RowBox[{"x", "+", RowBox[{"\[ImaginaryI]", " ", "y"}]}], "]"}], "]"}]], "Input", CellChangeTimes->{ 3.491059950515625*^9, {3.491059982328125*^9, 3.49106002703125*^9}, { 3.492471402578125*^9, 3.492471414828125*^9}}], Cell[TextData[{ "(", StyleBox["ComplexExpand", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " will give you the same thing there as ", StyleBox["Expand", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ".)" }], "Text", CellChangeTimes->{{3.492471430296875*^9, 3.49247145034375*^9}}], Cell[TextData[{ "We humans cannot directly visualize the graph\n\t", Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{"z", ",", RowBox[{ RowBox[{"f", "(", "z", ")"}], ":", RowBox[{"z", "\[Element]", "\[DoubleStruckCapitalC]"}]}]}], "}"}], TraditionalForm]]], "\nof ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", ":", "\[DoubleStruckCapitalC]"}], "\[Rule]", "\[DoubleStruckCapitalC]"}], TraditionalForm]]], " directly. Why not? You shall see in a moment." }], "Text", ShowCellTags->False, CellChangeTimes->{{3.491060047484375*^9, 3.491060073203125*^9}, { 3.492471467859375*^9, 3.492471479125*^9}}, ParagraphSpacing->{0.5, 0}], Cell[TextData[{ "Form the \"real part\" ", Cell[BoxData[ FormBox["u", TraditionalForm]]], " and \"imaginary part\" ", Cell[BoxData[ FormBox["v", TraditionalForm]]], " of the function ", Cell[BoxData[ FormBox["f", TraditionalForm]]], ". These are the functions ", Cell[BoxData[ FormBox[ RowBox[{"u", "(", RowBox[{"x", ",", "y"}], ")"}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{"v", "(", RowBox[{"x", ",", "y"}], ")"}], TraditionalForm]]], " for which\n \t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", RowBox[{"x", "+", RowBox[{"\[ImaginaryI]", " ", "y"}]}], ")"}], "=", RowBox[{ RowBox[{"u", "(", RowBox[{"x", ",", "y"}], ")"}], "+", RowBox[{"\[ImaginaryI]", " ", RowBox[{"v", "(", RowBox[{"x", ",", "y"}], ")"}]}]}]}], TraditionalForm]]], "\n when ", Cell[BoxData[ FormBox["x", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["y", TraditionalForm]]], " are real. In ", StyleBox["Mathematica:", FontSlant->"Italic"] }], "Text", ShowCellTags->False, CellChangeTimes->{{3.491060078734375*^9, 3.491060106171875*^9}, 3.492471503109375*^9, {3.494501539046875*^9, 3.494501539046875*^9}}, ParagraphSpacing->{0.5, 0}], Cell[BoxData[{ RowBox[{ RowBox[{"u", "[", RowBox[{"x_", ",", "y_"}], "]"}], " ", ":=", " ", RowBox[{"ComplexExpand", "@", RowBox[{"Re", "[", RowBox[{"f", "[", RowBox[{"x", "+", RowBox[{"\[ImaginaryI]", " ", "y"}]}], "]"}], "]"}]}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"v", "[", RowBox[{"x_", ",", "y_"}], "]"}], " ", ":=", RowBox[{"ComplexExpand", "@", RowBox[{"Im", "[", RowBox[{"f", "[", RowBox[{"x", "+", RowBox[{"\[ImaginaryI]", " ", "y"}]}], "]"}], "]"}]}]}]}], "Input", ShowCellTags->False, CellChangeTimes->{{3.49106011096875*^9, 3.49106017865625*^9}, 3.49106062825*^9, {3.4924715149375*^9, 3.49247151640625*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"u", "[", RowBox[{"x", ",", "y"}], "]"}], ",", RowBox[{"v", "[", RowBox[{"x", ",", "y"}], "]"}]}], "}"}]], "Input", CellChangeTimes->{{3.491060137453125*^9, 3.491060139546875*^9}, { 3.49106018453125*^9, 3.491060193203125*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Check that those are really the real and imaginary parts of ", Cell[BoxData[ FormBox["f", TraditionalForm]], FormatType->"TraditionalForm"], " by calculating ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"u", "(", RowBox[{"x", ",", "y"}], ")"}], "+", RowBox[{"\[ImaginaryI]", " ", RowBox[{"v", "(", RowBox[{"x", ",", "y"}], ")"}]}]}], TraditionalForm]], FormatType->"TraditionalForm"], " by hand and then with ", StyleBox["Mathematica", FontSlant->"Italic"], "." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.492471546515625*^9, 3.4924716679375*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Find the real and imaginary parts of ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"g", "(", "z", ")"}], "=", SuperscriptBox["z", "3"]}], TraditionalForm]], FormatType->"TraditionalForm"], " both by hand and with ", StyleBox["Mathematica", FontSlant->"Italic"], "." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.492471546515625*^9, 3.4924716679375*^9}, {3.492473554546875*^9, 3.492473581890625*^9}}], Cell[TextData[{ "Strictly in terms of real and imaginary parts, think of ", Cell[BoxData[ FormBox["f", TraditionalForm]]], " as the following function ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"F", ":", SuperscriptBox["\[DoubleStruckCapitalR]", "2"]}], "\[Rule]", SuperscriptBox["\[DoubleStruckCapitalR]", "2"]}], TraditionalForm]], FormatType->"TraditionalForm"], " from the plane to itself:" }], "Text", ShowCellTags->False, CellChangeTimes->{{3.49247169515625*^9, 3.492471752203125*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"F", "[", RowBox[{"x_", ",", "y_"}], "]"}], ":=", RowBox[{"{", RowBox[{ RowBox[{"u", "[", RowBox[{"x", ",", "y"}], "]"}], ",", RowBox[{"v", "[", RowBox[{"x", ",", "y"}], "]"}]}], "}"}]}]], "Input", ShowCellTags->False, CellChangeTimes->{{3.49106026234375*^9, 3.491060272609375*^9}}], Cell[BoxData[ RowBox[{"F", "[", RowBox[{"x", ",", "y"}], "]"}]], "Input", ShowCellTags->False, CellChangeTimes->{3.49106027665625*^9}], Cell[TextData[{ "Since ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"F", ":", SuperscriptBox["\[DoubleStruckCapitalR]", "2"]}], "\[Rule]", SuperscriptBox["\[DoubleStruckCapitalR]", "2"]}], TraditionalForm]]], ", then:\n\t", Cell[BoxData[{ FormBox[ RowBox[{ RowBox[{ RowBox[{"graph", "(", "F", ")"}], " ", "=", RowBox[{"{", " ", RowBox[{ RowBox[{"(", " ", RowBox[{ RowBox[{"(", RowBox[{"x", ",", "y"}], ")"}], ",", " ", RowBox[{"F", "(", RowBox[{"x", ",", "y"}], ")"}]}], " ", ")"}], " ", ":", " ", RowBox[{ RowBox[{"(", RowBox[{"x", ",", "y"}], ")"}], "\[Element]", "\[DoubleStruckCapitalC]"}]}], " ", "}"}]}], " "}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{" ", RowBox[{"=", " ", RowBox[{ RowBox[{"{", " ", RowBox[{ RowBox[{"(", " ", RowBox[{ RowBox[{"(", RowBox[{"x", ",", "y"}], "}"}], ",", " ", RowBox[{"(", " ", RowBox[{ RowBox[{"u", "(", RowBox[{"x", ",", "y"}], ")"}], ",", RowBox[{"v", "(", RowBox[{"x", ",", "y"}], ")"}]}], " ", ")"}]}], " ", ")"}], " ", ":", " ", RowBox[{ RowBox[{"(", RowBox[{"x", ",", "y"}], ")"}], " ", "\[Element]", " ", "\[DoubleStruckCapitalC]"}]}], " ", "}"}], " ", "\[Subset]", " ", SuperscriptBox["\[DoubleStruckCapitalR]", "4"]}]}]}], TraditionalForm]}]], " \nThus the graph of ", Cell[BoxData[ FormBox["F", TraditionalForm]], FormatType->"TraditionalForm"], " is a subset of 4-dimensional space. That's why you cannot directly \ visualize it." }], "Text", ShowCellTags->False, CellChangeTimes->{{3.49106029159375*^9, 3.491060364734375*^9}, { 3.491060447546875*^9, 3.491060568390625*^9}, {3.492471755921875*^9, 3.49247183078125*^9}}, ParagraphSpacing->{0.5, 0}], Cell[TextData[{ "However, you can directly visualize the graphs of\n\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"u", ":", SuperscriptBox["\[DoubleStruckCapitalR]", "2"]}], "\[Rule]", "\[DoubleStruckCapitalR]"}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"v", ":", SuperscriptBox["\[DoubleStruckCapitalR]", "2"]}], "\[Rule]", "\[DoubleStruckCapitalR]"}], TraditionalForm]]], ",\nbecause these are subsets of 3-dimensional space ", Cell[BoxData[ FormBox[ SuperscriptBox["\[DoubleStruckCapitalR]", "3"], TraditionalForm]], FormatType->"TraditionalForm"], " given by:\n\t ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"graph", "(", "u", ")"}], "=", RowBox[{ RowBox[{"{", " ", RowBox[{ RowBox[{"(", RowBox[{"x", ",", "y", ",", RowBox[{"u", "(", RowBox[{"x", ",", "y"}], ")"}]}], " ", ")"}], " ", ":", " ", RowBox[{ RowBox[{"(", RowBox[{"x", ",", "y"}], ")"}], "\[Element]", SuperscriptBox["\[DoubleStruckCapitalR]", "2"]}]}], " ", "}"}], "=", " ", RowBox[{ RowBox[{"{", " ", RowBox[{ RowBox[{"(", RowBox[{"x", ",", "y", ",", RowBox[{ SuperscriptBox["x", "2"], "-", SuperscriptBox["y", "2"]}]}], " ", ")"}], " ", ":", " ", RowBox[{ RowBox[{"(", RowBox[{"x", ",", "y"}], ")"}], "\[Element]", SuperscriptBox["\[DoubleStruckCapitalR]", "2"]}]}], " ", "}"}], " ", "\[Subset]", " ", SuperscriptBox["\[DoubleStruckCapitalR]", "3"]}]}]}], TraditionalForm]]], ",\n\t ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"graph", "(", "v", ")"}], "=", RowBox[{ RowBox[{"{", " ", RowBox[{ RowBox[{"(", RowBox[{"x", ",", "y", ",", RowBox[{"v", "(", RowBox[{"x", ",", "y"}], ")"}]}], " ", ")"}], " ", ":", " ", RowBox[{ RowBox[{"(", RowBox[{"x", ",", "y"}], ")"}], "\[Element]", SuperscriptBox["\[DoubleStruckCapitalR]", "2"]}]}], " ", "}"}], "=", " ", RowBox[{ RowBox[{"{", " ", RowBox[{ RowBox[{"(", RowBox[{"x", ",", "y", ",", RowBox[{"2", "x", " ", "y"}]}], " ", ")"}], " ", ":", " ", RowBox[{ RowBox[{"(", RowBox[{"x", ",", "y"}], ")"}], "\[Element]", SuperscriptBox["\[DoubleStruckCapitalR]", "2"]}]}], " ", "}"}], "\[Subset]", " ", SuperscriptBox["\[DoubleStruckCapitalR]", "3"]}]}]}], TraditionalForm]]] }], "Text", ShowCellTags->False, CellChangeTimes->{{3.49106029159375*^9, 3.491060364734375*^9}, { 3.491060447546875*^9, 3.491060568390625*^9}, {3.492471755921875*^9, 3.492471878140625*^9}, {3.49247287778125*^9, 3.492472882484375*^9}}, ParagraphSpacing->{0.5, 0}], Cell["These graphs will be drawn below.", "Text", ShowCellTags->False, CellChangeTimes->{{3.4910605763125*^9, 3.491060592*^9}, { 3.492472889515625*^9, 3.49247292165625*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Visualizing the real and imaginary parts of a function in terms of Cartesian \ coordinates\ \>", "Subsubsection", ShowGroupOpener->True, ShowCellTags->False, CellChangeTimes->{{3.4910606560625*^9, 3.491060657953125*^9}, { 3.49260732653125*^9, 3.49260737703125*^9}, {3.49260749271875*^9, 3.492607522375*^9}}, CellTags->"Plot3DuAndv"], Cell[TextData[{ "Here is the graphs of the function ", Cell[BoxData[ FormBox[ RowBox[{"u", "(", RowBox[{"x", ",", "y"}], ")"}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{"v", "(", RowBox[{"x", ",", "y"}], ")"}], TraditionalForm]]], " in ", Cell[BoxData[ FormBox[ SuperscriptBox["\[DoubleStruckCapitalR]", "3"], TraditionalForm]]], ", drawn using the built-in ", StyleBox["Mathematica", FontSlant->"Italic"], " function ", StyleBox["Plot3D", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], ". (The option ", StyleBox["BoxRatios", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], " is used so that the scale on the vertical axis appear to be the same as \ that on the ", Cell[BoxData[ FormBox["x", TraditionalForm]]], "- and ", Cell[BoxData[ FormBox["y", TraditionalForm]]], "-axes)." }], "Text", ShowCellTags->False, CellChangeTimes->{{3.491060659671875*^9, 3.49106068096875*^9}, { 3.491061614890625*^9, 3.491061638109375*^9}, 3.494441266390625*^9}], Cell[BoxData[ RowBox[{"uVSxy", "=", RowBox[{"Plot3D", "[", RowBox[{ RowBox[{"u", "[", RowBox[{"x", ",", "y"}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"BoxRatios", "\[Rule]", RowBox[{"{", RowBox[{"1", ",", "1", ",", "0.8`"}], "}"}]}]}], "]"}]}]], "Input", ShowCellTags->False, CellChangeTimes->{{3.49106068965625*^9, 3.491060696875*^9}, { 3.49106080709375*^9, 3.491060920015625*^9}, 3.491061468828125*^9, { 3.49106151440625*^9, 3.49106151909375*^9}}], Cell[BoxData[ RowBox[{"vVSxy", "=", RowBox[{"Plot3D", "[", RowBox[{ RowBox[{"v", "[", RowBox[{"x", ",", "y"}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"BoxRatios", "\[Rule]", RowBox[{"{", RowBox[{"1", ",", "1", ",", "0.8`"}], "}"}]}]}], "]"}]}]], "Input", ShowCellTags->False, CellChangeTimes->{{3.491060929296875*^9, 3.49106093609375*^9}, { 3.491061445171875*^9, 3.491061457828125*^9}, 3.4910615748125*^9, { 3.491061682359375*^9, 3.491061689765625*^9}}], Cell[TextData[{ "Although it\[CloseCurlyQuote]s not obvious, the ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"x", ",", "y"}], ")"}], TraditionalForm]]], " plane is parallel to the bottom of the box, and the ", Cell[BoxData[ FormBox["u", TraditionalForm]]], "- and ", Cell[BoxData[ FormBox["v", TraditionalForm]]], "-axes, respectively, run perpendicular to the bottom. Labeling the axes \ will help. So here are the same things but in each case with labels for the \ axes, as well as a label for the plot:" }], "Text", CellChangeTimes->{{3.49106147803125*^9, 3.49106150371875*^9}, { 3.4910617005*^9, 3.49106170421875*^9}, {3.491061790546875*^9, 3.49106191465625*^9}, {3.49247297871875*^9, 3.49247297921875*^9}}], Cell[BoxData[ RowBox[{"uVSxy", "=", RowBox[{"Plot3D", "[", RowBox[{ RowBox[{"u", "[", RowBox[{"x", ",", "y"}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"BoxRatios", "\[Rule]", RowBox[{"{", RowBox[{"1", ",", "1", ",", "0.8`"}], "}"}]}], ",", "\[IndentingNewLine]", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"TraditionalForm", "/@", RowBox[{"{", RowBox[{"x", ",", "y", ",", "u"}], "}"}]}]}], ",", RowBox[{"PlotLabel", "\[Rule]", RowBox[{"TraditionalForm", "[", RowBox[{"Re", "[", "f", "]"}], "]"}]}], ",", "\[IndentingNewLine]", RowBox[{"ImageSize", "\[Rule]", RowBox[{"3", " ", "72"}]}]}], "]"}]}]], "Input", ShowCellTags->False, CellChangeTimes->{{3.49106068965625*^9, 3.491060696875*^9}, { 3.49106080709375*^9, 3.491060920015625*^9}, 3.491061468828125*^9, { 3.4910615288125*^9, 3.491061589734375*^9}, {3.491061725515625*^9, 3.491061726296875*^9}, {3.491061932015625*^9, 3.4910619391875*^9}, { 3.49247308078125*^9, 3.49247308709375*^9}, 3.492606401234375*^9}], Cell["\<\ You can rotate such a 3D graph by moving the cursor over it and then, while \ holding down the mouse button, moving the mouse around.\ \>", "SmallText", CellChangeTimes->{{3.49106207359375*^9, 3.49106215234375*^9}, { 3.4924730143125*^9, 3.492473024703125*^9}}], Cell[TextData[{ "The value for the option ", StyleBox["ImageSize", FontFamily->"Courier"], " is specified in printer's points. 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You may still rotate each of the surfaces in the usual way.