(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of 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). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. 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[ 10505, 403]*) (*NotebookOutlinePosition[ 11658, 439]*) (* CellTagsIndexPosition[ 11614, 435]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData["Math 225: Calculus III Assignment 10"], "Subsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell[TextData["Name:"], "Subsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell[TextData["Section:"], "Subsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[ "\nI affirm that the solutions presented in this assignment are entirely my \ own work."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Signature:"], "Subsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Instructions"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "This assignment contains problems involving vector fields, line integrals, \ flow integrals, and Green's Theorem, see Chapter 15 in Finney & Thomas or \ Chapter 5 in the Lecture Notes. Before you begin, you may want to review the \ notebook ", StyleBox["AdvancedCalculusDemo", FontWeight->"Bold"], " in the Notebook folder or the back of the Lecture Notes. \n\nBe 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[TextData["<Automatic, InitializationCell->True, AspectRatioFixed->True], Cell[TextData["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 ", StyleBox["Div[F]", FontFamily->"Courier", FontWeight->"Bold"], ", ", StyleBox["Curl[F]", FontFamily->"Courier", FontWeight->"Bold"], ", and ", StyleBox["Div[Curl[F]]", FontFamily->"Courier", FontWeight->"Bold"], " for the following vector fields.\n", StyleBox["a)", FontSlant->"Italic"], " ", StyleBox["F = (x^2 y + z) i + (y^2 z + x) j + (z^2 x + y) k", FontFamily->"Courier", FontWeight->"Bold"], "\n", StyleBox["b)", FontSlant->"Italic"], " ", StyleBox["F = Cos[x+y] i + Sin[x-y] j", FontFamily->"Courier", FontWeight->"Bold"], "\n", StyleBox["c)", FontSlant->"Italic"], " ", StyleBox["F = y E^z i + z E^x j + x E^y k\n", FontFamily->"Courier", FontWeight->"Bold"], StyleBox["d)", FontSlant->"Italic"], " \"Prove\" using ", StyleBox["Mathematica", FontSlant->"Italic"], " that for an arbitrary vector field,\n", StyleBox[" F = m[x,y,z]i + n[x,y,z]j + p[x,y,z]k\n", FontFamily->"Courier", FontWeight->"Bold"], " we always get ", StyleBox["Div[Curl[F]] == 0", 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[{ "Let ", StyleBox["F = (y-z) i + (x-z) j - (x+y) k", FontFamily->"Courier", FontWeight->"Bold"], ". Compute the flow integral \"", StyleBox["F.dr", FontFamily->"Courier", FontWeight->"Bold"], "\" over each of the following curves by using the appropriate formula with \ ", StyleBox["Integrate[]", FontFamily->"Courier", FontWeight->"Bold"], ". Compare your answer to ", StyleBox["FlowIntegrate[]", FontFamily->"Courier", FontWeight->"Bold"], ":\n", StyleBox["a)", FontSlant->"Italic"], " ", StyleBox["r[t_] = (2t^3-t) i + (t^2+t)/2 j + t k, 0<= t <=1", FontFamily->"Courier", FontWeight->"Bold"], "\n", StyleBox["b)", FontSlant->"Italic"], " ", StyleBox["r[t_] = (1-Cos[t])/2 i + Sin[t/2]j + t/Pi k, 0<= t <= Pi\n", FontFamily->"Courier", FontWeight->"Bold"], StyleBox["c)", FontSlant->"Italic"], " Explain why the answers to ", StyleBox["a)", FontSlant->"Italic"], " and ", StyleBox["b)", FontSlant->"Italic"], " have to be the same, even though the curves are quite different." }], "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 3"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell[TextData[{ StyleBox["a)", FontSlant->"Italic"], " Integrate\n", StyleBox["(y^3 - y + z)dx + (3x y^2 - x + 2y z)dy + (y^2 + x + 2z)dz", FontFamily->"Courier", FontWeight->"Bold"], "\nover the curve ", StyleBox["C", FontFamily->"Courier", FontWeight->"Bold"], " defined by ", StyleBox["t^2 i + t j + (t^3-1)k, 0 <= t <= 1", FontFamily->"Courier", FontWeight->"Bold"], ",\n(using ", StyleBox["Integrate[]", FontFamily->"Courier", FontWeight->"Bold"], ", or ", StyleBox["FlowIntegrate[]", FontFamily->"Courier", FontWeight->"Bold"], ").\n", StyleBox["b)", FontSlant->"Italic"], " Find a function ", StyleBox["f[x,y,z]", FontFamily->"Courier", FontWeight->"Bold"], " such that\n", StyleBox[ "Grad[f[x,y,z]] == \n (y^3 - y + z)i + (3x y^2 - x + 2y z)j + (y^2 + x + \ 2z)k", FontFamily->"Courier", FontWeight->"Bold"], "\n", StyleBox["c)", FontSlant->"Italic"], " Use the Fundamental Theorem of Line Integrals to evaluate the the \ integral in ", StyleBox["a)", FontSlant->"Italic"], " using the result of ", StyleBox["b)", FontSlant->"Italic"], "." }], "Text", CellMargins->{{Inherited, 2}, {Inherited, Inherited}}, 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[{ "Let ", StyleBox["C", FontFamily->"Courier", FontWeight->"Bold"], " be the boundary of the \"triangular\" region ", StyleBox["R", FontFamily->"Courier", FontWeight->"Bold"], " in the first quadrant enclosed by the ", StyleBox["x", FontFamily->"Courier", FontWeight->"Bold"], "-axis, the line ", StyleBox["x==1", FontFamily->"Courier", FontWeight->"Bold"], ", and the curve ", StyleBox["x==y^3", FontFamily->"Courier", FontWeight->"Bold"], ". Use Green's Theorem to transform the integral of\n", StyleBox["2x y^2dx + x^2 y dy", FontFamily->"Courier", FontWeight->"Bold"], " over ", StyleBox["C", FontFamily->"Courier", FontWeight->"Bold"], " into a double integral over ", StyleBox["R", FontFamily->"Courier", FontWeight->"Bold"], " and then evaluate this double integral." }], "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 5"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell[TextData[ "Produce a plot of a surface that you think has interesting features. The \ surface does not have to be the graph of a function and can be done in any \ coordinate system you wish. Explain what features of the surface are \ interesting and why."], "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->{{28, Automatic}, {30, Automatic}}, 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. 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->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[1709, 49, 148, 3, 44, "Subsection", PageBreakWithin->Automatic], Cell[1860, 54, 115, 3, 44, "Subsection", PageBreakWithin->Automatic], Cell[1978, 59, 118, 3, 44, "Subsection", PageBreakWithin->Automatic], Cell[CellGroupData[{ Cell[2121, 66, 161, 4, 46, "Text"], Cell[2285, 72, 120, 3, 44, "Subsection", PageBreakWithin->Automatic] }, Open ]], Cell[CellGroupData[{ Cell[2442, 80, 92, 2, 44, "Subsection"], Cell[2537, 84, 753, 19, 160, "Text"], Cell[CellGroupData[{ Cell[3315, 107, 127, 3, 41, "Subsubsection", PageBreakWithin->Automatic], Cell[3445, 112, 130, 3, 27, "Input", PageBreakWithin->Automatic, InitializationCell->True], Cell[3578, 117, 155, 3, 27, "Input", PageBreakWithin->Automatic, InitializationCell->True] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[3782, 126, 123, 3, 41, "Subsubsection", PageBreakWithin->Automatic], Cell[3908, 131, 1240, 50, 133, "Text"], Cell[5151, 183, 121, 3, 41, "Subsubsection", PageBreakWithin->Automatic], Cell[5275, 188, 73, 2, 27, "Input", PageBreakWithin->Automatic] }, Open ]], Cell[CellGroupData[{ Cell[5385, 195, 122, 3, 41, "Subsubsection", PageBreakWithin->Automatic], Cell[5510, 200, 1166, 43, 132, "Text"], Cell[6679, 245, 121, 3, 41, "Subsubsection", PageBreakWithin->Automatic], Cell[6803, 250, 73, 2, 27, "Input", PageBreakWithin->Automatic] }, Open ]], Cell[CellGroupData[{ Cell[6913, 257, 122, 3, 41, "Subsubsection", PageBreakWithin->Automatic], Cell[7038, 262, 1293, 50, 167, "Text"], Cell[8334, 314, 121, 3, 41, "Subsubsection", PageBreakWithin->Automatic], Cell[8458, 319, 43, 1, 27, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[8538, 325, 122, 3, 41, "Subsubsection", PageBreakWithin->Automatic], Cell[8663, 330, 946, 36, 81, "Text"], Cell[9612, 368, 121, 3, 41, "Subsubsection", PageBreakWithin->Automatic], Cell[9736, 373, 73, 2, 27, "Input", PageBreakWithin->Automatic] }, Open ]], Cell[CellGroupData[{ Cell[9846, 380, 122, 3, 41, "Subsubsection", PageBreakWithin->Automatic], Cell[9971, 385, 318, 6, 62, "Text"], Cell[10292, 393, 121, 3, 41, "Subsubsection", PageBreakWithin->Automatic], Cell[10416, 398, 73, 2, 27, "Input", PageBreakWithin->Automatic] }, Open ]] } ] *) (*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)