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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 "Times" 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" -1 213 "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 214 24 "Computing Flux Integrals" }} {PARA 19 "" 0 "" {TEXT 215 38 "\2512006 Mike May, S.J.- maymk@slu.edu" }}{PARA 0 "" 0 "" {TEXT 200 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 200 504 "In looking at flux integral s, it seems worthwhile to go through using Maple to set up and evaluat e flux integrals over a parameterized surface. We will walk through a step by step procedure, then produce new procedures that do everythin g in one command. Finally we look at the Maple command that computes \+ the flux integral in a single step. We use the Student[VectorCalculus ] package and set the BasisFormat variable so that fields are written \+ as vectors rather than as linear combinations of a basis." }{TEXT 200 0 "" }}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 57 "restart:with(Studen t[VectorCalculus]);BasisFormat(false):" }{MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 216 22 "A step by step example" }{TEXT 216 0 "" }}{PARA 0 "" 0 "" {TEXT 200 287 "First we need to define a vector fiel d that we want to integrate, the parameterized surface we are integrat ing over, and the limits on the parameters. For the example we will s tart with the vector field [0, 0, 1]. Then the flux integral will be \+ the area of the shadow on the x-y plane." }{TEXT 200 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 "srange := s=0..2*Pi;" } {MPLTEXT 1 0 0 "" }}}{PARA 211 "" 0 "" {TEXT 204 62 "Evaluating the fl ux integral can be broken into a seven steps:" }}{PARA 211 "" 0 "" {TEXT 200 112 "1) Graph the vector field and the parameterized surfac e to see what it looks like. (Not required, but useful.)" }}{PARA 211 "" 0 "" {TEXT 200 29 "2) Set up the flux integral." }}{PARA 211 "" 0 "" {TEXT 200 91 "3) Replace x, y, and z in the vector field with the expressions in s and t on the surface." }}{PARA 211 "" 0 "" {TEXT 200 96 "4) Find the tangent vectors to the surface obtained by differ entiating with respect to s and t." }}{PARA 211 "" 0 "" {TEXT 200 91 " 5) Find a normal vector to the surface by taking the cross product of the tangent vectors." }}{PARA 211 "" 0 "" {TEXT 200 112 "6) Find the integrand of the flux integral by taking the dot product of the vecto r field and the normal vector." }}{PARA 211 "" 0 "" {TEXT 200 26 "7) \+ Evaluate the integral." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT 200 87 "The first step is to look at a graph o f the vector field and the parameterized surface." }{TEXT 200 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "vfieldplot := VectorField(vf ield,output=plot,view=[-5..5,-5..5,-5..5]):\n" }{MPLTEXT 1 0 57 "param surf := 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 200 50 "Our second step is to set up the flux integral. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "In t(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 200 139 "The third step is to replace x, y, and \+ z in the vector field with the parameterizations of x and y at the app ropriate point on the surface." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "paramfield := evalVF(VectorField(vfield),surface);\n" }{MPLTEXT 1 0 52 "print(`the funtion on the surface is `, paramfield);" } {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 200 161 "The fourth step is to take the derivative of the parameterized surface with respect to t he parameters s and t. This gives us two tangent vectors to the surfa ce." }{TEXT 200 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "tanve ct := diff(surface, t):\n" }{MPLTEXT 1 0 30 "tanvecs := diff(surface, \+ s):\n" }{MPLTEXT 1 0 60 "print(`the t-tangent vector 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 200 92 "The fifth step is to take the cross product of those tangent v ectors to get a normal vector." }{TEXT 200 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "normvector := (tanvect &x tanvecs);" }{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 200 200 "The sixth step is to evalua te the dot product in the integrand and simplify. At this point we ha ve reduced the flux integral over a surface to an ordinary iterated in tegral over a section of a plane." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "integrand := simplify(paramfield. normvector):\n" }{MPLTEXT 1 0 38 "print(`The integrand is `, integrand);" }{MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT 200 87 "Finally we can evaluate the integral. W e do have Maple evaluate the integral in steps." }{TEXT 200 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, trange), srange);" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 3 "" 0 "" {TEXT 216 9 "Exercise:" }}{PARA 0 "" 0 "" {TEXT 200 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:-mrow(Typesetting:-mi( \"z\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo( \"≤\", mathvariant = \"normal\", fence = \"false\", separator = \" false\", stretchy = \"false\", symmetric = \"false\", largeop = \"fals e\", movablelimits = \"false\", accent = \"false\", form = \"infix\", \+ lspace = \"thickmathspace\", rspace = \"thickmathspace\", minsize = \" 1\", maxsize = \"infinity\"), Typesetting:-mn(\"4\", mathvariant = \"n ormal\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I #miGF$6%Q\"zF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$60Q&& leq;F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF=/%)stretchyGF=/%*s ymmetricGF=/%(largeopGF=/%.movablelimitsGF=/%'accentGF=/%%formGQ&infix F'/%'lspaceGQ/thickmathspaceF'/%'rspaceGFO/%(minsizeGQ\"1F'/%(maxsizeG Q)infinityF'-I#mnGF$6$Q\"4F'F9" }{TEXT 200 183 ". (You probably want \+ to parameterize the surface in terms of r and theta.) Compute the flu x integral of F through S by hand. Then use Maple to check your calcu lations at each step." }{TEXT 200 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 216 21 "An automat ed approach" }{TEXT 216 0 "" }}{PARA 0 "" 0 "" {TEXT 200 241 "For conv enience we block the code for those seven steps into two procedures we can use, one for plotting, and one for setting up the integral and ev aluating. We also set up a black box procedure that evaluates without showing all the steps." }{TEXT 200 0 "" }}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 52 "surfaceplot := proc(vecfield, path, 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(sur face,trange, srange, color=blue):" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 57 " plots[display](\{vfieldplot, paramsurf\}, ax es=boxed);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 4 "e nd:" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 54 "fluxint egral:= proc(vecfield, surface, trange, srange)" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 69 " local intval, paramfield, tan vect, tanvecs, normvector, 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 fie ld on the 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(sur face, 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 vector to the surface is `, tanvecs);" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 38 " normvector := (tanvect &x tan vecs):" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 61 " p rint(`the normal vector to the surface is `, normvector);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 50 " integrand := simpli fy(paramfield . 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(inte grand, trange), 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(integrand, 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 200 114 "With these procedu res defined we can define a vector field and a parameterized surface a nd find the flux integral." }{TEXT 200 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 "sra nge := 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 "fluxintegral(vfield, surface, trange, srange);" }{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 216 9 "Exercises" }{TEXT 216 0 "" }}{PARA 0 "" 0 "" {TEXT 200 104 "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 200 54 "Find the flux of the vector field through the surface ." }{TEXT 200 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT 200 66 "3) Let F = [z, y, 2x] and S is the cone z = sqrt(x^2 + y^2) with " }{XPPEDIT 2 0 "Typesetting:-mrow(Typesetti ng:-mi(\"z\", italic = \"true\", mathvariant = \"italic\"), Typesettin g:-mo(\"≤\", mathvariant = \"normal\", fence = \"false\", separato r = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \+ \"false\", movablelimits = \"false\", accent = \"false\", form = \"inf ix\", lspace = \"thickmathspace\", rspace = \"thickmathspace\", minsiz e = \"1\", maxsize = \"infinity\"), Typesetting:-mn(\"2\", mathvariant = \"normal\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibG F'6%-I#miGF$6%Q\"zF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF $60Q&≤F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF=/%)stretchyG F=/%*symmetricGF=/%(largeopGF=/%.movablelimitsGF=/%'accentGF=/%%formGQ &infixF'/%'lspaceGQ/thickmathspaceF'/%'rspaceGFO/%(minsizeGQ\"1F'/%(ma xsizeGQ)infinityF'-I#mnGF$6$Q\"2F'F9" }{TEXT 200 7 ". Find" }{TEXT 200 50 " the flux of the vector field through the surface." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 200 108 " 4) The command linalg[curl](F, [x, y, z]); computes the curl of the v ector F with respect to x, y, and z.\n" }{TEXT 200 92 "Let S be the sp here of radius 2. Describe S as a parameterized surface. (Think sphe rical.)" }{TEXT 200 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 211 "" 0 "" {TEXT 204 57 "Let F = [xyz, x+y+z, x^3y^2z] and l et G be the curl of F." }}{PARA 211 "" 0 "" {TEXT 204 31 "Compute the \+ flux of G through S" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT 200 47 "Explain why this nice result is not surp rising." }{TEXT 200 0 "" }}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 216 22 "Just the result please" } {TEXT 216 0 "" }}{EXCHG {PARA 211 "" 0 "" {TEXT 204 155 "It is also us eful to have a procedure that simply evaluates the flux integral. Fo r that we use the Flux command from the Student[VectorCalculus] packag e." }{TEXT 204 0 "" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 20 "vfi eld := ;" }{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, tran ge, srange));\n" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 216 8 "Exercise" } }{PARA 0 "" 0 "" {TEXT 200 148 "Exercise 5) Repeat exercise 4 with a \+ closed surface of your choice and a vector field that is the curl of a nontrivial vector field of your choice." }{TEXT 200 0 "" }}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 205 "" 0 "" {TEXT 217 0 "" }}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 15 10 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }