(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 9265, 272] NotebookOptionsPosition[ 9008, 259] NotebookOutlinePosition[ 9353, 274] CellTagsIndexPosition[ 9310, 271] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[BoxData[{"#16", "\n", RowBox[{"\"\\"", ";"}], "\n", RowBox[{"\"\\"", ";", RowBox[{ RowBox[{"F", "[", RowBox[{"{", RowBox[{"x_", ",", "y_", ",", "z_"}], "}"}], "]"}], ":=", RowBox[{"{", RowBox[{"x", ",", "y", ",", "z"}], "}"}]}], ";"}], "\n", RowBox[{"\"\\"", ";", RowBox[{ RowBox[{"Surf", "[", RowBox[{"u_", ",", "v_"}], "]"}], ":=", RowBox[{"{", RowBox[{ RowBox[{"2", " ", RowBox[{"Cos", "[", "v", "]"}], " ", RowBox[{"Sin", "[", "u", "]"}]}], ",", RowBox[{"2", " ", RowBox[{"Sin", "[", "v", "]"}], " ", RowBox[{"Sin", "[", "u", "]"}]}], ",", RowBox[{"2", " ", RowBox[{"Cos", "[", "u", "]"}]}]}], "}"}]}], ";"}], "\n", RowBox[{"\"\\"", ";", RowBox[{"Au", "=", "0"}], ";", RowBox[{"Bu", "=", FractionBox["\[Pi]", "3"]}], ";", RowBox[{"Av", "=", "0"}], ";", RowBox[{"Bv", "=", RowBox[{"2", " ", "\[Pi]"}]}], ";", "\"\\"", ";", RowBox[{"Ornt", "=", "1"}], ";"}], "\n", RowBox[{"\"\\"", ";", RowBox[{"mu", "=", "15"}], ";", RowBox[{"mv", "=", "15"}], ";", RowBox[{"Su", "=", FractionBox[ RowBox[{"Bu", "-", "Au"}], "mu"]}], ";", RowBox[{"Sv", "=", FractionBox[ RowBox[{"Bv", "-", "Av"}], "mv"]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"Nr", "[", RowBox[{"u_", ",", "v_"}], "]"}], "=", RowBox[{"Ornt", " ", RowBox[{"Normalize", "[", RowBox[{ RowBox[{ SubscriptBox["\[PartialD]", "u"], RowBox[{"Surf", "[", RowBox[{"u", ",", "v"}], "]"}]}], "\[Cross]", RowBox[{ SubscriptBox["\[PartialD]", "v"], RowBox[{"Surf", "[", RowBox[{"u", ",", "v"}], "]"}]}]}], "]"}]}]}], ";"}], "\n", RowBox[{ RowBox[{"Integrand", "=", RowBox[{"Simplify", "[", RowBox[{ RowBox[{ RowBox[{"F", "[", RowBox[{"Surf", "[", RowBox[{"u", ",", "v"}], "]"}], "]"}], ".", RowBox[{ RowBox[{ SubscriptBox["\[PartialD]", "u"], RowBox[{"Surf", "[", RowBox[{"u", ",", "v"}], "]"}]}], "\[Cross]", RowBox[{ SubscriptBox["\[PartialD]", "v"], RowBox[{"Surf", "[", RowBox[{"u", ",", "v"}], "]"}]}]}]}], ",", RowBox[{"Assumptions", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"Av", "\[LessEqual]", "v", "\[LessEqual]", "Bv"}], "&&", RowBox[{"Au", "\[LessEqual]", "u", "\[LessEqual]", "Bu"}]}], "}"}]}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"FI", "=", RowBox[{"Ornt", " ", RowBox[{ SubsuperscriptBox["\[Integral]", "Au", "Bu"], RowBox[{ SubsuperscriptBox["\[Integral]", "Av", "Bv"], RowBox[{ RowBox[{ RowBox[{"F", "[", RowBox[{"Surf", "[", RowBox[{"u", ",", "v"}], "]"}], "]"}], ".", RowBox[{ RowBox[{ SubscriptBox["\[PartialD]", "u"], RowBox[{"Surf", "[", RowBox[{"u", ",", "v"}], "]"}]}], "\[Cross]", RowBox[{ SubscriptBox["\[PartialD]", "v"], RowBox[{"Surf", "[", RowBox[{"u", ",", "v"}], "]"}]}]}]}], RowBox[{"\[DifferentialD]", "v"}], RowBox[{"\[DifferentialD]", "u"}]}]}]}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"SDC", "=", RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"Surf", "[", RowBox[{"u", ",", "v"}], "]"}], ",", RowBox[{"{", RowBox[{"u", ",", "Au", ",", "Bu"}], "}"}], ",", RowBox[{"{", RowBox[{"v", ",", "Av", ",", "Bv"}], "}"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"P1", "=", RowBox[{"Flatten", "[", RowBox[{ RowBox[{"Table", "[", " ", RowBox[{ RowBox[{"Graphics3D", "[", RowBox[{"{", RowBox[{"Blue", ",", RowBox[{"Arrowheads", "\[Rule]", "Small"}], ",", " ", RowBox[{"Arrow", "[", RowBox[{"{", " ", RowBox[{ RowBox[{"Surf", "[", RowBox[{ RowBox[{"Au", "+", RowBox[{"Su", " ", "i"}]}], ",", RowBox[{"Av", "+", RowBox[{"Sv", " ", "j"}]}]}], "]"}], ",", RowBox[{ RowBox[{"Surf", "[", RowBox[{ RowBox[{"Au", "+", RowBox[{"Su", " ", "i"}]}], ",", RowBox[{"Av", "+", RowBox[{"Sv", " ", "j"}]}]}], "]"}], "+", RowBox[{"Nr", "[", RowBox[{ RowBox[{"Au", "+", RowBox[{"Su", " ", "i"}]}], ",", RowBox[{"Av", "+", RowBox[{"Sv", " ", "j"}]}]}], "]"}]}]}], "}"}], "]"}]}], " ", "}"}], " ", "]"}], ",", RowBox[{"{", RowBox[{"i", ",", "mu"}], "}"}], ",", RowBox[{"{", RowBox[{"j", ",", "mv"}], "}"}]}], "]"}], ",", "2"}], "]"}]}], ";"}], "\n", RowBox[{ RowBox[{ RowBox[{"P2", "=", RowBox[{"Flatten", "[", RowBox[{ RowBox[{"Table", "[", " ", RowBox[{ RowBox[{"Graphics3D", "[", RowBox[{"{", RowBox[{"Red", ",", " ", RowBox[{"Arrowheads", "\[Rule]", "Small"}], ",", " ", RowBox[{"Arrow", "[", RowBox[{"{", " ", RowBox[{ RowBox[{"Surf", "[", RowBox[{ RowBox[{"Au", "+", RowBox[{"Su", " ", "i"}]}], ",", RowBox[{"Av", "+", RowBox[{"Sv", " ", "j"}]}]}], "]"}], ",", RowBox[{ RowBox[{"Surf", "[", RowBox[{ RowBox[{"Au", "+", RowBox[{"Su", " ", "i"}]}], ",", RowBox[{"Av", "+", RowBox[{"Sv", " ", "j"}]}]}], "]"}], "+", RowBox[{"F", "[", RowBox[{"Surf", "[", RowBox[{ RowBox[{"Au", "+", RowBox[{"Su", " ", "i"}]}], ",", RowBox[{"Av", "+", RowBox[{"Sv", " ", "j"}]}]}], "]"}], "]"}]}]}], "}"}], "]"}]}], " ", "}"}], " ", "]"}], ",", RowBox[{"{", RowBox[{"i", ",", "mu"}], "}"}], ",", RowBox[{"{", RowBox[{"j", ",", "mv"}], "}"}]}], "]"}], ",", "2"}], "]"}]}], ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Print", "[", RowBox[{ RowBox[{"Style", "[", RowBox[{"\"\\"", ",", "Bold", ",", "16"}], "]"}], ",", " ", RowBox[{"Style", "[", RowBox[{"Integrand", ",", "Bold", ",", "16"}], "]"}], ",", "\[IndentingNewLine]", " ", RowBox[{"Style", "[", RowBox[{ "\"\< Flux=\[Integral]\[Integral]F\[CenterDot]ndS=\>\"", ",", "Bold", ",", "16"}], "]"}], ",", " ", RowBox[{"Style", "[", RowBox[{"FI", ",", "Bold", ",", "16"}], "]"}]}], "]"}], ";"}], "\n", RowBox[{ RowBox[{"Print", "[", RowBox[{"Style", "[", RowBox[{"\"\<\!\(\* StyleBox[\"Unit\", FontColor->RGBColor[1, 0, 0]]\)\!\(\* StyleBox[\" \", FontColor->RGBColor[1, 0, 0]]\)\!\(\* StyleBox[\"Normal\", FontColor->RGBColor[1, 0, 0]]\) \!\(\* StyleBox[\"Vector\", FontColor->RGBColor[0, 0, 1]]\)\!\(\* StyleBox[\" \", FontColor->RGBColor[0, 0, 1]]\)\!\(\* StyleBox[\"Field\", FontColor->RGBColor[0, 0, 1]]\)\>\"", ",", "Bold", ",", "16"}], "]"}], "]"}], ";"}], "\n", RowBox[{"Show", "[", RowBox[{"P1", ",", "P2", ",", "SDC", ",", RowBox[{"ImageSize", "\[Rule]", RowBox[{"{", RowBox[{"500", ",", "500"}], "}"}]}]}], "]"}]}], "Input", CellChangeTimes->{{3.411857277390625*^9, 3.4118573029375*^9}, { 3.411857341234375*^9, 3.411857350375*^9}, {3.411857406921875*^9, 3.4118574106875*^9}, {3.411857613296875*^9, 3.411857629359375*^9}, { 3.4118576796875*^9, 3.411857790515625*^9}, {3.4723528277131233`*^9, 3.4723529155256233`*^9}, {3.4723529455881233`*^9, 3.4723530709943733`*^9}, { 3.4723531029006233`*^9, 3.4723531183693733`*^9}, {3.4723531491506233`*^9, 3.4723531496662483`*^9}, {3.4723532761506233`*^9, 3.4723533144474983`*^9}, { 3.4723533458224983`*^9, 3.4723533782131233`*^9}, {3.4723534111193733`*^9, 3.4723534689162483`*^9}, {3.4723535287912483`*^9, 3.4723535761818733`*^9}, { 3.4723536232131233`*^9, 3.4723536674943733`*^9}}] }, WindowSize->{1272, 922}, WindowMargins->{{25, Automatic}, {Automatic, 9}}, 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, 8459, 237, 401, "Input"] } ] *) (* End of internal cache information *)