\ \>", "SmallText", ShowCellTags->False, CellChangeTimes->{{3.491061007734375*^9, 3.49106100875*^9}, 3.491061991859375*^9, {3.491062197265625*^9, 3.491062199609375*^9}, { 3.492518773953125*^9, 3.492518833484375*^9}, {3.494180303265625*^9, 3.494180358921875*^9}, 3.494441269984375*^9}, ParagraphSpacing->{0.5, 0}], Cell[TextData[{ "In the list argument of ", StyleBox["Row", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " above, the middle item, ", StyleBox["Spacer[20]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " inserts extra space; the argument of ", StyleBox["Spacer", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], " is in printer's points (at the usual 72 points per inch)." }], "Text", CellChangeTimes->{{3.4941796939375*^9, 3.49417978284375*^9}}], Cell[TextData[{ "Next, add an overall label. To do this, stack into a ", StyleBox["Column", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " the desired text as one item, and the same ", StyleBox["Row", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " as before as the second item." }], "Text", CellChangeTimes->{{3.494179822015625*^9, 3.494179880015625*^9}, 3.494180126265625*^9, {3.494180382953125*^9, 3.49418038434375*^9}}], Cell[BoxData[ RowBox[{"Column", "[", RowBox[{ RowBox[{"{", "\[IndentingNewLine]", RowBox[{ RowBox[{"Style", "[", RowBox[{ RowBox[{"TraditionalForm", "@", RowBox[{"HoldForm", "[", RowBox[{ RowBox[{"f", "[", "z", "]"}], "==", SuperscriptBox["z", "2"]}], "]"}]}], ",", RowBox[{"FontFamily", "\[Rule]", "\"\\""}]}], "]"}], ",", "\[IndentingNewLine]", RowBox[{"Row", "[", RowBox[{"{", RowBox[{"uVSxy", ",", RowBox[{"Spacer", "[", "20", "]"}], ",", "vVSxy"}], "}"}], "]"}]}], "\[IndentingNewLine]", "}"}], ",", RowBox[{"Alignment", "\[Rule]", "Center"}]}], "]"}]], "Input", ShowCellTags->False, CellChangeTimes->{{3.4910607494375*^9, 3.491060751484375*^9}, { 3.49106096540625*^9, 3.49106099071875*^9}, {3.4924731296875*^9, 3.492473155734375*^9}, {3.49417961353125*^9, 3.494179657609375*^9}, { 3.49417989228125*^9, 3.494179970125*^9}, {3.494180041421875*^9, 3.49418010696875*^9}}], Cell[TextData[{ "For cruder labeling, you could eliminate the ", StyleBox["Style", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " treatment of the text and omit the ", StyleBox["Alignment", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " option of ", StyleBox["Column", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "." }], "Text", CellChangeTimes->{{3.494180140921875*^9, 3.494180198421875*^9}}], Cell[TextData[{ "Another way to place two drawings next to one another is to use ", StyleBox["GraphicsRow", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], " instead of ", StyleBox["Row", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " and then just use a ", StyleBox["PlotLabel", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " option to label the entire figure:" }], "Text", CellChangeTimes->{{3.49417966859375*^9, 3.494179675140625*^9}, { 3.494180217859375*^9, 3.4941802375*^9}, {3.49418052025*^9, 3.4941805213125*^9}, {3.494433680671875*^9, 3.494433711125*^9}}], Cell[BoxData[ RowBox[{"GraphicsRow", "[", RowBox[{ RowBox[{"{", RowBox[{"uVSxy", ",", "vVSxy"}], "}"}], ",", RowBox[{"PlotLabel", "\[Rule]", RowBox[{"TraditionalForm", "@", RowBox[{"HoldForm", "[", RowBox[{ RowBox[{"f", "[", "z", "]"}], "==", SuperscriptBox["z", "2"]}], "]"}]}]}], ",", RowBox[{"ImageSize", "\[Rule]", RowBox[{"7", " ", "72"}]}]}], "]"}]], "Input", ShowCellTags->False, CellChangeTimes->{{3.4910607494375*^9, 3.491060751484375*^9}, { 3.49106096540625*^9, 3.491060971421875*^9}, {3.491061156015625*^9, 3.4910611724375*^9}, {3.49106121653125*^9, 3.49106123909375*^9}, { 3.491061323421875*^9, 3.491061375421875*^9}, 3.491061426390625*^9, { 3.492473051609375*^9, 3.4924730670625*^9}, 3.492473191703125*^9}], Cell["\<\ To interactively change the size of the that overall two-graph figure, click \ on it and then drag one of the \"handles\". But you cannot now interactively \ change the size of just one of the two drawings. However, you may still rotate each of the surfaces in the usual way.\ \>", "SmallText", ShowCellTags->False, CellChangeTimes->{{3.491061007734375*^9, 3.49106100875*^9}, 3.491061991859375*^9, {3.491062197265625*^9, 3.491062199609375*^9}, { 3.492518773953125*^9, 3.492518833484375*^9}, {3.494180421578125*^9, 3.49418050346875*^9}, {3.49418057715625*^9, 3.4941805778125*^9}, { 3.494441285*^9, 3.494441285609375*^9}}, ParagraphSpacing->{0.5, 0}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Similarly, visualize the real and imaginary parts of ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"g", "(", "z", ")"}], "=", SuperscriptBox["z", "3"]}], TraditionalForm]], FormatType->"TraditionalForm"], "." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.492473484328125*^9, 3.492473531171875*^9}, {3.492473590484375*^9, 3.492473591*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Visualize together ", Cell[BoxData[ FormBox[ RowBox[{"|", "z", "|"}], TraditionalForm]], FormatType->"TraditionalForm"], " and ", Cell[BoxData[ FormBox[ RowBox[{"Arg", "(", "z", ")"}], TraditionalForm]], FormatType->"TraditionalForm"], " by using ", StyleBox["Plot3D", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.492473484328125*^9, 3.492473531171875*^9}, {3.492473590484375*^9, 3.492473591*^9}, { 3.4935449468125*^9, 3.493544998046875*^9}, {3.493545107671875*^9, 3.49354513109375*^9}}], Cell[TextData[{ "Evidently, forming the expression with ", StyleBox["GraphicsRow", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " is easier than with ", StyleBox["Row", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "\[LongDash]in particular, in constructing the label for the overall figure. \ So why was it done first by the more complicated method, with ", StyleBox["Row", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "? For two reasons:\n\t\[FilledSmallCircle] ", StyleBox["GraphicsRow", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " forces both drawings to become the same size, whereas ", StyleBox["Row", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " keeps the original sizes of both. This behavior of ", StyleBox["GraphicsRow", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " may be undesirable behavior when you are trying to compare two drawings, \ e.g., when one shows the domain of a complex function and the other shows the \ image of that function.\n\t\[FilledSmallCircle] When the drawings are made \ dynamic, so as to depend on one or more control variables, within a ", StyleBox["Manipulate", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " expression, and when one of those controls is a ", StyleBox["Locator", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "\[LongDash]see ", ButtonBox["below", BaseStyle->"Hyperlink", ButtonData->"ImagingWithLocators"], "\[LongDash]then ", StyleBox["GraphicsRow", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " will not work correctly." }], "Text", CellChangeTimes->{{3.49418051409375*^9, 3.49418057290625*^9}, { 3.49418060953125*^9, 3.49418065109375*^9}, {3.494180802203125*^9, 3.494180803828125*^9}, {3.494180989015625*^9, 3.4941810141875*^9}}, ParagraphSpacing->{0.5, 0}] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Using Cartesian coordinates to plot real and imaginary parts of a function \ with ", StyleBox["Presentations", FontSlant->"Italic"] }], "Subsection", CellChangeTimes->{{3.49256197375*^9, 3.49256199425*^9}, { 3.492607645265625*^9, 3.492607659125*^9}, {3.49261647528125*^9, 3.492616476921875*^9}}], Cell[CellGroupData[{ Cell[TextData[{ "Using ", StyleBox["Presentations", FontSlant->"Italic"], " ", StyleBox["Draw3D", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], " items instead of built-in ", StyleBox["Mathematica", FontSlant->"Italic"], " ", StyleBox["Plot3D", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"] }], "Subsubsection", CellChangeTimes->{{3.4926077694375*^9, 3.492607803421875*^9}, 3.492616168390625*^9}], Cell[TextData[{ "As an alternative to using ", StyleBox["Mathematica", FontSlant->"Italic"], "'s built-in ", StyleBox["Plot3D", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", you may use the ", StyleBox["Presentations", FontSlant->"Italic"], " function ", StyleBox["Draw3DItems", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], ". This has two advantages:\n\t\[FilledSmallCircle] ", StyleBox["Draw3DItems", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " uses the same paradigm as ", StyleBox["Draw2D", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", where you just form a list of graphics directives and graphics objects \ (whereas to include additional objects with ", StyleBox["Plot3D", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", you need to use ", StyleBox["Mathematica", FontSlant->"Italic"], "'s ", StyleBox["Epilog", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " or ", StyleBox["Prolog", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " options).\n\t\[FilledSmallCircle] Among ", StyleBox["Draw3D", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " items you may include special three-dimensional complex objects involving \ the \"Riemann sphere\", as you shall see in the notebook ", StyleBox["RiemannSphere.nb", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "." }], "Text", CellChangeTimes->{{3.49256029909375*^9, 3.49256034646875*^9}, { 3.492560390015625*^9, 3.492560573390625*^9}, {3.492560636015625*^9, 3.492560638390625*^9}, {3.494441296328125*^9, 3.4944413101875*^9}}, ParagraphSpacing->{0.5, 0}], Cell[TextData[{ "At first we shall still need the expressions ", Cell[BoxData[ FormBox[ RowBox[{"u", "(", RowBox[{"x", ",", "y"}], ")"}], TraditionalForm]], FormatType->"TraditionalForm"], " and ", Cell[BoxData[ FormBox[ RowBox[{"v", "(", RowBox[{"x", ",", "y"}], ")"}], TraditionalForm]], FormatType->"TraditionalForm"], " for the real and imaginary parts of ", Cell[BoxData[ FormBox["f", TraditionalForm]], FormatType->"TraditionalForm"], " as functions of Cartesian coordinates ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"u", ",", "v"}], ")"}], TraditionalForm]], FormatType->"TraditionalForm"], "." }], "Text", CellChangeTimes->{{3.4926076758125*^9, 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3.4910611724375*^9}, {3.49106121653125*^9, 3.49106123909375*^9}, { 3.491061323421875*^9, 3.491061375421875*^9}, 3.491061426390625*^9, { 3.492473051609375*^9, 3.4924730670625*^9}, 3.492473191703125*^9}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Produce essentially the same figure as the preceding output but by using ", StyleBox["Row", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " instead of ", StyleBox["GraphicsRow", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ". (You will need to use a ", StyleBox["Column", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " in order to produce an overall label over the pair of drawings.)" }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.492473484328125*^9, 3.492473531171875*^9}, {3.492473590484375*^9, 3.492473591*^9}, { 3.4941811550625*^9, 3.494181232421875*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Similarly, visualize the real and imaginary parts of ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"g", "(", "z", ")"}], "=", SuperscriptBox["z", "3"]}], TraditionalForm]]], "." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.492473484328125*^9, 3.492473531171875*^9}, {3.492473590484375*^9, 3.492473591*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Visualize together", Cell[BoxData[ FormBox[ RowBox[{"|", "z", "|"}], TraditionalForm]], FormatType->"TraditionalForm"], " and ", Cell[BoxData[ FormBox[ RowBox[{"Arg", "(", "z", ")"}], TraditionalForm]], FormatType->"TraditionalForm"], " by using ", StyleBox["Draw3DItems", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.492473484328125*^9, 3.492473531171875*^9}, {3.492473590484375*^9, 3.492473591*^9}, { 3.4935449468125*^9, 3.493544998046875*^9}, {3.493545152796875*^9, 3.49354515525*^9}}] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Using ", StyleBox["Presentations", FontSlant->"Italic"], " ", StyleBox["ComplexCartesianSurface", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"] }], "Subsubsection", CellChangeTimes->{{3.492607852984375*^9, 3.492607875015625*^9}}], Cell[TextData[{ StyleBox["Presentations", FontSlant->"Italic"], " provides the function ", StyleBox["ComplexCartesianSurface", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], " as another way to plot the real and imaginary parts of a function ", Cell[BoxData[ FormBox[ RowBox[{"f", "\[Colon]", RowBox[{ "\[DoubleStruckCapitalC]", "\[Rule]", "\[DoubleStruckCapitalC]"}]}], TraditionalForm]]], ", that is, to plot the composite functions ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"Re", "\[ThinSpace]", "\[SmallCircle]", RowBox[{"f", ":", SuperscriptBox["\[DoubleStruckCapitalR]", "2"]}]}], "\[Rule]", "\[DoubleStruckCapitalR]"}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{" ", FormBox[ RowBox[{ RowBox[{"Im", "\[ThinSpace]", "\[SmallCircle]", RowBox[{"f", ":", SuperscriptBox["\[DoubleStruckCapitalR]", "2"]}]}], "\[Rule]", "\[DoubleStruckCapitalR]"}], TraditionalForm]}], TraditionalForm]]], ", so that you do ", StyleBox["not", FontSlant->"Italic"], " first have to express things in terms of Cartesian coordinates. In fact, \ with ", StyleBox["ComplexCartesianSurface", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", you can plot other composites ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"g", "\[ThinSpace]", "\[SmallCircle]", RowBox[{"f", ":", SuperscriptBox["\[DoubleStruckCapitalR]", "2"]}]}], "\[Rule]", "\[DoubleStruckCapitalR]"}], TraditionalForm]]], " where ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"g", ":", "\[DoubleStruckCapitalR]"}], "\[Rule]", "\[DoubleStruckCapitalR]"}], TraditionalForm]]], " is an arbitrary function! (For example, ", Cell[BoxData[ FormBox["g", TraditionalForm]]], " could be the modulus or principal argument function.)" }], "Text", CellChangeTimes->{{3.492606761046875*^9, 3.49260706359375*^9}, { 3.492607899296875*^9, 3.49260797225*^9}, {3.49443376278125*^9, 3.494433763421875*^9}}], Cell[TextData[{ "As you can discover by inserting a template for ", StyleBox["ComplexCartesianSurface", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " by using the ", StyleBox["Surfaces", FontFamily->"Helvetica", FontWeight->"Plain", FontSlant->"Plain"], " group in the ", StyleBox["ComplexGraphics", FontFamily->"Helvetica", FontWeight->"Plain", FontSlant->"Plain"], " section of ", StyleBox["PresentationsPalette", FontFamily->"Helvetica", FontWeight->"Plain", FontSlant->"Plain"], ", the syntax of ", StyleBox["ComplexCartesianSurface", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " is:\n\t", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["ComplexCartesianSurface", 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ButtonData->"createFinalCartesianGrid"], ". You may wish to look at that first and evaluate it, then return here to \ see the step-by-step process leading to it." }], "Text", CellChangeTimes->{{3.492531629140625*^9, 3.492531682984375*^9}, { 3.492531748390625*^9, 3.492531804015625*^9}}], Cell[TextData[{ "To draw such a grid involves two steps: first, constructing the grid using ", StyleBox["DrawCartesianMap", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], "; and second, displaying the grid using ", StyleBox["Draw2D", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], "." }], "Text", CellChangeTimes->{{3.49106282353125*^9, 3.4910629620625*^9}}], Cell[TextData[{ StyleBox["Step 1:", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" ", FontWeight->"Bold"], StyleBox["Construct the grid.", FontWeight->"Bold", FontSlant->"Italic"], " The first argument to ", StyleBox["DrawCartesianMap", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " is just ", StyleBox["z", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", indicating that the horizontal and vertical lines constituting the grid \ are not to be transformed in way. The second argument is a list of the form\n\ \t", StyleBox["{z,zmin,zmax}", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "\nwhere ", StyleBox["zmin", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " and ", StyleBox["zmax", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " are the complex numbers designating the lower-left corner and the \ upper-right corner, respectively, of the rectangle for the grid." }], "Text", CellChangeTimes->{{3.4910629835625*^9, 3.491063213390625*^9}, 3.491066709296875*^9, {3.491068144125*^9, 3.491068177359375*^9}, 3.49247345334375*^9, 3.49444135209375*^9}, ParagraphSpacing->{0.5, 0.}], Cell["Here is the grid we shall use:", "Text", CellChangeTimes->{{3.491068189984375*^9, 3.4910681958125*^9}}], Cell[BoxData[ RowBox[{" ", RowBox[{ RowBox[{"grid", "=", RowBox[{"DrawCartesianMap", "[", RowBox[{"z", ",", RowBox[{"{", RowBox[{"z", ",", RowBox[{ RowBox[{"-", "2"}], "-", RowBox[{"2", "\[ImaginaryI]"}]}], ",", RowBox[{"2", "+", RowBox[{"2", "\[ImaginaryI]"}]}]}], "}"}]}], "]"}]}], ";"}]}]], "Input", CellChangeTimes->{{3.491063009828125*^9, 3.491063011046875*^9}, { 3.491063378109375*^9, 3.4910633793125*^9}, 3.49247347171875*^9}], Cell["\<\ The output there was suppressed by the semicolon at the end. Here's just a \ part of it:\ \>", "Text", CellChangeTimes->{{3.4924736015*^9, 3.492473630359375*^9}}], Cell[BoxData[ RowBox[{"grid", "//", "Short"}]], "Input", CellChangeTimes->{{3.49247363171875*^9, 3.49247364421875*^9}}], Cell[TextData[{ "The ", StyleBox["<<898>>", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " and ", StyleBox["<<1>>", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " denote the 898 parts of ", StyleBox["grid", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " the one part at the end omitted from the output display." }], "SmallText", CellChangeTimes->{{3.492473723609375*^9, 3.49247375478125*^9}, { 3.49247378728125*^9, 3.492473824359375*^9}}], Cell[TextData[{ "The value of ", StyleBox["grid", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " has a rather complicated structure. Ordinarily you won\[CloseCurlyQuote]t \ want to see the value as output. But the essence of the value is that its \ first part is a list of coordinates of points where the horizontal and \ vertical lines of the grid intersect\[LongDash]along with points one the \ lines between those intersections so as to permit more precise drawing." }], "Text", CellChangeTimes->{{3.4910667195625*^9, 3.491067096796875*^9}, { 3.49106713678125*^9, 3.49106719653125*^9}, {3.491067247265625*^9, 3.491067353296875*^9}, {3.4910675504375*^9, 3.49106756796875*^9}, { 3.4910676155*^9, 3.491067636578125*^9}, {3.49106767803125*^9, 3.491067768390625*^9}, 3.49106787696875*^9, {3.491068069328125*^9, 3.4910681054375*^9}, {3.491068205265625*^9, 3.491068232421875*^9}, { 3.492473829984375*^9, 3.492473867265625*^9}}, ParagraphSpacing->{0.5, 0.}], Cell[TextData[{ StyleBox["Step 2: Display the grid.", FontWeight->"Bold", FontSlant->"Italic"], " For this, use ", StyleBox["Draw2D", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " as usual." }], "Text", CellChangeTimes->{ 3.491062979359375*^9, {3.4910633333125*^9, 3.4910633491875*^9}, { 3.491068246421875*^9, 3.49106825653125*^9}}], Cell[BoxData[ RowBox[{"Draw2D", "[", RowBox[{ RowBox[{"{", "\[IndentingNewLine]", RowBox[{"DrawCartesianMap", "[", RowBox[{"z", ",", RowBox[{"{", RowBox[{"z", ",", RowBox[{ RowBox[{"-", "2"}], "-", RowBox[{"2", "I"}]}], ",", RowBox[{"2", "+", RowBox[{"2", "I"}]}]}], "}"}]}], "]"}], "\[IndentingNewLine]", "}"}], ",", RowBox[{"ImageSize", "\[Rule]", RowBox[{"3", " ", "72"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.491062556890625*^9, 3.491062603515625*^9}, 3.491063439015625*^9, {3.491069822890625*^9, 3.491069841734375*^9}}], Cell[TextData[{ "It is OK to insert the ", StyleBox["DrawCartesianMap", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " expression like that directly into the ", StyleBox["Draw2D", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " argument. But when you add options to ", StyleBox["DrawCartesianMap", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " or include additional complex graphics objects in the ", StyleBox["Draw2D", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " argument, you will probably want to form the grid first and then refer to \ it by name inside ", StyleBox["Draw2D", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "\[LongDash]like this: " }], "Text", CellChangeTimes->{{3.49261715340625*^9, 3.492617308*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"grid", "=", RowBox[{"DrawCartesianMap", "[", RowBox[{"z", ",", RowBox[{"{", RowBox[{"z", ",", RowBox[{ RowBox[{"-", "2"}], "-", RowBox[{"2", "\[ImaginaryI]"}]}], ",", RowBox[{"2", "+", RowBox[{"2", "\[ImaginaryI]"}]}]}], "}"}]}], "]"}]}], ";"}], "\n"}], "\[IndentingNewLine]", RowBox[{"Draw2D", "[", RowBox[{ RowBox[{"{", "grid", "}"}], ",", "\[IndentingNewLine]", RowBox[{"ImageSize", "\[Rule]", RowBox[{"3", " ", "72"}]}]}], "]"}]}], "Input", CellChangeTimes->{{3.4926171263125*^9, 3.492617126984375*^9}, { 3.4926173135*^9, 3.492617355359375*^9}}], Cell[TextData[{ StyleBox["To change the number of grid lines in each direction", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], ", use the ", StyleBox["Mesh", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], " option to ", StyleBox["DrawCartesianMap", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ". And to do that, you may either specify the actual number of lines in each \ direction or, instead, specify the (horizontal or vertical) coordinates of \ those lines:" }], "Text", CellChangeTimes->{{3.4943487116875*^9, 3.4943488243125*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"grid", "=", RowBox[{"DrawCartesianMap", "[", RowBox[{"z", ",", RowBox[{"{", RowBox[{"z", ",", RowBox[{ RowBox[{"-", "2"}], "-", RowBox[{"2", "\[ImaginaryI]"}]}], ",", RowBox[{"2", "+", RowBox[{"2", "\[ImaginaryI]"}]}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"Mesh", "\[Rule]", RowBox[{"{", RowBox[{"5", ",", "5"}], "}"}]}]}], "]"}]}], ";"}], "\n"}], "\[IndentingNewLine]", RowBox[{"Draw2D", "[", RowBox[{ RowBox[{"{", "grid", "}"}], ",", "\[IndentingNewLine]", RowBox[{"ImageSize", "\[Rule]", RowBox[{"3", " ", "72"}]}]}], "]"}]}], "Input", CellChangeTimes->{{3.4926171263125*^9, 3.492617126984375*^9}, { 3.4926173135*^9, 3.492617355359375*^9}, {3.494348751375*^9, 3.494348762484375*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"grid", "=", RowBox[{"DrawCartesianMap", "[", RowBox[{"z", ",", RowBox[{"{", RowBox[{"z", ",", RowBox[{ RowBox[{"-", "2"}], "-", RowBox[{"2", "\[ImaginaryI]"}]}], ",", RowBox[{"2", "+", RowBox[{"2", "\[ImaginaryI]"}]}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"Mesh", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "1.5"}], ",", RowBox[{"-", "1"}], ",", "0", ",", "1", ",", "1.5"}], "}"}], ",", "5"}], "}"}]}]}], "]"}]}], ";"}], "\n"}], "\[IndentingNewLine]", RowBox[{"Draw2D", "[", RowBox[{ RowBox[{"{", "grid", "}"}], ",", "\[IndentingNewLine]", RowBox[{"ImageSize", "\[Rule]", RowBox[{"3", " ", "72"}]}]}], "]"}]}], "Input", CellChangeTimes->{{3.4926171263125*^9, 3.492617126984375*^9}, { 3.4926173135*^9, 3.492617355359375*^9}, {3.494348751375*^9, 3.494348762484375*^9}, {3.49434885015625*^9, 3.494348873296875*^9}}], Cell[TextData[{ "Probably the default gray background color of the grid itself is too dark.\n\ ", StyleBox["To change the background of a grid", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], ", in the ", StyleBox["Draw2D", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " argument precede the ", StyleBox["DrawCartesianMap", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " expression with a color directive." }], "Text", CellChangeTimes->{{3.4924738915*^9, 3.492473901484375*^9}, { 3.49247406684375*^9, 3.49247407709375*^9}, {3.49247422290625*^9, 3.49247425103125*^9}, 3.492518461265625*^9, {3.49251920228125*^9, 3.49251921175*^9}, {3.492519514984375*^9, 3.49251957059375*^9}}, ParagraphSpacing->{0.5, 0}], Cell[BoxData[ RowBox[{"Draw2D", "[", RowBox[{ RowBox[{"{", "\[IndentingNewLine]", RowBox[{ RowBox[{"HTML", "@", "Wheat"}], ",", "\[IndentingNewLine]", "grid"}], "\[IndentingNewLine]", "}"}], ",", RowBox[{"ImageSize", "\[Rule]", RowBox[{"3", " ", "72"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.491062556890625*^9, 3.491062603515625*^9}, 3.491063439015625*^9, 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But later, when those lines are mapped by ", Cell[BoxData[ FormBox["f", TraditionalForm]]], ", you will want to know which images are of the horizontal segments and \ which of the vertical segments." }], "Text", CellChangeTimes->{{3.4925182900625*^9, 3.49251831965625*^9}, { 3.4925186070625*^9, 3.492518612296875*^9}, {3.492519903671875*^9, 3.492520034546875*^9}, 3.4925200714375*^9, {3.492522294609375*^9, 3.492522308078125*^9}, {3.492531822921875*^9, 3.492531826046875*^9}, { 3.49253189053125*^9, 3.492531906890625*^9}, {3.494348417078125*^9, 3.494348428578125*^9}, 3.494441366640625*^9}], Cell[TextData[{ StyleBox["To distinguish the horizontal segments in the grid from the \ vertical ones", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], ", use the ", StyleBox["MeshStyle", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], " option to ", StyleBox["DrawCartesianMap", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ". " }], "Text", ShowCellTags->False, CellChangeTimes->{{3.4925318553125*^9, 3.4925320041875*^9}}], Cell["\<\ You may use different colors, or gray scale, or thickness, or dashing \ treatments, etc., for the horizontal and vertical segments. We shall use \ different colors.\ \>", "Text", ShowCellTags->False, CellChangeTimes->{{3.4925318553125*^9, 3.492531964375*^9}, { 3.4925320073125*^9, 3.492532050890625*^9}, 3.494441373953125*^9}], Cell[TextData[{ "Below is our final version of the Cartesian grid. To enhance it we:\n\t\ \[FilledSmallCircle] style the grid lines by using a ", StyleBox["MeshStyle", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], " option to ", StyleBox["DrawCartesianMap", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ";\n\t\[FilledSmallCircle] style the boundary of the rectangular region by \ using a ", StyleBox["BoundaryStyle", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], " option to ", StyleBox["DrawCartesianMap", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ";\n\t\[FilledSmallCircle] include axes;\n\t\[FilledSmallCircle] label the \ whole drawing.\nAnd we assign a name, ", StyleBox["domain", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", to the whole drawing, so that we can display it later alongside a drawing \ of the image of the grid under ", Cell[BoxData[ FormBox["f", TraditionalForm]]], "." }], "Text", CellChangeTimes->{{3.492520074375*^9, 3.492520104203125*^9}, { 3.492520227953125*^9, 3.4925203150625*^9}, {3.4925203856875*^9, 3.4925203866875*^9}, {3.492520528703125*^9, 3.492520529265625*^9}, { 3.492521444390625*^9, 3.4925215635625*^9}, {3.492617492015625*^9, 3.4926175506875*^9}, 3.492617647859375*^9, 3.49261774478125*^9, { 3.494348283921875*^9, 3.494348372375*^9}, {3.494348461796875*^9, 3.49434846234375*^9}, {3.494416666671875*^9, 3.494416667984375*^9}}, ParagraphSpacing->{0.5, 0}], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"grid", "=", RowBox[{"DrawCartesianMap", "[", RowBox[{"z", ",", RowBox[{"{", RowBox[{"z", ",", RowBox[{ RowBox[{"-", "2"}], "-", RowBox[{"2", "I"}]}], ",", RowBox[{"2", "+", RowBox[{"2", "I"}]}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"BoundaryStyle", "\[Rule]", RowBox[{"Directive", "[", RowBox[{"Thick", ",", "Black"}], "]"}]}], ",", "\[IndentingNewLine]", RowBox[{"MeshStyle", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"HTML", "@", "SeaGreen"}], ",", RowBox[{"Darker", "@", "Brown"}]}], "}"}]}]}], "\[IndentingNewLine]", "]"}]}], ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{"domain", "=", RowBox[{"Draw2D", "[", RowBox[{ RowBox[{"{", "\[IndentingNewLine]", RowBox[{ RowBox[{"Opacity", "[", RowBox[{"0.5", ",", RowBox[{"HTML", "@", "Wheat"}]}], "]"}], ",", "\[IndentingNewLine]", "grid"}], "\[IndentingNewLine]", "}"}], ",", "\[IndentingNewLine]", RowBox[{"Axes", "\[Rule]", "True"}], ",", "\[IndentingNewLine]", RowBox[{"PlotLabel", "\[Rule]", RowBox[{"Row", "[", RowBox[{"{", RowBox[{"\"\\"", ",", "z"}], "}"}], "]"}]}], ",", "\[IndentingNewLine]", RowBox[{"ImageSize", "\[Rule]", RowBox[{"3", " ", "72"}]}]}], "]"}]}]}], "Input", CellChangeTimes->{{3.491062556890625*^9, 3.491062603515625*^9}, 3.491063439015625*^9, {3.491069822890625*^9, 3.491069841734375*^9}, { 3.4924740950625*^9, 3.49247411615625*^9}, {3.4924741985625*^9, 3.492474214515625*^9}, {3.4925169065625*^9, 3.492516914125*^9}, { 3.492516954515625*^9, 3.492516976203125*^9}, {3.49251708578125*^9, 3.4925171014375*^9}, {3.492518219796875*^9, 3.4925182473125*^9}, { 3.4925183335625*^9, 3.49251838034375*^9}, {3.492519694125*^9, 3.492519717015625*^9}, {3.492519753796875*^9, 3.49251979634375*^9}, { 3.492520455046875*^9, 3.49252051496875*^9}, {3.492521568875*^9, 3.492521664265625*^9}, {3.492617440578125*^9, 3.492617481296875*^9}, { 3.4926177373125*^9, 3.49261773765625*^9}}, CellTags->{"createFinalCartesianGrid", "createDomain"}], Cell[TextData[{ "Constructing the ", StyleBox["PlotLabel", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " value as a ", StyleBox["Row", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " allows ", StyleBox["Mathematica", FontSlant->"Italic"], " to display the \"", StyleBox["z", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "\" automatically in a traditional mathematical style." }], "SmallText", CellChangeTimes->{{3.492521938171875*^9, 3.49252200975*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Try the same drawing except that, instead of axes, you put the tick marks \ along sides of a frame for the drawing. (In some of your own drawings, you \ may want to do that for clarity.)" }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.492520653359375*^9, 3.49252072346875*^9}, {3.49252075575*^9, 3.492520794265625*^9}}], Cell[TextData[{ "In your own drawings, you will doubtless want to use ", StyleBox["MeshStyle", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " to distinguish the horizontal from the vertical lines. But how far you \ want to go with changing other colors, etc., depends on what is necessary for \ clarity. And the more artistic your inclination, the nicer you can make the \ drawing look." }], "Text", CellChangeTimes->{{3.492518620234375*^9, 3.49251864090625*^9}, { 3.4925192389375*^9, 3.49251933615625*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["Image of a Cartesian grid under a function", "Subsection", ShowGroupOpener->True, ShowCellTags->False, CellChangeTimes->{{3.491063471078125*^9, 3.49106347178125*^9}, { 3.49252083775*^9, 3.49252084528125*^9}, {3.492521802265625*^9, 3.492521804515625*^9}}], Cell["Here again is the code to form the Cartesian grid:", "Text", CellChangeTimes->{{3.49443398728125*^9, 3.49443405678125*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"grid", "=", RowBox[{"DrawCartesianMap", "[", RowBox[{"z", ",", RowBox[{"{", RowBox[{"z", ",", RowBox[{ RowBox[{"-", "2"}], "-", RowBox[{"2", "I"}]}], ",", RowBox[{"2", "+", RowBox[{"2", "I"}]}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"BoundaryStyle", "\[Rule]", RowBox[{"Directive", "[", RowBox[{"Thick", ",", "Black"}], "]"}]}], ",", "\[IndentingNewLine]", RowBox[{"MeshStyle", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"HTML", "@", "SeaGreen"}], ",", RowBox[{"Darker", "@", "Brown"}]}], "}"}]}]}], "\[IndentingNewLine]", "]"}]}], ";"}]], "Input", CellChangeTimes->{{3.491062556890625*^9, 3.491062603515625*^9}, 3.491063439015625*^9, {3.491069822890625*^9, 3.491069841734375*^9}, { 3.4924740950625*^9, 3.49247411615625*^9}, {3.4924741985625*^9, 3.492474214515625*^9}, {3.4925169065625*^9, 3.492516914125*^9}, { 3.492516954515625*^9, 3.492516976203125*^9}, {3.49251708578125*^9, 3.4925171014375*^9}, {3.492518219796875*^9, 3.4925182473125*^9}, { 3.4925183335625*^9, 3.49251838034375*^9}, {3.492519694125*^9, 3.492519717015625*^9}, {3.492519753796875*^9, 3.49251979634375*^9}, { 3.492520455046875*^9, 3.49252051496875*^9}, {3.492521568875*^9, 3.492521664265625*^9}, {3.492617440578125*^9, 3.492617481296875*^9}, { 3.4926177373125*^9, 3.49261773765625*^9}, {3.49443407409375*^9, 3.494434074765625*^9}}, CellTags->{"createFinalCartesianGrid", "createDomain"}], Cell[TextData[{ "The set of images of the horizontal and vertical lines of a Cartesian grid \ under ", Cell[BoxData[ FormBox["f", TraditionalForm]]], " is included in the output of a ", StyleBox["DrawCartesianMap[f[z],", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], StyleBox["\[Ellipsis]", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " expression. Here is just the beginning of the output: " }], "Text", CellChangeTimes->{{3.491063482703125*^9, 3.491063566296875*^9}, 3.491065904390625*^9, {3.491068302796875*^9, 3.4910683165*^9}, { 3.492517848140625*^9, 3.492517896515625*^9}, {3.492518043171875*^9, 3.492518069984375*^9}, {3.49252204534375*^9, 3.492522046984375*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"DrawCartesianMap", "[", RowBox[{ RowBox[{"f", "[", "z", "]"}], ",", RowBox[{"{", RowBox[{"z", ",", RowBox[{ RowBox[{"-", "2"}], "-", RowBox[{"2", "I"}]}], ",", RowBox[{"2", "+", RowBox[{"2", "I"}]}]}], "}"}]}], "]"}], "//", RowBox[{ RowBox[{"Short", "[", RowBox[{"#", ",", "4"}], "]"}], "&"}]}]], "Input", CellChangeTimes->{{3.49251774153125*^9, 3.492517744171875*^9}, { 3.492517807578125*^9, 3.492517835828125*^9}, {3.4925179041875*^9, 3.492517931703125*^9}}], Cell[TextData[{ "The pure function ", StyleBox["Short[#,4]&", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " is a version of ", StyleBox["Short", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " with a second argument requesting approximately 4 lines of output." }], "SmallText", CellChangeTimes->{{3.49251794046875*^9, 3.492517999671875*^9}}], Cell[TextData[{ "For example, two points of the original grid are ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "2"}], ",", RowBox[{"-", "2"}]}], ")"}], TraditionalForm]], FormatType->"TraditionalForm"], " and, approximately, ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "1.71429"}], ",", RowBox[{"-", "2"}]}], ")"}], TraditionalForm]], FormatType->"TraditionalForm"], ". The values of ", Cell[BoxData[ FormBox["f", TraditionalForm]], FormatType->"TraditionalForm"], " at these points are:" }], "Text", CellChangeTimes->{{3.491068322*^9, 3.4910683333125*^9}, {3.49106843628125*^9, 3.4910685285625*^9}, {3.4910685820625*^9, 3.491068599453125*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"f", "[", RowBox[{ RowBox[{"-", "2"}], "-", RowBox[{"2", "\[ImaginaryI]"}]}], "]"}], ",", RowBox[{"f", "[", RowBox[{ RowBox[{"-", "1.71429"}], "-", RowBox[{"2", "\[ImaginaryI]"}]}], "]"}]}], "}"}]], "Input", CellChangeTimes->{{3.491068530984375*^9, 3.49106857028125*^9}}], Cell[TextData[{ "And those values have as coordinates the entries ", Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{"0.", ",", "8."}], "}"}], TraditionalForm]], FormatType->"TraditionalForm"], " and ", Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{ RowBox[{"-", "1.06121"}], ",", "6.85714"}], "}"}], TraditionalForm]], FormatType->"TraditionalForm"], " that you see at the beginning of the output from the ", StyleBox["DrawCartesianMap[f[z],", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], StyleBox["\[Ellipsis]", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " expression." }], "Text", CellChangeTimes->{{3.491068603796875*^9, 3.4910687100625*^9}}], Cell[TextData[{ "Then the images of the lines of the original Cartesian grid are as shown \ below. Notice that the only changes to the code for the Cartesian grid to \ this code is to change ", StyleBox["z", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " to ", StyleBox["f[z]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "and to change the ", StyleBox["PlotLabel", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "! Of course you could change various directives and options here. 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In a two-panel plot such as above, visualize how ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"g", "(", "z", ")"}], "=", SuperscriptBox["z", "3"]}], TraditionalForm]], FormatType->"TraditionalForm"], " maps the complex plane." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.49252117265625*^9, 3.492521274703125*^9}, {3.492617839734375*^9, 3.492617880640625*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], ". Repeat the preceding exercise but for ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"h", "(", "z", ")"}], "=", RowBox[{"1", "/", "z"}]}], TraditionalForm]], FormatType->"TraditionalForm"], "." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.49252117265625*^9, 3.492521274703125*^9}, {3.492617839734375*^9, 3.492617900421875*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], ". Encapsulate in a function ", StyleBox["TwoPanelCartesianPlot", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " the entire process of creating such a drawing of how a complex function \ maps a Cartesian grid. The arguments to ", StyleBox["TwoPanelCartesianPlot", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " should include at least:\n\t\[FilledSmallCircle] the variable name ", StyleBox["z", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], ";\n\t\[FilledSmallCircle] a function name ", StyleBox["f", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], "; and\n\t\[FilledSmallCircle] a list ", StyleBox["{", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], StyleBox["zmin", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[",", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], StyleBox["zmax", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["}", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " indicating the lower-left and upper-right corner of the grid.\n(You may \ wish to include additional arguments to specify grid segment coloring, etc.)\n\ For example, to reproduce the preceding drawing of the squaring function, \ after having defined ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ RowBox[{ StyleBox["f", FontSlant->"Plain"], "[", "z_", "]"}], FontFamily->"Courier"], ":=", SuperscriptBox["z", "2"]}], TraditionalForm]]], ", you would invoke:\n\t", Cell[BoxData[ FormBox[ RowBox[{"TwoPanelCartesianPlot", "[", RowBox[{ StyleBox["z", FontSlant->"Plain"], ",", StyleBox["f", FontSlant->"Plain"], ",", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"-", "2"}], "-", RowBox[{"2", "\[ImaginaryI]"}]}], ",", RowBox[{"2", "+", RowBox[{"2", "\[ImaginaryI]"}]}]}], "}"}]}], "]"}], TraditionalForm]], FontFamily->"Courier"] }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.49252117265625*^9, 3.492521274703125*^9}, {3.49253414253125*^9, 3.49253465153125*^9}, { 3.492534953*^9, 3.4925349615*^9}, {3.49253500421875*^9, 3.49253501234375*^9}, 3.492817070828125*^9}, ParagraphSpacing->{0.5, 0}, CellTags->"defineTwoPanelCartesianPlot"], Cell[TextData[{ StyleBox["When you print your work on a black-and-white printer", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], ", the color treatments are probably not going to show up very well. In that \ case, you will want to try something else, such as:\n\t\[FilledSmallCircle] \ for the grid background, include a ", StyleBox["White", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " color directive before grid in the ", StyleBox["Draw2D", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " argument; and\n\t\[FilledSmallCircle] for the mesh lines, use shades of \ gray\[LongDash]using the ", StyleBox["Gray", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " or ", StyleBox["GrayLevel", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " directives\[LongDash]or solid vs. dashing, using thickness or dashing \ directives.\nMoreover, you will want to keep in mind the width of paper when \ you set ", StyleBox["ImageSize", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ". (If you set an ", StyleBox["ImageSize", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " option in the final ", StyleBox["GraphicsRow", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " expression, then you don't need them in the individual ", StyleBox["Draw3D", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " expressions\[LongDash]although that will scale both panels to have the \ same overall size.)" }], "Text", CellChangeTimes->{{3.49262119140625*^9, 3.49262129803125*^9}, { 3.492621332109375*^9, 3.492621405234375*^9}, {3.4926214396875*^9, 3.49262150465625*^9}, {3.4926235633125*^9, 3.492623658984375*^9}, { 3.492623785390625*^9, 3.492623799828125*^9}, {3.49263373540625*^9, 3.49263390853125*^9}, {3.494107010484375*^9, 3.494107022203125*^9}, 3.494350265265625*^9}, ParagraphSpacing->{0.5, 0}], Cell[TextData[{ " For example, here is the code for the same visualization of ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", "z", ")"}], 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Modify your function ", StyleBox["TwoPanelCartesianPlot", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " from Exercise ", CounterBox["Exercise", "defineTwoPanelCartesianPlot"], " so as to produce output suitable for printing on a black-and-white \ printer. Perhaps name the modified function ", StyleBox["TwoPanelCartesianPlotPrintVersion", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ". \nOr instead, modify ", StyleBox["TwoPanelCartesianPlot", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " so as to include an optional argument specifying whether to produce output \ suitable for printing or, instead, ordinary output with all the pretty color \ treatments.\nFor example, to produce output for viewing on-screen, you would \ invoke ", StyleBox["TwoPanelCartesianPlot", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " exactly as before. But to produce output suitable for black-and-white \ printing, you would now invoke the function like this:\n\t", Cell[BoxData[ FormBox[ RowBox[{"TwoPanelCartesianPlot", "[", RowBox[{ StyleBox["z", FontSlant->"Plain"], ",", StyleBox["f", FontSlant->"Plain"], ",", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"-", "2"}], "-", RowBox[{"2", "\[ImaginaryI]"}]}], ",", RowBox[{"2", "+", RowBox[{"2", "\[ImaginaryI]"}]}]}], "}"}], ",", RowBox[{"Printing", "\[Rule]", "BW"}]}], "]"}], TraditionalForm]], FontFamily->"Courier"], "\n(For this alternative version with an optional argument, you'll need to \ consult the Documentation Center about how to handle optional arguments.)" }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.49252117265625*^9, 3.492521274703125*^9}, {3.49253414253125*^9, 3.49253465153125*^9}, { 3.492534953*^9, 3.4925349615*^9}, {3.49253500421875*^9, 3.49253501234375*^9}, {3.49263399125*^9, 3.492634274859375*^9}, { 3.49263430778125*^9, 3.49263449603125*^9}, 3.492634579609375*^9, 3.49281705828125*^9}, ParagraphSpacing->{0.5, 0}, CellTags->"defineTwoPanelCartesianPlotPrinting"] }, Closed]], Cell[CellGroupData[{ Cell["Highlighting geometric objects and their images", "Subsection", ShowGroupOpener->True, ShowCellTags->False, CellChangeTimes->{{3.492798412640625*^9, 3.49279842084375*^9}}, CellTags->"hilitePtsLinesCartesian"], Cell["Still the function is:", "Text", CellChangeTimes->{{3.49443708375*^9, 3.494437086765625*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"f", "[", "z_", "]"}], ":=", SuperscriptBox["z", "2"]}]], "Input", CellChangeTimes->{{3.49443708765625*^9, 3.494437092171875*^9}}], Cell["\<\ To better understand how a complex function maps the plane, you will often \ want to highlight some particular points or line segments and to see their \ images in a two-panel Cartesian plot. 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And the value of ", StyleBox["ComplexMap[", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], StyleBox["f", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " is then a ", StyleBox["Mathematica", FontSlant->"Italic"], " function which in turn takes an argument, the list ", StyleBox["lis", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], ".\nTo avoid the back-to-back brackets, it is common to use ", StyleBox["ComplexMap[", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], StyleBox["f", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " in postifx form:\n\t", StyleBox["{", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], StyleBox["lis", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["}//ComplexMap[", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], StyleBox["f", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"] }], "Text", CellChangeTimes->{{3.4925357079375*^9, 3.49253580596875*^9}, { 3.492537887015625*^9, 3.4925379260625*^9}, {3.492538282984375*^9, 3.49253829409375*^9}, {3.492538571125*^9, 3.49253861115625*^9}, { 3.492538658828125*^9, 3.492538661640625*^9}, {3.49253874578125*^9, 3.49253874765625*^9}, {3.492538786734375*^9, 3.492538805015625*^9}, { 3.49253885815625*^9, 3.49253886034375*^9}, {3.492623869109375*^9, 3.492623870859375*^9}}, ParagraphSpacing->{0.5, 0}], Cell[TextData[{ "From the ", ButtonBox["code to create ", BaseStyle->"Hyperlink", ButtonData->"createDomain"], StyleBox[ButtonBox["domain", BaseStyle->"Hyperlink", ButtonData->"createDomain"], FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", extract the list argument of ", StyleBox["Draw2D", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "\[LongDash]without any of the options to ", StyleBox["Draw2D", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " (output is suppressed below):" }], "Text", CellChangeTimes->{{3.4925365095*^9, 3.492536517703125*^9}, { 3.4925365698125*^9, 3.492536625*^9}, {3.4925388859375*^9, 3.492538925140625*^9}, {3.49262388725*^9, 3.49262388859375*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"grid", "=", RowBox[{"DrawCartesianMap", "[", RowBox[{"z", ",", RowBox[{"{", RowBox[{"z", ",", RowBox[{ RowBox[{"-", "2"}], "-", RowBox[{"2", "I"}]}], ",", RowBox[{"2", "+", RowBox[{"2", "I"}]}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"BoundaryStyle", "\[Rule]", RowBox[{"Directive", "[", RowBox[{"Thick", ",", "Black"}], "]"}]}], ",", "\[IndentingNewLine]", RowBox[{"MeshStyle", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"HTML", "@", "SeaGreen"}], ",", RowBox[{"Darker", "@", "Brown"}]}], "}"}]}]}], "]"}]}], ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Opacity", "[", RowBox[{"0.5", ",", RowBox[{"HTML", "@", "Wheat"}]}], "]"}], ",", "grid"}], "}"}], ";"}]}], "Input", CellChangeTimes->{{3.492538045078125*^9, 3.492538075796875*^9}, { 3.492538160140625*^9, 3.492538172890625*^9}, 3.492538215890625*^9, { 3.49253895746875*^9, 3.492538958265625*^9}, {3.4926181190625*^9, 3.492618155578125*^9}}], Cell[TextData[{ "Now we are going to include in that list a two points and the line segment \ from 0 to ", Cell[BoxData[ FormBox[ RowBox[{"1", "+", "\[ImaginaryI]"}], TraditionalForm]], FormatType->"TraditionalForm"], ". But there's a twist here: you need parameterize the line segment. (The \ reason for doing that will become clear later.)" }], "Text", CellChangeTimes->{{3.492538185640625*^9, 3.492538190390625*^9}, { 3.492538962453125*^9, 3.49253896990625*^9}, {3.492539908640625*^9, 3.492539994046875*^9}, {3.492618217515625*^9, 3.49261822115625*^9}, { 3.492618252*^9, 3.492618283671875*^9}}], Cell[TextData[{ "The line segment from 0 to ", Cell[BoxData[ FormBox[ RowBox[{"1", "+", "\[ImaginaryI]"}], TraditionalForm]]], ". is parametrized by ", Cell[BoxData[ FormBox[ RowBox[{"z", "=", RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"1", "-", "t"}], ")"}], RowBox[{"(", "0", ")"}]}], "+", RowBox[{"t", "(", RowBox[{"1", "+", "\[ImaginaryI]"}], ")"}]}], "=", RowBox[{"t", "(", RowBox[{"1", "+", "\[ImaginaryI]"}], ")"}]}]}], TraditionalForm]]], " for ", Cell[BoxData[ FormBox[ RowBox[{"0", "\[LessEqual]", "t", "\[LessEqual]", "1"}], TraditionalForm]]], ". 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(You may wish to see notebook ", StyleBox["CalculatingImages.nb", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ".)" }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.49252117265625*^9, 3.492521274703125*^9}, {3.492617839734375*^9, 3.492617880640625*^9}, { 3.49441535190625*^9, 3.494415354875*^9}, {3.494440720515625*^9, 3.49444086159375*^9}, {3.494441105234375*^9, 3.49444111121875*^9}}, ParagraphSpacing->{0.5, 0}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], ". 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Modify the definition of ", StyleBox["TwoPanelCartesianPlot", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " from Exercise ", CounterBox["Exercise", "defineTwoPanelCartesianPlot"], " so as to handle not just the grids, but additional graphical objects such \ as points, line segments, circles, and other curves that are to be \ highlighted." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.49252117265625*^9, 3.492521274703125*^9}, {3.49253414253125*^9, 3.49253465153125*^9}, { 3.492534953*^9, 3.4925349615*^9}, {3.49253500421875*^9, 3.49253501234375*^9}, {3.492536309625*^9, 3.492536348*^9}, { 3.492536390703125*^9, 3.4925364171875*^9}, {3.49253648115625*^9, 3.492536482375*^9}, {3.492634670296875*^9, 3.4926348015*^9}}, ParagraphSpacing->{0.5, 0}], Cell[TextData[{ "For the line segment, why was ", StyleBox["ComplexCurve", FontFamily->"Courier"], " used rather than the simpler ", StyleBox["ComplexLine", FontFamily->"Courier"], ", as in the following modification of the preceding code?\nSee what happens \ if in the preceding code you replace the ", StyleBox["ComplexCurve", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " expression with the simpler ", StyleBox["ComplexLine", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " expression:" }], "Text", ShowCellTags->False, CellChangeTimes->{{3.492624014140625*^9, 3.492624035421875*^9}, { 3.492624103046875*^9, 3.49262411021875*^9}, {3.4926244384375*^9, 3.4926245059375*^9}, {3.492624713921875*^9, 3.492624717453125*^9}, { 3.49262752075*^9, 3.492627530671875*^9}, {3.492627594296875*^9, 3.492627618234375*^9}, {3.492627676921875*^9, 3.4926277503125*^9}}, ParagraphSpacing->{0.5, 0}], Cell[BoxData[ RowBox[{" ", RowBox[{ RowBox[{ RowBox[{"domainObjects", "=", RowBox[{"{", "\[IndentingNewLine]", RowBox[{ RowBox[{"Opacity", "[", RowBox[{"0.5", ",", RowBox[{"HTML", "@", "Wheat"}]}], "]"}], ",", "\[IndentingNewLine]", "grid", ",", "\[IndentingNewLine]", RowBox[{"Directive", "[", RowBox[{"Thick", ",", "Red"}], "]"}], ",", RowBox[{"ComplexLine", "[", RowBox[{"{", RowBox[{"0", ",", RowBox[{"1", "+", "\[ImaginaryI]"}]}], "}"}], "]"}], ",", "\[IndentingNewLine]", RowBox[{"Directive", "[", RowBox[{ RowBox[{"PointSize", "@", "Large"}], ",", "Blue"}], "]"}], ",", RowBox[{"ComplexPoint", "/@", RowBox[{"{", RowBox[{ RowBox[{"1", "+", "\[ImaginaryI]"}], ",", "\[ImaginaryI]"}], "}"}]}]}], "\[IndentingNewLine]", "}"}]}], ";"}], "\n", RowBox[{ RowBox[{"domain", "=", RowBox[{"Draw2D", "[", RowBox[{ RowBox[{"{", "domainObjects", "}"}], ",", "\[IndentingNewLine]", RowBox[{"Axes", "\[Rule]", "True"}], ",", RowBox[{"PlotLabel", "\[Rule]", RowBox[{"Row", "[", RowBox[{"{", RowBox[{"\"\\"", ",", "z"}], "}"}], "]"}]}], ",", "\[IndentingNewLine]", RowBox[{"ImageSize", "\[Rule]", RowBox[{"3", " ", "72"}]}]}], "]"}]}], ";"}], "\n", RowBox[{ RowBox[{"image", "=", RowBox[{"Draw2D", "[", RowBox[{ RowBox[{"{", RowBox[{"domainObjects", "//", RowBox[{"ComplexMap", "[", "f", "]"}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"Axes", "\[Rule]", "True"}], ",", RowBox[{"PlotLabel", "\[Rule]", RowBox[{"Row", "[", RowBox[{"{", RowBox[{"\"\\"", ",", RowBox[{"f", "[", "z", "]"}]}], "}"}], "]"}]}], ",", "\[IndentingNewLine]", RowBox[{"ImageSize", "\[Rule]", RowBox[{"3", " ", "72"}]}]}], "]"}]}], ";"}], "\n", RowBox[{"GraphicsRow", "[", RowBox[{ RowBox[{"{", RowBox[{"domain", ",", "image"}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"PlotLabel", "\[Rule]", RowBox[{"Row", "[", RowBox[{"{", RowBox[{ "\"\\"", ",", "z", ",", "\"\< \[LongRightArrow] \>\"", ",", RowBox[{"f", "[", "z", "]"}]}], "}"}], "]"}]}]}], "]"}]}]}]], "Input", CellChangeTimes->{{3.4925371580625*^9, 3.492537158375*^9}, { 3.49253719003125*^9, 3.492537234453125*^9}, {3.492537271984375*^9, 3.492537328765625*^9}, {3.492537373703125*^9, 3.49253737671875*^9}, { 3.492537752890625*^9, 3.492537759546875*^9}, {3.4925390265*^9, 3.49253902771875*^9}, {3.4925392063125*^9, 3.49253921934375*^9}, { 3.49253956159375*^9, 3.492539565671875*^9}, {3.492539605234375*^9, 3.492539659578125*^9}, {3.492539735796875*^9, 3.492539767796875*^9}, { 3.492618196265625*^9, 3.492618197140625*^9}, {3.49261836684375*^9, 3.49261836875*^9}, 3.49262443109375*^9, {3.4926245438125*^9, 3.49262454540625*^9}, 3.4926246198125*^9, {3.492624880078125*^9, 3.49262489703125*^9}, {3.49434032796875*^9, 3.49434033290625*^9}}], Cell[TextData[{ "That seems to give exactly the same plot as when ", StyleBox["ComplexCurve", FontFamily->"Courier"], " was used to parameterize the line segment. So why all the bother to \ parameterize? The explanation comes in the ", ButtonBox["following subsubsection", BaseStyle->"Hyperlink", ButtonData->"whyParameterize"], "." }], "Text", ShowCellTags->False, CellChangeTimes->{{3.49262490940625*^9, 3.49262494178125*^9}, { 3.492627759453125*^9, 3.4926277624375*^9}, 3.492634841*^9}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], ". Modify the definition of ", StyleBox["TwoPanelCartesianPlot", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " from Exercise ", CounterBox["Exercise", "defineTwoPanelCartesianPlot"], " so as to handle not just the grids, but additional graphical objects such \ as points, line segments, circles, and other curves that are to be \ highlighted." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.49252117265625*^9, 3.492521274703125*^9}, {3.49253414253125*^9, 3.49253465153125*^9}, { 3.492534953*^9, 3.4925349615*^9}, {3.49253500421875*^9, 3.49253501234375*^9}, {3.492536309625*^9, 3.492536348*^9}, { 3.492536390703125*^9, 3.4925364171875*^9}, {3.49253648115625*^9, 3.492536482375*^9}, {3.492634670296875*^9, 3.4926348015*^9}}, ParagraphSpacing->{0.5, 0}], Cell[CellGroupData[{ Cell["Why parameterize the curves to be mapped?", "Subsubsection", CellChangeTimes->{{3.49262754446875*^9, 3.49262756315625*^9}, { 3.492634862296875*^9, 3.49263486659375*^9}}, CellTags->"whyParameterize"], Cell["\<\ The short answer to the question is:\ \>", "Text", CellChangeTimes->{{3.49262778209375*^9, 3.492627925875*^9}, { 3.4926280001875*^9, 3.4926280040625*^9}, {3.49263488440625*^9, 3.49263491103125*^9}, {3.492635327140625*^9, 3.492635360640625*^9}}], Cell[TextData[{ Cell[BoxData[ FormBox[ StyleBox["\[WarningSign]", FontSize->18, FontWeight->"Bold", FontColor->RGBColor[1, 0, 0], Background->RGBColor[1, 1, 0]], TraditionalForm]]], " Y", StyleBox["ou may get an incorrect image if you use ", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["ComplexLine", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain", FontColor->RGBColor[0, 0, 1]], StyleBox[" instead of ", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["ComplexCurve", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain", FontColor->RGBColor[0, 0, 1]], ". Using ", StyleBox["ComplexLine", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " will give a correct image only if the image of the line segment is, in \ fact, another line segment! And likewise for ", StyleBox["ComplexCircle", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "." }], "PinkComments", CellFrame->True, CellChangeTimes->{{3.49262778209375*^9, 3.492627925875*^9}, { 3.4926280001875*^9, 3.4926280040625*^9}, {3.49263488440625*^9, 3.49263491103125*^9}, {3.492635327140625*^9, 3.492635361765625*^9}, { 3.4926365091875*^9, 3.492636509203125*^9}}], Cell[TextData[{ "It just so happened that, as you saw, the image of the line segment from 0 \ to ", Cell[BoxData[ FormBox[ RowBox[{"1", "+", "\[ImaginaryI]"}], TraditionalForm]]], " is, in fact, another line segment. But try a different line segment\ \[LongDash]", StyleBox["do the following exercise", Background->RGBColor[0.941, 0.973, 1.]], "." }], "Text", CellChangeTimes->{{3.49262793046875*^9, 3.4926279826875*^9}, { 3.49263509103125*^9, 3.49263509428125*^9}, {3.49263540153125*^9, 3.49263540515625*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " (a) By direct calculation determine the image under ", Cell[BoxData[ FormBox[ RowBox[{" ", FormBox[ RowBox[{ RowBox[{"f", "(", "z", ")"}], "=", SuperscriptBox["z", "2"]}], TraditionalForm]}], TraditionalForm]], FormatType->"TraditionalForm"], " of the line segment from ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"-", "1"}], "+", "\[ImaginaryI]"}], TraditionalForm]], FormatType->"TraditionalForm"], " to ", Cell[BoxData[ FormBox[ RowBox[{"1", "+", "\[ImaginaryI]"}], TraditionalForm]], FormatType->"TraditionalForm"], ".\n(b) Draw a two-panel plot of ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", "z", ")"}], "=", SuperscriptBox["z", "2"]}], TraditionalForm]], FormatType->"TraditionalForm"], " again, but now also highlight and map that line segment as well as its \ endpoints and its midpoint ", Cell[BoxData[ FormBox["\[ImaginaryI]", TraditionalForm]], FormatType->"TraditionalForm"], ". Use ", StyleBox["ComplexCurve", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " to parameterize the line segment.\n(c) Repeat (b) but instead use ", StyleBox["ComplexLine", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " for the line segment.\n(d) Which output gives the correct image of the \ line segment, from (b) or from (c)?" }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.492634956953125*^9, 3.492635281359375*^9}}, ParagraphSpacing->{0.5, 0}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Repeat the preceding exercise but instead of the line segment, highlight \ the circle with center ", Cell[BoxData[ FormBox["\[ImaginaryI]", TraditionalForm]], FormatType->"TraditionalForm"], " and radius 1/2." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.492634956953125*^9, 3.492635281359375*^9}, {3.492635731140625*^9, 3.49263578753125*^9}}, ParagraphSpacing->{0.5, 0}], Cell[TextData[{ "Why you must parameterize line segments and circles, using ", StyleBox["ComplexCurve", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", to obtain correct images under mapping? Here's the long answer." }], "Text", CellChangeTimes->{{3.492635987703125*^9, 3.492636022921875*^9}, { 3.492636912578125*^9, 3.49263693246875*^9}}], Cell[TextData[{ "The answer lies in the way that ", StyleBox["ComplexMap[f]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " calculates the images of the complex graphics objects. In case those \ objects are defined by means of the graphics primitive ", StyleBox["ComplexLine", FontFamily->"Courier"], ", the function ", StyleBox["ComplexMap[f]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " merely takes the particular points determining the graphics objects (here, \ the endpoints of the lines); calculates the images of those particular \ points; and finally forms the same kind of primitive graphics objects\ \[LongDash]lines, again\[LongDash]using those image points. (But under the \ squaring function, the image of a horizontal or vertical line does ", StyleBox["not", FontSlant->"Italic"], " need to be another line!)" }], "Text", ShowCellTags->False, CellChangeTimes->{{3.49263614265625*^9, 3.49263633196875*^9}, { 3.492636390140625*^9, 3.492636408015625*^9}, {3.492636742375*^9, 3.492636746546875*^9}}], Cell[TextData[{ "The same sort of thing happens with ", StyleBox["ComplexMap[f]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " in case you use ", StyleBox["ComplexCircle", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ". " }], "Text", CellChangeTimes->{{3.4926363425*^9, 3.492636366296875*^9}, { 3.49263643084375*^9, 3.492636453109375*^9}}], Cell[TextData[{ "On the other hand, as you saw earlier, the result of a ", StyleBox["ComplexCurve", FontFamily->"Courier"], " expression involves a whole long list of points along the curve. Then ", StyleBox["ComplexMap[f]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " calculates the images of ", StyleBox["all", FontSlant->"Italic"], " those points and connects the image points to obtain a new curve. So the \ displayed image is correct." }], "Text", ShowCellTags->False, CellChangeTimes->{ 3.4926364198125*^9, 3.49263647434375*^9, {3.49263675828125*^9, 3.49263676171875*^9}, {3.492636839328125*^9, 3.49263688803125*^9}}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Interactively moving objects under a map", "Subsection", CellChangeTimes->{{3.49281199184375*^9, 3.492812009703125*^9}}], Cell["Still use the function:", "Text", CellChangeTimes->{{3.49281249871875*^9, 3.4928125163125*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"f", "[", "z_", "]"}], ":=", SuperscriptBox["z", "2"]}]], "Input", CellChangeTimes->{{3.492812505984375*^9, 3.492812509421875*^9}}], Cell["And use the same Cartesian grid as before:", "Text", CellChangeTimes->{{3.49281264375*^9, 3.4928126553125*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"grid", "=", RowBox[{"DrawCartesianMap", "[", RowBox[{"z", ",", RowBox[{"{", RowBox[{"z", ",", RowBox[{ RowBox[{"-", "2"}], "-", RowBox[{"2", "I"}]}], ",", RowBox[{"2", "+", RowBox[{"2", "I"}]}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"BoundaryStyle", "\[Rule]", RowBox[{"Directive", "[", RowBox[{"Thick", ",", "Black"}], "]"}]}], ",", "\[IndentingNewLine]", RowBox[{"MeshStyle", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"HTML", "@", "SeaGreen"}], ",", RowBox[{"Darker", "@", "Brown"}]}], "}"}]}]}], "]"}]}], ";"}]], "Input", CellChangeTimes->{{3.492538045078125*^9, 3.492538075796875*^9}, { 3.492538160140625*^9, 3.492538172890625*^9}, 3.492538215890625*^9, { 3.49253895746875*^9, 3.492538958265625*^9}, {3.4926181190625*^9, 3.492618155578125*^9}, {3.492812670921875*^9, 3.49281267215625*^9}}], Cell[TextData[{ "But now, instead of a fixed geometric object such as a point or a line, use \ a parameterized point and line and then change the drawings of the domain and \ the image as you vary the parameters. This will be implemented by using the ", StyleBox["Mathematica", FontSlant->"Italic"], " function ", StyleBox["Manipulate", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], "." }], "Text", CellChangeTimes->{{3.492812780375*^9, 3.492812856421875*^9}}], Cell[TextData[{ "For example, we are going to move a horizontal line segment upwards in the \ grid, from bottom to top. At a height ", StyleBox["b", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", such as segment is given by:\n\t", StyleBox["ComplexCurve[(1-t)(-2+b\[ThinSpace]\[ImaginaryI])+t(2+b\[ThinSpace]\ \[ImaginaryI]),{t,0,1}]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"] }], "Text", CellChangeTimes->{{3.49281286396875*^9, 3.492812911671875*^9}, { 3.492813012765625*^9, 3.492813049828125*^9}, {3.492813476140625*^9, 3.492813603234375*^9}}, ParagraphSpacing->{0.5, 0}], Cell[BoxData[ RowBox[{"Draw2D", "[", RowBox[{ RowBox[{"{", "\[IndentingNewLine]", RowBox[{ RowBox[{"Directive", "[", RowBox[{"Thick", ",", "Red"}], "]"}], ",", "\[IndentingNewLine]", RowBox[{"ComplexCurve", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"1", "-", "t"}], ")"}], RowBox[{"(", RowBox[{ RowBox[{"-", "2"}], "+", RowBox[{"b", " ", "\[ImaginaryI]"}]}], ")"}]}], "+", RowBox[{"t", RowBox[{"(", RowBox[{"2", "+", RowBox[{"b", " ", "\[ImaginaryI]"}]}], ")"}]}]}], "/.", RowBox[{"b", "\[Rule]", "0.5"}]}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "1"}], "}"}]}], "]"}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"Axes", "\[Rule]", "True"}], ",", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "2"}], ",", "2"}], "}"}]}], "}"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.4928137768125*^9, 3.4928137796875*^9}}], Cell[TextData[{ "Now just take out the particular