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Name" -1 35 "Times" 1 12 104 64 92 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Output" -1 20 "Times" 1 12 0 0 255 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Dictionary Hyperlink" -1 45 "Times" 1 12 147 0 15 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "Help Emphasized" -1 22 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Italic Bold" -1 40 "Tim es" 1 12 0 0 0 1 1 1 2 2 2 2 0 0 0 1 }{CSTYLE "LaTeX" -1 32 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Menus" -1 36 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Prompt" -1 1 "Courier" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined" -1 44 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined Italic" -1 43 "Times" 1 12 0 0 0 1 1 2 1 2 2 2 0 0 0 1 }{CSTYLE "_cstyle3" -1 214 " Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Bold" -1 5 " Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic" -1 3 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }} {SECT 0 {PARA 18 "" 0 "" {TEXT 215 24 "Computing Flux Integrals" }} {PARA 19 "" 0 "" {TEXT 216 38 "\2512006 Mike May, S.J.- maymk@slu.edu" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 201 505 "In considering flux integra ls, it seems worthwhile to go through using Maple to set up and evalua te flux integrals over a parameterized surface. We will walk through \+ a step by step procedure, then produce new procedures that do everythi ng in one command. Finally we look at the Maple command that computes the flux integral in a single step. We use the Student[VectorCalculu s] package and set the BasisFormat variable so that fields are written as vectors rather than as linear combinations of a basis." }{TEXT 201 0 "" }}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 57 "restart:with(St udent[VectorCalculus]);BasisFormat(false):" }{MPLTEXT 1 0 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT 217 22 "A step by step example" }{TEXT 217 0 "" }}{PARA 0 "" 0 "" {TEXT 201 302 "First we need to define a ve ctor field that we want to integrate, the parameterized surface we are integrating over, and the limits on the parameters. For the example \+ we will start with the vector field [0, 0, 1]. Then the flux integral will be the area of the shadow of the surface on the x-y plane." } {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "vfield := < 0,0,1>;\n" }{MPLTEXT 1 0 48 "surface := <4*t*cos(s), 2*t*sin(s), t*sin (s)>;\n" }{MPLTEXT 1 0 19 "trange := t=0..1;\n" }{MPLTEXT 1 0 20 "sran ge := s=0..2*Pi;" }{MPLTEXT 1 0 0 "" }}}{PARA 210 "" 0 "" {TEXT 205 62 "Evaluating the flux integral can be broken into a seven steps:" }} {PARA 210 "" 0 "" {TEXT 201 112 "1) Graph the vector field and the pa rameterized surface to see what it looks like. (Not required, but use ful.)" }}{PARA 210 "" 0 "" {TEXT 201 29 "2) Set up the flux integral. " }}{PARA 210 "" 0 "" {TEXT 201 91 "3) Replace x, y, and z in the vec tor field with the expressions in s and t on the surface." }}{PARA 210 "" 0 "" {TEXT 201 96 "4) Find the tangent vectors to the surface obt ained by differentiating with respect to s and t." }}{PARA 210 "" 0 "" {TEXT 201 91 "5) Find a normal vector to the surface by taking the c ross product of the tangent vectors." }}{PARA 210 "" 0 "" {TEXT 201 112 "6) Find the integrand of the flux integral by taking the dot pro duct of the vector field and the normal vector." }}{PARA 210 "" 0 "" {TEXT 201 26 "7) Evaluate the integral." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 201 87 "The first step is t o look at a graph of the vector field and the parameterized surface." }{TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "vfieldplot := VectorField(vfield,output=plot,view=[-5..5,-5..5,-5..5]):\n" } {MPLTEXT 1 0 57 "paramsurf := plot3d(surface,trange, srange,color=blue ):\n" }{MPLTEXT 1 0 54 "plots[display](\{vfieldplot, paramsurf\}, axes =boxed);" }{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 201 50 "Our seco nd step is to set up the flux integral. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Int(Int((vfield.[Diff(surface,t) " }{MPLTEXT 1 0 3 " *X*" }{MPLTEXT 1 0 37 " Diff(surface,s)]), trange), srange);" } {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 201 139 "The third step is \+ to replace x, y, and z in the vector field with the parameterizations \+ of x and y at the appropriate point on the surface." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "paramfield := evalVF(VectorField(vfield),su rface);\n" }{MPLTEXT 1 0 53 "print(`the function on the surface is `, \+ paramfield);" }{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 201 161 "The fourth step is to take the derivative of the parameterized surface wi th respect to the parameters s and t. This gives us two tangent vecto rs to the surface." }{TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "tanvect := diff(surface, t):\n" }{MPLTEXT 1 0 30 "tan vecs := diff(surface, s):\n" }{MPLTEXT 1 0 60 "print(`the t-tangent ve ctor to the surface is `, tanvect);\n" }{MPLTEXT 1 0 58 "print(`the s- tangent vector to the surface is `, tanvecs);" }{MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT 201 92 "The fifth step is to take the cross prod uct of those tangent vectors to get a normal vector." }{TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "normvector := (tanvect &x \+ tanvecs);" }{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 201 200 "The si xth step is to evaluate the dot product in the integrand and simplify. At this point we have reduced the flux integral over a surface to an ordinary iterated integral over a section of a plane." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "integrand := simplify(paramfield. n ormvector):\n" }{MPLTEXT 1 0 38 "print(`The integrand is `, integrand) ;" }{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 201 84 "Finally we can \+ evaluate the integral. We have Maple evaluate the integral in steps." }{TEXT 201 0 "" }}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 36 "Int(Int (integrand, trange), srange);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 36 "Int(int(integrand, trange), srange);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 36 "int(int(integrand, trang e), srange);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 217 9 "Exercise:" }}{PARA 0 "" 0 "" {TEXT 201 75 "1) Let F be the field [x, y, z]. Let S be the surface z = x^2 + y^2 with " }{XPPEDIT 2 0 "Typesetting:-mro w(Typesetting:-mi(\"z\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"≤\", mathvariant = \"normal\", fence = \"false \", separator = \"false\", stretchy = \"false\", symmetric = \"false\" , largeop = \"false\", movablelimits = \"false\", accent = \"false\", \+ form = \"infix\", lspace = \"thickmathspace\", rspace = \"thickmathspa ce\", minsize = \"1\", maxsize = \"infinity\"), Typesetting:-mn(\"4\", mathvariant = \"normal\"));" "-I%mrowG6#/I+modulenameG6\"I,Typesettin gGI(_syslibGF'6%-I#miGF$6%Q\"zF'/%'italicGQ%trueF'/%,mathvariantGQ'ita licF'-I#moGF$60Q&≤F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF= /%)stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelimitsGF=/%'accent GF=/%%formGQ&infixF'/%'lspaceGQ/thickmathspaceF'/%'rspaceGFO/%(minsize GQ\"1F'/%(maxsizeGQ)infinityF'-I#mnGF$6$Q\"4F'F9" }{TEXT 201 183 ". ( You probably want to parameterize the surface in terms of r and theta. ) Compute the flux integral of F through S by hand. Then use Maple t o check your calculations at each step." }{TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 217 21 "An automated approach" }{TEXT 217 0 "" }}{PARA 0 "" 0 "" {TEXT 201 241 "For convenience we block the code for those seven steps into two procedures we can use, one for plotting, and one for setting up t he integral and evaluating. We also set up a black box procedure that evaluates without showing all the steps." }{TEXT 201 0 "" }}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 52 "surfaceplot := proc(vecfield, pat h, trange, srange, " }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 30 " viewwindow)" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 31 " local vfieldplot, paramsurf;" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 29 " vfieldplot := VectorField(" }{MPLTEXT 1 0 8 "vecfield" }{MPLTEXT 1 0 18 ",output =plot,view=" }{MPLTEXT 1 0 10 "viewwindow" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 2 "):" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 59 " paramsurf := plot3d(surface,trange, srange, color=blue):" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 57 " plots[disp lay](\{vfieldplot, paramsurf\}, axes=boxed);" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 4 "end:" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 54 "fluxintegral:= proc(vecfield, su rface, trange, srange)" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 2 "\n" } {MPLTEXT 1 0 69 " local intval, paramfield, tanvect, tanvecs, normve ctor, integrand;" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 40 " print(`the vector field `, vecfield);" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 69 " print(`the surface `, surface , ` with `, trange, ` and `, srange);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 63 " print(Int(Int((vfield.[Diff(surface,t)*X *Diff(surface,s)]), " }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 2 "\n" } {MPLTEXT 1 0 27 " trange), srange));" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 17 " paramfield := " }{MPLTEXT 1 0 38 "evalVF(VectorField(vecfield),surface):" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 60 " print(`the vector field on th e surface is `, paramfield);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 31 " tanvect := diff(surface, t):" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 31 " tanvecs := diff(surface, s): " }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 61 " print(` the t-tangent vector to the surface is `, tanvect);" }{MPLTEXT 1 0 0 " " }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 61 " print(`the s-tangent vecto r to the surface is `, tanvecs);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 2 " \n" }{MPLTEXT 1 0 38 " normvector := (tanvect &x tanvecs):" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 61 " print(`the normal vector to the surface is `, normvector);" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 50 " integrand := simplify(paramfi eld . normvector):" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 41 " print(`The integrand is `, integrand);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 46 " print(Int(Int(integrand, tra nge), srange));" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 46 " print(Int(int(integrand, trange), srange));" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 49 " intval := int(int(integra nd, trange), srange);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 2 "\n" } {MPLTEXT 1 0 38 " print(`the integral is `, intval); " }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 4 "end:" }{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 201 114 "With these procedures defined we can define a vector field and a parameterized surface and find the flux i ntegral." }{TEXT 201 0 "" }}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 20 "vfield := ;" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 2 "\n" } {MPLTEXT 1 0 37 "surface := ;" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 20 "trange := t=0..2*Pi;" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 17 "srange := s=0 ..2;" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 45 "surfaceplot(vfield, surface, trange, srange, " }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 30 " [-3..3, -3..3, -1..5]);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 46 "fluxintegra l(vfield, surface, trange, srange);" }{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 217 9 "Exercises" }{TEXT 217 0 "" }}{PARA 0 "" 0 "" {TEXT 201 104 "2) Let F = [y, x, 0] and S(s,t) = [3sin(t), 3cos(t ), s+1] with 0 \262 t \262 2Pi and 0 \262 s \262 1. " }{TEXT 201 54 " Find the flux of the vector field through the surface." }{TEXT 201 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 201 66 "3) Let F = [z, y, 2x] and S is the cone z = sqrt(x^2 + \+ y^2) with " }{XPPEDIT 2 0 "Typesetting:-mrow(Typesetting:-mi(\"z\", it alic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"≤\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", s tretchy = \"false\", symmetric = \"false\", largeop = \"false\", movab lelimits = \"false\", accent = \"false\", form = \"infix\", lspace = \+ \"thickmathspace\", rspace = \"thickmathspace\", minsize = \"1\", maxs ize = \"infinity\"), Typesetting:-mn(\"2\", mathvariant = \"normal\")) ;" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6%Q \"zF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$60Q&≤F'/F3 Q'normalF'/%&fenceGQ&falseF'/%*separatorGF=/%)stretchyGF=/%*symmetricG F=/%(largeopGF=/%.movablelimitsGF=/%'accentGF=/%%formGQ&infixF'/%'lspa ceGQ/thickmathspaceF'/%'rspaceGFO/%(minsizeGQ\"1F'/%(maxsizeGQ)infinit yF'-I#mnGF$6$Q\"2F'F9" }{TEXT 201 7 ". Find" }{TEXT 201 50 " the flux of the vector field through the surface." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 201 108 "4) The command li nalg[curl](F, [x, y, z]); computes the curl of the vector F with respe ct to x, y, and z.\n" }{TEXT 201 92 "Let S be the sphere of radius 2. \+ Describe S as a parameterized surface. (Think spherical.)" }{TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 210 "" 0 "" {TEXT 205 57 "Let F = [xyz, x+y+z, x^3y^2z] and let G be the curl of F." }}{PARA 210 "" 0 "" {TEXT 205 31 "Compute the flux of G throug h S" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 201 47 "Explain why this nice result is not surprising." }{TEXT 201 0 "" }}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 217 22 "Just the result please" }{TEXT 217 0 "" }}{EXCHG {PARA 210 "" 0 "" {TEXT 205 154 "It is also useful to have a \+ procedure that simply evaluates the flux integral. For that we use th e Flux command from the Student[VectorCalculus] package." }{TEXT 205 0 "" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 20 "vfield := ;" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 37 "surface \+ := ;" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 20 "trange := t=0..2*Pi;" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 17 "srange := s=0..2;" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 78 "Flux(VectorField(vfield), \+ Surface(surface, trange, srange),output=integral);\n" }{MPLTEXT 1 0 62 "Flux(VectorField(vfield), Surface(surface, trange, srange));\n" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 3 "" 0 "" {TEXT 217 8 "Exercise" }}{PARA 0 "" 0 "" {TEXT 201 148 "Exercise 5) Repeat exercise 4 with a closed surface of your choice and a vector field that is the curl of a nontrivial vector fie ld of your choice." }{TEXT 201 0 "" }}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 205 "" 0 "" {TEXT 218 0 "" }}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 15 10 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }