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"Times" 1 18 0 0 0 1 2 1 2 2 2 2 0 0 0 1 } {PSTYLE "" -1 209 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 210 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 211 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "" -1 211 "Time s" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 212 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 213 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 214 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 215 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 } {CSTYLE "" -1 216 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }} {SECT 0 {PARA 208 "" 0 "" {TEXT 209 0 "" }{TEXT 210 0 "" }{TEXT 204 9 "Maple Lab" }}{PARA 209 "" 0 "" {TEXT 204 6 "Week 3" }}{PARA 210 "" 0 "" {TEXT 204 0 "" }{TEXT 204 0 "" }{TEXT 204 14 "Plotting in 3D" }} {PARA 211 "" 0 "" {TEXT 204 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 204 289 "The \"plots\" library is a module that needs to be \"plugged in\" in order to use Maple commands such as \"plot3d\" and \"spacecurve\". The first thing that should be done at the beginning of this lab is \+ to load the \"plots\" library using the \"with\" command. The syntax \+ is as follows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plot s):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 37 "(Go to the line above and hit Enter.)" }}{PARA 0 "" 0 "" {TEXT 204 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 440 "Notice first the use of a colon to end the above sta tement. Without the colon, the above command would output a very long list of exactly what it had done. The colon can be used this way to \+ suppress output. The next thing to note is the warning that was given . Nearly every library that is \"plugged in\" outputs a similar warni ng, and are usually of no consequence, unless you plan on using a chan ged commands such as \"changecoords\"." }}}{PARA 0 "" 0 "" {TEXT 217 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT 218 36 " Cartesian (rectangular) \+ coordinates" }}{EXCHG {PARA 0 "" 0 "" {TEXT 217 32 "To plot the graph \+ of a function " }{XPPEDIT 217 0 "Typesetting:-mrow(Typesetting:-mi(\"z \", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"= \", mathvariant = \"normal\", fence = \"false\", separator = \"false\" , stretchy = \"false\", symmetric = \"false\", largeop = \"false\", mo vablelimits = \"false\", accent = \"false\", form = \"infix\", lspace \+ = \"thickmathspace\", rspace = \"thickmathspace\", minsize = \"1\", ma xsize = \"infinity\"), Typesetting:-mrow(Typesetting:-mi(\"f\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"&ApplyFuncti on;\", mathvariant = \"normal\", fence = \"false\", separator = \"fals e\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", form = \"infix\", lspa ce = \"0em\", rspace = \"0em\", minsize = \"1\", maxsize = \"infinity \"), Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mi(\"x\", ita lic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\",\", math variant = \"normal\", fence = \"false\", separator = \"true\", stretch y = \"false\", symmetric = \"false\", largeop = \"false\", movablelimi ts = \"false\", accent = \"false\", form = \"infix\", lspace = \"0em\" , rspace = \"verythickmathspace\", minsize = \"1\", maxsize = \"infini ty\"), Typesetting:-mi(\"y\", italic = \"true\", mathvariant = \"itali c\")), mathvariant = \"normal\")), Typesetting:-mo(\"⁢ \", mathvariant = \"normal\", fence = \"false\", separator = \"false\" , stretchy = \"false\", symmetric = \"false\", largeop = \"false\", mo vablelimits = \"false\", accent = \"false\", form = \"\", lspace = \"0 em\", rspace = \"0em\", minsize = \"1\", maxsize = \"infinity\"));" "- I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6&-I#miGF$6%Q\"zF' /%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$60Q\"=F'/F3Q'normal F'/%&fenceGQ&falseF'/%*separatorGF=/%)stretchyGF=/%*symmetricGF=/%(lar geopGF=/%.movablelimitsGF=/%'accentGF=/%%formGQ&infixF'/%'lspaceGQ/thi ckmathspaceF'/%'rspaceGFO/%(minsizeGQ\"1F'/%(maxsizeGQ)infinityF'-F#6% -F,6%Q\"fF'F/F2-F660Q0⁡F'F9F;F>F@FBFDFFFHFJ/FNQ$0emF'/FQ F[oFRFU-I(mfencedGF$6$-F#6%-F,6%Q\"xF'F/F2-F660Q\",F'F9F;/F?F1F@FBFDFF FHFJFjn/FQQ3verythickmathspaceF'FRFU-F,6%Q\"yF'F/F2F9-F660Q1&Invisible Times;F'F9F;F>F@FBFDFFFH/FKQ!F'FjnF\\oFRFU" }{TEXT 204 16 "you can use the " }{TEXT 211 6 "plot3d" }{TEXT 204 9 " command." }}{PARA 0 "" 0 " " {TEXT 217 31 "Let's look at a few examples.\n" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 38 "plot3d(sin(x)*cos(y),x=-3..3,y=-3..3);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 204 34 "Let's change some of the settings :" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 0 38 "plot3d(sin(x)*cos(y) ,x=-7..7,y=-7..7);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 0 73 "plot3d(sin(x)*cos(y),x=-10..10,y=-10..10,axes=boxed,s caling=constrained);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 309 "Click on the plot; this will activate a new menu bar. T he angles theta and phi are the spherical coordinate angles,, they cha nge as you rotate the plot. The next buttons control how the plot is d isplayed (try the different settings), how the coordinate axes are plo tted and whether the plot is drawn to scale." }}{PARA 0 "" 0 "" {TEXT 204 1 " " }}}{EXCHG {PARA 206 "" 0 "" {TEXT 219 31 "Let us try some ot her function:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f:=sqrt(1- x^2-y^2);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot3d(f,x=-1..1,y=-1..1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 125 "How can you explain those strange \"spikes\" on the graph? (You m ay need this explanation for one of your homework problems.)" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 217 0 "" }}{PARA 0 "" 0 "" {TEXT 217 73 " What if you need to draw the graph of an implicit function, for instan ce " }{XPPEDIT 217 0 "Typesetting:-mrow(Typesetting:-msup(Typesetting: -mi(\"x\", italic = \"true\", mathvariant = \"italic\"), Typesetting:- mrow(Typesetting:-mn(\"2\", mathvariant = \"normal\")), superscriptshi ft = \"0\"), Typesetting:-mo(\"&uminus0;\", mathvariant = \"normal\", \+ fence = \"false\", separator = \"false\", stretchy = \"false\", symmet ric = \"false\", largeop = \"false\", movablelimits = \"false\", accen t = \"false\", form = \"infix\", lspace = \"mediummathspace\", rspace \+ = \"mediummathspace\", minsize = \"1\", maxsize = \"infinity\"), Types etting:-msup(Typesetting:-mi(\"y\", italic = \"true\", mathvariant = \+ \"italic\"), Typesetting:-mrow(Typesetting:-mn(\"2\", mathvariant = \" normal\")), superscriptshift = \"0\"), Typesetting:-mo(\"+\", mathvari ant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits \+ = \"false\", accent = \"false\", form = \"infix\", lspace = \"mediumma thspace\", rspace = \"mediummathspace\", minsize = \"1\", maxsize = \" infinity\"), Typesetting:-msup(Typesetting:-mi(\"z\", italic = \"true \", mathvariant = \"italic\"), Typesetting:-mrow(Typesetting:-mn(\"2\" , mathvariant = \"normal\")), superscriptshift = \"0\"), Typesetting:- mo(\"=\", mathvariant = \"normal\", fence = \"false\", separator = \"f alse\", stretchy = \"false\", symmetric = \"false\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", form = \"infix\", l space = \"thickmathspace\", rspace = \"thickmathspace\", minsize = \"1 \", maxsize = \"infinity\"), Typesetting:-mn(\"0.\", mathvariant = \"n ormal\"), Typesetting:-mo(\" \", mathvariant = \"normal\", fence = \"f alse\", separator = \"false\", stretchy = \"false\", symmetric = \"fal se\", largeop = \"false\", movablelimits = \"false\", accent = \"false \", form = \"\", lspace = \"0em\", rspace = \"0em\", minsize = \"1\", \+ maxsize = \"infinity\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI (_syslibGF'6*-I%msupGF$6%-I#miGF$6%Q\"xF'/%'italicGQ%trueF'/%,mathvari antGQ'italicF'-F#6#-I#mnGF$6$Q\"2F'/F6Q'normalF'/%1superscriptshiftGQ \"0F'-I#moGF$60Q*&uminus0;F'F>/%&fenceGQ&falseF'/%*separatorGFI/%)stre tchyGFI/%*symmetricGFI/%(largeopGFI/%.movablelimitsGFI/%'accentGFI/%%f ormGQ&infixF'/%'lspaceGQ0mediummathspaceF'/%'rspaceGFen/%(minsizeGQ\"1 F'/%(maxsizeGQ)infinityF'-F,6%-F/6%Q\"yF'F2F5F8F@-FD60Q\"+F'F>FGFJFLFN FPFRFTFVFYFfnFhnF[o-F,6%-F/6%Q\"zF'F2F5F8F@-FD60Q\"=F'F>FGFJFLFNFPFRFT FV/FZQ/thickmathspaceF'/FgnF_pFhnF[o-F;6$Q#0.F'F>-FD60Q\"~F'F>FGFJFLFN FPFRFT/FWQ!F'/FZQ$0emF'/FgnFjpFhnF[o" }{TEXT 217 22 "Then your best be t is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "implicitplot3d(x^2 -y^2+z^2=0,x=-3..3,y=-3..3,z=-3..3,grid=[30,30,30]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 218 38 " Cylindrical and sphe rical coordinates" }}{EXCHG {PARA 0 "" 0 "" {TEXT 204 124 "When doing \+ your homework, you may want to modify some of these examples. Keep in \+ mind that, similar to how Maple understands" }}{PARA 0 "" 0 "" {TEXT 204 5 "that " }{XPPEDIT 2 0 "Typesetting:-mrow(Typesetting:-mi(\"z\", \+ italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"=\", m athvariant = \"normal\", fence = \"false\", separator = \"false\", str etchy = \"false\", symmetric = \"false\", largeop = \"false\", movable limits = \"false\", accent = \"false\", form = \"infix\", lspace = \"t hickmathspace\", rspace = \"thickmathspace\", minsize = \"1\", maxsize = \"infinity\"), Typesetting:-mrow(Typesetting:-mi(\"f\", italic = \" true\", mathvariant = \"italic\"), Typesetting:-mo(\"⁡\" , mathvariant = \"normal\", fence = \"false\", separator = \"false\", \+ stretchy = \"false\", symmetric = \"false\", largeop = \"false\", mova blelimits = \"false\", accent = \"false\", form = \"infix\", lspace = \+ \"0em\", rspace = \"0em\", minsize = \"1\", maxsize = \"infinity\"), T ypesetting:-mfenced(Typesetting:-mrow(Typesetting:-mi(\"x\", italic = \+ \"true\", mathvariant = \"italic\"), Typesetting:-mo(\",\", mathvarian t = \"normal\", fence = \"false\", separator = \"true\", stretchy = \" false\", symmetric = \"false\", largeop = \"false\", movablelimits = \+ \"false\", accent = \"false\", form = \"infix\", lspace = \"0em\", rsp ace = \"verythickmathspace\", minsize = \"1\", maxsize = \"infinity\") , Typesetting:-mi(\"y\", italic = \"true\", mathvariant = \"italic\")) , mathvariant = \"normal\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+m odulenameG6\"I,TypesettingGI(_syslibGF'6&-I#miGF$6%Q\"zF'/%'italicGQ%t rueF'/%,mathvariantGQ'italicF'-I#moGF$60Q\"=F'/F3Q'normalF'/%&fenceGQ& falseF'/%*separatorGF=/%)stretchyGF=/%*symmetricGF=/%(largeopGF=/%.mov ablelimitsGF=/%'accentGF=/%%formGQ&infixF'/%'lspaceGQ/thickmathspaceF' /%'rspaceGFO/%(minsizeGQ\"1F'/%(maxsizeGQ)infinityF'-F#6%-F,6%Q\"fF'F/ F2-F660Q0⁡F'F9F;F>F@FBFDFFFHFJ/FNQ$0emF'/FQF[oFRFU-I(mfe ncedGF$6$-F#6%-F,6%Q\"xF'F/F2-F660Q\",F'F9F;/F?F1F@FBFDFFFHFJFjn/FQQ3v erythickmathspaceF'FRFU-F,6%Q\"yF'F/F2F9-F,6#Q!F'" }{TEXT 204 15 ", Ma ple treats " }{XPPEDIT 2 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), T ypesetting:-mrow(Typesetting:-mi(\"r\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \+ \"false\", form = \"infix\", lspace = \"thickmathspace\", rspace = \"t hickmathspace\", minsize = \"1\", maxsize = \"infinity\"), Typesetting :-mrow(Typesetting:-mi(\"f\", italic = \"true\", mathvariant = \"itali c\"), Typesetting:-mo(\"⁡\", mathvariant = \"normal\", f ence = \"false\", separator = \"false\", stretchy = \"false\", symmetr ic = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", form = \"infix\", lspace = \"0em\", rspace = \"0em\", mi nsize = \"1\", maxsize = \"infinity\"), Typesetting:-mfenced(Typesetti ng:-mrow(Typesetting:-mi(\"z\", italic = \"true\", mathvariant = \"ita lic\"), Typesetting:-mo(\",\", mathvariant = \"normal\", fence = \"fal se\", separator = \"true\", stretchy = \"false\", symmetric = \"false \", largeop = \"false\", movablelimits = \"false\", accent = \"false\" , form = \"infix\", lspace = \"0em\", rspace = \"verythickmathspace\", minsize = \"1\", maxsize = \"infinity\"), Typesetting:-mi(\"theta\", \+ italic = \"false\", mathvariant = \"normal\")), mathvariant = \"normal \")), Typesetting:-mi(\"\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+m odulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6&-F,6%Q\"rF '/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$60Q\"=F'/F8Q'norma lF'/%&fenceGQ&falseF'/%*separatorGFB/%)stretchyGFB/%*symmetricGFB/%(la rgeopGFB/%.movablelimitsGFB/%'accentGFB/%%formGQ&infixF'/%'lspaceGQ/th ickmathspaceF'/%'rspaceGFT/%(minsizeGQ\"1F'/%(maxsizeGQ)infinityF'-F#6 %-F,6%Q\"fF'F4F7-F;60Q0⁡F'F>F@FCFEFGFIFKFMFO/FSQ$0emF'/F VF`oFWFZ-I(mfencedGF$6$-F#6%-F,6%Q\"zF'F4F7-F;60Q\",F'F>F@/FDF6FEFGFIF KFMFOF_o/FVQ3verythickmathspaceF'FWFZ-F,6%Q&thetaF'/F5FBF>F>F+F+" } {TEXT 204 37 "in cylindrical coordinate system and " }{XPPEDIT 2 0 "Ty pesetting:-mrow(Typesetting:-mi(\"rho\", italic = \"false\", mathvaria nt = \"normal\"), Typesetting:-mo(\"=\", mathvariant = \"normal\", fen ce = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", form = \"infix\", lspace = \"thickmathspace\", rspace = \" thickmathspace\", minsize = \"1\", maxsize = \"infinity\"), Typesettin g:-mrow(Typesetting:-mi(\"f\", italic = \"true\", mathvariant = \"ital ic\"), Typesetting:-mo(\"⁡\", mathvariant = \"normal\", \+ fence = \"false\", separator = \"false\", stretchy = \"false\", symmet ric = \"false\", largeop = \"false\", movablelimits = \"false\", accen t = \"false\", form = \"infix\", lspace = \"0em\", rspace = \"0em\", m insize = \"1\", maxsize = \"infinity\"), Typesetting:-mfenced(Typesett ing:-mrow(Typesetting:-mi(\"theta\", italic = \"false\", mathvariant = \"normal\"), Typesetting:-mo(\",\", mathvariant = \"normal\", fence = \"false\", separator = \"true\", stretchy = \"false\", symmetric = \" false\", largeop = \"false\", movablelimits = \"false\", accent = \"fa lse\", form = \"infix\", lspace = \"0em\", rspace = \"verythickmathspa ce\", minsize = \"1\", maxsize = \"infinity\"), Typesetting:-mi(\"phi \", italic = \"false\", mathvariant = \"normal\")), mathvariant = \"no rmal\")), Typesetting:-mo(\"⁢\", mathvariant = \"normal \", fence = \"false\", separator = \"false\", stretchy = \"false\", sy mmetric = \"false\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", form = \"infix\", lspace = \"thickmathspace\", rspa ce = \"thickmathspace\", minsize = \"1\", maxsize = \"infinity\"));" " -I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6&-I#miGF$6%Q$rho F'/%'italicGQ&falseF'/%,mathvariantGQ'normalF'-I#moGF$60Q\"=F'F2/%&fen ceGF1/%*separatorGF1/%)stretchyGF1/%*symmetricGF1/%(largeopGF1/%.movab lelimitsGF1/%'accentGF1/%%formGQ&infixF'/%'lspaceGQ/thickmathspaceF'/% 'rspaceGFL/%(minsizeGQ\"1F'/%(maxsizeGQ)infinityF'-F#6%-F,6%Q\"fF'/F0Q %trueF'/F3Q'italicF'-F660Q0⁡F'F2F9F;F=F?FAFCFEFG/FKQ$0em F'/FNF\\oFOFR-I(mfencedGF$6$-F#6%-F,6%Q&thetaF'F/F2-F660Q\",F'F2F9/FF@FBFDFFFHFJ/FNQ$0emF'/FQFinFRFU" }{TEXT 204 1 " " }}{PARA 0 "" 0 "" {TEXT 204 60 "After you execute the command, try replacing the l imits for " }{TEXT 212 1 "z" }{TEXT 204 6 " with " }{TEXT 213 5 "-1..1 " }{TEXT 204 1 "." }}{PARA 0 "" 0 "" {TEXT 204 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "plot3d(z,theta=0..2*Pi,z=0..1,coords=cylind rical);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 65 "To illustrate an impo rtant point, let us repeat the same command:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "plot3d(z,theta=0..2*pi,z=0..1,coords=cylindrical) ;" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 50 "Can you \+ determine what happened and how to fix it?" }}{PARA 0 "" 0 "" {TEXT 204 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 25 "What if you need to p lot " }{XPPEDIT 2 0 "Typesetting:-mrow(Typesetting:-mi(\"theta\", ital ic = \"false\", mathvariant = \"normal\"), Typesetting:-mo(\"=\", math variant = \"normal\", fence = \"false\", separator = \"false\", stretc hy = \"false\", symmetric = \"false\", largeop = \"false\", movablelim its = \"false\", accent = \"false\", form = \"infix\", lspace = \"thic kmathspace\", rspace = \"thickmathspace\", minsize = \"1\", maxsize = \+ \"infinity\"), Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-m frac(Typesetting:-mn(\"1\", mathvariant = \"normal\"), Typesetting:-mn (\"2\", mathvariant = \"normal\"), linethickness = \"1\", denomalign = \"center\", numalign = \"center\", bevelled = \"false\"), Typesetting :-mo(\"⁢\", mathvariant = \"normal\", fence = \"false\" , separator = \"false\", stretchy = \"false\", symmetric = \"false\", \+ largeop = \"false\", movablelimits = \"false\", accent = \"false\", fo rm = \"infix\", lspace = \"0em\", rspace = \"0em\", minsize = \"1\", m axsize = \"infinity\"), Typesetting:-mrow(Typesetting:-mi(\"Pi\", ital ic = \"false\", mathvariant = \"normal\"), Typesetting:-mo(\"?\", math variant = \"normal\", fence = \"false\", separator = \"false\", stretc hy = \"false\", symmetric = \"false\", largeop = \"false\", movablelim its = \"false\", accent = \"false\", form = \"infix\", lspace = \"very thinmathspace\", rspace = \"verythinmathspace\", minsize = \"1\", maxs ize = \"infinity\"), Typesetting:-mo(\"⁢\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \" false\", symmetric = \"false\", largeop = \"false\", movablelimits = \+ \"false\", accent = \"false\", form = \"\", lspace = \"0em\", rspace = \"0em\", minsize = \"1\", maxsize = \"infinity\")), Typesetting:-mi( \"\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,Typeset tingGI(_syslibGF'6&-I#miGF$6%Q&thetaF'/%'italicGQ&falseF'/%,mathvarian tGQ'normalF'-I#moGF$60Q\"=F'F2/%&fenceGF1/%*separatorGF1/%)stretchyGF1 /%*symmetricGF1/%(largeopGF1/%.movablelimitsGF1/%'accentGF1/%%formGQ&i nfixF'/%'lspaceGQ/thickmathspaceF'/%'rspaceGFL/%(minsizeGQ\"1F'/%(maxs izeGQ)infinityF'-F#6'-F,6#Q!F'-I&mfracGF$6(-I#mnGF$6$FQF2-Fhn6$Q\"2F'F 2/%.linethicknessGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGFbo/%)beve lledGF1-F660Q1⁢F'F2F9F;F=F?FAFCFEFG/FKQ$0emF'/FNF[pFOFR -F#6%-F,6%Q#PiF'F/F2-F660Q\"?F'F2F9F;F=F?FAFCFEFG/FKQ2verythinmathspac eF'/FNFfpFOFR-F660FioF2F9F;F=F?FAFCFE/FHFYFjoF\\pFOFRFWFW" }{TEXT 204 18 " This should work:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "p lot3d([r,Pi/4,z],z=-1..1,r=0..1,coords=cylindrical,axes=normal);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 204 63 "For spherical coordinates, you ca n use the same syntax, except " }{TEXT 214 6 "coords" }{TEXT 204 11 " \+ should be " }{TEXT 215 9 "spherical" }{TEXT 204 12 ", of course:" }} {PARA 0 "" 0 "" {TEXT 204 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "plot3d([1,theta,phi],theta=0..Pi,phi=0..Pi/3,coords=spherical, s caling=constrained);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "plot3d([rho,theta,Pi/4],rho=0..3,theta=0..15*Pi/8,coo rds=spherical,scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 3 "" 0 "" {TEXT 218 13 " Space curves" }}{PARA 0 "" 0 "" {TEXT 204 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 204 128 "The \"spacecur ve\" function plots a curve or a set of curves in three-dimensional sp ace. The syntax that is need is as follows:" }}{PARA 0 "" 0 "" {TEXT 204 0 "" }}{PARA 0 "" 0 "" {TEXT 204 12 "spacecurve([" }{TEXT 204 22 " x expr, y expr, z expr" }{TEXT 204 24 "], var = a..b, options);" }} {PARA 0 "" 0 "" {TEXT 204 0 "" }}{PARA 0 "" 0 "" {TEXT 204 35 "Now let s try the following example:" }}{PARA 0 "" 0 "" {TEXT 204 0 "" }} {PARA 0 "" 0 "" {XPPEDIT 2 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesettin g:-mi(\"x\", italic = \"true\", mathvariant = \"italic\"), Typesetting :-mo(\"=\", mathvariant = \"normal\", fence = \"false\", separator = \+ \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"fa lse\", movablelimits = \"false\", accent = \"false\", form = \"infix\" , lspace = \"thickmathspace\", rspace = \"thickmathspace\", minsize = \+ \"1\", maxsize = \"infinity\"), Typesetting:-mrow(Typesetting:-mi(\"co s\", italic = \"false\", mathvariant = \"normal\"), Typesetting:-mo(\" ⁡\", mathvariant = \"normal\", fence = \"false\", separa tor = \"false\", stretchy = \"false\", symmetric = \"false\", largeop \+ = \"false\", movablelimits = \"false\", accent = \"false\", form = \"i nfix\", lspace = \"0em\", rspace = \"0em\", minsize = \"1\", maxsize = \"infinity\"), Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mi (\"t\", italic = \"true\", mathvariant = \"italic\")), mathvariant = \+ \"normal\")), Typesetting:-mi(\"\")), Typesetting:-mi(\"\")), Typesett ing:-mo(\";\", mathvariant = \"normal\", fence = \"false\", separator \+ = \"true\", stretchy = \"false\", symmetric = \"false\", largeop = \"f alse\", movablelimits = \"false\", accent = \"false\", form = \"infix \", lspace = \"0em\", rspace = \"thickmathspace\", minsize = \"1\", ma xsize = \"infinity\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_ syslibGF'6%-I#miGF$6#Q!F'-F#6%F+-F#6&-F,6%Q\"xF'/%'italicGQ%trueF'/%,m athvariantGQ'italicF'-I#moGF$60Q\"=F'/F:Q'normalF'/%&fenceGQ&falseF'/% *separatorGFD/%)stretchyGFD/%*symmetricGFD/%(largeopGFD/%.movablelimit sGFD/%'accentGFD/%%formGQ&infixF'/%'lspaceGQ/thickmathspaceF'/%'rspace GFV/%(minsizeGQ\"1F'/%(maxsizeGQ)infinityF'-F#6%-F,6%Q$cosF'/F7FDF@-F= 60Q0⁡F'F@FBFEFGFIFKFMFOFQ/FUQ$0emF'/FXFcoFYFfn-I(mfenced GF$6$-F#6#-F,6%Q\"tF'F6F9F@F+F+-F=60Q\";F'F@FB/FFF8FGFIFKFMFOFQFboFWFY Ffn" }}{PARA 0 "" 0 "" {XPPEDIT 2 0 "Typesetting:-mrow(Typesetting:-mi (\"\"), Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typ esetting:-mi(\"y\", italic = \"true\", mathvariant = \"italic\"), Type setting:-mo(\"=\", mathvariant = \"normal\", fence = \"false\", separa tor = \"false\", stretchy = \"false\", symmetric = \"false\", largeop \+ = \"false\", movablelimits = \"false\", accent = \"false\", form = \"i nfix\", lspace = \"thickmathspace\", rspace = \"thickmathspace\", mins ize = \"1\", maxsize = \"infinity\"), Typesetting:-mrow(Typesetting:-m i(\"sin\", italic = \"false\", mathvariant = \"normal\"), Typesetting: -mo(\"⁡\", mathvariant = \"normal\", fence = \"false\", \+ separator = \"false\", stretchy = \"false\", symmetric = \"false\", la rgeop = \"false\", movablelimits = \"false\", accent = \"false\", form = \"infix\", lspace = \"0em\", rspace = \"0em\", minsize = \"1\", max size = \"infinity\"), Typesetting:-mfenced(Typesetting:-mrow(Typesetti ng:-mi(\"t\", italic = \"true\", mathvariant = \"italic\")), mathvaria nt = \"normal\")), Typesetting:-mi(\"\")), Typesetting:-mi(\"\")), Typ esetting:-mo(\";\", mathvariant = \"normal\", fence = \"false\", separ ator = \"true\", stretchy = \"false\", symmetric = \"false\", largeop \+ = \"false\", movablelimits = \"false\", accent = \"false\", form = \"i nfix\", lspace = \"0em\", rspace = \"thickmathspace\", minsize = \"1\" , maxsize = \"infinity\"));" "-I%mrowG6#/I+modulenameG6\"I,Typesetting GI(_syslibGF'6%-I#miGF$6#Q!F'-F#6%F+-F#6&-F,6%Q\"yF'/%'italicGQ%trueF' /%,mathvariantGQ'italicF'-I#moGF$60Q\"=F'/F:Q'normalF'/%&fenceGQ&false F'/%*separatorGFD/%)stretchyGFD/%*symmetricGFD/%(largeopGFD/%.movablel imitsGFD/%'accentGFD/%%formGQ&infixF'/%'lspaceGQ/thickmathspaceF'/%'rs paceGFV/%(minsizeGQ\"1F'/%(maxsizeGQ)infinityF'-F#6%-F,6%Q$sinF'/F7FDF @-F=60Q0⁡F'F@FBFEFGFIFKFMFOFQ/FUQ$0emF'/FXFcoFYFfn-I(mfe ncedGF$6$-F#6#-F,6%Q\"tF'F6F9F@F+F+-F=60Q\";F'F@FB/FFF8FGFIFKFMFOFQFbo FWFYFfn" }}{PARA 0 "" 0 "" {XPPEDIT 2 0 "Typesetting:-mrow(Typesetting :-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow (Typesetting:-mi(\"z\", italic = \"true\", mathvariant = \"italic\"), \+ Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \"false\", se parator = \"false\", stretchy = \"false\", symmetric = \"false\", larg eop = \"false\", movablelimits = \"false\", accent = \"false\", form = \"infix\", lspace = \"thickmathspace\", rspace = \"thickmathspace\", \+ minsize = \"1\", maxsize = \"infinity\"), Typesetting:-mi(\"t\", itali c = \"true\", mathvariant = \"italic\")), Typesetting:-mi(\"\")), Type setting:-mo(\";\", mathvariant = \"normal\", fence = \"false\", separa tor = \"true\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", form = \"in fix\", lspace = \"0em\", rspace = \"thickmathspace\", minsize = \"1\", maxsize = \"infinity\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingG I(_syslibGF'6%-I#miGF$6#Q!F'-F#6%F+-F#6%-F,6%Q\"zF'/%'italicGQ%trueF'/ %,mathvariantGQ'italicF'-I#moGF$60Q\"=F'/F:Q'normalF'/%&fenceGQ&falseF '/%*separatorGFD/%)stretchyGFD/%*symmetricGFD/%(largeopGFD/%.movableli mitsGFD/%'accentGFD/%%formGQ&infixF'/%'lspaceGQ/thickmathspaceF'/%'rsp aceGFV/%(minsizeGQ\"1F'/%(maxsizeGQ)infinityF'-F,6%Q\"tF'F6F9F+-F=60Q \";F'F@FB/FFF8FGFIFKFMFOFQ/FUQ$0emF'FWFYFfn" }}{PARA 0 "" 0 "" {TEXT 204 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "spacecurve([cos( t),sin(t), t],t=0..4*Pi, axes=normal,color=blue);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 131 "Using the \"spacecurve\" command is a great way to visu alize some rather difficult parametric equations, such as intricate sp irals." }}{PARA 0 "" 0 "" {TEXT 204 0 "" }}{PARA 0 "" 0 "" {TEXT 204 126 "A useful thing that will be used in our course (possibly on the e xams too) is the simple way to transform geometric figures by" }} {PARA 0 "" 0 "" {TEXT 204 92 "vector operations. As an example, let us take the unit circle in the (x,y)-plane (so z = 0)." }}{PARA 0 "" 0 " " {TEXT 204 0 "" }}{PARA 0 "" 0 "" {TEXT 204 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "C:=[cos(t),sin(t),0];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "spacecurve(C,t=0..2*Pi,color=black,axes=normal) ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 38 "We can magnify it by a scal ar factor: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "C1:=C*2;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "spacecurve(C1,t=0..2*Pi,colo r=black,axes=normal, labels=[x,y,z]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 83 "And shift the resulting curve so that its center is at <-1,2, 3>, not at the origin:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "C 2:=C1+[-1,2,3];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "spacecur ve(C2,t=0..2*Pi,color=black,axes=normal, labels=[x,y,z]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 110 "The original curve C can be described a s <1,0,0> cos(t) + <0,1,0> sin(t). Let's see what happens if we replac e" }}{PARA 0 "" 0 "" {TEXT 204 46 "the vectors <1,0,0> and <0,1,0> by \+ some other " }{TEXT 216 16 "orthogonal unit " }{TEXT 204 8 "vectors." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "u:=[0,0,1]; v:=[-1/sqrt(2 ),1/sqrt(2),0];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 31 "This way, u*c os(t)+v*sin(t) is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "C3:=[ 0,0,cos(t)]+[-sin(t)/sqrt(2),sin(t)/sqrt(2),0];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "spacecurve(C3,t=0..2*Pi,colo r=black,axes=boxed, labels=[x,y,z],scaling=constrained);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 110 "Make a reasonable guess what kind of circle this is; and think ho w would you verify that your guess is right. " }}{PARA 0 "" 0 "" {TEXT 204 82 "You may want to use this example as a template for one of you r homework questions." }}{PARA 0 "" 0 "" {TEXT 204 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 3 "" 0 "" {TEXT 218 45 " Displaying several objects on t he same graph" }}{EXCHG {PARA 0 "" 0 "" {TEXT 204 89 "Sometimes it is \+ necessary to display several objects of different kind on the same gra ph." }}{PARA 0 "" 0 "" {TEXT 204 92 "For instance, the line with param etric equations r(t) = < 1+t, t, 1-t> intersects the plane " }}{PARA 0 "" 0 "" {TEXT 204 77 "x+y+z = 3 at the point <2,1,0>. Let's display t he plane, line, and the point." }}{PARA 0 "" 0 "" {TEXT 204 0 "" }} {PARA 0 "" 0 "" {TEXT 204 105 "First, we draw the objects separately, \+ keeping in mind that we are interested in points close to (2,1,0):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Plane:=plot3d(3-x-y,x=1..3, y=0..2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 108 "Maple should not dr aw the plot; rather it stores the whole graph, and gives you access wh enever you need it:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "Plane ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 126 "At the point (2,1,0), the \+ value of parameter t for the line is 1, so we draw the line with the v alue of t ranging from 0 to 2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Line:=spacecurve(1+t,t,1-t,t=0..2):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 112 "Don't ignore the error messages at this stage. Here Ma ple complains that you did not include the brackets around" }}{PARA 0 "" 0 "" {TEXT 204 25 "the parametric equations." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 37 "Line:=spacecurve([1+t,t,1-t],t=0..2):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Point:=pointplot3d([2,1,0],c olor=red):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 44 "Now we are ready t o display the three plots:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "display3d([Plane,Line,Point]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 94 "This plot is not too nice: you can barely see the red point, a nd the line is not very visible." }}{PARA 0 "" 0 "" {TEXT 204 55 "This is what all the options do in Maple. For instance:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "Plane:=implicitplot3d(3-x-y,x=1..3,y=0..2, z=-1..1,style=hidden):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "L ine:=spacecurve([1+t,t,1-t],t=0..2,color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "Point:=pointplot3d([2,1,0],color=red,symbol=c ircle,symbolsize=20):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "di splay3d([Plane,Line,Point]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT 204 0 "" }}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 15 10 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }