(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 17697, 434]*) (*NotebookOutlinePosition[ 53371, 1658]*) (* CellTagsIndexPosition[ 53297, 1652]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ StyleBox["Lab 5 - Applications of Integration", 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, gantner@math.umn.edu" }], "Text", TextAlignment->Center, FontColor->GrayLevel[1], Background->RGBColor[0, 0, 1]], Cell[CellGroupData[{ Cell["Introduction", "Section", Background->None], Cell["\<\ Our goal in this part of the lab is to find the volume of liquid in \ a cylindrical tank which is tilted at some angle from the horizontal. To be \ specific, we start with the cylinder with radius 2 m and length 10 m, whose \ axis is the vertical line through (2,0,0): we will put the bottom of the \ cylinder at z=0 and the top at z=10. A parametrization of this cylinder is \ given below: what should be the bounds on s and t?\ \>", "Text"], Cell[BoxData[ \(\(f[s_, t_] = {2 + 2 Cos[t], 2*Sin[t], s};\)\)], "Input"], Cell[TextData[{ "Evaluate the above cell to define f. Then, in the cell below, fill in the \ appropriate bounds for s and t, and evaluate the cell to plot the cylinder. \ [Notice that we name the plot ", StyleBox["cyl", FontWeight->"Bold"], ".] " }], "Text"], Cell[BoxData[ \(cyl = ParametricPlot3D[f[s, t], {s, , }, {t, , }, AxesLabel \[Rule] {X, Y, Z}]\)], "Input"], Cell["\<\ The surface of the liquid in our tank should be given by a \ horizontal plane: here is an example.\ \>", "Text"], Cell[BoxData[{ \(plane1 = ParametricPlot3D[{x, y, 4}, {x, 0, 4}, {y, \(-2\), 2}]\), "\[IndentingNewLine]", \(Show[cyl, plane1, AxesLabel \[Rule] {x, y, z}]\)}], "Input", CellTags->"zerodeg"], Cell[TextData[{ "In this case, the volume of the liquid is \[Pi]", Cell[BoxData[ \(TraditionalForm\`\(r\^2\) h\)]], " = 16\[Pi]. (See why?)" }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Tilting Cylinders Is Too Hard", "Section", Background->None], Cell["\<\ To find the volume of liquid when the cylinder is tilted, we will \ need to use integration. Evaluate the next cell to see an example of a \ tilted cylinder, with a horizontal plane representing the surface of a \ liquid.\ \>", "Text"], Cell[BoxData[{ \(\(u = Pi/4;\)\), "\[IndentingNewLine]", \(cyl2 = ParametricPlot3D[{Cos[u]*\((2*Cos[t] + 2)\) - Sin[u]*s, 2*Sin[t], Sin[u]*\((2*Cos[t] + 2)\) + Cos[u]*s}, {s, 0, 10}, {t, 0, 2 Pi}]\), "\[IndentingNewLine]", \(plane2 = ParametricPlot3D[{x, y, 4}, {x, \(-6\), 4}, {y, \(-2\), 2}]\), "\[IndentingNewLine]", \(Show[cyl2, plane2, \ AxesLabel \[Rule] {x, y, z}, ViewPoint -> {0, \ \(-3\), \ 0}, BoxRatios \[Rule] {1, 1, 1}]\)}], "Input"], Cell["\<\ This picture is not very useful for calculation: we don't know what \ to integrate, nor over what region! [It was also tricky to reparametrize the \ cylinder.]\ \>", "Text"], Cell[TextData[{ "We will take a different approach: again, we imagine the cylinder tilting, \ but to simplify the calculations, we tilt the plane instead, and reuse ", StyleBox["cyl", FontWeight->"Bold"], ", our original cylinder. To put this another way, we are tilting the \ coordinate axes along with the cylinder." }], "Text"], Cell[BoxData[{ \(plane3 = ParametricPlot3D[{x, y, 6 - x}, {x, \(-2\), 6}, {y, \(-2\), 2}]\), "\[IndentingNewLine]", \(Show[cyl, plane3, AxesLabel \[Rule] {x, y, z}, BoxRatios \[Rule] {1, 1, 1}, ViewPoint -> {0, \ \(-3\), \ 0}]\)}], "Input"], Cell["\<\ If you tilt your head,you can see that this graph shows the same \ part of the cylinder as in the tilted-cylinder graph above. If you're not \ sure about that, evaluate the next cell to see the cylinder in a \ LiveGraphics3D window. Tilt the picture until the plane is flat. (HINT: hold \ the shift button down, click on the picture, and move your mouse to the \ left.) It should look the same as the earlier picture.\ \>", "Text"], Cell[BoxData[ \(\(ShowLive[cyl, plane3, AxesLabel \[Rule] {x, y, z}, BoxRatios \[Rule] {1, 1, 1}, ViewPoint -> {0, \ \(-3\), \ 0}];\)\)], "Input"], Cell[TextData[{ "We like this new viewpoint because it makes calculation much easier: we \ can find the volume of liquid by integration. Specifically, we want the \ volume of the region below our plane, and above the bottom of the tank.\n\n\ Therefore, the region of integration should be the bottom of the tank, which \ is in the xy-plane, and given by the inequality ", Cell[BoxData[ \(TraditionalForm\`\((x - 2)\)\^2 + y\^2\)]], "\[LessEqual] 4, and the integrand should be the height of the plane." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["The Fluid Finds Its Level", "Section", Background->None], Cell["\<\ Notice that our plane doesn't move up or down in the y-direction. \ The next set of commands will draw the same picture as above, but without \ perspective, so that the y-direction vanishes:\ \>", "Text"], Cell[BoxData[{ \(graycyl = FilledPlot[ If[\((x \[GreaterEqual] 0)\) && \((x \[LessEqual] 4)\), 10, 0], {x, \(-2\), 6}]\), "\[IndentingNewLine]", \(line = ParametricPlot[{x, 6 - x}, {x, \(-2\), 6}]\), "\[IndentingNewLine]", \(Show[graycyl, line, AxesLabel \[Rule] {x, z}, AspectRatio \[Rule] 1]\)}], "Input"], Cell["\<\ This makes it easier to find the equation of the plane. If we \ don't show perspective in our plot, then our cylinder looks like a rectangle \ with base 4 and height 10, and our plane looks like a line in the xz-plane. \ We can find the equation of this line in terms of x and z, using nothing but \ algebra: the slope-intercept form gives z=-x+6. When we think in three \ dimensions, this same equation gives us the plane we want! Using this \ equation, we can calculate the desired volume (remember the explanation at \ the end of the last section).\ \>", "Text"], Cell[BoxData[ \(Integrate[\(-x\) + 6, {x, 0, 4}, {y, \(-Sqrt[4 - \((x - 2)\)^2]\), Sqrt[4 - \((x - 2)\)^2]}]\)], "Input"], Cell["\<\ In the following exercises, sketching \"side elevations\" by hand \ will help you to find the equations you'll need for the various \ surfaces.\ \>", "Text"], Cell[TextData[{ StyleBox["Order of Integration: ", FontWeight->"Bold"], StyleBox["In which order did ", FontVariations->{"CompatibilityType"->0}], StyleBox["Mathematica", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox[" perform the integration in the previous command? What happens \ when you switch the order of the variables as they are listed after the \ integrand?", FontVariations->{"CompatibilityType"->0}] }], "Text", CellFrame->True, Background->GrayLevel[0.849989]] }, Closed]], Cell[CellGroupData[{ Cell["Exercises", "Section", CellFrame->False, Background->None], Cell[TextData[{ StyleBox["Exercise 1", FontWeight->"Bold"], "\na) Suppose that the fluid level in the tank is 7 m on the left edge of \ the tank (where x=0) and 5 m on the right edge (where x=4). Find the \ equation of the plane of the liquid, and use a double integral to find the \ volume of liquid in the tank. [Hint: you should use a \"dy dx\" iterated \ integral, where the bounds on y depend on x, and are given by the equation of \ the base of the cylinder.] Also find the angle at which the tank is tipped \ from its upright position. For consistency, assume that the ", ButtonBox["tank", ButtonData:>"zerodeg", ButtonStyle->"Hyperlink"], " in the ", StyleBox["Introduction", FontWeight->"Bold"], " above is tipped at a 0\[Degree] angle. [Hint: use a \"side elevation\" \ sketch, and use trigonometry to find the angle between your plane and the \ plane z=5.]\n\nb) Suppose now that the fluid level is 7 m at the left edge of \ the tank, but that there isn't enough fluid to reach the right edge; instead, \ the fluid covers the base of the tank only out to the plane x=3. [You can \ produce a graph with the commands below: you'll have to paste them into \ another cell.]\n", Cell[BoxData[{ \(plane4 = ParametricPlot3D[{x, y, 7 - 7 x/3}, {x, 0, 3}, {y, \(-2\), 2}]\), "\n", \(Show[cyl, plane4, AxesLabel \[Rule] {x, y, z}, ViewPoint -> {0, \ \(-3\), \ 0}]\)}], "Input", CellFrame->False], "\nFind the equation of the plane of the liquid,and use a double integral \ to find the volume of liquid in the tank. Also find the angle at which the \ tank is tipped from its upright position. [", StyleBox["Warning", FontWeight->"Bold"], ": the region of integration has changed -- if you reuse the bounds from \ problem 1, you'll include the \"negative volume\" from the region where z<0.]\ \n\nc) Suppose that the tank is tipped so far that the liquid touches both \ ends of the tank, touching the top (at z=10) out to the plane x=1, and \ covering the bottom out to the plane x=3. Find the volume of liquid in the \ tank using two double integrals. Explain why you can't just use one double \ integral. Lastly, find the angle at which the tank is tipped from its \ upright position.\n\nd) Suppose, finally, that the liquid touches the top of \ the tank out to the plane x=3, and reaches a level of 3 m on the right edge \ of the tank. Find the volume of liquid in the tank using two double \ integrals, and find the angle at which the tank is tipped from its upright \ position.\n\ne) Suppose that our tank is tilted at an angle of 80\[Degree] \ from its upright position. Where could you mark the surface of the tank to \ indicate when the tank is exactly one-third full? [Hint: use trigonometry to \ decide which of the first four exercises most resembles this problem. \ Calculate the volume in terms of some unknown variable, and use ", StyleBox["Mathematica", FontSlant->"Italic"], "'s ", StyleBox["FindRoot", FontFamily->"Courier", FontWeight->"Bold"], " command, which you can look up in the help browser.]" }], "Text", CellFrame->True, Background->RGBColor[1, 0.501961, 0.501961]], Cell[BoxData[ \(\[IndentingNewLine]\)], "Input"], Cell["\<\ In exercise 2, rather than using the tank that was described in the \ introduction, we will use an oblong tank. When intersected with the \ xy-plane, the tank will be an ellipse which goes through the points (0,0,0), \ (4,2,0), (4,-2,0), and (8,0,0). This ellipse is centered at (4,0,0), and \ every point on the ellipse satisfies the equation (1/4)(x-4)^2 + y^2 = 4. We \ will assume that the tank has height 10. The new tank looks like\ \>", "Text"], Cell[BoxData[ \(tank\ = \ ContourPlot3D[\((1/4)\)*\((x - 4)\)^2\ + \ y^2\ - \ 4, \ {x, 0, 8}, \ {y, \(-2\), 2}, \ {z, 0, 10}, \ AxesLabel \[Rule] {x, y, z}, \ Axes \[Rule] True]\)], "Input"], Cell["\<\ We will continue to \"tip\" the tank towards the negative y \ direction. This is the same direction as in the examples above, but here it \ actually matters which direction we tip it, because the tank is no longer \ circular. \ \>", "Text"], Cell[TextData[{ StyleBox["Exercise 2\n", FontWeight->"Bold"], StyleBox["\na) Find a parametrization of the tank described in the \ preceding paragraph. Be sure to include your bounds on the parameters. \ [Hint: For the x and y coordinates, first divide the equation by 4 on both \ sides. If you make the substitution u = (1/4)*(x-4) and v = (1/2)*y, what is \ the resulting equation? How would you parametrize the equation in u and v?]", FontVariations->{"CompatibilityType"->0}], StyleBox["\n", FontWeight->"Bold"], "\nb) Suppose that the fluid level in the tank is 7 m on the left edge of \ the tank (where x=0) and 5 m on the right edge (where x=8). Find the \ equation of the plane of the liquid, and use a double integral to find the \ volume of liquid in the tank. [Hint: you should use a \"dy dx\" iterated \ integral, where the bounds on y depend on x, and are given by the equation of \ the base of the cylinder.] Also find the angle at which the tank is tipped \ from its upright position. For consistency, assume that the tank in the \ paragraph before part a) above is tipped at a 0\[Degree] angle. [Hint: use a \ \"side elevation\" sketch, and use trigonometry to find the angle between \ your plane and the plane z=5.]\n\nc) Suppose now that the fluid level is 7 m \ at the left edge of the tank, but that there isn't enough fluid to reach the \ right edge; instead, the fluid covers the base of the tank only out to the \ plane x=4. [You can produce a graph with the commands below: you'll have to \ paste them into another cell.]\n", Cell[BoxData[{ \(plane5 = ParametricPlot3D[{x, y, 7 - 7 x/4}, {x, 0, 4}, {y, \(-3\), 3}]\), "\n", \(Show[tank, plane4, AxesLabel \[Rule] {x, y, z}, ViewPoint -> {0, \ \(-4\), \ 0}]\)}], "Input", CellFrame->False], "\nFind the equation of the plane of the liquid,and use a double integral \ to find the volume of liquid in the tank. Also find the angle at which the \ tank is tipped from its upright position. [", StyleBox["Warning", FontWeight->"Bold"], ": the region of integration has changed -- if you reuse the bounds from \ problem 1, you'll include the \"negative volume\" from the region where z<0.]\ \n\nd) Suppose that the tank is tipped so far that the liquid touches both \ ends of the tank, touching the top (at z=10) out to the plane x=2, and \ covering the bottom out to the plane x=4. Find the volume of liquid in the \ tank using two double integrals. Explain why you can't just use one double \ integral. Lastly, find the angle at which the tank is tipped from its \ upright position.\n\ne) Suppose, finally, that the liquid touches the top of \ the tank out to the plane x=6, and reaches a level of 4 m on the right edge \ of the tank. Find the volume of liquid in the tank using two double \ integrals, and find the angle at which the tank is tipped from its upright \ position.\n\nf) Suppose that our tank is tilted at an angle of 80\[Degree] \ from its upright position. Where could you mark the surface of the tank to \ indicate when the tank is exactly one-third full? [Hint: use trigonometry to \ decide which of the first four exercises most resembles this problem. \ Calculate the volume in terms of some unknown variable, and use ", StyleBox["Mathematica", FontSlant->"Italic"], "'s ", StyleBox["FindRoot", FontFamily->"Courier", FontWeight->"Bold"], " command, which you can look up in the help browser.]" }], "Text", CellFrame->True, Background->RGBColor[1, 0.501961, 0.501961]], Cell[CellGroupData[{ Cell["Credits", "Subsection", Background->None], Cell["\<\ This lab was inspired by a lab written at Lafayette College in \ Pennsylvania. Cindy Kaus (cindy.kaus@metrostate.edu) wrote the first version \ that we used at the University of Minnesota. James Swenson \ (swenson@math.umn.edu) completely rewrote the text in Spring 2002 but kept \ the same exercises. In Spring 2004 Jonathan Rogness (rogness@math.umn.edu) \ went through and added a few Live commands and polished things up a bit; Ryan \ Gantner (gantner@math.umn.edu) added new exercises, where the cylinder is no \ longer circular. This lab (or portions thereof) copyrighted by their respective authors. \ We've all agreed to use 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/ 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"] }, Closed]] }, Closed]] }, FrontEndVersion->"5.0 for X", ScreenRectangle->{{0, 1024}, {0, 768}}, WindowSize->{906, 687}, WindowMargins->{{28, Automatic}, {Automatic, 27}}, PrintingPageRange->{Automatic, Automatic}, PrintingOptions->{"PaperSize"->{612, 792}, "PaperOrientation"->"Portrait", "PostScriptOutputFile":>FrontEnd`FileName[{"user002", "rogness"}, \ "Newlab.nb.ps", CharacterEncoding -> "iso8859-1"], "Magnification"->1}, CellLabelAutoDelete->True, 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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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, 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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, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonNote->ButtonData}], Cell[StyleData["Hyperlink", "Presentation"], FontSize->16], Cell[StyleData["Hyperlink", "Condensed"], FontSize->11], Cell[StyleData["Hyperlink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, 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, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "MainBook", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["MainBookLink", "Presentation"], FontSize->16], Cell[StyleData["MainBookLink", "Condensed"], FontSize->11], Cell[StyleData["MainBookLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["AddOnsLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "AddOns", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["AddOnsLink", "Presentation"], FontSize->16], Cell[StyleData["AddOnsLink", "Condensed"], FontSize->11], Cell[StyleData["AddOnsLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["RefGuideLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "RefGuide", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["RefGuideLink", "Presentation"], FontSize->16], Cell[StyleData["RefGuideLink", "Condensed"], FontSize->11], Cell[StyleData["RefGuideLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["GettingStartedLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "GettingStarted", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["GettingStartedLink", "Presentation"], FontSize->16], Cell[StyleData["GettingStartedLink", "Condensed"], FontSize->11], Cell[StyleData["GettingStartedLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["OtherInformationLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "OtherInformation", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["OtherInformationLink", "Presentation"], FontSize->16], Cell[StyleData["OtherInformationLink", "Condensed"], FontSize->11], Cell[StyleData["OtherInformationLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, 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[ 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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 ]] }] ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. *******************************************************************) (*CellTagsOutline CellTagsIndex->{ "zerodeg"->{ Cell[3281, 99, 219, 5, 43, "Input", CellTags->"zerodeg"]} } *) (*CellTagsIndex CellTagsIndex->{ {"zerodeg", 53203, 1645} } *) (*NotebookFileOutline Notebook[{ Cell[1754, 51, 382, 10, 114, "Text"], Cell[CellGroupData[{ Cell[2161, 65, 51, 1, 56, "Section"], Cell[2215, 68, 454, 7, 68, "Text"], Cell[2672, 77, 78, 1, 27, "Input"], Cell[2753, 80, 270, 7, 50, "Text"], Cell[3026, 89, 127, 3, 27, "Input"], Cell[3156, 94, 122, 3, 32, "Text"], Cell[3281, 99, 219, 5, 43, "Input", CellTags->"zerodeg"], Cell[3503, 106, 165, 5, 32, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[3705, 116, 68, 1, 36, "Section"], Cell[3776, 119, 246, 5, 50, "Text"], Cell[4025, 126, 528, 11, 91, "Input"], Cell[4556, 139, 184, 4, 32, "Text"], Cell[4743, 145, 341, 7, 50, "Text"], Cell[5087, 154, 281, 6, 43, "Input"], Cell[5371, 162, 444, 7, 68, "Text"], Cell[5818, 171, 168, 3, 27, "Input"], Cell[5989, 176, 527, 9, 104, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[6553, 190, 64, 1, 36, "Section"], Cell[6620, 193, 215, 4, 50, "Text"], Cell[6838, 199, 361, 8, 59, "Input"], Cell[7202, 209, 579, 9, 86, "Text"], Cell[7784, 220, 134, 2, 27, "Input"], Cell[7921, 224, 167, 4, 32, "Text"], Cell[8091, 230, 533, 14, 66, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[8661, 249, 68, 2, 36, "Section"], Cell[8732, 253, 3211, 60, 473, "Text"], Cell[11946, 315, 52, 1, 43, "Input"], Cell[12001, 318, 465, 7, 68, "Text"], Cell[12469, 327, 224, 4, 27, "Input"], Cell[12696, 333, 251, 6, 68, "Text"], Cell[12950, 341, 3578, 63, 563, "Text"], Cell[CellGroupData[{ Cell[16553, 408, 49, 1, 45, "Subsection"], Cell[16605, 411, 1064, 19, 194, "Text"] }, Closed]] }, Closed]] } ] *) (******************************************************************* End of Mathematica Notebook file. *******************************************************************)