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"" -1 362 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 363 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 364 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 365 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 366 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 367 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 368 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 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 "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "List Item" -1 14 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 3 3 1 0 1 0 2 2 14 5 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 0 1 } {PSTYLE "Title" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 16 "Lecture 8 Part b" }} {PARA 19 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 25 "Space curves and surfaces" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Exec ute the commands " }{TEXT 267 11 "with(plots)" }{TEXT -1 5 " and " } {TEXT 268 15 "with(plottools)" }{TEXT -1 115 " to see some of the many other plotting procedures in Maple. I will illustrate a few more belo w: First note that a " }{TEXT 271 5 "curve" }{TEXT -1 36 " in three sp ace may be specified by " }{TEXT 275 20 "parametric equations" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 30 "\n \+ " }{TEXT 269 22 "x=f(t), y=g(t), z=h(t)" }{TEXT -1 13 " \nor simply " }}{PARA 0 "" 0 "" {TEXT -1 7 " " }}{PARA 0 "" 0 "" {TEXT 270 44 " [f(t),g(t),h(t)]" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "where t i s in some interval: Here's a simple example:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "spacecurve([cos(t),sin(t),t],t=0..6*Pi,color=black, t hickness=3,axes=normal);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 112 "Note how different the syntax is from the way a curve is plotted via param etric equations in the plane. Compare:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot([2*cos(t),2*sin(t),t=0..2*Pi]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "\nA " }{TEXT 272 7 "surface" }{TEXT -1 66 " in three space may be defined by parametric equations of the form" } }{PARA 0 "" 0 "" {TEXT -1 26 " \n " }{TEXT 273 30 "x = f(s,t), y=g(s,t), z=h(s,t)" }{TEXT -1 2 " \n" }}{PARA 0 "" 0 " " {TEXT -1 74 "where s and t lies in some region in the plane: Here we map the rectangle " }{TEXT 274 15 "[0,Pi] x [-1,1]" }{TEXT -1 23 " to half of a cylinder:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "plot3d([cos (s),sin(s),t], s=0..Pi,t=-1..1, scaling=constrained,axes=normal, style =patchnogrid);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Here's a helico id:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "plot3d([r*cos(s),r*s in(s),s], r=0..1,s=0..6*Pi,grid=[15,45],style=patch, axes=box); " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 49 "Surfaces in cylindrica l and spherical coordinates" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "One can also use " }{TEXT 257 8 "spherica" }{TEXT -1 5 "l or " }{TEXT 258 23 "cylindrical coordinates" }{TEXT -1 10 " with the " }{TEXT 259 6 "plot3d" }{TEXT -1 46 " procedure: Here is an example of each: See " }{TEXT 264 7 "?coords" }{TEXT -1 6 " and " }{TEXT 265 15 "?plot3d[ coords]" }{TEXT -1 128 " for more details on each of the coordinate sy stems. There are over 30 different types of coordinate systems support ed by Maple." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Recall that in " }{TEXT 262 21 "spherical coordinates" }{TEXT -1 47 " there are three parameters usually denoted by " }{XPPEDIT 18 0 "rho;" "6#%$rhoG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "theta;" "6#%&thet aG" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 98 ". If you have forgotten look it up in your calculus book. Most co mmonly surfaces are of the form " }{XPPEDIT 18 0 "rho = f(theta,phi); " "6#/%$rhoG-%\"fG6$%&thetaG%$phiG" }{TEXT -1 17 ". In our example " } {XPPEDIT 18 0 "rho = 1;" "6#/%$rhoG\"\"\"" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "plo t3d(1, theta=0..2*Pi, phi=0..Pi,coords=spherical, scaling = constraine d, title=\"Sphere of Radius 1\");\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "For " }{TEXT 263 23 "cylindrical coordinates" }{TEXT -1 21 " the variable are r, " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 44 " , and z. Usually the surface is give by r = " }{XPPEDIT 18 0 "f(theta, z);" "6#-%\"fG6$%&thetaG%\"zG" }{TEXT -1 24 ". In our example r = 2. \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "plot3d(2,theta=0..2*Pi , z=-1..1, coords=cylindrical, scaling=constrained, title=\"Cylinder o f Radius 2\");" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "We can put these in the same plot using " }{TEXT 256 7 "display" }{TEXT -1 18 ": Remember to use " }{TEXT 260 7 "displa y" }{TEXT -1 14 " you need the " }{TEXT 261 5 "plots" }{TEXT -1 82 " p ackage which we already loaded above. To spice things up we add the pl ane z = 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 260 "P1:=plot3d(1, theta=0..2*Pi, phi=0..Pi,style=patch ,coords=spherical, scaling = constrained, color = red):\n\nP2:=plot3d( 2,theta=0..2*Pi, z=-1..1,style=patch, coords=cylindrical, scaling=cons trained, color = green):\n\nP3:=plot3d(0,x=-3..3,y=-3..3, color = whit e):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "display([P1,P2,P3] , title=\"Sphere Inside Cylinder Cut By A Plane\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 8 " Tubeplot" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 266 9 "Tubeplots" }{TEXT -1 100 " are plots of tubes about one o r more space curves: First let's make a unit circle in the x-y plane: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Curve:=[cos(theta),sin(theta),0],theta=0..2*Pi;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "spacecurve(Curve, axes = nor mal, color = black);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Now we pu t a tube of radius 1/4 around the circle to make a torus: Note that " }{TEXT 352 9 "numpoints" }{TEXT -1 5 " and " }{TEXT 353 10 "tubepoints " }{TEXT -1 145 " control the number of points plotted along the curve and the number of points around the tube. Change the values and you c an see what they mean." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 " tubeplot(Curve,radius=1/4, numpoints=20, tubepoints=10,scaling=constra ined, style=patch, axes = normal);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "The tube radius can be \+ made to depend on the curve's parameter as in the following example:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "tubeplot([cos(theta),sin (theta),0],theta=0..2*Pi,radius=theta/8, numpoints=40, tubepoints=20,s caling=constrained, style=patch, axes = none);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 32 "Thre e dimensional implicit plots" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "This refers to the plotting of sur faces that are solution sets of equations of the form " }}{PARA 0 "" 0 "" {TEXT -1 23 "\n " }{TEXT 291 11 "f(x,y,z)=0 \+ " }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 144 "that cannot be solv ed for one variable in terms of the other two. These are the three di mensional analogues of curves in the plane of the form " }{TEXT 324 10 "f(x,y) = 0" }{TEXT -1 24 " which are plotted with " }{TEXT 323 11 "implictplot" }{TEXT -1 65 " . For example let's see what the surface \n\n " }{TEXT 292 1 " " }{XPPEDIT 256 0 "x^2+y^2-z ^2 = 1;" "6#/,(*$%\"xG\"\"#\"\"\"*$%\"yGF'F(*$%\"zGF'!\"\"F(" }{TEXT -1 14 " \n\nlooks like:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " restart:\nwith(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 " implicitplot3d(x^2 + y^2 - z^2 = 1 ,x=-3..3,y=-3..3,z=-3..3, grid=[20 ,20,20],\nstyle=patchcontour, contours=20,axes=box);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 19 "Packages are tables" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Note that \+ you don't have to execute " }{TEXT 325 11 "with(plots)" }{TEXT -1 37 " to be able to use a command such as " }{TEXT 328 7 "display" }{TEXT -1 4 " or " }{TEXT 329 12 "implicitplot" }{TEXT -1 8 " in the " } {TEXT 326 5 "plots" }{TEXT -1 46 " package. You may instead use the lo ng forms " }{TEXT 330 14 "plots[display]" }{TEXT -1 4 " or " }{TEXT 331 19 "plots[implicitplot]" }{TEXT -1 62 " without loading plots. S imilarly you don't have to execute " }{TEXT 333 19 "with(LinearAlgebra )" }{TEXT -1 8 " to use " }{TEXT 354 11 "Determinant" }{TEXT -1 36 ", \+ you may instead use the long form " }{TEXT 355 26 "LinearAlgebra[Deter minant]" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 31 "Note that packages behave like " }{TEXT 327 6 "tab les" }{TEXT -1 178 " with the indices of the table consisting of the n ames of the procedures in the table and the corresponding entry is the actual procedure. Actually some packages are written as " }{TEXT 332 7 "modules" }{TEXT -1 13 " rather than " }{TEXT 356 6 "tables" } {TEXT -1 55 " but still can be accessed like tables. Some examples:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "type(display, procedure);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "type(plots, table);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "type(plots[display],procedure);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "A:=Matrix([[1,2],[3,4]]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Determinant(A);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "LinearAlgebra[Determinant](A );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "plots[implicitplot](x ^2-(y-1)*(y+1)*y = 0,x=-2..2,y=-2..2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Here's another example of the use of the long form of a p ackage procedure:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "combin at[powerset](\{1,2,3\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "The option filled = true" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 14 "" 0 "" {TEXT -1 7 "If the " }{TEXT 357 6 "filled" }{TEXT -1 220 " option is set to true, the area between the curve and the x-axis is given a solid color. This option is valid only with the following commands: plot, contourplot, implicitplot, li stcontplot, polarplot, and semilogplot. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plot(sin(x), x=0..2*Pi, filled = true, color=green); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Here's an example due to Vedr an Cacic." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "plot(arcsin(ab s(x)-1)-Pi/2, x=-3..3, y=-4..3,filled=true, color=red, scaling = const rained);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "This also due to Vedr an Cacic:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "plot(sqrt(2*ab s(x)-x^2), x=-3..3, y=-4..3,filled=true, color= red, scaling = constra ined);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "And Vedran's goal was t his " }{TEXT 358 6 "lovely" }{TEXT -1 9 " picture:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "plot(\{arcsin(abs(x)-1)-Pi/2,sqrt(2*abs(x) -x^2)\}, x=-3..3,\ny=-4..3,filled=true, color=red, scaling = constrain ed, axes=boxed, xtickmarks=[],ytickmarks=[], title=\"Happy Valentines \+ Day\");" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Actually it works like this for implicitplot:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 " plots[implicitplot](x^2+y^2=1, x=-2..2,y=-2..2, scaling = constrained, filled=true);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 36 "The package p lottools -- stop lights" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "The package \+ " }{TEXT 359 9 "plottools" }{TEXT -1 179 " has a number of useful proc edures. I will illustrate just a few of them, The commands are more or less self-explanatory. Use help if you want more details. Note that t he command " }{TEXT 293 7 "display" }{TEXT -1 56 " must be used to dis play some of the plots so generated." }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 16 "with(plottools);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Click and rotate as usual to get a better look." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "display(icosahedron([0,0,0], 0.8));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "f := icosahedron( [0,0,0],0.8), dodecahedron([1,1,1],0.5):\nplots[display](f,style=patch );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "DD:=disk([0,0],4, col or=green, scaling = constrained):\ndisplay(DD, axes = none);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "L:=[green,yellow,red]: dis play([seq(disk([1,i],.45,colour=L[i]),i=1..3)],scaling=constrained, ax es = boxed, xtickmarks=[], ytickmarks=[], title = \"Stop Lights\");" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 36 "The procedures animate and animate3d" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "Below I give several examples. In the first we have an expression " }{TEXT 314 6 "F(x,t)" }{TEXT -1 7 " where " } {TEXT 315 1 "t" }{TEXT -1 55 " is considered as a parameter. For each \+ fixed value of " }{TEXT 316 1 "t" }{TEXT -1 31 " we get a different fu ntion of " }{TEXT 317 1 "x" }{TEXT -1 3 ". " }{TEXT 360 30 "After exe cuting the procedure " }{TEXT 310 7 "animate" }{TEXT 361 4 " or " } {TEXT 311 9 "animate3d" }{TEXT 362 93 " one must click on the plot and then at the top of the worksheet various buttons will appear." } {TEXT -1 45 " If you select under the rightmost help menu " }{TEXT 309 13 "show balloons" }{TEXT -1 124 " and then run your cursor over \+ the various buttons it will tell what the most important ones are for. If you click on the " }{TEXT 318 14 "black triangle" }{TEXT -1 48 " \+ the animation will begin. If you click on the " }{TEXT 319 12 "black \+ square" }{TEXT -1 19 " it will stop. The " }{TEXT 320 23 "arrow going \+ in a circle" }{TEXT -1 56 " at the right is to make the animation run \+ continuously." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "animate( x^2*t,x=-1..1,t=-2..2,fram es=20, scaling=constrained,color=red, thickness = 3,axes=boxed);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "You may animate curves and surface s:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "animate( [u*sin(t),u *cos(t),t=-Pi..Pi],u=1..8,view=[-8..8,-8..8],\ncolor=blue, scaling = c onstrained, axes = none, thickness = 3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "animate3d([u*t*sin(x),u*t*cos(x),t],x=0..2*Pi,t=0..2, u=.25..4,frames = 20);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 199 "Here's an example to show how \+ the tangent line to y = x^2 changes as we move along the curve. Recall that the equation of the tangent line to y = f(x) at the point (a, f( a)) is y = f'(a)(x-a) + f(a)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "P1:=plot(x^2,x=-1..1, color = red, thickness = 3):\nP2:=animate (2*a*(x-a) + a^2, x = -2..2,a=-1..1, color = green, thickness = 3):\n display(\{P1,P2\}, scaling = constrained);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "To make the above reverse itself we may replace the varia ble a by sin(b) and let b go from 0 to " }{XPPEDIT 18 0 "2*Pi;" "6#*& \"\"#\"\"\"%#PiGF%" }{TEXT -1 209 ". Then sin(b) will go from 0 to 1 , 1 to -1 and back to 0. Note that in both cases the domain of the tang ent line is made larger than the domain of the parabola to keep the ta ngent line from looking too short." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "a:=sin(b);\nP2:=animate(2*a*(x-a) + a^2, x = -2..2,b =0..2*Pi, color = green, thickness = 3):\ndisplay(\{P1,P2\}, scaling \+ = constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 21 "Look at the help for " }{TEXT 312 7 "anim ate" }{TEXT -1 5 " and " }{TEXT 313 9 "animate3d" }{TEXT -1 19 " for m ore examples." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 39 "Use of matrixpl ot and insequence = true" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 35 "with(plots): \nwith(LinearAlgebra):\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 334 13 "matrixplot(A)" }{TEXT -1 97 " displays m atrix in a graphical form. Here's an example. First let A be a random \+ 10 by 10 matrix." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "A:=Rand omMatrix(10,10):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Now we displa y the matrix as a surface." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "matrixplot(A, axes=boxed);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Or it can be done as a \"histogram\" as follows:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 43 "matrixplot(A,axes=boxed,heights=histogram); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "Here's an interesting animat ion of a collection of matrices. First we create a table of matrixplot s. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "for j from 1 to 30 do \n P[j]:=matrixplot(matrix(5,5, [seq(sin(i)*j,i=1..25)]), axes=box , heights=histogram):\nod:\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "N ext we display them with " }{TEXT 335 17 "insequence = true" }{TEXT -1 151 ". If you click on the plot you will see that it is an animatio n and can be started by clicking on the triangle and stopped by clicki ng on the square. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "disp lay([seq(P[j],j=1..30)],insequence=true);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "More generally given any sequence of plots, say,seq( P[i ], i=a..b), then one may animate them by using the command " }{TEXT 336 50 "display ( [seq(P[i],i=a..b)], insequence = true) ." }{TEXT -1 90 " Sometimes one can form an animation this way that is difficult to form using the command " }{TEXT 363 7 "animate" }{TEXT -1 4 " or " } {TEXT 364 9 "animate3d" }{TEXT -1 26 ".\n\nHere's another example:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 333 "with(plottools):\n\nTheGra ss:=plot(0,x=-11..11, color=green,thickness=3):\n\nfor i from -10 to 1 0 do\nPq[i]:=disk([i,(10^2-i^2)/10],.5, color=black, scaling = constra ined):\nod:\n\nTheBall:=display([seq(Pq[i],i=-10..10)],scaling = const rained, insequence=true,axes=none,title=\"Click on cannon ball animati on:\"):\n\ndisplay(\{TheBall,TheGrass\});" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 22 "plotting \+ vector fields" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 14 "The procedure " }{TEXT 294 9 "fieldplot" }{TEXT -1 35 " allows one to draw a picture of a " }{TEXT 295 12 "vector field" } {TEXT -1 59 " in the plane: Recall that a vector field assigns a vecto r " }{TEXT 296 14 "[f(x,y),g(x,y)" }{TEXT -1 16 "] to each point " } {TEXT 297 5 "(x,y)" }{TEXT -1 173 " in the plane: Here are some simple examples: The first is a constant vector field. This may be thought o f as a steady wind blowing across the plain (or plane, if you wish)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:\nwith(plots):" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "fieldplot( [1,1],x=-10..10 , y=-10..10, axes = boxed);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 202 "T he next example describes a fluid, say, moving in a counter-clockwise \+ manner. Note that each vector has the same length. So if the vectors i ndicate velocity, all particles are moving at the same speed." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "fieldplot([-y/sqrt(x^2+y^2), x/sqrt(x^2+y^2)], x=-5..5,y=-5..5, axes = boxed);" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 14 "The procedure " }{TEXT 304 8 "gradplot" }{TEXT -1 11 " plots the " }{TEXT 305 8 "gradient" }{TEXT -1 48 " of the functio n at each point. Recall that the " }{TEXT 306 8 "gradient" }{TEXT -1 8 " is the " }{TEXT 307 12 "vector field" }{TEXT -1 355 " obtained by \+ taking the partial derivatives with respect to x for the first compone nt and the partial with respect to y for the second component. The gra dient at each point (x,y) points in the direction of greatest rate of \+ increase. So you should follow the arrow to go up, as they say. \n\nTo see any plot better, just use the mouse to increase its size. " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "p:=exp(-(x^2+y^2))*(5-x^2-y^ 2)+ exp(-(x-2)^2-(y-2)^2)*((x-2)^2 + (y-2)^2-4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "gradplot(p,x=-3..4,y=-3..4);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 189 "We can see in this next plot what the su rface looks like. Note that from examining the plot of the gradient ab ove we can identify one peak and one valley. This is visible in the pl ot below." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "plot3d(p, x = \+ -3..4,y=-3..4, axes = boxed);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 34 "Plotting level curves \+ or contours" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Maple also can plo t " }{TEXT 298 12 "level curves" }{TEXT -1 4 " or " }{TEXT 299 8 "cont ours" }{TEXT -1 74 " of a function of two variables. Here we use for \+ an example the function " }{TEXT 308 9 "f(x,y) = " }{XPPEDIT 257 0 "x^ 2-y^2;" "6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF&!\"\"" }{TEXT -1 1 "." }{TEXT 365 92 " The contours are colored from yellow to red. The yellow lines being the greatest in value. " }{TEXT -1 66 "We first plot the graph \+ of the function to see what it looks like:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 21 "restart:\nwith(plots):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 68 "plot3d(x^2-y^2,x=-3..3,y=-3..3, axes = boxed, style = patchcontour);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "The procedur e " }{TEXT 300 11 "contourplot" }{TEXT -1 16 " projects these " } {TEXT 301 12 "level curves" }{TEXT -1 4 " or " }{TEXT 302 8 "contours " }{TEXT -1 16 " onto the plane." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "contourplot(x^2-y^2,x=-3..3,y=-3..3, filled = true); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 303 13 "contourplot3d" }{TEXT -1 46 " raises the contours to the appropria te level:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "contourplot3d( x^2-y^2,x=-3..3,y=-3..3, filled = true, axes = boxed);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "To see more about " }{TEXT 337 11 "contou rplot" }{TEXT -1 5 " and " }{TEXT 338 13 "contourplot3d" }{TEXT -1 21 " see the help pages. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 37 "Assignment 8 Due Tuesday, Nov ember 5\n" }}{EXCHG {PARA 0 "" 0 "" {TEXT 290 9 "Problem 1" }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 339 7 "delto id" }{TEXT -1 99 " plane curve with parameter a is the set of all poi nts (x,y) in the plane satisfying the equation " }}{PARA 0 "" 0 "" {TEXT -1 6 " " }}{PARA 0 "" 0 "" {TEXT -1 11 " (1) " } {XPPEDIT 18 0 "(x^2+y^2)^2-8*a*x*(x^2-3*y^2)+18*a^2*(x^2+y^2) = 27*a^4 ;" "6#/,(*$,&*$%\"xG\"\"#\"\"\"*$%\"yGF)F*F)F***\"\")F*%\"aGF*F(F*,&*$ F(F)F**&\"\"$F**$F,F)F*!\"\"F*F5*(\"#=F**$F/F)F*,&*$F(F)F**$F,F)F*F*F* *&\"#FF**$F/\"\"%F*" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 64 "and the same curve may be described by th e parametric equations:" }}{PARA 0 "" 0 "" {TEXT -1 13 "\n (2) \+ " }{XPPEDIT 18 0 "x = a*(2*cos(t)+cos(2*t));" "6#/%\"xG*&%\"aG\"\"\",& *&\"\"#F'-%$cosG6#%\"tGF'F'-F,6#*&F*F'F.F'F'F'" }{TEXT -1 7 ", " }{XPPEDIT 18 0 "y = a*(2*sin(t)-sin(2*t));" "6#/%\"yG*&%\"aG\"\"\",&*& \"\"#F'-%$sinG6#%\"tGF'F'-F,6#*&F*F'F.F'!\"\"F'" }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "(a) Using equation (1) and the command " }{TEXT 340 12 "implicitplot" }{TEXT -1 7 " graph " }{TEXT 349 19 "in the same picture" }{TEXT -1 86 " the \+ deltoid curves with parameters a = 1, 2, and 3.\n\n(b) Using equations (2) and the " }{TEXT 341 4 "plot" }{TEXT -1 16 " command graph " } {TEXT 350 19 "in the same picture" }{TEXT -1 52 " the deltoid curves w ith parameters a = 1, 2, and 3." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 83 "You will need to do some experimenting wi th the ranges of the plots and the option " }{TEXT 342 9 "numpoints" } {TEXT -1 174 " in question (a) to get a decent picture. Note that you \+ can copy and paste equations (1) and (2) and with some judicious editi ng can save yourself the trouble of typing them." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 343 9 "Problem 2" } {TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 30 "(a) The graph of the f unction " }{XPPEDIT 18 0 "g(x) = 10/(x^3-10*x-10*x^2+100);" "6#/-%\"gG 6#%\"xG*&\"#5\"\"\",**$F'\"\"$F**&F)F*F'F*!\"\"*&F)F**$F'\"\"#F*F/\"$+ \"F*F/" }{TEXT -1 174 " has three vertical asymptotes at the three ro ots of the denominator. Use solve to find the location of these three asymptotes and then graph the function using the option " }{TEXT 344 14 "discont = true" }{TEXT -1 125 " to remove the vertical asymptotic \+ lines. Choose horizontal as well as vertical ranges to give a nice pic ture of the curves. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 26 "(b) Plot the graph of the " }{TEXT 346 18 "Nephroid of \+ Freeth" }{TEXT -1 20 ", the curve given " }{TEXT 351 20 "in polar co ordinates" }{TEXT -1 17 " by the equation " }{XPPEDIT 345 0 "r = 1+2*s in(theta/2);" "6#/%\"rG,&\"\"\"F&*&\"\"#F&-%$sinG6#*&%&thetaGF&F(!\"\" F&F&" }{TEXT -1 36 " . Note that you will need to take " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 19 " in the range 0 to " } {XPPEDIT 18 0 "4*Pi;" "6#*&\"\"%\"\"\"%#PiGF%" }{TEXT -1 26 " to get t he right picture." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 276 10 "Problem 3." }{TEXT -1 100 " Use Maple to make th e following picture. [The 13 stars in the original flag of the United \+ States.]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 " " } {GLPLOT2D 154 142 142 {PLOTDATA 2 "64-%'CURVESG6%7(7$$\"\"!F)$\"\")F)7 $$\"39+++CD&y(e!#=$\"3s*****f+$)4>'!#<7$$!3/+++l^c5&*F/$\"3F+++%*p,4tF 27$$\"3/+++l^c5&*F/F67$$!39+++CD&y(eF/F0F'-%'COLOURG6&%$RGBG$\"#5!\"\" F(F(-%&STYLEG6#%%LINEG-F$6%7(7$$\"3#)*****H?iID$F2$\"3Q+++!=#>)>(F27$$ \"3-+++bu%3%QF2$\"37+++'=v\"*Q&F27$$\"3A+++'o0?I#F2$\"3w*****R<4s]'F27 $$\"3J+++?(=T?%F2FY7$$\"31+++^pFlEF2FTFLF>FE-F$6%7(7$$\"37+++iq)3w&F2$ \"3I+++EKXw\\F27$$\"3))*****RJs'[jF2$\"3-+++KiVnJF27$$\"3r*****faI)4[F 2$\"3m******>-Z&G%F27$$\"3j*****zdV>r'F2F[p7$$\"3M+++5=5t^F2FfoF^oF>FE -F$6%7(7$$\"3%*******=@'*[pF2$\"3-+++jnvV=F27$$\"3q*****4PZn`(F2$\"3s* *****\\o(RZ$!#>7$$\"3M+++-c!z*fF2$\"31+++dPx_6F27$$\"3V+++O'=+!zF2Fdq7 $$\"3=+++noFE-F$6%7(7$$\"3#********p8^a'F2$!3%******z?MA[ \"F27$$\"3o*****>&*)*G8(F2$!3?+++-7D\"H$F27$$\"3S+++%=dSf&F2$!37+++9s@ t@F27$$\"3K+++;-<'\\(F2F\\s7$$\"3;+++[%Gt&fF2FgrF_rF>FE-F$6%7(7$$\"3!) ******3'e=k%F2$!3I+++O_dRUF27$$\"3c*****4'QkH_F2$!3o******HAf[gF27$$\" 3=+++#4-3p$F2$!3.+++U#e0$\\F27$$\"3G+++E^\"Hf&F2Fdt7$$\"3/+++dL2aSF2F_ tFgsF>FE-F$6%7(7$$\"3)*******\\'4_n\"F2$!3u*****>s#f'z&F27$$\"3)****** >!\\*HE#F2$!3-+++;(4cg(F27$$\"3!******\\LJ:C(F/$!3Q+++Gdd(['F27$$\"3=+ ++mhEEEF2F\\v7$$\"3-+++)RCu3\"F2FguF_uF>FE-F$6%7(7$$!3)*******\\'4_n\" F2Fbu7$$!3-+++)RCu3\"F2Fgu7$$!3=+++mhEEEF2F\\v7$$!3!******\\LJ:C(F/F\\ v7$$!3)******>!\\*HE#F2FguFgvF>FE-F$6%7(7$$!3!)******3'e=k%F2Fjs7$$!3/ +++dL2aSF2F_t7$$!3G+++E^\"Hf&F2Fdt7$$!3=+++#4-3p$F2Fdt7$$!3c*****4'QkH _F2F_tFiwF>FE-F$6%7(7$$!3#********p8^a'F2Fbr7$$!3;+++[%Gt&fF2Fgr7$$!3K +++;-<'\\(F2F\\s7$$!3S+++%=dSf&F2F\\s7$$!3o*****>&*)*G8(F2FgrF[yF>FE-F $6%7(7$$!3%*******=@'*[pF2Fip7$$!3=+++noFE-F$6%7(7$$!37+++iq)3w&F 2Fao7$$!3M+++5=5t^F2Ffo7$$!3j*****zdV>r'F2F[p7$$!3r*****faI)4[F2F[p7$$ !3))*****RJs'[jF2FfoF_[lF>FE-F$6%7(7$$!3#)*****H?iID$F2FO7$$!31+++^pFl EF2FT7$$!3J+++?(=T?%F2FY7$$!3A+++'o0?I#F2FY7$$!3-+++bu%3%QF2FTFa\\lF>F EF#-%(SCALINGG6#%,CONSTRAINEDG-%*AXESSTYLEG6#%%NONEG-%+AXESLABELSG6%Q! 6\"F[^l-%%FONTG6#%(DEFAULTG-%%VIEWG6$F`^lF`^l" 1 2 0 1 10 0 2 9 1 1 1 1.000000 47.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 366 19 "Here are some hints" }{TEXT -1 46 ": Test each step by p lotting the points using " }{TEXT 284 13 "style = point" }{TEXT -1 8 " and or " }{TEXT 285 12 "style = line" }{TEXT -1 161 " before continu ing to the next step.\n\n1. The following procedure will produce a lis t of n equally spaced points on a circle of radius r centered at the o rigin :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "with(plots):\nC irclePoints:=proc(n,r) \nevalf([seq(r*[sin(i*(2*Pi/n)), cos(i*(2*Pi/n) )], i=0..n)]); \nend proc:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "plot(CirclePoints(5,1),style = point, symbol = circle, axes = no ne, color = black);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "2. Use the above procedure to make a list " }{TEXT 286 1 "L" }{TEXT -1 62 " of 5 points on the unit circle. Then you may form a new list " }{TEXT 277 21 "W:=[L[1], L[3], ...,]" }{TEXT -1 36 " which when plotted with the \+ option " }{TEXT 289 12 "style = line" }{TEXT -1 146 " will produce a s tar. Number the points in the original circle and by hand trace out th e star. This will show you how to complete the list for W. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "3. The followin g procedure can be used to translate the star (given by " }{TEXT 281 1 "W" }{TEXT -1 33 ") from the origin to a new point " }{TEXT 282 1 "P " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "translate:=proc(W,P)\n \+ map(x->x+P,W);\nend proc:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "4. \+ Now create the list , say, " }{TEXT 280 1 "X" }{TEXT -1 57 " of 13 poi nts in a suitably sized circle. For each point " }{TEXT 279 8 "P = X[i ]" }{TEXT -1 32 " in the list translate the star " }{TEXT 287 1 "W" } {TEXT -1 18 " to the new point " }{TEXT 283 1 "P" }{TEXT -1 48 " and c reate a plot of this star. Call it, say, " }{TEXT 288 7 "Plot[i]" } {TEXT -1 39 ". Do this using a do loop. Finally use " }{TEXT 278 7 "di splay" }{TEXT -1 72 " to show all these plots at the same time to crea te the above picture.\n\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }} }{EXCHG {PARA 0 "" 0 "" {TEXT 321 10 "Problem 4." }{TEXT -1 6 " \nUse \+ " }{TEXT 348 14 "implicitplot3d" }{TEXT -1 65 " to graph the 6 surface s in three space with equations given by " }{XPPEDIT 18 0 "x^2+a*y^2+ b*z^2 = 1;" "6#/,(*$%\"xG\"\"#\"\"\"*&%\"aGF(*$%\"yGF'F(F(*&%\"bGF(*$% \"zGF'F(F(F(" }{TEXT -1 80 " where a is 1 or -1 and b is 0, 1 or -1. \+ Make a title for each using the option" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 46 "title=convert(x ^2 + a*y^2 + b*z^2 = 1,string)\n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "with the appropriate value of a and b inserted. Use also \+ the option " }{MPLTEXT 1 0 12 "axes = boxed" }{TEXT -1 2 ". " }{TEXT 367 8 "You can " }{TEXT 347 10 "and should" }{TEXT 368 34 " do all 6 w ith a pair of do loops." }{TEXT -1 75 " And don't forget to shrink eac h plot before printing. You may need to use " }{MPLTEXT 1 0 19 "print( display(...))" }{TEXT -1 54 " to get display to work properly inside t wo do loops. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 322 0 "" }}}}{MARK "12 0 0" 36 } {VIEWOPTS 0 0 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }