(************** Content-type: application/mathematica ************** 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[ 26245, 744]*) (*NotebookOutlinePosition[ 26985, 769]*) (* CellTagsIndexPosition[ 26941, 765]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ "First read the End of Chapter Additional Exercises #31 & #32.\nGo through \ Part I, answering the two questions posed.\nDo the You Try It for Part I. \n\ Go through Part II and then do the You Try It for Part II. \nGo through Part \ III and then look up the You Try It for Part III. \nI will want to see your \ printouts WITH COMMENTS THROUGHOUT THE PRINTOUTS, as well as a summary of \ what you have accomplished in each of the You Try It for Parts I and III and \ what you found out in the You Try It for Part II.\nBe prepared to ask me \ questions about the ", StyleBox["Mathematica", FontSlant->"Italic"], " commands in the labs." }], "Text"], Cell[CellGroupData[{ Cell["\<\ Volumes That You Can Use and Plotting in Cylindrical and Spherical \ Coordinates\ \>", "Title"], Cell[TextData[StyleBox["Chapter 12, End of Chapter Additional Exercises and \ Plotting in Cylindrical and Spherical Coordinates", FontFamily->"Arial", FontSize->16, FontWeight->"Bold"]], "Text"], Cell[CellGroupData[{ Cell["Introduction", "Section"], Cell["\<\ OBJECTIVE: Use the concept of volume to solve practical problems involving \ rain catchers and satellite dishes.\ \>", "Text"], Cell["\<\ How can you measure precipitation if you collect rain in a container that is \ narrower at the bottom than at the top? At what angle must a satellite dish \ be tilted for rain not to collect in it? \ \>", "Text"], Cell[CellGroupData[{ Cell["Technology Guidelines", "Subsection", CellDingbat->"\[LightBulb]"], Cell[TextData[{ StyleBox["NOTE: If you have just finished a module, restart ", CellFrame->True, Background->None], StyleBox["Mathematica", CellFrame->True, FontSlant->"Italic", Background->None], StyleBox[" before executing a new module.\nTO OPEN CELLS, put your cursor \ on the right cell bracket and double click.", CellFrame->True, Background->None], "\nTO STOP AN EXECUTION\n\tSelect the ", StyleBox["Kernel", FontSlant->"Italic"], " pull-down menu and click on ", StyleBox["Abort Evaluation.\n", FontSlant->"Italic"], "ORDER OF EXECUTION\n\tExecute cells in the order given. Do not skip any \ Input cells within a given notebook.\nSAVING NOTEBOOKS\n\tYou can save \ anytime to any directory you choose, and it is wise to save often.\n\t\ However, before you do your final save, delete all your output by selecting \ the \n\t ", StyleBox["Delete All Output", FontSlant->"Italic"], " selection under the ", StyleBox["Kernel", FontSlant->"Italic"], " pull-down menu.\nEXPERIENCING MAJOR PROBLEMS\n\tSave if appropriate, and \ then shut down ", StyleBox["Mathematica", FontSlant->"Italic"], " and start it up again." }], "Text"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Part I: A Hemispherical Rain Gauge ", "Section"], Cell[TextData[StyleBox["End of Chapter Additional Exercises, Variation of \ Exercise 31", FontWeight->"Bold"]], "Text"], Cell[TextData[{ "Suppose you are trying to determine how to use the height of water in a \ hemispherical bowl of radius and depth 10 inches to measure the amount of \ rain that has fallen. Why do we use the volume of a cylinder of radius 10 \ inches filled to a particular height (", StyleBox["k", FontSlant->"Italic"], ")? What does that have to do with the volume of water that will collect in \ the bowl? The key idea is that the volume in each is the same; the only \ difference is in the height of the water in the two different containers." }], "Text"], Cell[TextData[{ "We will write our formulas using a general radius (rad), so that we can go \ back and change it out for hemispherical bowls with different radii. Since \ the bowl is considered to be sitting on top of the ", StyleBox["xy", FontSlant->"Italic"], "-plane with the axis of symmetry through the ", StyleBox["z", FontSlant->"Italic"], " axis, it is easiest to use cylindrical coordinates. The equation of the \ sphere is ", Cell[BoxData[ \(TraditionalForm\`r\^2\)]], "+ ", Cell[BoxData[ \(TraditionalForm\`\((z\ - rad)\)\^2\)]], "= ", Cell[BoxData[ \(TraditionalForm\`rad\^2\)]], "." }], "Text"], Cell[BoxData[{ \(Off[General::spell]\ \), "\n", \(Off[General::spell1]\ \), "\n", \(Clear[r, \[Theta], z, h, rad, k]\), "\n", \(Print["\", volcylinderht[k_] = \[Pi]\ \ rad\^2\ k]\), "\n", \(Print["\", volbowlht[h_] = Integrate[ r, {\[Theta], 0, 2 \[Pi]}, {r, 0, Sqrt[2 h\ rad\ - h\^2]}, {z, rad - Sqrt[rad\^2 - r\^2], h}] // Simplify]\)}], "Input"], Cell[TextData[{ "If there is 1 inch of rain in the 10-inch radius cylinder, we will \ determine the height of water in the bowl. To do this, we equate the volume \ in the cylinder to the volume in the bowl. ", StyleBox["Why?", FontColor->RGBColor[1, 0, 0]] }], "Text"], Cell[BoxData[{ \(\(rad = 10;\)\), "\n", \(one = NSolve[volbowlht[h] == volcylinderht[1], h]\)}], "Input"], Cell["\<\ The answer of 3.3555 is the only reasonable one and we extract that as \ follows.\ \>", "Text"], Cell[BoxData[ \(Print["\", one[\([2, 1, 2]\)], "\< inch level.\>"]\)], "Input"], Cell["\<\ What if there are 3 inches of water in the bowl? How many inches of water \ would this produce in the cylinder?\ \>", "Text"], Cell[BoxData[{ \(\(rad = 10;\)\), "\n", \(\(three = NSolve[volbowlht[3] == volcylinderht[k], k];\)\), "\n", \(Print["\", three[\([1, 1, 2]\)], "\< inches.\>"]\)}], "Input"], Cell[TextData[{ "Because of multiple roots, it is somewhat difficult to solve explicitly \ for the height in the bowl as a function of the actual amount of rain (height \ of water in the cylinder). However, it is easy to determine the relationship \ the other way around; that is, if we know the height of water in the bowl, we \ can easily find the amount of precipitation by dividing the volume of water \ in the bowl by \[Pi] ", Cell[BoxData[ \(TraditionalForm\`rad\^2\)]], ", which is the cross-sectional area of the cylinder with the same size \ top. This procedure is employed below to compute the actual precipitation \ amount." }], "Text"], Cell["\<\ We look at some tables of values associating the actual precipitation and the \ height of the water in the bowl corresponding to different values of the \ radius. \ \>", "Text"], Cell[BoxData[{ \(Clear[rad, precip, preciplist]\), "\n", \(\(precip[h_, rad_] = volbowlht[h]/\((rad\^2\ \[Pi])\) // Simplify;\)\), "\n", \(\(preciplist[rad_] = Table[{h, precip[h, rad] // N}, {h, .5, 9.5}];\)\)}], "Input"], Cell["\<\ We can apply these formulas to create some tables of values that associate \ the height of water in the bowl with the actual precipitation.\ \>", "Text"], Cell[BoxData[{ \(TableForm[preciplist[10], TableHeadings -> {None, {"\", \ "\"}}]\), "\n", \(TableForm[preciplist[15], TableHeadings -> {None, {"\", \ "\"}}]\), "\n", \(TableForm[preciplist[20], TableHeadings -> {None, {"\", \ "\"}}]\)}], "Input"], Cell[TextData[{ "To get a visual perspective of this data, plot the sets using ", StyleBox["MultipleListPlot", FontWeight->"Bold"], ". To do this, first read in a package." }], "Text"], Cell[BoxData[ \(<< Graphics`MultipleListPlot`\)], "Input"], Cell[BoxData[ \(\(MultipleListPlot[preciplist[10], preciplist[15], preciplist[20], AxesLabel -> {height, precipitation}, PlotJoined -> True, PlotStyle -> {RGBColor[1, 0, 0], RGBColor[0, 1, 0], RGBColor[0, 0, 1]}];\)\)], "Input"], Cell[TextData[StyleBox["Which plot goes with which radius?", FontColor->RGBColor[1, 0, 0]]], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["You Try It: Part I", FontColor->RGBColor[1, 0, 0]]], "Section"], Cell[TextData[{ "Now, select a different shaped rain-collector, and determine the \ relationship between the height of water in your collector and the actual \ precipitation. \n", Cell[BoxData[ FormBox[ RowBox[{\(Suppose\ that\ the\ bowl\ is\ parabolic\), ",", " ", RowBox[{ RowBox[{"say", " ", "with", " ", "the", " ", RowBox[{"equation", ":", StyleBox[" ", FontSize->14], StyleBox["z", FontSlant->"Italic"]}]}], StyleBox["=", FontSlant->"Italic"], StyleBox[\(0.5\ r\^2\), FontSlant->"Italic"]}], ",", " ", \(with\ maximum\ radius\ 10\ inches\)}], TextForm]]], "." }], "Text"], Cell[BoxData[ RowBox[{\(volofbowlht[h_]\), "=", RowBox[{ RowBox[{"Integrate", "[", RowBox[{ "r", ",", \({\[Theta], 0, 2 Pi}\), ",", \({r, 0, Sqrt[2 h]}\), ",", RowBox[{"{", RowBox[{"z", ",", RowBox[{ StyleBox[".5", FontColor->RGBColor[1, 0, 0]], SuperscriptBox[ StyleBox["r", FontColor->RGBColor[1, 0, 0]], StyleBox["2", FontColor->RGBColor[1, 0, 0]]]}], ",", "h"}], "}"}]}], "]"}], "//", "Simplify"}]}]], "Input"], Cell[TextData[{ "For a bowl of radius 10, the ", StyleBox["volofbowlht[h]", FontWeight->"Bold"], " can be divided by 100\[Pi] to give the precipitation level for height ", StyleBox["h", FontSlant->"Italic"], " in your bowl as follows." }], "Text"], Cell[BoxData[{ \(\(rad = 10;\)\), "\n", \(newprecip[h_] = volofbowlht[h]/rad\^2\ \[Pi] // Simplify\)}], "Input"], Cell["\<\ Find the precipiation level if the height in the bowl is 2.5 inches.\ \>", "Text"], Cell[BoxData[ \(newprecip[2.5]\)], "Input"], Cell["Repeat these steps for containers having different shapes.", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Part II: Water in a Satellite Dish", "Section"], Cell[TextData[StyleBox[ "End of Chapter Additional Exercises, Exercise 32 and extension", FontWeight->"Bold"]], "Text"], Cell[TextData[{ "In this exercise, you are asked to determine the smallest tilt of a \ satellite dish so that it holds no water. In selecting a convenient \ coordinate system to solve this problem, instead of tilting the bowl, let us \ picture the bowl upright in a rectangular coordinate system and think of the \ water level as tilted.\nWe will let the bowl be centered on the ", StyleBox["z", FontSlant->"Italic"], " axis and rest on top of the ", StyleBox["xy", FontSlant->"Italic"], "-plane. The equation of the plane that represents the water level can be \ written as ", Cell[BoxData[ \(TraditionalForm\`z = m\ y + b\)]], ". In each case, since the water level plane must pass through the top edge \ of the bowl with coordinate points ", Cell[BoxData[ \(TraditionalForm\`z = 0.5\)]], " when ", Cell[BoxData[ \(TraditionalForm\`y = 1\)]], ", the intercept ", StyleBox["b", FontSlant->"Italic"], " can be written as", Cell[BoxData[ \(TraditionalForm\`\(\ \ 0.5\ - \ m\)\)]], "." }], "Text"], Cell[BoxData[{ \(xyzbowl[x_, y_] := .5 \((x\^2 + y\^2)\)\), "\n", \(xyzplane[y_, m_] := m\ y\ + \ \(( .5 - m)\)\), "\n", \(\(plotbowl = Plot[{xyzbowl[0, y], xyzplane[y, 0]}, {y, \(-1\), 1}, PlotStyle -> {RGBColor[1, 0, 0], RGBColor[0, 1, 0]}, DisplayFunction -> Identity];\)\), "\n", \(\(plot1 = Plot[xyzplane[y, .25], {y, \(- .5\), 1}, PlotStyle -> RGBColor[0, 1, 1], DisplayFunction -> Identity];\)\), "\n", \(\(plot2 = Plot[xyzplane[y, .5], {y, 0, 1}, PlotStyle -> RGBColor[0, 0, 1], DisplayFunction -> Identity];\)\), "\n", \(\(Show[plotbowl, plot1, plot2, DisplayFunction :> $DisplayFunction, AspectRatio -> .5, AxesLabel -> {"\", "\"}];\)\)}], "Input"], Cell[TextData[{ "Let us begin by drawing some scenarios as the slope of the plane (value of \ ", StyleBox["m", FontSlant->"Italic"], ") goes from 0 to .75. We can integrate to find the volume in each case \ first by finding the boundary curve for ", StyleBox["x", FontSlant->"Italic"], " and ", StyleBox["y", FontSlant->"Italic"], " along which the plane and the paraboloid intersect and then by leaving \ our integral written as a function of the slope of the water level plane." }], "Text"], Cell[BoxData[{ \(Clear[x, y, z, m, volume]\), "\n", \(sol[m_] = Solve[xyzplane[y, m] == xyzbowl[x, y], {y}] // Simplify\)}], "Input"], Cell[TextData[{ "Note that the equation of intersection of the plane and the paraboloid is \ a circle, centered at ", Cell[BoxData[ \(TraditionalForm\`x\ = \ 0\)]], " and ", Cell[BoxData[ \(TraditionalForm\`y\ = \ m\)]], " and with radius ", Cell[BoxData[ \(TraditionalForm\`1 - m\)]], ", so we will let ", Cell[BoxData[ \(TraditionalForm\`z\)]], " go from the paraboloid to the plane and integrate ", StyleBox["x", FontSlant->"Italic"], " and ", StyleBox["y", FontSlant->"Italic"], " over the appropriate circle." }], "Text"], Cell[BoxData[ \(volume[m_] := NIntegrate[ 1, {x, m - 1, 1 - m}, {y, \(sol[m]\)[\([1, 1, 2]\)], \(sol[m]\)[\([2, 1, 2]\)]}, {z, xyzbowl[x, y], xyzplane[y, m]}]\)], "Input"], Cell[TextData[{ "Now, let's visualize the results for a few values of ", StyleBox["m", FontSlant->"Italic"], ". We first read in a package to help us plot the region filled with water. \ Then we let the slope take on the values of .25, 1, and .75, each time \ evaluating the volume of water in the satellite." }], "Text"], Cell[BoxData[ \(<< \ Graphics`FilledPlot`\)], "Input"], Cell[BoxData[{ \(\(p0 = FilledPlot[{xyzplane[y, 0], xyzbowl[0, y]}, {y, \(-1\), 1}, AxesLabel -> {y, z}, AxesOrigin\ -> \ {0, 0}, PlotLabel -> "\"];\)\), "\n", \(Print["\", volume[0], "\< cubic meters\>"]\), "\n", \(\(p1 = FilledPlot[{xyzplane[y, .25], xyzbowl[0, y]}, {y, \(- .5\), 1}, AxesLabel -> {y, z}, PlotLabel -> "\"];\)\), "\n", \(Print["\", volume[ .25], "\< cubic meters\>"]\), "\n", \(\(p2 = FilledPlot[{xyzplane[y, .5], xyzbowl[0, y]}, {y, 0, 1}, AxesLabel -> {y, z}, AxesOrigin\ -> \ {0, 0}, PlotLabel -> "\"];\)\), "\n", \(Print["\", volume[ .5], "\< cubic meters\>"]\), "\n", \(\(p3 = FilledPlot[{xyzplane[y, .75], xyzbowl[0, y]}, {y, .5, 1}, AxesLabel -> {y, z}, AxesOrigin\ -> \ {0, 0}, PlotLabel -> "\", DisplayFunction -> Identity];\)\), "\n", \(\(p4 = Plot[xyzbowl[0, y], {y, 0, .5}, AxesLabel -> {y, z}, AxesOrigin\ -> \ {0, 0}, DisplayFunction -> Identity];\)\), "\n", \(\(Show[p3, p4, DisplayFunction :> $DisplayFunction];\)\), "\n", \(Print["\", volume[ .75], "\< cubic meters\>"]\)}], "Input"], Cell[TextData[{ "As you recognize that the slope of the water level represents the tangent \ of the angle through which the bowl is tilted, you will also notice that the \ volume is approaching 0 as ", StyleBox["m", FontSlant->"Italic"], " is approaching 1, and, hence, as the angle of tilt is approaching 45\ \[Degree]." }], "Text"], Cell[BoxData[ \(volume[1]\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["You Try It: Part II", FontColor->RGBColor[1, 0, 0]]], "Section"], Cell["\<\ Check out satellite dishes that are installed at residences or at \ communication facilities. See if their shape is parabolic, and find out what \ their tilt is. Does it match up to what you expected?\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Part III: Using Moments to Determine Acceptable Heights", "Section"], Cell[TextData[StyleBox["Section 12.2, Extension of Exercise 43 (tipping up to \ an angle of 45\[Degree] problem)", FontWeight->"Bold"]], "Text"], Cell["\<\ Suppose that you have a buoy in the form of a solid hemisphere of radius 40 \ centimeters and uniform density that is topped with a solid cylinder of the \ same material and radius 40 centimeters. This buoy floats with the center of \ the hemisphere above the water surface. Determine the maximum height that the \ cylinder may attain that permits the buoy to continue floating upright. To \ picture this, we load two packages and plot the shape of the body.\ \>", "Text"], Cell[BoxData[{ \(<< Graphics`ParametricPlot3D`\), "\n", \(<< Graphics`Graphics3D`\)}], "Input"], Cell[BoxData[{ \(Clear[x, y, z, u, v, t, r, \[Theta]]\), "\n", \(\(hemisplot = CylindricalPlot3D[\(-\@\(1600 - r\^2\)\), {r, 0, 40}, {\[Theta], 0, 2 \[Pi]}, AxesLabel -> {x, y, z}, DisplayFunction -> Identity];\)\), "\n", \(\(cylpts\ = \ Table[{40 Cos[t]\ , 40\ Sin[t], \n\ \ u}, \ {t, \ 0, \ 2 Pi, \ Pi/12}, \n\ \ {u, \ 0, 40}];\)\), "\n", \(\(cylplot = ListSurfacePlot3D[cylpts, DisplayFunction -> Identity];\)\), "\n", \(\(topplot = CylindricalPlot3D[40, {r, 0, 40}, {\[Theta], 0, 2 \[Pi]}, AxesLabel -> {x, y, z}, DisplayFunction -> Identity];\)\), "\n", \(\(flagpolepts = Table[{0, 0, t}, {t, 35, 80}];\)\), "\n", \(\(flagpts = Table[{0, u, v}, {u, 0, 20}, {v, 60, 80}];\)\), "\n", \(\(fp1 = ScatterPlot3D[flagpolepts, DisplayFunction -> Identity];\)\), "\n", \(\(fp2 = ListSurfacePlot3D[flagpts, DisplayFunction -> Identity];\)\), "\n", \(\(Show[hemisplot, cylplot, topplot, fp1, fp2, DisplayFunction :> $DisplayFunction];\)\)}], "Input"], Cell[TextData[{ "You may assume that the flag on top is weightless. \n\nThe method of \ moments can help us. It can be determined that the buoyancy force exerted by \ the body of water due to the amount of water displaced will be in a direction \ through the point that would be the midpoint of the sphere, as long as the \ object is upright or tilted so that no part of the cylinder is immersed. \ Hence, if the center of gravity of the solid is at that point or closer to to \ the bottom of the hemisphere, the object will stay upright. If the center of \ gravity is in the region of the cylinder, the object will tilt. \n\nWe first \ determine the ", StyleBox["z", FontSlant->"Italic"], "-coordinate of the center of gravity of the hemisphere. Recall that the \ volume of a hemisphere with radius of ", StyleBox["r", FontSlant->"Italic"], " is 2/3 \[Pi] ", Cell[BoxData[ \(TraditionalForm\`r\^3\)]], "." }], "Text"], Cell[BoxData[{ \(\(volhemis = 2/3\ \[Pi]\ 40\^3;\)\), "\n", \(\(hemis = \(-\@\(1600 - x\^2 - y\^2\)\);\)\), "\n", \(zbar = Integrate[ z\ , {x, \(-40\), 40}, {y, \(-\@\(1600 - x\^2\)\), \@\(1600 - x\^2\)}, {z, hemis, 0}]/volhemis\)}], "Input"], Cell[TextData[{ "The ", StyleBox["z", FontSlant->"Italic"], "-coordinate of the center of mass of the cylinder is ", StyleBox["h/2", FontSlant->"Italic"], ", so if we equate the first moment of the hemisphere to the first moment \ of the cylinder, we can solve for ", StyleBox["h", FontSlant->"Italic"], ". Since the radius is 40 here, the volume of the cylinder is ", "1600", StyleBox[" ", FontSlant->"Italic"], "\[Pi] ", StyleBox["h", FontSlant->"Italic"], "." }], "Text"], Cell[BoxData[{ \(solh = Solve[\(-zbar\)*volhemis == h/2*\((1600 \[Pi]\ h)\), h]\), "\n", \(Print["\", solh[\([2, 1, 2]\)], "\< or \>", N[solh[\([2, 1, 2]\)]], "\< centimeters.\>"]\)}], "Input"], Cell[TextData[{ "In general, if the radius were ", StyleBox["R", FontSlant->"Italic"], ", the maximum height of the cylinder could be at most ", Cell[BoxData[ \(TraditionalForm\`\(R \@ 2\)\/2\)]], "." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["You Try It: Part III", FontColor->RGBColor[1, 0, 0]]], "Section"], Cell[TextData[{ StyleBox["Experiment with some of the 3-D plot functions in cylindrical and \ spherical coordinates. ", FontColor->RGBColor[1, 0, 0]], "\n\nBegin by practicing with a cylindrical plot. It is understood in these \ plots that ", StyleBox["z", FontSlant->"Italic"], " is a function of ", StyleBox["r", FontSlant->"Italic"], " and \[Theta] . ", StyleBox["Try some of your own functions. ", FontColor->RGBColor[1, 0, 0]], "The packages used in the above plots need to be read in if you have not \ already done that. Change the terms in red." }], "Text"], Cell[BoxData[{ \(<< Graphics`ParametricPlot3D`\), "\n", \(<< Graphics`Graphics3D`\)}], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"z", "=", StyleBox[\(r\ \ Cos[\[Theta]]\^2\), FontColor->RGBColor[1, 0, 0]]}], ";", "\n", RowBox[{"newcyl", "=", RowBox[{"CylindricalPlot3D", "[", RowBox[{"z", ",", RowBox[{"{", RowBox[{"r", ",", "0", ",", StyleBox["10", FontColor->RGBColor[1, 0, 0]]}], "}"}], ",", RowBox[{"{", RowBox[{"\[Theta]", ",", "0", StyleBox[",", FontColor->GrayLevel[0]], \(2 \[Pi]\)}], "}"}], ",", \(AxesLabel -> {x, y, z}\)}], "]"}]}], ";"}]], "Input"], Cell[TextData[{ "You can create spherical plots with the same package you used to create \ cylindrical plots. In these plots, it is assumed that \[Rho] is a function of \ \[Phi] and \[Theta]. ", StyleBox["Try some of your own by changing the terms in red. ", FontColor->RGBColor[1, 0, 0]], "Not everything will give satisfying results." }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{"\[Rho]", "=", StyleBox[\(\[Theta]\ Cos[\[Phi]]\), FontColor->RGBColor[1, 0, 0]]}], ";", "\n", \(SphericalPlot3D[\[Rho], {\[Phi], 0, \[Pi]}, {\[Theta], 0, 2 \[Pi]}, AxesLabel -> {x, y, z}]\), ";"}]], "Input"], Cell["\<\ You may have noticed that to draw the flag we had to first generate points \ and then plot the points. The difference between the two was that the pole \ could be thought of as a line, whereas, the flag was more like a surface. \ Experiment with the following commands and draw something else by changing \ the terms in red. Remember to leave one parameter in the first expression and \ two parameters in the second experession.\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"newflagpolepts", "=", RowBox[{"Table", "[", RowBox[{ RowBox[{"{", StyleBox[\(0, 0, t\), FontColor->RGBColor[1, 0, 0]], "}"}], ",", RowBox[{"{", StyleBox[\(t, 0, 30, .1\), FontColor->RGBColor[1, 0, 0]], "}"}]}], "]"}]}], ";", "\n", RowBox[{"newflagpts", "=", RowBox[{"Table", "[", RowBox[{ RowBox[{"{", StyleBox[\(0, u, v\), FontColor->RGBColor[1, 0, 0]], "}"}], ",", RowBox[{"{", StyleBox[\(u, 0, 20\), FontColor->RGBColor[1, 0, 0]], "}"}], ",", RowBox[{"{", StyleBox[\(v, 15, 30\), FontColor->RGBColor[1, 0, 0]], "}"}]}], "]"}]}], ";", "\n", \(fp1 = ScatterPlot3D[newflagpolepts]\), ";", "\n", \(fp2 = ListSurfacePlot3D[newflagpts]\), ";", "\n", \(Show[fp1, fp2, Boxed -> False, Axes -> False]\), ";"}]], "Input"] }, Closed]] }, Open ]] }, FrontEndVersion->"4.1 for Microsoft Windows", ScreenRectangle->{{0, 1024}, {0, 695}}, WindowSize->{603, 505}, WindowMargins->{{14, Automatic}, {Automatic, 5}}, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic}, StyleDefinitions -> "Default.nb" ] (******************************************************************* Cached data follows. 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