{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 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 0 1 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 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 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 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Times" 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 "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 18 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 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 49 "Plotting in Cylindrical and Spherical Coordinates" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "There are two basic 3d-pl otters which will be introduced in this section. The first is \221" } {TEXT 257 12 "cylinderplot" }{TEXT -1 22 " \222 and the second is \221 " }{TEXT 258 11 "sphereplot " }{TEXT -1 238 "\222which use cylindrical coordinates and spherical coordinates as their bases respectively. T he syntax we'll use is for the parametric form (there is another less \+ useful form too). Both cylinderplot and sphereplot require first exec uting:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 272 23 "C ylindrical Coordinate:" }}{PARA 0 "" 0 "" {TEXT -1 83 "For now we will concentrate on cylinderplot . Suppose the surface is defined by " }{XPPEDIT 18 0 "z = f(r,theta);" "6#/%\"zG-%\"fG6$%\"rG%&thetaG" } {TEXT -1 8 ", for " }{XPPEDIT 18 0 "r;" "6#%\"rG" }{TEXT -1 11 " be tween " }{XPPEDIT 18 0 "g(theta);" "6#-%\"gG6#%&thetaG" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "h(theta);" "6#-%\"hG6#%&thetaG" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 9 " between " } {XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b;" " 6#%\"bG" }{TEXT -1 21 " . For example, say " }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 18 "Instead of using " }{XPPEDIT 18 0 "[r, theta, z];" "6# 7%%\"rG%&thetaG%\"zG" }{TEXT -1 61 ", we use [r, t, z] (only because ' t' is easier to enter than " }{XPPEDIT 18 0 "theta" "6#%&thetaG" } {TEXT -1 13 " in Maple.) " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "cylinderplot([2,t,z],t=0..2*Pi,z=-2..2,axes=BOXED);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "Notice that we let the range of z go fro m -2 to 2 here. If we leave blank here, we will get a syntax error." } }{PARA 0 "" 0 "" {TEXT -1 37 "Now let's plot the surface of z= r^2." } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "cylinderplot([r,t,r^2],r=0 ..3,t=0..2*Pi,axes=BOXED);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "cylinderplot([r,Pi/3,z],r=0..3,z=-2..2,axes=boxed);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 10 "Example 1:" }{TEXT -1 56 " (1) Use cyli ndrical coordinates to sketch the sphere " }{XPPEDIT 18 0 "x^2+y^2+z^ 2 = 4;" "6#/,(*$%\"xG\"\"#\"\"\"*$%\"yGF'F(*$%\"zGF'F(\"\"%" }{TEXT -1 110 " that lies in the first octant. (Assume that the package \+ \221plots\222 has been included above in the worksheet.)" }}{PARA 0 " " 0 "" {TEXT -1 138 "(2) Use cylindrical coordinates to sketch the fol lowing equation that is given in rectangular coordinate: x^2 +y^2 -z^2 =16 (page 699 #22)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "(1)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "cylinderplot([r,t,sqrt(4-r^2)],r=0..2,t=0..Pi/2,axes=BOXED);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 135 "( 2) First we convert the given equation to the following two equations \+ in cylindrical coordinate: z = sqrt(16-r^2) and z=-sqrt(16-r^2)." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "cylinderplot(\{[r,t,sqrt(16- r^2)],[r,t,-sqrt(16-r^2)]\},r=0..4,t=0..2*Pi,axes=BOXED);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 273 20 "Spherica l Coordinate" }}{PARA 0 "" 0 "" {TEXT -1 88 " Now we turn our atte ntion to spherical coordinates where a point is identified by " } {XPPEDIT 18 0 "[rho, theta, phi];" "6#7%%$rhoG%&thetaG%$phiG" }{TEXT -1 17 " . The command " }{TEXT 260 55 "sphereplot(f(theta,phi),theta =g(phi)..h(phi),phi=a..b);" }{TEXT -1 15 " is used when " }{XPPEDIT 18 0 "rho = f(theta,phi);" "6#/%$rhoG-%\"fG6$%&thetaG%$phiG" }{TEXT -1 208 ". It is a little easier to use sphereplot in the parametric f orm where one of the variables is a function of the other two. For ex ample, to obtain all of a magenta sphere of radius 2 centered at the o rigin," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "sphereplot([2,the ta,phi],theta=0..2*Pi,phi=0..Pi,color=magenta);" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Th e value of " }{XPPEDIT 18 0 "rho;" "6#%$rhoG" }{TEXT -1 59 " is the distance to the point from the origin. The angle " }{XPPEDIT 18 0 "t heta;" "6#%&thetaG" }{TEXT -1 77 " is the same as that used in cylind rical coordinates. Remember, the angle " }{XPPEDIT 18 0 "phi;" "6#%$ phiG" }{TEXT -1 29 " is measured down from the " }{TEXT 271 1 "z" } {TEXT -1 200 "-axis to the point. We will begin with the example that we just completed,but will use spherical coordinates. All the variab les will lie between constants. Please think about this last statemen t. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 126 "First, let\222s consider the consequences of holding each of t he variables constant in spherical coordinates. In a system with " } {XPPEDIT 18 0 "[rho, theta, phi];" "6#7%%$rhoG%&thetaG%$phiG" }{TEXT -1 43 " , a portion of a sphere of fixed radius " }{XPPEDIT 18 0 "rh o;" "6#%$rhoG" }{TEXT -1 15 " occurs when " }{XPPEDIT 18 0 "rho;" "6 #%$rhoG" }{TEXT -1 75 " is a constant. Second, a part of a vertical \+ plane containing the usual z" }{TEXT 261 0 "" }{TEXT -1 77 "-axis is d etermined. It would look like a piece of pie on its side. Last, " } {XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 66 " constant determines a \+ cone with the center axis along the usual " }{TEXT 262 1 "z" }{TEXT -1 17 "-axis. If 0 < " }{XPPEDIT 18 0 "phi <= Pi/2;" "6#1%$phiG*&%# PiG\"\"\"\"\"#!\"\"" }{TEXT -1 59 " , the cone opens upward and will \+ hold water, while if " }{XPPEDIT 18 0 "Pi/2 <= phi;" "6#1*&%#PiG\" \"\"\"\"#!\"\"%$phiG" }{TEXT -1 3 " < " }{XPPEDIT 18 0 "Pi;" "6#%#PiG " }{TEXT -1 49 " , it opens downward, like a traffic cone/pylon." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 274 7 "E xample" }{TEXT -1 159 ": page 699 number 16. Identify the surface by c onverting the equation to a recognizable equation by hand first and co mpare your answer with the following plot." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 59 "sphereplot([2/sin(phi),theta,phi],theta=0..2*Pi,phi =0..Pi);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 275 7 "Example" }{TEXT -1 159 ": page 699 number 18. Identify the surface by converting the equa tion to a recognizable equation by hand first and compare your answer \+ with the following plot." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "sphereplot([2*cos(phi),theta,phi],theta=0..2*Pi,phi=0..Pi);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 263 8 "Example:" }{TEXT -1 76 " Use spherical coordinates to plot the solid that lies inside \+ the sphere " }{XPPEDIT 18 0 "x^2+y^2+z^2 = 9;" "6#/,(*$%\"xG\"\"#\" \"\"*$%\"yGF'F(*$%\"zGF'F(\"\"*" }{TEXT -1 13 " , above the " }{TEXT 264 2 "xy" }{TEXT -1 54 "-plane, and the x and y coordinates are also \+ positive." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "sphereplot([3, theta,phi],theta=0..Pi/2,phi=0..Pi/2,color=red);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "sphereplot([rho,Pi/3,phi],rho=0..2,phi=0..Pi, axes=boxed);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 276 8 "Exercise" }{TEXT -1 71 ": Convert theta=Pi/3 into an equation in rectangular coordinate system." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "implicitplot3d(4*x^2=(x ^2+3*x^2+z^2)*sin(arctan(2*x)/z),x=0..1,y=0..1,z=0..2,axes=boxed);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "cone3:=sphereplot([rho,thet a,Pi/4],theta=Pi/2..2*Pi,rho=0..3,color=blue):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 265 9 "Example: " }{TEXT -1 45 " Plot the solid that lies i nside the sphere " }{XPPEDIT 18 0 "x^2+y^2+z^2 = 4;" "6#/,(*$%\"xG\" \"#\"\"\"*$%\"yGF'F(*$%\"zGF'F(\"\"%" }{TEXT -1 9 " and is " }{TEXT 266 7 "outside" }{TEXT -1 15 " the cylinder " }{XPPEDIT 18 0 "x^2+y^2 = 1;" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F(F(" }{TEXT -1 4 " .\n" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "A three-quarter view of this solid results from:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "spherep lot([2,theta,phi],theta=Pi/2..2*Pi,phi=Pi/6..5*Pi/6);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "sphereplot([csc(phi),theta,phi],the ta=Pi/2..2*Pi,phi=Pi/6..5*Pi/6);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "sphereplot([rho,0,phi],rho=csc(phi)..2,phi=Pi/6..5*Pi /6);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "sphereplot([rho,Pi/ 2,phi],rho=csc(phi)..2,phi=Pi/6..5*Pi/6);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 31 "display(sph4,cyl4,side5,side6);" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "sph4:=sphereplot([2,theta,ph i],theta=Pi/2..2*Pi,phi=Pi/6..5*Pi/6):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "cyl4:=sphereplot([csc(phi),theta,phi],theta=Pi/2..2*P i,phi=Pi/6..5*Pi/6):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "sid e5:=sphereplot([rho,0,phi],rho=csc(phi)..2,phi=Pi/6..5*Pi/6):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "side6:=sphereplot([rho,Pi/2, phi],rho=csc(phi)..2,phi=Pi/6..5*Pi/6):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "display(sph4,cyl4,side5,side6,axes=BOXED);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 21 "Challenging Pr oblems " }{TEXT -1 63 " Use cylinderplot and/or sphereplot to plot the solids listed:\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 2 "1." }{TEXT -1 25 " Q is bounded above by " }{XPPEDIT 18 0 "z = 4;" "6#/%\"zG\" \"%" }{TEXT -1 15 " , below by " }{XPPEDIT 18 0 "z = x^2+y^2;" "6#/ %\"zG,&*$%\"xG\"\"#\"\"\"*$%\"yGF(F)" }{TEXT -1 24 " , and on the side by " }{XPPEDIT 18 0 "x^2+y^2 = 1;" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yGF 'F(F(" }{TEXT -1 3 " ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 269 1 "2" } {TEXT -1 54 ". R is a \223slice of cheese\224. It is bounded above by " }{XPPEDIT 18 0 "z = 2;" "6#/%\"zG\"\"#" }{TEXT -1 13 " , below by " }{XPPEDIT 18 0 "z = 0;" "6#/%\"zG\"\"!" }{TEXT -1 20 " , on the si des by " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 8 " and \+ " }{XPPEDIT 18 0 "y = x;" "6#/%\"yG%\"xG" }{TEXT -1 27 " , and on the outside by " }{XPPEDIT 18 0 "x^2+y^2 = 4;" "6#/,&*$%\"xG\"\"#\"\"\" *$%\"yGF'F(\"\"%" }{TEXT -1 3 " ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 270 2 "3." }{TEXT -1 30 " S is inside the the sphere " }{XPPEDIT 18 0 "x^2+y^2+z^2 = 9;" "6#/,(*$%\"xG\"\"#\"\"\"*$%\"yGF'F(*$%\"zGF'F(\" \"*" }{TEXT -1 24 " and above the cone " }{XPPEDIT 18 0 "3*z^2 = x ^2+y^2;" "6#/*&\"\"$\"\"\"*$%\"zG\"\"#F&,&*$%\"xGF)F&*$%\"yGF)F&" } {TEXT -1 3 " ." }}}}{MARK "44" 0 }{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }