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2 0 0 0 1 }} {SECT 0 {PARA 18 "" 0 "" {TEXT 216 24 "Computing Flux Integrals" }} {PARA 19 "" 0 "" {TEXT 217 35 "\302\2512006 Mike May, S.J.- maymk@slu. edu" }}{PARA 0 "" 0 "" {TEXT 202 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 202 505 "In considering flu x integrals, it seems worthwhile to go through using Maple to set up a nd evaluate flux integrals over a parameterized surface. We will walk through a step by step procedure, then produce new procedures that do everything in one command. Finally we look at the Maple command that computes the flux integral in a single step. We use the Student[Vect orCalculus] package and set the BasisFormat variable so that fields ar e written as vectors rather than as linear combinations of a basis." } {TEXT 202 0 "" }}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 58 "restart:w ith(Student[VectorCalculus]); BasisFormat(false):" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "7RI#&xG6\"I\"*G%*protectedGI\"+GF&I\"- GF$I\".GF$I$<,>GF$I$<|gr>GF$I,BasisFormatGF$I)BinormalGF$I-CrossProduc tGF$I%CurlGF$I*CurvatureGF$I\"DGF$I$DelGF$I+DivergenceGF$I+DotProductG F$I)FlowLineGF$I%FluxGF$I/GetCoordinatesGF$I)GradientGF$I*LaplacianGF$ I(LineIntGF$I+MapToBasisGF$I&NablaGF$I%NormGF$I*NormalizeGF$I(PathIntG F$I/PositionVectorGF$I0PrincipalNormalGF$I2RadiusOfCurvatureGF$I-Roote dVectorGF$I0ScalarPotentialGF$I/SetCoordinatesGF$I+SpaceCurveGF$I0Spac eCurveTutorGF$I+SurfaceIntGF$I)TNBFrameGF$I.TangentVectorGF$I(TorsionG F$I'VectorGF$I,VectorFieldGF$I1VectorFieldTutorGF$I0VectorPotentialGF$ I%diffGF&I'evalVFGF$I$intGF$I&limitGF$I'seriesGF&" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 218 22 "A step by step example" }{TEXT 218 0 "" }}{PARA 0 "" 0 "" {TEXT 202 302 "First we need to define a vector field that w e want to integrate, the parameterized surface we are integrating over , and the limits on the parameters. For the example we will start wit h 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 202 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "vfield := <0,0,1>;\n" } {MPLTEXT 1 0 47 "surface := <4*t*cos(s), 2*t*sin(s), t*sin(s)>;\n" } {MPLTEXT 1 0 18 "trange := t=0..1;\n" }{MPLTEXT 1 0 20 "srange := s=0. .2*Pi;" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "-&I'VectorG 6$%*protectedGI(_syslibG6\"6#I'columnGF(6#/I$%idGF(\"(Km8#" }}{PARA 11 "" 1 "" {XPPMATH 20 "-&I'VectorG6$%*protectedGI(_syslibG6\"6#I'column GF(6#/I$%idGF(\"(Kf_#" }}{PARA 11 "" 1 "" {XPPMATH 20 "/I\"tG6\";\"\"! \"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "/I\"sG6\";\"\"!,$I#PiG%*protec tedG\"\"#" }}}{PARA 211 "" 0 "" {TEXT 206 62 "Evaluating the flux inte gral can be broken into a seven steps:" }}{PARA 211 "" 0 "" {TEXT 202 112 "1) Graph the vector field and the parameterized surface to see w hat it looks like. (Not required, but useful.)" }}{PARA 211 "" 0 "" {TEXT 202 29 "2) Set up the flux integral." }}{PARA 211 "" 0 "" {TEXT 202 91 "3) Replace x, y, and z in the vector field with the expressi ons in s and t on the surface." }}{PARA 211 "" 0 "" {TEXT 202 96 "4) \+ Find the tangent vectors to the surface obtained by differentiating wi th respect to s and t." }}{PARA 211 "" 0 "" {TEXT 202 91 "5) Find a n ormal vector to the surface by taking the cross product of the tangent vectors." }}{PARA 211 "" 0 "" {TEXT 202 112 "6) Find the integrand o f the flux integral by taking the dot product of the vector field and \+ the normal vector." }}{PARA 211 "" 0 "" {TEXT 202 26 "7) Evaluate the integral." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 202 87 "The first step is to look at a graph of the vector field and the parameterized surface." }{TEXT 202 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "vfieldplot := VectorField(vfield,output=p lot,view=[-5..5,-5..5,-5..5]):\n" 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0012C00EE8DD4748BF0A3FE0907DC19306C13FD0907DC19306C1C00FFFFFFFFFFFFC3C FD1A62633145BC3CED1A62633145BCC00EE8DD4748BF19BFE0907DC1930651BFD0907D C1930651C00BB67AE8584CB9BFEFFFFFFFFFFFC0BFDFFFFFFFFFFFC0C006A09E667F3B E4BFF6A09E667F3BAFBFE6A09E667F3BAFC00000000000001EBFFBB67AE8584C95BFEB B67AE8584C95BFF0907DC19306D9BFFEE8DD4748BF07BFEEE8DD4748BF07BD14D3C9CA 64F44DBFFFFFFFFFFFFFFCBFEFFFFFFFFFFFFC3FF0907DC1930639BFFEE8DD4748BF1C BFEEE8DD4748BF1C3FFFFFFFFFFFFFAABFFBB67AE8584CBFBFEBB67AE8584CBF4006A0 9E667F3BA6BFF6A09E667F3BEDBFE6A09E667F3BED400BB67AE8584C8DBFF000000000 002BBFE000000000002B400EE8DD4748BF02BFE0907DC19306F9BFD0907DC19306F940 0FFFFFFFFFFFFCBD0D1A62633145BCBCFD1A62633145BC-%&COLORG6&%$RGBG$\"\"!! \"\"$\"\"!!\"\"$\"#5!\"\"-%&TITLEG6#Q!6\"-%%VIEWG6%;$!#]!\"\"$\"#]!\" \";$!#]!\"\"$\"#]!\"\";$!#]!\"\"$\"#]!\"\"" 1 2 5 1 10 1 2 1 1 2 2 1.0 47.0 74.0 1 0 "Curve 1" "Curve 2" }}{TEXT 219 0 "" }}}{PARA 0 "" 0 "" {TEXT 202 50 "Our second step is to set up the flux integral. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Int(Int((vfield.[Diff(surfac e,t) " }{MPLTEXT 1 0 3 "*X*" }{MPLTEXT 1 0 37 " Diff(surface,s)]), tra nge), srange);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I$ IntG6$%*protectedGI(_syslibG6\"6$-F#6$-I\".GF$6$-&I'VectorGF$6#I'colum nGF'6#/I$%idGF'\"(Km8#7#*(-I%DiffGF$6$-F/6#/F5\"(Kf_#I\"tGF'\"\"\"I\"X GF'FA-F:6$F " 0 "" {MPLTEXT 1 0 51 "paramfi eld := evalVF(VectorField(vfield),surface);\n" }{MPLTEXT 1 0 53 "print (`the function on the surface is `, paramfield);" }{MPLTEXT 1 0 0 "" } }{PARA 11 "" 1 "" {XPPMATH 20 "-&I'VectorG6$%*protectedGI(_syslibG6\"6 #I'columnGF(6#/I$%idGF(\"(?@W%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$I@th e~function~on~the~surface~is~G6\"-&I'VectorG6$%*protectedGI(_syslibGF$ 6#I'columnGF$6#/I$%idGF$\"(?@W%" }}}{PARA 0 "" 0 "" {TEXT 202 161 "The fourth step is to take the derivative of the parameterized surface wi th respect to the parameters s and t. This gives us two tangent vecto rs to the surface." }{TEXT 202 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "tanvect := diff(surface, t):\n" }{MPLTEXT 1 0 29 "tan vecs := diff(surface, s):\n" }{MPLTEXT 1 0 59 "print(`the t-tangent ve ctor to the surface is `, tanvect);\n" }{MPLTEXT 1 0 58 "print(`the s- tangent vector to the surface is `, tanvecs);" }{MPLTEXT 1 0 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$IHthe~t-tangent~vector~to~the~surface~ is~G6\"-&I'VectorG6$%*protectedGI(_syslibGF$6#I'columnGF$6#/I$%idGF$\" (SAa#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$IHthe~s-tangent~vector~to~the ~surface~is~G6\"-&I'VectorG6$%*protectedGI(_syslibGF$6#I'columnGF$6#/I $%idGF$\"(?Z2&" }}}{PARA 0 "" 0 "" {TEXT 202 92 "The fifth step is to \+ take the cross product of those tangent vectors to get a normal vector ." }{TEXT 202 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "normvec tor := (tanvect &x tanvecs);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "-&I'VectorG6$%*protectedGI(_syslibG6\"6#I'columnGF(6#/I$% idGF(\"(Ki8&" }}}{PARA 0 "" 0 "" {TEXT 202 200 "The sixth step is to e valuate the dot product in the integrand and simplify. At this point \+ we have reduced the flux integral over a surface to an ordinary iterat ed integral over a section of a plane." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "integrand := simplify(paramfield. normvector):\n" } {MPLTEXT 1 0 38 "print(`The integrand is `, integrand);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$I2The~integrand~is~G6\",$I\"t GF$\"\")" }}}{PARA 0 "" 0 "" {TEXT 202 84 "Finally we can evaluate the integral. We have Maple evaluate the integral in steps." }{TEXT 202 0 "" }}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 36 "Int(Int(integrand, \+ trange), srange);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 36 "Int(int(integrand, trange), srange);" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 36 "int(int(integrand, trange), sran ge);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I$IntG6$%*pr otectedGI(_syslibG6\"6$-F#6$,$I\"tGF'\"\")/F,;\"\"!\"\"\"/I\"sGF';F0,$ I#PiGF%\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I$IntG6$%*protectedGI(_ syslibG6\"6$\"\"%/I\"sGF';\"\"!,$I#PiGF%\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$I#PiG%*protectedG\"\")" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 218 9 "Exercise:" }}{PARA 0 "" 0 "" {TEXT 202 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\", \+ lspace = \"0.2777778em\", rspace = \"0.2777778em\"), Typesetting:-mn( \"4\", mathvariant = \"normal\"));" "-I%mrowG6#/I+modulenameG6\"I,Type settingGI(_syslibGF'6%-I#miGF$6%Q\"zF'/%'italicGQ%trueF'/%,mathvariant GQ'italicF'-I#moGF$6-Q&≤F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separa torGF=/%)stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelimitsGF=/%' accentGF=/%'lspaceGQ,0.2777778emF'/%'rspaceGFL-I#mnGF$6$Q\"4F'F9" } {TEXT 202 183 ". (You probably want to parameterize the surface in te rms of r and theta.) Compute the flux integral of F through S by hand . Then use Maple to check your calculations at each step." }{TEXT 202 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 218 21 "An automated approach" }{TEXT 218 0 "" } }{PARA 0 "" 0 "" {TEXT 202 241 "For convenience we block the code for \+ those seven steps into two procedures we can use, one for plotting, an d one for setting up the integral and evaluating. We also set up a bl ack box procedure that evaluates without showing all the steps." } {TEXT 202 0 "" }}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 52 "surfacepl ot := proc(vecfield, path, trange, srange, " }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 30 " viewwindow)" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 31 " local vfi eldplot, paramsurf;" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\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 1 "\n" }{MPLTEXT 1 0 59 " paramsurf := plot3d(surface,trange, sran ge, color=blue):" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 55 " plots[display](\{vfieldplot, paramsurf\}, axes=boxed);" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 4 "end:" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 54 "fluxintegral: = proc(vecfield, surface, trange, srange)" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 69 " local intval, paramfield, tan vect, tanvecs, normvector, integrand;" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 40 " print(`the vector field `, vecfield);" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 69 " print(`the surface `, surface, ` with `, trange, ` and `, srange);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 63 " print(Int(Int((vfield .[Diff(surface,t)*X*Diff(surface,s)]), " }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 27 " trange), srange));" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 17 " paramfield := " } {MPLTEXT 1 0 38 "evalVF(VectorField(vecfield),surface):" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 60 " print(`the vector fie ld on the surface is `, paramfield);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 31 " tanvect := diff(surface, t):" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 31 " tanvecs := diff(sur face, s):" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 61 " \+ print(`the t-tangent vector to the surface is `, tanvect);" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 61 " print(`the s-tangent vector to the surface is `, tanvecs);" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 38 " normvector := (tanvect &x tan vecs):" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 61 " p rint(`the normal vector to the surface is `, normvector);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 50 " integrand := simpli fy(paramfield . normvector):" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 41 " print(`The integrand is `, integrand);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 46 " print(Int(Int(inte grand, trange), srange));" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 46 " print(Int(int(integrand, trange), srange));" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 49 " intval := \+ int(int(integrand, trange), srange);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 38 " print(`the integral is `, intval); " } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 4 "end:" } {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 202 114 "With these procedu res defined we can define a vector field and a parameterized surface a nd find the flux integral." }{TEXT 202 0 "" }}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 20 "vfield := ;" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 37 "surface := ;" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 20 "trange := t=0 ..2*Pi;" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 17 "sra nge := s=0..2;" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "-&I 'VectorG6$%*protectedGI(_syslibG6\"6#I'columnGF(6#/I$%idGF(\"(g61(" }} {PARA 11 "" 1 "" {XPPMATH 20 "-&I'VectorG6$%*protectedGI(_syslibG6\"6# I'columnGF(6#/I$%idGF(\"('*31(" }}{PARA 11 "" 1 "" {XPPMATH 20 "/I\"tG 6\";\"\"!,$I#PiG%*protectedG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "/I \"sG6\";\"\"!\"\"#" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 45 "sur faceplot(vfield, surface, trange, srange, " }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 30 " [-3..3, -3..3, -1..5]);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 46 "fluxintegral (vfield, surface, trange, srange);" }{MPLTEXT 1 0 0 "" }}{PARA 13 "" 1 "" {TEXT 219 0 "" }{GLPLOT3D 400 400 400 {PLOTDATA 3 "6(-%+AXESLABELS G6%Q\"x6\"Q\"y6\"Q\"z6\"-%*AXESSTYLEG6#%$BOXG-%'CURVESG6jr7%7%$!3gR;v; #yol#!#<$!3gR;v;#yol#!#<$!3eka]A2Ec))!#=7%$!3Sg$[Ky@JM$!#<$!3Sg$[Ky@JM $!#<$!3a`%\\x#RP96!#<7%$!3#*\\)[/?y>4G$!# <$\"3#e=V8#GCC`!#=7%7%$!3gR;v;#yol#!#<$!3gR;v;#yol#!#<$\"3#H4,X9_7x\"! 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bGF$6#I'columnGF$6#/I$%idGF$\"('*31(I'~with~GF$/I\"tGF$;\"\"!,$I#PiGF) \"\"#I&~and~GF$/I\"sGF$;F5F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I$IntG6 $%*protectedGI(_syslibG6\"6$-F#6$-I\".GF$6$-&I'VectorGF$6#I'columnGF'6 #/I$%idGF'\"(g61(7#*(-I%DiffGF$6$-F/6#/F5\"('*31(I\"tGF'\"\"\"I\"XGF'F A-F:6$F " 0 " " {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 202 66 "3) Let F = [z, y , 2x] and S is the cone z = sqrt(x^2 + y^2) with " }{XPPEDIT 2 0 "Type setting:-mrow(Typesetting:-mi(\"z\", italic = \"true\", mathvariant = \+ \"italic\"), Typesetting:-mo(\"≤\", mathvariant = \"normal\", fenc e = \"false\", separator = \"false\", stretchy = \"false\", symmetric \+ = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \+ \"false\", lspace = \"0.2777778em\", rspace = \"0.2777778em\"), Typese tting:-mn(\"2\", mathvariant = \"normal\"));" "-I%mrowG6#/I+modulename G6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6%Q\"zF'/%'italicGQ%trueF'/%,m 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Find" }{TEXT 202 50 " the flux of the vector fi eld through the surface." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 202 107 "4) The command linalg[curl](F, [x, y, z]); computes the curl of the vector F with respect to x, y, and z .\n" }{TEXT 202 92 "Let S be the sphere of radius 2. Describe S as a \+ parameterized surface. (Think spherical.)" }{TEXT 202 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 211 "" 0 "" {TEXT 206 57 "Let F = [xyz, x+y+z, x^3y^2z] and let G be the curl of F." }}{PARA 211 "" 0 "" {TEXT 206 31 "Compute the flux of G through S" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 202 47 "E xplain why this nice result is not surprising." }{TEXT 202 0 "" }} {EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 218 22 "Just the result please" }{TEXT 218 0 "" }}{EXCHG {PARA 211 "" 0 "" {TEXT 206 154 "It is also useful to have a procedure that simply evaluates the flux integral. For that we use the Flux co mmand from the Student[VectorCalculus] package." }{TEXT 206 0 "" }}} {EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 20 "vfield := ;" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 37 "surface := ;" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 20 "trange := t=0..2*Pi;" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 17 "srange := s=0..2;" }{MPLTEXT 1 0 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "-&I'VectorG6$%*protectedGI(_syslibG6\"6# I'columnGF(6#/I$%idGF(\"(o)*o%" }}{PARA 11 "" 1 "" {XPPMATH 20 "-&I'Ve ctorG6$%*protectedGI(_syslibG6\"6#I'columnGF(6#/I$%idGF(\"(of&G" }} {PARA 11 "" 1 "" {XPPMATH 20 "/I\"tG6\";\"\"!,$I#PiG%*protectedG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "/I\"sG6\";\"\"!\"\"#" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 77 "Flux(VectorField(vfield), Surface (surface, trange, srange),output=integral);\n" }{MPLTEXT 1 0 61 "Flux( VectorField(vfield), Surface(surface, trange, srange));\n" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I$IntG6$%*protectedGI(_sysli bG6\"6$-F#6$,(*&I\"sGF'\"\"$-I$cosGF$6#I\"tGF'\"\"#F3*&F-F.-I$sinGF$F1 F3F3*&F-F3,&*&F-\"\"\"F5F3!\"\"*&F-F:F/F3F;F:F:/F-;\"\"!F3/F2;F?,$I#Pi GF%F3" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$I#PiG%*protectedG\"\")" }}} {EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 218 8 "Exercise" }}{PARA 0 "" 0 "" {TEXT 202 148 "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 202 0 "" }}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 205 "" 0 "" {TEXT 220 0 "" }}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 15 10 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }