(* 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[ 86509, 2384] NotebookOptionsPosition[ 41826, 1421] NotebookOutlinePosition[ 84624, 2328] CellTagsIndexPosition[ 84581, 2325] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[TextData[{ StyleBox["Lab 7B - Divergence 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["\<\ In last week's lab, we recalled the six different 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fields \ to surface integrals of vector fields via the curl. This week, we will work \ with the Divergence Theorem, which links surface integrals of vector fields \ to triple integrals via the divergence.\ \>", "Text"], Cell[TextData[{ StyleBox["Divergence Theorem", FontWeight->"Bold"], ": Suppose S is a solid with smooth boundary \[PartialD]S, and let ", StyleBox["F", FontWeight->"Bold"], " be a vector field which is smooth on and \"near\" S. Then ", Cell[BoxData[ RowBox[{ RowBox[{ SubscriptBox["\[DoubleContourIntegral]", RowBox[{"\[PartialD]", "S"}]], RowBox[{ RowBox[{ StyleBox["F", FontWeight->"Bold"], "\[CenterDot]", StyleBox["n", FontWeight->"Bold"]}], " ", RowBox[{"\[DifferentialD]", "\[Sigma]"}]}]}], "=", RowBox[{"\[Integral]", RowBox[{"\[Integral]", RowBox[{ SubscriptBox["\[Integral]", "S"], RowBox[{"div", " ", StyleBox["F", FontWeight->"Bold"], RowBox[{"\[DifferentialD]", "V"}]}]}]}]}]}]], "Text"], "." }], "Text", CellFrame->True, FontWeight->"Plain", Background->GrayLevel[0.849989]] }, Closed]], Cell[CellGroupData[{ Cell["Using the Divergence Theorem: Example 1", "Section", Background->None], Cell["\<\ Triple integrals of scalar functions are almost always easier to compute than \ surface integrals of vector fields. Therefore, we usually apply the \ Divergence Theorem to replace a surface integral with a triple integral. Of \ course, this only works if our surface is the boundary of some solid!\ \>", "Text"], Cell[CellGroupData[{ Cell["Example 1", "Subsection"], Cell[TextData[{ "Let ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ StyleBox["F", FontWeight->"Bold", FontSlant->"Plain"], "(", RowBox[{"x", ",", "y", ",", "z"}], ")"}], "=", RowBox[{"(", RowBox[{ RowBox[{"sinx", " ", SuperscriptBox["cos", "2"], "y"}], ",", " ", RowBox[{ SuperscriptBox["sin", "3"], "y", " ", SuperscriptBox["cos", "4"], "z"}], ",", " ", RowBox[{ SuperscriptBox["sin", "5"], "z", " ", SuperscriptBox["cos", "6"], "x"}]}], ")"}]}], TraditionalForm]]], ", and let M be the surface of the cube in the first octant bounded by the \ coordinate planes and the planes ", Cell[BoxData[ RowBox[{ StyleBox["x", FontSlant->"Italic"], "=", FractionBox["\[Pi]", "2"]}]], "Text"], ", ", Cell[BoxData[ RowBox[{ StyleBox["y", FontSlant->"Italic"], "=", FractionBox["\[Pi]", "2"]}]]], ", and ", Cell[BoxData[ RowBox[{ StyleBox["z", FontSlant->"Italic"], "=", FractionBox["\[Pi]", "2"]}]]], ". Using the Divergence Theorem, calculate the outward flux of ", StyleBox["F", FontWeight->"Bold"], " across M." }], "Text"], Cell[TextData[{ "We need to calculate div ", StyleBox["F", FontWeight->"Bold"], "; we can do this by hand, or use ", StyleBox["Mathematica", FontSlant->"Italic"], ". Let's use ", StyleBox["Mathematica", FontSlant->"Italic"], ":" }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{"F", "[", RowBox[{"x_", ",", "y_", ",", "z_"}], "]"}], "=", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"Sin", "[", "x", "]"}], ")"}], "*", RowBox[{"(", RowBox[{ RowBox[{"Cos", "[", "y", "]"}], "^", "2"}], ")"}]}], ",", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"Sin", "[", "y", "]"}], "^", "3"}], ")"}], "*", RowBox[{"(", RowBox[{ RowBox[{"Cos", "[", "z", "]"}], "^", "4"}], ")"}]}], ",", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"Sin", "[", "z", "]"}], "^", "5"}], ")"}], "*", RowBox[{"(", RowBox[{ RowBox[{"Cos", "[", "x", "]"}], "^", "6"}], ")"}]}]}], "}"}]}]], "Input"], Cell[TextData[{ "Remember: ", StyleBox["Mathematica", FontSlant->"Italic"], " will not understand our shorthand notation ", StyleBox["Sin^5[x]", FontFamily->"Courier", FontWeight->"Bold"], StyleBox["!", FontFamily->"Courier"] }], "Text", CellFrame->True, Background->GrayLevel[0.849989]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " can compute the divergence of a vector field using the command ", StyleBox["Div", FontWeight->"Bold"], ":" }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{"d", "[", RowBox[{"x_", ",", "y_", ",", "z_"}], "]"}], "=", RowBox[{"Div", "[", RowBox[{"F", "[", RowBox[{"x", ",", "y", ",", "z"}], "]"}], "]"}]}]], "Input"], Cell["\<\ Now, by the Divergence Theorem, we can just take the triple integral of our \ result.\ \>", "Text"], Cell[BoxData[ RowBox[{"Integrate", "[", RowBox[{ RowBox[{"d", "[", RowBox[{"x", ",", "y", ",", "z"}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", RowBox[{"Pi", "/", "2"}]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", "0", ",", RowBox[{"Pi", "/", "2"}]}], "}"}], ",", RowBox[{"{", RowBox[{"z", ",", "0", ",", RowBox[{"Pi", "/", "2"}]}], "}"}]}], "]"}]], "Input"], Cell["\<\ If we didn't know the Divergence Theorem, we'd have to parametrize all six \ faces individually, check the normal vectors to be sure they pointed outward, \ evaluate the six integrals, and sum the values. If you'd like, you can do \ this to check our result.\ \>", "Text"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Using the Divergence Theorem: Example 2", "Section", Background->None], Cell["\<\ Sometimes, we can use the Divergence Theorem to simplify a surface integral, \ even when the surface isn't the boundary of any solid. The idea is to invent \ another, simpler surface which completes the boundary of a solid. Then the \ Divergence Theorem will give the surface integral over the two surfaces \ together; we can calculate the simpler integral and subtract it from the \ total to get our solution. This is tricky, but it's an important idea. Re-read that last paragraph \ until it makes sense, or ask your TA to help you figure it out.\ \>", "Text"], Cell[CellGroupData[{ Cell["Example 2", "Subsection"], Cell[TextData[{ "Let ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ StyleBox["F", FontWeight->"Bold", FontSlant->"Plain"], "(", RowBox[{"x", ",", "y", ",", "z"}], ")"}], "=", RowBox[{"(", RowBox[{ RowBox[{ SuperscriptBox["y", "2"], "-", RowBox[{"2", "y", " ", "z"}]}], ",", " ", RowBox[{ SuperscriptBox["z", "2"], "-", RowBox[{"2", "x", " ", "z"}]}], ",", " ", RowBox[{ SuperscriptBox["x", "2"], "-", RowBox[{"2", "x", " ", "y"}]}]}], ")"}]}], TraditionalForm]]], " and let M be the surface parametrized by ", StyleBox["f", FontWeight->"Bold"], "(r,\[Theta])=(r cos(\[Theta]), r sin(\[Theta]), cos(5\[Pi](1-r))-1, 0\ \[LessEqual]r\[LessEqual]1, 0\[LessEqual]\[Theta]\[LessEqual]2\[Pi]. Find \ the flux of ", StyleBox["F", FontWeight->"Bold"], " through M, using the upward normal vector." }], "Text"], Cell["This surface is terrible! Here's a graph:", "Text"], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"f", "[", RowBox[{"r_", ",", "theta_"}], "]"}], "=", RowBox[{"{", RowBox[{ RowBox[{"r", "*", RowBox[{"Cos", "[", "theta", "]"}]}], ",", RowBox[{"r", "*", RowBox[{"Sin", "[", "theta", "]"}]}], ",", RowBox[{ RowBox[{"Cos", "[", RowBox[{"Sin", "[", RowBox[{"5", "Pi", "*", RowBox[{"(", RowBox[{"1", "-", "r"}], ")"}]}], "]"}], "]"}], "-", "1"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"M", "=", RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"f", "[", RowBox[{"r", ",", "theta"}], "]"}], ",", RowBox[{"{", RowBox[{"r", ",", "0", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"theta", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}]}], "]"}]}]}], "Input", CellChangeTimes->{{3.41235673531692*^9, 3.412356739767358*^9}}], Cell[TextData[{ "Of course, M is not the boundary of a solid. However, the boundary of M is \ the unit circle in the ", StyleBox["xy-", FontSlant->"Italic"], "plane (check this!). Therefore, M and the unit disk (call it \ \[CapitalDelta]) ", StyleBox["do", FontSlant->"Italic"], " form the boundary of a solid (call it S): try looking at the following \ graph from a few different points of view." }], "Text"], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"g", "[", RowBox[{"r_", ",", "theta_"}], "]"}], "=", RowBox[{"{", RowBox[{ RowBox[{"r", "*", RowBox[{"Cos", "[", "theta", "]"}]}], ",", RowBox[{"r", "*", RowBox[{"Sin", "[", "theta", "]"}]}], ",", "0"}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"\[CapitalDelta]", "=", RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"g", "[", RowBox[{"r", ",", "theta"}], "]"}], ",", RowBox[{"{", RowBox[{"r", ",", "0", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"theta", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}], ",", RowBox[{"Mesh", "->", "10"}]}], "]"}]}], "\[IndentingNewLine]", RowBox[{"S", "=", RowBox[{"Show", "[", RowBox[{"M", ",", "\[CapitalDelta]"}], "]"}]}]}], "Input", CellChangeTimes->{{3.412356767110247*^9, 3.412356796059209*^9}}], Cell["\<\ Roughly speaking, in English here's what we're going to do in order to \ calculate the surface integral of our original surface:\ \>", "Text"], Cell[TextData[{ StyleBox["1. (Surface Integral over our original surface and the disk) = \ (triple integral of div ", FontFamily->"Helvetica"], StyleBox["F", FontFamily->"Helvetica", FontWeight->"Bold"], StyleBox[" over the solid region enclosed by the two surfaces)", FontFamily->"Helvetica"], "\t", StyleBox["[By the Div Them]", FontSlant->"Italic"], "\n\n2. ", StyleBox["(Surface Integral over original surface) + (Surface integral over \ disk) = (triple integral over the solid region)", FontFamily->"Helvetica"], "\t", StyleBox["[Computing the two surface integrals separately]", FontSlant->"Italic"], "\n\n3. ", StyleBox["(Surface Integral over original surface) = (triple integral over \ solid region) - (Surface Integral over disk)", FontFamily->"Helvetica"], "\t\t", StyleBox["[Solving for our original integral]", FontSlant->"Italic"] }], "Text", Background->RGBColor[1, 1, 0.733333]], Cell[TextData[{ "Remember, in the Divergence Theorem we use the ", StyleBox["outward", FontSlant->"Italic"], " pointing normal vector. In the problem, we're asked to find the integral \ over M with the upward pointing normal vector, but when we combine it with \ the unit disk, the outward pointing normal is the ", StyleBox["downward", FontSlant->"Italic"], " pointing normal vector. That means we have to replace M with -M in our \ calculations. (As you should recall from lecture, the only real difference \ is that it will multiply our answer by -1.)\n\nLet's rewrite the three \ equations above mathematically." }], "Text"], Cell[TextData[{ StyleBox["1. ", FontFamily->"Helvetica"], Cell[BoxData[ StyleBox[ RowBox[{ RowBox[{ SubscriptBox["\[DoubleContourIntegral]", RowBox[{"\[CapitalDelta]", "+", RowBox[{"(", RowBox[{"-", "M"}], ")"}]}]], RowBox[{ RowBox[{ StyleBox["F", FontWeight->"Bold"], "\[CenterDot]", StyleBox["n", FontWeight->"Bold"]}], RowBox[{"\[DifferentialD]", "\[Sigma]"}]}]}], "=", RowBox[{"\[Integral]", RowBox[{"\[Integral]", RowBox[{ SubscriptBox["\[Integral]", "S"], RowBox[{"div", " ", StyleBox["F", FontWeight->"Bold"], " ", RowBox[{"\[DifferentialD]", "V"}]}]}]}]}]}], FontSize->14]], "Text"], "\t\t\t", StyleBox["[By the Div Them; -M because we've switched the normal]", FontSlant->"Italic"], "\n\n2. ", StyleBox[" ", FontFamily->"Helvetica"], Cell[BoxData[ StyleBox[ RowBox[{ RowBox[{ RowBox[{"\[Integral]", RowBox[{ SubscriptBox["\[Integral]", "\[CapitalDelta]"], RowBox[{ RowBox[{ StyleBox["F", FontWeight->"Bold"], "\[CenterDot]", StyleBox["n", FontWeight->"Bold"]}], RowBox[{"\[DifferentialD]", "\[Sigma]"}]}]}]}], "+", RowBox[{"\[Integral]", RowBox[{ SubscriptBox["\[Integral]", RowBox[{"-", "M"}]], RowBox[{ RowBox[{ StyleBox["F", FontWeight->"Bold"], "\[CenterDot]", StyleBox["n", FontWeight->"Bold"]}], RowBox[{"\[DifferentialD]", "\[Sigma]"}]}]}]}]}], "=", RowBox[{"\[Integral]", RowBox[{"\[Integral]", RowBox[{ SubscriptBox["\[Integral]", "S"], RowBox[{"div", " ", StyleBox["F", FontWeight->"Bold"], " ", RowBox[{"\[DifferentialD]", "V"}]}]}]}]}]}], FontSize->14]], "Text"], "\t", StyleBox["[Computing the two surface integrals separately]", FontSlant->"Italic"], "\n\n3. ", Cell[BoxData[ RowBox[{ StyleBox[ RowBox[{"\[Integral]", RowBox[{ SubscriptBox["\[Integral]", RowBox[{"-", "M"}]], RowBox[{ RowBox[{ StyleBox["F", FontWeight->"Bold"], "\[CenterDot]", StyleBox["n", FontWeight->"Bold"]}], RowBox[{"\[DifferentialD]", "\[Sigma]"}]}]}]}], FontSize->14], StyleBox["=", FontSize->14], RowBox[{ StyleBox[ RowBox[{"\[Integral]", RowBox[{"\[Integral]", RowBox[{ SubscriptBox["\[Integral]", "S"], RowBox[{"div", " ", StyleBox["F", FontWeight->"Bold"], " ", RowBox[{"\[DifferentialD]", "V"}]}]}]}]}], FontSize->14], StyleBox["-", FontSize->14], RowBox[{"\[Integral]", RowBox[{ SubscriptBox["\[Integral]", "\[CapitalDelta]"], RowBox[{ RowBox[{ StyleBox["F", FontWeight->"Bold"], "\[CenterDot]", StyleBox["n", FontWeight->"Bold"]}], RowBox[{"\[DifferentialD]", "\[Sigma]"}]}]}]}]}]}]], "Text"], "\t\t", StyleBox["[Solving for our original integral; here I've multiplied by -1 to \ get the correct sign]", FontSlant->"Italic"] }], "Text", Background->RGBColor[1, 1, 0.733333]], Cell["\<\ We're interested in the integral over M, not -M, so let's fix that:\ \>", "Text"], Cell[TextData[{ "4(a). -", Cell[BoxData[ RowBox[{ StyleBox[ RowBox[{"\[Integral]", RowBox[{ SubscriptBox["\[Integral]", "M"], RowBox[{ RowBox[{ StyleBox["F", FontWeight->"Bold"], "\[CenterDot]", StyleBox["n", FontWeight->"Bold"]}], RowBox[{"\[DifferentialD]", "\[Sigma]"}]}]}]}], FontSize->14], StyleBox["=", FontSize->14], RowBox[{ StyleBox[ RowBox[{"\[Integral]", RowBox[{"\[Integral]", RowBox[{ SubscriptBox["\[Integral]", "S"], RowBox[{"div", " ", StyleBox["F", FontWeight->"Bold"], " ", RowBox[{"\[DifferentialD]", "V"}]}]}]}]}], FontSize->14], StyleBox["-", FontSize->14], RowBox[{"\[Integral]", RowBox[{ SubscriptBox["\[Integral]", "\[CapitalDelta]"], RowBox[{ RowBox[{ StyleBox["F", FontWeight->"Bold"], "\[CenterDot]", StyleBox["n", FontWeight->"Bold"]}], RowBox[{"\[DifferentialD]", "\[Sigma]"}]}]}]}]}]}]], "Text"], "\t\t", StyleBox["[integral over -M = - (integral over M)]\n\n", FontSlant->"Italic"], "4(b). ", Cell[BoxData[ RowBox[{ StyleBox[ RowBox[{"\[Integral]", RowBox[{ SubscriptBox["\[Integral]", "M"], RowBox[{ RowBox[{ StyleBox["F", FontWeight->"Bold"], "\[CenterDot]", StyleBox["n", FontWeight->"Bold"]}], RowBox[{"\[DifferentialD]", "\[Sigma]"}]}]}]}], FontSize->14], StyleBox["=", FontSize->14], RowBox[{ RowBox[{"\[Integral]", RowBox[{ SubscriptBox["\[Integral]", "\[CapitalDelta]"], RowBox[{ RowBox[{ StyleBox["F", FontWeight->"Bold"], "\[CenterDot]", StyleBox["n", FontWeight->"Bold"]}], RowBox[{"\[DifferentialD]", "\[Sigma]"}]}]}]}], StyleBox["-", FontSize->14], StyleBox[ RowBox[{"\[Integral]", RowBox[{"\[Integral]", RowBox[{ SubscriptBox["\[Integral]", "S"], RowBox[{"div", " ", StyleBox["F", FontWeight->"Bold"], " ", RowBox[{"\[DifferentialD]", "V"}]}]}]}]}], FontSize->14]}]}]], "Text"], "\t\t\t", StyleBox["[Multiplying by -1]", FontSlant->"Italic"] }], "Text", Background->RGBColor[1, 1, 0.733333]], Cell[TextData[{ "Now there's another step which makes this particular problem much easier. \ It turns out that ", StyleBox["F", FontWeight->"Bold"], " is a ", StyleBox["very", FontSlant->"Italic"], " nice vector field: evaluate the cell below." }], "Text"], Cell[BoxData[{ RowBox[{ RowBox[{"F", "[", RowBox[{"x_", ",", "y_", ",", "z_"}], "]"}], "=", RowBox[{"{", RowBox[{ RowBox[{ SuperscriptBox["y", "2"], "-", RowBox[{"2", "y", " ", "z"}]}], ",", " ", RowBox[{ SuperscriptBox["z", "2"], "-", RowBox[{"2", "x", " ", "z"}]}], ",", RowBox[{ SuperscriptBox["x", "2"], "-", RowBox[{"2", "x", " ", "y"}]}]}], "}"}]}], "\[IndentingNewLine]", RowBox[{"Div", "[", RowBox[{"F", "[", RowBox[{"x", ",", "y", ",", "z"}], "]"}], "]"}]}], "Input"], Cell[TextData[{ "Hence the triple intregral of div ", StyleBox["F", FontWeight->"Bold"], " is 0, and w", "e can conclude that ", Cell[BoxData[ RowBox[{ RowBox[{"\[Integral]", RowBox[{ SubscriptBox["\[Integral]", "\[CapitalDelta]"], RowBox[{ RowBox[{ StyleBox["F", FontWeight->"Bold"], "\[CenterDot]", StyleBox["n", FontWeight->"Bold"]}], RowBox[{"\[DifferentialD]", "\[Sigma]"}]}]}]}], "-", RowBox[{"\[Integral]", RowBox[{ SubscriptBox["\[Integral]", "M"], RowBox[{ RowBox[{ StyleBox["F", FontWeight->"Bold"], "\[CenterDot]", StyleBox["n", FontWeight->"Bold"]}], RowBox[{"\[DifferentialD]", "\[Sigma]"}]}]}]}]}]]], " = 0, so the integrals are equal! This means that we can work with the \ unit disk in place of our terrible surface M.\n", "\n", "Let's finish off the calculation on \[CapitalDelta]: ", Cell[BoxData[ RowBox[{"\[Integral]", RowBox[{ SubscriptBox["\[Integral]", "\[CapitalDelta]"], RowBox[{ RowBox[{ StyleBox["F", FontWeight->"Bold"], "\[CenterDot]", StyleBox["n", FontWeight->"Bold"]}], RowBox[{"\[DifferentialD]", "\[Sigma]"}]}]}]}]]], "=", Cell[BoxData[ RowBox[{"\[Integral]", RowBox[{ SubsuperscriptBox["\[Integral]", "R", " "], RowBox[{ StyleBox["F", FontWeight->"Bold"], RowBox[{ RowBox[{"(", RowBox[{ StyleBox["g", FontWeight->"Bold"], RowBox[{"(", RowBox[{"r", ",", "\[Theta]"}], ")"}]}], ")"}], "\[CenterDot]", RowBox[{"(", RowBox[{ FractionBox[ RowBox[{"\[PartialD]", StyleBox["g", FontWeight->"Bold"]}], RowBox[{"\[PartialD]", "r"}]], "\[Times]", FractionBox[ RowBox[{"\[PartialD]", StyleBox["g", FontWeight->"Bold"]}], RowBox[{"\[PartialD]", "\[Theta]"}]]}], ")"}]}], RowBox[{"\[DifferentialD]", "A"}]}]}]}]], "Text"], ", so first, we substitute ", StyleBox["g", FontWeight->"Bold"], " into ", StyleBox["F", FontWeight->"Bold"], ":" }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{"v1", "[", RowBox[{"r_", ",", "theta_"}], "]"}], "=", RowBox[{"Apply", "[", RowBox[{"F", ",", RowBox[{"g", "[", RowBox[{"r", ",", "theta"}], "]"}]}], "]"}]}]], "Input"], Cell[TextData[{ "Next, we cross the partial derivatives of ", StyleBox["g", FontWeight->"Bold"], "." }], "Text"], Cell[BoxData[{ RowBox[{ RowBox[{"dgdr", "[", RowBox[{"r_", ",", "theta_"}], "]"}], "=", RowBox[{"D", "[", RowBox[{ RowBox[{"g", "[", RowBox[{"r", ",", "theta"}], "]"}], ",", "r"}], "]"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"dgdtheta", "[", RowBox[{"r_", ",", "theta_"}], "]"}], "=", RowBox[{"D", "[", RowBox[{ RowBox[{"g", "[", RowBox[{"r", ",", "theta"}], "]"}], ",", "theta"}], "]"}]}], "\[IndentingNewLine]", RowBox[{"Cross", "[", RowBox[{ RowBox[{"dgdr", "[", RowBox[{"r", ",", "theta"}], "]"}], ",", RowBox[{"dgdtheta", "[", RowBox[{"r", ",", "theta"}], "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"v2", "[", RowBox[{"r_", ",", "theta_"}], "]"}], "=", RowBox[{"Simplify", "[", "%", "]"}]}]}], "Input"], Cell["Lastly, we dot our results together and integrate.", "Text"], Cell[BoxData[{ RowBox[{ RowBox[{"v1", "[", RowBox[{"r", ",", "theta"}], "]"}], ".", RowBox[{"v2", "[", RowBox[{"r", ",", "theta"}], "]"}]}], "\[IndentingNewLine]", RowBox[{"Integrate", "[", RowBox[{"%", ",", RowBox[{"{", RowBox[{"r", ",", "0", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"theta", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}]}], "]"}]}], "Input"], Cell[TextData[{ "So the flux of ", StyleBox["F", FontWeight->"Bold"], " through \[CapitalDelta] is ", Cell[BoxData[ FormBox[ FractionBox["\[Pi]", "4"], TraditionalForm]]], ". Notice that this is the flux ", StyleBox["up", FontSlant->"Italic"], " through \[CapitalDelta]: the normal vector above is (0,0,", StyleBox["r", FontSlant->"Italic"], "), and ", StyleBox["r", FontSlant->"Italic"], " is positive. By the Divergence Theorem, the flux of ", StyleBox["F", FontWeight->"Bold"], " ", StyleBox["down", FontSlant->"Italic"], " through M is ", Cell[BoxData[ FormBox[ FractionBox[ RowBox[{"-", "\[Pi]"}], "4"], TraditionalForm]]], "; the flux of ", StyleBox["F", FontWeight->"Bold"], " ", StyleBox["up", FontSlant->"Italic"], " through M is ", Cell[BoxData[ FormBox[ FractionBox["\[Pi]", "4"], TraditionalForm]]], ". Remember, when we apply the Divergence Theorem, our combined surface \ integral has to use the outward normal vector; this is the downward normal to \ M and the upward normal to \[CapitalDelta]." }], "Text", FontSize->14] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Exercises", "Section", Background->None], Cell[TextData[{ StyleBox["Exercise 1", FontWeight->"Bold"], "\n\nLet ", StyleBox["r", FontWeight->"Bold"], "(x,y,z)=(x,y,z), and ", StyleBox["F", FontWeight->"Bold"], "(x,y,z)=\[DoubleVerticalBar] ", StyleBox["r", FontWeight->"Bold"], "\[DoubleVerticalBar] ", StyleBox["r", FontWeight->"Bold"], ", where the double bars stand for magnitude, and let M be the surface of \ the solid bounded by the paraboloid ", Cell[BoxData[ FormBox[ RowBox[{"z", "=", RowBox[{"25", "-", SuperscriptBox["x", "2"], "-", SuperscriptBox["y", "2"]}]}], TraditionalForm]]], " and the ", StyleBox["xy", FontSlant->"Italic"], "-plane. Use the Divergence Theorem to compute the outward flux of ", StyleBox["F", FontWeight->"Bold"], " through M. [Hint: ", StyleBox["Mathematica", FontSlant->"Italic"], " can't compute the integral exactly; you should use ", StyleBox["NIntegrate", FontFamily->"Courier", FontWeight->"Bold"], ".]\n\n", StyleBox["Exercise 2", FontWeight->"Bold"], "\n\nLet ", StyleBox["r", FontWeight->"Bold"], "(x,y,z)=(x,y,z), and ", StyleBox["F", FontWeight->"Bold"], "(x,y,z) = ", StyleBox["r", FontWeight->"Bold"], " / ", "\[DoubleVerticalBar] ", StyleBox["r", FontWeight->"Bold"], "\[DoubleVerticalBar] , where the double bars stand for magnitude, and let M \ be the surface of the solid bounded by the paraboloid ", Cell[BoxData[ FormBox[ RowBox[{"z", "=", RowBox[{"26", "-", SuperscriptBox["x", "2"], "-", SuperscriptBox["y", "2"]}]}], TraditionalForm]]], " and the plane ", Cell[BoxData[ FormBox[ RowBox[{"z", "=", "1"}], TraditionalForm]]], ". Use the Divergence Theorem to compute the outward flux of ", StyleBox["F", FontWeight->"Bold"], " through M. [Hint: ", StyleBox["Mathematica", FontSlant->"Italic"], " can't compute the integral exactly; you should use ", StyleBox["NIntegrate", FontFamily->"Courier", FontWeight->"Bold"], ".]\n\n", StyleBox["Exercise 3", FontWeight->"Bold"], "\n\nUse the Divergence Theorem to calculate the outward flux of ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ RowBox[{"F", "(", RowBox[{"x", ",", "y", ",", "z"}], ")"}], "=", FormBox[ RowBox[{"(", RowBox[{ RowBox[{ SuperscriptBox["z", "2"], "x"}], ",", " ", RowBox[{ RowBox[{ FractionBox["1", "3"], SuperscriptBox["y", "3"]}], "+", RowBox[{"tan", "(", "z", ")"}]}], ",", " ", RowBox[{ RowBox[{ SuperscriptBox["x", "2"], "z"}], "+", SuperscriptBox["y", "2"]}]}], ")"}], TraditionalForm]}], FontSize->14], TraditionalForm]]], " through the top half of the sphere ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"], "+", SuperscriptBox["z", "2"]}], "=", "1"}], TraditionalForm]]], ". [Hint: the surface is not closed; you need a closed surface to apply the \ Divergence Theorem. Make a closed surface by adding a flat bottom to the \ hemisphere -- and think carefully!]\n\n", StyleBox["Exercise 4", FontWeight->"Bold"], "\n\nUse the Divergence Theorem to calculate the outward flux of ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ RowBox[{"F", "(", RowBox[{"x", ",", "y", ",", "z"}], ")"}], "=", FormBox[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["y", "2"], "x"}], "+", SuperscriptBox["e", "yz"]}], ",", " ", RowBox[{ RowBox[{ SuperscriptBox["x", "2"], "y"}], "+", RowBox[{"tan", "(", "z", ")"}]}], ",", " ", RowBox[{ RowBox[{ SuperscriptBox["z", "3"], "/", "3"}], "+", RowBox[{ SuperscriptBox["x", "2"], "/", "5"}], "+", RowBox[{"9", SuperscriptBox["y", "2"]}]}]}], ")"}], "/", "3"}], TraditionalForm]}], FontSize->14], TraditionalForm]]], " through the bottom half of the sphere ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"], "+", SuperscriptBox["z", "2"]}], "=", "4"}], TraditionalForm]]], ". [Hint: the surface is not closed; you need a closed surface to apply the \ Divergence Theorem. Make a closed surface by adding a flat top to the \ hemisphere -- and think carefully!]\n\n", StyleBox["Exercise 5", FontWeight->"Bold"], "\n\nIn this problem, we will prove Archimedes's Principle: the buoyant \ force on a submerged object equals the weight of the fluid it displaces. We \ choose a system of coordinates so that z represents the depth below the fluid \ surface; that is, the fluid surface is the xy-plane and the z-axis points \ down into the fluid. At every point where z\[GreaterEqual]0, the pressure is \ given by the scalar function ", StyleBox["p(x,y,z)=\[Delta]gz", FontSlant->"Italic"], ", where \[Delta] is the density of the fluid (assumed constant) and ", StyleBox["g", FontSlant->"Italic"], " is the acceleration due to gravity (also assumed constant).\n\n Set up an \ integral to calculate the weight of the fluid displaced by S, where S is any \ solid completely submerged in fluid (i.e. z\[GreaterEqual]0 at every point of \ S). This means that you are to write a general integral that works for every \ S, not just to choose some particular S. [Hint: how much fluid does S \ displace?]\n\nb) Loosely, buoyant force is the integral of pressure. More \ precisely, the buoyant force is given by the vector integral ", Cell[BoxData[ RowBox[{"B", "=", RowBox[{"-", RowBox[{ SubscriptBox["\[DoubleContourIntegral]", RowBox[{"\[PartialD]", "S"}]], RowBox[{ RowBox[{"(", StyleBox[ RowBox[{"p", StyleBox["n", FontWeight->"Bold"]}]], ")"}], RowBox[{"\[DifferentialD]", "\[Sigma]"}]}]}]}]}]], "Text"], ". Write a paragraph explaining why this is a reasonable formula. [Hint: \ think of the integral as a sum of many little pieces. Don't forget to \ explain the minus sign!]\n\nc) By direct calculations, we can see that ", Cell[BoxData[ RowBox[{ StyleBox[ RowBox[{"p", StyleBox["n", FontWeight->"Bold"]}]], "=", RowBox[{ RowBox[{"p", RowBox[{"(", RowBox[{ SubscriptBox["n", "1"], ",", SubscriptBox["n", "2"], ",", SubscriptBox["n", "3"]}], ")"}]}], "=", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ SubscriptBox["pn", "1"], ",", "0", ",", "0"}], ")"}], "+", RowBox[{"(", RowBox[{"0", ",", SubscriptBox["pn", "2"], ",", "0"}], ")"}], "+", RowBox[{"(", RowBox[{"0", ",", "0", ",", SubscriptBox["pn", "3"]}], ")"}]}], "=", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ StyleBox[ RowBox[{"p", StyleBox["i", FontWeight->"Bold"]}]], "\[CenterDot]", StyleBox["n", FontWeight->"Bold"]}], ")"}], StyleBox["i", FontWeight->"Bold"]}], "+", RowBox[{ RowBox[{"(", RowBox[{ StyleBox[ RowBox[{"p", StyleBox["j", FontWeight->"Bold"]}]], "\[CenterDot]", StyleBox["n", FontWeight->"Bold"]}], ")"}], StyleBox["j", FontWeight->"Bold"]}], "+", RowBox[{ RowBox[{"(", RowBox[{ StyleBox[ RowBox[{"p", StyleBox["k", FontWeight->"Bold"]}]], "\[CenterDot]", StyleBox["n", FontWeight->"Bold"]}], ")"}], StyleBox["k", FontWeight->"Bold"]}]}]}]}]}]], "Text"], ". Use this substitution, rearrange, and use the Divergence Theorem (three \ times) to prove Archimedes's Principle." }], "Text", CellFrame->True, FontSize->14, Background->RGBColor[1, 0.490196, 0.490196]], Cell[CellGroupData[{ Cell["Credits", "Subsection"], Cell[TextData[{ "This lab was written by James Swenson (swenson@math.umn.edu) in 2002. In \ Spring 2004 Jonathan Rogness (rogness@math.umn.edu) went through and updated \ a few minor things to reflect the use of our math2374.nb file; he also added \ some text (such as the yellow boxes) to example 2 and added a few exercises. \ Minor updates in Spring 2008 for ", StyleBox["Mathematica", FontSlant->"Italic"], " 6.0.\n\nThis lab is copyright 2002, 2004 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\nAlthough 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->{{3.412358292944512*^9, 3.412358297990698*^9}}] }, Closed]] }, Closed]] }, ScreenStyleEnvironment->"Working", WindowSize->{960, 707}, WindowMargins->{{73, Automatic}, {36, Automatic}}, PrintingPageRange->{Automatic, Automatic}, PrintingOptions->{"Magnification"->1, "PaperOrientation"->"Portrait", "PaperSize"->{612, 792}, "PostScriptOutputFile":>FrontEnd`FileName[{"user002", "rogness"}, "Newlab.nb.ps", CharacterEncoding -> "iso8859-1"]}, 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. 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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, 341, 10, 113, "Text"], Cell[CellGroupData[{ Cell[934, 35, 77, 1, 52, "Section"], Cell[1014, 38, 161, 3, 29, "Text"], Cell[1178, 43, 8916, 288, 381, "Text"], Cell[10097, 333, 299, 5, 47, "Text"], Cell[10399, 340, 912, 31, 69, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[11348, 376, 77, 1, 32, "Section"], Cell[11428, 379, 323, 5, 47, "Text"], Cell[CellGroupData[{ Cell[11776, 388, 31, 0, 46, "Subsection"], Cell[11810, 390, 1166, 43, 57, "Text"], Cell[12979, 435, 254, 11, 29, "Text"], Cell[13236, 448, 750, 26, 28, "Input"], Cell[13989, 476, 302, 12, 46, "Text"], Cell[14294, 490, 190, 7, 29, "Text"], Cell[14487, 499, 208, 6, 28, "Input"], Cell[14698, 507, 109, 3, 29, "Text"], Cell[14810, 512, 427, 13, 28, "Input"], Cell[15240, 527, 283, 5, 47, "Text"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[15572, 538, 77, 1, 32, "Section"], Cell[15652, 541, 576, 10, 101, "Text"], Cell[CellGroupData[{ Cell[16253, 555, 31, 0, 46, "Subsection"], Cell[16287, 557, 917, 30, 47, "Text"], Cell[17207, 589, 58, 0, 29, "Text"], Cell[17268, 591, 913, 28, 49, "Input"], Cell[18184, 621, 420, 11, 47, "Text"], Cell[18607, 634, 894, 26, 84, "Input"], Cell[19504, 662, 152, 3, 29, "Text"], Cell[19659, 667, 927, 27, 174, "Text"], Cell[20589, 696, 640, 13, 101, "Text"], Cell[21232, 711, 3267, 116, 138, "Text"], Cell[24502, 829, 91, 2, 29, "Text"], Cell[24596, 833, 2357, 88, 95, "Text"], Cell[26956, 923, 267, 9, 29, "Text"], Cell[27226, 934, 546, 17, 64, "Input"], Cell[27775, 953, 2177, 78, 93, "Text"], Cell[29955, 1033, 224, 7, 43, "Input"], Cell[30182, 1042, 117, 5, 29, "Text"], Cell[30302, 1049, 815, 26, 104, "Input"], Cell[31120, 1077, 66, 0, 29, "Text"], Cell[31189, 1079, 405, 12, 64, "Input"], Cell[31597, 1093, 1098, 42, 78, "Text"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[32744, 1141, 47, 1, 32, "Section"], Cell[32794, 1144, 7981, 251, 918, "Text"], Cell[CellGroupData[{ Cell[40800, 1399, 29, 0, 46, "Subsection"], Cell[40832, 1401, 966, 16, 173, "Text"] }, Closed]] }, Closed]] } ] *) (* End of internal cache information *)