value of ", StyleBox["b", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " there, wrap the entire expression with ", StyleBox["Manipulate", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", and let ", StyleBox["b", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " vary from ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{"-", StyleBox["2", FontSlant->"Plain"]}], FontFamily->"Courier"], TraditionalForm]], FormatType->"TraditionalForm"], " to ", Cell[BoxData[ FormBox[ StyleBox["2", FontFamily->"Courier", FontSlant->"Plain"], TraditionalForm]], FormatType->"TraditionalForm"], "." }], "Text", CellChangeTimes->{{3.492813787546875*^9, 3.492813831578125*^9}, { 3.492813897234375*^9, 3.492813902625*^9}}], Cell[BoxData[ RowBox[{"Manipulate", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"Draw2D", "[", RowBox[{ RowBox[{"{", "\[IndentingNewLine]", RowBox[{ RowBox[{"Directive", "[", RowBox[{"Thick", ",", "Red"}], "]"}], ",", "\[IndentingNewLine]", RowBox[{"ComplexCurve", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"1", "-", "t"}], ")"}], RowBox[{"(", RowBox[{ RowBox[{"-", "2"}], "+", RowBox[{"b", " ", "\[ImaginaryI]"}]}], ")"}]}], "+", RowBox[{"t", RowBox[{"(", RowBox[{"2", "+", RowBox[{"b", " ", "\[ImaginaryI]"}]}], ")"}]}]}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "1"}], "}"}]}], "]"}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"Background", "->", RowBox[{"Opacity", "[", RowBox[{"0.5", ",", RowBox[{"HTML", "@", 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), InterpretationFunction -> (RowBox[{"Row", "[", RowBox[{"{", RowBox[{#, ",", #2}], "}"}], "]"}]& )], TraditionalForm], PlotRange->Automatic, PlotRangeClipping->True]], CellChangeTimes->{3.494433622609375*^9}]], "Text", Editable->False, Deletable->False, CellChangeTimes->{3.4944349248125*^9}, TextAlignment->Center], Cell[TextData[{ StyleBox["To create a polar grid", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], ", use the ", StyleBox["Presentations", FontSlant->"Italic"], " function ", StyleBox["DrawPolarMap", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], ". To specify a polar grid radiating from the origin, you give two points in \ polar form, using ", StyleBox["ComplexPolar", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ": \n\t", Cell[BoxData[ RowBox[{ RowBox[{"ComplexPolar", "[", RowBox[{ SubscriptBox["r", "mim"], ",", SubscriptBox["\[Theta]", "min"]}], "]"}], ",", " ", RowBox[{"ComplexPolar", "[", RowBox[{ SubscriptBox["r", "max"], ",", SubscriptBox["\[Theta]", "max"]}], "]"}]}]], CellChangeTimes->{{3.492638388140625*^9, 3.492638468953125*^9}}], "\nThen the grid will cover the region described in polar coordinates ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"r", ",", "\[Theta]"}], ")"}], TraditionalForm]], FormatType->"TraditionalForm"], " by\n\t", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["r", "min"], "\[LessEqual]", "r", "\[LessEqual]", SubscriptBox["r", "max"]}], TraditionalForm]], FormatType->"TraditionalForm"], ", ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["\[Theta]", "min"], "\[LessEqual]", "\[Theta]", "\[LessEqual]", SubscriptBox["\[Theta]", "max"]}], TraditionalForm]], FormatType->"TraditionalForm"], "\n a point on an inner circular arc bounding the region; a point on an \ outer circular arc bounding the region." }], "Text", CellChangeTimes->{{3.492637605609375*^9, 3.492637626734375*^9}, { 3.49263772321875*^9, 3.492637730953125*^9}, {3.492637848890625*^9, 3.492638005171875*^9}, {3.492638075328125*^9, 3.49263812446875*^9}, { 3.49263829340625*^9, 3.492638384859375*^9}, {3.492638481359375*^9, 3.492638688359375*^9}}, ParagraphSpacing->{0.5, 0}], Cell[TextData[{ "For example, here is a polar grid consisting of the disk inside the circle ", Cell[BoxData[ FormBox[ RowBox[{"r", "=", "2"}], TraditionalForm]], FormatType->"TraditionalForm"], ":" }], "Text", CellChangeTimes->{{3.492638702125*^9, 3.492638743875*^9}, { 3.492638809421875*^9, 3.4926388098125*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"polargrid", "=", RowBox[{"DrawPolarMap", "[", RowBox[{"z", ",", RowBox[{"{", RowBox[{"z", ",", RowBox[{"ComplexPolar", "[", RowBox[{"0", ",", "0"}], "]"}], ",", RowBox[{"ComplexPolar", "[", RowBox[{"2", ",", RowBox[{"2", "\[Pi]"}]}], "]"}]}], "}"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"Draw2D", "[", RowBox[{ RowBox[{"{", "polargrid", "}"}], ",", RowBox[{"Axes", "\[Rule]", "True"}], ",", RowBox[{"ImageSize", "\[Rule]", RowBox[{"3", " ", "72"}]}]}], "]"}]}], "Input", CellChangeTimes->{{3.491063009828125*^9, 3.491063011046875*^9}, { 3.491063378109375*^9, 3.4910633793125*^9}, 3.49247347171875*^9, { 3.492637712359375*^9, 3.49263771734375*^9}, {3.492637773328125*^9, 3.492637816859375*^9}, {3.492638040828125*^9, 3.492638056375*^9}, { 3.492638180046875*^9, 3.492638183328125*^9}, {3.492638757015625*^9, 3.492638765265625*^9}, {3.49280937925*^9, 3.492809401296875*^9}, { 3.49280951553125*^9, 3.49280952096875*^9}}], Cell["\<\ Here is a polar grid consisting of just the portion of that disk that lies in \ the first quadrant:\ \>", "Text", CellChangeTimes->{{3.492638774671875*^9, 3.492638820703125*^9}, 3.494343790265625*^9}], Cell[BoxData[{ RowBox[{ RowBox[{"polargrid", "=", RowBox[{"DrawPolarMap", "[", RowBox[{"z", ",", RowBox[{"{", RowBox[{"z", ",", RowBox[{"ComplexPolar", "[", RowBox[{"0", ",", "0"}], "]"}], ",", RowBox[{"ComplexPolar", "[", RowBox[{"2", ",", RowBox[{"\[Pi]", "/", "2"}]}], "]"}]}], "}"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"Draw2D", "[", RowBox[{ RowBox[{"{", "polargrid", "}"}], ",", RowBox[{"Axes", "\[Rule]", "True"}], ",", RowBox[{"ImageSize", "\[Rule]", RowBox[{"3", " ", "72"}]}]}], "]"}]}], "Input", CellChangeTimes->{{3.491063009828125*^9, 3.491063011046875*^9}, { 3.491063378109375*^9, 3.4910633793125*^9}, 3.49247347171875*^9, { 3.492637712359375*^9, 3.49263771734375*^9}, {3.492637773328125*^9, 3.492637816859375*^9}, {3.492638040828125*^9, 3.492638056375*^9}, { 3.492638180046875*^9, 3.492638183328125*^9}, {3.492638757015625*^9, 3.492638765265625*^9}, {3.492638832859375*^9, 3.492638838875*^9}, { 3.4928093831875*^9, 3.4928093976875*^9}, {3.49280952790625*^9, 3.492809536640625*^9}}], Cell["\<\ For a grid that is an annulus, or part of an annulus, do something like this:\ \>", "Text", CellChangeTimes->{{3.49263884990625*^9, 3.49263887803125*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"polargrid", "=", RowBox[{"DrawPolarMap", "[", RowBox[{"z", ",", RowBox[{"{", RowBox[{"z", ",", RowBox[{"ComplexPolar", "[", RowBox[{"1", ",", "0"}], "]"}], ",", RowBox[{"ComplexPolar", "[", RowBox[{"2", ",", RowBox[{"2", "\[Pi]"}]}], "]"}]}], "}"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"Draw2D", "[", RowBox[{ RowBox[{"{", "polargrid", "}"}], ",", RowBox[{"Axes", "\[Rule]", "True"}], ",", RowBox[{"ImageSize", "\[Rule]", RowBox[{"3", " ", "72"}]}]}], "]"}]}], "Input", CellChangeTimes->{{3.491063009828125*^9, 3.491063011046875*^9}, { 3.491063378109375*^9, 3.4910633793125*^9}, 3.49247347171875*^9, { 3.492637712359375*^9, 3.49263771734375*^9}, {3.492637773328125*^9, 3.492637816859375*^9}, {3.492638040828125*^9, 3.492638056375*^9}, { 3.492638180046875*^9, 3.492638183328125*^9}, {3.492638757015625*^9, 3.492638765265625*^9}, {3.492638832859375*^9, 3.492638838875*^9}, { 3.49263888809375*^9, 3.4926389124375*^9}, 3.492809393171875*^9, { 3.492809541296875*^9, 3.492809547265625*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Draw a polar grid consisting of the portion of the preceding annulus that \ lies in the upper half-plane." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.492638922765625*^9, 3.4926389545625*^9}}], Cell["\<\ Usually for understanding how functions map the plane in polar coordinates, \ you will want to use a polar region radiating from the origin. But to specify \ a polar grid radiating from some point other than the origin, you include \ that point as an optional third item in the specification list. For example: \ \>", "Text", CellChangeTimes->{{3.492637605609375*^9, 3.492637626734375*^9}, { 3.49263772321875*^9, 3.492637730953125*^9}, {3.492637848890625*^9, 3.492638005171875*^9}, {3.492638075328125*^9, 3.49263816446875*^9}, 3.49263827209375*^9, {3.492639000296875*^9, 3.492639122453125*^9}, 3.494441462859375*^9}], Cell[BoxData[{ RowBox[{ RowBox[{"polargrid", "=", RowBox[{"DrawPolarMap", "[", RowBox[{"z", ",", RowBox[{"{", RowBox[{"z", ",", RowBox[{"ComplexPolar", "[", RowBox[{"0", ",", "0"}], "]"}], ",", RowBox[{"ComplexPolar", "[", RowBox[{"2", ",", RowBox[{"2", "\[Pi]"}]}], "]"}], ",", RowBox[{"1", "+", "\[ImaginaryI]"}]}], "}"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"Draw2D", "[", RowBox[{ RowBox[{"{", "polargrid", "}"}], ",", RowBox[{"Axes", "\[Rule]", "True"}], ",", RowBox[{"AxesOrigin", "\[Rule]", RowBox[{"{", RowBox[{"0", ",", "0"}], "}"}]}], ",", RowBox[{"ImageSize", "\[Rule]", RowBox[{"3", " ", "72"}]}]}], "]"}]}], "Input", CellChangeTimes->{{3.491063009828125*^9, 3.491063011046875*^9}, { 3.491063378109375*^9, 3.4910633793125*^9}, 3.49247347171875*^9, { 3.492637712359375*^9, 3.49263771734375*^9}, {3.492637773328125*^9, 3.492637816859375*^9}, {3.492638040828125*^9, 3.492638056375*^9}, { 3.492638180046875*^9, 3.492638249046875*^9}, {3.492639136703125*^9, 3.4926392239375*^9}, {3.492809409015625*^9, 3.49280941090625*^9}, { 3.4928095524375*^9, 3.492809553046875*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Draw a polar grid consisting of the portion of the preceding disk whose \ points ", Cell[BoxData[ FormBox["z", TraditionalForm]], FormatType->"TraditionalForm"], " have ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"Im", "(", "z", ")"}], "\[GreaterEqual]", "1"}], TraditionalForm]], FormatType->"TraditionalForm"], "." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.492638922765625*^9, 3.4926389545625*^9}, {3.492639240875*^9, 3.492639285734375*^9}}], Cell["\<\ As with Cartesian grids, to change the number or location of the grid rays \ and arcs, use the Mesh option to DrawPolarMap. For example:\ \>", "Text", CellChangeTimes->{{3.494348951140625*^9, 3.49434899159375*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"polargrid", "=", RowBox[{"DrawPolarMap", "[", RowBox[{"z", ",", RowBox[{"{", RowBox[{"z", ",", RowBox[{"ComplexPolar", "[", RowBox[{"0", ",", "0"}], "]"}], ",", RowBox[{"ComplexPolar", "[", RowBox[{"2", ",", RowBox[{"\[Pi]", "/", "2"}]}], "]"}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"Mesh", "\[Rule]", RowBox[{"{", RowBox[{"5", ",", "4"}], "}"}]}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"Draw2D", "[", RowBox[{ RowBox[{"{", "polargrid", "}"}], ",", RowBox[{"Axes", "\[Rule]", "True"}], ",", RowBox[{"ImageSize", "\[Rule]", RowBox[{"3", " ", "72"}]}]}], "]"}]}], "Input", CellChangeTimes->{{3.491063009828125*^9, 3.491063011046875*^9}, { 3.491063378109375*^9, 3.4910633793125*^9}, 3.49247347171875*^9, { 3.492637712359375*^9, 3.49263771734375*^9}, {3.492637773328125*^9, 3.492637816859375*^9}, {3.492638040828125*^9, 3.492638056375*^9}, { 3.492638180046875*^9, 3.492638183328125*^9}, {3.492638757015625*^9, 3.492638765265625*^9}, {3.492638832859375*^9, 3.492638838875*^9}, { 3.4928093831875*^9, 3.4928093976875*^9}, {3.49280952790625*^9, 3.492809536640625*^9}, {3.494349012265625*^9, 3.494349020140625*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"polargrid", "=", RowBox[{"DrawPolarMap", "[", RowBox[{"z", ",", RowBox[{"{", RowBox[{"z", ",", RowBox[{"ComplexPolar", "[", RowBox[{"0", ",", "0"}], "]"}], ",", RowBox[{"ComplexPolar", "[", RowBox[{"2", ",", RowBox[{"\[Pi]", "/", "2"}]}], "]"}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"Mesh", "\[Rule]", RowBox[{"{", RowBox[{"5", ",", RowBox[{"{", RowBox[{ FractionBox["\[Pi]", "8"], ",", FractionBox["\[Pi]", "4"], ",", FractionBox[ RowBox[{"3", "\[Pi]"}], "8"]}], "}"}]}], "}"}]}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"Draw2D", "[", RowBox[{ RowBox[{"{", "polargrid", "}"}], ",", RowBox[{"Axes", "\[Rule]", "True"}], ",", RowBox[{"ImageSize", "\[Rule]", RowBox[{"3", " ", "72"}]}]}], "]"}]}], "Input", CellChangeTimes->{{3.491063009828125*^9, 3.491063011046875*^9}, { 3.491063378109375*^9, 3.4910633793125*^9}, 3.49247347171875*^9, { 3.492637712359375*^9, 3.49263771734375*^9}, {3.492637773328125*^9, 3.492637816859375*^9}, {3.492638040828125*^9, 3.492638056375*^9}, { 3.492638180046875*^9, 3.492638183328125*^9}, {3.492638757015625*^9, 3.492638765265625*^9}, {3.492638832859375*^9, 3.492638838875*^9}, { 3.4928093831875*^9, 3.4928093976875*^9}, {3.49280952790625*^9, 3.492809536640625*^9}, {3.494349012265625*^9, 3.49434909315625*^9}}], Cell[TextData[{ "Often you will want to use coloring for a polar grid. Do that essentially \ the same way as for a Cartesian grid:\n\t\[FilledSmallCircle] change the \ style of the radial line segments and concentric circles by using a ", StyleBox["MeshStyle", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], " option to ", StyleBox["DrawPolarMap", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ";\n\t\[FilledSmallCircle] change the style of the boundary of the polar \ region by using a ", StyleBox["BoundaryStyle", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], " option to ", StyleBox["DrawPolarMap", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "; and\n\t\[FilledSmallCircle] change the style of the background by a color \ directive in ", StyleBox["Draw2D", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "." }], "Text", CellChangeTimes->{{3.49263930534375*^9, 3.4926393215625*^9}, { 3.492639437578125*^9, 3.492639442078125*^9}, {3.494343839953125*^9, 3.494343878296875*^9}, 3.494413884546875*^9, 3.494441470578125*^9}, ParagraphSpacing->{0.5, 0.}], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"polargrid", "=", RowBox[{"DrawPolarMap", "[", RowBox[{"z", ",", RowBox[{"{", RowBox[{"z", ",", RowBox[{"ComplexPolar", "[", RowBox[{"0", ",", "0"}], "]"}], ",", RowBox[{"ComplexPolar", "[", RowBox[{"2", ",", RowBox[{"\[Pi]", "/", "2"}]}], "]"}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"BoundaryStyle", "\[Rule]", RowBox[{"Directive", "[", RowBox[{"Thick", ",", "Black"}], "]"}]}], ",", "\[IndentingNewLine]", RowBox[{"MeshStyle", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"Legacy", "@", "AlizarinCrimson"}], ",", RowBox[{"Legacy", "@", "CobaltGreen"}]}], "}"}]}]}], "]"}]}], ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{"polardomain", "=", RowBox[{"Draw2D", "[", RowBox[{ RowBox[{"{", "\[IndentingNewLine]", RowBox[{ RowBox[{"Opacity", "[", RowBox[{"0.5", ",", RowBox[{"HTML", "@", "Wheat"}]}], "]"}], ",", "\[IndentingNewLine]", "polargrid"}], "\[IndentingNewLine]", "}"}], ",", "\[IndentingNewLine]", RowBox[{"Axes", "\[Rule]", "True"}], ",", "\[IndentingNewLine]", RowBox[{"PlotLabel", "\[Rule]", RowBox[{"Row", "[", RowBox[{"{", RowBox[{"\"\\"", ",", "z"}], "}"}], "]"}]}], ",", "\[IndentingNewLine]", RowBox[{"ImageSize", "\[Rule]", RowBox[{"3", " ", "72"}]}]}], "]"}]}]}], "Input", CellChangeTimes->{{3.492639341234375*^9, 3.492639426171875*^9}, { 3.4926394631875*^9, 3.49263946778125*^9}, {3.49263949846875*^9, 3.492639499109375*^9}, {3.492639616953125*^9, 3.492639620875*^9}, { 3.4926397391875*^9, 3.492639742546875*^9}, {3.492639880796875*^9, 3.492639920546875*^9}, {3.49280936334375*^9, 3.4928093714375*^9}, { 3.492809560484375*^9, 3.492809561796875*^9}, {3.49280959778125*^9, 3.492809708125*^9}, {3.494348645140625*^9, 3.49434864621875*^9}, 3.494350460765625*^9}] }, Closed]], Cell[CellGroupData[{ Cell["Image of a polar grid under a function", "Subsection", CellChangeTimes->{{3.492639545625*^9, 3.492639550609375*^9}}], Cell["Still we use:", "Text", CellChangeTimes->{{3.492639791703125*^9, 3.49263979371875*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"f", "[", "z_", "]"}], ":=", SuperscriptBox["z", "2"]}]], "Input", CellChangeTimes->{{3.492639773078125*^9, 3.49263978271875*^9}}], Cell[TextData[{ "Mapping a polar grid by a function works essentially the same way as did \ mapping a Cartesian grid: in the ", StyleBox["DrawPolarMap", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " expression, replace the first argument ", StyleBox["z", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " by ", StyleBox["f[z]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ". That's all!" }], "Text", CellChangeTimes->{{3.492639637828125*^9, 3.49263969765625*^9}, { 3.492807816453125*^9, 3.492807822828125*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"imagepolargrid", "=", RowBox[{"DrawPolarMap", "[", RowBox[{ RowBox[{"f", "[", "z", "]"}], ",", RowBox[{"{", RowBox[{"z", ",", RowBox[{"ComplexPolar", "[", RowBox[{"0", ",", "0"}], "]"}], ",", RowBox[{"ComplexPolar", "[", RowBox[{"2", ",", RowBox[{"\[Pi]", "/", "2"}]}], "]"}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"BoundaryStyle", "\[Rule]", RowBox[{"Directive", "[", RowBox[{ RowBox[{"Opacity", "[", "1", "]"}], ",", "Thick", ",", "Black"}], "]"}]}], ",", "\[IndentingNewLine]", RowBox[{"MeshStyle", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"Legacy", "@", "AlizarinCrimson"}], ",", RowBox[{"Legacy", "@", "CobaltGreen"}]}], "}"}]}]}], "]"}]}], ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{"polarimage", "=", RowBox[{"Draw2D", "[", RowBox[{ RowBox[{"{", "\[IndentingNewLine]", RowBox[{ 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function maps a grid", "Subsection", CellChangeTimes->{{3.4925209805625*^9, 3.492521005890625*^9}, { 3.49253486421875*^9, 3.49253487584375*^9}, {3.494435060828125*^9, 3.49443506165625*^9}}], Cell[BoxData[ RowBox[{"GraphicsRow", "[", RowBox[{ RowBox[{"{", RowBox[{"polardomain", ",", "polarimage"}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"PlotLabel", "\[Rule]", RowBox[{"Row", "[", RowBox[{"{", RowBox[{ "\"\\"", ",", "z", ",", "\"\< \[LongRightArrow] \>\"", ",", RowBox[{"f", "[", "z", "]"}]}], "}"}], "]"}]}], ",", RowBox[{"PlotRange", "\[Rule]", "All"}]}], "]"}]], "Input", CellChangeTimes->{{3.491081526234375*^9, 3.491081533125*^9}, { 3.491081639546875*^9, 3.49108165778125*^9}, {3.49253241796875*^9, 3.49253247021875*^9}, 3.4925328145625*^9, {3.4925328485625*^9, 3.492532883734375*^9}, {3.492532954484375*^9, 3.49253299821875*^9}, { 3.492533084609375*^9, 3.49253318315625*^9}, {3.49253335209375*^9, 3.492533393296875*^9}, {3.492533499859375*^9, 3.492533523015625*^9}, { 3.4925335788125*^9, 3.49253361*^9}, {3.492533690109375*^9, 3.492533692609375*^9}, {3.4925337429375*^9, 3.492533834*^9}, { 3.492533869625*^9, 3.492533877078125*^9}, {3.49253390959375*^9, 3.492533913734375*^9}, {3.4925339531875*^9, 3.492534019953125*^9}, { 3.492640384828125*^9, 3.492640389421875*^9}, {3.492809828078125*^9, 3.49280983228125*^9}}], Cell[TextData[{ "Once again, using ", StyleBox["GraphicsRow", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " forces different scales in the two drawings because it makes the widths of \ the two drawings the same." }], "Text", CellChangeTimes->{{3.494349239484375*^9, 3.494349284328125*^9}, { 3.4943503021875*^9, 3.4943503110625*^9}}], Cell[TextData[{ "There are two ways ", StyleBox["to preserve the same scale for the drawings", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], ": \n\t(1) use ", StyleBox["Row", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], " instead of ", StyleBox["GraphicsRow", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ".\n\t(2) still use ", StyleBox["GraphicsRow", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", but specify the same PlotRange for both drawings." }], "Text", CellChangeTimes->{{3.494413953078125*^9, 3.4944140536875*^9}, { 3.494414176015625*^9, 3.494414177015625*^9}}, ParagraphSpacing->{0.5, 0}], Cell[TextData[{ "In our example, use ", StyleBox["Row", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " instead of ", StyleBox["GraphicsRow", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ":" }], "Text", CellChangeTimes->{{3.49441406746875*^9, 3.49441412865625*^9}}], Cell[BoxData[ RowBox[{"Row", "[", RowBox[{"{", RowBox[{"polardomain", ",", "polarimage"}], "}"}], "]"}]], "Input", CellChangeTimes->{{3.49434933084375*^9, 3.49434933284375*^9}}], Cell[TextData[{ "Then to label the entire figure, stack the pair of drawings and a label \ into a ", StyleBox["Column", FontFamily->"Courier", 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Repeat the preceding exercise but for ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"h", "(", "z", ")"}], "=", RowBox[{"1", "/", "z"}]}], TraditionalForm]]], "." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.49252117265625*^9, 3.492521274703125*^9}, {3.492617839734375*^9, 3.492617900421875*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], ". Encapsulate in a function ", StyleBox["TwoPanelPolarPlot", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " the entire process of creating such a drawings such as above of how a \ complex function maps a polar grid. This function should be an analog of the \ function ", StyleBox["TwoPanelCartesianPlot", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " requested in Exercise ", CounterBox["Exercise", "defineTwoPanelCartesianPlot"], " should include at least:\n\t\[FilledSmallCircle] the variable name ", StyleBox["z", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], ";\n\t\[FilledSmallCircle] a function name ", StyleBox["f", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], "; and\n\t\[FilledSmallCircle] a list ", StyleBox["{", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], StyleBox["zmin", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[",", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], StyleBox["zmax", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["}", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " indicating the lower-left and upper-right corner of the grid.\n(You may \ wish to include additional arguments to specify grid segment coloring, etc.)\n\ For example, to reproduce the preceding drawing of the squaring function, \ after having defined ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ RowBox[{ StyleBox["f", FontSlant->"Plain"], "[", "z_", "]"}], FontFamily->"Courier"], ":=", SuperscriptBox["z", "2"]}], TraditionalForm]]], ", you would invoke:\n\t", Cell[BoxData[ FormBox[ RowBox[{"TwoPanelCartesianPlot", "[", RowBox[{ StyleBox["z", FontSlant->"Plain"], ",", StyleBox["f", FontSlant->"Plain"], ",", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"-", "2"}], "-", RowBox[{"2", "\[ImaginaryI]"}]}], ",", RowBox[{"2", "+", RowBox[{"2", "\[ImaginaryI]"}]}]}], "}"}]}], "]"}], TraditionalForm]], FontFamily->"Courier"] }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.49252117265625*^9, 3.492521274703125*^9}, {3.49253414253125*^9, 3.49253465153125*^9}, { 3.492534953*^9, 3.4925349615*^9}, {3.49253500421875*^9, 3.49253501234375*^9}, {3.492808889796875*^9, 3.492808958546875*^9}, { 3.492812209578125*^9, 3.4928122115*^9}, 3.494350478109375*^9}, ParagraphSpacing->{0.5, 0}, CellTags->{"defineTwoPanelCartesianPlot", "defineTwoPanelPolarPlot"}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], ". Modify your function ", StyleBox["TwoPanelPolarPlot", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " from Exercise ", CounterBox["Exercise", "defineTwoPanelPolarPlot"], " so as to produce output suitable for printing on a black-and-white \ printer. Perhaps name the modified function ", StyleBox["TwoPanelPolarPlotPrintVersion", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ". 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We'll define the additional objects first." }], "Text", CellChangeTimes->{{3.492810483578125*^9, 3.49281058215625*^9}, { 3.4928106864375*^9, 3.492810701140625*^9}, 3.4943505425*^9, { 3.494350770828125*^9, 3.494350772*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"polargrid", "=", RowBox[{"DrawPolarMap", "[", RowBox[{"z", ",", RowBox[{"{", RowBox[{"z", ",", RowBox[{"ComplexPolar", "[", RowBox[{"0", ",", "0"}], "]"}], ",", RowBox[{"ComplexPolar", "[", RowBox[{"2", ",", RowBox[{"\[Pi]", "/", "2"}]}], "]"}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"BoundaryStyle", "\[Rule]", RowBox[{"Directive", "[", RowBox[{"Thick", ",", "Black"}], "]"}]}], ",", "\[IndentingNewLine]", RowBox[{"MeshStyle", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"Legacy", "@", "AlizarinCrimson"}], ",", RowBox[{"Legacy", "@", "CobaltGreen"}]}], "}"}]}]}], "]"}]}], ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"ray", "=", RowBox[{"ComplexCurve", "[", RowBox[{ RowBox[{"t", RowBox[{"(", RowBox[{"2", RowBox[{"Exp", "[", RowBox[{"\[ImaginaryI]", " ", RowBox[{"\[Pi]", "/", "3"}]}], "]"}]}], ")"}]}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "1"}], "}"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"arc", "=", RowBox[{"ComplexCurve", "[", RowBox[{ RowBox[{"1.5", " ", RowBox[{"Exp", "[", RowBox[{"\[ImaginaryI]", " ", "\[Theta]"}], "]"}]}], ",", RowBox[{"{", RowBox[{"\[Theta]", ",", "0", ",", RowBox[{"\[Pi]", "/", "2"}]}], "}"}]}], "]"}]}], ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"polardomainObjects", "=", RowBox[{"{", "\[IndentingNewLine]", RowBox[{ RowBox[{"Opacity", "[", RowBox[{"0.5", ",", RowBox[{"HTML", "@", "Wheat"}]}], "]"}], ",", "\[IndentingNewLine]", "polargrid", ",", "\[IndentingNewLine]", RowBox[{"Directive", "[", RowBox[{"Thick", ",", RowBox[{"Darker", "@", RowBox[{"Legacy", "@", "CobaltGreen"}]}]}], "]"}], ",", "arc", ",", "\[IndentingNewLine]", RowBox[{"Directive", "[", RowBox[{"Thick", ",", RowBox[{"Legacy", "@", "CadmiumOrange"}]}], "]"}], ",", "ray"}], "\[IndentingNewLine]", "}"}]}], ";"}], "\n"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"polardomain", "=", RowBox[{"Draw2D", "[", RowBox[{ RowBox[{"{", "polardomainObjects", "}"}], ",", "\[IndentingNewLine]", RowBox[{"Axes", "\[Rule]", "True"}], ",", RowBox[{"PlotLabel", "\[Rule]", RowBox[{"Row", "[", RowBox[{"{", RowBox[{"\"\\"", ",", "z"}], "}"}], "]"}]}], ",", "\[IndentingNewLine]", RowBox[{"ImageSize", "\[Rule]", RowBox[{"3", " ", "72"}]}]}], "]"}]}], ";"}], "\n"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"polarimage", "=", RowBox[{"Draw2D", "[", RowBox[{ RowBox[{"{", RowBox[{"polardomainObjects", "//", RowBox[{"ComplexMap", "[", "f", "]"}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"Axes", "\[Rule]", "True"}], ",", RowBox[{"PlotLabel", "\[Rule]", RowBox[{"Row", "[", RowBox[{"{", RowBox[{"\"\\"", ",", "z"}], "}"}], "]"}]}], ",", "\[IndentingNewLine]", RowBox[{"ImageSize", "\[Rule]", RowBox[{"3", " ", "72"}]}]}], "]"}]}], ";"}], "\n"}], "\[IndentingNewLine]", RowBox[{"GraphicsRow", "[", RowBox[{ RowBox[{"{", RowBox[{"polardomain", ",", "polarimage"}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"PlotLabel", "\[Rule]", RowBox[{"Row", "[", RowBox[{"{", RowBox[{ "\"\\"", ",", "z", ",", "\"\< \[LongRightArrow] \>\"", ",", RowBox[{"f", "[", "z", "]"}]}], "}"}], "]"}]}]}], "]"}]}], "Input", CellChangeTimes->{{3.492810704203125*^9, 3.49281109115625*^9}, { 3.494362537703125*^9, 3.49436253825*^9}, {3.494435194140625*^9, 3.494435195515625*^9}}], Cell[TextData[{ "Again, remember that ", StyleBox["you ", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["must", FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" use ", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["ComplexCurve", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain", FontColor->RGBColor[0, 0, 1]], StyleBox[", and ", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["not", FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" ", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["ComplexLine", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain", FontColor->RGBColor[0, 0, 1]], StyleBox[" or ", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["ComplexCircle", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain", FontColor->RGBColor[0, 0, 1]], ", in order that the images of the highlighted objects be calculated \ correctly." }], "Text", CellChangeTimes->{{3.492811130546875*^9, 3.492811206453125*^9}, { 3.49435056059375*^9, 3.494350565609375*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], ". (a) Use the two-panel polar plot output to describe what the images of \ rays from the origin and circles centered at the origin are under ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", "z", ")"}], "=", SuperscriptBox["z", "2"]}], TraditionalForm]], FormatType->"TraditionalForm"], " seem to be.\n(b) Through appropriate calculations, with or without ", StyleBox["Mathematica", FontSlant->"Italic"], ", make your description quantitatively precise and prove that it is \ correct. 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In a two-panel polar plot with highlighting of a ray and an arc such as \ above, visualize how ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"g", "(", "z", ")"}], "=", SuperscriptBox["z", "3"]}], TraditionalForm]]], " maps the complex plane." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.49252117265625*^9, 3.492521274703125*^9}, {3.492617839734375*^9, 3.492617880640625*^9}, { 3.49441535190625*^9, 3.494415354875*^9}, {3.494415556640625*^9, 3.494415557703125*^9}, {3.494415638875*^9, 3.494415648234375*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], ". Repeat the preceding exercise but for ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"h", "(", "z", ")"}], "=", RowBox[{"1", "/", "z"}]}], TraditionalForm]]], "." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.49252117265625*^9, 3.492521274703125*^9}, {3.492617839734375*^9, 3.492617900421875*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["Interactively moving objects under a map", "Subsection", CellChangeTimes->{{3.49281199184375*^9, 3.492812009703125*^9}, { 3.4943506230625*^9, 3.49435062359375*^9}, 3.49441567034375*^9}], Cell["Still use the function:", "Text", CellChangeTimes->{{3.49281249871875*^9, 3.4928125163125*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"f", "[", "z_", "]"}], ":=", SuperscriptBox["z", "2"]}]], "Input", CellChangeTimes->{{3.492812505984375*^9, 3.492812509421875*^9}}], Cell["And use the same polar grid as before:", "Text", CellChangeTimes->{{3.49281264375*^9, 3.4928126553125*^9}, {3.494350648625*^9, 3.494350649203125*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"polargrid", "=", RowBox[{"DrawPolarMap", "[", RowBox[{"z", ",", RowBox[{"{", RowBox[{"z", ",", RowBox[{"ComplexPolar", "[", RowBox[{"0", ",", "0"}], "]"}], ",", RowBox[{"ComplexPolar", "[", RowBox[{"2", ",", RowBox[{"\[Pi]", "/", "2"}]}], "]"}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"BoundaryStyle", "\[Rule]", RowBox[{"Directive", "[", RowBox[{"Thick", ",", "Black"}], "]"}]}], ",", "\[IndentingNewLine]", RowBox[{"MeshStyle", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"Legacy", "@", "AlizarinCrimson"}], ",", RowBox[{"Legacy", "@", "CobaltGreen"}]}], "}"}]}]}], "]"}]}], ";"}]], "Input", CellChangeTimes->{{3.49435112959375*^9, 3.494351130328125*^9}, 3.494351237921875*^9}], Cell[TextData[{ "But now, instead of a fixed geometric object such as a line segment or a \ circular arc, use a parameterized point and line and then change the drawings \ of the domain and the image as you vary the parameters. To implement this, \ use the ", StyleBox["Mathematica", FontSlant->"Italic"], " function ", StyleBox["Manipulate", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], "." }], "Text", CellChangeTimes->{{3.492812780375*^9, 3.492812856421875*^9}, { 3.49435066834375*^9, 3.494350684515625*^9}, {3.49435073834375*^9, 3.494350761546875*^9}, 3.4943514625625*^9}], Cell[TextData[{ "We are going to rotate a ray counterclockwise around the origin and move \ circular arcs outward from the origin. 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In a two-panel polar plot with dynamic highlighting of rays and arcs such \ as above, visualize how ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"g", "(", "z", ")"}], "=", SuperscriptBox["z", "3"]}], TraditionalForm]]], " maps the complex plane." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.49252117265625*^9, 3.492521274703125*^9}, {3.492617839734375*^9, 3.492617880640625*^9}, { 3.49441535190625*^9, 3.494415354875*^9}, {3.494415608609375*^9, 3.494415621765625*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], ". 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Include other suitable geometric objects so as to \ better indicate how ", Cell[BoxData[ FormBox["f", TraditionalForm]], FormatType->"TraditionalForm"], " maps the complex plane." }], "Exercise", ShowCellTags->False, CellChangeTimes->{ 3.492520123078125*^9, {3.49443548134375*^9, 3.494435489109375*^9}, { 3.494439998453125*^9, 3.49444033025*^9}, {3.494440363421875*^9, 3.494440444234375*^9}}, CellTags->"plotConjugate"], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Using the figures you produce in Exercise ", CounterBox["Exercise", "plotConjugate"], ", repeat Exercise ", CounterBox["Exercise", "describeMultiplyByI"], " for the function ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", "z", ")"}], "=", RowBox[{"z", "\[Conjugate]"}]}], TraditionalForm]], FormatType->"TraditionalForm"], "." }], "Exercise", ShowCellTags->False, CellChangeTimes->{ 3.492520123078125*^9, {3.49443548134375*^9, 3.494435489109375*^9}, { 3.494439998453125*^9, 3.49444033025*^9}, {3.494440363421875*^9, 3.494440444234375*^9}, {3.49444048609375*^9, 3.494440541984375*^9}, 3.49444057646875*^9, 3.49444061553125*^9}, CellTags->"plotConjugate"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Appendix: Visualizing by using the built-in ", StyleBox["ParametricPlot", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"] }], "Section", ShowGroupOpener->True, ShowCellTags->False, CellChangeTimes->{{3.492293254*^9, 3.492293264359375*^9}, { 3.492293326140625*^9, 3.492293338890625*^9}, 3.49279435390625*^9, { 3.492797921875*^9, 3.4927979403125*^9}, {3.49279951325*^9, 3.49279951765625*^9}}, CellTags->"appendix 1"], Cell[TextData[{ "There is an alternative way to visualize complex maps that does not use ", StyleBox["Presentations", FontSlant->"Italic"], " at all, with built-in ", StyleBox["Mathematica", FontSlant->"Italic"], " functions only." }], "Text", ShowCellTags->False, CellChangeTimes->{{3.492293430203125*^9, 3.49229349425*^9}, { 3.492293680109375*^9, 3.492293690453125*^9}, {3.4922937481875*^9, 3.492293821828125*^9}, {3.49279499959375*^9, 3.49279507396875*^9}}], Cell[TextData[{ "Although this may be a handy way to do things when you don't have access to \ ", StyleBox["Presentations", FontSlant->"Italic"], ", it is clumsier to handle because you must deal directly with the real and \ imaginary parts of complex numbers. Moreover, you must also handle \ highlighted objects by using ", StyleBox["Prolog", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " (or ", StyleBox["Epilog", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ") options, or through other means of combining graphics." }], "Text", ShowCellTags->False, CellChangeTimes->{{3.492293430203125*^9, 3.49229349425*^9}, { 3.492293680109375*^9, 3.492293690453125*^9}, {3.4922937481875*^9, 3.492293821828125*^9}, {3.49279499959375*^9, 3.49279510725*^9}, { 3.4944353883125*^9, 3.494435389796875*^9}}], Cell[CellGroupData[{ Cell[TextData[{ "Mapping in terms of Cartesian coordinates\[LongDash]using ", StyleBox["ParametricPlot", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], " " }], "Subsection", ShowGroupOpener->True, ShowCellTags->False, CellChangeTimes->{{3.492293834046875*^9, 3.4922938433125*^9}}, CellTags->"CartesianParametrically"], Cell["Return here to the function of example 1:", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"f", "[", "z_", "]"}], " ", ":=", " ", SuperscriptBox["z", "2"]}]], "Input"], Cell[TextData[{ "First we create a Cartesian grid using the built-in ", StyleBox["Mathematica", FontSlant->"Italic"], " function ", StyleBox["ParametricPlot", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], ". The parameters will be ", StyleBox["x", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " and ", StyleBox["y", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " when ", StyleBox["z", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " is expressed in Cartesian form as ", StyleBox["x+y\[ThinSpace]\[ImaginaryI]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "." }], "Text", CellChangeTimes->{{3.492294585*^9, 3.49229461440625*^9}, 3.492520123171875*^9, 3.492794488328125*^9, {3.492794729671875*^9, 3.492794795609375*^9}}], Cell[BoxData[ RowBox[{"dom", "=", RowBox[{"ParametricPlot", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Re", "[", RowBox[{"x", "+", RowBox[{"y", " ", "\[ImaginaryI]"}]}], "]"}], ",", RowBox[{"Im", "[", RowBox[{"x", "+", RowBox[{"y", " ", "\[ImaginaryI]"}]}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", "All"}], ",", RowBox[{"ImageSize", "\[Rule]", RowBox[{"3", " ", "72"}]}]}], "]"}]}]], "Input", CellChangeTimes->{{3.492293562171875*^9, 3.492293634484375*^9}, { 3.492294481046875*^9, 3.492294497703125*^9}}], Cell[TextData[{ "You can shorten the code a bit by using ", StyleBox["Mathematica", FontSlant->"Italic"], "'s ", StyleBox["Through", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " function. For two functions ", StyleBox["g", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " and ", StyleBox["h", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", the expression ", StyleBox["Through[{g,h}][arg]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " means the same thing as ", StyleBox["{g[arg],h[arg]}", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ". Thus ", StyleBox["Through", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " avoids repeating the same argument. For example:" }], "Text", CellChangeTimes->{{3.492293873546875*^9, 3.492293956109375*^9}, { 3.492294371828125*^9, 3.492294419015625*^9}, 3.492294451203125*^9}], Cell[BoxData[ RowBox[{"Through", "[", RowBox[{ RowBox[{"{", RowBox[{"Exp", ",", "Sin"}], "}"}], "[", RowBox[{"\[Pi]", " ", "\[ImaginaryI]"}], "]"}], "]"}]], "Input", CellChangeTimes->{{3.49229396140625*^9, 3.492294001515625*^9}, { 3.492294035515625*^9, 3.492294089578125*^9}}], Cell[TextData[{ "The way we shall exploit it is with ", StyleBox["{Re,Im}", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ". For example:" }], "Text", CellChangeTimes->{{3.4922940146875*^9, 3.4922940286875*^9}, { 3.492294093296875*^9, 3.492294161984375*^9}, {3.492294321765625*^9, 3.4922943400625*^9}, {3.492294443484375*^9, 3.492294447609375*^9}}], Cell[BoxData[ RowBox[{"Through", "[", RowBox[{ RowBox[{"{", RowBox[{"Re", ",", "Im"}], "}"}], "[", RowBox[{ SuperscriptBox["\[ExponentialE]", "2"], "+", RowBox[{"3", SuperscriptBox["\[ImaginaryI]", "3"]}]}], "]"}], "]"}]], "Input", CellChangeTimes->{{3.49229416528125*^9, 3.492294186375*^9}, { 3.492294220046875*^9, 3.49229424296875*^9}, {3.492294277859375*^9, 3.49229429496875*^9}, {3.492294344375*^9, 3.492294348765625*^9}}], Cell["\<\ To avoid parentheses nested too deeply, we shall use prefix syntax, with @. \ For example:\ \>", "Text", CellChangeTimes->{{3.4922947115625*^9, 3.492294784*^9}}], Cell[BoxData[ RowBox[{"Through", "@", RowBox[{ RowBox[{"{", RowBox[{"Re", ",", "Im"}], "}"}], "[", RowBox[{ SuperscriptBox["\[ExponentialE]", "2"], "+", RowBox[{"3", SuperscriptBox["\[ImaginaryI]", "3"]}]}], "]"}]}]], "Input", CellChangeTimes->{{3.49229416528125*^9, 3.492294186375*^9}, { 3.492294220046875*^9, 3.49229424296875*^9}, {3.492294277859375*^9, 3.49229429496875*^9}, {3.492294344375*^9, 3.492294348765625*^9}, { 3.492294770265625*^9, 3.4922947720625*^9}}], Cell[BoxData[ RowBox[{"dom", "=", RowBox[{"ParametricPlot", "[", RowBox[{ RowBox[{"Through", "@", RowBox[{ RowBox[{"{", RowBox[{"Re", ",", "Im"}], "}"}], "[", RowBox[{"x", "+", RowBox[{"y", " ", "\[ImaginaryI]"}]}], "]"}]}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", "All"}], ",", RowBox[{"PlotLabel", "\[Rule]", "\"\\""}], ",", RowBox[{"ImageSize", "\[Rule]", RowBox[{"3", " ", "72"}]}]}], "]"}]}]], "Input", CellChangeTimes->{{3.492293562171875*^9, 3.492293634484375*^9}, { 3.492294481046875*^9, 3.492294497703125*^9}, {3.492294530765625*^9, 3.49229454534375*^9}, {3.492294790390625*^9, 3.492294793421875*^9}, { 3.4922949183125*^9, 3.492294927125*^9}}], Cell[TextData[{ "Now form the image merely by replacing ", StyleBox["x+y\[ThinSpace]\[ImaginaryI]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " with ", StyleBox["f[x+\[ImaginaryI]\[ThinSpace]y]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " in the input for the domain." }], "Text", CellChangeTimes->{{3.49279450815625*^9, 3.49279457203125*^9}, 3.49279481534375*^9}], Cell[BoxData[ RowBox[{"image", "=", RowBox[{"ParametricPlot", "[", RowBox[{ RowBox[{"Through", "@", RowBox[{ RowBox[{"{", RowBox[{"Re", ",", "Im"}], "}"}], "[", RowBox[{"f", "[", RowBox[{"x", "+", RowBox[{"y", " ", "\[ImaginaryI]"}]}], "]"}], "]"}]}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", "All"}], ",", RowBox[{"PlotLabel", "\[Rule]", "\"\\""}], ",", RowBox[{"ImageSize", "\[Rule]", RowBox[{"3", " ", "72"}]}]}], "]"}]}]], "Input", CellChangeTimes->{{3.492794431375*^9, 3.49279446096875*^9}}], Cell[TextData[{ "As when ", StyleBox["Presentations", FontSlant->"Italic"], " is used, you may now use a ", StyleBox["GraphicsRow", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " to display the two drawings side-by-side" }], "Text", CellChangeTimes->{{3.49279458934375*^9, 3.49279462259375*^9}}] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Mapping in terms of polar coordinates\[LongDash]using ", StyleBox["ParametricPlot", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"] }], "Subsection", ShowGroupOpener->True, ShowCellTags->False, CellChangeTimes->{{3.492794686234375*^9, 3.4927946906875*^9}, { 3.492795215671875*^9, 3.492795217*^9}}], Cell["Still use here to the function of example 1:", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"f", "[", "z_", "]"}], " ", ":=", " ", SuperscriptBox["z", "2"]}]], "Input"], Cell[TextData[{ "The trick is to use the trig form ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"x", "+", RowBox[{"y", " ", "\[ImaginaryI]"}]}], "=", RowBox[{ RowBox[{"r", " ", "cos", " ", "\[Theta]"}], " ", "+", " ", RowBox[{"\[ImaginaryI]", " ", "r", " ", "sin", " ", "\[Theta]"}]}]}], TraditionalForm]], FormatType->"TraditionalForm"], " for complex numbers and then regard the latter as Cartesian coordinates \ parameterized by ", Cell[BoxData[ FormBox["x", TraditionalForm]], FormatType->"TraditionalForm"], " and ", Cell[BoxData[ FormBox["y", TraditionalForm]], FormatType->"TraditionalForm"], ". 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Recall that in ", StyleBox["Mathematica,", FontSlant->"Italic"], " the functions ", Cell[BoxData[ FormBox["U", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["V", TraditionalForm]]], " are given by:\t" }], "Text", CellChangeTimes->{{3.492797515109375*^9, 3.492797586125*^9}, { 3.49279774653125*^9, 3.49279775334375*^9}, {3.492798646765625*^9, 3.492798665703125*^9}, {3.492798811078125*^9, 3.492798823171875*^9}, { 3.49279937940625*^9, 3.49279940540625*^9}, {3.494441528140625*^9, 3.49444153290625*^9}}, ParagraphSpacing->{0.5, 0}], Cell[BoxData[{ RowBox[{ RowBox[{"U", "[", RowBox[{"r_", ",", "\[Theta]_"}], "]"}], ":=", RowBox[{"ComplexExpand", "@", RowBox[{"Re", "[", RowBox[{"f", "[", RowBox[{"r", " ", RowBox[{"Exp", "[", RowBox[{"\[ImaginaryI]", " ", "\[Theta]"}], "]"}]}], "]"}], "]"}]}]}], "\n", RowBox[{ RowBox[{"V", "[", RowBox[{"r_", ",", "\[Theta]_"}], "]"}], ":=", RowBox[{"ComplexExpand", "@", RowBox[{"Im", "[", RowBox[{"f", "[", RowBox[{"r", " ", RowBox[{"Exp", "[", RowBox[{"\[ImaginaryI]", " ", "\[Theta]"}], "]"}]}], "]"}], "]"}]}]}], "\[IndentingNewLine]", RowBox[{"{", RowBox[{ RowBox[{"U", "[", RowBox[{"r", ",", "\[Theta]"}], "]"}], ",", RowBox[{"V", "[", RowBox[{"r", ",", "\[Theta]"}], "]"}]}], "}"}]}], "Input", ShowCellTags->False, CellChangeTimes->{{3.492637317828125*^9, 3.492637347421875*^9}, { 3.492799114171875*^9, 3.49279912803125*^9}}], Cell[TextData[{ "Now use the values of ", Cell[BoxData[ FormBox["U", TraditionalForm]], FormatType->"TraditionalForm"], " and ", Cell[BoxData[ FormBox["V", TraditionalForm]], FormatType->"TraditionalForm"], " as the ", Cell[BoxData[ FormBox["x", TraditionalForm]], FormatType->"TraditionalForm"], "- and ", Cell[BoxData[ FormBox["y", TraditionalForm]], FormatType->"TraditionalForm"], "-coordinates for the parametric plot of the image:" }], "Text", CellChangeTimes->{{3.492799425609375*^9, 3.4927994739375*^9}}], Cell[BoxData[ RowBox[{"ParametricPlot", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"U", "[", RowBox[{"r", ",", "\[Theta]"}], "]"}], ",", RowBox[{"V", "[", RowBox[{"r", ",", "\[Theta]"}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{"\[Theta]", ",", "0", ",", RowBox[{"\[Pi]", "/", "2"}]}], "}"}], ",", RowBox[{"{", RowBox[{"r", ",", "0", ",", "2"}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", "All"}], ",", RowBox[{"PlotLabel", "\[Rule]", "\"\\""}], ",", RowBox[{"ImageSize", "\[Rule]", RowBox[{"3", " ", "72"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.492799290921875*^9, 3.492799340375*^9}, 3.4927994811875*^9}], Cell["\<\ Actually, there is no need to define U and V first, as above. Instead, do it \ directly:\ \>", "Text", CellChangeTimes->{{3.49280712878125*^9, 3.4928071810625*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"ComplexExpand", "@", RowBox[{"Re", "[", RowBox[{"f", "[", RowBox[{"r", " ", RowBox[{"Exp", "[", RowBox[{"\[ImaginaryI]", " ", "\[Theta]"}], "]"}]}], "]"}], "]"}]}], ",", RowBox[{"ComplexExpand", "@", RowBox[{"Im", "[", RowBox[{"f", "[", RowBox[{"r", " ", RowBox[{"Exp", "[", RowBox[{"\[ImaginaryI]", " ", "\[Theta]"}], "]"}]}], "]"}], "]"}]}]}], "}"}]], "Input", CellChangeTimes->{{3.492807186515625*^9, 3.492807228078125*^9}}], Cell[TextData[{ "That expression is rather long, though. As in the preceding subsection, ", StyleBox["Mathematica", FontSlant->"Italic"], "'s ", StyleBox["Through", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " can shorten that:" }], "Text", CellChangeTimes->{{3.492807233953125*^9, 3.49280726575*^9}}], Cell[BoxData[ RowBox[{"ComplexExpand", "/@", RowBox[{"Through", "[", RowBox[{ RowBox[{"{", RowBox[{"Re", ",", "Im"}], "}"}], "[", RowBox[{"f", "[", RowBox[{"r", " ", RowBox[{"Exp", "[", RowBox[{"\[ImaginaryI]", " ", "\[Theta]"}], "]"}]}], "]"}], "]"}], "]"}]}]], "Input", CellChangeTimes->{{3.49280727090625*^9, 3.492807341296875*^9}, { 3.492807397796875*^9, 3.492807427078125*^9}}], Cell[BoxData[ RowBox[{"ParametricPlot", "[", RowBox[{ RowBox[{"ComplexExpand", "/@", RowBox[{"Through", "[", RowBox[{ RowBox[{"{", RowBox[{"Re", ",", "Im"}], "}"}], "[", RowBox[{"f", "[", RowBox[{"r", " ", RowBox[{"Exp", "[", RowBox[{"\[ImaginaryI]", " ", "\[Theta]"}], "]"}]}], "]"}], "]"}], "]"}]}], ",", RowBox[{"{", RowBox[{"\[Theta]", ",", 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