{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 "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 262 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 272 "" 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 1 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "MyWork" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 14 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 "R3 Font 0" -1 257 1 {CSTYLE "" -1 -1 "Helvetica" 1 12 255 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 2" -1 258 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 128 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" -1 259 1 {CSTYLE "" -1 -1 "Comic Sans MS" 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" -1 260 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 } {PSTYLE "Normal" -1 261 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 1 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" -1 262 1 {CSTYLE "" -1 -1 "Times" 1 14 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" -1 263 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 1 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" -1 264 1 {CSTYLE "" -1 -1 "Times" 1 24 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 }{PSTYLE "Normal" -1 265 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 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT 256 27 "CURVES GIVEN PARAMETRICALLY " }{TEXT 261 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 22 "Two dimensional plane:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "x1:=t->cos(t);y1:=t->sin(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot([x1(t),y1(t), t=0..2*Pi],label s=[x,y]); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "I" }{TEXT 259 87 "t is important to notice that this curve also has an orientation. As th e parameter \"t\" " }}{PARA 261 "" 0 "" {TEXT -1 96 "increases the cir cumference \"C\" is traced in the counterclockwise direction. We can v erify this " }}{PARA 261 "" 0 "" {TEXT -1 61 "by restricting the lengh t of the interval where t is varying." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "plot([x1(t),y1(t), t=0. .3*Pi/2],labels=[x,y]); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 24 "Anot her example: Spirals" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "x2:=t->exp(-t/4)*cos(t);y2:=t->exp(-t/4)* sin(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "plot([x2(t),y2(t ), t=-20..20],labels=[x,y]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 262 "" 0 "" {TEXT -1 10 "Exercises:" }}{PARA 262 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "1.- Do the above graph s represent real-valued functions of one variable?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "2.- Use MAPLE to find the orientation on the above spiral." }}{PARA 0 "" 0 "" {TEXT -1 58 "What can you do to reverse this orientation on the spiral?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 160 "3.- Redifine the pa rametric equation for the circle in such a way that t traces the circ umference in a clockwise manner. Restrict the range and run MAPLE plo t" }}{PARA 0 "" 0 "" {TEXT -1 33 "subroutine to confirm the change." } }}{EXCHG {PARA 265 "" 0 "" {TEXT -1 0 "" }}{PARA 265 "" 0 "" {TEXT -1 46 "We can also represent curves in the 3-D space." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 19 "Curves in 3-D space" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Example 1 : Circumference in parallel planes to the x-y plane." }}{PARA 0 "" 0 " " {TEXT -1 105 "Construct a circle of radius one that lies the the pla ne z = 1, and another that lies in the plane z = 0." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "i :=vector([1,0,0]); j:=vector([0,1,0]);k:=vector([0,0,1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "r1:=t->cos(t)*i+sin(t)*j+1*k;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " with(plots): \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 " spacecurve(evalm(r1(t)), t=0..4*Pi , color=blue,axes=NORMAL, orientation=[33,68],labels=[x,y,z]); \n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "r2 := t -> cos(t)*i + sin(t) *j + 0*k;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "spacecurve(eva lm(r2(t)), t=0..4*Pi, color=blue,axes=NORMAL,orientation=[33,68],label s=[x,y,z]); \n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 11 "Example 2: " }{TEXT 268 14 "Circular Helix" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 115 "When the z value is allowed to vary, the equation will yield a helix instead o f a circle that lies in a flat plane." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "r4 := t -> 3*cos(t)*i + 3*sin(t)*j + t*k;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "spac ecurve(evalm(r4(t)), t=0..4*Pi, color=blue,axes=framed,orientation=[-1 30,52],labels=[x,y,z],numpoints=200); \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "r5 := t -> (4*t^2/2+t+10)*i + (t^2/2+2*t+3)*j + (t^2/ 2+5*t+5)*k;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "spacecurve( evalm(r5(t)), t=-10..10, color=blue,axes=framed, orientation=[-40,71], labels=[x,y,z],numpoints=200); \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "r6 := t -> (4*t^2/2+8*t-1)*i + (t^2/2+2*t+4)*j + (t^2 /2+2*t+10)*k;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "spacecurve(evalm( r6(t)), t=-10..10, color=blue,axes=framed, orientation=[-40,71],labels =[x,y,z],numpoints=200); \n" }}}{EXCHG {PARA 263 "" 0 "" {TEXT -1 0 " " }}{PARA 263 "" 0 "" {TEXT 260 48 "Unit Tangent, Principal Normal, Os culating Plane" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 126 "Find the unit tangent, principal \+ normal, and the osculating plane for the helix curve given above at Po corresponding to t=ti." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "The parametric function r4(t) is not a vector functi on for MAPLE yet." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 16 "type(r4,vector);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "However, it can be easily converted into it." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "r4v: =t->evalm(r4(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "type(r 4v(t),vector);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Point Po where the osculating plane will be construc ted." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "ti:=3*Pi;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Po:=r4v(ti);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 27 "First, we want to obtain a " }{TEXT 264 60 "graph \+ for the unit tangent vector of r4(t) at the point Po" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "Stora ge of the helix curve at the variable SC." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "SC:=spacecurve(eval m(r4(t)), t=0..4*Pi, color=blue,axes=framed,orientation=[-130,52],labe ls=[x,y,z]):\n" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 141 "In order to obtain the unit tangent vector at certain point Po, we ne ed to obtain the derivative of the function r4(t) at t=ti and normaliz e." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "r4p:=D(r4);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 91 "Therefore, the operator D, gives us t he function which is the derivative of r4(t) directly." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 " At this point, we n eed to load a set of linear algebra subroutines called linalg." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 29 "Unit tangent vector function." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Tv:=t->no rmalize(r4p(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "type(Tv (t),vector);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 24 "Now, we can compute the " }{TEXT 265 25 "unit tangent v ector at Po" }{TEXT -1 64 " by evaluating the unit tangent vector func tion, T(t) at t = ti." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "Tv(ti);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 20 "type(Tv(ti),vector);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 313 "The graph for the helix curve corresponding to the vecto r function r4v(t) was already obtained an stored in the variable SC. N ow, we will obtain the graph for the unit tangent vector at Po and wil l store this as the variable l1, then with display3d both will be show ed in the same 3-D cartesian coordinate system." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "l1 := arrow (Po, Tv(ti), width = [0.053], head_length=[0.08], color=black):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "AR:=plots[display](l1,axes=f ramed):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "display3d([SC,AR ],axes=framed,orientation=[131,53]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 269 9 "Exercise:" }}{PARA 0 "" 0 "" {TEXT -1 75 "Find and graph the unit tangent vector for the given curv e r4(t) at t=2*Pi." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 165 "The principal normal vector is perpendicular to \+ the unit tangent and is found by taking the derivative of the tangent \+ vector function r4p(t) and then normalizing it." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "r4pp:=D(r4p );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Nv:=t->normalize(r4pp (t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 30 "Principal Normal vector at Po." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "Nv(ti);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "l2 := arrow(r4v(ti),Nv(ti), \+ width = [0.053,relative], head_length=[0.08,relative], color=red):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "ARO:=plots[display](l2,axes =framed):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "display3d([ARO ,AR,SC],axes=framed,orientation=[131,53]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 270 9 "Exercise:" }}{PARA 0 "" 0 "" {TEXT -1 78 "Find and graph \+ the Principal Normal vector for the given curve r4(t) at t=2*Pi" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 272 39 "Equation of the Osculating Plane at Po." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 188 "The osculating p lane is the plane containing both the Unit Tangent Vector, and the Pri ncipal Normal. Therefore, its normal can be obtained by taking the cro ss product of these two vectors." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "PlaneNor:=crossprod(Tv(ti),N v(ti));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "Now, we can obtain the equation for the plane using the \+ normal, PlaneNor and the point Po over the plane." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "PoP:=vector ([x,y,z])-Po;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "type(PoP,v ector);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "P:=(x,y,z)->dotp rod(PlaneNor,PoP);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 39 "Equation of the Osculating Plane at Po." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "P(x,y,z)=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "OP:=im plicitplot3d(P(x,y,z)=0,x=-5..-1, y=-7..2,z=5..10,axes=boxed, style=wi reframe):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "display3d([ARO ,AR,SC,OP],axes=framed,orientation=[131,53]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 271 9 "Exercise: " }}{PARA 0 "" 0 "" {TEXT -1 80 "Obtain the equation for the osculatin g plane of the given curve r4(t) at t=2*Pi." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "74" 0 }{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }