{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 1 14 0 0 0 0 2 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 257 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 1 14 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 275 "" 1 14 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 277 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 280 "" 1 14 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 284 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Ti mes" 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output " -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Norma l" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 20 "Vectors and Graphics" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 560 "This se ssion will examine several graphics routines with vector Calculus, the n show how Maple can be applied to differential equations. Data is fit to a cubic polynomial, then graphed. 3-D surfaces are plotted, then t he volume and surface integrals are analyzed. The Divergence and Stoke s' Theorems are visualized, then special functions, including gradient and curl are applied. We include the latex command to show how to gen erate latex code for good document presentation. The graphics ends wit h animations of Fourier series and a parametrized 3-D surface. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 284 7 " Vectors" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "We begin with some basic vector operations. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "a := [2, -1, 5]; b := [1, 3, -4]; c := Vector[row](3,-2);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "Adding vectors is easy, but different for ms may require the evalm (matrix evaluation) command." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "a+b;evalm(a+b-c);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "Many of the o perations with vectors require one of the Maple special packages, lina lg or LinearAlgebra." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "wit h(linalg):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Below we show the d ot or inner product and the cross product." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 29 "dotprod(a,b); innerprod(a,c);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "crossprod(a ,b);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "Next we enter a vector function, then take the divergence and curl of this function." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "F := (x,y,z) -> [x^2*cos(y*z),-x^2*z*exp(y^2),ln(x)/(z*y)];" }} {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "diverge(F(x,y,z),[x,y,z]);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "curl(F(x,y,z),[x,y,z]);" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 9 "Graphics " }}{PARA 0 "" 0 "" {TEXT -1 137 "Maple's graphics capabil ities cover a variety areas using several special packages. These requ ire special tools and some are shown below." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 8 "restart:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 259 12 "Fitting Data" }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 156 "Suppose we want to fit data to a cubic model and visua lize both the data and the model. Our example comes from the oxygen co nsumption of the \"kissing bug,\" " }{TEXT 256 19 "Triatoma phyllosoma " }{TEXT 276 3 ", " }{TEXT -1 177 "during the fourth instar stage (Da ta from Boyd Collier). This bug carries chagas disease. The time data \+ (in hours) is stored as td, while the oxygen consumption is stored as \+ yd." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "td := [1, 2, 3, 4, \+ 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]: yd := [116.6, 120.1, 114.9, 12 9.9, 116.5, 107.7, 99.0, 104.0, 100.7, 87.5, 82.7, 53.8, 54.0, 72.4, 8 1.1]: " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 169 "We will use the statis tics package (for least squares), the statistics plotting package (for scatterplot), and the plotting package (for displaying two types of g raphs)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "with(stats): wit h(stats[statplots]): with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warn ing, the name changecoords has been redefined\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 118 "Graph the data with circles for points, then store \+ this graph for later. First view the graph, then store the picture." } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "plots[display](scatterplot (td,yd),symbol=circle);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "Data := plots[display](scatterplot( td,yd),symbol=circle):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Create \+ a general cubic model." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "m odel := a*t^3 + b*t^2 + c*t + d;" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Find the least squares best fit of the cubic model to the data." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "eqn := fit[leastsquare[[t,y], y=model, \{a,b,c,d\}]]([td, yd]); " }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Make " }{TEXT 257 1 "y" }{TEXT -1 30 " into a function depending on " }{TEXT 258 1 "t" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "y := unapply(rhs(eqn),t);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "Store the graph of the \+ model, then display it with the graph of the data." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Y := plot(y(t),t=0..15):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "display(\{Data,Y\});" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Select individual coefficients from the model." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "b := coeff(y(t),t,2);" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 275 24 "Volume and Surface Area " }}{PARA 0 "" 0 "" {TEXT -1 197 "Maple provi des a good tool for visualizing the 3-D surfaces from vector Calculus. It can be readily used to find the surface area and volume of these 3 -D objects. Consider the following paraboloid." }}{PARA 0 "" 0 "" {TEXT -1 9 " S = \{(" }{TEXT 260 1 "x" }{TEXT -1 2 ", " }{TEXT 261 1 "y" }{TEXT -1 2 ", " }{TEXT 262 1 "z" }{TEXT -1 3 ")| " }{TEXT 263 1 "z" }{TEXT -1 8 " = 4 - 4" }{TEXT 264 1 "x" }{TEXT -1 5 "^2 - " } {TEXT 265 1 "y" }{TEXT -1 4 "^2, " }{TEXT 266 1 "z" }{TEXT -1 1 " " } {TEXT 267 1 ">" }{TEXT -1 3 " 0\}" }}{PARA 0 "" 0 "" {TEXT -1 64 "To v isualize this surface in Maple, we need to parameterize the " }{TEXT 268 1 "x" }{TEXT -1 2 ", " }{TEXT 269 1 "y" }{TEXT -1 6 ", and " } {TEXT 270 1 "z" }{TEXT -1 136 " components into 2 parameters. I want t he graph centered on the origin (axes = NORMAL) and the picture unscal ed (scaling = CONSTRAINED)." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "r := [u*cos(v)/2,u*sin(v),4-u^2];" }}{PARA 11 "" 1 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "plot3d([r[1],r[2],r[3] ],u=0..2,v=0..2*Pi,axes = NORMAL, labels = [x,y,z], scaling = CONSTRAI NED, orientation = [40,70]);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "We find the volume under this surf ace (using symmetry)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "4* Int(Int(4-4*x^2-y^2,y=0..sqrt(4-4*x^2)),x=0..1);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "The formula can b e inserted into a LaTeX document by letting Maple generate the code." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "latex(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Below we compute the double integral." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "4*int(int(4-4*x^2-y^2,y=0.. sqrt(4-4*x^2)),x=0..1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "We fin d the surface area of this surface (again using symmetry). A reminder \+ that if " }{TEXT 271 1 "z" }{TEXT -1 3 " = " }{TEXT 272 1 "f" }{TEXT -1 1 "(" }{TEXT 273 1 "x" }{TEXT -1 2 ", " }{TEXT 274 1 "y" }{TEXT -1 60 "), then the formula for computing the area of the surface is" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f := (x,y) -> 4 - 4*x^2 - y^2;" }}{PARA 11 "" 1 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "4*Int(Int(sqrt((diff(f( x,y),x))^2+(diff(f(x,y),y))^2+1),y=0..sqrt(4-4*x^2)),x=0..1);" }} {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "4*int(int(sqrt((diff(f(x,y),x))^2+(diff(f(x,y),y))^2+1),y=0..sqr t(4-4*x^2)),x=0..1);" }}{PARA 12 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 78 "An alternate method for computing surface integrals use s the following formula" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and trace have been red efined and unprotected\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "We cr eated the parametrized surface by the function " }{TEXT 281 1 "r" } {TEXT -1 1 "(" }{TEXT 282 1 "u" }{TEXT -1 1 "," }{TEXT 283 1 "v" } {TEXT -1 152 ") given above. By taking the crossproduct of the partial derivatives of the parametrized surface function, we obtain the outwa rd normal to the surface. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "ruxrv := crossprod(diff(r,u),diff(r,v));" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "absruxrv := sqrt(ruxrv[1]^2 + ruxrv[2]^2 + ruxrv[3]^2);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 215 "The area of the \+ surface is the integral over the range of the parameters of the magnit ude of the outward normal. One could use the norm function, but Maple \+ has difficulty integrating quantities with absolute values." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "int(int(absruxrv, u = 0..2), v = 0..2*Pi);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "Often when the \+ integral cannot be evaluated exactly, the evalf function allows numeri cal solution of the integral." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 5 " " 0 "" {TEXT 280 27 "Gauss' and Stokes' Theorems" }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 267 "Two of the major theorems used for fluid flow are the Divergence or Gauss' Theorem and Stokes' Theorem. The Di vergence Theorem provides an means of determining the flow through a b ounded region in 3-space. Stokes' Theorem gives information on the rot ation of a fluid." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 131 "Let us consider these theorems for a fluid flowing throu gh a hemisphere. We begin by visualizing the fluid flow and the hemisp here." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "View the fluid flow, using Maple's fieldplot." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "F := [x^2*y, y*z, 2*z-1];" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 171 "fieldplot3d(F,x=-4..4,y=-4..4,z=-1..4,grid=[8 ,8,5],axes = NORMAL, thickness = 2);\nFlow := fieldplot3d(F,x=-4..4,y= -4..4,z=-1..4,grid=[8,8,5],axes = NORMAL, thickness = 2):" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 111 "Graph th e hemisphere, which is the surface bounding the region for applying th e Divergence and Stokes' Theorem." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "r := [3*cos(u)*cos(v),3*sin(u)*cos(v),3*sin(v)];" }} {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 301 "plot3d([r[1],r[2],r[3]],u=0..2*Pi,v=0..Pi/2,axes = NORMAL, labe ls = [x,y,z], orientation = [40,70], style = WIREFRAME, color = BLACK) ; \nHemi := plot3d([r[1],r[2],r[3]],u=0..2*Pi,v=0..Pi/2,axes = NORMAL, labels = [x,y,z], scaling = CONSTRAINED, orientation = [40,70], style = WIREFRAME, color = BLACK):" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "display3d(\{Flow,Hemi\});" } }{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "The Divergence Theorem is given by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "Here we compute the divergence of the ve ctor flow field, then integrate this quantity over the volume." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "divF := diverge(F,[x,y,z]); " }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "Int(Int(Int(divF,z = 0..sqrt(9-x^2-y^2)),y = -sqrt(9- x^2)..sqrt(9-x^2)),x = -3..3);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "int(int(int(divF,z = 0..sqrt (9-x^2-y^2)),y = -sqrt(9-x^2)..sqrt(9-x^2)),x = -3..3);" }}{PARA 11 " " 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf (%);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "The above integral is more easily written in spherical co ordinates and given by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 " Int(Int(Int((2*rho^2*cos(theta)*sin(theta)*sin(phi)^2 + rho*cos(phi) + 2)*rho^2*sin(phi),rho = 0..3),theta = 0..2*Pi),phi = 0..Pi/2);" }} {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "int(int(int((2*rho^2*cos(theta)*sin(theta)*sin(phi)^2 + rho*cos (phi) + 2)*rho^2*sin(phi),rho = 0..3),theta = 0..2*Pi),phi = 0..Pi/2); " }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Stokes' Theorem measures the circulation \+ around any curve " }{TEXT 277 1 "C" }{TEXT -1 63 ". If we consider the curve where the hemisphere intersects the " }{TEXT 278 2 "xy" }{TEXT -1 21 "-plane with the flow " }{TEXT 279 1 "F" }{TEXT -1 46 " given ab ove, then Stokes' Theorem states that" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 103 "Here we compute the curl of the vect or field, then evaluate the double integral over the circle in the " } {TEXT 256 2 "xy" }{TEXT -1 30 "-plane beneath the hemisphere." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "curlF := curl(F,[x,y,z]);" } }{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "curlFn := dotprod(curlF,[0,0,1]);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "Int(Int(curlFn,y \+ = -sqrt(9-x^2)..sqrt(9-x^2)),x = -3..3);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "int(int(curlFn,y = - sqrt(9-x^2)..sqrt(9-x^2)),x = -3..3);" }}{PARA 11 "" 1 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "In polar coordinates..." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "int(int(-R^3*cos(theta)^2,R \+ = 0..3),theta = 0..2*Pi);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Animat ions" }}{PARA 0 "" 0 "" {TEXT -1 86 "Maple has programs that allow vis ualization of 2 and 3-D graphics through animations. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 306 "We begin with a repea t of our Fourier series expansion from the previous session. However, \+ this time, the evolution of the partial sums of the Fourier series is \+ viewed through animation, which allows one to easily see how the Fouri er series is converging. The animation program requires the plots pack age. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "bi := int(x*sin(i*Pi*x),x=-1 ..1);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "S := n -> sum(bi*sin(i*Pi*x), i = 1..n);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "ani mate(S(n), x=-3..3,n=1..20,frames=20,numpoints=500, color = blue);" }} {PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 200 "After invoking the animation, a series of buttons appears on the menu bar. These allow the user to view the movie of the graph evolving eit her forward or backward, cycle through, or go frame by frame." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "Below we simply took one of the 3-D examples from Maple's Help. In the 3-D cas e, the default uses just 8 frames." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "animate3d(cos(t*x)*sin(t*y),x=-Pi..Pi, y=-Pi..Pi,t=1. .2,color=cos(x*y));" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {MARK "1 0 0" 7 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }