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Before you \ begin, you may want to review the notebooks ", StyleBox["MultipleIntegrals", FontWeight->"Bold"], ", ", StyleBox["SphericalCoords", FontWeight->"Bold"], " and ", StyleBox["AdvancedCalculusDemo", FontWeight->"Bold"], " in the Notebook folder or the back of the Lecture Notes. There are also \ many examples of how to do these problems in the Quizzes folder. In ", StyleBox["Problem 1", FontWeight->"Bold"], ", Jacobian determinants can be computed using the function ", StyleBox["J[]", FontFamily->"Courier", FontWeight->"Bold"], ":\n", StyleBox["?J\n", FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ "J[f,g,{u,v}] gives the Jacobian determinant of the coordinate\n \ transformation x = f(u,v), y = g(u,v). J[f,g,h,{u,v,w}]\n gives the \ Jacobian determinant of the coordinate\n transformation x = f(u,v,w), y = \ g(u,v,w), z = h(u,v,w).\n", FontFamily->"Courier"], "To create plots in cylindrical coordinates in ", StyleBox["Problem 4", FontWeight->"Bold"], ", make a substitution like\n", StyleBox["{x,y,z} /. {x -> r Cos[theta], y -> r Sin[theta]}\n", FontFamily->"Courier", FontWeight->"Bold"], "and plug it into ", StyleBox["ParametricPlot3D[..,{r,a,b},{t,c,d}].\n", FontFamily->"Courier", FontWeight->"Bold"], "In ", StyleBox["Problem 5", FontWeight->"Bold"], ", use a similar procedure for spherical coordinates:", StyleBox["\n", FontFamily->"Courier"], StyleBox[ "{x,y,z} /.\n {x->rho Cos[theta]Sin[phi],\n y->rho \ Sin[theta]Sin[phi],\n z->rho Cos[phi]}\n", FontFamily->"Courier", FontWeight->"Bold"], "and plug it into ", StyleBox["ParametricPlot3D[..,{theta,a,b},{phi,c,d}].\n", FontFamily->"Courier", FontWeight->"Bold"], StyleBox["\n", FontFamily->"Courier"], "Be sure to type in comments explaining what you are doing.\n\nRemember to \ uncheck ", StyleBox["Show In/Out Names", FontFamily->"Courier", FontWeight->"Bold"], " in the ", StyleBox["Kernel", FontFamily->"Courier", FontWeight->"Bold"], " menu and close this group before printing." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["Initialization"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell["<Automatic, InitializationCell->True, AspectRatioFixed->True], Cell["i = {1, 0, 0}; j = {0, 1, 0}; k = {0, 0, 1};", "Input", PageBreakWithin->Automatic, InitializationCell->True, AspectRatioFixed->True] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Problem 1 "], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell[TextData[{ "Compute the Jacobian determinant of the following coordinate \ transformations. Note that you must first solve for ", StyleBox["x", FontFamily->"Courier", FontWeight->"Bold"], " and ", StyleBox["y", FontFamily->"Courier", FontWeight->"Bold"], " as functions of ", StyleBox["u", FontFamily->"Courier", FontWeight->"Bold"], " and ", StyleBox["v", FontFamily->"Courier", FontWeight->"Bold"], " (use\n", StyleBox["Solve[{u==..., v==...},{u,v}]", FontFamily->"Courier", FontWeight->"Bold"], "), then apply ", StyleBox["D[]", FontFamily->"Courier", FontWeight->"Bold"], " and", StyleBox[" Det[]", FontFamily->"Courier", FontWeight->"Bold"], " in an appropriate way. Finally compute the Jacobian using ", StyleBox["J[]", FontFamily->"Courier", FontWeight->"Bold"], ".\n", StyleBox["a)", FontSlant->"Italic"], " ", StyleBox["u == x + y, v == x^2 - y^2", FontFamily->"Courier", FontWeight->"Bold"], "\n", StyleBox["b)", FontSlant->"Italic"], " ", StyleBox["u == 3x - y + z, v == x + 2y + z, w == -x + y + z", FontFamily->"Courier", FontWeight->"Bold"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Solution"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell["", "Input", PageBreakWithin->Automatic, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Problem 2"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell[TextData[{ "Plot the following vector fields using ", StyleBox["VectorField[F,{x,-5,5},{y,-5,5}]", FontFamily->"Courier", FontWeight->"Bold"], ".\n", StyleBox["a)", FontSlant->"Italic"], " ", StyleBox["F = { 0.2 x, 0.1 y}", FontFamily->"Courier", FontWeight->"Bold"], "\n", StyleBox["b) ", FontSlant->"Italic"], StyleBox["F = { 0.1 y, 0.2 x}", FontFamily->"Courier", FontWeight->"Bold"], "\n", StyleBox["c) ", FontSlant->"Italic"], StyleBox["F = {-0.1 y, 0.2 x}", FontFamily->"Courier", FontWeight->"Bold"] }], "Text", CellMargins->{{Inherited, 37}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Solution"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell["", "Input", PageBreakWithin->Automatic, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Problem 3"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell[TextData[{ "Let ", StyleBox["C", FontFamily->"Courier", FontWeight->"Bold"], " be the curve defined by\n", StyleBox[" r[t_] = t(t^2-1) i + 2t j - t^2 k\n", FontFamily->"Courier", FontWeight->"Bold"], "for ", StyleBox["-1<= t <= 2", FontFamily->"Courier", FontWeight->"Bold"], ".\nIntegrate the function ", StyleBox["f[x_,y_,z_] = z^2 - x y", FontFamily->"Courier", FontWeight->"Bold"], " over ", StyleBox["C", FontFamily->"Courier", FontWeight->"Bold"], " by using the appropriate formula with ", StyleBox["NIntegrate[]", FontFamily->"Courier", FontWeight->"Bold"], ". Compare the result with ", StyleBox["LineIntegrate[]//N", FontFamily->"Courier", FontWeight->"Bold"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Solution"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell["", "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Problem 4"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell[TextData[{ "Use cylindrical coordinates and ", StyleBox["ParamtericPlot3D[]", FontFamily->"Courier", FontWeight->"Bold"], " to plot the surfaces\n", StyleBox["a)", FontSlant->"Italic"], " ", StyleBox["z = theta, 0 <= theta <= 2Pi, -2 <= r <= 2", FontFamily->"Courier", FontWeight->"Bold"], "\n", StyleBox["b)", FontSlant->"Italic"], " ", StyleBox["z = E^(-r^2/4)Cos[Pi r], 0 <= theta <= 2Pi,\n 0 <= r <= 3", FontFamily->"Courier", FontWeight->"Bold"] }], "Text", CellMargins->{{Inherited, 72}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Solution"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell["", "Input", PageBreakWithin->Automatic, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Problem 5"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell[TextData[{ "Use spherical coordinates and ", StyleBox["ParamtericPlot3D[]", FontFamily->"Courier", FontWeight->"Bold"], " to plot the surface\n", StyleBox[ "rho = 1/(phi^2(phi-Pi)^2), Pi/4 <= phi <=3Pi/4,\n0 <= theta <= 2Pi", FontFamily->"Courier", FontWeight->"Bold"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Solution"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell["", "Input", PageBreakWithin->Automatic, AspectRatioFixed->True] }, Open ]] }, FrontEndVersion->"Macintosh 3.0", ScreenRectangle->{{0, 1024}, {0, 748}}, AutoGeneratedPackage->None, WindowToolbars->{"RulerBar", "EditBar"}, CellGrouping->Manual, WindowSize->{520, 509}, WindowMargins->{{5, Automatic}, {Automatic, 12}}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, 128}}, ShowCellLabel->True, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, CharacterEncoding->"MacintoshAutomaticEncoding", MacintoshSystemPageSetup->"\<\ 00<0001804P000000]P2:?oQon82n@960dL5:0?l0080001804P000000]P2:001 0000I00000400`<300000BL?00400@00000000000000060801T1T00000000000 00000000000000000000000000000000\>" ] (*********************************************************************** Cached data follows. 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