(* 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[ 78696, 2123] NotebookOptionsPosition[ 34154, 1163] NotebookOutlinePosition[ 76939, 2070] CellTagsIndexPosition[ 76896, 2067] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[TextData[{ StyleBox["Lab 7A - Stokes's Theorem", FontSize->24, FontWeight->"Bold", FontVariations->{"Underline"->True}], "\nMath 2374 - University of Minnesota\nhttp://www.math.umn.edu/math2374\n\ Questions to: swenson@math.umn.edu" }], "Text", TextAlignment->Center, FontColor->GrayLevel[1], Background->RGBColor[0, 0, 1]], Cell[CellGroupData[{ Cell["Integration in More than Two Dimensions", "Section", Background->None], Cell[TextData[{ "This semester, we have learned to use six new kinds of integrals. Here's a \ quick review: as usual, vector-valued objects are in ", StyleBox["boldface", FontWeight->"Bold"], "." }], "Text"], Cell[BoxData[GridBox[{ { StyleBox["Type", FontVariations->{"Underline"->True}], StyleBox["Symbol", FontVariations->{"Underline"->True}], StyleBox["Region", FontVariations->{"Underline"->True}], StyleBox["Integrand", FontVariations->{"Underline"->True}], StyleBox[ RowBox[{"How", " ", "to", " ", "Do", " ", "It"}], FontVariations->{"Underline"->True}]}, {"double", RowBox[{"\[Integral]", RowBox[{ SubsuperscriptBox["\[Integral]", "R", " "], " ", RowBox[{"u", " ", RowBox[{"\[DifferentialD]", "A"}]}]}]}], RowBox[{"R", ",", " ", RowBox[{"a", " ", "region"}]}], RowBox[{"scalar", " ", "function", " ", "u"}], RowBox[{"\[Integral]", RowBox[{ RowBox[{"(", RowBox[{"\[Integral]", RowBox[{"u", RowBox[{"(", RowBox[{"x", ",", "y"}], ")"}], RowBox[{"\[DifferentialD]", "y"}]}]}], ")"}], RowBox[{"\[DifferentialD]", "x"}]}]}]}, {"triple", RowBox[{"\[Integral]", RowBox[{"\[Integral]", RowBox[{ SubsuperscriptBox["\[Integral]", "S", " "], RowBox[{"u", " ", RowBox[{"\[DifferentialD]", "V"}]}]}]}]}], RowBox[{"S", ",", " ", RowBox[{"a", " ", "solid"}]}], RowBox[{"scalar", " ", "function", " ", "u"}], RowBox[{"\[Integral]", RowBox[{ RowBox[{"(", RowBox[{"\[Integral]", RowBox[{ RowBox[{"(", RowBox[{"\[Integral]", RowBox[{"u", RowBox[{"(", RowBox[{"x", ",", "y", ",", "z"}], ")"}], RowBox[{"\[DifferentialD]", "z"}]}]}], ")"}], RowBox[{"\[DifferentialD]", "y"}]}]}], ")"}], RowBox[{"\[DifferentialD]", "x"}]}]}]}, { RowBox[{"line", " ", RowBox[{"(", "path", ")"}]}], RowBox[{ SubsuperscriptBox["\[Integral]", "C", " "], RowBox[{"u", " ", RowBox[{"\[DifferentialD]", "s"}], " ", "or", " ", RowBox[{ SubscriptBox["\[ContourIntegral]", "C"], RowBox[{"u", " ", RowBox[{"\[DifferentialD]", "s"}]}]}]}]}], RowBox[{ RowBox[{"C", ",", " ", RowBox[{"a", " ", "curve", " ", "with"}]}], "\[IndentingNewLine]", "parametrization", "\[IndentingNewLine]", RowBox[{ RowBox[{ StyleBox["f", FontWeight->"Bold"], StyleBox[ RowBox[{"(", "t", ")"}], FontWeight->"Plain"]}], StyleBox[",", FontWeight->"Plain"], StyleBox[" ", FontWeight->"Plain"], RowBox[{ StyleBox["a", FontWeight->"Plain"], "\[LessEqual]", "t", "\[LessEqual]", "b"}]}]}], RowBox[{"scalar", " ", "function", " ", "u"}], RowBox[{ RowBox[{ SubsuperscriptBox["\[Integral]", "a", "b"], RowBox[{"u", RowBox[{"(", RowBox[{ StyleBox["f", FontWeight->"Bold"], RowBox[{"(", "t", ")"}]}], ")"}]}]}], "\[DoubleVerticalBar]", RowBox[{ RowBox[{ StyleBox["f", FontWeight->"Bold"], "'"}], RowBox[{"(", "t", ")"}]}], "\[DoubleVerticalBar]", " ", RowBox[{"\[DifferentialD]", "t"}]}]}, { RowBox[{"line", " ", RowBox[{"(", "path", ")"}]}], RowBox[{ SubsuperscriptBox["\[Integral]", "C", " "], RowBox[{ RowBox[{ StyleBox["F", FontWeight->"Bold"], "\[CenterDot]", RowBox[{"\[DifferentialD]", StyleBox["s", FontSize->14, FontWeight->"Bold"]}]}], StyleBox[" ", FontWeight->"Bold"], "or", " ", RowBox[{ SubscriptBox["\[ContourIntegral]", "C"], RowBox[{ StyleBox["F", FontWeight->"Bold"], "\[CenterDot]", RowBox[{"\[DifferentialD]", StyleBox["s", FontSize->14, FontWeight->"Bold"]}]}]}]}]}], RowBox[{ RowBox[{"C", ",", " ", RowBox[{"a", " ", "curve", " ", "with"}]}], "\[IndentingNewLine]", "parametrization", "\[IndentingNewLine]", RowBox[{ RowBox[{ StyleBox["f", FontWeight->"Bold"], StyleBox[ RowBox[{"(", "t", ")"}], FontWeight->"Plain"]}], StyleBox[",", FontWeight->"Plain"], StyleBox[" ", FontWeight->"Plain"], RowBox[{ StyleBox["a", FontWeight->"Plain"], "\[LessEqual]", "t", "\[LessEqual]", "b"}]}]}], RowBox[{"vector", " ", "field", " ", StyleBox["F", FontWeight->"Bold"]}], RowBox[{ SubsuperscriptBox["\[Integral]", "a", "b"], RowBox[{ StyleBox["F", FontWeight->"Bold"], RowBox[{ RowBox[{"(", RowBox[{ StyleBox["f", FontWeight->"Bold"], RowBox[{"(", "t", ")"}]}], ")"}], "\[CenterDot]", RowBox[{ StyleBox["f", FontWeight->"Bold"], "'"}]}], RowBox[{"(", "t", ")"}], RowBox[{"\[DifferentialD]", "t"}]}]}]}, {"surface", RowBox[{"\[Integral]", RowBox[{ SubsuperscriptBox["\[Integral]", "S", " "], RowBox[{"f", RowBox[{"\[DifferentialD]", "S"}], " ", "or", " ", RowBox[{ SubscriptBox["\[DoubleContourIntegral]", "S"], RowBox[{"f", RowBox[{"\[DifferentialD]", "S"}]}]}]}]}]}], RowBox[{ RowBox[{"S", ",", RowBox[{"a", " ", "surface", " ", "with"}]}], "\[IndentingNewLine]", RowBox[{"parametrization", " ", "\[CapitalPhi]", RowBox[{"(", RowBox[{"u", ",", "v"}], ")"}]}], "\[IndentingNewLine]", RowBox[{"and", " ", "parameter", " ", "domain", " ", "R"}], "\[IndentingNewLine]"}], RowBox[{"scalar", " ", "function", " ", "f"}], RowBox[{ RowBox[{"\[Integral]", RowBox[{ SubsuperscriptBox["\[Integral]", "R", " "], RowBox[{"f", RowBox[{"(", RowBox[{"\[CapitalPhi]", RowBox[{"(", RowBox[{"u", ",", "v"}], ")"}]}], ")"}]}]}]}], "\[DoubleVerticalBar]", RowBox[{ FractionBox[ RowBox[{"\[PartialD]", "\[CapitalPhi]"}], RowBox[{"\[PartialD]", "u"}]], "\[Times]", FractionBox[ RowBox[{"\[PartialD]", "\[CapitalPhi]"}], RowBox[{"\[PartialD]", "v"}]]}], "\[DoubleVerticalBar]", RowBox[{ RowBox[{"\[DifferentialD]", "u"}], RowBox[{"\[DifferentialD]", "v"}]}]}]}, { RowBox[{"surface", " ", RowBox[{"(", "flux", ")"}]}], RowBox[{"\[Integral]", RowBox[{ SubscriptBox["\[Integral]", "S"], RowBox[{ RowBox[{ StyleBox["F", FontWeight->"Bold"], "\[CenterDot]", RowBox[{"\[DifferentialD]", StyleBox["S", FontSize->14, FontWeight->"Bold"]}]}], " ", "or", " ", RowBox[{ SubscriptBox["\[DoubleContourIntegral]", "S"], RowBox[{ StyleBox["F", FontWeight->"Bold"], "\[CenterDot]", RowBox[{"\[DifferentialD]", StyleBox["S", FontSize->14, FontWeight->"Bold"], " "}]}]}]}]}]}], RowBox[{ RowBox[{"S", ",", RowBox[{"a", " ", "surface", " ", "with"}]}], "\[IndentingNewLine]", RowBox[{"parametrization", " ", "\[CapitalPhi]", RowBox[{"(", RowBox[{"u", ",", "v"}], ")"}]}], "\[IndentingNewLine]", StyleBox[ RowBox[{"and", " ", "parameter", " ", "domain", " ", "R"}], FontWeight->"Plain"], "\[IndentingNewLine]"}], RowBox[{"vector", " ", "field", " ", StyleBox["F", FontWeight->"Bold"]}], RowBox[{"\[Integral]", RowBox[{ SubsuperscriptBox["\[Integral]", "R", " "], RowBox[{ StyleBox["F", FontWeight->"Bold"], RowBox[{ RowBox[{"(", " ", RowBox[{"\[CapitalPhi]", RowBox[{"(", RowBox[{"u", ",", "v"}], ")"}]}], ")"}], "\[CenterDot]", RowBox[{"(", RowBox[{ FractionBox[ RowBox[{"\[PartialD]", "\[CapitalPhi]"}], RowBox[{"\[PartialD]", "u"}]], "\[Times]", FractionBox[ RowBox[{"\[PartialD]", "\[CapitalPhi]"}], RowBox[{"\[PartialD]", "v"}]]}], ")"}]}], RowBox[{"\[DifferentialD]", "u"}], RowBox[{ RowBox[{"\[DifferentialD]", "v"}], " "}]}]}]}]} }]], "Text", CellFrame->True, TextAlignment->Left, FontSize->13, Background->GrayLevel[0.849989]], Cell["\<\ Notice that each kind of integral has its own definition, and there is no \ obvious way to convert one kind to another. Stokes's Theorem provides the \ missing link between surface integrals of vector fields and line integrals of \ vector fields.\ \>", "Text"], Cell[TextData[{ StyleBox["Stokes's Theorem", FontWeight->"Bold"], ": Suppose M is a smooth surface with a chosen orientation, and let ", StyleBox["F", FontWeight->"Bold"], " be a vector field which has continuous partial derivatives on and \"near\" \ M. Then ", Cell[BoxData[ RowBox[{ SubsuperscriptBox["\[Integral]", RowBox[{"\[PartialD]", "S"}], " "], RowBox[{ StyleBox["F", FontWeight->"Bold"], "\[CenterDot]", RowBox[{"\[DifferentialD]", StyleBox["s", FontWeight->"Bold"]}]}]}]], "Text"], " = ", Cell[BoxData[ RowBox[{"\[Integral]", RowBox[{ SubsuperscriptBox["\[Integral]", "S", " "], RowBox[{ RowBox[{"(", RowBox[{"curl", " ", StyleBox["F", FontWeight->"Bold"]}], StyleBox[")", FontWeight->"Bold"]}], "\[CenterDot]", RowBox[{"\[DifferentialD]", StyleBox["S", FontWeight->"Bold"]}]}]}]}]], "Text"], ", where \[PartialD]S is the positively-oriented boundary of S." }], "Text", CellFrame->True, FontWeight->"Plain", Background->GrayLevel[0.849989]] }, Closed]], Cell[CellGroupData[{ Cell["Using Stokes's Theorem: Example 1", "Section", Background->None], Cell[TextData[{ "Applying Stokes's Theorem can be tricky: each problem has its own kind of \ challenge. For example, we might want to calculate the surface integral of \ some vector field ", StyleBox["G", FontWeight->"Bold"], " over some complicated surface with a \"nice\" boundary. We can use \ Stokes's Theorem to avoid working with the surface, if we can find a vector \ field ", StyleBox["F", FontWeight->"Bold"], " whose curl is ", StyleBox["G", FontWeight->"Bold"], ". This can be hard." }], "Text"], Cell[CellGroupData[{ Cell["Example 1", "Subsection"], Cell[TextData[{ "Calculate ", Cell[BoxData[ RowBox[{"\[Integral]", RowBox[{ SubsuperscriptBox["\[Integral]", "M", " "], RowBox[{ StyleBox["G", FontWeight->"Bold"], "\[CenterDot]", RowBox[{"\[DifferentialD]", StyleBox["S", FontWeight->"Bold"]}]}]}]}]], "Text"], ", where ", StyleBox["G", FontWeight->"Bold"], "(x,y,z)=(0,0,2), and M is the surface parametrized by ", StyleBox["\[CapitalPhi]", FontWeight->"Bold"], "(r,\[Theta])=(r cos \[Theta], r sin \[Theta], sin(4\[Pi]r), with 0\ \[LessEqual]r\[LessEqual]1, 0\[LessEqual]\[Theta]\[LessEqual]2\[Pi]. [Use \ the upward-pointing normal vector.]" }], "Text"], Cell[BoxData[{ RowBox[{ RowBox[{"\[CapitalPhi]", "[", RowBox[{"r_", ",", "t_"}], "]"}], "=", RowBox[{"{", RowBox[{ RowBox[{"r", " ", RowBox[{"Cos", "[", "t", "]"}]}], ",", RowBox[{"r", " ", RowBox[{"Sin", "[", "t", "]"}]}], ",", RowBox[{"Sin", "[", RowBox[{"4", " ", "\[Pi]", " ", "r"}], "]"}]}], "}"}]}], "\n", RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"\[CapitalPhi]", "[", RowBox[{"r", ",", "t"}], "]"}], ",", RowBox[{"{", RowBox[{"r", ",", "0", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"2", " ", "\[Pi]"}]}], "}"}]}], "]"}]}], "Input", CellChangeTimes->{{3.41235635720215*^9, 3.412356363813921*^9}}], Cell[TextData[{ "This surface is intimidating, but its boundary is very nice. The boundary \ is just the part of the surface where r=1; we get its parametrization by \ substituting r=1 in ", StyleBox["\[CapitalPhi]", FontWeight->"Bold"], "." }], "Text"], Cell[BoxData[{ RowBox[{ RowBox[{"g", "[", "t_", "]"}], "=", RowBox[{"\[CapitalPhi]", "[", RowBox[{"1", ",", "t"}], "]"}]}], "\[IndentingNewLine]", RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"g", "[", "t", "]"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}]}], "]"}]}], "Input", CellChangeTimes->{{3.412356386889156*^9, 3.412356395306496*^9}}], Cell[TextData[{ "We need ", StyleBox["F", FontWeight->"Bold"], " so that curl ", StyleBox["F", FontWeight->"Bold"], "=", StyleBox["G", FontWeight->"Bold"], "; then our integral will be the right-hand side of the equation in Stokes's \ Theorem. One possible choice is ", StyleBox["F", FontWeight->"Bold"], "(x,y,z)=(-y,x,0). (Check this! Also note that, given ", StyleBox["G", FontWeight->"Bold"], ", there is no specific way for you to find an ", StyleBox["F", FontWeight->"Bold"], " such that curl ", StyleBox["F", FontWeight->"Bold"], "=", StyleBox["G", FontWeight->"Bold"], ". In this case we just \"pulled it out of the hat.\")", " If we use this ", StyleBox["F", FontWeight->"Bold"], ", then the left-hand side of the equation in Stokes's Theorem is almost \ (not quite) easy enough to do in your head:" }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{"\[Integral]", RowBox[{ SubsuperscriptBox["\[Integral]", "S", " "], RowBox[{ StyleBox["G", FontWeight->"Bold"], "\[CenterDot]", RowBox[{"\[DifferentialD]", StyleBox["S", FontWeight->"Bold"]}]}]}]}], "=", RowBox[{ RowBox[{ SubsuperscriptBox["\[Integral]", StyleBox[ RowBox[{"\[PartialD]", "S"}], FontSize->14], " "], RowBox[{ StyleBox["F", FontWeight->"Bold"], "\[CenterDot]", RowBox[{"\[DifferentialD]", StyleBox["s", FontWeight->"Bold"]}]}]}], StyleBox["=", FontWeight->"Bold"], RowBox[{ RowBox[{ SubsuperscriptBox["\[Integral]", "0", RowBox[{"2", "\[Pi]"}]], RowBox[{ StyleBox["F", FontWeight->"Bold"], RowBox[{ RowBox[{"(", RowBox[{ StyleBox["g", FontWeight->"Bold"], StyleBox[ RowBox[{"(", "t", ")"}], FontWeight->"Plain"]}], ")"}], "\[CenterDot]", RowBox[{ StyleBox["g", FontWeight->"Bold"], StyleBox["'", FontWeight->"Plain"]}]}], RowBox[{"(", "t", ")"}], RowBox[{"\[DifferentialD]", "t"}]}]}], "=", RowBox[{ RowBox[{ SubsuperscriptBox["\[Integral]", "0", RowBox[{"2", "\[Pi]"}]], RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", "sin"}], " ", "t"}], ",", RowBox[{"cos", " ", "t"}], ",", " ", "0"}], ")"}], "\[CenterDot]", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", "sin"}], " ", "t"}], ",", RowBox[{"cos", " ", "t"}], ",", " ", "0"}], ")"}]}], RowBox[{"\[DifferentialD]", "t"}]}]}], "=", RowBox[{ RowBox[{ SubsuperscriptBox["\[Integral]", "0", RowBox[{"2", "\[Pi]"}]], RowBox[{"1", RowBox[{"\[DifferentialD]", "t"}]}]}], "=", RowBox[{"2", RowBox[{"\[Pi]", "."}]}]}]}]}]}]}]], "Text"], Cell["\<\ Re-read this example to make sure you followed the process, and ask your TA \ if you have any questions about what we did or why. These problems can be \ tricky!\ \>", "Text"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Using Stokes's Theorem: Example 2", "Section", Background->None], Cell[TextData[{ "More often, we use Stokes's Theorem in the other direction: we have a \ tricky line integral, and try to replace it with an easy surface integral. \ This means we don't have to invent a new vector field with a given curl; \ instead, we have to invent a surface with a given boundary. Any surface with \ the right boundary will be good enough, just like in the last example, where \ any ", StyleBox["F ", FontWeight->"Bold"], "with the right curl was good enough.\n\nIf you'd like to see an example of \ many different surfaces which have the same boundary, copy the following \ address and paste it into your web browser:\n\n\ http://www.math.umn.edu/~rogness/multivar/multiple_surfaces.shtml" }], "Text"], Cell[CellGroupData[{ Cell["Example 2", "Subsection"], Cell[TextData[{ "Calculate ", Cell[BoxData[ RowBox[{ SubsuperscriptBox["\[Integral]", "C", " "], RowBox[{ StyleBox["F", FontWeight->"Bold"], "\[CenterDot]", RowBox[{"\[DifferentialD]", StyleBox["s", FontWeight->"Bold"]}]}]}]], "Text"], " , where ", StyleBox["F", FontWeight->"Bold"], "(x,y,z)=(", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "2"], "z"}], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["xy", "2"], ","}], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ SuperscriptBox["z", "2"], TraditionalForm]]], "), and C is the intersection of the plane ", StyleBox["x+y+z", FontSlant->"Italic"], "=4 with the cylinder ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]}], "=", "16"}], TraditionalForm]]], "." }], "Text"], Cell["Here is a plot of the two surfaces intersecting. ", "Text", CellChangeTimes->{{3.412356445748001*^9, 3.412356446368008*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"plane", "=", RowBox[{"Plot3D", "[", RowBox[{ RowBox[{"4", "-", "x", "-", "y"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "5"}], ",", "5"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "5"}], ",", "5"}], "}"}], ",", RowBox[{"Mesh", "->", "5"}], ",", RowBox[{"PlotStyle", "->", RowBox[{"Opacity", "[", "0.8", "]"}]}]}], "]"}]}], ";"}], "\n", RowBox[{ RowBox[{"cyl", "=", RowBox[{"ContourPlot3D", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"x", "^", "2"}], "+", RowBox[{"y", "^", "2"}]}], "==", "16"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "4"}], ",", "4"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "4"}], ",", "4"}], "}"}], ",", RowBox[{"{", RowBox[{"z", ",", RowBox[{"-", "4"}], ",", "12"}], "}"}], ",", RowBox[{"ContourStyle", "->", RowBox[{"Opacity", "[", "0.8", "]"}]}]}], "]"}]}], ";"}], "\n", RowBox[{"Show", "[", RowBox[{ RowBox[{"{", RowBox[{"plane", ",", "cyl"}], "}"}], ",", RowBox[{"BoxRatios", "\[Rule]", RowBox[{"{", RowBox[{"1", ",", "1", ",", "1.6"}], "}"}]}]}], "]"}]}], "Input", CellChangeTimes->{{3.41235645053652*^9, 3.412356534228788*^9}}], Cell["\<\ Let's parametrize the curve C, so we can graph it. Thinking in cylindrical \ coordinates, every point on C is on the cylinder, so has r=4. Also, every \ point on C is on the plane, so has z=4-x-y. Letting \[Theta] range from 0 to \ 2\[Pi], we get the following parametrization.\ \>", "Text"], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"f", "[", "t_", "]"}], "=", RowBox[{"{", RowBox[{ RowBox[{"4", RowBox[{"Cos", "[", "t", "]"}]}], ",", RowBox[{"4", RowBox[{"Sin", "[", "t", "]"}]}], ",", RowBox[{"4", "-", RowBox[{"4", RowBox[{"Cos", "[", "t", "]"}]}], "-", RowBox[{"4", RowBox[{"Sin", "[", "t", "]"}]}]}]}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"curve", "=", RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"f", "[", "t", "]"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}]}], "]"}]}]}], "Input"], Cell[TextData[{ "What is the orientation of this curve -- \"clockwise\" or \ \"counterclockwise\"? Check your answer using ", StyleBox["PathAnimate3D", FontFamily->"Courier", FontWeight->"Bold"], " or ", StyleBox["PathTangentAnimate3D", FontFamily->"Courier", FontWeight->"Bold"], " from Lab 4." }], "Text", CellFrame->True, Background->GrayLevel[0.849989]], Cell[TextData[{ "To use Stokes's Theorem, we need a surface whose boundary is C. We have an \ ideal surface available: it's just the part of the plane which lies inside \ the cylinder. [", StyleBox["Check", FontWeight->"Bold"], ": does C have the right orientation to be the boundary of this surface?] \ We can parametrize this surface just as we did C; the only difference is that \ instead of r=4, we have 0\[LessEqual]r\[LessEqual]4. " }], "Text"], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"\[CapitalPhi]", "[", RowBox[{"r_", ",", "t_"}], "]"}], "=", RowBox[{"{", RowBox[{ RowBox[{"r", "*", RowBox[{"Cos", "[", "t", "]"}]}], ",", RowBox[{"r", "*", RowBox[{"Sin", "[", "t", "]"}]}], ",", RowBox[{"4", "-", RowBox[{"r", "*", RowBox[{"Cos", "[", "t", "]"}]}], "-", RowBox[{"r", "*", RowBox[{"Sin", "[", "t", "]"}]}]}]}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"surface", "=", RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"\[CapitalPhi]", "[", RowBox[{"r", ",", "t"}], "]"}], ",", RowBox[{"{", RowBox[{"r", ",", "0", ",", "4"}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}]}], "]"}]}]}], "Input"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " gives us an oblique view of this surface; you should rotate it and compare \ to the earlier picture (with the plane and the cylinder) and convince \ yourself that our surface is exactly the part of the plane contained within \ the cylinder." }], "Text", CellChangeTimes->{3.41235658898673*^9}], Cell[TextData[{ "Finally, we can use Mathematica to calculate our integral: ", Cell[BoxData[ RowBox[{ RowBox[{ SubsuperscriptBox["\[Integral]", "C", " "], RowBox[{ StyleBox["F", FontWeight->"Bold"], "\[CenterDot]", RowBox[{"\[DifferentialD]", StyleBox["s", FontWeight->"Bold"]}]}]}], "=", RowBox[{"\[Integral]", RowBox[{ SubsuperscriptBox["\[Integral]", "S", " "], RowBox[{"curl", " ", RowBox[{ StyleBox["F", FontWeight->"Bold"], "\[CenterDot]", RowBox[{"\[DifferentialD]", StyleBox["S", FontWeight->"Bold"]}]}]}]}]}]}]], "Text"], " = ", Cell[BoxData[ RowBox[{"\[Integral]", RowBox[{ SubsuperscriptBox["\[Integral]", "R", " "], RowBox[{ RowBox[{"(", RowBox[{"curl", " ", StyleBox["F", FontWeight->"Bold"]}], ")"}], RowBox[{ RowBox[{"(", RowBox[{"\[CapitalPhi]", RowBox[{"(", RowBox[{"r", ",", "t"}], ")"}]}], ")"}], "\[CenterDot]", RowBox[{"(", RowBox[{ FractionBox[ RowBox[{"\[PartialD]", "\[CapitalPhi]"}], RowBox[{"\[PartialD]", "r"}]], "\[Times]", FractionBox[ RowBox[{"\[PartialD]", "\[CapitalPhi]"}], RowBox[{"\[PartialD]", "t"}]]}], ")"}]}], RowBox[{"\[DifferentialD]", "A"}]}]}]}]], TextAlignment->Left, FontSize->13], ", so we need the following calculations. First, we define ", StyleBox["F", FontWeight->"Bold"], ", ", "take its curl,", " and plug in the parametrization of our surface:" }], "Text"], Cell[BoxData[{ RowBox[{ RowBox[{"F", "[", RowBox[{"x_", ",", "y_", ",", "z_"}], "]"}], "=", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"x", "^", "2"}], "*", "z"}], ",", RowBox[{"x", "*", RowBox[{"y", "^", "2"}]}], ",", RowBox[{"z", "^", "2"}]}], "}"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"curlF", "[", RowBox[{"x_", ",", "y_", ",", "z_"}], "]"}], "=", RowBox[{"Curl", "[", RowBox[{"F", "[", RowBox[{"x", ",", "y", ",", "z"}], "]"}], "]"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"v1", "[", RowBox[{"r_", ",", "t_"}], "]"}], "=", RowBox[{"Apply", "[", RowBox[{"curlF", ",", RowBox[{"\[CapitalPhi]", "[", RowBox[{"r", ",", "t"}], "]"}]}], "]"}]}]}], "Input"], Cell[TextData[{ "Remember the ", StyleBox["Apply", FontFamily->"Courier", FontWeight->"Bold"], " command? We've used it a few times now; ask your TA for help if you don't \ remember it." }], "Text", CellFrame->True, Background->GrayLevel[0.849989]], Cell["\<\ Next, we take the partial derivatives of our parametrization and take the \ cross product:\ \>", "Text"], Cell[BoxData[{ RowBox[{ RowBox[{"d\[CapitalPhi]dr", "[", RowBox[{"r_", ",", "t_"}], "]"}], "=", RowBox[{"D", "[", RowBox[{ RowBox[{"\[CapitalPhi]", "[", RowBox[{"r", ",", "t"}], "]"}], ",", "r"}], "]"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"d\[CapitalPhi]dt", "[", RowBox[{"r_", ",", "t_"}], "]"}], "=", RowBox[{"D", "[", RowBox[{ RowBox[{"\[CapitalPhi]", "[", RowBox[{"r", ",", "t"}], "]"}], ",", "t"}], "]"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"v2", "[", RowBox[{"r_", ",", "t_"}], "]"}], "=", RowBox[{"Cross", "[", RowBox[{ RowBox[{"d\[CapitalPhi]dr", "[", RowBox[{"r", ",", "t"}], "]"}], ",", RowBox[{"d\[CapitalPhi]dt", "[", RowBox[{"r", ",", "t"}], "]"}]}], "]"}]}], "\[IndentingNewLine]", RowBox[{"Simplify", "[", "%", "]"}]}], "Input"], Cell["\<\ Finally, we take the dot product of our results, and integrate the result.\ \>", "Text"], Cell[BoxData[{ RowBox[{ RowBox[{"v1", "[", RowBox[{"r", ",", "t"}], "]"}], ".", RowBox[{"v2", "[", RowBox[{"r", ",", "t"}], "]"}]}], "\n", RowBox[{"Integrate", "[", RowBox[{"%", ",", RowBox[{"{", RowBox[{"r", ",", "0", ",", "4"}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}]}], "]"}]}], "Input"], Cell[TextData[{ "It's probably good to check our answer by calculating the line integral \ directly: ", Cell[BoxData[ RowBox[{ SubsuperscriptBox["\[Integral]", "C", " "], RowBox[{ StyleBox["F", FontWeight->"Bold"], "\[CenterDot]", RowBox[{"\[DifferentialD]", StyleBox["s", FontWeight->"Bold"]}]}]}]]], " = ", Cell[BoxData[ RowBox[{ SubsuperscriptBox["\[Integral]", "0", RowBox[{"2", "\[Pi]"}]], RowBox[{ StyleBox["F", FontWeight->"Bold"], RowBox[{ RowBox[{"(", RowBox[{ StyleBox["f", FontWeight->"Bold"], RowBox[{"(", "t", ")"}]}], ")"}], "\[CenterDot]", RowBox[{ StyleBox["f", FontWeight->"Bold"], "'"}]}], RowBox[{"(", "t", ")"}], RowBox[{"\[DifferentialD]", "t"}]}]}]], "Text"], ", so we get:" }], "Text"], Cell[BoxData[{ RowBox[{ RowBox[{"w1", "[", "t_", "]"}], "=", RowBox[{"Apply", "[", RowBox[{"F", ",", RowBox[{"f", "[", "t", "]"}]}], "]"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"w2", "[", "t_", "]"}], "=", RowBox[{"D", "[", RowBox[{ RowBox[{"f", "[", "t", "]"}], ",", "t"}], "]"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"w1", "[", "t", "]"}], ".", RowBox[{"w2", "[", "t", "]"}]}], "\[IndentingNewLine]", RowBox[{"Simplify", "[", "%", "]"}], "\[IndentingNewLine]", RowBox[{"Integrate", "[", RowBox[{"%", ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}]}], "]"}]}], "Input"], Cell["\<\ If this looks easier to you than the surface integral, take a careful look at \ the functions you're integrating. If the line integral still looks easier, \ you should try doing both by hand!\ \>", "Text"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Exercises", "Section", Background->None], Cell[TextData[{ "1) Graph the cylinder ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]}], "=", "1"}], TraditionalForm]]], "and the hyperbolic paraboloid ", Cell[BoxData[ FormBox[ RowBox[{"z", "=", RowBox[{ SuperscriptBox["y", "2"], "-", SuperscriptBox["x", "2"]}]}], TraditionalForm]]], "on the same set of axes, and parametrize the curve C where they intersect. \ Verify Stokes's Theorem for the vector field ", StyleBox["F", FontWeight->"Bold"], "(x,y,z)=(", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "2"], "y"}], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ RowBox[{ FractionBox["1", "3"], SuperscriptBox["x", "3"]}], TraditionalForm]]], ", ", StyleBox["xy", FontSlant->"Italic"], ") and the curve C. [i.e. Show that the line integral of ", StyleBox["F", FontWeight->"Bold"], " along C equals the appropriate surface integral. You will need to produce \ your own surface.]\n\n2) Now consider the vector field ", StyleBox["F", FontWeight->"Bold"], "(x,y,z)=(", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "2"], "y"}], TraditionalForm]]], ", ", Cell[BoxData[ FormBox["x", TraditionalForm]]], ", ", StyleBox["xy", FontSlant->"Italic"], ").\na) Calculate the line integral of ", StyleBox["F", FontWeight->"Bold"], " along the circle parametrized by ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ SubscriptBox["f", "1"], "(", "t", ")"}], "=", " ", RowBox[{"(", RowBox[{ RowBox[{"cos", "(", "t", ")"}], ",", " ", RowBox[{"sin", "(", "t", ")"}], ",", " ", "1"}], ")"}]}], ",", " ", RowBox[{ RowBox[{"for", " ", "0"}], "<", "t", "<", RowBox[{"2", RowBox[{"\[Pi]", "."}]}]}]}], TraditionalForm]]], "\nb) Calculate the line integral of ", StyleBox["F", FontWeight->"Bold"], " along the circle parametrized by ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ SubscriptBox["f", "2"], "(", "t", ")"}], "=", " ", RowBox[{"(", RowBox[{ RowBox[{"cos", "(", "t", ")"}], ",", " ", RowBox[{"sin", "(", "t", ")"}], ",", " ", "0"}], ")"}]}], ",", " ", RowBox[{ RowBox[{"for", " ", "0"}], "<", "t", "<", RowBox[{"2", RowBox[{"\[Pi]", "."}]}]}]}], TraditionalForm]]], "\nc) Calculate the surface integral of curl(", StyleBox["F", FontWeight->"Bold"], ") over the surface parametrized by ", StyleBox["\[CapitalPhi]", FontSlant->"Italic"], "(", StyleBox["s", FontSlant->"Italic"], ",", StyleBox["t", FontSlant->"Italic"], ")=(cos(", StyleBox["t", FontSlant->"Italic"], "),sin(", StyleBox["t", FontSlant->"Italic"], "),", StyleBox["s), ", FontSlant->"Italic"], "for", StyleBox[" ", FontSlant->"Italic"], "0", StyleBox[""Italic"], "2 \[Pi]", StyleBox[" ", FontSlant->"Italic"], "and 0<", StyleBox["s", FontSlant->"Italic"], "<1.\nd) Explain your answers in terms of Stokes's Theorem.\n\n3) Use \ Stokes's Theorem to evaluate the line integral of the vector field ", StyleBox["F", FontWeight->"Bold"], "(x,y,z)=(", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"y", "+", RowBox[{"sin", "(", "x", ")"}]}], ",", RowBox[{ SuperscriptBox["z", "2"], "+", RowBox[{"cos", "(", "y", ")"}]}], ",", " ", SuperscriptBox["x", "3"]}], TraditionalForm]]], "), where C is the curve parametrized by ", StyleBox["f", FontWeight->"Bold"], "(t)=(sin t, cos t, sin(2t)), for 0\[LessEqual]t\[LessEqual]2\[Pi].\n\n4) \ Consider the vector field ", StyleBox["G", FontWeight->"Bold"], "(x,y,z)=(x,y,z). Use Stokes's Theorem to prove, by contradiction, that ", StyleBox["G", FontWeight->"Bold"], " is not the curl of any smooth vector field ", StyleBox["F", FontWeight->"Bold"], ". [Hint: Let M be the unit sphere centered at the origin. The path \ integral over \[PartialD]M of every smooth vector field ", StyleBox["F", FontWeight->"Bold"], " equals 0 -- explain why!]" }], "Text", CellFrame->True, Background->RGBColor[1, 0.501961, 0.501961]], Cell[CellGroupData[{ Cell["Credits", "Subsection"], Cell[TextData[{ "This lab was written by James Swenson in 2002. In Spring 2004 Jonathan \ Rogness went through and updated a few minor things to reflect the use of our \ math2374.nb file. (A few more minor updates in Spring 2008 for ", StyleBox["Mathematica", FontSlant->"Italic"], " 6.0.)\n\nThis lab is copyright 2002 by James Swenson \ (swenson@math.umn.edu) and is protected by the Creative Commons \ Attribution-NonCommercial-ShareAlike License. You can find more information \ on this license at http://creativecommons.org/licenses/by-nc-sa/1.0/. \n\n\ Although it's not specifically required by the license, I'd appreciate it if \ you let me know at rogness@math.umn.edu if you use parts of our labs, just so \ I can keep track of it. Please send me any questions or comments!" }], "Text", CellChangeTimes->{{3.41235663639472*^9, 3.412356646878349*^9}}] }, Open ]] }, Closed]] }, WindowToolbars->{}, WindowSize->{959, 594}, WindowMargins->{{20, Automatic}, {Automatic, 29}}, PrintingPageRange->{Automatic, Automatic}, PrintingOptions->{"Magnification"->1, "PaperOrientation"->"Portrait", "PaperSize"->{612, 792}, "PostScriptOutputFile":>FrontEnd`FileName[{"user002", "rogness"}, "Newlab.nb.ps", CharacterEncoding -> "WindowsANSI"]}, FrontEndVersion->"6.0 for Linux x86 (32-bit) (April 20, 2007)", StyleDefinitions->Notebook[{ Cell[ CellGroupData[{ Cell["Style Definitions", "Subtitle"], Cell[ "Modify the definitions below to change the default appearance of all \ cells in a given style. Make modifications to any definition using commands \ in the Format menu.", "Text"], Cell[ CellGroupData[{ Cell["Style Environment Names", "Section"], Cell[ StyleData[All, "Working"], PageWidth -> WindowWidth, CellLabelMargins -> {{12, Inherited}, {Inherited, Inherited}}, ScriptMinSize -> 9], Cell[ StyleData[All, "Presentation"], PageWidth -> WindowWidth, CellLabelMargins -> {{24, Inherited}, {Inherited, Inherited}}, ScriptMinSize -> 12], Cell[ StyleData[All, "Condensed"], PageWidth -> WindowWidth, CellLabelMargins -> {{8, Inherited}, {Inherited, Inherited}}, ScriptMinSize -> 8], Cell[ StyleData[All, "Printout"], PageWidth -> PaperWidth, CellLabelMargins -> {{2, Inherited}, {Inherited, Inherited}}, ScriptMinSize -> 5, PrivateFontOptions -> {"FontType" -> "Outline"}]}, Closed]], Cell[ CellGroupData[{ Cell["Notebook Options", "Section"], Cell[ "The options defined for the style below will be used at the \ Notebook level.", "Text"], Cell[ StyleData["Notebook"], PageHeaders -> {{ Cell[ TextData[{ CounterBox["Page"]}], "PageNumber"], None, Cell[ TextData[{ ValueBox["FileName"]}], "Header"]}, { Cell[ TextData[{ ValueBox["FileName"]}], "Header"], None, Cell[ TextData[{ CounterBox["Page"]}], "PageNumber"]}}, CellFrameLabelMargins -> 6, StyleMenuListing -> None]}, Closed]], Cell[ CellGroupData[{ Cell["Styles for Headings", "Section"], Cell[ CellGroupData[{ Cell[ StyleData["Title"], CellMargins -> {{12, Inherited}, {20, 40}}, CellGroupingRules -> {"TitleGrouping", 0}, PageBreakBelow -> False, DefaultNewInlineCellStyle -> "None", InputAutoReplacements -> {"TeX" -> StyleBox[ RowBox[{"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "LaTeX" -> StyleBox[ RowBox[{"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-0.36, -0.1}, {0, 0}}, BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma" -> "Mathematica", "Mma" -> "Mathematica", "MMA" -> "Mathematica"}, LanguageCategory -> "NaturalLanguage", CounterIncrements -> "Title", CounterAssignments -> {{"Section", 0}, {"Equation", 0}, { "Figure", 0}, {"Subtitle", 0}, {"Subsubtitle", 0}}, FontFamily -> "Helvetica", FontSize -> 36, FontWeight -> "Bold"], Cell[ StyleData["Title", "Presentation"], CellMargins -> {{24, 10}, {20, 40}}, LineSpacing -> {1, 0}, FontSize -> 44], Cell[ StyleData["Title", "Condensed"], CellMargins -> {{8, 10}, {4, 8}}, FontSize -> 20], Cell[ StyleData["Title", "Printout"], CellMargins -> {{2, 10}, {12, 30}}, FontSize -> 24]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["Subtitle"], CellMargins -> {{12, Inherited}, {20, 15}}, CellGroupingRules -> {"TitleGrouping", 10}, PageBreakBelow -> False, DefaultNewInlineCellStyle -> "None", InputAutoReplacements -> {"TeX" -> StyleBox[ RowBox[{"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "LaTeX" -> StyleBox[ RowBox[{"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-0.36, -0.1}, {0, 0}}, BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma" -> "Mathematica", "Mma" -> "Mathematica", "MMA" -> "Mathematica"}, LanguageCategory -> "NaturalLanguage", CounterIncrements -> "Subtitle", CounterAssignments -> {{"Section", 0}, {"Equation", 0}, { "Figure", 0}, {"Subsubtitle", 0}}, FontFamily -> "Helvetica", FontSize -> 24], Cell[ StyleData["Subtitle", "Presentation"], CellMargins -> {{24, 10}, {20, 20}}, LineSpacing -> {1, 0}, FontSize -> 36], Cell[ StyleData["Subtitle", "Condensed"], CellMargins -> {{8, 10}, {4, 4}}, FontSize -> 14], Cell[ StyleData["Subtitle", "Printout"], CellMargins -> {{2, 10}, {12, 8}}, FontSize -> 18]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["Subsubtitle"], CellMargins -> {{12, Inherited}, {20, 15}}, CellGroupingRules -> {"TitleGrouping", 20}, PageBreakBelow -> False, DefaultNewInlineCellStyle -> "None", InputAutoReplacements -> {"TeX" -> StyleBox[ RowBox[{"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "LaTeX" -> StyleBox[ RowBox[{"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-0.36, -0.1}, {0, 0}}, BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma" -> "Mathematica", "Mma" -> "Mathematica", "MMA" -> "Mathematica"}, LanguageCategory -> "NaturalLanguage", CounterIncrements -> "Subsubtitle", CounterAssignments -> {{"Section", 0}, {"Equation", 0}, { "Figure", 0}}, FontFamily -> "Helvetica", FontSize -> 14, FontSlant -> "Italic"], Cell[ StyleData["Subsubtitle", "Presentation"], CellMargins -> {{24, 10}, {20, 20}}, LineSpacing -> {1, 0}, FontSize -> 24], Cell[ StyleData["Subsubtitle", "Condensed"], CellMargins -> {{8, 10}, {8, 8}}, FontSize -> 12], Cell[ StyleData["Subsubtitle", "Printout"], CellMargins -> {{2, 10}, {12, 8}}, FontSize -> 14]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["Section"], CellDingbat -> "\[FilledSquare]", CellMargins -> {{25, Inherited}, {8, 24}}, CellGroupingRules -> {"SectionGrouping", 30}, PageBreakBelow -> False, DefaultNewInlineCellStyle -> "None", InputAutoReplacements -> {"TeX" -> StyleBox[ RowBox[{"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "LaTeX" -> StyleBox[ RowBox[{"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-0.36, -0.1}, {0, 0}}, BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma" -> "Mathematica", "Mma" -> "Mathematica", "MMA" -> "Mathematica"}, LanguageCategory -> "NaturalLanguage", CounterIncrements -> "Section", CounterAssignments -> {{"Subsection", 0}, {"Subsubsection", 0}}, FontFamily -> "Helvetica", FontSize -> 16, FontWeight -> "Bold", Background -> RGBColor[1, 0, 0]], Cell[ StyleData["Section", "Presentation"], CellMargins -> {{40, 10}, {11, 32}}, LineSpacing -> {1, 0}, FontSize -> 24], Cell[ StyleData["Section", "Condensed"], CellMargins -> {{18, Inherited}, {6, 12}}, FontSize -> 12], Cell[ StyleData["Section", "Printout"], CellMargins -> {{13, 0}, {7, 22}}, FontSize -> 14]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["Subsection"], CellDingbat -> "\[FilledSmallSquare]", CellMargins -> {{22, Inherited}, {8, 20}}, CellGroupingRules -> {"SectionGrouping", 40}, PageBreakBelow -> False, DefaultNewInlineCellStyle -> "None", InputAutoReplacements -> {"TeX" -> StyleBox[ RowBox[{"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "LaTeX" -> StyleBox[ RowBox[{"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-0.36, -0.1}, {0, 0}}, BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma" -> "Mathematica", "Mma" -> "Mathematica", "MMA" -> "Mathematica"}, LanguageCategory -> "NaturalLanguage", CounterIncrements -> "Subsection", CounterAssignments -> {{"Subsubsection", 0}}, FontFamily -> "Times", FontSize -> 14, FontWeight -> "Bold"], Cell[ StyleData["Subsection", "Presentation"], CellMargins -> {{36, 10}, {11, 32}}, LineSpacing -> {1, 0}, FontSize -> 22], Cell[ StyleData["Subsection", "Condensed"], CellMargins -> {{16, Inherited}, {6, 12}}, FontSize -> 12], Cell[ StyleData["Subsection", "Printout"], CellMargins -> {{9, 0}, {7, 22}}, FontSize -> 12]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["Subsubsection"], CellDingbat -> "\[FilledSmallSquare]", CellMargins -> {{22, Inherited}, {8, 18}}, CellGroupingRules -> {"SectionGrouping", 50}, PageBreakBelow -> False, DefaultNewInlineCellStyle -> "None", InputAutoReplacements -> {"TeX" -> StyleBox[ RowBox[{"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "LaTeX" -> StyleBox[ RowBox[{"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-0.36, -0.1}, {0, 0}}, BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma" -> "Mathematica", "Mma" -> "Mathematica", "MMA" -> "Mathematica"}, LanguageCategory -> "NaturalLanguage", CounterIncrements -> "Subsubsection", FontFamily -> "Times", FontWeight -> "Bold"], Cell[ StyleData["Subsubsection", "Presentation"], CellMargins -> {{34, 10}, {11, 26}}, LineSpacing -> {1, 0}, FontSize -> 18], Cell[ StyleData["Subsubsection", "Condensed"], CellMargins -> {{17, Inherited}, {6, 12}}, FontSize -> 10], Cell[ StyleData["Subsubsection", "Printout"], CellMargins -> {{9, 0}, {7, 14}}, FontSize -> 11]}, Closed]]}, Closed]], Cell[ CellGroupData[{ Cell["Styles for Body Text", "Section"], Cell[ CellGroupData[{ Cell[ StyleData["Text"], CellMargins -> {{12, 10}, {7, 7}}, InputAutoReplacements -> {"TeX" -> StyleBox[ RowBox[{"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "LaTeX" -> StyleBox[ RowBox[{"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-0.36, -0.1}, {0, 0}}, BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma" -> "Mathematica", "Mma" -> "Mathematica", "MMA" -> "Mathematica"}, Hyphenation -> True, LineSpacing -> {1, 3}, CounterIncrements -> "Text"], Cell[ StyleData["Text", "Presentation"], CellMargins -> {{24, 10}, {10, 10}}, LineSpacing -> {1, 5}, FontSize -> 16], Cell[ StyleData["Text", "Condensed"], CellMargins -> {{8, 10}, {6, 6}}, LineSpacing -> {1, 1}, FontSize -> 11], Cell[ StyleData["Text", "Printout"], CellMargins -> {{2, 2}, {6, 6}}, TextJustification -> 0.5, FontSize -> 10]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["SmallText"], CellMargins -> {{12, 10}, {6, 6}}, DefaultNewInlineCellStyle -> "None", Hyphenation -> True, LineSpacing -> {1, 3}, LanguageCategory -> "NaturalLanguage", CounterIncrements -> "SmallText", FontFamily -> "Helvetica", FontSize -> 9], Cell[ StyleData["SmallText", "Presentation"], CellMargins -> {{24, 10}, {8, 8}}, LineSpacing -> {1, 5}, FontSize -> 12], Cell[ StyleData["SmallText", "Condensed"], CellMargins -> {{8, 10}, {5, 5}}, LineSpacing -> {1, 2}, FontSize -> 9], Cell[ StyleData["SmallText", "Printout"], CellMargins -> {{2, 2}, {5, 5}}, TextJustification -> 0.5, FontSize -> 7]}, Closed]]}, Closed]], Cell[ CellGroupData[{ Cell["Styles for Input/Output", "Section"], Cell[ "The cells in this section define styles used for input and output \ to the kernel. Be careful when modifying, renaming, or removing these \ styles, because the front end associates special meanings with these style \ names. Some attributes for these styles are actually set in FormatType Styles \ (in the last section of this stylesheet). ", "Text"], Cell[ CellGroupData[{ Cell[ StyleData["Input"], CellMargins -> {{45, 10}, {5, 7}}, Evaluatable -> True, CellGroupingRules -> "InputGrouping", CellHorizontalScrolling -> True, PageBreakWithin -> False, GroupPageBreakWithin -> False, DefaultFormatType -> DefaultInputFormatType, HyphenationOptions -> { "HyphenationCharacter" -> "\[Continuation]"}, AutoItalicWords -> {}, LanguageCategory -> "Formula", FormatType -> InputForm, ShowStringCharacters -> True, NumberMarks -> True, LinebreakAdjustments -> {0.85, 2, 10, 0, 1}, CounterIncrements -> "Input", FontWeight -> "Bold"], Cell[ StyleData["Input", "Presentation"], CellMargins -> {{72, Inherited}, {8, 10}}, LineSpacing -> {1, 0}, FontSize -> 16], Cell[ StyleData["Input", "Condensed"], CellMargins -> {{40, 10}, {2, 3}}, FontSize -> 11], Cell[ StyleData["Input", "Printout"], CellMargins -> {{39, 0}, {4, 6}}, LinebreakAdjustments -> {0.85, 2, 10, 1, 1}, FontSize -> 9]}, Closed]], Cell[ StyleData["InputOnly"], Evaluatable -> True, CellGroupingRules -> "InputGrouping", CellHorizontalScrolling -> True, DefaultFormatType -> DefaultInputFormatType, HyphenationOptions -> {"HyphenationCharacter" -> "\[Continuation]"}, AutoItalicWords -> {}, LanguageCategory -> "Formula", FormatType -> InputForm, ShowStringCharacters -> True, NumberMarks -> True, LinebreakAdjustments -> {0.85, 2, 10, 0, 1}, CounterIncrements -> "Input", StyleMenuListing -> None, FontWeight -> "Bold"], Cell[ CellGroupData[{ Cell[ StyleData["Output"], CellMargins -> {{47, 10}, {7, 5}}, CellEditDuplicate -> True, CellGroupingRules -> "OutputGrouping", CellHorizontalScrolling -> True, PageBreakWithin -> False, GroupPageBreakWithin -> False, GeneratedCell -> True, CellAutoOverwrite -> True, DefaultFormatType -> DefaultOutputFormatType, HyphenationOptions -> { "HyphenationCharacter" -> "\[Continuation]"}, AutoItalicWords -> {}, LanguageCategory -> "Formula", FormatType -> InputForm, CounterIncrements -> "Output"], Cell[ StyleData["Output", "Presentation"], CellMargins -> {{72, Inherited}, {10, 8}}, LineSpacing -> {1, 0}, FontSize -> 16], Cell[ StyleData["Output", "Condensed"], CellMargins -> {{41, Inherited}, {3, 2}}, FontSize -> 11], Cell[ StyleData["Output", "Printout"], CellMargins -> {{39, 0}, {6, 4}}, FontSize -> 9]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["Message"], CellMargins -> {{45, Inherited}, {Inherited, Inherited}}, CellGroupingRules -> "OutputGrouping", PageBreakWithin -> False, GroupPageBreakWithin -> False, GeneratedCell -> True, CellAutoOverwrite -> True, ShowCellLabel -> False, DefaultFormatType -> DefaultOutputFormatType, HyphenationOptions -> { "HyphenationCharacter" -> "\[Continuation]"}, AutoItalicWords -> {}, FormatType -> InputForm, CounterIncrements -> "Message", StyleMenuListing -> None, FontSize -> 11, FontColor -> RGBColor[0, 0, 1]], Cell[ StyleData["Message", "Presentation"], CellMargins -> {{72, Inherited}, {Inherited, Inherited}}, LineSpacing -> {1, 0}, FontSize -> 16], Cell[ StyleData["Message", "Condensed"], CellMargins -> {{41, Inherited}, {Inherited, Inherited}}, FontSize -> 11], Cell[ StyleData["Message", "Printout"], CellMargins -> {{39, Inherited}, {Inherited, Inherited}}, FontSize -> 7, FontColor -> GrayLevel[0]]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["Print"], CellMargins -> {{45, Inherited}, {Inherited, Inherited}}, CellGroupingRules -> "OutputGrouping", CellHorizontalScrolling -> True, PageBreakWithin -> False, GroupPageBreakWithin -> False, GeneratedCell -> True, CellAutoOverwrite -> True, ShowCellLabel -> False, DefaultFormatType -> DefaultOutputFormatType, HyphenationOptions -> { "HyphenationCharacter" -> "\[Continuation]"}, AutoItalicWords -> {}, FormatType -> InputForm, CounterIncrements -> "Print", StyleMenuListing -> None], Cell[ StyleData["Print", "Presentation"], CellMargins -> {{72, Inherited}, {Inherited, Inherited}}, LineSpacing -> {1, 0}, FontSize -> 16], Cell[ StyleData["Print", "Condensed"], CellMargins -> {{41, Inherited}, {Inherited, Inherited}}, FontSize -> 11], Cell[ StyleData["Print", "Printout"], CellMargins -> {{39, Inherited}, {Inherited, Inherited}}, FontSize -> 8]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["Graphics"], CellMargins -> {{4, Inherited}, {Inherited, Inherited}}, CellGroupingRules -> "GraphicsGrouping", CellHorizontalScrolling -> True, PageBreakWithin -> False, GeneratedCell -> True, CellAutoOverwrite -> True, ShowCellLabel -> False, DefaultFormatType -> DefaultOutputFormatType, LanguageCategory -> None, FormatType -> InputForm, CounterIncrements -> "Graphics", ImageMargins -> {{43, Inherited}, {Inherited, 0}}, StyleMenuListing -> None, FontFamily -> "Courier", FontSize -> 10], Cell[ StyleData["Graphics", "Presentation"], ImageMargins -> {{62, Inherited}, {Inherited, 0}}], Cell[ StyleData["Graphics", "Condensed"], ImageMargins -> {{38, Inherited}, {Inherited, 0}}, Magnification -> 0.6], Cell[ StyleData["Graphics", "Printout"], ImageMargins -> {{30, Inherited}, {Inherited, 0}}, Magnification -> 0.8]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["CellLabel"], StyleMenuListing -> None, FontFamily -> "Helvetica", FontSize -> 9, FontColor -> RGBColor[0, 0, 1]], Cell[ StyleData["CellLabel", "Presentation"], FontSize -> 12], Cell[ StyleData["CellLabel", "Condensed"], FontSize -> 9], Cell[ StyleData["CellLabel", "Printout"], FontFamily -> "Courier", FontSize -> 8, FontSlant -> "Italic", FontColor -> GrayLevel[0]]}, Closed]]}, Closed]], Cell[ CellGroupData[{ Cell["Inline Formatting", "Section"], Cell[ "These styles are for modifying individual words or letters in a \ cell exclusive of the cell tag.", "Text"], Cell[ StyleData["RM"], StyleMenuListing -> None, FontWeight -> "Plain", FontSlant -> "Plain"], Cell[ StyleData["BF"], StyleMenuListing -> None, FontWeight -> "Bold"], Cell[ StyleData["IT"], StyleMenuListing -> None, FontSlant -> "Italic"], Cell[ StyleData["TR"], StyleMenuListing -> None, FontFamily -> "Times", FontWeight -> "Plain", FontSlant -> "Plain"], Cell[ StyleData["TI"], StyleMenuListing -> None, FontFamily -> "Times", FontWeight -> "Plain", FontSlant -> "Italic"], Cell[ StyleData["TB"], StyleMenuListing -> None, FontFamily -> "Times", FontWeight -> "Bold", FontSlant -> "Plain"], Cell[ StyleData["TBI"], StyleMenuListing -> None, FontFamily -> "Times", FontWeight -> "Bold", FontSlant -> "Italic"], Cell[ StyleData["MR"], StyleMenuListing -> None, FontFamily -> "Courier", FontWeight -> "Plain", FontSlant -> "Plain"], Cell[ StyleData["MO"], StyleMenuListing -> None, FontFamily -> "Courier", FontWeight -> "Plain", FontSlant -> "Italic"], Cell[ StyleData["MB"], StyleMenuListing -> None, FontFamily -> "Courier", FontWeight -> "Bold", FontSlant -> "Plain"], Cell[ StyleData["MBO"], StyleMenuListing -> None, FontFamily -> "Courier", FontWeight -> "Bold", FontSlant -> "Italic"], Cell[ StyleData["SR"], StyleMenuListing -> None, FontFamily -> "Helvetica", FontWeight -> "Plain", FontSlant -> "Plain"], Cell[ StyleData["SO"], StyleMenuListing -> None, FontFamily -> "Helvetica", FontWeight -> "Plain", FontSlant -> "Italic"], Cell[ StyleData["SB"], StyleMenuListing -> None, FontFamily -> "Helvetica", FontWeight -> "Bold", FontSlant -> "Plain"], Cell[ StyleData["SBO"], StyleMenuListing -> None, FontFamily -> "Helvetica", FontWeight -> "Bold", FontSlant -> "Italic"], Cell[ CellGroupData[{ Cell[ StyleData["SO10"], StyleMenuListing -> None, FontFamily -> "Helvetica", FontSize -> 10, FontWeight -> "Plain", FontSlant -> "Italic"], Cell[ StyleData["SO10", "Printout"], StyleMenuListing -> None, FontFamily -> "Helvetica", FontSize -> 7, FontWeight -> "Plain", FontSlant -> "Italic"], Cell[ StyleData["SO10", "EnhancedPrintout"], StyleMenuListing -> None, FontFamily -> "Futura", FontSize -> 7, FontWeight -> "Plain", FontSlant -> "Italic"]}, Closed]]}, Closed]], Cell[ CellGroupData[{ Cell["Formulas and Programming", "Section"], Cell[ CellGroupData[{ Cell[ StyleData["InlineFormula"], CellMargins -> {{10, 4}, {0, 8}}, CellHorizontalScrolling -> True, HyphenationOptions -> { "HyphenationCharacter" -> "\[Continuation]"}, LanguageCategory -> "Formula", ScriptLevel -> 1, SingleLetterItalics -> True], Cell[ StyleData["InlineFormula", "Presentation"], CellMargins -> {{24, 10}, {10, 10}}, LineSpacing -> {1, 5}, FontSize -> 16], Cell[ StyleData["InlineFormula", "Condensed"], CellMargins -> {{8, 10}, {6, 6}}, LineSpacing -> {1, 1}, FontSize -> 11], Cell[ StyleData["InlineFormula", "Printout"], CellMargins -> {{2, 0}, {6, 6}}, FontSize -> 10]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["DisplayFormula"], CellMargins -> {{42, Inherited}, {Inherited, Inherited}}, CellHorizontalScrolling -> True, DefaultFormatType -> DefaultInputFormatType, HyphenationOptions -> { "HyphenationCharacter" -> "\[Continuation]"}, LanguageCategory -> "Formula", ScriptLevel -> 0, SingleLetterItalics -> True, UnderoverscriptBoxOptions -> {LimitsPositioning -> True}], Cell[ StyleData["DisplayFormula", "Presentation"], LineSpacing -> {1, 5}, FontSize -> 16], Cell[ StyleData["DisplayFormula", "Condensed"], LineSpacing -> {1, 1}, FontSize -> 11], Cell[ StyleData["DisplayFormula", "Printout"], FontSize -> 10]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["Program"], CellFrame -> {{0, 0}, {0.5, 0.5}}, CellMargins -> {{10, 4}, {0, 8}}, CellHorizontalScrolling -> True, Hyphenation -> False, LanguageCategory -> "Formula", ScriptLevel -> 1, FontFamily -> "Courier"], Cell[ StyleData["Program", "Presentation"], CellMargins -> {{24, 10}, {10, 10}}, LineSpacing -> {1, 5}, FontSize -> 16], Cell[ StyleData["Program", "Condensed"], CellMargins -> {{8, 10}, {6, 6}}, LineSpacing -> {1, 1}, FontSize -> 11], Cell[ StyleData["Program", "Printout"], CellMargins -> {{2, 0}, {6, 6}}, FontSize -> 9]}, Closed]]}, Closed]], Cell[ CellGroupData[{ Cell["Hyperlink Styles", "Section"], Cell[ "The cells below define styles useful for making hypertext \ ButtonBoxes. The \"Hyperlink\" style is for links within the same Notebook, \ or between Notebooks.", "Text"], Cell[ CellGroupData[{ Cell[ StyleData["Hyperlink"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, FontVariations -> {"Underline" -> True}, FontColor -> RGBColor[0, 0, 1], ButtonBoxOptions -> { Active -> True, ButtonFunction :> (FrontEndExecute[{ FrontEnd`NotebookLocate[#2]}]& ), ButtonNote -> ButtonData}], Cell[ StyleData["Hyperlink", "Presentation"], FontSize -> 16], Cell[ StyleData["Hyperlink", "Condensed"], FontSize -> 11], Cell[ StyleData["Hyperlink", "Printout"], FontSize -> 10, FontVariations -> {"Underline" -> False}, FontColor -> GrayLevel[0]]}, Closed]], Cell[ "The following styles are for linking automatically to the on-line \ help system.", "Text"], Cell[ CellGroupData[{ Cell[ StyleData["MainBookLink"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, FontVariations -> {"Underline" -> True}, FontColor -> RGBColor[0, 0, 1], ButtonBoxOptions -> { Active -> True, ButtonFrame -> "None", ButtonFunction :> (FrontEndExecute[{ FrontEnd`HelpBrowserLookup["MainBook", #]}]& )}], Cell[ StyleData["MainBookLink", "Presentation"], FontSize -> 16], Cell[ StyleData["MainBookLink", "Condensed"], FontSize -> 11], Cell[ StyleData["MainBookLink", "Printout"], FontSize -> 10, FontVariations -> {"Underline" -> False}, FontColor -> GrayLevel[0]]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["AddOnsLink"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, FontFamily -> "Courier", FontVariations -> {"Underline" -> True}, FontColor -> RGBColor[0, 0, 1], ButtonBoxOptions -> { Active -> True, ButtonFrame -> "None", ButtonFunction :> (FrontEndExecute[{ FrontEnd`HelpBrowserLookup["AddOns", #]}]& )}], Cell[ StyleData["AddOnsLink", "Presentation"], FontSize -> 16], Cell[ StyleData["AddOnsLink", "Condensed"], FontSize -> 11], Cell[ StyleData["AddOnsLink", "Printout"], FontSize -> 10, FontVariations -> {"Underline" -> False}, FontColor -> GrayLevel[0]]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["RefGuideLink"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, FontFamily -> "Courier", FontVariations -> {"Underline" -> True}, FontColor -> RGBColor[0, 0, 1], ButtonBoxOptions -> { Active -> True, ButtonFrame -> "None", ButtonFunction :> (FrontEndExecute[{ FrontEnd`HelpBrowserLookup["RefGuide", #]}]& )}], Cell[ StyleData["RefGuideLink", "Presentation"], FontSize -> 16], Cell[ StyleData["RefGuideLink", "Condensed"], FontSize -> 11], Cell[ StyleData["RefGuideLink", "Printout"], FontSize -> 10, FontVariations -> {"Underline" -> False}, FontColor -> GrayLevel[0]]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["GettingStartedLink"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, FontVariations -> {"Underline" -> True}, FontColor -> RGBColor[0, 0, 1], ButtonBoxOptions -> { Active -> True, ButtonFrame -> "None", ButtonFunction :> (FrontEndExecute[{ FrontEnd`HelpBrowserLookup["GettingStarted", #]}]& )}], Cell[ StyleData["GettingStartedLink", "Presentation"], FontSize -> 16], Cell[ StyleData["GettingStartedLink", "Condensed"], FontSize -> 11], Cell[ StyleData["GettingStartedLink", "Printout"], FontSize -> 10, FontVariations -> {"Underline" -> False}, FontColor -> GrayLevel[0]]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["OtherInformationLink"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, FontVariations -> {"Underline" -> True}, FontColor -> RGBColor[0, 0, 1], ButtonBoxOptions -> { Active -> True, ButtonFrame -> "None", ButtonFunction :> (FrontEndExecute[{ FrontEnd`HelpBrowserLookup["OtherInformation", #]}]& )}], Cell[ StyleData["OtherInformationLink", "Presentation"], FontSize -> 16], Cell[ StyleData["OtherInformationLink", "Condensed"], FontSize -> 11], Cell[ StyleData["OtherInformationLink", "Printout"], FontSize -> 10, FontVariations -> {"Underline" -> False}, FontColor -> GrayLevel[0]]}, Closed]]}, Closed]], Cell[ CellGroupData[{ Cell["Styles for Headers and Footers", "Section"], Cell[ StyleData["Header"], CellMargins -> {{0, 0}, {4, 1}}, DefaultNewInlineCellStyle -> "None", LanguageCategory -> "NaturalLanguage", StyleMenuListing -> None, FontSize -> 10, FontSlant -> "Italic"], Cell[ StyleData["Footer"], CellMargins -> {{0, 0}, {0, 4}}, DefaultNewInlineCellStyle -> "None", LanguageCategory -> "NaturalLanguage", StyleMenuListing -> None, FontSize -> 9, FontSlant -> "Italic"], Cell[ StyleData["PageNumber"], CellMargins -> {{0, 0}, {4, 1}}, StyleMenuListing -> None, FontFamily -> "Times", FontSize -> 10]}, Closed]], Cell[ CellGroupData[{ Cell["Palette Styles", "Section"], Cell[ "The cells below define styles that define standard ButtonFunctions, \ for use in palette buttons.", "Text"], Cell[ StyleData["Paste"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, ButtonBoxOptions -> {ButtonFunction :> (FrontEndExecute[{ FrontEnd`NotebookApply[ FrontEnd`InputNotebook[], #, After]}]& )}], Cell[ StyleData["Evaluate"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, ButtonBoxOptions -> {ButtonFunction :> (FrontEndExecute[{ FrontEnd`NotebookApply[ FrontEnd`InputNotebook[], #, All], SelectionEvaluate[ FrontEnd`InputNotebook[], All]}]& )}], Cell[ StyleData["EvaluateCell"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, ButtonBoxOptions -> {ButtonFunction :> (FrontEndExecute[{ FrontEnd`NotebookApply[ FrontEnd`InputNotebook[], #, All], FrontEnd`SelectionMove[ FrontEnd`InputNotebook[], All, Cell, 1], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[], All]}]& )}], Cell[ StyleData["CopyEvaluate"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, ButtonBoxOptions -> {ButtonFunction :> (FrontEndExecute[{ FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[], #, All], FrontEnd`SelectionEvaluate[ FrontEnd`InputNotebook[], All]}]& )}], Cell[ StyleData["CopyEvaluateCell"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, ButtonBoxOptions -> {ButtonFunction :> (FrontEndExecute[{ FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[], #, All], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[], All]}]& )}]}, Closed]], Cell[ CellGroupData[{ Cell["Placeholder Styles", "Section"], Cell[ "The cells below define styles useful for making placeholder objects \ in palette templates.", "Text"], Cell[ CellGroupData[{ Cell[ StyleData["Placeholder"], Placeholder -> True, StyleMenuListing -> None, FontSlant -> "Italic", FontColor -> RGBColor[0.890623, 0.864698, 0.384756], TagBoxOptions -> { Editable -> False, Selectable -> False, StripWrapperBoxes -> False}], Cell[ StyleData["Placeholder", "Presentation"]], Cell[ StyleData["Placeholder", "Condensed"]], Cell[ StyleData["Placeholder", "Printout"]]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["PrimaryPlaceholder"], StyleMenuListing -> None, DrawHighlighted -> True, FontSlant -> "Italic", Background -> RGBColor[0.912505, 0.891798, 0.507774], TagBoxOptions -> { Editable -> False, Selectable -> False, StripWrapperBoxes -> False}], Cell[ StyleData["PrimaryPlaceholder", "Presentation"]], Cell[ StyleData["PrimaryPlaceholder", "Condensed"]], Cell[ StyleData["PrimaryPlaceholder", "Printout"]]}, Closed]]}, Closed]], Cell[ CellGroupData[{ Cell["FormatType Styles", "Section"], Cell[ "The cells below define styles that are mixed in with the styles of \ most cells. If a cell's FormatType matches the name of one of the styles \ defined below, then that style is applied between the cell's style and its \ own options. This is particularly true of Input and Output.", "Text"], Cell[ StyleData["CellExpression"], PageWidth -> Infinity, CellMargins -> {{6, Inherited}, {Inherited, Inherited}}, ShowCellLabel -> False, ShowSpecialCharacters -> False, AllowInlineCells -> False, Hyphenation -> False, AutoItalicWords -> {}, StyleMenuListing -> None, FontFamily -> "Courier", FontSize -> 12, Background -> GrayLevel[1]], Cell[ StyleData["InputForm"], InputAutoReplacements -> {}, AllowInlineCells -> False, Hyphenation -> False, StyleMenuListing -> None, FontFamily -> "Courier"], Cell[ StyleData["OutputForm"], PageWidth -> Infinity, TextAlignment -> Left, LineSpacing -> {0.6, 1}, StyleMenuListing -> None, FontFamily -> "Courier"], Cell[ StyleData["StandardForm"], InputAutoReplacements -> { "->" -> "\[Rule]", ":>" -> "\[RuleDelayed]", "<=" -> "\[LessEqual]", ">=" -> "\[GreaterEqual]", "!=" -> "\[NotEqual]", "==" -> "\[Equal]", Inherited}, LineSpacing -> {1.25, 0}, StyleMenuListing -> None, FontFamily -> "Courier"], Cell[ StyleData["TraditionalForm"], InputAutoReplacements -> { "->" -> "\[Rule]", ":>" -> "\[RuleDelayed]", "<=" -> "\[LessEqual]", ">=" -> "\[GreaterEqual]", "!=" -> "\[NotEqual]", "==" -> "\[Equal]", Inherited}, LineSpacing -> {1.25, 0}, SingleLetterItalics -> True, TraditionalFunctionNotation -> True, DelimiterMatching -> None, StyleMenuListing -> None], Cell[ "The style defined below is mixed in to any cell that is in an \ inline cell within another.", "Text"], Cell[ StyleData["InlineCell"], TextAlignment -> Left, ScriptLevel -> 1, StyleMenuListing -> None], Cell[ StyleData["InlineCellEditing"], StyleMenuListing -> None, Background -> RGBColor[1, 0.749996, 0.8]]}, Closed]], Cell[ CellGroupData[{ Cell["Automatic Styles", "Section"], Cell[ "The cells below define styles that are used to affect the display \ of certain types of objects in typeset expressions. For example, \ \"UnmatchedBracket\" style defines how unmatched bracket, curly bracket, and \ parenthesis characters are displayed (typically by coloring them to make them \ stand out).", "Text"], Cell[ StyleData["UnmatchedBracket"], StyleMenuListing -> None, FontColor -> RGBColor[0.760006, 0.330007, 0.8]]}, Closed]]}, Open]]}, Visible -> False, FrontEndVersion -> "6.0 for Linux x86 (32-bit) (April 20, 2007)", StyleDefinitions -> "Default.nb"] ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[568, 21, 339, 10, 113, "Text"], Cell[CellGroupData[{ Cell[932, 35, 77, 1, 52, "Section"], Cell[1012, 38, 213, 6, 29, "Text"], Cell[1228, 46, 8383, 262, 381, "Text"], Cell[9614, 310, 271, 5, 47, "Text"], Cell[9888, 317, 1085, 37, 66, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[11010, 359, 71, 1, 32, "Section"], Cell[11084, 362, 522, 15, 65, "Text"], Cell[CellGroupData[{ Cell[11631, 381, 31, 0, 46, "Subsection"], Cell[11665, 383, 664, 21, 50, "Text"], Cell[12332, 406, 727, 21, 49, "Input"], Cell[13062, 429, 259, 7, 29, "Text"], Cell[13324, 438, 413, 11, 49, "Input"], Cell[13740, 451, 864, 32, 65, "Text"], Cell[14607, 485, 2029, 71, 46, "Text"], Cell[16639, 558, 186, 4, 29, "Text"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[16874, 568, 71, 1, 32, "Section"], Cell[16948, 571, 729, 13, 137, "Text"], Cell[CellGroupData[{ Cell[17702, 588, 31, 0, 46, "Subsection"], Cell[17736, 590, 904, 39, 32, "Text"], Cell[18643, 631, 131, 1, 29, "Text"], Cell[18777, 634, 1357, 41, 69, "Input"], Cell[20137, 677, 304, 5, 47, "Text"], Cell[20444, 684, 656, 22, 49, "Input"], Cell[21103, 708, 373, 13, 46, "Text"], Cell[21479, 723, 459, 9, 65, "Text"], Cell[21941, 734, 825, 26, 49, "Input"], Cell[22769, 762, 362, 8, 47, "Text"], Cell[23134, 772, 1598, 54, 54, "Text"], Cell[24735, 828, 751, 24, 69, "Input"], Cell[25489, 854, 258, 9, 46, "Text"], Cell[25750, 865, 114, 3, 29, "Text"], Cell[25867, 870, 852, 26, 89, "Input"], Cell[26722, 898, 98, 2, 29, "Text"], Cell[26823, 902, 376, 12, 49, "Input"], Cell[27202, 916, 840, 32, 34, "Text"], Cell[28045, 950, 670, 20, 109, "Input"], Cell[28718, 972, 216, 4, 47, "Text"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[28983, 982, 47, 1, 32, "Section"], Cell[29033, 985, 4165, 154, 302, "Text"], Cell[CellGroupData[{ Cell[33223, 1143, 29, 0, 46, "Subsection"], Cell[33255, 1145, 871, 14, 155, "Text"] }, Open ]] }, Closed]] } ] *) (* End of internal cache information *)