(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 9481, 267] NotebookOptionsPosition[ 9107, 251] NotebookOutlinePosition[ 9451, 266] CellTagsIndexPosition[ 9408, 263] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[BoxData[{ RowBox[{ RowBox[{"Helix", "=", RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"4", RowBox[{"Cos", "[", RowBox[{"5", " ", "t"}], "]"}]}], ",", RowBox[{"4", RowBox[{"Sin", "[", RowBox[{"5", " ", "t"}], "]"}]}], ",", "t"}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"3", "Pi"}]}], "}"}], ",", RowBox[{"PlotStyle", "\[Rule]", "Red"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Cylind", "=", RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"4", RowBox[{"Cos", "[", "u", "]"}]}], ",", RowBox[{"4", RowBox[{"Sin", "[", "u", "]"}]}], ",", "v"}], "}"}], ",", RowBox[{"{", RowBox[{"u", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}], ",", RowBox[{"{", RowBox[{"v", ",", RowBox[{"-", "1"}], ",", "10"}], "}"}]}], "]"}]}], ";", "\"\<\!\(\*SuperscriptBox[\(x\), \(2\)]\)+\!\(\*SuperscriptBox[\(y\), \ \(2\)]\)=16\>\"", ";"}], "\[IndentingNewLine]", RowBox[{"GraphicsRow", "[", RowBox[{ RowBox[{"{", RowBox[{"Helix", ",", "Cylind", ",", RowBox[{"Show", "[", RowBox[{"Helix", ",", "Cylind"}], "]"}]}], "}"}], ",", RowBox[{"ImageSize", "\[Rule]", "1000"}]}], "]"}]}], "Input", CellChangeTimes->{{3.411598822015625*^9, 3.411598889578125*^9}, { 3.47017133359375*^9, 3.470171338578125*^9}, {3.47017155015625*^9, 3.470171550625*^9}, {3.47017161790625*^9, 3.470171819609375*^9}, { 3.470171904203125*^9, 3.47017213246875*^9}}], Cell[BoxData[{ RowBox[{"\"\<#35 Paraboloid z=\!\(\*SuperscriptBox[\(x\), \ \(2\)]\)+\!\(\*SuperscriptBox[\(y\), \(2\)]\) and Plane x - y = 0 \>\"", " "}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Paraboloid", "=", RowBox[{"Plot3D", "[", RowBox[{ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "3"}], ",", "3"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "3"}], ",", "3"}], "}"}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{"\"\\"", ",", "\"\\"", ",", "\"\\""}], "}"}]}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Paraboloid1", "=", RowBox[{"Plot3D", "[", RowBox[{ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "10"}], ",", "10"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "10"}], ",", "10"}], "}"}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{"\"\\"", ",", "\"\\"", ",", "\"\\""}], "}"}]}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Plane", "=", RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"{", RowBox[{"u", ",", "u", ",", "v"}], "}"}], ",", RowBox[{"{", RowBox[{"u", ",", RowBox[{"-", "3"}], ",", "3"}], "}"}], ",", RowBox[{"{", RowBox[{"v", ",", RowBox[{"-", "1"}], ",", "20"}], "}"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"Plane1", "=", RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"{", RowBox[{"u", ",", "u", ",", "v"}], "}"}], ",", RowBox[{"{", RowBox[{"u", ",", "0", ",", "10"}], "}"}], ",", RowBox[{"{", RowBox[{"v", ",", "0", ",", "10"}], "}"}]}], "]"}]}], ";"}], " "}], "\n", RowBox[{ RowBox[{"Curve", "=", RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"{", RowBox[{"t", ",", "t", ",", RowBox[{"2", " ", SuperscriptBox["t", "2"]}]}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", RowBox[{"-", "3"}], ",", "3"}], "}"}], ",", RowBox[{"PlotStyle", "\[Rule]", RowBox[{"{", RowBox[{"Red", ",", "Thick"}], "}"}]}]}], "]"}]}], ";"}], "\n", RowBox[{"GraphicsRow", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Show", "[", RowBox[{"Paraboloid", ",", "Plane", ",", "Curve", ",", RowBox[{"BoxRatios", "\[Rule]", "1"}]}], "]"}], ",", "Curve"}], "}"}], ",", RowBox[{"ImageSize", "\[Rule]", "1000"}]}], "]"}]}], "Input", CellChangeTimes->{{3.411598822015625*^9, 3.411598889578125*^9}, { 3.47017133359375*^9, 3.470171338578125*^9}, {3.47017155015625*^9, 3.470171550625*^9}, {3.47017161790625*^9, 3.470171819609375*^9}, { 3.470171904203125*^9, 3.470171982109375*^9}, {3.470172162328125*^9, 3.470172210453125*^9}, {3.470172257109375*^9, 3.47017229584375*^9}, { 3.4701723589375*^9, 3.47017238059375*^9}, {3.47017242009375*^9, 3.470172481890625*^9}, {3.470172567953125*^9, 3.470172570875*^9}, { 3.470172675125*^9, 3.4701726823125*^9}, {3.470172714515625*^9, 3.4701727498125*^9}, {3.470172792875*^9, 3.470172811984375*^9}, { 3.470172851984375*^9, 3.4701728656875*^9}, {3.470172903484375*^9, 3.470172904234375*^9}, {3.470172941421875*^9, 3.470172950453125*^9}, { 3.470172989875*^9, 3.47017301909375*^9}, {3.470173085515625*^9, 3.470173176609375*^9}}], Cell[BoxData[{"\"\< #47 c) \>\"", "\[IndentingNewLine]", RowBox[{"GraphicsRow", "[", RowBox[{ RowBox[{"{", "\[IndentingNewLine]", RowBox[{ RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Sin", "[", "t", "]"}], ",", RowBox[{"Cos", "[", "t", "]"}], ",", RowBox[{"Sin", "[", RowBox[{"2", " ", "t"}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"2", " ", "\[Pi]"}]}], "}"}], ",", " ", RowBox[{"PlotStyle", "\[Rule]", RowBox[{"{", RowBox[{"Cyan", ",", "Thick"}], "}"}]}]}], "]"}], ",", RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Sin", "[", "t", "]"}], ",", RowBox[{"Cos", "[", "t", "]"}], ",", RowBox[{"Sin", "[", RowBox[{"2", " ", "t"}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"2", " ", "\[Pi]"}]}], "}"}], ",", " ", RowBox[{"PlotStyle", "\[Rule]", RowBox[{"{", RowBox[{"Cyan", ",", "Thick", ",", RowBox[{"Tube", "[", "0.05", "]"}]}], "}"}]}]}], "]"}]}], "}"}], ",", RowBox[{"ImageSize", "\[Rule]", "1000"}]}], "]"}]}], "Input", CellChangeTimes->{{3.411598822015625*^9, 3.411598889578125*^9}, { 3.47017133359375*^9, 3.470171338578125*^9}, {3.47017155015625*^9, 3.470171550625*^9}, {3.47017161790625*^9, 3.470171819609375*^9}, { 3.470171904203125*^9, 3.470171982109375*^9}, {3.470172162328125*^9, 3.47017216640625*^9}, {3.470172637*^9, 3.470172641140625*^9}, { 3.470173221359375*^9, 3.470173303421875*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"p", "=", "2"}], ";", RowBox[{"q", "=", "3"}], ";"}], "\[IndentingNewLine]", RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"2", "+", RowBox[{"Cos", "[", RowBox[{"2", " ", "q", " ", "\[Pi]", " ", "t"}], "]"}]}], ")"}], " ", RowBox[{"Cos", "[", RowBox[{"2", " ", "p", " ", "\[Pi]", " ", "t"}], "]"}]}], ",", RowBox[{ RowBox[{"(", RowBox[{"2", "+", RowBox[{"Cos", "[", RowBox[{"2", " ", "q", " ", "\[Pi]", " ", "t"}], "]"}]}], ")"}], " ", RowBox[{"Sin", "[", RowBox[{"2", " ", "p", " ", "\[Pi]", " ", "t"}], "]"}]}], ",", RowBox[{"2", " ", RowBox[{"Sin", "[", RowBox[{"2", " ", "q", " ", "\[Pi]", " ", "t"}], "]"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "1"}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"ImageSize", "\[Rule]", RowBox[{"{", RowBox[{"1000", ",", "600"}], "}"}]}], ",", RowBox[{"PlotStyle", "\[Rule]", RowBox[{"{", RowBox[{"Cyan", ",", "Thick", ",", RowBox[{"Tube", "[", "0.12", "]"}]}], "}"}]}]}], "]"}]}], "Input", CellChangeTimes->{{3.411598822015625*^9, 3.411598889578125*^9}, { 3.47017133359375*^9, 3.470171338578125*^9}, {3.47017155015625*^9, 3.470171550625*^9}, {3.47017161790625*^9, 3.470171819609375*^9}, { 3.470171904203125*^9, 3.470171982109375*^9}, {3.470172162328125*^9, 3.47017216640625*^9}, {3.470172637*^9, 3.470172641140625*^9}}] }, WindowSize->{1272, 922}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, FrontEndVersion->"7.0 for Microsoft Windows (32-bit) (February 18, 2009)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[545, 20, 1649, 46, 72, "Input"], Cell[2197, 68, 3635, 98, 160, "Input"], Cell[5835, 168, 1704, 41, 92, "Input"], Cell[7542, 211, 1561, 38, 72, "Input"] } ] *) (* End of internal cache information *)