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We must determine a portion of the plane to se rve as the domain of a continuous function. Then we must determine th at function that maps this domain onto the surface we are parameterizi ng in a nice way. Later we will learn why this is of value. You will find that we will lean heavily on " }{TEXT 258 9 "z=f (x,y)" }{TEXT -1 2 ", " }{TEXT 259 8 "y=g(x,z)" }{TEXT -1 5 ", or " }{TEXT 260 8 "x= h(y,z)" }{TEXT -1 257 " in the rectangular case. But cylindrical or s pherical coordinates will be of equal value in accomplishing this task . Following the syntax of Maple, we'll use square brackets, [ ], inst ead of the angle brackets, < >, used in the text to represent vectors. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "If " }{TEXT 261 1 "D" }{TEXT -1 13 " is a set in " }{XPPEDIT 18 0 "R^2" "6#*$%\"RG\"\"#" }{TEXT -1 20 " and g is defined as" }}{PARA 258 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "g:=D->R^3" "6#>%\"gGR6#%\"DG7\"6$%)operatorG%&arrowG6\" *$%\"RG\"\"$F,F,F," }{TEXT -1 13 " where " }{TEXT 303 1 "g" } {TEXT -1 1 "(" }{TEXT 304 3 "u,v" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "[ x,y,z]" "6#7%%\"xG%\"yG%\"zG" }{TEXT -1 8 " and " }{XPPEDIT 18 0 "x =g1(u,v)" "6#/%\"xG-%#g1G6$%\"uG%\"vG" }{TEXT -1 5 " , " }{XPPEDIT 18 0 "y=g2(u,v)" "6#/%\"yG-%#g2G6$%\"uG%\"vG" }{TEXT -1 5 " , " } {XPPEDIT 18 0 "z=g3(u,v)" "6#/%\"zG-%#g3G6$%\"uG%\"vG" }{TEXT -1 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "then \+ our surface is defined as the image " }{TEXT 262 6 "g(D)=S" }{TEXT -1 60 " . This looks worse than it really is. For one thing, the " } {TEXT 305 5 "x,y,z" }{TEXT -1 43 " are normally replaced by the functi ons of " }{TEXT 264 1 "u" }{TEXT -1 5 " and " }{TEXT 263 1 "v" }{TEXT -1 123 ". They were used this time to emphasize that the entry for th ose positions determine the values of the x,y,z coordinates.\n" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "Suppose " }{TEXT 265 9 "z=f (x,y)" }{TEXT -1 26 " with domain D . Define " }{TEXT 266 8 "g(u,v )=" } {TEXT -1 1 "[" }{TEXT 281 10 "u,v,f(u,v)" }{TEXT -1 1 "]" }{TEXT 282 3 " w" }{TEXT -1 24 "here it is obvious that " }{TEXT 267 1 "u" } {TEXT -1 5 " and " }{TEXT 268 1 "v" }{TEXT -1 18 " play the role of " }{TEXT 269 1 "x" }{TEXT -1 5 " and " }{TEXT 270 3 "y. " }{TEXT -1 13 " Or, just use " }{TEXT 271 1 "x" }{TEXT -1 5 " and " }{TEXT 272 1 "y" } {TEXT -1 251 " as the independent variables. After a few examples thi s will seem easier. Note how the parameterization ties in nicely with the plotting of the surface. While we could easily use cylinderplot \+ or sphereplot in certain problems, having the function " }{TEXT 273 1 "g" }{TEXT -1 302 " makes it much easier to just use plot3d . One of \+ the tricks to ease parameterization is to ask yourself this question: \+ \223Is the surface constant for any of the variables in any of the thr ee coordinate systems that we use?\224 If so, using the other two var iables is probably the easiest way to proceed.\n" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 379 "In each example that follows we will define the f unction in Maple and then use plot3d to display it. The first step ma kes us get used to the idea that we are defining a function and the se cond forces us to define the domain of the function. When we set up t he plot3d restrictions on the variables we are defining the domain. T hese skills will be essential later in the course." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 274 12 "Example 1. " } {TEXT -1 42 "Parameterize the portion of the surface " }{XPPEDIT 18 0 "z=x^2+y^2" "6#/%\"zG,&*$%\"xG\"\"#\"\"\"*$%\"yGF(F)" }{TEXT -1 46 " that lies above the triangle with vertices " }{TEXT 275 24 "P(0 ,0) , Q(2,0), R(2 ,2)" }{TEXT -1 10 ". Because " }{TEXT 276 9 "z=f (x,y)" }{TEXT -1 54 ", we just use the obvious approach as described above." }}}{EXCHG {PARA 259 "" 0 "" {XPPEDIT 18 0 "g(x,y)=[x,y,x^2+y^2]" "6#/- %\"gG6$%\"xG%\"yG7%F'F(,&*$F'\"\"#\"\"\"*$F(F,F-" }{TEXT -1 38 " \+ where 0<=y<=x and 0<=x<=2 ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Define the function in Maple:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "g:=(x,y)->[x,y,x^2+y^2];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "To plot this surface use:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot3d(g(x,y),y=0..x,x=0..2,color=red);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 277 9 "Example 2" }{TEXT -1 44 ". Parameterize that portion of the plane " }{XPPEDIT 18 0 "x+3y+4z=12" "6#/,(%\"xG \"\"\"*&\"\"$F&%\"yGF&F&*&\"\"%F&%\"zGF&F&\"#7" }{TEXT -1 37 " that l ies in the first octant with " }{TEXT 278 1 "x" }{TEXT -1 28 " as the \+ dependent variable.\n" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 10 "g(y,z) = " }{XPPEDIT 18 0 "[12-3y-4z,y,z]" "6#7%,(\"#7\"\"\"*&\"\"$F&%\"yGF &!\"\"*&\"\"%F&%\"zGF&F*F)F-" }{TEXT -1 12 " where " }{TEXT 279 10 "0 <= y <= " }{XPPEDIT 18 0 "4-4z/3" "6#,&\"\"%\"\"\"*(F$F%%\"zGF% \"\"$!\"\"F)" }{TEXT -1 6 " and " }{TEXT 280 11 "0 <= z <= 3" }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Define the function i n Maple:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "g:=(y,z)->[12-3 *y-4*z,y,z];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "And to plot this \+ surface:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "plot3d(g(y,z),y =0..4-4*z/3,z=0..3,color=blue,axes=NORMAL);" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 84 "You may be puzzled by this plot. Pay attention to the \+ axes. Because the bounds on " }{TEXT 283 1 "y" }{TEXT -1 38 " were li sted first, Maple thinks that " }{TEXT 284 1 "y" }{TEXT -1 26 " belong s where we put the " }{TEXT 285 1 "x" }{TEXT -1 79 "-axis. It is very important that you realize that the domain of this function " }{TEXT 286 1 "g" }{TEXT -1 24 " is the triangle in the " }{TEXT 287 3 "yz-" } {TEXT -1 21 "plane bounded by the " }{TEXT 288 1 "y" }{TEXT -1 5 " and " }{TEXT 289 1 "z" }{TEXT -1 20 " axes and the plane " }{TEXT 290 14 "x +3y +4z =12 " }{TEXT -1 2 ".\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "Before we do this next example we remind you of the cylindrical co ordinate system.\n" }}}{EXCHG {PARA 261 "" 0 "" {TEXT -1 5 " " } {XPPEDIT 18 0 "x=r*cos(theta)" "6#/%\"xG*&%\"rG\"\"\"-%$cosG6#%&thetaG F'" }{TEXT -1 5 " , " }{XPPEDIT 18 0 "y=r*sin(theta)" "6#/%\"yG*&%\" rG\"\"\"-%$sinG6#%&thetaGF'" }{TEXT -1 5 " , " }{XPPEDIT 18 0 "z=z" "6#/%\"zGF$" }{TEXT -1 2 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 291 10 " Example 3." }{TEXT -1 47 " The solid in the first octant lies betwee n " }{XPPEDIT 18 0 "x^2+y^2=1" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F(F( " }{TEXT -1 8 " and " }{XPPEDIT 18 0 "x^2+y^2 = 4;" "6#/,&*$%\"xG\" \"#\"\"\"*$%\"yGF'F(\"\"%" }{TEXT -1 11 " , above " }{XPPEDIT 18 0 " z=0" "6#/%\"zG\"\"!" }{TEXT -1 29 " and below the paraboloid " } {XPPEDIT 18 0 "z=9-x^2-y^2" "6#/%\"zG,(\"\"*\"\"\"*$%\"xG\"\"#!\"\"*$% \"yGF*F+" }{TEXT -1 219 " . We are going to parameterize the surfac es that we can see and show how to plot them in Maple. This is easies t when done from the viewpoint of cylindrical coordinates. We will be gin with the surface on the left, " }{XPPEDIT 18 0 "y=0" "6#/%\"yG\"\" !" }{TEXT -1 8 " or " }{XPPEDIT 18 0 "theta=0" "6#/%&thetaG\"\"!" }{TEXT -1 23 " . Note that having " }{XPPEDIT 18 0 "theta=0" "6#/%& thetaG\"\"!" }{TEXT -1 16 " lets us use " }{TEXT 292 1 "r" }{TEXT -1 5 " and " }{TEXT 293 1 "z" }{TEXT -1 19 " . Observe that " } {XPPEDIT 18 0 "cos(0)=1" "6#/-%$cosG6#\"\"!\"\"\"" }{TEXT -1 8 " and " }{XPPEDIT 18 0 "sin(0) = 0;" "6#/-%$sinG6#\"\"!F'" }{TEXT -1 67 " \+ . To see the surface now, execute the Maple commands that follow." }} }{EXCHG {PARA 262 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "g1(r,z)=[r, 0,z]" "6#/-%#g1G6$%\"rG%\"zG7%F'\"\"!F(" }{TEXT -1 41 " where 1 < = r <= 2 and 0 <= z <= " }{XPPEDIT 18 0 "9-r^2" "6#,&\"\"*\"\"\"*$% \"rG\"\"#!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "The function th at parameterizes the top is:" }}}{EXCHG {PARA 263 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "g2(r,theta)=[r*cos(theta),r*sint(theta),9-r^2] " "6#/-%#g2G6$%\"rG%&thetaG7%*&F'\"\"\"-%$cosG6#F(F+*&F'F+-%%sintG6#F( F+,&\"\"*F+*$F'\"\"#!\"\"" }{TEXT -1 23 " where 1 <= r <= " } {XPPEDIT 18 0 "Pi/2" "6#*&%#PiG\"\"\"\"\"#!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "The domain is the annular region between the circles in the first quadrant." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Now we will parameterize the outside wall. In this case, " }{TEXT 294 1 "r " }{TEXT -1 50 " is constant. What determines the upper bound on " } {TEXT 295 1 "z" }{TEXT -1 2 " ?" }}}{EXCHG {PARA 264 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 " g3(theta,z)=[2*cos(theta),2*sin(theta),z]" "6# /-%#g3G6$%&thetaG%\"zG7%*&\"\"#\"\"\"-%$cosG6#F'F,*&F+F,-%$sinG6#F'F,F (" }{TEXT -1 15 " where 0 <= " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 4 " <= " }{XPPEDIT 18 0 "Pi/2" "6#*&%#PiG\"\"\"\"\"#!\"\"" } {TEXT -1 12 " and 0 <= " }{TEXT 296 1 "z" }{TEXT -1 9 " <= 5 . " }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Define the functions in Maple:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "g1:=(r,z)->[r,0,z];" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 38 "g2:=(r,t)->[r*cos(t),r*sin(t),9-r^2];\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "g3:=(t,z)->[2*cos(t),2*sin(t),z];\n " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "And to plot the surfaces:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "A1:=plot3d(g1(r,z),r=1..2,z=0..9-r^ 2,color=cyan):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "A2:=plot3 d(g2(r,t),r=1..2,t=0..Pi/2,color=magenta):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 59 "A3:=plot3d(g3(t,z),t=0..Pi/2,z=0..5,color=red,axes= NORMAL):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "display(A1,A2,A 3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "Before we do this last example we remind you of the spherical c oordinate system." }}}{EXCHG {PARA 265 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "x=rho*sin(phi)*cos(theta)" "6#/%\"xG*(%$rhoG\"\"\"-%$si nG6#%$phiGF'-%$cosG6#%&thetaGF'" }{TEXT -1 8 " , " }{XPPEDIT 18 0 "y=rho*sin(phi)*sin(theta)" "6#/%\"yG*(%$rhoG\"\"\"-%$sinG6#%$phiGF' -F)6#%&thetaGF'" }{TEXT -1 7 " , " }{XPPEDIT 18 0 "z=rho*cos(phi) " "6#/%\"zG*&%$rhoG\"\"\"-%$cosG6#%$phiGF'" }{TEXT -1 4 " " }}} {EXCHG {PARA 0 "" 0 "" {TEXT 297 11 "Example 4. " }{TEXT -1 58 " The \+ solid in the first octant lies between the spheres " }{XPPEDIT 18 0 " x^2+y^2+z^2=1" "6#/,(*$%\"xG\"\"#\"\"\"*$%\"yGF'F(*$%\"zGF'F(F(" } {TEXT -1 8 " and " }{XPPEDIT 18 0 "x^2+y^2+z^2=9" "6#/,(*$%\"xG\"\" #\"\"\"*$%\"yGF'F(*$%\"zGF'F(\"\"*" }{TEXT -1 5 " , (" }{TEXT 298 1 " x" }{TEXT -1 8 " >= 0 , " }{TEXT 299 1 "y" }{TEXT -1 8 " >= 0 , " } {TEXT 300 3 "z >" }{TEXT -1 26 "= 0) and below the cone " }{XPPEDIT 18 0 "z=sqrt(3*(x^2+y^2))" "6#/%\"zG-%%sqrtG6#*&\"\"$\"\"\",&*$%\"xG\" \"#F**$%\"yGF.F*F*" }{TEXT -1 149 " . We are going to parameterize th e four surfaces that we can see in the figure, beginning with the smal l spherical surface to the lower left where " }{XPPEDIT 18 0 "rho=1" " 6#/%$rhoG\"\"\"" }{TEXT -1 49 " . The cone on the top is produced by letting " }{XPPEDIT 18 0 "phi=Pi/6" "6#/%$phiG*&%#PiG\"\"\"\"\"'!\" \"" }{TEXT -1 69 " . (To see the surface now, execute the Maple comma nds that follow.)" }}}{EXCHG {PARA 266 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "h1(theta,phi)=[sin(phi)*cos(theta),sin(phi)*sin(theta), cos(phi)]" "6#/-%#h1G6$%&thetaG%$phiG7%*&-%$sinG6#F(\"\"\"-%$cosG6#F'F .*&-F,6#F(F.-F,6#F'F.-F06#F(" }{TEXT -1 18 " where 0 <= " } {XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 4 " <= " }{XPPEDIT 18 0 " Pi/2" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 8 " and " }{XPPEDIT 18 0 "Pi/6" "6#*&%#PiG\"\"\"\"\"'!\"\"" }{TEXT -1 4 " <= " }{XPPEDIT 18 0 "phi" "6#%$phiG" }{TEXT -1 4 " <= " }{XPPEDIT 18 0 "Pi/2" "6#*&%#PiG \"\"\"\"\"#!\"\"" }{TEXT -1 6 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "By adjusting " }{XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 47 " to be 3 we have the outside spherical surface:" }}}{EXCHG {PARA 267 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "h2(theta,phi)=[3*sin(phi)*c os(theta),3*sin(phi)*sin(theta),3*cos(phi)]" "6#/-%#h2G6$%&thetaG%$phi G7%*(\"\"$\"\"\"-%$sinG6#F(F,-%$cosG6#F'F,*(F+F,-F.6#F(F,-F.6#F'F,*&F+ F,-F16#F(F," }{TEXT -1 15 " where 0 <= " }{XPPEDIT 18 0 "theta" "6# %&thetaG" }{TEXT -1 4 " <= " }{XPPEDIT 18 0 "Pi/2" "6#*&%#PiG\"\"\"\" \"#!\"\"" }{TEXT -1 8 " and " }{XPPEDIT 18 0 "Pi/6" "6#*&%#PiG\"\" \"\"\"'!\"\"" }{TEXT -1 4 " <= " }{XPPEDIT 18 0 "phi" "6#%$phiG" } {TEXT -1 4 " <= " }{XPPEDIT 18 0 "Pi/2" "6#*&%#PiG\"\"\"\"\"#!\"\"" } {TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "The side closest to the viewer occurs when " }{XPPEDIT 18 0 "theta=0" "6#/%&thetaG\"\" !" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "cos(0)=1" "6#/-%$cosG6#\"\"!\"\" \"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "sin(0)=0" "6#/-%$sinG6#\"\"!F'" } {TEXT -1 1 " " }}}{EXCHG {PARA 268 "" 0 "" {XPPEDIT 18 0 "h3(rho,phi)= [rho*sin(phi),0,rho*cos(phi)]" "6#/-%#h3G6$%$rhoG%$phiG7%*&F'\"\"\"-%$ sinG6#F(F+\"\"!*&F'F+-%$cosG6#F(F+" }{TEXT -1 15 " where 1 <= " } {XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 4 " <= " }{XPPEDIT 18 0 "3*Pi " "6#*&\"\"$\"\"\"%#PiGF%" }{TEXT -1 8 " and " }{XPPEDIT 18 0 "Pi/6 " "6#*&%#PiG\"\"\"\"\"'!\"\"" }{TEXT -1 4 " <= " }{XPPEDIT 18 0 "phi" "6#%$phiG" }{TEXT -1 4 " <= " }{XPPEDIT 18 0 "Pi/2" "6#*&%#PiG\"\"\"\" \"#!\"\"" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "The t op comes from the cone, " }{XPPEDIT 18 0 "phi = Pi/6;" "6#/%$phiG*&%# PiG\"\"\"\"\"'!\"\"" }{TEXT -1 13 " , where " }{XPPEDIT 18 0 "sin( Pi/6) = 1/2;" "6#/-%$sinG6#*&%#PiG\"\"\"\"\"'!\"\"*&F)F)\"\"#F+" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "cos(Pi/6) = sqrt(3)/2;" "6#/-%$co sG6#*&%#PiG\"\"\"\"\"'!\"\"*&-%%sqrtG6#\"\"$F)\"\"#F+" }{TEXT -1 4 " \+ .\n" }}}{EXCHG {PARA 269 "" 0 "" {XPPEDIT 18 0 "h4(rho,theta) = [rho*[ 1/2]*cos(theta), rho*[1/2]*cos(theta), rho*sqrt(3)/2];" "6#/-%#h4G6$%$ rhoG%&thetaG7%*(F'\"\"\"7#*&F+F+\"\"#!\"\"F+-%$cosG6#F(F+*(F'F+7#*&F+F +F.F/F+-F16#F(F+*(F'F+-%%sqrtG6#\"\"$F+F.F/" }{TEXT -1 17 " where \+ 1 <= " }{XPPEDIT 18 0 "rho;" "6#%$rhoG" }{TEXT -1 4 " <= " }{XPPEDIT 18 0 "3*Pi;" "6#*&\"\"$\"\"\"%#PiGF%" }{TEXT -1 8 " and " } {XPPEDIT 18 0 "0;" "6#\"\"!" }{TEXT -1 4 " <= " }{XPPEDIT 18 0 "theta; " "6#%&thetaG" }{TEXT -1 4 " <= " }{XPPEDIT 18 0 "Pi/2" "6#*&%#PiG\"\" \"\"\"#!\"\"" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 141 "We show h4 in an unsimplified form to \+ make the process more transparent. Now we translate this into Maple a nd show how to plot the surfaces." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "h1:=(t,phi)->[sin(phi)*cos(t),sin(phi)*sin(t),cos(phi )];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "h2:=(t,phi)->[3*sin( phi)*cos(t),3*sin(phi)*sin(t),3*cos(phi)];" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 45 "h3:=(rho,phi)->[rho*sin(phi),0,rho*cos(phi)];" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "h4:=(rho,t)->[rho*(1/2)*cos( t),rho*(1/2)*sin(t),rho*(sqrt(3)/2)];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Now for the plot:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "B1:=plot3d(h1(t,phi),t=0..Pi/2,phi=Pi/6..Pi/2,color=blue):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "B2:=plot3d(h2(t,phi),t=0..P i/2,phi=Pi/6..Pi/2,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "B3:=plot3d(h3(rho,phi),rho=1..3,phi=Pi/6..Pi/2,color=cyan):" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "B4:=plot3d(h4(rho,t),rho=1 ..3,t=0..Pi/2,color=magenta,axes=NORMAL):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 21 "display(B1,B2,B3,B4);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 301 15 "C3M4 Problems: " }{TEXT -1 53 "Use Maple to parameterize and to plot using plot3d .\n" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 29 " 1. S , the cylinder " } {XPPEDIT 18 0 "x^2+y^2 = 3;" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F(\"\"$ " }{TEXT -1 13 " for 0 <= " }{XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT -1 18 " <= 2 and its top." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 " \+ 2. T , the triangular plate " }{XPPEDIT 18 0 "3*x+y+4*z = 12;" "6 #/,(*&\"\"$\"\"\"%\"xGF'F'%\"yGF'*&\"\"%F'%\"zGF'F'\"#7" }{TEXT -1 26 " in the first octant (" }{XPPEDIT 18 0 "0 <= x;" "6#1\"\"!%\"xG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "0 <= y;" "6#1\"\"!%\"yG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "0 <= z;" "6#1\"\"!%\"zG" }{TEXT -1 2 ")." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 78 " 3. U , the portion of the s phere of radius 3 that lies above the cone " }{XPPEDIT 18 0 "x^2+y^2 \+ = z^2;" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F(*$%\"zGF'" }{TEXT -1 27 " , and in the half-plane " }{XPPEDIT 18 0 "0 <= y;" "6#1\"\"!%\"yG" } {TEXT -1 30 " . And include the cone with " }{XPPEDIT 18 0 "0 <= y;" "6#1\"\"!%\"yG" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "0 <= z;" "6#1\"\" !%\"zG" }{TEXT -1 3 " ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 " \+ 4. For " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 100 " a constant as signed by your instructor (see the preface), plot the part of the surf ace given by " }{XPPEDIT 18 0 "z = k+5-sqrt(x)-sqrt(y);" "6#/%\"zG,* %\"kG\"\"\"\"\"&F'-%%sqrtG6#%\"xG!\"\"-F*6#%\"yGF-" }{TEXT -1 37 " t hat lies above the region in the " }{TEXT 302 2 "xy" }{TEXT -1 34 "-pl ane bounded by the graphs of " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\" !" }{TEXT -1 4 " , " }{XPPEDIT 18 0 "y = 0;" "6#/%\"yG\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "x+y = k+1;" "6#/,&%\"xG\"\"\"%\"yGF&,&% \"kGF&F&F&" }{TEXT -1 48 " . Include the coordinate axes in your pict ure." }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }