{VERSION 2 3 "SUN SPARC SOLARIS" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "2 D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "Help Headin g" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE " " -1 259 "" 0 1 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 0 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "H eading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 " " 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 } {PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Helvetica" 1 12 0 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 0 0 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 3" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 0 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 4" -1 259 1 {CSTYLE "" -1 -1 "He lvetica" 1 12 0 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 5" -1 260 1 {CSTYLE "" -1 -1 "Helvetica" 1 12 0 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R 3 Font 6" -1 261 1 {CSTYLE "" -1 -1 "Helvetica" 1 12 0 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 7" -1 262 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 0 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 8" -1 263 1 {CSTYLE "" -1 -1 "Helvetica" 1 12 0 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 9" -1 264 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 0 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R 3 Font 10" -1 265 1 {CSTYLE "" -1 -1 "Helvetica" 1 12 0 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 11" -1 266 1 {CSTYLE "" -1 -1 "Helvetica" 1 12 0 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 269 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 18 "" 0 "" {TEXT -1 42 "mvcal2: Maple commands for vector calculus" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT -1 85 "Based o n the calcplot-based worksheet, examplesR3.ms, prepared by Tim Murdoch , 1994)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 268 "" 0 "" {TEXT -1 36 "Douglas B. Meade (meade@math.sc.edu)" }}{PARA 269 "" 0 "" {TEXT -1 11 "17 Feb 1997" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with( plots ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 " setoptions( labels=[`x`,`y`] ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 " setoptions3d( labels=[`x`,`y`,`z`] ):" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 22 "Note: availability of " }{TEXT 19 6 "mvcal2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "This worksheet utilizes the " }{TEXT 19 6 "mvcal2" }{TEXT -1 109 " package. Users on the Mathematics Network at USC can load this p ackage as illustrated below -- be sure that " }{TEXT 19 7 "libname" } {TEXT -1 85 " is appropriately modified. Users of other systems will h ave to find a local copy of " }{TEXT 19 6 "mvcal2" }{TEXT -1 67 " or i nstall the package on their own. Additional information about " } {TEXT 19 6 "mvcal2" }{TEXT -1 24 " can be obtained by FTP." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "libname := `/m1/faculty/meade/computer/symbolic/maple /lib-mVr4/mvcal2/`,libname;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with ( mvcal2 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 25 "Plotting of Vector Fields" }}{SECT 1 {PARA 4 "" 0 "" {TEXT 26 3 "1. " }{TEXT 19 15 "plotvectorfield" } {TEXT 26 5 " and " }{TEXT 19 16 "plots[fieldplot]" }{TEXT 26 1 " " }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 156 "Two-dimensional vectors can be di splayed by specifying the two components of F, the ranges for the two \+ variables, and the spacing between vectors (see also " }{HYPERLNK 17 " plots[fieldplot]" 2 "fieldplot" "" }{TEXT -1 2 ")." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plotvectorfield( [x,y], x=-5..5, y=-5..5, 2 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "fieldplot( [x,y], x= -5..5, y=-5..5, grid=[6,6] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "plotvectorfield( [-y,x], x=-6..6, y=-6..6, 3 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "fieldplot( [-y,x], x=-6..6, y=-6..6 , grid=[5,5] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT 26 3 "2. " }{TEXT 19 17 "plotvectorfield 3d" }{TEXT 26 5 " and " }{TEXT 19 18 "plots[fieldplot3d]" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "plotvectorfield3d( [2*x,2*y,1], x=- 1..1, y=-1..1, 0.4, z=-1..1, layers=5 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "fieldplot3d([2*x,2*y,1],x=-1..1,y=-1..1,z=-1..1,grid= [5,5,5]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "plotvectorfiel d3d([y*z,-x*z,x*y],x=-5..5,y=-5..5,0.5,z=0..1,layers=4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "3. " }{TEXT 19 11 "drawvfgraph" }{TEXT -1 5 " and " }{TEXT 19 13 "drawvfsurface" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "drawvfgraph(z=sqrt(1-x^2-y^2),[x,y,z],x=-1..1,y=-1..1 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "drawvfsurface([p*cos( q),p*sin(q),p^2],[y,x+2*y,z-2*x],p=0..2,q=0..2*Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 29 "Curves in 2- and 3-dimensions" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "4. " }{TEXT 19 10 "tracecurve" }{TEXT -1 38 " (animated graphing of a curve in 2-D)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "tracecur ve( [cos(t),sin(t), t=0..2*Pi] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "tracecurve( [sin(t),cos(t), t=0..2*Pi], scaling=CONST RAINED, color=BLUE );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "tr acecurve( [t^2-1,t^3-t, t=-1.5..1.5], scaling=CONSTRAINED, color=red); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 3 "5. " }{TEXT 19 15 "tracespacecurve" }{TEXT -1 38 " \+ (animated graphing of a curve in 3-D)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "tracespacecurve( [cos(t),sin(t),t^2], t=0..2*Pi, axes =BOXED);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "tracespacecurv e( [(4+sin(3*t))*cos(2*t),(4+sin(3*t))*sin(2*t),cos(3*t)], t=0..2*Pi, \+ color=RED, axes=BOXED );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "6. " }{TEXT 19 12 "velocityd emo" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "veloc itydemo( [t*cos(t),t*sin(t),t], t=0..2*Pi );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "velocitydemo([t*cos(t),t*sin(t),t],t=0..2*Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "velocitydemo([t*cos(t),t* sin(t),t],t=0..2*Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "ve locitydemo([t*cos(t),t*sin(t),t],t=0..2*Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "velocitydemo( [t^2-1,t^3-t,0], t=-1.5..1.5, scal ing=CONSTRAINED, color=red, orientation=[0,180]);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 54 "T wo-Dimensional Regions and Surfaces above 2-D Regions" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "7. " }{TEXT 19 8 "dxdyplot" }{TEXT -1 7 " \+ and " }{TEXT 19 8 "dydxplot" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Us e " }{TEXT 19 8 "dxdyplot" }{TEXT -1 11 " to plot a " }{TEXT 257 19 " horizontally-simple" }{TEXT -1 23 " domain of the form: " }{XPPEDIT 18 0 "phi[1](y)" "-&%$phiG6#\"\"\"6#%\"yG" }{TEXT -1 5 " <= " } {XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 5 " <= " }{XPPEDIT 18 0 "phi[2 ](y)" "-&%$phiG6#\"\"#6#%\"yG" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "a" " I\"aG6\"" }{TEXT -1 6 " <= " }{XPPEDIT 18 0 "y" "I\"yG6\"" }{TEXT -1 5 " <= " }{XPPEDIT 18 0 "b" "I\"bG6\"" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 5 "Use " }{TEXT 19 8 "dydxplot" }{TEXT -1 11 " to plo t a " }{TEXT 256 17 "vertically-simple" }{TEXT -1 23 " domain of the f orm: " }{XPPEDIT 18 0 "phi[1](x)" "-&%$phiG6#\"\"\"6#%\"xG" }{TEXT -1 5 " <= " }{XPPEDIT 18 0 "y" "I\"yG6\"" }{TEXT -1 5 " <= " } {XPPEDIT 18 0 "phi[2](x)" "-&%$phiG6#\"\"#6#%\"xG" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "a" "I\"aG6\"" }{TEXT -1 6 " <= " }{XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 5 " <= " }{XPPEDIT 18 0 "b" "I\"bG6\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "dydxplot( y=x^2. .x+1, x=0..1 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "dxdyplot ( x=0..sqrt(y), y=0..1 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "dydxplot( y=x^2..1, x=0..1 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "dydxplot( y=0..2*x+1, x=0..1, view=[-1..2,0..4], titl e=`A trapezoidal region` );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "8. " }{TEXT 19 8 "drdtplo t" }{TEXT -1 7 " and " }{TEXT 19 8 "dtdrplot" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Use " }{TEXT 19 8 "drdtplot" }{TEXT -1 12 " to plot a " }{TEXT 258 15 "radially-simple" }{TEXT -1 22 " domain of the form: \+ " }{XPPEDIT 18 0 "R[1](theta)" "-&%\"RG6#\"\"\"6#%&thetaG" }{TEXT -1 4 " <= " }{XPPEDIT 18 0 "r" "I\"rG6\"" }{TEXT -1 4 " <= " }{XPPEDIT 18 0 "R[2](theta)" "-&%\"RG6#\"\"#6#%&thetaG" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "theta[0]" "&%&thetaG6#\"\"!" }{TEXT -1 4 " <= " } {XPPEDIT 18 0 "theta" "I&thetaG6\"" }{TEXT -1 4 " <= " }{XPPEDIT 18 0 "theta[1]" "&%&thetaG6#\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Use " }{TEXT 19 8 "dtdrplot" }{TEXT -1 33 " to plot a do main of the form: " }{XPPEDIT 18 0 "Theta[1](r)" "-&%&ThetaG6#\"\"\"6 #%\"rG" }{TEXT -1 4 " <= " }{XPPEDIT 18 0 "theta" "I&thetaG6\"" } {TEXT -1 4 " <= " }{XPPEDIT 18 0 "Theta[2](r)" "-&%&ThetaG6#\"\"#6#%\" rG" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "r[0]" "&%\"rG6#\"\"!" }{TEXT -1 4 " <= " }{XPPEDIT 18 0 "r" "I\"rG6\"" }{TEXT -1 4 " <= " }{XPPEDIT 18 0 "r[1]" "&%\"rG6#\"\"\"" }{TEXT -1 6 ". (" }{TEXT 259 53 "Does \+ anyone have a good name for regions of this type" }{TEXT -1 2 "?)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "drdtplot( r=1..1+cos(theta), theta=-Pi/2..Pi/2 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "dt drplot( theta=0..r, r=Pi/2..Pi );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "plots[display]( \{ drdtplot( r=sin(3*theta)..sin(thet a), theta=Pi/4..Pi/3 )," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 " \+ drdtplot( r=0..sin(theta), theta=Pi/3..Pi/2 ) \}," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 96 " scaling=constrained, titl e=`A region between r=sin(3*theta) and r=sin(theta)` );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "9. " }{TEXT 19 9 "drawsolid" }{TEXT -1 4 " ( " }{TEXT 19 11 "xy graphplot" }{TEXT 18 7 " and " }{TEXT 19 11 "yxgraphplot" }{TEXT 18 14 " in calcplot )" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Use " } {TEXT 19 9 "drawsolid" }{TEXT -1 39 " to plot a solid formed by a sur face " }{XPPEDIT 18 0 "z=f(x,y)" "/%\"zG-%\"fG6$%\"xG%\"yG" }{TEXT -1 55 " and a horizontally- or vertically-simple region (see " } {TEXT 19 8 "dxdyplot" }{TEXT -1 5 " and " }{TEXT 19 8 "dydxplot" } {TEXT -1 2 ")." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "drawsolid ( x*y, y=0..sqrt(1-x^2), x=-1..1 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "drawsolid( sin(Pi*(x^2+y^2)), y=x^2..x^(1/4), x=0..1 \+ );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "drawsolid( sin(Pi*(x^ 2+y^2)), y=-sqrt(4-x^2)..sqrt(4-x^2), x=-2..2 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "drawsolid( x*y, x=-sqrt(1-y^2)..sqrt(1-y^2), \+ y=0..1 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 25 "Three-Dimensional Regions" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "10. " }{TEXT 19 10 "dxdydzplot" }{TEXT -1 2 ", " }{TEXT 19 10 "dxdzdyplot" }{TEXT -1 2 ", " }{TEXT 19 10 "dydxdz plot" }{TEXT -1 2 ", " }{TEXT 19 10 "dydzdxplot" }{TEXT -1 2 ", " } {TEXT 19 10 "dzdxdyplot" }{TEXT -1 6 ", and " }{TEXT 19 10 "dzdydxplot " }{TEXT -1 1 " " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "Plotting of t hree-dimesional solids that can be described by one of the six orderin g of the Cartesian coordinates (" }{XPPEDIT 18 0 "x" "I\"xG6\"" } {TEXT -1 1 "," }{XPPEDIT 18 0 "y" "I\"yG6\"" }{TEXT -1 1 "," } {XPPEDIT 18 0 "z" "I\"zG6\"" }{TEXT -1 2 ")." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "dydzdxplot( y=-sqrt(1-x^2)..1-z, z=0..1, x=-1..1 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "dzdydxplot( z=0..8-2 *x^2, y=x^2..8-x^2, x=-2..2, title=`Square cross-sections`);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "dzdydxplot( z=0..sqrt((4-x^ 2)^2-(y-4)^2), y=x^2..8-x^2, x=-2..2, title=`Semi-circular cross-secti ons (almost)` );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "dzdydxp lot( z=0..((4-x^2)^2-(y-4)^2)^(0.5), y=x^2..(8-x^2), x=-2..2, title=`F ull semi-circles`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "11. " }{TEXT 19 10 "drdtdzplot" } {TEXT -1 2 ", " }{TEXT 19 10 "drdzdtplot" }{TEXT -1 2 ", " }{TEXT 19 10 "dtdrdzplot" }{TEXT -1 2 ", " }{TEXT 19 10 "dtdzdrplot" }{TEXT -1 2 ", " }{TEXT 19 10 "dzdrdtplot" }{TEXT -1 6 ", and " }{TEXT 19 10 "dz dtdrplot" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "Plotting of three-dim esional solids that can be described by one of the six ordering of the cylindrical coordinates (" }{XPPEDIT 18 0 "r" "I\"rG6\"" }{TEXT -1 1 "," }{XPPEDIT 18 0 "theta" "I&thetaG6\"" }{TEXT -1 1 "," }{XPPEDIT 18 0 "z" "I\"zG6\"" }{TEXT -1 14 "). Note that " }{TEXT 19 1 "t" }{TEXT -1 13 " represents " }{XPPEDIT 18 0 "theta" "I&thetaG6\"" }{TEXT -1 24 " in the procedure names." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "drdtdzplot( r = 2..3, theta = Pi/4..Pi/2, z = 0..1 );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "dzdtdrplot( z = r^2..2+r*cos (theta), theta=0..2*Pi, r = 0..1 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "dzdrdtplot( z=0..(1-r^2)^(0.5),r=0..cos(theta), theta =0..Pi, grid=[40,15] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 " dzdtdrplot( z=-(1-r^2*cos(theta)^2)..(1-r^2*cos(theta)^2), theta=0..2* Pi, r=0..1 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "dzdrdtplot ( z=0..sqrt(4-r^2), r=0..1+cos(theta), theta=0..2*Pi );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "p1:=dzdrdtplot( z=0..sqrt(1-r^2), r =0..1, theta=-Pi/2..Pi/2 ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "p2:= dzdrdtplot( z=0..sqrt(1-r^2), r=0..1+cos(theta), theta=Pi/2..3*Pi/2 ): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "plots[display]( \{ p1, p2 \} ); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 4 "12. " }{TEXT 19 14 "drhodtdphiplot" }{TEXT -1 2 ", \+ " }{TEXT 19 14 "drhodphidtplot" }{TEXT -1 2 ", " }{TEXT 19 14 "dphidrh odtplot" }{TEXT -1 2 ", " }{TEXT 19 14 "dphidtdrhoplot" }{TEXT -1 2 ", " }{TEXT 19 14 "dtdphidrhoplot" }{TEXT -1 6 ", and " }{TEXT 19 14 "dt drhodphiplot" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "Plotting of three -dimesional solids that can be described by one of the six ordering of the spherical coordinates (" }{XPPEDIT 18 0 "rho" "I$rhoG6\"" }{TEXT -1 1 "," }{XPPEDIT 18 0 "theta" "I&thetaG6\"" }{TEXT -1 1 "," } {XPPEDIT 18 0 "phi" "I$phiG6\"" }{TEXT -1 76 "). Note that p represent s rho and t represents theta in the procedure names." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "dphidrhodtplot( phi=0..Pi/3, rho=0..1, th eta=0..2*Pi );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "P1:=drhod tdphiplot( rho=sec(phi)..2/(cos(phi)+sin(phi)), theta=0..2*Pi, phi=0.. Pi/4 ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "P2:=drhodtdphiplot( rho= 0..cot(phi)*csc(phi), theta=0..2*Pi, phi=Pi/4..Pi/2 ):" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 46 "plots[display]( \{ P1, P2 \}, style=wireframe \+ );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 41 "Double Integrals: Riemann sums and slices" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "13. " }{TEXT 19 8 "blockapp" } {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "f := 3*x^2 \+ + (y-2)^2 + 2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "blockapp( f, x=0..1, y=0..3, 5, 5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "blockapp( f, x=0..1, y=0..3, 10, 10);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "14. " } {TEXT 19 9 "xsliceapp" }{TEXT -1 5 " and " }{TEXT 19 9 "ysliceapp" } {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "f := (x,y) \+ -> 3*x^2 + (y-2)^2 + 2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 " xsliceapp( f(x,y), x=0..1, y=0..3 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "ysliceapp( f(x,y), x=0..1, y=0..3 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "15. " }{TEXT 19 11 "approxint2d" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f := 3*x^2 + (y-2)^2 +2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "approxint2d( f, x=0..1, y=0..3, 5, \+ 5 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "approxint2d( f, x=0 ..1, y=0..3, 10, 10 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "a pproxint2d( f, x=0..1, y=0..3, 100, 100 );" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "And, to tes t the limit as the order of the partition increases without bound:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "approxint2d( f, x=0..1, y=0 ..3, n, n );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "limit( \", \+ n=infinity );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 18 "Other Commands in " }{TEXT 19 6 " mvcal2" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "For a full list of comma nds and additional information on " }{HYPERLNK 17 "mvcal2" 2 "mvcal2" "" }{TEXT -1 40 ", consult the links to the on-line help." }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }