(* 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[ 748355, 19581] NotebookOptionsPosition[ 692403, 17883] NotebookOutlinePosition[ 697293, 18016] CellTagsIndexPosition[ 696017, 17984] WindowFrame->Normal ContainsDynamic->True *) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"2", "-", "2"}]], "Input", CellChangeTimes->{{3.3949044134974327`*^9, 3.394904422720999*^9}}], Cell[BoxData["0"], "Output", CellChangeTimes->{3.395004135038413*^9, 3.395160912296954*^9}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Physicist's Introduction to ", StyleBox["Mathematica", FontSlant->"Italic"], " 6" }], "Title", CellChangeTimes->{{3.394579470096292*^9, 3.3945794805215673`*^9}, 3.3959264266064987`*^9}, FontColor->RGBColor[0, 0, 1]], Cell[TextData[StyleBox["Third Edition 2007", FontSlant->"Italic"]], "Subtitle", CellChangeTimes->{{3.394579519605548*^9, 3.3945795336356897`*^9}, { 3.394579849753665*^9, 3.3945798497607594`*^9}}], Cell["PHYSICS 200", "Subtitle", CellChangeTimes->{{3.394579873314529*^9, 3.3945798826114264`*^9}}], Cell[CellGroupData[{ Cell["Nicholas Wheeler", "Author", CellChangeTimes->{{3.39457972270331*^9, 3.3945797314311037`*^9}}], Cell["REED COLLEGE", "Institution", CellChangeTimes->{{3.3945797341591387`*^9, 3.39457973632885*^9}, { 3.394579793724296*^9, 3.3945797937303047`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["", "Author", CellChangeTimes->{{3.3950041788207006`*^9, 3.395004178907696*^9}}], Cell["", "Institution", CellChangeTimes->{{3.395004178988262*^9, 3.395004178999803*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["", "Author", CellChangeTimes->{{3.395004194158606*^9, 3.3950041941734962`*^9}}], Cell["", "Institution"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Laboratory 3", FontColor->RGBColor[1, 0, 0]]], "Subtitle", CellChangeTimes->{{3.394581250479834*^9, 3.394581255896365*^9}, 3.394810647542058*^9, 3.394904595337902*^9, 3.3950042110191517`*^9}], Cell[CellGroupData[{ Cell["Some Useful Standard Packages", "Section", CellChangeTimes->{{3.3950050750221233`*^9, 3.3950050875924597`*^9}}], Cell[TextData[{ "The power of ", StyleBox["Mathematica", FontSlant->"Italic"], " is greatly enhanced by the large number of ", StyleBox["Standard Packages", FontWeight->"Bold"], " that are tucked into its memory, and stand available for activation on an \ as-needed basis. 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Here I can \ attempt only to illustrate the kind of material that lies hidden away in \ packages of various descriptions.", FontColor->GrayLevel[0]] }], "Text", CellChangeTimes->{{3.3959689370896893`*^9, 3.395969062430229*^9}, { 3.395969131617908*^9, 3.395969194049079*^9}, {3.395969264082231*^9, 3.395969596006513*^9}, {3.395969642999258*^9, 3.3959696816321*^9}}], Cell[CellGroupData[{ Cell["Physical Constants", "Subsection", CellChangeTimes->{{3.3950051068552723`*^9, 3.3950051106110487`*^9}}, FontColor->RGBColor[1, 0, 0]], Cell[TextData[{ "Go to the Documentation Center and search for ", StyleBox["PhysicalConstants/tutorial/PhysicalConstants", FontColor->RGBColor[0, 0, 1]], ". 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To learn about it, search for ", StyleBox["Units/tutorial/Units. 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To activate the package, command" }], "Text", CellChangeTimes->{{3.395007362047785*^9, 3.395007400376161*^9}, { 3.395007430645227*^9, 3.395007479695931*^9}}], Cell[BoxData[ RowBox[{"Needs", "[", "\"\\"", "]"}]], "Input", "Deemphasis", CellID->518473573], Cell["\<\ and I now provide quick illustration of its use. My friend Oya was born on 15 \ March 1944. \ \>", "Text", CellChangeTimes->{{3.395008252923325*^9, 3.395008290110097*^9}, { 3.395008331966585*^9, 3.395008363307438*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"DayOfWeek", "[", RowBox[{"{", RowBox[{"1944", ",", "3", ",", "15"}], "}"}], "]"}]], "Input"], Cell[BoxData["Wednesday"], "Output", CellChangeTimes->{3.395007629803487*^9}] }, Open ]], Cell["\<\ we learn that she was born on a Wednesday. I am writing on 31 July 2007, and \ from\ \>", "Text", CellChangeTimes->{{3.395008296726348*^9, 3.395008314588402*^9}, { 3.395008369386921*^9, 3.395008375213924*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"DaysBetween", "[", RowBox[{ RowBox[{"{", RowBox[{"1944", ",", "3", ",", "15"}], "}"}], ",", RowBox[{"{", RowBox[{"2007", ",", "7", ",", "31"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{ 3.395007653471483*^9, {3.3950083813483543`*^9, 3.395008383812313*^9}}], Cell[BoxData["23148"], "Output", CellChangeTimes->{3.395007661748118*^9, 3.395008388451579*^9}] }, Open ]], Cell["we learn that she is today 23,148 days old. That comes to", "Text", CellChangeTimes->{{3.395008403010977*^9, 3.3950084343969193`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Convert", "[", RowBox[{ RowBox[{"23148", " ", "Day"}], ",", "Second"}], "]"}]], "Input", CellChangeTimes->{3.395007693540101*^9, 3.3950084410945997`*^9}], Cell[BoxData[ RowBox[{"1999987200", " ", "Second"}]], "Output", CellChangeTimes->{3.39500771807992*^9, 3.395008445622986*^9}] }, Open ]], Cell["\<\ so she will\[LongDash]early tomorrow\[LongDash]be able to celebrate a moment \ when she has become precisely 2 billions seconds old:\ \>", "Text", CellChangeTimes->{{3.395008459697598*^9, 3.395008494112233*^9}, 3.395969765529847*^9, {3.395969799973284*^9, 3.395969896404031*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Convert", "[", RowBox[{ RowBox[{"2000000000", "Second"}], ",", "Day"}], "]"}], "//", "N"}]], "Input"], Cell[BoxData[ RowBox[{"23148.14814814815`", " ", "Day"}]], "Output", CellChangeTimes->{3.3950077540066643`*^9, 3.3950085044142723`*^9}] }, Open ]], Cell["", "Text", CellChangeTimes->{{3.395012163876793*^9, 3.395012163966812*^9}}], Cell["", "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Elementary Algebra", "Section", CellChangeTimes->{{3.3950121838728333`*^9, 3.395012188620791*^9}}], Cell[CellGroupData[{ Cell["Short List of Basic Operations", "Subsection", CellChangeTimes->{{3.395014255877482*^9, 3.395014266116695*^9}}, FontColor->RGBColor[1, 0, 0]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " has lots to say about algebraic subjects of all descriptions, many quite \ esoteric. My simple intent here is to discuss the resources that come into \ view when one goes to ", StyleBox["Palettes > AlgebraicManipulation", FontWeight->"Bold"], ", which can be read about by consulting on or another of the eleven \ tutorial notebooks to which ", StyleBox["tutorial/AlgebraicManipulationOverview", FontColor->RGBColor[0, 0, 1]], " provides links. " }], "Text", CellChangeTimes->{{3.3950122392668247`*^9, 3.3950123599779453`*^9}, { 3.395012411695631*^9, 3.395012439856256*^9}, {3.395012475594976*^9, 3.395012514177067*^9}}], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " looks upon ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"x", "+", "y"}], ")"}], "7"], TraditionalForm]]], "as a primitive object. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ SuperscriptBox[ RowBox[{"(", RowBox[{"x", "+", "y"}], ")"}], "7"]], "Input"], Cell[BoxData[ SuperscriptBox[ RowBox[{"(", RowBox[{"x", "+", "y"}], ")"}], "7"]], "Output", CellChangeTimes->{3.3950125996299458`*^9}] }, Open ]], Cell[" But it will expand that binomial if you ask it to:", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"Expand", "[", SuperscriptBox[ RowBox[{"(", RowBox[{"x", "+", "y"}], ")"}], "7"], "]"}], "\[IndentingNewLine]", RowBox[{"Expand", "[", RowBox[{ RowBox[{"(", RowBox[{"2", "+", RowBox[{"3", "x"}], "+", RowBox[{"17", SuperscriptBox["x", "2"]}]}], ")"}], RowBox[{"(", RowBox[{ SuperscriptBox["x", "2"], "-", "9"}], ")"}], RowBox[{"(", RowBox[{ SuperscriptBox["x", "2"], "+", "25"}], ")"}]}], "]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Expand", "[", RowBox[{ RowBox[{"(", RowBox[{"2", "+", RowBox[{"3", "x"}], "+", RowBox[{"17", SuperscriptBox["x", "2"]}]}], ")"}], RowBox[{"(", RowBox[{ SuperscriptBox["x", "2"], "-", "9"}], ")"}], RowBox[{"(", RowBox[{ SuperscriptBox["x", "2"], "+", "25"}], ")"}]}], "]"}], "//", StyleBox["TraditionalForm", FontColor->RGBColor[1, 0, 0]]}]}], "Input", CellChangeTimes->{{3.395012647336035*^9, 3.3950126609048357`*^9}}], Cell[BoxData[ RowBox[{ SuperscriptBox["x", "7"], "+", RowBox[{"7", " ", SuperscriptBox["x", "6"], " ", "y"}], "+", RowBox[{"21", " ", SuperscriptBox["x", "5"], " ", SuperscriptBox["y", "2"]}], "+", RowBox[{"35", " ", SuperscriptBox["x", "4"], " ", SuperscriptBox["y", "3"]}], "+", RowBox[{"35", " ", SuperscriptBox["x", "3"], " ", SuperscriptBox["y", "4"]}], "+", RowBox[{"21", " ", SuperscriptBox["x", "2"], " ", SuperscriptBox["y", "5"]}], "+", RowBox[{"7", " ", "x", " ", SuperscriptBox["y", "6"]}], "+", SuperscriptBox["y", "7"]}]], "Output", CellChangeTimes->{3.395012670448962*^9}], Cell[BoxData[ RowBox[{ RowBox[{"-", "450"}], "-", RowBox[{"675", " ", "x"}], "-", RowBox[{"3793", " ", SuperscriptBox["x", "2"]}], "+", RowBox[{"48", " ", SuperscriptBox["x", "3"]}], "+", RowBox[{"274", " ", SuperscriptBox["x", "4"]}], "+", RowBox[{"3", " ", SuperscriptBox["x", "5"]}], "+", RowBox[{"17", " ", SuperscriptBox["x", "6"]}]}]], "Output", CellChangeTimes->{3.3950126706019907`*^9}], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"17", " ", SuperscriptBox["x", "6"]}], "+", RowBox[{"3", " ", SuperscriptBox["x", "5"]}], "+", RowBox[{"274", " ", SuperscriptBox["x", "4"]}], "+", RowBox[{"48", " ", SuperscriptBox["x", "3"]}], "-", RowBox[{"3793", " ", SuperscriptBox["x", "2"]}], "-", RowBox[{"675", " ", "x"}], "-", "450"}], TraditionalForm]], "Output", CellChangeTimes->{3.3950126707115793`*^9}] }, Open ]], Cell[TextData[{ "Standardly, the powers ", Cell[BoxData[ FormBox[ SuperscriptBox["x", "n"], TraditionalForm]]], " are presented in ascending order. The command ", StyleBox["//TraditionalForm", "Input"], " reverses the order and adjusts the font." }], "Text", CellChangeTimes->{{3.395012714235118*^9, 3.395012727495455*^9}, { 3.395969981527754*^9, 3.395969983742799*^9}}], Cell[TextData[{ "\t\nThe action of ", StyleBox["Together", "Input"], " is similarly self-explanatory, and opposite to that of ", StyleBox["Apart", "Input"], ":" }], "Text", CellChangeTimes->{3.3959700028370647`*^9}], Cell[BoxData[ RowBox[{"Clear", "[", RowBox[{"a", ",", "b", ",", "c", ",", "d"}], "]"}]], "Input"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Together", "[", RowBox[{ FractionBox["a", "b"], "+", FractionBox["c", "d"]}], "]"}]], "Input"], Cell[BoxData[ FractionBox[ RowBox[{ RowBox[{"b", " ", "c"}], "+", RowBox[{"a", " ", "d"}]}], RowBox[{"b", " ", "d"}]]], "Output", CellChangeTimes->{3.39501283990945*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Apart", "[", FractionBox[ RowBox[{ RowBox[{"b", " ", "c"}], "+", RowBox[{"a", " ", "d"}]}], 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[REMARK: In ", StyleBox["Mathematica", FontSlant->"Italic"], " 5 one could achieve the same result from the keyboard, by typing ", StyleBox["\[ShiftKey]\[CloverLeaf]", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], StyleBox["C", FontColor->RGBColor[1, 0, 0]], StyleBox[", but in ", FontColor->GrayLevel[0]], StyleBox["Mathematica", FontSlant->"Italic", FontColor->GrayLevel[0]], " 6 that capability has been lost.] A menu will appear, where you will be \ asked to state what it is you desire (select amongst several options). Do so, \ then hit ", StyleBox["OK", FontWeight->"Bold"], ". You will have created something like this:" }], "Text", CellChangeTimes->{{3.3950718154923*^9, 3.39507195106129*^9}, { 3.395072016945484*^9, 3.395072072149311*^9}, {3.3950725335010843`*^9, 3.395072602048036*^9}, {3.395072634594079*^9, 3.395072733570517*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(", "\[NoBreak]", GridBox[{ {"\[Placeholder]", "\[Placeholder]", "\[Placeholder]", "\[Placeholder]"}, {"\[Placeholder]", "\[Placeholder]", "\[Placeholder]", "\[Placeholder]"}, {"\[Placeholder]", "\[Placeholder]", "\[Placeholder]", "\[Placeholder]"}, {"\[Placeholder]", "\[Placeholder]", "\[Placeholder]", "\[Placeholder]"} }], "\[NoBreak]", ")"}]}]], "Input", CellChangeTimes->{3.395072752822461*^9}], Cell[TextData[{ "Place the cursor on the 11 box, type ", StyleBox["a", "Input", FontColor->GrayLevel[0]], ", then type \[TabKey] ", StyleBox["b", "Input"], " \[TabKey] ", StyleBox["c", "Input"], " \[TabKey] ", StyleBox["d", "Input"], "\[Ellipsis] to produce the following:" }], "Text", CellChangeTimes->{{3.395073506599523*^9, 3.395073556485259*^9}, { 3.395073590541357*^9, 3.395073656616807*^9}}], Cell[BoxData[ RowBox[{"Clear", "[", RowBox[{ "a", ",", "b", ",", "c", ",", "d", ",", "e", ",", "f", ",", "g", ",", "h", ",", "i", ",", "j", ",", "k", ",", "l", ",", "m", ",", "n", ",", "o", ",", "p"}], "]"}]], "Input", CellChangeTimes->{{3.395073364274342*^9, 3.395073376178479*^9}}], Cell[BoxData[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"a", "b", "c", "d"}, {"e", "f", "g", "h"}, {"i", "j", "k", "l"}, {"m", "n", "o", "p"} }], "\[NoBreak]", ")"}]], "Input", CellChangeTimes->{{3.395073729334745*^9, 3.3950737405506897`*^9}}], Cell[TextData[{ "The ", StyleBox["column-adding", FontWeight->"Bold"], " effect of placing the cursor between a and b and typing \ \[ControlKey]+comma is shown below" }], "Text", CellChangeTimes->{{3.3950737972276983`*^9, 3.395073887946064*^9}}], Cell[BoxData[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"a", "\[Placeholder]", "b", "c", "d"}, {"e", "\[Placeholder]", "f", "g", "h"}, {"i", "\[Placeholder]", "j", "k", "l"}, {"m", "\[Placeholder]", "n", "o", "p"} }], "\[NoBreak]", ")"}]], "Input", CellChangeTimes->{{3.39507332631212*^9, 3.395073343192123*^9}, 3.395073390416491*^9, 3.395073440977947*^9}], Cell[TextData[{ "while the following matrix shows the ", StyleBox["row-adding", FontWeight->"Bold"], " effect of placing the cursor as before, but typing \[ControlKey]+\ \[ReturnKey]" }], "Text", CellChangeTimes->{{3.39507390411134*^9, 3.3950739403867292`*^9}, { 3.395073972034189*^9, 3.39507400137948*^9}}], Cell[BoxData[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"a", "b", "c", "d"}, {"\[Placeholder]", "\[Placeholder]", "\[Placeholder]", "\[Placeholder]"}, {"e", "f", "g", "h"}, {"i", "j", "k", "l"}, {"m", "n", "o", "p"} }], "\[NoBreak]", ")"}]], "Input", CellChangeTimes->{3.395073453154648*^9}], Cell[TextData[{ "To ", StyleBox["remove/modify an element", FontWeight->"Bold"], ", select that element and hit \[DeleteKey]; thus" }], "Text", CellChangeTimes->{{3.395074627595551*^9, 3.395074705918724*^9}, { 3.3950748031375217`*^9, 3.39507480380018*^9}}], Cell[BoxData[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"a", "b", "c", "d"}, {"e", "f", "\[Placeholder]", "h"}, {"i", "j", "k", "l"}, {"m", "n", "o", "p"} }], "\[NoBreak]", ")"}]], "Input", CellChangeTimes->{ 3.3950745019722967`*^9, {3.395074562029235*^9, 3.39507457449723*^9}, 3.395074648830245*^9}], Cell["\<\ To remove a row (or column), select that row/column and hit \[DeleteKey]; \ thus\ \>", "Text", CellChangeTimes->{{3.395074726117154*^9, 3.395074797680108*^9}}], Cell[BoxData[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"a", "c", "d"}, {"e", "g", "h"}, {"i", "k", "l"}, {"m", "o", "p"} }], "\[NoBreak]", ")"}]], "Input", CellChangeTimes->{3.395074828952903*^9}], Cell["", "Text", CellChangeTimes->{{3.395074868487809*^9, 3.395074868568277*^9}}], Cell["", "Text"], Cell[TextData[{ StyleBox["TIP", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], ": It rapidly becomes tedious to return again and again to the Table/Matrix \ menu to create successive matrices of similar structure. In preparing the \ preceding paragraph, I proceeded in the manner just described to create" }], "Text", CellChangeTimes->{{3.395074874249799*^9, 3.395074877512659*^9}, { 3.395074932146352*^9, 3.395074947898986*^9}, {3.395074984211903*^9, 3.3950750770024557`*^9}, {3.395075268869068*^9, 3.3950753337181883`*^9}, 3.3950757075911703`*^9, {3.395972193376903*^9, 3.395972212159256*^9}}], Cell[TextData[{ Cell[BoxData[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"\[Placeholder]", "\[Placeholder]", "\[Placeholder]", "\[Placeholder]"}, {"\[Placeholder]", "\[Placeholder]", "\[Placeholder]", "\[Placeholder]"}, {"\[Placeholder]", "\[Placeholder]", "\[Placeholder]", "\[Placeholder]"}, {"\[Placeholder]", "\[Placeholder]", "\[Placeholder]", "\[Placeholder]"} }], "\[NoBreak]", ")"}]]], "\[LongDash]actually, to create ", Cell[BoxData[ RowBox[{"(", "\[NoBreak]", GridBox[{ { StyleBox["a", FontColor->RGBColor[0, 0, 1]], StyleBox["b", FontColor->RGBColor[0, 0, 1]], StyleBox["c", FontColor->RGBColor[0, 0, 1]], StyleBox["d", FontColor->RGBColor[0, 0, 1]]}, { StyleBox["e", FontColor->RGBColor[0, 0, 1]], StyleBox["f", FontColor->RGBColor[0, 0, 1]], StyleBox["g", FontColor->RGBColor[0, 0, 1]], StyleBox["h", FontColor->RGBColor[0, 0, 1]]}, { StyleBox["i", FontColor->RGBColor[0, 0, 1]], StyleBox["j", FontColor->RGBColor[0, 0, 1]], StyleBox["k", FontColor->RGBColor[0, 0, 1]], StyleBox["l", FontColor->RGBColor[0, 0, 1]]}, { StyleBox["m", FontColor->RGBColor[0, 0, 1]], StyleBox["n", FontColor->RGBColor[0, 0, 1]], StyleBox["o", FontColor->RGBColor[0, 0, 1]], StyleBox["p", FontColor->RGBColor[0, 0, 1]]} }], "\[NoBreak]", ")"}]]], "\[LongDash]and then used ", StyleBox["Copy/Paste", FontWeight->"Bold"], "." }], "Text", CellChangeTimes->{ 3.395075097235528*^9, {3.395075133575519*^9, 3.395075180922544*^9}, { 3.395075213957736*^9, 3.395075222851695*^9}}], Cell["", "Text", CellChangeTimes->{{3.3950753642010098`*^9, 3.3950753642950773`*^9}}], Cell["", "Text"], Cell["\<\ Notice that options available on the Table/Matrix menu permit one to create \ matrices with (for example) the structure\ \>", "Text", CellChangeTimes->{{3.3950756882500153`*^9, 3.395075738606811*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0", "0", "0"}, {"0", "1", "0", "0"}, {"0", "0", "1", "0"}, {"0", "0", "0", "1"} }], "\[NoBreak]", ")"}]}]], "Input", CellChangeTimes->{3.395075779478869*^9}], Cell["with a single keystroke. Alternatively, one might command", "Text", CellChangeTimes->{{3.395075789376585*^9, 3.395075825625228*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"IdentityMatrix", "[", "4", "]"}], "//", "MatrixForm"}]], "Input", CellChangeTimes->{{3.3950758308407393`*^9, 3.395075843684519*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0", "0", "0"}, {"0", "1", "0", "0"}, {"0", "0", "1", "0"}, {"0", "0", "0", "1"} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{3.395075845344722*^9}] }, Open ]], Cell["", "Text", CellChangeTimes->{{3.395075969695436*^9, 3.395075969785475*^9}}], Cell[TextData[{ StyleBox["REMARK", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], ": When using \[TabKey] to fill in the elements of ", Cell[BoxData[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"\[Placeholder]", "\[Placeholder]", "\[Placeholder]", "\[Placeholder]"}, {"\[Placeholder]", "\[Placeholder]", "\[Placeholder]", "\[Placeholder]"}, {"\[Placeholder]", "\[Placeholder]", "\[Placeholder]", "\[Placeholder]"}, {"\[Placeholder]", "\[Placeholder]", "\[Placeholder]", "\[Placeholder]"} }], "\[NoBreak]", ")"}]]], " it is useful to remember that the control keys \[UpArrow], \[DownArrow], \ \[RightArrow] and \[LeftArrow] act to relocate the \"selected box\" in \ obvious ways. " }], "Text", CellChangeTimes->{{3.395075979796988*^9, 3.395075995379548*^9}, { 3.395076034570326*^9, 3.395076212975309*^9}, {3.395972264861225*^9, 3.395972265868813*^9}}], Cell["", "Text", CellChangeTimes->{{3.395076247578076*^9, 3.395076247671185*^9}}], Cell["", "Text"], Cell[TextData[{ StyleBox["Matrices can constructed at the keyboard", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], ", and this is sometimes the fastest way to go. An example serves to \ describe the simple routing. In ", StyleBox["Input", "Input"], " mode, type ()\[LeftArrow] to create" }], "Text", CellChangeTimes->{{3.395972298438583*^9, 3.395972338672225*^9}, { 3.395972446813102*^9, 3.3959724988026323`*^9}, {3.395972700606626*^9, 3.3959727133294477`*^9}}], Cell[BoxData[ StyleBox[ RowBox[{ RowBox[{"(", ")"}], " ", "with", " ", "the", " ", "cursor", " ", "blinking", " ", "inside"}], "Output", FontColor->GrayLevel[0.5]]], "Input", CellChangeTimes->{{3.395972508032771*^9, 3.395972536497348*^9}}], Cell[TextData[{ "Now type ", StyleBox["a", "Input"], "\[ControlKey]", StyleBox[",b", "Input"], "\[ControlKey]", StyleBox[",c", "Input"], "\[ControlKey]", StyleBox[",d", "Input"], " (NOTE: \[ControlKey]", StyleBox[",", "Input"], " means \"type comma with the \[ControlKey] key depressed\") to produce" }], "Text", CellChangeTimes->{{3.395972542295145*^9, 3.3959726822059717`*^9}, { 3.3959727585194902`*^9, 3.395972761247864*^9}}], Cell[BoxData[ StyleBox[ RowBox[{ RowBox[{"(", GridBox[{ {"a", "b", "c", "d"} }], ")"}], " ", "with", " ", "the", " ", "cursor", " ", "blinking", " ", "after", " ", "the", " ", "d"}], "Output", FontColor->GrayLevel[0.5]]], "Input", CellChangeTimes->{{3.39597277333603*^9, 3.395972795315454*^9}}], Cell["Now type \[ControlKey]\[ReturnKey] to produce", "Text", CellChangeTimes->{{3.3959728080418777`*^9, 3.395972837015656*^9}}], Cell[BoxData[ StyleBox[ RowBox[{"(", GridBox[{ {"a", "b", "c", "d"}, {"\[Placeholder]", "\[Placeholder]", "\[Placeholder]", "\[Placeholder]"} }], ")"}], "Output", FontColor->GrayLevel[0.5]]], "Input", CellChangeTimes->{3.3959728665415487`*^9}], Cell["\<\ Fill in the new row. How rows can be lengthened/shortened, new rows can be \ created and old ones elminated should by now be clear.\ \>", "Text", CellChangeTimes->{{3.39597288313548*^9, 3.395972985825747*^9}}], Cell["", "Text", CellChangeTimes->{{3.3959736043306093`*^9, 3.3959736044322567`*^9}}], Cell["", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Subscripts & Superscripts/Exponents", "Subsection", CellChangeTimes->{{3.395076875692977*^9, 3.395076887184964*^9}, { 3.395076939642641*^9, 3.395076942655102*^9}}, FontColor->RGBColor[1, 0, 0]], Cell[TextData[{ "Subscripts are useful in ", StyleBox["man", FontSlant->"Italic"], "y contexts, but in matrix algebra they are especially important, since they \ serve very efficiently to make it clear who stands where, and to clarify the \ relatinships that develop. For information, search for ", StyleBox["tutorial/EnteringTwoDimensionalExpressionsOverview ", FontColor->RGBColor[0, 0, 1]], StyleBox["and select ", FontColor->GrayLevel[0]], StyleBox["Typing Superscripts", FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox[".", FontColor->RGBColor[0, 0, 1]], " " }], "Text", CellChangeTimes->{{3.395077054504857*^9, 3.395077208991252*^9}, { 3.39507848545321*^9, 3.395078495657143*^9}, {3.39507852629272*^9, 3.3950785803282137`*^9}, {3.395973701847967*^9, 3.395973759327652*^9}}], Cell["", "Text", CellChangeTimes->{{3.3950786689890623`*^9, 3.395078669007105*^9}}], Cell["To create", "Text", CellChangeTimes->{{3.395078607869309*^9, 3.395078609937071*^9}}], Cell[BoxData[ SubscriptBox["a", "1"]], "Input", CellChangeTimes->{{3.395077216624302*^9, 3.395077226637143*^9}}], Cell[TextData[{ "one might type a and then hit the ", Cell[BoxData[ FormBox[ SubscriptBox["\[Placeholder]", "\[Placeholder]"], TraditionalForm]]], " button near the bottom of the ", StyleBox["BasicMathInput", FontWeight->"Bold"], " palette. I find it quicker, however, to type ", StyleBox["a", "Input"], "\[ControlKey] ", StyleBox["_1", "Input"], "\[ControlKey]\[ShiftKey], where the final \[ControlKey]\[ShiftKey] is \ needed to get the cursor out of the basement." }], "Text", CellChangeTimes->{{3.395077237260805*^9, 3.3950774058568373`*^9}, { 3.395077439206026*^9, 3.395077440755028*^9}, {3.395077490987906*^9, 3.3950775608508263`*^9}, {3.395077610068873*^9, 3.395077623326932*^9}, { 3.39507862237647*^9, 3.395078641296916*^9}, {3.395080848134993*^9, 3.395080874182971*^9}, 3.395080904738921*^9, {3.395080984901205*^9, 3.395080984927519*^9}, 3.395081052704793*^9, {3.395973804559465*^9, 3.395973837335743*^9}}], Cell["", "Text", CellChangeTimes->{{3.395078687762783*^9, 3.395078687780343*^9}}], Cell["Double superscripts are a bit more ticklish. To create", "Text", CellChangeTimes->{{3.39507869993147*^9, 3.395078728125153*^9}}], Cell[CellGroupData[{ Cell[BoxData[ SubscriptBox["a", "12"]], "Input", CellChangeTimes->{{3.395077655867876*^9, 3.395077661767622*^9}}], Cell[BoxData[ SubscriptBox["a", "12"]], "Output", CellChangeTimes->{3.395078056969029*^9}] }, Open ]], Cell[TextData[{ "you might proceed essentially as before: type ", StyleBox["a", "Input"], "\[ControlKey] _12\[ControlKey]\[ShiftKey]. And for many purposes that is \ sufficient, as I demonstrate:" }], "Text", CellChangeTimes->{{3.395077671050658*^9, 3.395077727625155*^9}, { 3.395078745760091*^9, 3.3950787830546103`*^9}, {3.395081021843672*^9, 3.3950810481077147`*^9}, 3.3950811383515673`*^9}], Cell[BoxData[{ RowBox[{ RowBox[{"A", "=", RowBox[{"(", "\[NoBreak]", GridBox[{ { SubscriptBox["a", "11"], SubscriptBox["a", "12"]}, { SubscriptBox["a", "21"], SubscriptBox["a", "22"]} }], "\[NoBreak]", ")"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"B", "=", RowBox[{"(", "\[NoBreak]", GridBox[{ { SubscriptBox["b", "11"], SubscriptBox["b", "12"]}, { SubscriptBox["b", "21"], SubscriptBox["b", "22"]} }], "\[NoBreak]", ")"}]}], ";"}]}], "Input", CellChangeTimes->{{3.395078795656261*^9, 3.395078878631805*^9}}], Cell[CellGroupData[{ Cell[BoxData[{"A", "\[IndentingNewLine]", "B"}], "Input", CellChangeTimes->{{3.395078889904276*^9, 3.3950788923811083`*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ SubscriptBox["a", "11"], ",", SubscriptBox["a", "12"]}], "}"}], ",", RowBox[{"{", RowBox[{ SubscriptBox["a", "21"], ",", SubscriptBox["a", "22"]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.395078893792226*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ SubscriptBox["b", "11"], ",", SubscriptBox["b", "12"]}], "}"}], ",", RowBox[{"{", RowBox[{ SubscriptBox["b", "21"], ",", SubscriptBox["b", "22"]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.3950788939046097`*^9}] }, Open ]], Cell[TextData[{ StyleBox["NOTE", FontWeight->"Bold"], " that ", StyleBox["Mathematica", FontSlant->"Italic"], " insists upon thinking of matrices as lists of lists. 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To discover those we might command" }], "Text", CellChangeTimes->{ 3.395095337442828*^9, {3.3950973365515633`*^9, 3.3950973390868597`*^9}}], Cell[BoxData[ StyleBox[ RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{"CharacteristicPolynomial", "[", RowBox[{"matrix", ",", "\[Lambda]"}], "]"}], "\[Equal]", "0"}], ",", "\[Lambda]"}], "]"}], FontColor->GrayLevel[0.500008]]], "Input", CellChangeTimes->{{3.395097470259025*^9, 3.3950974818327312`*^9}}], Cell[TextData[{ "For the 2\[Times]2 ", StyleBox["M", "Input"], "-matrix introduced just above this would give" }], "Text", CellChangeTimes->{{3.3950973195052433`*^9, 3.395097321508873*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{"CharacteristicPolynomial", "[", RowBox[{"M", ",", "\[Lambda]"}], "]"}], "\[Equal]", "0"}], ",", "\[Lambda]"}], "]"}]], "Input", CellChangeTimes->{{3.395097487577396*^9, 3.395097491493383*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"\[Lambda]", "\[Rule]", RowBox[{ FractionBox["1", "2"], " ", RowBox[{"(", RowBox[{"a", "+", "d", "-", SqrtBox[ RowBox[{ SuperscriptBox["a", "2"], "+", RowBox[{"4", " ", "b", " ", "c"}], "-", RowBox[{"2", " ", "a", " ", "d"}], "+", SuperscriptBox["d", "2"]}]]}], ")"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"\[Lambda]", "\[Rule]", RowBox[{ FractionBox["1", "2"], " ", RowBox[{"(", RowBox[{"a", "+", "d", "+", SqrtBox[ RowBox[{ SuperscriptBox["a", "2"], "+", RowBox[{"4", " ", "b", " ", "c"}], "-", RowBox[{"2", " ", "a", " ", "d"}], "+", SuperscriptBox["d", "2"]}]]}], ")"}]}]}], "}"}]}], "}"}]], "Output",\ CellChangeTimes->{3.3950954155215282`*^9, 3.39509749578906*^9}] }, Open ]], Cell[TextData[{ "but ", StyleBox["Mathematica", FontSlant->"Italic"], " provides a better way:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", StyleBox["Eigenvalues", FontColor->RGBColor[1, 0, 0]]}]], "Input"], Cell[BoxData[ RowBox[{ StyleBox["\<\"\\!\\(\\*RowBox[{\\\"Eigenvalues\\\", \\\"[\\\", \ StyleBox[\\\"m\\\", \\\"TI\\\"], \\\"]\\\"}]\\) gives a list of the \ eigenvalues of the square matrix m. \\n\\!\\(\\*RowBox[{\\\"Eigenvalues\\\", \ \\\"[\\\", RowBox[{\\\"{\\\", RowBox[{StyleBox[\\\"m\\\", \\\"TI\\\"], \ \\\",\\\", StyleBox[\\\"a\\\", \\\"TI\\\"]}], \\\"}\\\"}], \\\"]\\\"}]\\) \ gives the generalized eigenvalues of m with respect to a. \ \\n\\!\\(\\*RowBox[{\\\"Eigenvalues\\\", \\\"[\\\", \ RowBox[{StyleBox[\\\"m\\\", \\\"TI\\\"], \\\",\\\", StyleBox[\\\"k\\\", \ \\\"TI\\\"]}], \\\"]\\\"}]\\) gives the first k eigenvalues of m. \ \\n\\!\\(\\*RowBox[{\\\"Eigenvalues\\\", \\\"[\\\", \ RowBox[{RowBox[{\\\"{\\\", RowBox[{StyleBox[\\\"m\\\", \\\"TI\\\"], \ \\\",\\\", StyleBox[\\\"a\\\", \\\"TI\\\"]}], \\\"}\\\"}], \\\",\\\", \ StyleBox[\\\"k\\\", \\\"TI\\\"]}], \\\"]\\\"}]\\) gives the first k \ generalized eigenvalues.\"\>", "MSG"], " ", ButtonBox[ StyleBox["\[RightSkeleton]", "SR"], Active->True, BaseStyle->"Link", ButtonData->"paclet:ref/Eigenvalues"]}]], "Print", "PrintUsage", CellChangeTimes->{3.395095465792385*^9}, CellTags->"Info3395070263-5769222"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Eigenvalues", "[", "M", "]"}], "//", "TableForm"}]], "Input"], Cell[BoxData[ TagBox[ TagBox[GridBox[{ { RowBox[{ FractionBox["1", "2"], " ", RowBox[{"(", RowBox[{"a", "+", "d", "-", SqrtBox[ RowBox[{ SuperscriptBox["a", "2"], "+", RowBox[{"4", " ", "b", " ", "c"}], "-", RowBox[{"2", " ", "a", " ", "d"}], "+", SuperscriptBox["d", "2"]}]]}], ")"}]}]}, { RowBox[{ FractionBox["1", "2"], " ", RowBox[{"(", RowBox[{"a", "+", "d", "+", SqrtBox[ RowBox[{ SuperscriptBox["a", "2"], "+", RowBox[{"4", " ", "b", " ", "c"}], "-", RowBox[{"2", " ", "a", " ", "d"}], "+", SuperscriptBox["d", "2"]}]]}], ")"}]}]} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.5599999999999999]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], Column], Function[BoxForm`e$, TableForm[BoxForm`e$]]]], "Output", CellChangeTimes->{3.3950954956413403`*^9}] }, Open ]], Cell["", "Text", CellChangeTimes->{{3.395098780467811*^9, 3.395098780528349*^9}}], Cell["", "Text"], Cell[TextData[{ "Thus far we have asked ", StyleBox["Mathematica", FontSlant->"Italic"], " to do matrix-theoretic things we could as easily have done by hand. 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" }], "Text", CellChangeTimes->{{3.395101956679072*^9, 3.3951020045041857`*^9}, { 3.396010818178816*^9, 3.3960108370115957`*^9}, {3.396010876748515*^9, 3.396010897590768*^9}}], Cell["", "Text", CellChangeTimes->{{3.395102111134881*^9, 3.395102111164494*^9}}], Cell["", "Text"], Cell[TextData[{ "The ", StyleBox["eigen", FontColor->RGBColor[0, 0, 1]], StyleBox["vectors", FontVariations->{"Underline"->True}, FontColor->RGBColor[0, 0, 1]], " of a matrix arise by solution of an equation of the form" }], "Text", CellChangeTimes->{3.3951021199972982`*^9}], Cell[BoxData[ StyleBox[ RowBox[{ RowBox[{"matrix", ".", "eigenvector"}], "\[Equal]", RowBox[{"number", "*", "eigenvector"}]}], FontColor->GrayLevel[0.500008]]], "Equation"], Cell["\<\ which is possible iff \"number\" is in fact one or another of the eigenvalues \ of the matrix. 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\\n\\!\\(\\*RowBox[{\\\"Eigensystem\\\", \\\"[\\\", \ RowBox[{RowBox[{\\\"{\\\", RowBox[{StyleBox[\\\"m\\\", \\\"TI\\\"], \ \\\",\\\", StyleBox[\\\"a\\\", \\\"TI\\\"]}], \\\"}\\\"}], \\\",\\\", \ StyleBox[\\\"k\\\", \\\"TI\\\"]}], \\\"]\\\"}]\\) gives the first \ \\!\\(\\*StyleBox[\\\"k\\\", \\\"TI\\\"]\\) generalized eigenvalues and \ eigenvectors.\"\>", "MSG"], " ", ButtonBox[ StyleBox["\[RightSkeleton]", "SR"], Active->True, BaseStyle->"Link", ButtonData->"paclet:ref/Eigensystem"]}]], "Print", "PrintUsage", CellChangeTimes->{3.395102434238882*^9}, CellTags->"Info3395077233-7935860"] }, Open ]], Cell["\<\ The following simple example is designed to show how this works:\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"S", "=", RowBox[{"(", GridBox[{ {"1", "2"}, {"2", "3"} }], ")"}]}], ";"}]], "Input"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Eigensystem", "[", "S", "]"}]], "Input", CellChangeTimes->{3.3951025482567377`*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ 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Integrals that it can't \ do analytically it is always prepared to do numerically." }], "Text", CellChangeTimes->{{3.3951516538025084`*^9, 3.3951516982047377`*^9}, { 3.395151738169362*^9, 3.395151741172028*^9}, {3.3951517879532423`*^9, 3.395151867515482*^9}}], Cell[TextData[{ StyleBox["Integral commands can be entered in three distinct ways: in \ \"telegraphic style\" from the keyboard; with 2D symbols constructed by \ keyboard codes; by means of the ", FontSlant->"Plain"], StyleBox["BasicMathInput", FontWeight->"Bold", FontSlant->"Plain"], StyleBox[" palette. I will illustrate those three methods in turn.", FontSlant->"Plain"] }], "Text", CellChangeTimes->{{3.3951518755610247`*^9, 3.395152072773642*^9}}, FontSlant->"Italic"], Cell["", "Text", CellChangeTimes->{{3.395152201462154*^9, 3.395152201507783*^9}}], Cell[CellGroupData[{ Cell[TextData[StyleBox["Examples of Indefinite Integrals", FontColor->RGBColor[0, 0, 1]]], "Subsubsection", CellChangeTimes->{{3.395152216297344*^9, 3.395152227295451*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Integrate", "[", RowBox[{ SuperscriptBox[ RowBox[{"Sech", "[", "x", "]"}], "2"], ",", "x"}], "]"}]], "Input", CellChangeTimes->{{3.395152262944717*^9, 3.3951522960205307`*^9}}], Cell[BoxData[ RowBox[{"Tanh", "[", "x", "]"}]], "Output", CellChangeTimes->{3.395152308710224*^9}] }, Open ]], Cell[TextData[{ "That was a \"telegraphic\" command. We might alternatively have used the ", StyleBox["BasicMathInput", FontWeight->"Bold"], " palette (being very careful of its \"unfortunate quirk\") or typed \ \[EscapeKey]intt\[EscapeKey] to produce" }], "Text", CellChangeTimes->{{3.395152455414338*^9, 3.395152555549779*^9}, 3.3960114274542103`*^9, {3.396011883253471*^9, 3.396011970715911*^9}}], Cell[BoxData[ RowBox[{"\[Integral]", RowBox[{"\[Placeholder]", RowBox[{"\[DifferentialD]", "\[Placeholder]"}]}]}]], "Input", CellChangeTimes->{{3.396011349645233*^9, 3.3960113513542643`*^9}}], Cell["and then filled in the boxes to obtain", "Text", CellChangeTimes->{{3.396011993829689*^9, 3.3960120397405157`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[Integral]", RowBox[{ SuperscriptBox[ RowBox[{"Sech", "[", "x", "]"}], "2"], RowBox[{"\[DifferentialD]", "x"}]}]}]], "Input", CellChangeTimes->{{3.3951524117812843`*^9, 3.395152440359116*^9}}], Cell[BoxData[ RowBox[{"Tanh", "[", "x", "]"}]], "Output", 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CellChangeTimes->{{3.395161325040822*^9, 3.395161325191888*^9}}], Cell[TextData[{ StyleBox["REMARK", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], ": To create fractions\[LongDash]like, for example," }], "Text", CellChangeTimes->{{3.3951613489206142`*^9, 3.3951613654179497`*^9}}], Cell[BoxData[ FractionBox["x", "y"]], "Input", CellChangeTimes->{{3.395161375187581*^9, 3.395161381641942*^9}}], Cell[TextData[{ "\[LongDash]you can either use the ", StyleBox["BasicMathInput", FontWeight->"Bold"], " palette (being again very careful of its \"unfortunate quirk\") or\ \[LongDash]much more economically\[LongDash]type ", StyleBox["x", "Input"], "\[ControlKey]/", StyleBox["y", "Input"], ", as instructed in ", StyleBox["tutorial/EnteringTwoDimensionalInput", FontColor->RGBColor[0, 0, 1]], "." }], "Text", CellChangeTimes->{{3.395161393326371*^9, 3.395161430831718*^9}, { 3.3951615244878063`*^9, 3.395161549480988*^9}, {3.396012188089871*^9, 3.3960122414251833`*^9}, {3.396012312150837*^9, 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"Text", CellChangeTimes->{{3.3951620961565228`*^9, 3.395162096175033*^9}, 3.396013197760282*^9}], Cell[TextData[{ StyleBox["REMARK", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], ": To create the box to receive the lower limit, type \[ControlKey]- \ (control + hypen). 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Cell[BoxData[ FractionBox[ SqrtBox["\[Pi]"], SqrtBox["a"]]], "Output", CellChangeTimes->{3.39601438553649*^9}] }, Open ]], Cell["", "Text", CellChangeTimes->{{3.3960131920464087`*^9, 3.396013197910544*^9}}], Cell["", "Text", CellChangeTimes->{3.396013197911928*^9}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Functions Defined by Integrals", FontColor->RGBColor[1, 0, 0]]], "Subsection", CellChangeTimes->{{3.396013233739985*^9, 3.3960132450699883`*^9}}], Cell[TextData[{ "It is often the case even of integrals with simple integrands that they can \ be described only in terms of fancy functions (which in many case were \ historically ", StyleBox["defined", FontSlant->"Italic"], " by integrals): thus" }], "Text", CellChangeTimes->{{3.395162820669806*^9, 3.395162857840667*^9}, { 3.395162888161901*^9, 3.3951629274973392`*^9}, 3.396013197913287*^9}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"\[Integral]", RowBox[{ FractionBox[ RowBox[{"Log", "[", RowBox[{"1", "-", 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