(* Content-type: application/vnd.wolfram.cdf.text *) (*** Wolfram CDF File ***) (* http://www.wolfram.com/cdf *) (* CreatedBy='Mathematica 11.0' *) (*************************************************************************) (* *) (* The Mathematica License under which this file was created prohibits *) (* restricting third parties in receipt of this file from republishing *) (* or redistributing it by any means, including but not limited to *) (* rights management or terms of use, without the express consent of *) (* Wolfram Research, Inc. For additional information concerning CDF *) (* licensing and redistribution see: *) (* *) (* www.wolfram.com/cdf/adopting-cdf/licensing-options.html *) (* *) (*************************************************************************) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 1064, 20] NotebookDataLength[ 598329, 11066] NotebookOptionsPosition[ 593441, 10906] NotebookOutlinePosition[ 596047, 10985] CellTagsIndexPosition[ 595442, 10967] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell["Potential Flow Over an Airfoil", "Text", CellChangeTimes->{{3.6276464268948593`*^9, 3.627646468243655*^9}, { 3.680788806485434*^9, 3.680788817947734*^9}}, TextAlignment->Right], Cell[CellGroupData[{ Cell["1 Review of Vector Fields", "Title", CellChangeTimes->{{3.585302745672161*^9, 3.5853027546686754`*^9}, { 3.5853030932160397`*^9, 3.5853030962252116`*^9}, {3.5939638853139668`*^9, 3.5939638857049894`*^9}, 3.616837368743743*^9, 3.682256098489386*^9}, CellTags->"review vector fields"], Cell["\<\ Richard L. Fearn (rlf@ufl.edu) Department of Mechanical and Aerospace Engineering University of Florida January 15, 2017\ \>", "Text", CellChangeTimes->{{3.596196010741573*^9, 3.5961960186030226`*^9}, { 3.596281573065505*^9, 3.5962815739135537`*^9}, {3.596447151691945*^9, 3.596447155900186*^9}, {3.5980023721244545`*^9, 3.598002392541622*^9}, { 3.598197345221053*^9, 3.5981973459130926`*^9}, {3.5981974064065523`*^9, 3.5981974339781294`*^9}, {3.5981974853800693`*^9, 3.598197541846299*^9}, { 3.5982624337007723`*^9, 3.5982624545329638`*^9}, {3.59860761717274*^9, 3.598607618310805*^9}, {3.5987858514906406`*^9, 3.5987858521876802`*^9}, { 3.609067640735259*^9, 3.609067644388441*^9}, {3.616837122425612*^9, 3.616837127471208*^9}, {3.623665571101694*^9, 3.6236655994494457`*^9}, { 3.6276633879142523`*^9, 3.627663399808323*^9}, {3.680012512431391*^9, 3.680012532028631*^9}, 3.680013846534143*^9, {3.6801015967866907`*^9, 3.6801015975682364`*^9}, {3.680699726907361*^9, 3.6806997285938177`*^9}, { 3.6812260998994617`*^9, 3.681226100521598*^9}, {3.682256133750959*^9, 3.68225614179663*^9}, {3.68225675988122*^9, 3.682256761807423*^9}, 3.682257599390752*^9, {3.686308974715517*^9, 3.686308979832296*^9}, { 3.686402662063026*^9, 3.6864026812361298`*^9}, {3.686668246572433*^9, 3.6866682475216017`*^9}, {3.686748356289061*^9, 3.6867483575820827`*^9}, { 3.686830285037448*^9, 3.686830285986972*^9}, {3.686840432639433*^9, 3.686840435126247*^9}, {3.687445846867556*^9, 3.687445861271638*^9}, { 3.68865546861644*^9, 3.688655472604047*^9}, {3.6925418495793343`*^9, 3.6925418668724422`*^9}, {3.693328701878461*^9, 3.693328702547462*^9}}], Cell["", "Text", CellChangeTimes->{{3.68070035728701*^9, 3.680700359527088*^9}, 3.6812261065697603`*^9}], Cell[TextData[ButtonBox["Open Table of Contents", BaseStyle->"Hyperlink", ButtonData->{"TableOfContents.cdf", "table of contents"}]], "Text", CellChangeTimes->{{3.686318804142681*^9, 3.6863188197167387`*^9}, { 3.686318889367898*^9, 3.686318889370879*^9}}, TextAlignment->Right], Cell[CellGroupData[{ Cell["1.1 Introduction", "Section", CellChangeTimes->{{3.5853876378066697`*^9, 3.585387642924962*^9}, { 3.5939638882471347`*^9, 3.5939638887061605`*^9}, 3.616837368829814*^9}, CellTags->"introduction"], Cell[TextData[{ "Many quantities of interest in physics and engineering can be represented \ as vector fields. For example, a force field or a velocity field. Consider a \ vector field represented by ", Cell[BoxData[ FormBox[ RowBox[{ OverscriptBox["V", "\[RightVector]"], "(", OverscriptBox["r", "\[RightVector]"], ")"}], TraditionalForm]]], " in Cartesian coordinates, where ", Cell[BoxData[ FormBox[ RowBox[{ OverscriptBox["V", "\[RightVector]"], "=", RowBox[{"(", RowBox[{ SubscriptBox["v", "x"], ",", SubscriptBox["v", "y"], ",", SubscriptBox["v", "z"]}], ")"}]}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{ OverscriptBox["r", "\[RightVector]"], "=", RowBox[{"(", RowBox[{"x", ",", "y", ",", "z"}], ")"}]}], TraditionalForm]]], ". This vector field can be visualized by plotting a collection of arrows, \ each arrow having components ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ SubscriptBox["v", "x"], ",", SubscriptBox["v", "y"], ",", SubscriptBox["v", "z"]}], ")"}], TraditionalForm]]], " with the tail of the vector at the location ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"x", ",", "y", ",", "z"}], ")"}], TraditionalForm]]], ". For physical problems of interest, vector fields are usually three \ dimensional and complicated, making visualization a challenge." }], "Text", CellChangeTimes->{{3.585387670911563*^9, 3.5853877161911535`*^9}, { 3.585387841110298*^9, 3.5853879817583427`*^9}, {3.585388012515102*^9, 3.58538821826087*^9}, {3.5853882545019426`*^9, 3.5853885120506735`*^9}, { 3.585388576305349*^9, 3.5853886431081696`*^9}, {3.585388694710121*^9, 3.5853887434779105`*^9}, {3.5855579921987143`*^9, 3.5855580826838903`*^9}, 3.5855581248593025`*^9, {3.585558157878191*^9, 3.5855582115072584`*^9}, { 3.585558250796506*^9, 3.5855582723737392`*^9}, {3.5855583085768104`*^9, 3.5855583241767025`*^9}, {3.5855583600937567`*^9, 3.585558378166791*^9}, 3.58555954833072*^9, {3.585559604401927*^9, 3.5855596268082085`*^9}, { 3.5855761395161953`*^9, 3.585576139981222*^9}, {3.586077799496566*^9, 3.5860778213218145`*^9}, {3.586079211618335*^9, 3.586079507205241*^9}, { 3.5861624755261545`*^9, 3.5861624772092505`*^9}, {3.586162516868519*^9, 3.5861625244099503`*^9}, {3.586162605900611*^9, 3.5861626378794403`*^9}, { 3.587199573968856*^9, 3.5871996310601215`*^9}, {3.5871996701553574`*^9, 3.587199707405488*^9}, {3.5873734124869585`*^9, 3.5873734133970103`*^9}, { 3.587373467654114*^9, 3.5873734713423243`*^9}, 3.587375686562028*^9, 3.587377491354256*^9, {3.593962426008499*^9, 3.593962451302946*^9}, { 3.616837368841547*^9, 3.616837368854089*^9}, {3.6277267491056213`*^9, 3.62772675059156*^9}, {3.6277268362803802`*^9, 3.6277268461519403`*^9}, { 3.627726880917091*^9, 3.627726890022416*^9}, {3.627726933649589*^9, 3.627727027626017*^9}, {3.6807004873307457`*^9, 3.680700519704932*^9}, { 3.680700561159114*^9, 3.680700567982828*^9}, {3.6807006029333477`*^9, 3.680700625436623*^9}, {3.6874458900707808`*^9, 3.687445892645969*^9}, { 3.692541892960525*^9, 3.692541922612604*^9}}], Cell[TextData[{ StyleBox["The purpose of this brief review is to remind you of some \ properties of vector fields used in this collection of documents, and to \ introduce some basic graphical tools of Mathematica.", FontWeight->"Bold"], " If you don\[CloseCurlyQuote]t need this review skip it. Links to online \ resources are used to augment information presented. If you need a \ comprehensive textbook review, consider Chapter 2 of Principles of \ Ideal-Fluid Aerodynamics by ", ButtonBox["Karamcheti", BaseStyle->"Hyperlink", ButtonData->"karamcheti"], "." }], "Text", CellChangeTimes->{{3.609070201547068*^9, 3.609070513191552*^9}, { 3.609070553779406*^9, 3.6090705958903847`*^9}, {3.6090706988041687`*^9, 3.6090709220596523`*^9}, {3.609075595653989*^9, 3.609075697825884*^9}, { 3.609076628617579*^9, 3.6090766522657413`*^9}, 3.609077033179454*^9, { 3.609077248297659*^9, 3.609077271396769*^9}, {3.616837368865819*^9, 3.616837368902309*^9}, {3.6168375328101397`*^9, 3.616837534055697*^9}, { 3.6277283381852293`*^9, 3.627728365587421*^9}, 3.680012584531362*^9, { 3.680012618121463*^9, 3.680012648272863*^9}, {3.680700678354682*^9, 3.680700723744689*^9}, {3.68631924696465*^9, 3.6863192477476797`*^9}, { 3.686748458210726*^9, 3.6867484622025623`*^9}, 3.68744596152195*^9, 3.692542007942827*^9, {3.692957989572414*^9, 3.692957996600882*^9}}, CellTags->"introduction: karamcheti"] }, Open ]], Cell[CellGroupData[{ Cell["1.2 Visualizing Vector Fields with Mathematica", "Section", CellChangeTimes->{{3.5855585623073225`*^9, 3.5855585872457495`*^9}, { 3.5939638907632785`*^9, 3.5939638909722905`*^9}, 3.6168373689139147`*^9}, CellTags->"visualizing"], Cell["\<\ Mathematica provides several graphics functions to plot vector fields. The \ following examples are taken from the documentation for Mathematica. Many \ options for each plotting function enable the user to customize the graphic \ presentation.\ \>", "Text", CellChangeTimes->{{3.585390247169917*^9, 3.5853903154078197`*^9}, 3.585558501322835*^9, {3.5855587787447023`*^9, 3.5855588125096335`*^9}, { 3.5855592664145956`*^9, 3.585559270562833*^9}, {3.585559432250081*^9, 3.585559477816687*^9}, {3.5860781016038456`*^9, 3.586078119410864*^9}, 3.616837368939959*^9, {3.6236657255564947`*^9, 3.623665727385976*^9}, { 3.687445991703829*^9, 3.6874460168142633`*^9}}], Cell[TextData[{ "The functions ", StyleBox[ButtonBox["VectorPlot", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/mathematica/ref/VectorPlot.html?q=\ VectorPlot&lang=en"], None}, ButtonNote-> "http://reference.wolfram.com/mathematica/ref/VectorPlot.html?q=VectorPlot&\ lang=en"], "Input"], ", ", StyleBox[ButtonBox["VectorPlot3D", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/mathematica/ref/VectorPlot3D.html?q=\ VectorPlot3D&lang=en"], None}, ButtonNote-> "http://reference.wolfram.com/mathematica/ref/VectorPlot3D.html?q=\ VectorPlot3D&lang=en"], "Input"], ", ", StyleBox[ButtonBox["ListVectorPlot", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/mathematica/ref/ListVectorPlot.html?q=\ ListVectorPlot&lang=en"], None}, ButtonNote-> "http://reference.wolfram.com/mathematica/ref/ListVectorPlot.html?q=\ ListVectorPlot&lang=en"], "Input"], " and ", StyleBox[ButtonBox["ListVectorPlot3D", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/mathematica/ref/ListVectorPlot3D.html?q=\ ListVectorPlot3D&lang=en"], None}, ButtonNote-> "http://reference.wolfram.com/mathematica/ref/ListVectorPlot3D.html?q=\ ListVectorPlot3D&lang=en"], "Input"], " provide basic graphs for plotting two- and three-dimensional vector \ fields. The functions with the prefix List are used with collections of data \ points." }], "Text", CellChangeTimes->{{3.5855596820123663`*^9, 3.58555972019255*^9}, { 3.5855598089006243`*^9, 3.5855598089036245`*^9}, {3.5855598882681637`*^9, 3.5855598882701635`*^9}, {3.585559952221822*^9, 3.585559955487008*^9}, { 3.585559993602188*^9, 3.5855599936041884`*^9}, {3.585560035824603*^9, 3.585560231117773*^9}, {3.5855603015798035`*^9, 3.5855603017878156`*^9}, { 3.5855605669809837`*^9, 3.5855607008986435`*^9}, {3.5855608110209417`*^9, 3.5855608665561185`*^9}, {3.58556092304935*^9, 3.5855610729729247`*^9}, { 3.5855611122211695`*^9, 3.585561113773258*^9}, {3.5855771639017873`*^9, 3.585577215558742*^9}, {3.587199761368574*^9, 3.5871997905322423`*^9}, { 3.587199940255806*^9, 3.587199940258806*^9}, {3.587200015838129*^9, 3.5872000313240147`*^9}, 3.616837368952504*^9, {3.686319357080945*^9, 3.686319358086915*^9}, {3.6867485614806833`*^9, 3.6867486247723627`*^9}}], Cell[TextData[{ "Two examples are shown below. On the left is a two-dimensional vector plot \ using functions to specify the components. 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Arrow3DBox[{{-0.9584448411081425, -0.8780390827760998, \ -0.12402652113707142`}, {-1.1844123017490003`, -0.8362466315096146, \ -0.1616877645772144}}]}, { Arrowheads[{{0.01853379502568317, 1.}}, Appearance -> "Projected"], Arrow3DBox[{{-0.9584448411081425, -0.8780390827760998, 0.12402652113707133`}, {-1.1844123017490003`, -0.8362466315096146, 0.16168776457721426`}}]}, { Arrowheads[{{0.02038087179863818, 1.}}, Appearance -> "Projected"], Arrow3DBox[{{-0.9584448411081425, -0.8780390827760998, 0.37207956341121406`}, {-1.1844123017490003`, -0.8362466315096146, 0.4850632937316429}}]}, { Arrowheads[{{0.023646071337740198`, 1.}}, Appearance -> "Projected"], Arrow3DBox[{{-0.9584448411081425, -0.8780390827760998, 0.6201326056853568}, {-1.1844123017490003`, -0.8362466315096146, 0.8084388228860715}}]}, { Arrowheads[{{0.0278347233031866, 1.}}, Appearance -> "Projected"], Arrow3DBox[{{-0.9584448411081425, -0.8780390827760998, 0.8681856479594996}, {-1.1844123017490003`, -0.8362466315096146, 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Arrow3DBox[{{-1.1090898148687143`, 0.264818060081043, 0.37207956341121406`}, {-1.0337673279884285`, 0.3066105113475282, 0.4850632937316429}}]}, { Arrowheads[{{0.016480994534176365`, 1.}}, Appearance -> "Projected"], Arrow3DBox[{{-1.1090898148687143`, 0.264818060081043, 0.6201326056853568}, {-1.0337673279884285`, 0.30661051134752815`, 0.8084388228860715}}]}, { Arrowheads[{{0.022073928342979904`, 1.}}, Appearance -> "Projected"], Arrow3DBox[{{-1.1090898148687143`, 0.264818060081043, 0.8681856479594996}, {-1.0337673279884285`, 0.30661051134752815`, 1.1318143520405002`}}]}, { Arrowheads[{{0.02783472330318659, 1.}}, Appearance -> "Projected"], Arrow3DBox[{{-1.1844123017490003`, 0.8362466315096144, -0.8681856479594997}, {-0.9584448411081425, 0.8780390827760995, -1.1318143520405002`}}]}, { Arrowheads[{{0.023646071337740198`, 1.}}, Appearance -> "Projected"], Arrow3DBox[{{-1.1844123017490003`, 0.8362466315096144, -0.6201326056853569}, {-0.9584448411081425, 0.8780390827760995, 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Arrowheads[{{0.030307236360346896`, 1.}}, Appearance -> "Projected"], Arrow3DBox[{{-1.2597347886292862`, 1.4076752029381858`, -0.12402652113707142`}, {-0.8831223542278567, 1.449467654204671, -0.16168776457721437`}}]}, { Arrowheads[{{0.030307236360346896`, 1.}}, Appearance -> "Projected"], Arrow3DBox[{{-1.2597347886292862`, 1.4076752029381858`, 0.12402652113707133`}, {-0.8831223542278567, 1.449467654204671, 0.16168776457721426`}}]}, { Arrowheads[{{0.03147073168867257, 1.}}, Appearance -> "Projected"], Arrow3DBox[{{-1.2597347886292862`, 1.4076752029381858`, 0.37207956341121406`}, {-0.8831223542278567, 1.449467654204671, 0.4850632937316429}}]}, { Arrowheads[{{0.03367734709648895, 1.}}, Appearance -> "Projected"], Arrow3DBox[{{-1.2597347886292862`, 1.4076752029381858`, 0.6201326056853569}, {-0.8831223542278567, 1.449467654204671, 0.8084388228860715}}]}, { Arrowheads[{{0.03673960858682158, 1.}}, Appearance -> "Projected"], Arrow3DBox[{{-1.2597347886292862`, 1.4076752029381858`, 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Appearance -> "Projected"], Arrow3DBox[{{-1.335057275509572, 1.9791037743667572`, 0.12402652113707133`}, {-0.8077998673475708, 2.0208962256332423`, 0.16168776457721426`}}]}, { Arrowheads[{{0.04304587645341027, 1.}}, Appearance -> "Projected"], Arrow3DBox[{{-1.335057275509572, 1.9791037743667572`, 0.37207956341121406`}, {-0.8077998673475708, 2.0208962256332423`, 0.4850632937316429}}]}, { Arrowheads[{{0.04468449657408298, 1.}}, Appearance -> "Projected"], Arrow3DBox[{{-1.335057275509572, 1.9791037743667572`, 0.6201326056853569}, {-0.8077998673475708, 2.0208962256332423`, 0.8084388228860715}}]}, { Arrowheads[{{0.0470355117515978, 1.}}, Appearance -> "Projected"], Arrow3DBox[{{-1.335057275509572, 1.9791037743667572`, 0.8681856479594996}, {-0.8077998673475708, 2.0208962256332423`, 1.1318143520405002`}}]}, { Arrowheads[{{0.047580566072261965`, 1.}}, Appearance -> "Projected"], Arrow3DBox[{{-0.37922843877614215`, -1.9502813941829746`, \ -0.8681856479594997}, {-0.9064858469381434, 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Entering \ \[OpenCurlyDoubleQuote]vector field\[CloseCurlyDoubleQuote] in the search box \ at this web site returns a collection of demonstrations on this topic.", " ", "If you want to explore this resource, a reasonable starting point is ", ButtonBox["Vector Fields: Plot Examples", BaseStyle->"Hyperlink", ButtonData->{ URL["http://demonstrations.wolfram.com/VectorFieldsPlotExamples/"], None}, ButtonNote->"http://demonstrations.wolfram.com/VectorFieldsPlotExamples/"], ", by Gosia Konwerska, a developer at Wolfram Research." }], "Text", CellChangeTimes->{{3.587374170688325*^9, 3.587374340955064*^9}, { 3.587374520167314*^9, 3.5873746139136763`*^9}, {3.587374724735015*^9, 3.587374765523348*^9}, {3.5873748921085877`*^9, 3.5873749754003515`*^9}, { 3.587375015033619*^9, 3.5873750459983897`*^9}, {3.5873751879895115`*^9, 3.587375382277624*^9}, {3.5873754695586157`*^9, 3.587375469561616*^9}, { 3.587375546198*^9, 3.587375546204*^9}, {3.616837369207595*^9, 3.616837369249836*^9}, {3.687446251835142*^9, 3.687446262241189*^9}}], Cell[TextData[{ ButtonBox["Remove", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/language/ref/Remove.html?q=Remove"], None}, ButtonNote-> "http://reference.wolfram.com/language/ref/Remove.html?q=Remove"], " global variables to prevent accidental misuse." }], "Text", CellChangeTimes->{{3.6869232884413013`*^9, 3.686923334166301*^9}, 3.688053896755134*^9}], Cell[BoxData[ RowBox[{"Remove", "[", RowBox[{"g1", ",", "g2", ",", "vectors", ",", "vf"}], "]"}]], "Input", CellChangeTimes->{{3.686579411905676*^9, 3.686579420777915*^9}, { 3.686579451049851*^9, 3.6865794572164288`*^9}, {3.687446297048429*^9, 3.687446299110712*^9}, 3.687446417897434*^9, {3.688053762309214*^9, 3.688053763949211*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["1.3 Calculus of Vector Fields", "Section", CellChangeTimes->{{3.58600174388059*^9, 3.5860017795956326`*^9}, { 3.5860787704301004`*^9, 3.5860787728232374`*^9}, {3.5939638975476665`*^9, 3.5939638979016867`*^9}, 3.6168373692791224`*^9}, CellTags->"calculus"], Cell[CellGroupData[{ Cell["1.3.1 Integrals", "Subsection", CellChangeTimes->{{3.596209383856471*^9, 3.5962093862756095`*^9}, { 3.5962094621729507`*^9, 3.596209464770099*^9}}, CellTags->"calculus: integrals"], Cell[TextData[{ "Line integrals and surface integrals of vector fields occur in many \ disciplines of physics and engineering. ", StyleBox["The work done by a force field and the flux of a velocity field \ are chosen as the exemplars for line integrals and surface integrals. 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Rewrite \ the work integral.\n\t\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"W", "(", RowBox[{ OverscriptBox["V", "\[RightVector]"], ",", "\[ScriptCapitalC]"}], ")"}], "=", RowBox[{ SubscriptBox["\[Integral]", "\[ScriptCapitalC]"], RowBox[{ RowBox[{ OverscriptBox["V", "\[RightVector]"], "\[CenterDot]", OverscriptBox["t", "^"]}], RowBox[{"\[DifferentialD]", "\[ScriptS]"}]}]}]}], TraditionalForm]]], "\nThe curve ", Cell[BoxData[ FormBox["\[ScriptCapitalC]", TraditionalForm]]], " can be open or closed. If the curve is closed and the integral is zero, no \ net work is done in returning the particle to its starting point. 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"The ", ButtonBox["curl", BaseStyle->"Hyperlink", ButtonData->{ URL["http://mathworld.wolfram.com/Curl.html"], None}, ButtonNote->"http://mathworld.wolfram.com/Curl.html"], " of a vector field ", Cell[BoxData[ FormBox[ RowBox[{ OverscriptBox["V", "\[RightVector]"], "(", OverscriptBox["r", "\[RightVector]"], ")"}], TraditionalForm]]], " is itself a vector field. Let the point located by ", Cell[BoxData[ FormBox[ OverscriptBox["r", "\[RightVector]"], TraditionalForm]]], " lie on a small open surface ", Cell[BoxData[ FormBox["\[CapitalDelta]\[ScriptCapitalS]", TraditionalForm]]], " bounded by a closed curve ", Cell[BoxData[ FormBox["\[ScriptCapitalC]", TraditionalForm]]], "; the unit normal to ", Cell[BoxData[ FormBox["\[CapitalDelta]\[ScriptCapitalS]", TraditionalForm]]], " is denoted by ", Cell[BoxData[ FormBox[ OverscriptBox["n", "^"], TraditionalForm]]], ". The component of the vector ", Cell[BoxData[ FormBox[ RowBox[{"curl", " ", OverscriptBox["V", "\[RightVector]"]}], TraditionalForm]]], " in the direction of ", Cell[BoxData[ FormBox[ OverscriptBox["n", "^"], TraditionalForm]]], " is defined as the work integral around ", Cell[BoxData[ FormBox["C", TraditionalForm]]], " per unit area in the limit as ", Cell[BoxData[ FormBox["\[CapitalDelta]\[ScriptCapitalS]", TraditionalForm]]], " becomes vanishingly small.\t\t\n\t\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"curl", " ", OverscriptBox["V", "\[RightVector]"]}], ")"}], "\[CenterDot]", OverscriptBox["n", "^"]}], "=", RowBox[{ UnderscriptBox["lim", RowBox[{"\[CapitalDelta]\[ScriptCapitalS]", "\[RightArrow]", "0"}]], RowBox[{ FractionBox["1", "\[CapitalDelta]\[ScriptCapitalS]"], RowBox[{ SubscriptBox["\[ContourIntegral]", "\[ScriptCapitalC]"], RowBox[{ RowBox[{ OverscriptBox["V", "\[RightVector]"], "\[CenterDot]", OverscriptBox["t", "^"]}], " ", RowBox[{"\[DifferentialD]", "\[ScriptS]"}]}]}]}]}]}], TraditionalForm]]], "\n", StyleBox["If we are restricting a line integral to a closed contour, a more \ intuitive exemplar might be the circulation associated with a velocity field.", FontWeight->"Bold"], " In the terminology fluid mechanics, the closed line integral of the \ velocity field is named the circulation. The curl of the velocity field also \ has a name, vorticity. \n", StyleBox["The curl of a force field can be thought of as work density in the \ same sense that the vorticity of a velocity field can be thought of as the \ circulation density.", FontWeight->"Bold"], " An operational expression for the curl is obtained by specifying three \ small surfaces containing the point of interest with normals in the direction \ of each orthogonal axis, then expanding ", Cell[BoxData[ FormBox[ OverscriptBox["V", "\[RightVector]"], TraditionalForm]]], " in Taylor series to write expressions for each component of the curl. \n\t\ \t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"curl", " ", OverscriptBox["V", "\[RightVector]"]}], "=", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{ SubscriptBox["\[PartialD]", "y"], SubscriptBox["v", "z"]}], "-", RowBox[{ SubscriptBox["\[PartialD]", "z"], SubscriptBox["v", "y"]}]}], ",", RowBox[{ RowBox[{ SubscriptBox["\[PartialD]", "z"], SubscriptBox["v", "x"]}], "-", RowBox[{ 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operator ", ButtonBox["del", BaseStyle->"Hyperlink", ButtonData->{ URL["http://mathworld.wolfram.com/Del.html"], None}, ButtonNote->"http://mathworld.wolfram.com/Del.html"], ", denoted by the symbol ", Cell[BoxData[ FormBox["\[Del]", TraditionalForm]]], ".\n\t\t", Cell[BoxData[ FormBox[ RowBox[{"\[Del]", "=", RowBox[{"(", RowBox[{ SubscriptBox["\[PartialD]", "x"], RowBox[{",", RowBox[{ SubscriptBox["\[PartialD]", "y"], RowBox[{",", SubscriptBox["\[PartialD]", "z"]}]}]}]}], ")"}]}], TraditionalForm]]], "\nWrite the divergence and curl of a vector field using the del operator.\n\ \t\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"div", " ", OverscriptBox["V", "\[RightVector]"]}], "=", RowBox[{"\[Del]", "\[CenterDot]", OverscriptBox["V", "\[RightVector]"]}]}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"curl", " ", OverscriptBox["V", "\[RightVector]"]}], "=", RowBox[{"\[Del]", "\[Cross]", OverscriptBox["V", "\[RightVector]"]}]}], TraditionalForm]]] }], "Text", 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OverscriptBox["V", "\[RightVector]"], "(", OverscriptBox["r", "\[RightVector]"], ")"}], TraditionalForm]]], " lead to some important relations between surface and volume integrals \ known as the divergence and curl theorems." }], "Text", CellGroupingRules->{GroupTogetherGrouping, 10000.}, CellChangeTimes->{{3.586768115668233*^9, 3.586768250164926*^9}, { 3.5867682959765463`*^9, 3.58676843188932*^9}, {3.586768614204748*^9, 3.5867687074040785`*^9}, 3.5949793819104643`*^9}], Cell[TextData[{ "Construct a ", ButtonBox["Riemann integral", BaseStyle->"Hyperlink", ButtonData->{ URL["http://mathworld.wolfram.com/RiemannIntegral.html"], None}, ButtonNote->"http://mathworld.wolfram.com/RiemannIntegral.html"], " for the divergence of a vector field over a finite region ", Cell[BoxData[ FormBox["\[ScriptCapitalR]", TraditionalForm]]], " bounded by the surface ", Cell[BoxData[ FormBox["\[ScriptCapitalS]", TraditionalForm]]], ". \n\t\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["\[Integral]", "\[ScriptCapitalR]"], RowBox[{"div", " ", OverscriptBox["V", "\[RightVector]"], " ", RowBox[{"\[DifferentialD]", "\[Tau]"}]}]}], "=", RowBox[{ UnderscriptBox["lim", RowBox[{ SubscriptBox["\[CapitalDelta]\[Tau]", "k"], "\[RightArrow]", "0"}]], RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"k", "=", "1"}], "n"], RowBox[{ SubscriptBox[ RowBox[{"(", RowBox[{"div", " ", OverscriptBox["V", "\[RightVector]"]}], ")"}], "k"], SubscriptBox["\[CapitalDelta]\[Tau]", "k"]}]}]}]}], TraditionalForm]]], "\nThink of the finite volume ", Cell[BoxData[ FormBox["\[ScriptCapitalR]", TraditionalForm]]], " being constructed of many small volume elements ", Cell[BoxData[ FormBox[ SubscriptBox["\[CapitalDelta]\[Tau]", "k"], TraditionalForm]]], ", each element being bounded by surface ", Cell[BoxData[ FormBox[ SubscriptBox["\[CapitalDelta]\[ScriptCapitalS]", "k"], TraditionalForm]]], ". Use the concept of the divergence of ", Cell[BoxData[ FormBox[ OverscriptBox["V", "\[RightVector]"], TraditionalForm]]], " associated with each element as the flux of ", Cell[BoxData[ FormBox[ OverscriptBox["V", "\[RightVector]"], TraditionalForm]]], " across ", Cell[BoxData[ FormBox[ SubscriptBox["\[CapitalDelta]\[ScriptCapitalS]", "k"], TraditionalForm]]], " per unit volume.\n\t\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["\[Integral]", "\[ScriptCapitalR]"], RowBox[{"div", " ", OverscriptBox["V", "\[RightVector]"], " ", RowBox[{"\[DifferentialD]", "\[Tau]"}]}]}], "=", RowBox[{ UnderscriptBox["lim", RowBox[{ SubscriptBox["\[CapitalDelta]\[Tau]", "k"], "\[Rule]", "0"}]], RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"k", "=", "1"}], "n"], RowBox[{ SubscriptBox["\[ContourIntegral]", SubscriptBox["\[CapitalDelta]\[ScriptCapitalS]", "k"]], RowBox[{ RowBox[{ SubscriptBox[ OverscriptBox["V", "\[RightVector]"], "k"], "\[CenterDot]", SubscriptBox[ OverscriptBox["n", "^"], "k"]}], RowBox[{"\[DifferentialD]", "\[Sigma]"}]}]}]}]}]}], TraditionalForm]]], "\nThe key idea is that adjoining volume elements interior to ", Cell[BoxData[ FormBox["\[ScriptCapitalS]", TraditionalForm]]], " will share surfaces, and that the flux out of one element will be the flux \ into the adjoining element. 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