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"" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 238 "" 0 "Chap4" {TEXT 1838 2 "4:" }{TEXT 1839 1 " " }{TEXT 1840 37 "VISUALIZING DATA, ABSTRACT DATA TYPES" }{TEXT 1841 33 "; Electric Fields of Multipoles" }{TEXT 1842 0 "" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}}{SECT 0 {PARA 240 "" 0 "" {TEXT 1844 1 " " }{TEXT 1845 5 "4.1 " }{TEXT 1846 12 "Introduction" }{TEXT 1847 0 "" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 317 "" 0 "" {TEXT 1849 1495 "One of the most rewarding uses of computers is visualizing the r esults of calculations. This is done with 2-D and 3-D plots (especiall y with colored surfaces), with contour maps, and with animations. Thes e types of visualization are sometimes breathtakingly beautiful and of ten provide deep insight into a problem by letting you see and ``handl e'' the functions with which you are working. Visualization also assis ts the program debugging process, the development of physical and math ematical intuition, and the all-around enjoyment of your work. Some of the reasons for this may arise from the fact that some large fraction (approximartely 50%) of our brain gets involved in visual processing, and if you are able to use this extra brainpower in your scientific w ork, then you have extended what was otherwise possible with solely lo gical abilities.\n\nTraditionally, visualization of a scientific probl em was the last step in problem solving. After studying tables of numb ers for hours and gaining confidence that they are right, a scientist \+ might then go to the trouble of making a bunch of 2-D plots to examine various aspects of the data. Well, in present times computational sci entists have demonstrated how much there is to be gained by going beyo nd 2-D plots. Now it is regular practice to use surface plots, volume \+ rendering (dicing and slicing), and animations (movies). In this chapt er we use some of these techniques within the context of visualizing t he electric field around charges." }{TEXT 1849 0 "" }}{PARA 241 "" 0 " " {TEXT 1848 0 "" }}}{SECT 0 {PARA 243 "" 0 "" {TEXT 1850 2 " " } {TEXT 1851 4 "4.2 " }{TEXT 1850 2 " " }{TEXT 1852 41 "Problem: Stable Points in Electric Fields" }{TEXT 1850 0 "" }}{PARA 244 "" 0 "" {TEXT 1853 1 " " }}{PARA 245 "" 0 "" {TEXT 1854 161 " \+ \+ O O \+ O " }}{PARA 245 "" 0 "" {TEXT 1854 155 " \+ O \+ O O O O" }}{PARA 245 "" 0 "" {TEXT 1854 150 " \+ Figure 4.1 Static configurations for two, three, and four charge configurations" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 243 "" 0 "" {TEXT 1850 261 "You are given some simpl e configurations of two, three and four charge systems, as shown in Fi g, 4.1. The two charges are fixed on a line at coordinates (1,0), (-1, 0); the three charges are fixed to the corners of an equilateral trian gle at coordinates (0,1), (" }{XPPEDIT 456 0 "sqrt(3);" "6#-%%sqrtG6# \"\"$" }{TEXT 1850 14 " /2, -1/2), (-" }{XPPEDIT 457 0 "sqrt(3)" "6#-% %sqrtG6#\"\"$" }{TEXT 1850 176 "/2, -1/2); and the four charges are f ixed to the corners of a square at coordinates (1,1), (1,-1), (-1,-1), (-1,1). The origin is at the center of each geometric figure. Your " }{TEXT 1855 7 "problem" }{TEXT 1850 352 " is to determine the electric potential at the point (x,y) and see if there might be some points in space at which we can place a charge that is free to move and have it remain there even if perturbed. For the equivalent gravitational prob lem these stable points are known as Lagrange points and are the loca tion of asteroids for the earth-sun system." }}{PARA 246 "" 0 "" {TEXT 1856 0 "" }}}{SECT 0 {PARA 247 "" 0 "" {TEXT 1857 1 " " }{TEXT 1858 6 " 4.3 " }{TEXT 1859 47 "Theory: Stability Criteria for Potenti al Energy" }{TEXT 1860 0 "" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }} {PARA 243 "" 0 "" {TEXT 1850 89 "Coulomb's law tells us that if we hav e a charge q at the origin, then the electric field " }{TEXT 1855 1 "E " }{TEXT 1850 64 " (the force per unit charge) at a distance r from th at charge is" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 248 "" 0 "" {TEXT 1861 1 "E" }{TEXT 1862 5 " = " }{XPPEDIT 18 0 "k[e]*q/(r^2);" "6#*(&%\"kG6#%\"eG\"\"\"%\"qGF(*$%\"rG\"\"#!\"\"" }{TEXT 1862 2 " " } {XPPEDIT 453 0 "e[r];" "6#&%\"eG6#%\"rG" }{TEXT 1862 0 "" }}{PARA 249 "" 0 "" {TEXT 1863 1 " " }}{PARA 250 "" 0 "" {TEXT 1864 6 "where " } {XPPEDIT 18 0 "e[r];" "6#&%\"eG6#%\"rG" }{TEXT 1865 1 " " }{TEXT 1864 24 "is a unit vector in the" }{TEXT 1866 2 " r" }{TEXT 1864 18 " dire ctions. Here " }{XPPEDIT 18 0 "k[e];" "6#&%\"kG6#%\"eG" }{TEXT 1864 10 " = 8.9875 " }{XPPEDIT 450 0 "10^9;" "6#*$\"#5\"\"*" }{TEXT 1864 2 "N " }{XPPEDIT 449 0 "m^2;" "6#*$%\"mG\"\"#" }{TEXT 1864 1 "/" } {XPPEDIT 447 0 "C^2;" "6#*$%\"CG\"\"#" }{TEXT 1864 60 " is Coulomb's \+ constant in SI units, and the electric force " }{TEXT 1866 1 "E" } {TEXT 1864 53 " is directed radially away from the charge. Because " }{TEXT 1866 1 "E" }{TEXT 1864 218 " is a vector, the electric force fi eld about a charge is a vector field with both magnitude and direction at each point. However, no information is lost, and it is much simple r, if, instead of the electric force field " }{TEXT 1866 1 "E" }{TEXT 1864 42 ", we consider the electric potential field" }{TEXT 1865 0 "" }}{PARA 251 "" 0 "" {XPPEDIT 18 0 "V = k[e]*q/r;" "6#/%\"VG*(&%\"kG6#% \"eG\"\"\"%\"qGF*%\"rG!\"\"" }{TEXT 1867 4 " = " }{XPPEDIT 18 0 "q/r; " "6#*&%\"qG\"\"\"%\"rG!\"\"" }{TEXT 1867 0 "" }}{PARA 241 "" 0 "" {TEXT 1868 75 "In the second form of this equation we have left off th e electric constant " }{XPPEDIT 18 0 "k[e];" "6#&%\"kG6#%\"eG" }{TEXT 1848 1 " " }{TEXT 1868 149 "for simplicity; since this affects just t he magnitudes of the graphs and not their shapes, it will not change t he conclusions we draw. We see that " }{TEXT 1869 4 "V(r)" }{TEXT 1868 29 " falls off less rapidly than " }{TEXT 1870 1 "E" }{TEXT 1868 86 " and is a scalar, that is, has no direction associated with it.on \+ associated with it. " }{TEXT 1848 0 "" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 242 "" 0 "" {TEXT 1849 257 "Our problem requires us to de termine the potentials for two- and three-charge systems, and then to \+ look for stable points in these potentials. To determine the potential for two charges, we use Pythagoras's theorem to determine the distan ce to the charges," }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 251 " " 0 "" {XPPEDIT 18 0 "r1 = sqrt((x-a)^2+y^2);" "6#/%#r1G-%%sqrtG6#,&*$ ,&%\"xG\"\"\"%\"aG!\"\"\"\"#F,*$%\"yGF/F," }{TEXT 1867 1 " " }}{PARA 251 "" 0 "" {TEXT 1871 5 "and " }{XPPEDIT 18 0 "r2 = sqrt((x+a)^2+y^2 );" "6#/%#r2G-%%sqrtG6#,&*$,&%\"xG\"\"\"%\"aGF,\"\"#F,*$%\"yGF.F," } {TEXT 1867 1 " " }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 242 "" 0 "" {TEXT 1849 65 "and then just add up the potentials from the individ ual charges: " }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 251 "" 0 " " {XPPEDIT 18 0 "V[2](x,y) = q1/sqrt((x-a)^2+y^2)+q2/sqrt((x+a)^2+y^2) ;" "6#/-&%\"VG6#\"\"#6$%\"xG%\"yG,&*&%#q1G\"\"\"-%%sqrtG6#,&*$,&F*F/% \"aG!\"\"F(F/*$F+F(F/F7F/*&%#q2GF/-F16#,&*$,&F*F/F6F/F(F/*$F+F(F/F7F/ " }{TEXT 1867 1 " " }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 241 " " 0 "" {TEXT 1848 0 "" }}{PARA 241 "" 0 "" {TEXT 1868 91 "For three ch arges at the corners of the equilateral triangle, we know the coordin ates are " }{TEXT 1869 6 "(0, a)" }{TEXT 1868 9 ", (a cos" }{XPPEDIT 428 0 "theta;" "6#%&thetaG" }{TEXT 1868 9 " , -a cos" }{XPPEDIT 427 0 "theta;" "6#%&thetaG" }{TEXT 1868 9 "), where " }{XPPEDIT 426 0 "theta ;" "6#%&thetaG" }{TEXT 1848 3 " = " }{XPPEDIT 424 0 "30^o;" "6#)\"#I% \"oG" }{TEXT 1868 103 ". Again we use Pythagoras's theorem and add the potentials from the individual charges to obtain " }{TEXT 1848 0 "" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 251 "" 0 "" {TEXT 1871 1 " " }{XPPEDIT 18 0 "V[3](x,y) = q1/sqrt(x^2+(y-a)^2)+q2/sqrt((x -a*cos*theta)^2+(y+a*sin*theta)^2)+q3/sqrt((x+a*cos*theta)^2+(y+a*sin* theta)^2);" "6#/-&%\"VG6#\"\"$6$%\"xG%\"yG,(*&%#q1G\"\"\"-%%sqrtG6#,&* $F*\"\"#F/*$,&F+F/%\"aG!\"\"F5F/F9F/*&%#q2GF/-F16#,&*$,&F*F/*(F8F/%$co sGF/%&thetaGF/F9F5F/*$,&F+F/*(F8F/%$sinGF/FCF/F/F5F/F9F/*&%#q3GF/-F16# ,&*$,&F*F/*(F8F/FBF/FCF/F/F5F/*$,&F+F/*(F8F/FGF/FCF/F/F5F/F9F/" } {TEXT 1867 1 " " }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 252 "" 0 "" {TEXT 1850 171 "These equations for the electric potentials are what we wish to visualize. To make ou r work simpler, but not change the physics any, we will take a = 1 and substitute for " }{XPPEDIT 423 0 "theta;" "6#%&thetaG" }{TEXT 1850 0 "" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 251 "" 0 "" {XPPEDIT 18 0 "V[1](x,y) = q1/sqrt(x^2+y^2);" "6#/-&%\"VG6#\"\"\"6$%\"xG%\"yG*& %#q1GF(-%%sqrtG6#,&*$F*\"\"#F(*$F+F3F(!\"\"" }{TEXT 1867 1 " " }} {PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 251 "" 0 "" {XPPEDIT 18 0 "V [2](x,y) = q1/sqrt((x-1)^2+y^2)+q2/sqrt((x+1)^2+y^2);" "6#/-&%\"VG6#\" \"#6$%\"xG%\"yG,&*&%#q1G\"\"\"-%%sqrtG6#,&*$,&F*F/F/!\"\"F(F/*$F+F(F/F 6F/*&%#q2GF/-F16#,&*$,&F*F/F/F/F(F/*$F+F(F/F6F/" }{TEXT 1867 1 " " }} {PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 251 "" 0 "" {XPPEDIT 18 0 "V[3](x,y) = q1/sqrt(x^2+(y-1)^2)+q2 /sqrt((x-sqrt(3)/2)^2+(y+1/2)^2)+q3/sqrt((x+sqrt(3)/2)^2+(y+1/2)^2);" "6#/-&%\"VG6#\"\"$6$%\"xG%\"yG,(*&%#q1G\"\"\"-%%sqrtG6#,&*$F*\"\"#F/*$ ,&F+F/F/!\"\"F5F/F8F/*&%#q2GF/-F16#,&*$,&F*F/*&-F16#F(F/F5F8F8F5F/*$,& F+F/*&F/F/F5F8F/F5F/F8F/*&%#q3GF/-F16#,&*$,&F*F/*&-F16#F(F/F5F8F/F5F/* $,&F+F/*&F/F/F5F8F/F5F/F8F/" }{TEXT 1867 1 " " }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 242 "" 0 "" {TEXT 1849 922 "Owing to its two d imensional nature, a purely mathematical solution for the equilibrium \+ points in these potential gets complicated. Instead, we will solve it \+ graphically and rely on our intuitive understanding of how balls roll \+ under the action of gravity. Specifically, we know that a ball rele ased on a surface rolls downhill, and that if the ball is placed in a \+ concave depression, it will remain there. Because the gravitational p otential near the Earth's surface is proportional to height, our d escription of the ball on a surface is equivalent to a description of \+ how a particle behaves in a potential energy field. It therefore foll ows that charges will ``roll down'' the electric potential surface, \+ and will find a stable position at the concave minimum of the potenti al. So our problem translates into drawing pictures of the electric po tential surfaces and looking for minima at the bottom of hills.\n" }}} {SECT 0 {PARA 247 "" 0 "" {TEXT 1857 3 " " }{TEXT 1858 4 "4.4 " } {TEXT 1859 21 "Basic 2-D Plots: plot" }{TEXT 1860 0 "" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 252 "" 0 "" {TEXT 1850 364 "Before we get to Maple's plotting commands, let us examine some general principles. First, keep in mind that the point of visualization is to make the sc ience clearer and to better communicate your work to others. So it fol lows that when you produce a figure you should look at it and think if there are some better choices of units, ranges of axes colors, style, " }{TEXT 1872 9 "et cetera" }{TEXT 1850 815 " that might get the mess age across better and provide better insight. Taking into account that we are dealing with the complexity of human perception and cognition, there may not be one definite way to do things, and some trial and er ror is necessary to see what looks best.\n\nOur general recommendation for visualization is to make each figure as clear, informative and se lf-explanatory as possible. This means labels for various curves and d ata points, a title, and labels on the axes. We know, you are thinking that this is really a lot of work for a lousy assignment or report, a nd that you do not need all those time-consuming extras to comprehend \+ what is going on. Yet the more often you do it, the quicker and better you get at it, and the more useful will your work be to others (and y ourself in the future)." }{TEXT 1850 72 "\n\nThe convention when plott ing is to have the independent variable, say " }{TEXT 1872 1 "x" } {TEXT 1850 72 ", along the abscissa (horizontal axis), and the depende nt variable, say " }{TEXT 1872 8 "y = f(x)" }{TEXT 1850 369 ", along t he ordinate. (Remember that your mouth spreads horizontally across whe n you say ``abscissa'', and\nthat it puckers vertically up when you s ay ``ordinate''.) If you have trouble deciding which variable is indep endent, think of an experiment in which you measure the position or ve locity of a ball as a function of time. Because you are free to pick t he times at" }{TEXT 1850 226 "\nwhich you make the measurement, time i s an independent variable. However, once you have chosen the time, nat ure picks what the\nposition of the ball is at that particular time, s o position and velocity are dependent variables." }{TEXT 1850 0 "" }} {PARA 241 "" 0 "" {TEXT 1848 0 "" }}}{SECT 0 {PARA 253 "" 0 "" {TEXT 1873 12 " " }{TEXT 1874 7 " 4.4.1" }{TEXT 1875 3 " " } {TEXT 1876 4 "The " }{TEXT 1877 5 "plot " }{TEXT 1876 7 "command" } {TEXT 1873 0 "" }}{PARA 253 "" 0 "" {TEXT 1873 1 " " }}{PARA 254 "" 0 "" {TEXT 1871 303 "Maple excels at easily producing graphs of all sort s, and indeed, visualization is one of the most valuable aspects of Ma ple. Although we will discuss and give examples of a number of possibl e plots, Maple affords more options than we discuss, and we recommend \+ you look at the commands listed after the " }{TEXT 1878 11 "with(plots )" }{TEXT 1871 27 " statement, and browse the " }{TEXT 1878 4 "help" } {TEXT 1871 181 " pages to create just the graph you want. We will firs t make a simple plot and then embellish it with things like labels and colors. In the execution group below define a function " }{XPPEDIT 405 0 "f(x) = x*sin(x);" "6#/-%\"fG6#%\"xG*&F'\"\"\"-%$sinG6#F'F)" } {TEXT 1867 1 " " }{TEXT 1871 1 ":" }{TEXT 1867 0 "" }}{PARA 241 "" 0 " " {TEXT 1848 0 "" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 8 "restart;" } }{PARA 254 "> " 0 "" {MPLTEXT 1 1880 26 "with(plots); " } {TEXT 1871 29 "# Loads tools for you to try " }{TEXT 1867 0 "" }} {PARA 252 "> " 0 "" {MPLTEXT 1 1881 55 " \+ " }{TEXT 1850 41 "# Place your function defin iton f(x) here" }}{PARA 256 "> " 0 "" {MPLTEXT 1 1882 1 " " }}{PARA 243 "" 0 "" {TEXT 1850 31 "We see that in response to the " }{TEXT 1872 11 "with(plots)" }{TEXT 1850 111 " command, Maple displays all of the plotting commands that are available with this package. (We some times use " }{TEXT 1872 12 "with(plots):" }{TEXT 1850 146 " with a co lon rather than semicolon to avoid the listing.) Now that we have the \+ tools, let us look at the electric potential for a single charge:" }} {PARA 257 "" 0 "" {TEXT 1883 0 "" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 17 "V :=(r) -> 1/r;" }}{PARA 254 "> " 0 "" {MPLTEXT 1 1880 34 " plot(V(r), r = 0..0.2); " }{TEXT 1871 34 "# Plot previously \+ defined function" }{MPLTEXT 1 1884 1 " " }{TEXT 1867 0 "" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 25 "plot(V(r), r = 1/50 ..1);" }}{PARA 258 " > " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 243 "" 0 "" {TEXT 1850 67 "You w ill observe from the first figure that second argument to the " } {TEXT 1872 4 "plot" }{TEXT 1850 53 " command gives the range of values for the\nabscissa (" }{TEXT 1872 1 "r" }{TEXT 1850 43 " in this case) . Our interest is really for " }{TEXT 1872 1 "r" }{TEXT 1850 9 " betwe en " }{TEXT 1872 1 "0" }{TEXT 1850 180 " and infinity, but this does n ot produce such a useful result since we primarily see the repulsive p eak at the origin. In view of that, we get a more revealing plot by no t letting " }{TEXT 1872 1 "r" }{TEXT 1850 105 " get quite so close to \+ the origin. The second plot eliminates the part of the graph with the \+ infinity at " }{TEXT 1872 3 "r=0" }{TEXT 1850 198 ", and so does not f ully convey the image that the potential is infinite there. However, w e tailor our plot more to our liking by giving some limits to the ordi nate (also works if called the generic " }{TEXT 1872 1 "y" }{TEXT 1850 2 "):" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 251 "> " 0 "" {MPLTEXT 1 1880 33 "plot(V(r), r = 0..1, V = 0..10); " }{TEXT 1871 34 " # Limit the y range (cute, huh!)" }{TEXT 1867 0 "" }}{PARA 251 "> \+ " 0 "" {MPLTEXT 1 1880 33 "plot( min(V(r),10), r = 0..1); " }{TEXT 1871 41 "# Keep ordinate less than 10, another way" }{TEXT 1867 0 "" } }{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 251 "" 0 "" {TEXT 1871 111 "As an alternative, it is possible to tell Maple that you wan t to see the full dependence of the potential from " }{TEXT 1878 3 "r= 0" }{TEXT 1871 4 " to " }{XPPEDIT 18 0 "infinity" "6#%)infinityG" } {TEXT 1871 2 ", " }{TEXT 1878 27 "plot(V(r), r = 0..infinity)" }{TEXT 1871 106 ", but then you lose some details. Try leaving off the range \+ for the abscissa to test Maple's capabilities:" }{TEXT 1867 0 "" }} {PARA 258 "> " 0 "" {MPLTEXT 1 1879 28 "plot(V(r), r = 0..infinity);" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 243 "" 0 "" {TEXT 1850 52 "You see that because Maple was not given a range of " }{TEXT 1872 1 "r" }{TEXT 1850 59 " values to plot, it does not think it has a nything to plot " }{TEXT 1872 10 "empty plot" }{TEXT 1850 54 ".\n\nThe plot above shows the basic physics. If we view " }{TEXT 1872 4 "V(r) " }{TEXT 1850 331 " as an equivalent gravitational potential, a small \+ positive charge (mass) placed near the fixed positive charge will be r epelled (roll downhill) out to infinity. There are not any locations w here a charge remains at rest in equilibrium. If we had fixed a nega tive charge at the origin, the potential would have the opposite sign: " }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 34 "plot(-V(r), r = 0..1, V = \+ 0..-10);" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 243 "" 0 "" {TEXT 1850 197 "This shows that, regardless of where we place it, o ur positive test charge will fall into the hole at the origin. As we h ave just seen by placing a minus sign in front of the first argument t o the " }{TEXT 1872 4 "plot" }{TEXT 1850 89 " command, it is allowable to have the argument be an expression and not just a function:" }} {PARA 259 "" 0 "" {TEXT 1885 0 "" }}{PARA 241 "> " 0 "" {MPLTEXT 1 1886 53 "plot(-5/r, r = 0..5,V = 0..-20); " } {TEXT 1887 25 "# Plot explict expression" }{TEXT 1848 0 "" }}{PARA 256 "> " 0 "" {MPLTEXT 1 1882 1 " " }}{PARA 252 "" 0 "" {TEXT 1850 38 "In summary, the first argument to the " }{TEXT 1872 4 "plot" }{TEXT 1850 262 " command is the name of function or expression to be plotted along the ordinate, that is, the dependent variable. The second argum ent is the range of values for abscissa, that is, the independent vari able. The double period .. is used to specify the range, so " } {TEXT 1872 8 "-10 ..10" }{TEXT 1850 12 " means from " }{TEXT 1872 3 "- 10" }{TEXT 1850 4 " to " }{TEXT 1872 3 "+10" }{TEXT 1850 61 ". If the \+ upper end of the range is a decimal value, say like " }{TEXT 1872 2 ". 5" }{TEXT 1850 56 ", then it is clearer to enter it with a leading zer o as " }{TEXT 1872 3 "0.5" }{TEXT 1850 31 ", so that the range looks l ike " }{TEXT 1872 8 "-10..0.5" }{TEXT 1850 53 ", and not the confusing (to the reader and to Maple) " }{TEXT 1872 7 "-10...5" }{TEXT 1850 40 ".\n\nBefore we get on to embellishing the " }{TEXT 1872 4 "plot" } {TEXT 1850 71 " command, let us have some fun with the pretty graph yo u just produced:" }}{PARA 260 "" 0 "" {TEXT 1888 32 "Click on the grap h to select it." }}{PARA 261 "" 0 "" {TEXT 1889 135 "Note how a box is formed around the selected graph and that there are dark little nodes at the corners and in the middle of the sides. " }}{PARA 262 "" 0 "" {TEXT 1890 347 "Use your mouse to resize the graph by grabbing one of \+ the nodes, and then dragging it with the mouse button still depressed. Notice that when a node is selected, a little arrow appears to show y ou the direction in which the frame can be resized. The resizing can b e done diagonally along the corners or horizontally and vertically alo ng the edges." }}{PARA 262 "" 0 "" {TEXT 1890 135 "Select the graph, c opy it (it is placed on the ``clipboard''), and then paste it back to \+ the worksheet so that now you have two graphs." }}{PARA 263 "" 0 "" {TEXT 1850 152 "Make one of your graphs into a long-narrow one and the other into a high tall one. Notice how the tall one emphasizes the va riation in the magnitude of " }{TEXT 1872 4 "V(r)" }{TEXT 1850 63 ", w hile the long one emphasizes the range of r values to which" }{TEXT 1872 5 " V(r)" }{TEXT 1850 159 " extends. Both are perfectly legitimat e ways to view a function, one emphasizes the singular nature near the origin, the other the long range of the potential." }}{PARA 263 "" 0 "" {TEXT 1850 172 "Another way to view a function, especially one that has orders of magnitude variation in value, is with a semilog plot or log-log. (Note, however, that you cannot take the " }{TEXT 1872 3 "lo g" }{TEXT 1850 3 " of" }{TEXT 1872 2 " 0" }{TEXT 1850 41 ".) In the ex ecution group below, use the " }{TEXT 1872 5 "log10" }{TEXT 1850 69 " \+ function to see how semilog plot changes the appearance of the same " }{TEXT 1872 4 "f(x)" }{TEXT 1850 24 " we have been viewing. " }} {PARA 263 "" 0 "" {TEXT 1850 44 "Next try the explicit semilog plot fu nction " }{TEXT 1872 7 "logplot" }{TEXT 1850 57 ", following the instr uction on the comments fields below:" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 1884 2 " " }{TEXT 1871 42 " \+ " }{TEXT 1891 29 "# Repeat plot \+ with log(V(r))" }{TEXT 1867 0 "" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 25 "plot(log10(x^2),x=0..10);" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 12 "with(plots):" }}{PARA 241 "> " 0 "" {MPLTEXT 1 1886 22 "logpl ot(x^2,x=0..10); " }{TEXT 1887 42 "# Explicit semilog plot; log(1st ar gument)" }{TEXT 1848 0 "" }}{PARA 241 "> " 0 "" {MPLTEXT 1 1892 18 " \+ " }{TEXT 1887 23 "# Try log-log plot here" }{TEXT 1848 0 "" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}}{SECT 0 {PARA 243 "" 0 "" {TEXT 1850 15 " " }{TEXT 1851 6 "4.4.2 " } {TEXT 1850 3 " " }{TEXT 1852 38 "Labels and Titles (the plot thicke ns)" }{TEXT 1850 0 "" }}{PARA 264 "" 0 "" {TEXT 1893 0 "" }}{PARA 265 "" 0 "" {TEXT 1894 149 "Any plot worth looking at is worth explaining. This is done by placing labels along the axes and by placing a title \+ above the curves. Here, try this:" }}{PARA 264 "" 0 "" {TEXT 1893 0 " " }}{PARA 258 "> " 0 "" {MPLTEXT 1 1895 113 "plot(1/r, r =1/10..5, lab els=[`radius r (natural units)`,`V(r)`], title=`Potential for point ch arge at origin`); " }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }} {PARA 243 "" 0 "" {TEXT 1850 206 "Take stock of how we just added a \+ comma after the range and then added the options, separated by commas, to produce the labels and title. Because there are two axes, the labe ls field has an entry for the " }{TEXT 1872 1 "x" }{TEXT 1850 19 " axi s and then the " }{TEXT 1872 1 "y" }{TEXT 1850 93 " axis. The labels a nd title are enclosed in back quotes in order to delimit the expressi ons:" }}{PARA 243 "> " 0 "" {MPLTEXT 1 1881 21 " \+ " }{TEXT 1850 53 " # Enter previous plot command using ordinary accent s" }{MPLTEXT 1 1881 1 " " }{TEXT 1850 0 "" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 265 "" 0 "" {TEXT 1894 178 "Modify the pr evious command so that the words ``abscissa'' and ``ordinate'' appear \+ in the appropriate places and so that the actual expression being plot ted appears in the title:" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }} {PARA 243 "> " 0 "" {MPLTEXT 1 1881 12 " " }{TEXT 1850 67 " # Plot with your modified labels and title" } }{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 243 "" 0 "" {TEXT 1850 43 "You may have noticed that we have placed " }{TEXT 1872 1 "r " }{TEXT 1850 13 " along the ``" }{TEXT 1872 1 "x" }{TEXT 1850 12 "-ax is'' and " }{TEXT 1872 4 "V(r)" }{TEXT 1850 13 " along the ``" }{TEXT 1872 1 "y" }{TEXT 1850 46 "'' axis. In fact, there may be cases in whi ch " }{TEXT 1872 1 "y" }{TEXT 1850 118 " is the independent variable. \+ Thus you see why it may be better to use the words ``ordinate'' and `` abscissa'' than ``" }{TEXT 1872 1 "x" }{TEXT 1850 9 "'' and ``" } {TEXT 1872 1 "y" }{TEXT 1850 8 "'' axes." }}{PARA 264 "" 0 "" {TEXT 1893 0 "" }}}{SECT 0 {PARA 266 "" 0 "" {TEXT 1896 17 " 4.4.3 \+ " }{TEXT 1897 2 " " }{TEXT 1898 50 "Compound (Abstract) Data Types: \+ [Lists] and \{Sets\}" }{TEXT 1897 0 "" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 243 "" 0 "" {TEXT 1850 94 "As we proceed with our exercis es in visualization, you will see how to enter arguments to the " } {TEXT 1872 4 "plot" }{TEXT 1850 854 " commands using different types o f parentheses. We recognize that some users may prefer just following \+ the rules without questioning them. Nevertheless, the commands will ma ke more sense, and will be easier to generalize, if you have some unde rstanding of the method behind the madness. And so we now take a littl e excursion in which we define some terms that are frequently used in \+ mathematics and computer science and employed by Maple commands.\n\nWe have already seen a number of ways in which Maple displays data. At t imes there is just a single symbol, sometimes there is a bunch of symb ols separated by commas, sometimes there is a bunch of things in paren thesis, and sometime the symbols are in quotes. To illustrate, in Chap . 5 we you will see that when you solve an equation that has several \+ solutions, Maple separates the solutions with commas:" }{TEXT 1850 0 " " }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{EXCHG {PARA 258 "> " 0 "" {MPLTEXT 1 1895 16 "solve(x^4-1, x);" }}{PARA 243 "> " 0 "" {MPLTEXT 1 1899 23 "solve(x^4-1, x)[1]; " }{TEXT 1850 23 "# Solve for only 1 root" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}}{PARA 242 "" 0 " " {TEXT 1849 91 "Take note of two types of parentheses here and the di fferent forms given for the solutions." }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 243 "" 0 "" {TEXT 1855 8 "Abstract" }{TEXT 1850 4 " \+ or " }{TEXT 1855 8 "Compound" }{TEXT 1850 1 " " }{TEXT 1855 10 "Data T ypes" }{TEXT 1850 5 ": An " }{TEXT 1872 6 "object" }{TEXT 1850 87 " in Computer Science denotes a data type with multiple parts. It may also be called an " }{TEXT 1872 8 "abstract" }{TEXT 1850 17 " data type, o r a " }{TEXT 1872 8 "compound" }{TEXT 1850 233 " data type.Here we use the word ``abstract'' to mean that there is more to something than me ets the eye, that is, the data type may contain multiple parts. Many o f the individual symbols or variables used in Maple can be\nreplaced b y " }{TEXT 1872 7 "objects" }{TEXT 1850 82 ". We will discuss objects in more depth when we study Java, which is known as an " }{TEXT 1872 15 "object oriented" }{TEXT 1850 11 " language.\n" }{TEXT 1855 11 "\nD ata types" }{TEXT 1850 59 ": In specifying labels for the plot command , we placed the " }{TEXT 1872 1 "x" }{TEXT 1850 5 " and " }{TEXT 1872 1 "y" }{TEXT 1850 34 " labels in square parenthesis [..]" }{TEXT 1872 1 "." }{TEXT 1850 74 " These are called ``brackets''. Maple also uses the standard parenthesis " }{TEXT 1872 4 "(..)" }{TEXT 1850 24 ", and curly parenthesis " }{TEXT 1872 4 "\{..\}" }{TEXT 1850 154 " called \+ ``braces''. Brackets and braces are used to construct abstract data \+ types or objects from the more elementary data types we have already s een.\n" }{TEXT 1855 9 "\nSequence" }{TEXT 1850 84 ": A collection of v ariables (objects or data types) separated by commas is called a " } {TEXT 1872 8 "sequence" }{TEXT 1850 49 ". As a case in point, the argu ments given to the " }{TEXT 1872 5 "solve" }{TEXT 1850 352 " command a bove, and indeed to most Maple commands, are variables separated by co mmas, and, hence, form sequences. While we are speaking of sequences w e may as well indicate that when we give arguments to a Maple command \+ as a comma-separated list, the parenthesis indicate sequence. It is o ften convenient to let Maple form a sequence for you with the " } {TEXT 1872 3 "seq" }{TEXT 1850 9 " command:" }}{PARA 264 "" 0 "" {TEXT 1893 0 "" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 19 "seq(2*n-1, n =1..4);" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 239 "" 0 " " {TEXT 1843 0 "" }}{PARA 243 "" 0 "" {TEXT 1855 6 "List: " }{TEXT 1850 114 " A list is a sequence of numbers, or abstract data types, se parated by commas and placed within square brackets. T" }{TEXT 1872 27 "he order matters for a list" }{TEXT 1850 26 ", while it does not f or a " }{TEXT 1872 3 "set" }{TEXT 1850 55 " (to be defined soon).We us e a list when we issue the " }{TEXT 1872 4 "plot" }{TEXT 1850 10 " co mmand:\n" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1895 113 "plot(1/r, r =1/10 ..5, labels=[`radius r (natural units)`,`V(r)`], title=`Potential for \+ point charge at origin`); " }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 " " }}{PARA 257 "" 0 "" {TEXT 1883 78 "Now we try creating the plot agai n, this time changing the order of the list:" }}{PARA 257 "> " 0 "" {MPLTEXT 1 1900 5 " " }{TEXT 1883 27 "# Plot with reordered list " }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 267 "" 0 "" {TEXT 1901 154 "Clearly order matters here as we would not want thelab els for the abscissa and ordinates interchanged! You may enter the ele ments of a list by hand, as in" }}{PARA 253 "" 0 "" {TEXT 1873 0 "" }} {PARA 258 "> " 0 "" {MPLTEXT 1 1895 19 "list1 := [2,4,6,8];" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1895 0 "" }}{PARA 264 "" 0 "" {TEXT 1893 22 " Or you can use Maples " }{TEXT 1902 3 "seq" }{TEXT 1893 51 " command t o help generate the elements of the list:" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 30 "list2 := [seq(2*n- 1, n=1..4)];" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 264 " " 0 "" {TEXT 1893 130 "The individual elements of a list can be refere nced via Maple's square bracket notation (a standard way of indicating subscripts):" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 39 "list2[1]; list1[1]; list2[2]; list1[2];" }} {PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 243 "" 0 "" {TEXT 1855 3 "Set" }{TEXT 1850 167 ": A set is a well-defined collection of related objects or elements with \+ no repeated element [Fral 76] In contrast to a list, the order of the \+ elements in a set does " }{TEXT 1872 3 "not" }{TEXT 1850 375 " matter. Sets are also used as arguments to Maple commands, but only in cases \+ where the order of the elements does matters, for example, a set of eq uations to be solved. Sets are usually described by enumerating their \+ elements, separated by commas, as a sequence within braces. To prove t he point, the sets of equations and solutions used when solving simult aneous equations:" }}{PARA 264 "" 0 "" {TEXT 1893 0 "" }}{PARA 264 "> \+ " 0 "" {MPLTEXT 1 1903 50 "solve( \{a+3*b+4*c=41, x5*a+6*b+7*c=20\}, \+ \{a,b\}); " }{TEXT 1904 50 "# The order within braces does not matte r for a se" }{TEXT 1893 1 "t" }}{PARA 264 "> " 0 "" {MPLTEXT 1 1903 50 "solve( \{x5*a+6*b+7*c=20, a+3*b+4*c=41\}, \{b,a\}); " }{TEXT 1904 51 "# The order within braces does not matter for a set" }{TEXT 1893 0 "" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 267 "" 0 "" {TEXT 1901 34 "Maple uses braces to denote sets:" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 21 "NiceSet := \{0,2,4,6\};" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 267 "" 0 "" {TEXT 1901 0 "" }}{PARA 267 "" 0 "" {TEXT 1901 363 "Seeing that lists and sets both contain comma-separated sequences within the m, we emphasize that it is legal for the same element to occur more th an once in a list, with the order of the elements in the list part of \+ its definition. As an example, if we define a set with repeated elemen ts in arbitrary order, then Maple will remove the repeats and reorder \+ for us:" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 26 "MessySet := \{6,4,0,4,2,0\};" }}{PARA 258 "> " 0 " " {MPLTEXT 1 1879 0 "" }}{PARA 268 "" 0 "" {TEXT 1905 67 "In contrast, Maple does not change the order or elements of a list:" }}{PARA 239 " " 0 "" {TEXT 1843 0 "" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 27 "Messy List := [6,4,0,4,2,0];" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }} {PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 243 "" 0 "" {TEXT 1855 6 "Ar rays" }{TEXT 1850 93 ": Another data type related to vectors and matri ces are arrays. We discuss them in Chapter 7." }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}}{SECT 0 {PARA 269 "" 0 "" {TEXT 1844 5 " " } {TEXT 1845 5 "4.5 " }{TEXT 1844 1 " " }{TEXT 1846 33 " Several Curves on One Plot, Sets" }{TEXT 1847 0 "" }}{PARA 264 "" 0 "" {TEXT 1893 0 "" }}{PARA 243 "" 0 "" {TEXT 1850 45 " We have seen that the first arg ument to the " }{TEXT 1872 4 "plot" }{TEXT 1850 147 " command is the f unction to be plotted. As a case in point, imagine that as part of our charge problem, we want to compare, in a single plot, the " }{TEXT 1872 1 "r" }{TEXT 1850 203 " dependence of the potential and the magni tude of the electric field due to both positive and negative charges. \+ Seeing that Maple treats the argument as an object, we substitute the \+ set \{ 1/r, -1/r , 1/" }{XPPEDIT 18 0 "r^2;" "6#*$%\"rG\"\"#" }{TEXT 1906 1 " " }{TEXT 1850 7 " , -1/ " }{XPPEDIT 18 0 "r^2;" "6#*$%\"rG\" \"#" }{TEXT 1906 1 " " }{TEXT 1850 31 "\} as the first argument to the " }{TEXT 1872 4 "plot" }{TEXT 1850 95 " command. The fact that we use a set as the object to plot rather than a list [1/r, -1/r , 1/" } {XPPEDIT 18 0 "r^2;" "6#*$%\"rG\"\"#" }{TEXT 1906 1 " " }{TEXT 1850 7 " , -1/ " }{XPPEDIT 18 0 "r^2;" "6#*$%\"rG\"\"#" }{TEXT 1906 1 " " } {TEXT 1850 118 "] makes sense since order does not matter and there i s no point in plotting identical functions on top of each other:" }} {PARA 270 "" 0 "" {TEXT 1907 0 "" }}{PARA 243 "> " 0 "" {MPLTEXT 1 1899 78 "plot(\{1/r, -1/r, 1/r^2, -1/r^2, r\},r=1/10..5, y= -20..20,la bels=[`r`,`V, E`]);" }{TEXT 1850 32 " # Set as 1st argument " }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 267 "" 0 "" {TEXT 1901 190 " Notice that \+ Maple has chosen a different color for each of the functions. Experime nt now with using a list as the first argument and noting how the \+ colors assigned to the curves differ:" }}{PARA 243 "> " 0 "" {MPLTEXT 1 1881 33 " " }{TEXT 1850 50 " \+ # Use a list [...] for function argument" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}}{SECT 0 {PARA 271 "" 0 "" {TEXT 1908 15 " \+ " }{TEXT 1909 7 "4.5.1 " }{TEXT 1910 24 "Using the Figure T oolbar" }{TEXT 1911 0 "" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 272 "" 0 "" {TEXT 1912 352 "The colors and line formats that Maple pic ks for graphs, such as the one below, may look great on your screen, \+ but may not print out or project (green and yellow are often barely vi sible). In the next subsection we discuss how to customize the colors to your preference. In this subsection we'll explore some of the opt ions using the figure toolbar." }}{PARA 273 "" 0 "" {TEXT 1913 338 "Se lect the graph with your mouse (you should notice a box appearing abou t the graph after it is selected). When the graph is selected, observe that the second line of the toolbar at the top of your screen now con tains icons for graphical options. Explore what each of these options \+ do. This is both useful and fun. We'll get you started. " }}{PARA 273 "" 0 "" {TEXT 1913 145 "Go to the Style pull-down menu and, under Line Width, select Broad. In particular, note the difference it makes for the yellow and green curves." }}{PARA 273 "" 0 "" {TEXT 1913 56 "Go t o the Legend pull-down menu and enable Show Legend. " }}{PARA 262 "" 0 "" {TEXT 1890 58 "Again go to the Legend pull-down, and select Edit \+ Legend. " }}{PARA 263 "" 0 "" {TEXT 1850 36 "Change the legends so tha t they are " }{TEXT 1872 25 "V(r), -V(r), E(r), -E(r)," }{TEXT 1850 35 " and r for the appropriate curves." }}{PARA 272 "" 0 "" {TEXT 1912 86 "Finally, explore how the different buttons controlling the pl acement of the axes work." }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}} {SECT 0 {PARA 274 "" 0 "" {TEXT 1914 10 " " }{TEXT 1915 8 "4. 5.2 " }{TEXT 1916 33 "Customizing Colors and Line Types" }{TEXT 1915 0 "" }}{PARA 264 "" 0 "" {TEXT 1893 0 "" }}{PARA 243 "" 0 "" {TEXT 1850 119 "Maple automatically chooses different color multi-fu nction plots. You control the color of your graphs by adding the " } {TEXT 1872 5 "color" }{TEXT 1850 26 " option to the end of the " } {TEXT 1872 4 "plot" }{TEXT 1850 9 " command:" }}{PARA 264 "" 0 "" {TEXT 1893 0 "" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1895 56 "plot([1/r, 1 /r^2, r], r = 0..1.5, y=0..25, color=black);" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 267 "" 0 "" {TEXT 1901 156 "Taking into a ccount that options are objects with multiple parts permitted, you ent er a list (order matters) of colors to specify the colors of each cur ve:" }}{PARA 246 "" 0 "" {TEXT 1856 0 "" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1895 69 " plot([1/r, 1/r^2, r], r = 0..1.5, y=0..25,color= [red,blue,maroon]);" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }} {PARA 275 "" 0 "" {TEXT 1850 152 "Likewise, you choose different sty les for each curve (like dashed and solid) to help tell the curves apa rt, even in basic black. You do that with the " }{TEXT 1872 9 "linesty le" }{TEXT 1850 49 " option, which may also be a list for each curve: " }}{PARA 276 "" 0 "" {TEXT 1917 96 " \+ n=1, solid; n=2, dotted; n=3, dashed, and n=4 dash-dotted. " }} {PARA 275 "> " 0 "" {MPLTEXT 1 1881 10 " " }{TEXT 1850 69 "# \+ Plot again, but with different linestyles as a list and color=black" } }{PARA 275 "> " 0 "" {MPLTEXT 1 1881 10 " " }{TEXT 1850 76 "# Plot again, but with different linestyles as a list and the default c olors" }{MPLTEXT 1 1881 6 " " }{TEXT 1850 0 "" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 275 "" 0 "" {TEXT 1850 134 "An espec ially effective way to distinguish different curves on the same plot w ithout the use of color, is to draw them with different " }{TEXT 1872 11 "thicknesses" }{TEXT 1850 29 ". Apply this change with the " } {TEXT 1872 11 "thickness=n" }{TEXT 1850 87 " option, where again, a li st for the curves is a legal option. The possible values for " }{TEXT 1872 9 "thickness" }{TEXT 1850 6 " are: " }{TEXT 1872 10 "n= 0, 1, 2" }{TEXT 1850 6 ", and " }{TEXT 1872 1 "3" }{TEXT 1850 8 ", where " } {TEXT 1872 1 "0" }{TEXT 1850 26 " is the default thickness." }}{PARA 275 "> " 0 "" {MPLTEXT 1 1881 19 " " }{TEXT 1850 59 "# Replot the graph, with different thickness for each curve" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}}{SECT 0 {PARA 269 "" 0 "" {TEXT 1844 13 " " }{TEXT 1845 9 " 4.5.3 " }{TEXT 1846 26 "Lege nds, Titles and Labels" }{TEXT 1847 0 "" }}{PARA 264 "" 0 "" {TEXT 1893 0 "" }}{PARA 276 "" 0 "" {TEXT 1917 842 "Legends explain to the r eader just what is being plotted with each curve. They are invaluable \+ and do wonders for your presentation. When presenting several curves i n one plot, it is important that the viewer not only be able to tell t hem apart by the different color or line style used for each, but also be given information as to what the difference curves represent. It i s good practice to explain in the caption below a graph what each curv e means, as well as in the text (or in your talk) when the graph gets \+ referenced. However, it is also good practice to have a legend in the \+ plot itself explaining what each curve means. Captions and your explan ations may get removed, but it is a lot harder to remove a legend.\n\n The legend option specifies a single or a list of strings in the same order as the curves with a legend for each curve:" }{TEXT 1917 0 "" } }{PARA 277 "> " 0 "" {MPLTEXT 1 1918 64 "plot([1/r,1/r^2], r = 0..1.5, y=0..25, legend=[ `V(r)`,`E(r)`]);" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}}{SECT 0 {PARA 269 "" 0 "" {TEXT 1844 13 " " } {TEXT 1845 9 " 4.5.4 " }{TEXT 1846 13 "Other Options" }{TEXT 1847 0 "" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 252 "" 0 "" {TEXT 1850 73 "As we have said, there are many ways to customize your graph and M aple's " }{TEXT 1872 5 "Help " }{TEXT 1850 282 " pages are a good plac e to find out about them. Once in a while your graph may not show the \+ features you want because one function gets very large and Maple autom atically adjusts the ordinate range to accommodate that. Here are a nu mber of ways to limit the range of the ordinates:" }}{PARA 278 "" 0 " " {TEXT 1919 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 1880 39 "plot([sin( x), tan(x),x], x = -Pi..Pi); " }{TEXT 1871 92 " \+ # tan(x) overshadows sin(x) " }{TEXT 1867 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 1880 51 "plot([sin (x), tan(x),x], x = -Pi..Pi, y= -10..10); " }{TEXT 1871 83 " \+ # y limits the verti cal" }{TEXT 1867 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 1880 48 "plot([ sin(x), min(10,tan(x)), x], x = -Pi..Pi); " }{TEXT 1871 95 " \+ # Limit + values to 1 0, not - values" }{TEXT 1867 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 1880 48 "plot([sin(x), max(-10,tan(x)), x], x = -Pi..Pi);" }{TEXT 1871 98 " # \+ Limits - values to -10, not + values" }{TEXT 1867 0 "" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 252 "" 0 "" {TEXT 1850 129 "There are occasions when a function falls off slowly, and so you might want to see its behavior for values of its argument from -" }{XPPEDIT 1770 0 "infinity;" "6#%)infinityG" }{TEXT 1850 6 " to + " }{XPPEDIT 1768 0 "infinity;" "6#%)infinityG" }{TEXT 1850 140 ". It is clear that Maple is not afraid of big numbers, yet it is nice to see that it m akes this type of graph in a finite amount of space:" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 57 "plot(ex p(-x^2), x=-infinity..infinity, title=`Gaussian`);" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 66 "plot(\{1/r,1/r^2\}, r=0..infinity, title=`V a nd E of Point Charge`);" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }} {PARA 279 "" 0 "" {TEXT 1920 126 "Look at the second plot and its \+ labels along the axes, rather than horizontal. This was accomplish ed with the commands" }}{PARA 278 "" 0 "" {TEXT 1919 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 1880 66 "plot([1/r,1/r^2,r], r = 0..1.5, y=0..25, labels=[`r`, `E, V, r`]);" }{TEXT 1871 55 " \+ " }{TEXT 1891 13 " # usual way" } {MPLTEXT 1 1884 1 " " }{TEXT 1867 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 1880 109 "plot([1/r,1/r^2,r], r = 0..1.5, y=0..25, labels=[`r`, `E, \+ V, r`], labeldirections = [horizontal, vertical]); " }{TEXT 1871 42 " \+ # rotate ordinate label" }{TEXT 1867 0 "" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}}{SECT 0 {PARA 247 "" 0 "" {TEXT 1857 3 " " }{TEXT 1858 6 "4.6 " }{TEXT 1859 40 "3D (Surface) Plots of Analytic Functions" }{TEXT 1860 0 "" }}{PARA 252 "" 0 "" {TEXT 1850 91 "We have examined the potential field V(r)=1/r surrounding a \+ single charge as a function of" }{TEXT 1872 2 " r" }{TEXT 1850 76 ". A 2-D plot is fine for this since there is only one independent variab le " }{TEXT 1872 1 "r" }{TEXT 1850 91 ". However, when the same poten tial is expresses as a function of the Cartesian coordinate " }{TEXT 1872 1 "x" }{TEXT 1850 5 " and " }{TEXT 1872 1 "y" }{TEXT 1850 2 ", " }}{PARA 252 "" 0 "" {XPPEDIT 517 0 "V(x,y) = 1/sqrt(x^2+y^2);" "6#/-% \"VG6$%\"xG%\"yG*&\"\"\"F*-%%sqrtG6#,&*$F'\"\"#F**$F(F0F*!\"\"" } {TEXT 1850 1 " " }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 252 "" 0 "" {TEXT 1850 35 "we have two independent variables, " }{TEXT 1872 1 " x" }{TEXT 1850 5 " and " }{TEXT 1872 1 "y" }{TEXT 1850 81 ", and so ne ed a 3-D visualization. We get that by creating a world in which the \+ " }{TEXT 1872 1 "z" }{TEXT 1850 64 " dimension (mountain height) is th e value of the potential, and " }{TEXT 1872 1 "x" }{TEXT 1850 5 " and \+ " }{TEXT 1872 1 "y" }{TEXT 1850 324 " lie on a flat plane below the mo untain. Because the surface we are creating is a 3-D object it is \+ not possible to draw it on a flat screen, and so different technique s are used to give the impression of three dimensions to our brains. 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As seen in the second plot, we get a more useful visualization if we limit the maximum value of " }{TEXT 1872 1 "V" }{TEXT 1850 4 " to " }{TEXT 1872 4 " 2.5" }{TEXT 1850 19 " and add labels: " }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 43 " plot3d( min(2.5,V(x,y)), x=-4..4, y=-4..4);" }}{PARA 254 "> " 0 "" {MPLTEXT 1 1880 87 "plot3d( min(2.5,V(x,y)), x=-4..4, y=-4..4, axes = \+ BOXED, labels=[`x`, `y`, `V(x,y)`]); " }{TEXT 1871 33 " \+ # With options" }{TEXT 1867 0 "" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 243 "" 0 "" {TEXT 1850 205 "We try to make this plot more intuitively informative by making the color red correspond to th e highest values of the potential, blue smaller, and green cooler stil l. Color may be specified as the option " }{TEXT 1872 9 "color=red" } {TEXT 1850 95 " or with a number. Consequently, we try to be clever an d use the actual value of the potential " }{TEXT 1872 5 "V(x,y" } {TEXT 1850 33 ") as the color of the graph with " }{TEXT 1872 14 "colo r = V(x,y)" }{TEXT 1850 50 " option (yet we minimize the maximum value of the " }{TEXT 1872 1 "V" }{TEXT 1850 158 " in order to keep the sin gularity from confusing the color function). Seeing as how the potenti al varies continuously, this means that the color will as well:" }} {PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 96 " plot3d( min(2.5,V(x,y)), x=-4..4, y=-4..4, color=min(2.5,V(x ,y)), labels=[`x`, `y`, `V(x,y)`]);" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 252 "" 0 "" {TEXT 1850 63 "Even though there are man y options you can give as part of the " }{TEXT 1872 6 "plot3d" }{TEXT 1850 139 " command, it is both easier and more fun to first make a bas ic plot and then use Maple's graphical user interface (GUI) to modify \+ the plot." }}{PARA 280 "" 0 "" {TEXT 1921 103 "Select this surface wit h your mouse (a box with filled little squares on the perimeter shoul d appear)." }}{PARA 280 "" 0 "" {TEXT 1921 273 "Grab the surface by de pressing your left (or only) mouse button and holding it down. Now as \+ you move your mouse, the surface will rotate in three dimensions. Make sure to move both right to left and up and down and take note how the object begins to look three dimensional." }}{PARA 280 "" 0 "" {TEXT 1921 319 "Make sure that the surface is still selected and notice the \+ extra buttons that appear on the control panel. You should experiment \+ and try to make sense out of all of the buttons. Remember that if you hold down a button, a message with the purpose of the button appears \+ at the bottom of the screen. In particular, note:" }}{PARA 261 "" 0 " " {TEXT 1889 38 " the different ways to draw axes," }}{PARA 261 " " 0 "" {TEXT 1889 47 " the different ways to render the surface, " }}{PARA 261 "" 0 "" {TEXT 1889 55 " how contour plots compare t o the actual surface. " }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}}{SECT 0 {PARA 281 "" 0 "" {TEXT 1844 14 " " }{TEXT 1845 7 " 4.6 .1 " }{TEXT 1846 35 "Contours and Equipotential Surfaces" }{TEXT 1847 0 "" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 252 "" 0 "" {TEXT 1850 155 "To further help your mind understand that different colors m ean different potential values, and that the surface is three dimensi onal, we now include the " }{TEXT 1872 20 "style = patchcontour" } {TEXT 1850 77 " option. This adds contour lines that show different le vels of the potential:" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 23 "restart; with(plots): " }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 33 "V := (x,y) -> 1/sqrt(x^2 + y^2);" } }{PARA 255 "> " 0 "" {MPLTEXT 1 1879 133 "plot3d( min(2.5,V(x,y)), x=- 4..4, y=-4..4, axes = BOXED, labels = [`x`, `y`, `V(x,y)`], color=min( 2.5,V(x,y)), style = patchcontour);" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 252 "" 0 "" {TEXT 1850 243 "In analogy to gravity, c ontours lines may be viewed as lines of equal elevation, which means \+ that walking along a contour line does not change your elevation. For \+ our electrical potential, the contours are called equipotential surfac es.\n\nThe " }{TEXT 1872 11 "contourplot" }{TEXT 1850 275 " command \+ also supports the option of making only a 2-D contour plot of the surf ace (we prefer the 3-D contours to be shown soon). Because the contou rs are not being projected onto a curved surface, or being viewed obli quely, these have the potential of being more precise:" }}{PARA 239 " " 0 "" {TEXT 1843 0 "" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 38 "conto urplot(V(x,y), x=-4..4, y=-4..4);" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 252 "" 0 "" {TEXT 1850 373 "However, we see in the l eft plot that the equipotential surfaces appear as ellipses, and not t he circles to be expected for the symmetric case of a single charge. T he reason is that the viewing screen tends to be broader than higher, \+ and so plots are spread out that way. To get a plot with the same scal es along the ordinate and abscissa, as done on the right, we add the \+ " }{TEXT 1872 19 "scaling=constrained" }{TEXT 1850 8 " option:" }} {PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 59 "contourplot(V(x,y), x=-4..4, y=-4..4, scaling=constrained);" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 252 "" 0 "" {TEXT 1850 195 "This produces a more symmetrical figure, but still not round . Apparently, the rapid rise of the potential near the origin is not being handled well by Maple, and so we exclude it by use of the " } {TEXT 1872 3 "min" }{TEXT 1850 10 " function:" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 66 "contourplot( min(1,V(x,y)), x=-4..4, y=-4..4, scaling=constrained);" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 272 "" 0 "" {TEXT 1912 246 " Thi s looks better, but is clearly rather grainy. That is a clue that Mapl e's algorithm for contours is probably not evaluating the function on \+ a fine enough grid (the default is 25 by 25). So let us try a finer gr id (this also takes more time):" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" } }{PARA 255 "> " 0 "" {MPLTEXT 1 1879 80 "contourplot(min(2,V(x,y)), x= -4..4, y=-4..4, grid=[75,75], scaling=constrained);" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 252 "" 0 "" {TEXT 1850 65 "The conto urs we have just drawn have all been projected onto the " }{TEXT 1872 3 "x-y" }{TEXT 1850 20 " plane. The command " }{TEXT 1872 13 "contourp lot3d" }{TEXT 1850 153 " draws the same contours on a 3-D surface that can be rotated as well for better visualization (looking straight dow n produces 2-D contours). Actually, " }{TEXT 1872 13 "contourplot3d" }{TEXT 1850 42 " is faster and more accurate than the 2-D " }{TEXT 1872 11 "contourplot" }{TEXT 1850 65 " as a consequence of it being wr itten in the compiled language C:" }}{PARA 241 "" 0 "" {TEXT 1848 0 " " }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 48 "contourplot3d(min(3,V(x,y) ), x=-3..3, y=-3..3 );" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 68 "cont ourplot3d(min(3,V(x,y)), x=-3..3, y=-3..3, scaling=constrained);" }} {PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}}{SECT 0 {PARA 247 "" 0 "" {TEXT 1858 6 " 4.7 " }{TEXT 1857 1 " " }{TEXT 1859 38 "Solution: Dipo le and Quadrupole Fields" }{TEXT 1860 0 "" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 242 "" 0 "" {TEXT 1849 570 "We have used a number of visualization tools to examine the potential field surrounding a sing le positive and a single negative charge. The tools only showed us wha t we probably knew already, namely that the potential field does not h ave concave areas in which a charge may remain stably at rest. This wa s as intended; it is a good idea to learn about new tools on problems \+ for which you know the right answer. We are now in the position to fin ally investigate the electric potential due to a dipole and tripole. W e start with the dipole configuration shown in Fig. 4.2:" }}{PARA 251 "" 0 "" {XPPEDIT 512 0 "V[2](x,y) = q1/sqrt((x-1)^2+y^2)+q2/sqrt((x+1) ^2+y^2);" "6#/-&%\"VG6#\"\"#6$%\"xG%\"yG,&*&%#q1G\"\"\"-%%sqrtG6#,&*$, &F*F/F/!\"\"F(F/*$F+F(F/F6F/*&%#q2GF/-F16#,&*$,&F*F/F/F/F(F/*$F+F(F/F6 F/" }{TEXT 1867 1 " " }}{PARA 242 "" 0 "" {TEXT 1849 37 "We start by d efining a Maple function" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 64 "V2 :=(x,y,q1,q2) -> q1/sqrt((x-1)^2+ y^2) + q2/sqrt((x+1)^2+y^2);" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 282 "" 0 "" {TEXT 1922 69 "Now we visualize a dipole with \+ one positive and one negative charge:" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 18 "q1 := 1; q2 := -1;" }} {PARA 255 "> " 0 "" {MPLTEXT 1 1879 116 "plot3d(V2(x,y,1,-1), x=-3.5.. 3.5, y=-3.5..3.5, color=min(3,V2(x,y,1,-1)), labels=[`x`, `y`, `V2(x,y )`], axes=boxed);" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 242 "" 0 "" {TEXT 1849 264 "If you grab and rotate this plot you will \+ see that wherever you place a charge it will either roll downhill away from the positive charge, or fall into the hole of the negative charg e. There is no stable equilibrium. So let us look at two charges of th e like sign:" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 252 "> " 0 " " {MPLTEXT 1 1881 24 " # " }{TEXT 1850 55 "Enter \+ command to make 3-D plot for two positive charges" }}{PARA 252 "> " 0 "" {MPLTEXT 1 1881 21 " " }{TEXT 1850 58 " # Enter command to make 3-D plot for two negative charges" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 252 "" 0 "" {TEXT 1850 256 "The figu re resulting from these commands looks like a saddle. There is a regi on between the two peaks where it appears that a charge will remain a t rest, and where it will roll back towards the midpoint if it is dis placed along the positive or negative " }{TEXT 1872 1 "x" }{TEXT 1850 69 " axes. However, due to the saddle, if the charge is displaced in t he " }{TEXT 1872 1 "y" }{TEXT 1850 76 " direction, then it rolls awa y to infinity. The charge is thus stable for " }{TEXT 1872 1 "x" } {TEXT 1850 33 " displacements, but unstable for " }{TEXT 1872 1 "y" } {TEXT 1850 61 " displacements. This type of equilibrium point is known as a " }{TEXT 1872 14 "saddle point, " }{TEXT 1850 240 "and occurs \+ for two positive or two negative charges. If the charges have unequal \+ values but the same sign, then the shape gets distorted, but still has the same property.\n\nAs a check on our analysis, look at the contour s for this surface:" }{TEXT 1850 0 "" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 252 "> " 0 "" {MPLTEXT 1 1881 1 " " }{TEXT 1850 30 " # Cr eate a contourplot3d here" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 242 "" 0 "" {TEXT 1849 361 "You should see the saddle point structure as a single point where two equipotential surfaces cross.\n\nNow we look at the quadrupole po tential (we leave the tripole for you). Our intuition tells us that th e high degree of symmetry here must lead to a stable position at the c enter. We define the potential and then we plot it as a 3-D surface an d as 3-D contours:" }{TEXT 1849 0 "" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 123 "V4 :=(x,y) -> 1/sqrt((x -1)^2+(y-1)^2) + 1/sqrt((x-1)^2 + (y+1)^2) + 1/sqrt((x+1)^2 + (y-1)^2) + 1/sqrt((x+1)^2 + (y+1)^2);" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 106 "plot3d(V4(x,y), x=-3.5..3.5, y=-3.5..3.5, color=min(4.5,V4(x,y)), labels=[`x`, `y`, `V4(x,y)`], axes=box);" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 72 "contourplot3d (min(4.5,V4(x,y)), x=-3..3, y=-3..3, scaling=constrained);" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }} {PARA 242 "" 0 "" {TEXT 1849 178 "Yes, we do indeed see a large centra l, flat region surrounded by a lip to hold the charge in. We can check this out further by looking at some slices through the central regio n:" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 47 "plot(V4(x,0), x=-3..3, title=`V4(x,y=0) vs x`);" } }{PARA 254 "> " 0 "" {MPLTEXT 1 1880 56 "plot(min(4,V4(x,x)), x=-3..3, title=`V4(x=y) vs x`); " }{TEXT 1871 46 "# Use min to cut off hill tops and see details." }{TEXT 1867 0 "" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 242 "" 0 "" {TEXT 1849 239 "So we see tha t the central portion of this potential is indeed a flat region with a ``lip'' all around it. Consequently, a charge placed there will be ha ve no force on it (the flatness) and will be stable for small displace ments (the lip)." }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}}{SECT 0 {PARA 247 "" 0 "" {TEXT 1857 3 " " }{TEXT 1858 5 "4.8 " }{TEXT 1859 24 "Exploration: The Tripole" }{TEXT 1860 0 "" }}{PARA 239 "" 0 " " {TEXT 1843 0 "" }}{PARA 242 "" 0 "" {TEXT 1849 265 "Repeat the analy sis carried out for the dipole and quadruple now for the tripole. Be s ure to make a 3-D plot with labels and title, as well as including con tours. You will also need to slice your plot through its center to ver ify that you have found a stable point." }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}}{SECT 0 {PARA 247 "" 0 "" {TEXT 1857 3 " " }{TEXT 1858 4 "4.9 " }{TEXT 1857 1 " " }{TEXT 1859 21 " Other Types of Plots" } {TEXT 1860 0 "" }}}{SECT 0 {PARA 247 "" 0 "" {TEXT 1857 13 " \+ " }{TEXT 1858 7 " 4.9.1 " }{TEXT 1859 15 " 2-D Animations" }{TEXT 1860 0 "" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 252 "" 0 "" {TEXT 1850 670 "We have just seen that surface-rendering techniques pe rmit us to create images from mathematical functions that give the imp ression of viewing a true three-dimensional object. This literally giv es a new dimension to our visualizations. In addition, if we have plot s that show the behavior of some quantity as a function of space,\nand if this behavior changes gradually with time, then the observation of a sequence of plots of the spatial dependencies, each one for a sligh tly different time, gives the impression of a continuous evolution of \+ the spatial function in time. The function appears to be alive and, in deed, creating a series of snapshots in time is known as " }{TEXT 1872 9 "animation" }{TEXT 1850 172 ".\n\nTo produce animations we add the dimension of time to our 2-D plots. To name an instance, let us s ay we wanted to show the changing temperature distributions along the \+ " }{TEXT 1872 1 "x" }{TEXT 1850 59 " direction of a metal bar as it co ols with increasing time " }{TEXT 1872 1 "t" }{TEXT 1850 16 ". We coul d plot " }{XPPEDIT 1700 0 "T[1](x);" "6#-&%\"TG6#\"\"\"6#%\"xG" } {TEXT 1850 10 " and then " }{XPPEDIT 1679 0 "T[2](x);" "6#-&%\"TG6#\" \"#6#%\"xG" }{TEXT 1850 117 " and so forth, where the subscript indica tes the time. It is more elegant and concise to envision a single fun ction " }{TEXT 1872 6 "T(x,t)" }{TEXT 1850 167 " that contains both t he space and time dependencies.\n\nBy way of example, if the bar was i nitially hot in the center, a possible temperature distribution is [K rey 88]" }{TEXT 1850 0 "" }}{PARA 254 "" 0 "" {TEXT 1871 47 " \+ T = T(x,t) = " }{XPPEDIT 1678 0 "sin(x)*exp(- t)-sin(3*x)*exp(-9*t)/9;" "6#,&*&-%$sinG6#%\"xG\"\"\"-%$expG6#,$%\"tG! \"\"F)F)*(-F&6#*&\"\"$F)F(F)F)-F+6#,$*&\"\"*F)F.F)F/F)F9F/F/" }{TEXT 1867 1 " " }}{PARA 242 "" 0 "" {TEXT 1849 80 "Here is a 3-D surface pl ot to show the variation of T with both space and time:" }}{PARA 241 "" 0 "" {GLPLOT3D 243 120 120 {PLOTDATA 3 "6'-%%GRIDG6%;$\"\"!F($\"0z* e`EfTJ!#9;F'$\"#?F(X,6\"6\"%)anythingG[gl'!%\"!!#\\bm\":\":00000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 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xF 0Q\"tF0Q'T(x,t)F0-%*AXESSTYLEG6#%$BOXG-%*LINESTYLEG6#F(-%%FONTG6$%*HEL VETICAG\"#5" 1 2 5 0 10 1 2 1 1 2 2 1.000000 45.000000 45.000000 1 0 " Curve 1" }}{TEXT 1848 0 "" }}{PARA 242 "" 0 "" {TEXT 1849 222 "In many ways an animation is more natural in displaying the time dependence. \+ In any case, an animation provides another way to visualize a function of two variables, and it is worth looking at to see if it is illumina ting." }}{PARA 250 "" 0 "" {TEXT 1865 0 "" }}{PARA 283 "" 0 "" {TEXT 1923 70 "We make a 3-D surface plot of this function from t =0 to 20, and for " }{TEXT 1924 6 "x = 0" }{TEXT 1923 5 " to " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT 1925 1 ":" }}{PARA 241 "" 0 "" {TEXT 1848 0 " " }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 118 "plot3d( sin(x)*exp(-0.3*t ) - sin(3*x)*exp(-9*0.3*t)/9, x = 0..Pi, t = 0..20, axes=BOXED, labels =[`x`, `t`,`T(x,t)`] );" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }} {PARA 284 "" 0 "" {TEXT 1926 240 "Observe that it is important to labe l the axes so that you know which variable is time and which is space. Go ahead and grab, enlarge, and rotate the left plot.\n\nThe plot on \+ the right is an animation of this same function produced with the " } {TEXT 1927 7 "animate" }{TEXT 1926 9 " command:" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 1880 48 "with(plots): with(plottools): " }{TEXT 1871 30 " # This loads th e needed tools" }{TEXT 1867 0 "" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 102 "animate( sin(x)*exp(-0.3*t) - sin(3*x)*exp(-9*0.3*t)/9, x = \+ 0..Pi, t = 0..20, labels=[`x`,`T(x,t)`] );" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 285 "" 0 "" {TEXT 1913 1 " " }}{PARA 243 "" 0 "" {TEXT 1850 340 "Whereas the plot does not look different from a static 2-D plot, if you are reading the electronic version of this \+ book, then you make it come alive by selecting it with your mouse. The n a bunch of buttons appear on the control bar on top that permit you \+ to ``play'' the animation as you might a CD. Do it now! Investigate th e buttons for " }{TEXT 1872 4 "stop" }{TEXT 1850 2 ", " }{TEXT 1872 7 "forward" }{TEXT 1850 2 ", " }{TEXT 1872 7 "reverse" }{TEXT 1850 3 ", \+ f" }{TEXT 1872 11 "ast forward" }{TEXT 1850 6 ", and " }{TEXT 1872 12 "fast reverse" }{TEXT 1850 254 ". However there is also a button that \+ let us the animation loop around and thus play continuously. We recomm end that.\n\nAn animation works by displaying (flipping through) a seq uence of slightly modified images. In movie parlance, these images are called " }{TEXT 1872 6 "frames" }{TEXT 1850 79 ", and the more you ha ve of them the smoother and slower will be your animation:" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 252 "> " 0 "" {MPLTEXT 1 1881 52 " # " }{TEXT 1850 52 " Include a frames = 100 option in the animate command" }}{PARA 258 "> \+ " 0 "" {MPLTEXT 1 1879 0 "" }}}{SECT 0 {PARA 247 "" 0 "" {TEXT 1857 12 " " }{TEXT 1858 8 " 4.9.2 " }{TEXT 1859 15 " 3-D Anima tion" }{TEXT 1860 0 "" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 242 "" 0 "" {TEXT 1849 444 "Well, if you have been reading and executi ng up to this point, it is fairly clear what 3-D animations are about. If you have a function of two space coordinates that also varies in t ime, then you make a 3-D surface plot to visualize the space dependenc e at any one time, or an animation to visualize the time dependence of the surface. For example, assume the temperature distribution is now \+ a function of two space coordinates as well as time:" }}{PARA 286 "" 0 "" {TEXT 1928 0 "" }}{PARA 243 "" 0 "" {XPPEDIT 510 0 "(sin(x)*exp(- .3*t)-sin(3*x)*exp(-9*.9*t)/9)*(sin(y)*exp(-.3*t)-sin(3*y)*exp(-9*.9*t )/9);" "6#*&,&*&-%$sinG6#%\"xG\"\"\"-%$expG6#,$*&-%&FloatG6$\"\"$!\"\" F*%\"tGF*F4F*F**(-F'6#*&F3F*F)F*F*-F,6#,$*(\"\"*F*-F16$F>F4F*F5F*F4F*F >F4F4F*,&*&-F'6#%\"yGF*-F,6#,$*&-F16$F3F4F*F5F*F4F*F**(-F'6#*&F3F*FEF* F*-F,6#,$*(F>F*-F16$F>F4F*F5F*F4F*F>F4F4F*" }{TEXT 1850 1 " " }}{PARA 287 "" 0 "" {TEXT 1929 0 "" }}{PARA 252 "" 0 "" {TEXT 1850 121 " This \+ is complicated enough that we will define a Maple function rather than try to squeeze the long expression into the " }{TEXT 1872 9 "animate3 d" }{TEXT 1850 36 " command. Other than using the name " }{TEXT 1872 9 "animate3d" }{TEXT 1850 78 ", the format of the command is the same \+ as before. We start with the function " }{TEXT 1872 8 "T(x,y,t)" } {TEXT 1850 62 ", then give the ranges for each variable, and then the \+ labels:" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 1880 33 "with(plots): " }{TEXT 1871 84 " # Lo ad needed tools" }{TEXT 1867 0 "" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 16 "with(plottools):" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 109 " T :=(x,y,t) -> (sin(x)*exp(-0.3*t) - sin(3*x)*exp(-9.9*t)/9) * (sin(y) *exp(-0.3*t) - sin(3*y)*exp(-9.9*t)/9);" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 82 "animate3d( T(x,y,t), x = 0..Pi, y = 0..Pi, t = 0.. 20, labels=[`x`,`y`, `T(x,y)`]);" }}{PARA 256 "> " 0 "" {MPLTEXT 1 1882 2 " " }}{PARA 257 "" 0 "" {TEXT 1883 102 "Select this plot, play it, and rotate it while it is vibrating (it looks like a vibrating d rum head)." }}}{SECT 0 {PARA 247 "" 0 "" {TEXT 1857 10 " " } {TEXT 1858 9 " 4.9.3 " }{TEXT 1859 32 " Phase Space (Parametric) Plo ts " }{TEXT 1860 0 "" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 243 "" 0 "" {TEXT 1850 138 "In science we often encounter several physical quantities that are simultaneous functions of the same variable. For \+ example, the position " }{TEXT 1872 4 "x(t)" }{TEXT 1850 12 ", veloci ty " }{TEXT 1872 4 "v(t)" }{TEXT 1850 19 ", and acceleration " }{TEXT 1872 5 "a(t) " }{TEXT 1850 86 " of a mass undergoing simple harmonic m otion are all trigonometric functions of time: " }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 243 "" 0 "" {TEXT 1850 23 " \+ " }{XPPEDIT 1640 0 "x(t) = sin(omega*t);" "6#/-%\"xG6#%\"tG-%$sinG 6#*&%&omegaG\"\"\"F'F-" }{TEXT 1850 7 " , " }{XPPEDIT 1638 0 "v(t) = -omega*cos(omega*t);" "6#/-%\"vG6#%\"tG,$*&%&omegaG\"\"\"-%$cosG6#* &F*F+F'F+F+!\"\"" }{TEXT 1872 1 " " }{TEXT 1850 10 " , " } {XPPEDIT 1614 0 "a(t) = -omega^2*sin(omega*t);" "6#/-%\"aG6#%\"tG,$*&% &omegaG\"\"#-%$sinG6#*&F*\"\"\"F'F0F0!\"\"" }{TEXT 1930 1 " " }{TEXT 1850 0 "" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 242 "" 0 "" {TEXT 1849 76 "We can easily plot the position and velocity on the sam e graph; for example," }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 73 "plot( [sin(wt), -2*cos(wt)], wt= 0. . 8*Pi, legend=[`x(wt)`, `v(wt)`] );" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 252 "" 0 "" {TEXT 1850 81 "We observe the position and velocity as out of phase, but\nwith the same period." }{TEXT 1850 72 "\n\nA more direct way to observe the relation of two dependen t variables (" }{TEXT 1872 1 "x" }{TEXT 1850 5 " and " }{TEXT 1872 1 " v" }{TEXT 1850 64 " in our example) as a function of the same independ ent variable " }{TEXT 1872 1 "t" }{TEXT 1850 15 " is known as a " } {TEXT 1872 11 "phase space" }{TEXT 1850 4 " or " }{TEXT 1872 15 "param etric plot" }{TEXT 1850 249 ". These types of plots have now proven\nt hemselves to be highly illuminating and valuable. Phase space is an ex tension of the usual space of position that also includes velocity as \+ if it were a new dimension, along with position. Explicitly, we plot \+ " }{TEXT 1872 4 "x(t)" }{TEXT 1850 63 " along the abscissa as if it we re the independent variable and " }{TEXT 1872 4 "v(t)" }{TEXT 1850 71 " along the ordinate. In a sense then, a phase-space plot is a plot of " }{TEXT 1872 4 "v(x)" }{TEXT 1850 341 ".\n\nRecognizing that it m ight be impossible, or very complicated, to analytically eliminate the time dependencies permitting two functions to be expressed in terms \+ of each other, it is still fairly easy to do this graphically. Explici tly, Maple breaks up the total time interval into a number of steps, a nd then records the pair of values " }{TEXT 1872 5 "(x,v)" }{TEXT 1850 54 " for each time step. These values then get plotted as " } {TEXT 1872 4 "v(t)" }{TEXT 1850 8 " versus " }{TEXT 1872 4 "x(t)" } {TEXT 1850 1 ":" }}{PARA 284 "" 0 "" {TEXT 1926 0 "" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 75 "plot( [sin(wt), -2*cos(wt), wt = 0..8*Pi], la bels=[`Position`,`Velocity`]);" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 252 "" 0 "" {TEXT 1850 218 "Observe that all we have done here is use a list (square brackets) as the function argument, and in cluded the range of time values as part of the list. The general synta x for a two-dimensional (2-D) parametric plot is " }{TEXT 1872 37 "plo t([x(t), y(t), t\n= a..b], options)" }{TEXT 1850 8 ", where " }{TEXT 1872 1 "t" }{TEXT 1850 62 " is known as the parametric variable or sim ply parameter, and " }{TEXT 1872 4 "x(t)" }{TEXT 1850 5 " and " } {TEXT 1872 4 "y(t)" }{TEXT 1850 697 " denote the horizontal and vertic al functions, respectively.\n\nAs far as the output goes, this phase-s pace plot looks like an ellipse. Yet it has properties that agree with the observations we have made before: when the mass is at its maximum positions, at the extreme right and left edges of the ellipse, the ve locity is zero. When the mass has it maximum speed, at the top and bot tom of the ellipse, it has zero position, that is, it is passing throu gh its equilibrium position. So while a harmonic oscillator is describ ed via a complicated set of position, velocity, and acceleration funct ions versus time, in phase space its motion is an elliptical orbit on \+ which the mass passes over and over." }{TEXT 1850 90 "\n\nYou may be w ondering in looking at this graph, just how we hit upon the exact rang e of " }{TEXT 1872 2 "wt" }{TEXT 1850 361 " values for which the grap h exactly closes on itself. Well, we really did not. In fact, our plot covers two full cycles and plots them on top of each other (which you cannot see). If, on the other hand, your phase-space plot did not for m a closed figure, then you would need to run for more values of the \+ time. Try out smaller and smaller ranges for the phase " }{TEXT 1872 2 "wt" }{TEXT 1850 8 " in the " }{TEXT 1872 5 "plot " }{TEXT 1850 40 " command until you are able to generate " }{TEXT 1872 4 "1/2 " }{TEXT 1850 4 "and " }{TEXT 1872 3 "1/4" }{TEXT 1850 25 " of an elliptical or bit:\n" }}{PARA 239 "> " 0 "" {MPLTEXT 1 1879 7 " " }{TEXT 1931 36 "# Create some phase space plots here" }{TEXT 1843 0 "" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 252 "" 0 "" {TEXT 1850 392 "As you see from looking at the monitor in front of you, visual disp lays tend to be broader than they are high. For this reason graphs ten d to get stretched horizontally (``scaled'') in order to fill the scre en. As long as the graph looks good this is not normally a concern, y et it is if you are trying to determine the proper shape of a geometr ical figure. For this reason Maple has the " }{TEXT 1872 19 "scaling=c onstrained" }{TEXT 1850 34 " option to avoid undue stretching:" }} {PARA 241 "> " 0 "" {MPLTEXT 1 1892 9 " " }{TEXT 1868 35 " \+ " }{TEXT 1848 1 " " }{TEXT 1887 63 "# \+ Add the scaling=constrained option to the preceeding command" }{TEXT 1848 0 "" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}}{SECT 0 {PARA 288 "" 0 "" {TEXT 1845 6 "4.9.4 " }{TEXT 1846 39 " Energy Conservation and Implicit Plots" }{TEXT 1847 0 "" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 243 "" 0 "" {TEXT 1850 65 "In our discussion of parametri c plots, we looked at the position " }{TEXT 1872 4 "x(t)" }{TEXT 1850 14 " and velocity " }{TEXT 1872 4 "v(t)" }{TEXT 1850 84 " of an oscill ator as a functions of time. Maple solves numerically for the function " }{TEXT 1872 4 "x(v)" }{TEXT 1850 4 " or " }{TEXT 1872 4 "v(x)" } {TEXT 1850 93 ". There may also be cases where you know some functiona l relation between two variables, say " }{TEXT 1872 1 "x" }{TEXT 1850 5 " and " }{TEXT 1872 1 "v" }{TEXT 1850 29 ", and wish to make a plot \+ of " }{TEXT 1872 1 "x" }{TEXT 1850 1 " " }{TEXT 1872 6 "versus" } {TEXT 1850 1 " " }{TEXT 1872 1 "v" }{TEXT 1850 127 ". To cite an insta nce, let us say that we have a spring with a nonlinear force law so th at the potential energy stored in it is" }}{PARA 243 "" 0 "" {XPPEDIT 505 0 "V(x) = k*x^6;" "6#/-%\"VG6#%\"xG*&%\"kG\"\"\"*$F'\"\"'F*" } {TEXT 1850 1 " " }}{PARA 289 "" 0 "" {TEXT 1932 0 "" }}{PARA 242 "" 0 "" {TEXT 1849 68 "The kinetic energy of a mass attached to this spring is, as always, " }}{PARA 243 "" 0 "" {XPPEDIT 502 0 "K = m*v^2/2;" "6 #/%\"KG*(%\"mG\"\"\"*$%\"vG\"\"#F'F*!\"\"" }{TEXT 1850 1 " " }}{PARA 242 "" 0 "" {TEXT 1849 146 "We know that the sum of kinetic plus poten tial energy is conserved, this means we have an implicit relation betw een position and velocity, namely," }}{PARA 290 "" 0 "" {TEXT 1933 0 " " }}{PARA 243 "" 0 "" {XPPEDIT 499 0 "E = V+K;" "6#/%\"EG,&%\"VG\"\"\" %\"KGF'" }{TEXT 1850 1 " " }}{PARA 243 "" 0 "" {XPPEDIT 496 0 "E = k*x ^6+m*v^2/2;" "6#/%\"EG,&*&%\"kG\"\"\"*$%\"xG\"\"'F(F(*(%\"mGF(*$%\"vG \"\"#F(F0!\"\"F(" }{TEXT 1850 1 " " }}{PARA 291 "" 0 "" {TEXT 1934 0 " " }}{PARA 252 "" 0 "" {TEXT 1850 106 "This last equation permits us to make parametric plot even though we do not have explicit solutions \+ for " }{TEXT 1872 4 "x(t)" }{TEXT 1850 5 " and " }{TEXT 1872 4 "v(t)" }{TEXT 1850 20 ". We do it with the " }{TEXT 1872 12 "implicitplot" } {TEXT 1850 20 " command that plots " }{TEXT 1872 8 "x versus" }{TEXT 1850 1 " " }{TEXT 1872 1 "v" }{TEXT 1850 33 " given equation relating \+ the two:" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 12 "with(plots):" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 54 "implicitplot(5*x^6 + (13/2)*v^2 = 1, x=-1..1,v=-1..1);" }} {PARA 242 "" 0 "" {TEXT 1849 91 " See how this phase space plot change s if the potential energy varies as the 5th power of x" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}}{SECT 0 {PARA 292 "" 0 "" {TEXT 1935 5 " " }{TEXT 1936 5 " " }{TEXT 1937 6 "4.9.5 " }{TEXT 1938 15 "Vector Fields: " }{TEXT 1939 9 "fieldplot" }{TEXT 1938 5 " and " } {TEXT 1939 12 "fieldplot3d*" }{TEXT 1935 0 "" }}{PARA 243 "" 0 "" {TEXT 1850 352 "Consider again the electric dipole shown in Fig. 4.2. \+ The problem we examined dealt with the electric potential for this typ e of system. Even though potentials are easier to compute and visualiz e than fields, it is usually fields that are related to the forces and measured in experiments. As an extension to our charge problem, we \+ now visualize the " }{TEXT 1872 14 "electric field" }{TEXT 1850 1 " " }{TEXT 1940 2 "E(" }{TEXT 1872 5 "x,y) " }{TEXT 1850 14 "for the dipol e" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 252 "" 0 "" {TEXT 1855 4 "E = " }{XPPEDIT 1369 0 "q[1]*(r-r[1])/(abs(r-r[1])^2)+q[2]*(r-r[2]) /(abs(r-r[2])^2);" "6#,&*(&%\"qG6#\"\"\"F(,&%\"rGF(&F*6#F(!\"\"F(*$-%$ absG6#,&F*F(&F*6#F(F-\"\"#F-F(*(&F&6#F5F(,&F*F(&F*6#F5F-F(*$-F06#,&F*F (&F*6#F5F-F5F-F(" }{TEXT 1850 0 "" }}{PARA 239 "" 0 "" {TEXT 1843 0 " " }}{PARA 243 "" 0 "" {TEXT 1850 10 "where the " }{TEXT 1872 1 "E" } {TEXT 1850 5 " and " }{TEXT 1872 1 "r" }{TEXT 1850 286 "'s in this equ ation are all vector quantities. Mathematically, we determine the elec tric field as the derivative of the potential using the techniques of \+ vector calculus, and then plot the individual components.This is comp licated, so let us have Maple do the work.\n\nWe start with the " } {TEXT 1872 1 "x" }{TEXT 1850 5 " and " }{TEXT 1872 1 "y" }{TEXT 1850 34 " components of the electric field " }{TEXT 1872 1 "E" }{TEXT 1850 1 ":" }}{PARA 257 "" 0 "" {TEXT 1883 0 "" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1895 57 "Ex := (x,y) -> (x+1)/((x+1)^2+y^2) - (x-1)/((x-1)^ 2+y^2);" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1895 49 "Ey := (x,y) -> y/(( x+1)^2+y^2) - y/((x-1)^2+y^2);" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1895 0 "" }}{PARA 257 "" 0 "" {TEXT 1883 66 "We can visualize these individ ual components in two surface plots:" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1895 30 "with (plots): with(plot tools);" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1895 82 "plot3d(Ex(x,y), x=- 3.5..3.5, y=-3.5..3.5, labels=[`x`,`y`,`Ex(x,y)`], axes=boxed);" }} {PARA 258 "> " 0 "" {MPLTEXT 1 1895 82 "plot3d(Ey(x,y), x=-3.5..3.5, y =-3.5..3.5, labels=[`x`,`y`,`Ex(x,y)`], axes=boxed);" }}{PARA 258 "> \+ " 0 "" {MPLTEXT 1 1895 0 "" }}{PARA 257 "" 0 "" {TEXT 1883 222 "While \+ this does show the force components at each point in space, it is rath er hard to get a good feel for the magnitude and direction of the forc e acting on a test charge at each point in space. This is obtained wit h the " }{TEXT 1941 18 "fieldplot command:" }{TEXT 1883 0 "" }}{PARA 257 "" 0 "" {TEXT 1883 0 "" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1895 86 " fieldplot( [Ex(x,y), Ey(x,y)], x=-3.5..3.5, y=-3.5..3.5, color=Ex(x,y) , arrows=THICK);" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1895 0 "" }}{PARA 257 "" 0 "" {TEXT 1883 147 "This creates a plot that shows the direct ion of the field as given by direction of arrows, and the magnitude as given\nby the length of the arrows." }{TEXT 1883 155 "\n\nYou may not know it from elementary physics classes, but the real world is actual ly three dimensional. Because of this the electric field actually has \+ a " }{TEXT 1941 1 "z" }{TEXT 1883 19 " component as well:" }}{PARA 257 "" 0 "" {TEXT 1883 0 "" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1895 67 " Ex := (x,y,z) -> (x+1)/((x+1)^2+y^2+z^2) - (x-1)/((x-1)^2+y^2+z^2);" } }{PARA 258 "> " 0 "" {MPLTEXT 1 1895 59 "Ey := (x,y,z) -> y/((x+1)^2+y ^2+z^2) - y/((x-1)^2+y^2+z^2);" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1895 59 "Ez := (x,y,z) -> z/((x+1)^2+y^2+z^2) - z/((x-1)^2+y^2+z^2);" }} {PARA 258 "> " 0 "" {MPLTEXT 1 1895 0 "" }}{PARA 257 "" 0 "" {TEXT 1883 46 "Here we visualize a 3-D vector field with the " }{TEXT 1941 11 "fieldplot3d" }{TEXT 1883 9 " command:" }}{PARA 257 "" 0 "" {TEXT 1883 0 "" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1895 101 "fieldplot3d( [Ex( x,y,z), Ey(x,y,z), Ez(x,y,z)], x=-3.5..3.5, y=-3.5..3.5, z=-3.5..3.5, \+ arrows=THICK);" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1895 0 "" }}{PARA 257 "" 0 "" {TEXT 1883 109 "Regardless of how good this plot looks on paper, it comes alive when you grab and rotate it electronically.\n" }}}{SECT 0 {PARA 247 "" 0 "" {TEXT 1857 9 " " }{TEXT 1858 10 " 4.9.6 " }{TEXT 1859 12 " Polar Plots" }{TEXT 1860 0 "" }}{PARA 283 "" 0 "" {TEXT 1925 59 " Polar coordinates describe a 2-D plane in \+ terms of radius " }{TEXT 1942 1 "r" }{TEXT 1925 17 " and orientation \+ " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT 1925 52 ", rather than t he more common Cartesian coordinates " }{TEXT 1942 1 "x" }{TEXT 1925 5 " and " }{TEXT 1942 1 "y" }{TEXT 1925 52 ". A polar plot is a repre sentation of the function " }{XPPEDIT 18 0 "r(theta);" "6#-%\"rG6#%&th etaG" }{TEXT 1925 39 " created by placing a point a distance " } {XPPEDIT 18 0 "r(theta);" "6#-%\"rG6#%&thetaG" }{TEXT 1925 32 " from t he origin for each angle " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" } {TEXT 1925 212 ". They are illuminating because the angle on the plot \+ corresponds to the actual angle of the function's argument, and so the shape of the plot lets you visualize the variation of the function in actual space. If " }{TEXT 1942 1 "r" }{TEXT 1925 21 " were independen t of " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT 1925 107 ", then th e polar plot would be a circle. Other dependencies are less obvious.\n \nMaking polar plots with the " }{TEXT 1942 4 "plot" }{TEXT 1925 66 " \+ command is similar to making parametric plots, only now we add a " } {TEXT 1942 13 "coord = polar" }{TEXT 1925 93 " option and use theta in stead of time as the parametric variable. Or do it directly with the \+ " }{TEXT 1942 9 "polarplot" }{TEXT 1925 9 " command:" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 241 "" 0 "" {TEXT 1848 32 " \+ " }{TEXT 1943 55 "plot( [r(theta), theta, theta=0..2 *Pi], coords=polar);" }{TEXT 1848 0 "" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 241 "" 0 "" {TEXT 1848 31 " \+ " }{TEXT 1943 53 " polarplot( [r(theta), theta(theta), theta=0.. 2*Pi]);" }{TEXT 1848 0 "" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 241 "" 0 "" {TEXT 1848 95 "To get a feel of how this works, let us mak e a polar plot of a function that is independent of " }{XPPEDIT 18 0 " theta;" "6#%&thetaG" }{TEXT 1848 31 ", and then one of the familiar " }{XPPEDIT 18 0 "sin(theta);" "6#-%$sinG6#%&thetaG" }{TEXT 1848 1 ":" } }{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 16 "with(plots): " }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 6 "r \+ :=1;" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 49 "plot( [r , theta, thet a=0..2*Pi], coords=polar );" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 58 "plot( [sin(theta), theta, theta=0..2*Pi], coords=polar );" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 252 "" 0 "" {TEXT 1850 17 " And now with the " }{TEXT 1872 9 "polarplot" }{TEXT 1850 9 " command: " }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 43 "polarplot( [ r , theta, theta=0..2*Pi ] );" }}{PARA 254 "> " 0 "" {MPLTEXT 1 1880 44 "polarplot( [ cos(t), sin(t), t=0..4*Pi ] ) ; " }{TEXT 1871 49 " # Using parametric form " }{TEXT 1867 0 "" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 272 "" 0 "" {TEXT 1912 166 "As a more realistic example, consider the \+ expression for the intensity of low-energy X-rays scattered off a ref lecting sphere as a function of the scattering angle: " }}{PARA 254 " " 0 "" {TEXT 1871 64 " \+ " }{XPPEDIT 493 0 "sigma(theta) = 3+2*cos(theta)^4+2*c os(theta);" "6#/-%&sigmaG6#%&thetaG,(\"\"$\"\"\"*&\"\"#F**$-%$cosG6#F' \"\"%F*F**&F,F*-F/6#F'F*F*" }{TEXT 1867 1 " " }}{PARA 293 "" 0 "" {TEXT 1944 1 " " }}{PARA 243 "" 0 "" {TEXT 1850 34 "Visualize this wit h a 2-D plot of " }{XPPEDIT 18 0 "sigma(theta)" "6#-%&sigmaG6#%&thetaG " }{TEXT 1906 1 " " }{TEXT 1850 71 "versus the scattering angle. You s hould get a plot like the one below: " }}{PARA 252 "> " 0 "" {MPLTEXT 1 1881 8 " " }{TEXT 1850 20 "# Use plot(...) here" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 250 "" 0 "" {TEXT 1865 0 "" }} {PARA 243 "" 0 "" {TEXT 1850 116 "It may not be obvious that this grap h means that there is a strong peak in the forward direction along the beam at " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT 1906 1 " " } {TEXT 1850 77 " = 0, and that there is a weaker amount of scattering b ack into the beam at " }{XPPEDIT 18 0 "theta = Pi;" "6#/%&thetaG%#PiG " }{TEXT 1850 88 ". However, if we plot this function in a polar plot, then these features become evident:" }}{PARA 278 "" 0 "" {TEXT 1919 0 "" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 75 "plot([3+2*cos(theta)^4 \+ + 2*cos(theta), theta, theta=0..2*Pi],coords=polar);" }}{PARA 258 "> \+ " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 285 "" 0 "" {TEXT 1913 74 "You in tuitively feel where the scattering is large and where it is small." } }{PARA 239 "" 0 "" {TEXT 1843 0 "" }}}{SECT 0 {PARA 254 "" 0 "" {TEXT 1871 12 " " }{TEXT 1945 8 " 4.9.7 " }{TEXT 1946 36 " Surfa ce Plots of Complex Functions*" }{TEXT 1867 0 "" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 242 "" 0 "" {TEXT 1849 183 "In Chap. 14 we giv e a review of complex numbers in preparation for representing them as \+ objects in Java. If may want to review that review in order to better \+ understand this section. " }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }} {PARA 252 "" 0 "" {TEXT 1850 23 "Maple has the commands " }{TEXT 1872 11 "complexplot" }{TEXT 1850 6 " and " }{TEXT 1872 13 "complexplot3d " }{TEXT 1850 67 " that are useful for visualizing complex functions. We have found " }{TEXT 1872 13 "complexplot3d" }{TEXT 1850 147 " the \+ more useful of the two, and will discuss only it. To describe how thi s works, let us say we have a complex function of the complex argument " }{TEXT 1872 11 "z = x + iy," }{TEXT 1850 0 "" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 252 "" 0 "" {TEXT 1850 33 " \+ " }{TEXT 1872 20 "f(z) = f(x + i y) = " }{TEXT 1850 1 "R " }{TEXT 1872 9 "e f + i " }{TEXT 1850 2 "Im" }{TEXT 1872 2 " f" } {TEXT 1850 0 "" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 252 "" 0 " " {TEXT 1850 12 "The command " }{TEXT 1872 13 "complexplot3d" }{TEXT 1850 147 " will make a 3D visualization of a function or an expression , with the form of the visualization differing if the input is an expl icit function of " }{TEXT 1872 1 " " }{TEXT 1850 20 "the complex argum ent" }{TEXT 1872 2 " z" }{TEXT 1850 59 ", or an explicit function of t he real and imaginary parts " }{TEXT 1872 1 "x" }{TEXT 1850 6 " and \+ " }{TEXT 1872 1 "y" }{TEXT 1850 59 ". If we input a function of the r eal and imaginary parts, " }{TEXT 1872 7 "(x,y), " }{TEXT 1850 5 "then " }{TEXT 1872 13 "complexplot3d" }{TEXT 1850 14 " plots the Re " } {TEXT 1872 1 "f" }{TEXT 1850 43 " while coloring the graphic using th e Im " }{TEXT 1872 1 "f" }{TEXT 1850 47 ". As an instance, consider t he complex function" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 252 " " 0 "" {TEXT 1850 44 " " } {TEXT 1872 36 "z^2 = (x+iy)^2 = x^2 - y^2 + 2ixy" }{TEXT 1850 0 "" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 242 "" 0 "" {TEXT 1849 42 "We enter via its real and imaginary parts:" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 12 "with(plots): " }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 56 "complexplot3d( [x^2 - y^2, 2*x*y], x = -2..2, y= -2..2);" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 294 "" 0 "" {TEXT 1947 137 "To understand what you see he re, select the figure, place some axes on it, and rotate it. You can a lso get the axes via a command option:" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 69 "complexplot3d( [x^ 2 - y^2, 2*x*y], x = -2..2, y= -2..2, axes=framed);" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 252 "" 0 "" {TEXT 1850 44 "Here the height \+ of the plot is the value Re " }{TEXT 1872 15 "f = x^2 - y^2" }{TEXT 1850 31 ", changing sign along the line " }{TEXT 1872 5 "x=y, " } {TEXT 1850 38 "while the color is controlled by Im " }{TEXT 1872 9 " f = 2xy. " }{TEXT 1850 56 " We obtain the same plot using the explicit function of " }{TEXT 1872 1 "x" }{TEXT 1850 5 " and " }{TEXT 1872 1 " y" }{TEXT 1850 1 ":" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 255 " > " 0 "" {MPLTEXT 1 1879 24 "Ref := (x,y)-> x^2-y^2; " }}{PARA 255 "> \+ " 0 "" {MPLTEXT 1 1879 22 "Imf := (x,y) -> 2*x*y;" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 74 "complexplot3d( [ Ref, Imf] , -2..2, -2..2, axes =framed, labels=[x,y,Ref]);" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 " " }}{PARA 252 "" 0 "" {TEXT 1850 37 "Note that the vertical axis gives Re " }{TEXT 1872 3 "f, " }{TEXT 1850 30 "and it changes sign along t he" }{TEXT 1872 1 " " }{TEXT 1850 5 "line " }{TEXT 1872 5 "x=y. " } {TEXT 1850 102 "The other approach to visualizing complex functions is to use complex numbers directly. In this case, " }{TEXT 1872 13 "comp lexplot3d" }{TEXT 1850 90 " plots the magnitude of the function while \+ coloring the resulting surface using the phase " }{XPPEDIT 1322 0 "the ta;" "6#%&thetaG" }{TEXT 1850 3 " = " }{XPPEDIT 1310 0 "tan^(-1);" "6# )%$tanG,$\"\"\"!\"\"" }{TEXT 1850 3 "(Re" }{TEXT 1872 2 " f" }{TEXT 1850 4 "/Im " }{TEXT 1872 2 "f)" }{TEXT 1850 17 " of the function:" }} {PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 59 "complexplot3d( z^2, z = -2 - 2*I .. 2 + 2*I, axes=framed );" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 242 "" 0 "" {TEXT 1849 48 "A more realistic example is the complex function" }}{PARA 251 "" 0 "" {XPPEDIT 18 0 "f(E) = 1/(E-2+I);" "6#/-%\"fG6#%\"EG*&\"\" \"F),(F'F)\"\"#!\"\"%\"IGF)F," }{TEXT 1867 1 " " }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 239 "" 0 "" {TEXT 1948 18 "that represents a \+ " }{TEXT 1949 9 "resonance" }{TEXT 1948 18 " as a function of " } {TEXT 1949 2 "E " }{TEXT 1948 21 "that may occur in an " }{TEXT 1949 3 "RLC" }{TEXT 1948 129 " circuit or in setting up standing waves in a \ntube. Regardless of the fact that experiments are done only at pure \+ real values of " }{TEXT 1949 1 "E" }{TEXT 1948 35 ", this function has a pole (equals " }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT 1948 22 ") at a complex energy " }{TEXT 1949 8 "E = 2 -i" }{TEXT 1948 51 ". To help understand the visualization, we rewrite " }{TEXT 1949 1 "f" }{TEXT 1948 120 " in a form with a real denominator (multiply th e numerator and denominator by the complex conjugate of the denominato r):" }{TEXT 1843 0 "" }}{PARA 243 "" 0 "" {XPPEDIT 492 0 "f(z) = (x-2- I*(y+1))/((x-2)^2+(y+1)^2);" "6#/-%\"fG6#%\"zG*&,(%\"xG\"\"\"\"\"#!\" \"*&%\"IGF+,&%\"yGF+F+F+F+F-F+,&*$,&F*F+F,F-F,F+*$,&F1F+F+F+F,F+F-" } {TEXT 1850 1 " " }}{PARA 242 "" 0 "" {TEXT 1849 46 "The squared magnit ude of this fountain is just" }}{PARA 251 "" 0 "" {XPPEDIT 490 0 "fmod (z) = 1/((x-2)^2+(y+1)^2);" "6#/-%%fmodG6#%\"zG*&\"\"\"F),&*$,&%\"xGF) \"\"#!\"\"F.F)*$,&%\"yGF)F)F)F.F)F/" }{TEXT 1867 1 " " }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 252 "" 0 "" {TEXT 1850 13 "We visualize \+ " }{TEXT 1872 4 "f(z)" }{TEXT 1850 31 " using the different forms of \+ " }{TEXT 1872 13 "complexplot3d" }{TEXT 1850 2 ": " }}{PARA 241 "" 0 " " {TEXT 1848 0 "" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 62 "complexplo t3d(1/(z-2+I), z = 1 - 3*I .. 3 + 2*I, axes=framed);" }}{PARA 258 "> \+ " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 252 "" 0 "" {TEXT 1850 173 "Inasmu ch as there are no sign changes here, it must be the modulus of the fu nction that is being plotted. If we look above we see that the modulu s does have a maximum when " }{TEXT 1872 3 "x=2" }{TEXT 1850 10 " and \+ when " }{TEXT 1872 6 "y = -1" }{TEXT 1850 178 ", and that the modulus \+ falls off in value as we get away from the pole position. As expected, we see those features on the plot. We visualize the real and imag inary parts of " }{TEXT 1872 4 "f(E)" }{TEXT 1850 12 " separately:" }} {PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 39 "Ref := (x,y)-> (x-2)/((x-2)^2+(1+y)^2);" }}{PARA 255 "> " 0 " " {MPLTEXT 1 1879 42 "Imf := (x,y) -> -(y +1)/((x-2)^2+(1+y)^2);" }} {PARA 255 "> " 0 "" {MPLTEXT 1 1879 77 "complexplot3d( [ Ref, Imf] , 1 ..3, -3..2, axes=framed, labels=[ReE,ImE,Ref]);" }}{PARA 258 "> " 0 " " {MPLTEXT 1 1879 0 "" }}{PARA 252 "" 0 "" {TEXT 1850 49 "As we see in the plot, there is a sign change at " }{TEXT 1872 2 "2 " }{TEXT 1850 56 "along one of the axes. Consequently, that must be the " }{TEXT 1872 1 "x" }{TEXT 1850 39 " axis and the function plotted must be " } {TEXT 1872 4 "Re f" }{TEXT 1850 2 " (" }{TEXT 1872 5 "Im f " }{TEXT 1850 46 "does not have a sign change). Check next that " }{TEXT 1872 4 "Re f" }{TEXT 1850 39 " does not change sign as a function of " } {TEXT 1872 1 "y" }{TEXT 1850 29 ", but does have a maximum at " } {TEXT 1872 6 "y = -1" }{TEXT 1850 46 " where the denominator gets smal l. In fact at " }{TEXT 1872 14 "(x,y) = (2,-1)" }{TEXT 1850 37 " there is a pole, yet it has residue " }{TEXT 1872 1 "0" }{TEXT 1850 46 ", w hich we see as a very large oscillation in " }{TEXT 1872 6 "Re f. " } {TEXT 1850 15 "To see how the " }{TEXT 1872 4 "Im f" }{TEXT 1850 104 " varies as a function of complex energy, we reverse the real and ima ginary parts in the argument call:" }}{PARA 239 "" 0 "" {TEXT 1843 0 " " }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 76 "complexplot3d( [ Imf, Ref] , 1..3, -3..2, axes=framed,labels=[ReE,ImE,Imf]);" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 243 "" 0 "" {TEXT 1850 54 "If we tak e this plot and rotate so we see a plot of Im" }{TEXT 1872 1 "f" } {TEXT 1850 1 " " }{TEXT 1872 7 "versus " }{TEXT 1850 2 "Re" }{TEXT 1872 3 "E, " }{TEXT 1850 35 "we see the classic resonance shape:" }} {PARA 248 "" 0 "" {GLPLOT3D 235 83 83 {PLOTDATA 3 "6&-%%GRIDG6&;$\"\" \"\"\"!$\"\"$F);$!\"$F)$\"\"#F)X,6\"6\"%)anythingG[gl'!%\"!!#\\bm\":\" :3FD999999999999A3FDB3C7D69E23AE93FDCE521F6E0103A3FDE717863A1E7173FDF9 F9F9F9F9FA03FDFF8961FF896203FDEB851EB851EB73FDACD7CBC4E58A43FD33333333 3332F3FBF81F81F81F810BFB52FAB4152FAC7BFD13404EA4A8C19BFD999999999999CB 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6``r%$RGBG$\"2wppo&yg>5!#<$\"#5!\"\"$\"1EtWp#R!)4*!#;F9F<$\"1O!*pLlqFAF9F<$\"1!\\,ffF'ovFAF9F<$\"0H4#>,++!)FIF9F<$\"1=+ eep:#R)FAF9F<$\"1E&Rhummm)FAF9F<$\"1k>w\\3'>!*)FAF9F<$\"1a-^&eB)e!*FAF 9F<$\"1W&e7K'o:#*FAF9F<$\"1k(pIPLLL*FAF9F<$\"13R%4uk!\\*FAF9F<$\"/Apwuio&*FFF9F<$\"/WQ`\\DP\"*FFF9F<$\"13hjZqB)eqF;F9F<$\"1MA&QRTHN'FAF9F<$\"1C\\)H3fqk&FAF9F<$\"1_+uv3Zw^FAF9F <$\"2%yH!=>bs8&F;F9F\">THvF;F9F<$\"1ij9.e@R!)FAF9F0?!\\ D*FAF9F<$\"1Oo+d!\\DP*FAF9F<$\"1#)4)[U!)4X*FAF9F<$\"1E^v#z6%H&*FAF9F!\\&)FAF9F<$\"0ed\\ &G'o:)FIF9F<$\"2sq7xkuio(F;F9F'FAF9F<$\"1U$)[6OLL` FAF9F<$\"1a>ZqB)eq%FAF9F<$\"1#)[`'ommm%FAF9F!*FAF9F<$\"1s 9KP1Zw\"*FAF9F<$\"0pK\"*o!\\$FAF9F<$\"23[6m6\\DP$F;F9F<$\"/$Q]@THN%FFF9F<$\"1 )*>#pwuio&FAF9F%F;F9 F<$\"2%)42=7^ui#F;F9F<$\"2G75Jbs8V#F;F9F<$\"2ONcQKPJ%QF;F9F!pFAF9F<$\"1aoe:gqkxFAF9F!\\&FAF9Fmo:#>F;F)FA F9FM3y6%H(FIF9F " 0 "" {MPLTEXT 1 1895 36 "seque nce = seq( [i,i^2], i=0..10 );" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 267 "" 0 "" {TEXT 1901 117 "If we store this sequenc e in a list, then the order is preserved, yet if we store it as a set, then the order is not:" }}{PARA 264 "" 0 "" {TEXT 1893 0 "" }}{PARA 243 "> " 0 "" {MPLTEXT 1 1899 43 "List := [ seq( [i,i^2], i=0..10 ) ] ; " }{TEXT 1850 33 "# Generate list, order preserved " }}{PARA 264 "> " 0 "" {MPLTEXT 1 1903 43 "Set := \{ seq( [i,i^2], i=0..10 ) \+ \}; " }{TEXT 1904 36 "# Generate set, order not preserved" } {MPLTEXT 1 1903 34 " " }{TEXT 1893 0 "" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 243 "" 0 "" {TEXT 1850 71 "The difference becomes evident when we plot the set and list using the " }{TEXT 1872 9 "pointplot" }{TEXT 1850 17 " command a nd the " }{TEXT 1872 12 "connect=true" }{TEXT 1850 172 " option to con nects the points. If the points are sequentially ordered, then the lis t should yield a single-valued function, otherwise the curve will loop back upon itself:" }}{PARA 264 "" 0 "" {TEXT 1893 0 "" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1895 12 "with(plots):" }}{PARA 243 "> " 0 "" {MPLTEXT 1 1899 34 "pointplot(List, connect=true); " }{TEXT 1850 23 " # Continuous with list" }}{PARA 264 "> " 0 "" {MPLTEXT 1 1903 34 "pointplot(Set, connect=true); " }{TEXT 1904 30 "# Loops back occ urs with list" }{TEXT 1893 0 "" }}{EXCHG {PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}}}{SECT 0 {PARA 247 "" 0 "" {TEXT 1857 7 " " } {TEXT 1858 11 " 4.9.9 " }{TEXT 1859 27 " Creating Simple Figures: " }{TEXT 1951 9 "pointplot" }{TEXT 1859 5 " and " }{TEXT 1951 11 "poi ntplot3d" }{TEXT 1859 1 "*" }{TEXT 1860 0 "" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 252 "" 0 "" {TEXT 1850 42 "Here we give some e xamples of the use of " }{TEXT 1872 9 "pointplot" }{TEXT 1850 5 " and " }{TEXT 1872 11 "pointplot3d" }{TEXT 1850 241 " to create simple fig ures of a barbell (we use\nthe figures in Chap. 7). There are three st eps involved. First we plot just the data points using circles for t he points. Then we plot lines connecting the data points.Finally we us e Maple's " }{TEXT 1872 7 "display" }{TEXT 1850 135 " command to disp lay both the\npoints and lines on the same graph. In 2-D, the left plot below, we give points as a list of doublets " }{TEXT 1872 6 "[x, y,]" }{TEXT 1850 1 ":" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 12 "with(plots):" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 85 "p1 := pointplot( \{[-0.7, 0.7], [0.7, -0.7]\}, co lor=black, thickness=2, style=LINE );" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 135 "p2 := pointplot( \{[-0.7, 0.7], [sqrt(2)/2, -0.7]\}, colo r=red, symbol=CIRCLE, thickness=2, symbolsize=45, axes=normal, labels =[x,y] );" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 18 "display( p1, p2 ) ;" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 242 "" 0 "" {TEXT 1849 56 "In 3-D, we give he points as a list of triplets [x,y,z] :" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 12 "with(plots):" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 91 "p1 \+ := pointplot3d( \{[-0.7,0,0.7], [0.7, 0, -0.7]\}, color=black, thickn ess=2, style=LINE );" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 137 "p2 := pointplot3d( \{[-0.7,0,0.7], [0.7, 0, -0.7]\}, color=red, symbol=CIR CLE, thickness=2, symbolsize=100, axes=normal, labels=[x,y,z] );" }} {PARA 255 "> " 0 "" {MPLTEXT 1 1879 16 "display(p1, p2);" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}}{SECT 0 {PARA 296 "" 0 "" {TEXT 1952 15 " " }{TEXT 1953 7 "4.9.10 " }{TEXT 1954 18 "Plot ting Vectors: " }{TEXT 1955 5 "arrow" }{TEXT 1953 1 "*" }{TEXT 1956 0 "" }}{PARA 252 "" 0 "" {TEXT 1850 8 "Maple's " }{TEXT 1872 13 "LinearA lgebra" }{TEXT 1850 282 " package is discussed in Chap. 7. That packa ge permits us to define vectors, matrices, and arrays of arbitrary siz es and dimensions (although matrices are always 2-D). Maple provides s ome easy-to-use tools for visualizing vectors and matrices, in particu lar, arrow, and matrixplot." }}{PARA 250 "" 0 "" {GLPLOT3D 349 295 295 {PLOTDATA 3 "6*-%)POLYGONSG6&7&7%$\"\"!F)$!+++++]!#6F(7%F($\"+++++ ]F,F(7%F(F.$\"+++++;!\"*7%F(F*F17%7%F($\"+++++5!#5F17%F(F($\"+++++?F37 %F($!+++++5F9F1-%&STYLEG6#%,PATCHNOGRIDG-%&COLORG6&%$RGBG$F)!\"\"FHFH- F$697&7%$\"+)*******\\F,$!+/p^D5!#?$!+********\\F,7%$\"++++];F3$!+$y0U Q$!#>$\"++++]:F37%$\"+]3XS;F3$!+6%pi:%F,$\"+]\"\\&f:F37%$\"+p\\3XSF,$! +y$pi:%F,$!+r\\3XSF,7&F^oFgn7%$\"+]3X:;F3$!+_^)\\s'F,$\"+]\"\\Xe\"F37% $\"+o\\3X:F,$!+>^)\\s'F,$!+p\\3X:F,7&F]pFfo7%F[pFioFgo7%$!+q\\3X:F,F`p $\"+p\\3X:F,7&FfpFep7%F\\o$!+4%pi:%F,Fhn7%Fco$!+w$pi:%F,$\"+r\\3XSF,7& F_qF\\q7%Fen$!+.C5zJFZFV7%$!+)*******\\F,$\"+/p^D5FR$\"+********\\F,7& FhqFeq7%F\\o$\"+X$pi:%F,Fhn7%$!+p\\3XSF,$\"+y$pi:%F,Fbq7&FcrF`r7%F[p$ \"+'3&)\\s'F,Fgo7%$!+o\\3X:F,$\"+>^)\\s'F,Fip7&F\\sFir7%FgoFjrF[p7%$\" +q\\3X:F,F_sFbp7&FcsFbs7%Fhn$\"+V$pi:%F,F\\o7%Fbq$\"+w$pi:%F,Fco7&FjsF gsFUFM7%7%$\"+++++:F37%F[uFft7%$\"+*p,4j\"F3$!+Fq*\\M\" F9$\"+,$)4p:F37%FcuFft7%FhuFfuFdu7%F[vFft7%F`u$!+'yQDJ)F,F\\u7%F]vFft7 %Fdt$!+72bwIFZF`t7%FavFft7%F`u$\"+C(QDJ)F,F\\u7%FevFft7%Fhu$\"+@q*\\M \"F9Fdu7%FivFft7%FduFjvFhu7%F]wFft7%F\\u$\"+?(QDJ)F,F`u7%F_wFftF_t7,F_ tF[uFcuF[vF]vFavFevFivF]wF_w7,FMF^oF]pFfpF_qFhqFcrF\\sFcsFjsF@-%&TITLE G6$Q " 0 "" {MPLTEXT 1 1879 34 " with(plots): with(LinearAlgebra):" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 27 " Omega := Vector([1,-3,6]);" }}{PARA 255 "> " 0 " " {MPLTEXT 1 1879 22 " L := Vector([6,0,6]);" }}{PARA 254 "> " 0 "" {MPLTEXT 1 1880 46 " arrow(Omega,color=black,shape=double_arrow); " } {TEXT 1871 48 "# Plot Omega as black arrow (grab it and rotate)" } {TEXT 1867 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 1880 11 " arrow(L); \+ " }{TEXT 1871 1 " " }{MPLTEXT 1 1884 1 " " }{TEXT 1871 46 "# Plot L as colored arrow (grab it and rotate)" }{TEXT 1867 0 "" }}{PARA 258 "> \+ " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 254 "" 0 "" {TEXT 1871 42 "Once we have visualizations of both L and " }{XPPEDIT 1226 0 "Omega;" "6#%&Om egaG" }{TEXT 1867 1 " " }{TEXT 1871 147 " that we can grab and rotate. We place them on the same graph by assigning objects (named variables ) to each arrow, and then displaying the arrows:" }{TEXT 1867 0 "" }} {PARA 242 "" 0 "" {TEXT 1849 2 " " }}{PARA 254 "> " 0 "" {MPLTEXT 1 1880 60 "w := arrow( Omega, color=black, shape=double_arrow ); \+ " }{TEXT 1871 33 "# Assign object to arrow of Omega" }{TEXT 1867 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 1880 18 "L1 := arrow( L ):" }{TEXT 1871 46 " # assign object to arrow of L " }{TEXT 1867 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 1880 94 "display( w, L1,axes=BOX ED,scaling=constrained, labels=[x,y,z], title=`Omega (black) and L` ); " }{TEXT 1871 18 "# Display together" }{TEXT 1867 0 "" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 24 "Omega := Vector([1,-3]);" }}{PARA 258 "> \+ " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 242 "" 0 "" {TEXT 1849 151 "We can also use the arrow command to plot several arrows (or sequences of ar rows) at one time. To illustrate, here are the familiar three unit vec tors:" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 39 " arrow( \{[0,0,1], [0,1,0], [1,0,0]\});" }} {PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 241 "" 0 "" {TEXT 1848 96 "Study our creation of arrows with the options to control the \+ color and shape of the arrows (see " }{TEXT 1943 10 "Help arrow" } {TEXT 1848 67 " for\nother properties). Although there are limited opt ions for the " }{TEXT 1943 5 "arrow" }{TEXT 1848 14 " command, the " } {TEXT 1943 7 "display" }{TEXT 1848 41 " command supports the\nusual on es such as " }{TEXT 1943 6 "labels" }{TEXT 1848 5 " and " }{TEXT 1943 7 "titles." }{TEXT 1848 0 "" }}{PARA 239 "" 0 "" {TEXT 1843 1 " " }}} {SECT 0 {PARA 247 "" 0 "" {TEXT 1857 3 " " }{TEXT 1858 5 "4.10 " } {TEXT 1859 26 "Visualizing Numerical Data" }{TEXT 1860 0 "" }}}{SECT 0 {PARA 247 "" 0 "" {TEXT 1857 1 " " }{TEXT 1858 6 " " }{TEXT 1857 5 " " }{TEXT 1858 7 "4.10.1 " }{TEXT 1859 17 " 2D Plots of Da ta" }{TEXT 1860 0 "" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 252 " " 0 "" {TEXT 1850 564 "Most realistic computations in science produce \+ numerical output, not analytic functions. Possibly in view of the fact that there is more work involved in plotting numerical data than the re is for an analytic function, Maple has a number of ways to make 2 -D plots of numerical data. (We thank David McIntyre for assistance wi th methods for plotting data.) In fact, the various packages that are used to extend the basic Maple capabilities, such as those for statis tics or linear algebra, often have their own plotting techniques. Here we demonstrate the use of " }{TEXT 1872 8 "listplot" }{TEXT 1850 10 " from the " }{TEXT 1872 4 "plot" }{TEXT 1850 13 " package and " } {TEXT 1872 11 "scatterplot" }{TEXT 1850 10 " from the " }{TEXT 1872 11 "statistical" }{TEXT 1850 9 " package." }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}}{SECT 0 {PARA 247 "" 0 "" {TEXT 1857 15 " \+ " }{TEXT 1858 7 "4.10.2 " }{TEXT 1859 18 " Numerical Plots: " }{TEXT 1951 8 "listplot" }{TEXT 1860 0 "" }}{PARA 242 "" 0 "" {TEXT 1849 2 " \+ " }}{PARA 252 "" 0 "" {TEXT 1850 5 "The " }{TEXT 1872 8 "listplot" } {TEXT 1850 89 " command creates a 2-dimensional plot from a list of n umerical data values. If only the " }{TEXT 1872 1 "y" }{TEXT 1850 60 " values of the data are given, say as the four-element list " }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 272 "" 0 "" {TEXT 1912 60 " \+ Ydata := [1, 8, 27, 100] ," }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 252 "" 0 "" {TEXT 1850 5 "then " }{TEXT 1872 8 "listplot" }{TEXT 1850 13 " will assign " }{TEXT 1872 1 "x" } {TEXT 1850 25 " values by counting from " }{TEXT 1872 1 "1" }{TEXT 1850 4 " to " }{TEXT 1872 1 "4" }{TEXT 1850 27 ", as if the data point s are" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 272 "" 0 "" {TEXT 1912 54 " (1,1), (2,8), (3,27), (4,400)" }} {PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 252 "" 0 "" {TEXT 1850 29 "T he same plot is obtained if " }{TEXT 1872 1 "y" }{TEXT 1850 22 " is en tered as a list:" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 241 "> \+ " 0 "" {MPLTEXT 1 1886 30 "with( plots ): " }{TEXT 1887 14 "# Load package" }{TEXT 1848 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 1880 29 "listplot( [1, 8, 27, 100] ); " }{TEXT 1871 51 " \+ # Plot these y values with x as order number" }{TEXT 1867 0 "" }} {PARA 254 "> " 0 "" {MPLTEXT 1 1880 27 "Ydata := [1, 8, 27, 100];" } {TEXT 1871 35 " # Enter y data into list Ydata" }{TEXT 1867 0 "" } }{PARA 254 "> " 0 "" {MPLTEXT 1 1880 16 "listplot(Ydata);" }{TEXT 1871 26 " " }{TEXT 1891 51 " # Plot data ob ject Ydata with x as order number" }{TEXT 1867 0 "" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 293 "" 0 "" {TEXT 1944 166 "If you w ant to give explicit x values to your data, then you can place each ( x_i, y_i) value in its own 2-element list, and make a big list up of \+ these data points:" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 293 " " 0 "" {TEXT 1944 89 " [ [x1 , y1], [x2, y2], [x3, y3], [x4, y4] ]" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 1880 60 "listplot( [[0, 1], [sqrt(3)/2, 1/2], [-sqrt(3)/2, 1/2] ] ); " }{TEXT 1871 47 " \+ # Plot (x_i,y_i) values in list" }{TEXT 1867 0 "" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 70 "listplot( [ [0, 1], [sqrt(3)/2, 1/2], [-sq rt(3)/2, 1/2], [0, 1] ]); " }}{PARA 241 "> " 0 "" {MPLTEXT 1 1886 71 "XYdata := [ [0, 1], [sqrt(3)/2, 1/2], [-sqrt(3)/2, 1/2], [0, 1] \+ ] ; " }{TEXT 1887 36 "# Enter (x,y) data into list XYdata" }{TEXT 1848 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 1880 20 "listplot( XYdata ) ; " }{TEXT 1871 1 " " }{TEXT 1891 14 " # Plot Ydata " }{TEXT 1867 0 " " }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 252 "" 0 "" {TEXT 1850 130 "These are the basic commands. There are options to the m for plotting only points, for controlling their size and color, and \+ (with " }{TEXT 1872 12 "connect=true" }{TEXT 1850 68 ") to connect the points in the true order in which they are plotted:" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 1880 63 "listplo t( XYdata, color=blue, axes=none, symbol=circle ); " }{TEXT 1871 24 " # Add color" }{TEXT 1867 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 1880 59 "listplot( XYdata, style=point, color=red, symbolsi ze=35 ); " }{TEXT 1871 37 " # Do not connect points" } {TEXT 1867 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 1880 21 " display( \{ %, %%\} ); " }{TEXT 1871 80 " # Put \+ two previous plots (% and %%) together" }{TEXT 1867 0 "" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 297 "" 0 "" {TEXT 1957 182 " To get the plot to close on itself and thereby form a geometric figure \+ we repeat the first point; to make the geometric figure without inter nal lines, the order of points matters:" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 241 "> " 0 "" {MPLTEXT 1 1886 53 "plot_data:=[ [4, 8 ], [2, 1], [6, 27], [8, 100] ]; " }{TEXT 1887 21 "# Out of order po ints" }{TEXT 1848 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 1880 24 " list plot( plot_data ); " }{TEXT 1871 60 " \+ # Out of order plot" }{TEXT 1867 0 "" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}}{SECT 0 {PARA 288 "" 0 "" {TEXT 1845 15 " \+ 4.10.3 " }{TEXT 1846 18 " Numerical Plots: " }{TEXT 1958 11 "scatte rplot" }{TEXT 1846 1 "*" }{TEXT 1847 0 "" }}{PARA 252 "" 0 "" {TEXT 1850 85 "A scatter plot is a plot of points in which the points are no t connected. To use the " }{TEXT 1872 11 "scatterplot" }{TEXT 1850 86 " command, we form a list of x values of our data and a corresponding \+ list of y values:" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 254 "> \+ " 0 "" {MPLTEXT 1 1880 36 "with(plots): with(stats[statplots]);" } {TEXT 1871 50 " # Load statistics package " } {TEXT 1867 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 1880 36 "Xdata := [0, sqrt(3.)/2,-sqrt(3.)/2];" }{TEXT 1871 61 " \+ # [list] of all x values" }{TEXT 1867 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 1880 23 "Ydata := [1,-1/2,-1/2];" }{TEXT 1871 23 " \+ " }{TEXT 1891 34 "# [list] of corresponding y value s" }{TEXT 1867 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 1880 27 "scatterp lot( Xdata, Ydata);" }{TEXT 1871 4 " " }{TEXT 1891 21 "# Plot poin ts, basic" }{TEXT 1867 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 1880 53 " scatterplot( Xdata, Ydata, color=red, symbolsize=35);" }{TEXT 1871 30 " #Plot points, +options" }{TEXT 1867 0 "" }}{PARA 256 "> " 0 " " {MPLTEXT 1 1882 1 " " }}}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{SECT 0 {PARA 292 "" 0 "" {TEXT 1935 6 " " }{TEXT 1937 7 "4.10.4 " } {TEXT 1935 1 " " }{TEXT 1938 23 "Histograms and Boxplots" }{TEXT 1935 0 "" }}}{PARA 298 "" 0 "" {TEXT 1959 0 "" }}{PARA 299 "" 0 "" {TEXT 1960 65 "Here we make a histogram and then a box plot of some random d ata:" }}{EXCHG {PARA 298 "" 0 "" {TEXT 1959 0 "" }}{PARA 258 "> " 0 " " {MPLTEXT 1 1879 70 "data := [-2., -.8, 2., .0, -.5, -.5, 1.6,.8,.5,- .5,-.2, -.2, .2, -.1];" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 28 "hist ogram(data, area=count);" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 14 "bo xplot(data);" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}}{PARA 257 "" 0 "" {TEXT 1883 131 "A histogram is seen to create bins or bars, w ith the height of the bar proportional to the number of events that oc cur within the " }{TEXT 1941 1 "x" }{TEXT 1883 182 " values covered by the bar. A box plot is seen to contain a central line showing the med ian, a lower line showing the first quartile, and an upper line show ing the third quartile. " }}{PARA 257 "" 0 "" {TEXT 1883 1 " " }} {SECT 0 {PARA 292 "" 0 "" {TEXT 1935 4 " " }{TEXT 1936 6 " " } {TEXT 1937 7 "4.10.5 " }{TEXT 1938 24 " Surface Plots of Data: " } {TEXT 1939 10 "listplot3d" }{TEXT 1935 1 "*" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 243 "" 0 "" {TEXT 1850 54 "We have already see n how to make 3-D or surface plots " }{TEXT 1872 10 "z = f(x,y)" } {TEXT 1850 236 " of analytic functions of two variables. Here we descr ibe how to make the same kind of plot from a table of numbers. In Chap . 11 we describe how to use the free plotting program gnuplot to make surface plots of numerical data (we find " }{TEXT 1872 7 "gnuplot" } {TEXT 1850 461 " easier.)\n\nMost often, realistic calculations produc e numerical data rather than analytic functions as their output. This \+ type of output is a consequence of the real world being less simple th an the assumptions made in those models which lead to purely analytic \+ answers. In realistic cases, the resulting equations may still be solv ed, only they must be solved numerically. Trying to understand if ther e is meaning in long lists of numbers obtained as output is" }{TEXT 1850 92 "\nquite the challenge, yet this point is exactly where visual ization tools are most valuable." }{TEXT 1850 228 "\n\nWhile yet to be discussed, (we do in Chap. 7), the use of matrices and arrays makes \+ the bookkeeping for multi-dimensional data rather straightforward. Hen ce, you may want to review matrices and then return to this subsection ." }{TEXT 1850 141 "\n\nAs with all 3-D visualizations, we want to cre ate a surface in 3-D space that represents our data. Picture data desc ribing the temperature " }{TEXT 1872 4 "T(x)" }{TEXT 1850 27 " as a fu nction of distance " }{TEXT 1872 1 "x" }{TEXT 1850 198 " along a one d imensional bar. However, the bar is cooling as a function of time, so there is a temperature distribution for each time that we put togethe r as a function of both time and position " }{TEXT 1872 9 "T(t,\nx). " }{TEXT 1850 38 "Consequently, we will make a graph of " }{TEXT 1872 6 "T(t,x)" }{TEXT 1850 6 " with " }{TEXT 1872 1 "T" }{TEXT 1850 61 " as the vertical distance above the plane formed by the time " } {TEXT 1872 1 "t" }{TEXT 1850 49 " as one horizontal coordinate and wi th position " }{TEXT 1872 1 "x" }{TEXT 1850 157 " as the other\nhorizo ntal coordinate. The fact that this 3-D surface does not truly exist \+ in nature is one reason this approach is called ``visualization''." } {TEXT 1850 193 "\n\nThe plotting of numerical data is done in two step s. First we read the data into a matrix, with one index for position a nd other for time. Then we plot the matrix. We have placed some sample " }{TEXT 1872 6 "T(t,x)" }{TEXT 1850 18 " data in the file " }{TEXT 1872 12 "EqHeat_z.dat" }{TEXT 1850 125 " on the CD, and you should rea d that file into a convenient location now.\n\nIt seem obvious that th e data file should contain " }{TEXT 1872 1 "x" }{TEXT 1850 5 " and " } {TEXT 1872 1 "t" }{TEXT 1850 28 " values with the associated" }{TEXT 1872 7 " T(t,x)" }{TEXT 1850 64 " value. Nonetheless, the problem is m ade simpler by just giving " }{TEXT 1872 1 "T" }{TEXT 1850 44 " values (what is indicated by the subscript " }{TEXT 1872 2 "_z" }{TEXT 1850 14 " in the file " }{TEXT 1872 12 "EqHeat_z.dat" }{TEXT 1850 31 ". Th e assumption is that these " }{TEXT 1872 1 "T" }{TEXT 1850 215 " value s correspond to a rectangular array of uniformly spaced $x$ and $t$ va lues, and we do not need to know the actually spacing to make the visu alization. To see how this works, here are the first three lines in \+ " }{TEXT 1872 12 "EqHeat_z.dat" }{TEXT 1850 2 ":\n" }}{PARA 242 "" 0 " " {TEXT 1849 255 "0.0 100.0 100.0 100.0 10 0.0 100.0 100.0 100.0 100.0 100.0 \+ 0.0 \n0.0 32.5 59.8 78.9 89.5 \+ 92.8 89.5 78.9 59.8 32.5 0.0" } {TEXT 1849 123 "\n0.0 22.7 43.0 59.0 69.1 \+ 72.5 69.1 59.0 43.0 22.7 0.0 " }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 252 "" 0 "" {TEXT 1850 266 "Think of each line of data as a row of a big matrix. Whereas the \+ number of digits used for each number may differ, and so the length of each line may differ, each row has the same number of entries or colu mns. These data are values of the temperature for increasing " }{TEXT 1872 1 "x" }{TEXT 1850 212 " values along the rod. Explicit values of \+ the time are not given, however, subsequent lines (rows of the matrix) contain the temperature distributions for later and later times. In o ther words, the full matrix is" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }} {PARA 252 "" 0 "" {TEXT 1850 29 " " } {TEXT 1872 6 "T(x,t)" }{TEXT 1850 3 " = " }{TEXT 1872 1 "T" }{TEXT 1850 16 "(column, row) = " }{TEXT 1872 1 "z" }{TEXT 1850 14 "(column, \+ row)," }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 252 "" 0 "" {TEXT 1850 291 "where each new row corresponds to the next value of the time . As no explicit values are given for the time or the position, the pl otting program will assume uniform spacing and time steps, and will as sign each integer values, 1, 2, 3 ... for the plot. Think of these as \+ ``the first, second, " }{TEXT 1872 9 "et cetera" }{TEXT 1850 34 " time s'' and the ``first, second, " }{TEXT 1872 9 "et cetera" }{TEXT 1850 54 " '' positions.\n\nThe data are read into Maple with the " }{TEXT 1872 8 "readdata" }{TEXT 1850 66 " command. We view each row of the ma trix as a list (order matters)" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }} {PARA 242 "" 0 "" {TEXT 1849 69 " [value _1, value_2, ..., value_numcols]" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 252 "" 0 "" {TEXT 1850 47 "and the collection of rows as a lis t of lists (" }{TEXT 1872 8 "listlist" }{TEXT 1850 2 "):" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 242 "" 0 "" {TEXT 1849 94 " \+ [[row 1 data], [row 2 data], ... [row n umrows data]]" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 241 "" 0 " " {TEXT 1848 0 "" }}{PARA 242 "" 0 "" {TEXT 1849 280 "Once we know how many rows or columns there are, it is easy to build the matrix with a ll the elements in their correct rows and columns. While we will show \+ you how to make an explicit matrix up out of these data soon, first we will work with just the z values as a list of lists. " }}{PARA 241 " " 0 "" {TEXT 1848 0 "" }}{PARA 252 "" 0 "" {TEXT 1850 20 "We use the v ariable " }{TEXT 1872 7 "numcols" }{TEXT 1850 170 " to describe the nu mber of elements (columns) in each row. We must tell Maple the number \+ of columns in each input row, but can then use Maple to count the numb er of rows:" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 255 "> " 0 " " {MPLTEXT 1 1879 8 "restart;" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 14 "with(plots): " }}{PARA 254 "> " 0 "" {MPLTEXT 1 1880 28 "with(Lin earAlgebra): # " }{TEXT 1871 50 "Load linear algebra package to d eal with matrices." }{TEXT 1867 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 1880 14 "numcols := 11;" }{TEXT 1871 9 " " }{TEXT 1891 43 " # \+ The number of elements (columns) per row" }{TEXT 1867 0 "" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 252 "" 0 "" {TEXT 1850 20 " The data (values of " }{TEXT 1872 1 "z" }{TEXT 1850 91 ") are read int o the variable (object or abstract data type) that we name data with M aple's " }{TEXT 1872 8 "readdata" }{TEXT 1850 300 " command. Because f iles are stored on your individual computer using the local operating \+ system's conventions for files, just how you specify the file name dep ends somewhat on the computer system you are on, and where the file is on that system. For all but the simple Unix version, we have placed a " }{TEXT 1872 1 "#" }{TEXT 1850 329 " sign in front of the\ncommand \+ so that Maple will treat the command as a comment. Examine the use of \+ left quote or accent grave, `as opposed to the normal', in the command to delineate the name of the data file. To see what works for you, tr y deleting the comment symbol and seeing if the command completes with no error message:" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 252 "> " 0 "" {MPLTEXT 1 1881 44 "#data:=readdata(`EqHeat_z.dat`, numcols); \+ " }{TEXT 1850 62 " # Unix, file in same directory from which Maple \+ was started." }}{PARA 252 "> " 0 "" {MPLTEXT 1 1881 55 "#data:=readdat a(\"Mac G3 HD:DMc:EqHeat_z.dat\",numcols): " }{TEXT 1850 18 " # Apple \+ MacIntosh" }}{PARA 252 "> " 0 "" {MPLTEXT 1 1881 74 "#data:=readdata( \"C:\\\\My Documents\\\\Rubin\\\\Books\\\\EqHeat_z.dat\", numcols): " }{TEXT 1850 24 " # Windows needs extra \\" }}{PARA 254 "> " 0 "" {MPLTEXT 1 1880 81 "data:=readdata(\"C:/My Documents/Rubin/Books/Intro /Maple/EqHeat_z.dat\", numcols); " }{TEXT 1871 24 " # Works also on Wi ndows" }{TEXT 1867 0 "" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }} {PARA 252 "" 0 "" {TEXT 1850 52 "As you see here from the Maple output , the variable " }{TEXT 1872 4 "data" }{TEXT 1850 36 " is a data objec t composed of a list" }{TEXT 1872 6 " [...]" }{TEXT 1850 24 " containi ng other lists " }{TEXT 1872 20 "[[...], [...], ....]" }{TEXT 1850 165 ". Each sublist is the temperature all along the bar at a differen t time. To look at any individual part of this list, for example the s econd row (the second list):" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }} {PARA 258 "> " 0 "" {MPLTEXT 1 1879 8 "data[2];" }}{PARA 258 "> " 0 " " {MPLTEXT 1 1879 0 "" }}{PARA 257 "" 0 "" {TEXT 1883 140 "In case you have some trouble with reading these files, you may also want to try \+ inputting the data by hand with the list of lists format, " }{TEXT 1941 30 "data := [[...], [...], ....]." }{TEXT 1883 0 "" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}}{SECT 0 {PARA 292 "" 0 "" {TEXT 1935 5 " \+ " }{TEXT 1937 6 "4.10.6" }{TEXT 1961 1 " " }{TEXT 1938 11 "listplot \+ 3d" }{TEXT 1935 0 "" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 252 " " 0 "" {TEXT 1850 4 "The " }{TEXT 1872 10 "listplot3d" }{TEXT 1850 126 " command creates a 3D plot of a list of lists of numeric values. \+ Remaining arguments are interpreted as plot options. See the " }{TEXT 1872 6 "plot3d" }{TEXT 1850 44 " help menu for a list of possible opti ons. " }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 12 "with(plots):" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 17 "listplot3d(data);" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 60 " listplot3d(data,style=patch,orientation=[-55,30],shading=z);" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}}{SECT 0 {PARA 292 "" 0 "" {TEXT 1937 7 " 4.11 " }{TEXT 1938 19 "Plotting a Matrix: " }{TEXT 1939 10 " matrixplot" }{TEXT 1935 1 "*" }}{PARA 252 "" 0 "" {TEXT 1850 43 "Anoth er way to make a 3-D plot is with the " }{TEXT 1872 10 "matrixplot" } {TEXT 1850 442 " command. Though this is an easy command to use, we f irst need to place the data in a matrix. In Chap. 7 we discuss matri ces and give examples of plotting matrices and vectors as well. If y ou have trouble following the procedure below, you may want to learn s ome more about matrices in Chap. 7.\n\nWe start by reading in the data file. In order to figure out how many rows there are in the matrix wi thout us doing the counting, we use the " }{TEXT 1872 4 "nops" }{TEXT 1850 30 " (number of operands) command:" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 21 "restart; with(plot s):" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 22 "with(linalg): numcols; " }}{PARA 252 "> " 0 "" {MPLTEXT 1 1881 44 "#data:=readdata(`EqHeat_z. dat`, numcols); " }{TEXT 1850 53 " # Unix, file in same directory w here Maple started." }}{PARA 252 "> " 0 "" {MPLTEXT 1 1881 55 "#data:= readdata(\"Mac G3 HD:DMc:EqHeat_z.dat\",numcols): " }{TEXT 1850 18 " # Apple MacIntosh" }}{PARA 252 "> " 0 "" {MPLTEXT 1 1881 74 "#data:=rea ddata(\"C:\\\\My Documents\\\\Rubin\\\\Books\\\\EqHeat_z.dat\", numcol s): " }{TEXT 1850 24 " # Windows needs extra \\" }}{PARA 254 "> " 0 " " {MPLTEXT 1 1880 81 "data:=readdata(\"C:/My Documents/Rubin/Books/Int ro/Maple/EqHeat_z.dat\", numcols); " }{TEXT 1871 24 " # Works also on \+ Windows" }{TEXT 1867 0 "" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 24 "nu mrows := nops( data );" }}{PARA 256 "> " 0 "" {MPLTEXT 1 1882 1 " " }} {PARA 252 "" 0 "" {TEXT 1850 30 "We now convert the list named " } {TEXT 1872 4 "data" }{TEXT 1850 21 " to the matrix named " }{TEXT 1872 11 "data_matrix" }{TEXT 1850 15 " by using the " }{TEXT 1872 7 " convert" }{TEXT 1850 9 " command:" }}{PARA 239 "" 0 "" {TEXT 1843 0 " " }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 34 "data_matrix:=convert(data, matrix);" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 242 "" 0 "" {TEXT 1849 126 "As a check, you should determine the individual ele ment in row 5, column 2. Here we do it for the 5th elements in the 2nd row:" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 1880 19 "data_matrix[2, 5]; " }{TEXT 1871 4 " # " }{TEXT 1891 26 "Element in row 2, column 5" }{TEXT 1867 0 "" }}{PARA 252 "> \+ " 0 "" {MPLTEXT 1 1881 21 " " }{TEXT 1850 40 " # D etermine elements in row 5, column 2" }{MPLTEXT 1 1881 6 " " } {TEXT 1850 0 "" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}}{SECT 0 {PARA 296 "" 0 "" {TEXT 1952 6 " " }{TEXT 1953 7 "4.11.1 " } {TEXT 1952 1 " " }{TEXT 1954 30 "Surface Plots with matrixplot*" } {TEXT 1956 0 "" }}{PARA 250 "" 0 "" {GLPLOT3D 267 156 156 {PLOTDATA 3 "6(-%%GRIDG6%;\"\"\"\"\"$F&7%7%$\"3A+++++++5!#<$\"\"!F/$F'F/7%F.$\"\"# F/F.7%F0F.$\"2y***************F--%&TITLEG6$Q:Inertia~tensor~of~barbell 6\"-%%FONTG6$%*HELVETICAG\"#5-%+AXESLABELSG6%Q$rowF;Q'columnF;Q!F;-%*A XESSTYLEG6#%$BOXG-%*LINESTYLEG6#F/F<" 1 2 5 0 10 1 2 1 1 2 2 1.000000 55.000000 143.000000 1 0 "Curve 1" }}{TEXT 1865 1 " " }}{PARA 241 "" 0 "" {TEXT 1848 57 "An entire 2-D matrix is visualized in one step wi th the " }{TEXT 1943 10 "matrixplot" }{TEXT 1848 153 " command. This p rovides a plot that may be grabbed and rotated, which is useful for fi nding missing elements. The options are much the same as those for " } {TEXT 1943 7 "plot3-D" }{TEXT 1848 1 ":" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 34 "with(LinearAlgebra ): with(plots):" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 96 "Slant := M atrix ( < <1,2,3,4> | <5,6,7,8> | <9,10,11,12> > ); # Set up matrix wi th three columns" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 72 "ID := Iden tityMatrix(6); # The identity matrix of specified dimension" }} {PARA 255 "> " 0 "" {MPLTEXT 1 1879 137 "matrixplot(ID,axes=boxed,titl e=`Identity Matrix`,style=point,symbol=CIRCLE, thickness=2,symbolsize= 20); # Plot points only" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 66 "matrixplot(ID,axes=boxed,title=`Identity Matrix`); # Plot surfaces" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 65 "matrixplot (Slant,axes=boxed,title=`Slant Matrix`); # A new slant" }}{PARA 258 " > " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 243 "" 0 "" {TEXT 1850 98 "When \+ working with data, the hard part is clearly getting the matrix into Ma ple. Plotting with the " }{TEXT 1872 10 "matrixplot" }{TEXT 1850 91 " \+ command is easy. For our data, different rows correspond to different \+ values for the time " }{TEXT 1872 1 "t" }{TEXT 1850 59 ", and differen t columns corresponds to different positions " }{TEXT 1872 2 "x " } {TEXT 1850 51 "along the bar. The rows will be plotting along the " } {TEXT 1872 1 "x" }{TEXT 1850 5 " and " }{TEXT 1872 1 "y" }{TEXT 1850 435 " axis in the plot, that is along the base of the solid figure. Th e temperature will be plotted as the height of the surface above the b ase. To help with the visualization, we will also use the color of the surface to convey the temperature\n(ideally with red being hot and bl ue cold). 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BeKFbbr$\"+l/npMFbbr$\"+&G5um$Fbbr$\"+6ln]QFbbr$\"+xeu=SFbbr$\"+*3b4<% Fbbr$\"+YMq1VFbbr$\"+8_XDWFbbr$\"+H " 0 "" {MPLTEXT 1 1879 149 "matrixplot( data_matrix, axes=boxed, labels=[Time , Position, Temp], title=`Cooling of Hot Bar`,orientation=[-20,70] ,st yle=patchnogrid,shading=zhue);" }}{PARA 256 "> " 0 "" {MPLTEXT 1 1882 1 " " }}{PARA 273 "" 0 "" {TEXT 1913 388 "You have the figure above, w hich is rather complete, but not necessarily an effective visualizatio n. You are probably the best judge of effectiveness, and to do that yo u need to try out the options and see what works. So, another command- line version of matrixplot with many options for things like tick mark s at informative places, contours, and better fonts and sizes for the \+ labels is:" }}{PARA 283 "" 0 "" {TEXT 1925 0 "" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 247 "matrixplot( data_matrix, axes=boxed, labels=[Time , Position, Temp], style=patchcontour, labelfont=[HELVETICA,14], axesf ont=[HELVETICA,9], tickmarks=[4,4,4], title=`Cooling of Hot Bar`, titl efont=[HELVETICA,18], orientation=[-20,70] ,shading=zhue);" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 300 "" 0 "" {TEXT 1871 170 "Below, try including and removing some of these options in order to see the \+ effects. Go back to the simplest version of the matrix plot command, m atrixplot( data_matrix )" }{TEXT 1962 2 ", " }{TEXT 1871 193 "and see \+ if you can obtain similar options from within Maple's graphical user i nterface. In particular, notice how the 3D effects depends on having m ade a good choice for the orientation angle. " }{TEXT 1867 0 "" }} {PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}}{SECT 0 {PARA 292 "" 0 "" {TEXT 1935 2 " " }{TEXT 1937 6 "4.11.2" }{TEXT 1961 1 " " }{TEXT 1938 34 "Non Uniform x-y Grid with surfdata" }{TEXT 1935 0 "" }}{PARA 301 "" 0 "" {TEXT 1850 4 "The " }{TEXT 1872 10 "matrixplot" }{TEXT 1850 64 " command assumes the grid spacing is uniform, that is, that t he " }{TEXT 1872 1 "x" }{TEXT 1850 5 " and " }{TEXT 1872 1 "y" }{TEXT 1850 217 " values are uniformly spaced. If they are not, then you need to provide information about how the grid spacing depends on grid loc ation (column and row number). An alternative approach is to read in t he set of values " }{TEXT 1872 9 "(x, y, z)" }{TEXT 1850 43 " so that \+ you effectively have the fungction" }{TEXT 1872 8 "\nz(x, y)" }{TEXT 1850 6 ". The " }{TEXT 1872 8 "surfdata" }{TEXT 1850 49 " command is u sed to plot a surface from a set of " }{TEXT 1872 9 "(x, y, z)" } {TEXT 1850 79 " data.\n\nTo make a surface plot of data we need data. To start we make a list " }{TEXT 1872 9 "data_list" }{TEXT 1850 12 " of uniform " }{TEXT 1872 5 "(x,y)" }{TEXT 1850 35 " values using the \+ sequence command " }{TEXT 1872 3 "seq" }{TEXT 1850 33 ". As before, we make each set of " }{TEXT 1872 6 "(x, y)" }{TEXT 1850 237 " values in to a list, and then place all of these sublists into a list of lists . For our temperature example, this placement means that we will repre sent time with the row index, position\nwith the column index, and tem perature with the " }{TEXT 1872 1 "z" }{TEXT 1850 47 " coordinate. Rat her than read in a big file of " }{TEXT 1872 9 "(x, y, z)" }{TEXT 1850 29 " values we will add explicit " }{TEXT 1872 6 "(x, y)" }{TEXT 1850 54 " values to our list of data values. We start with\nthe " } {TEXT 1872 3 "seq" }{TEXT 1850 73 " command that creates a sequence of integers. First we get a sequence of " }{TEXT 1872 2 "x " }{TEXT 1850 12 "values from " }{TEXT 1872 1 "0" }{TEXT 1850 4 " to " }{TEXT 1872 4 "100 " }{TEXT 1850 12 "in steps of " }{TEXT 1872 2 "50" }{TEXT 1850 25 ", and then a sequence of " }{TEXT 1872 5 "[x,t]" }{TEXT 1850 6 " with " }{TEXT 1872 1 "x" }{TEXT 1850 14 " running from " }{TEXT 1872 1 "1" }{TEXT 1850 4 " to " }{TEXT 1872 4 "1000" }{TEXT 1850 13 " \+ in\nsteps of " }{TEXT 1872 3 "100" }{TEXT 1850 1 ":" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 1880 36 "seq( [(j- 1)*10], j=1..numcols ); " }{TEXT 1871 50 " # A sequ ence of column (x) values" }{TEXT 1867 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 1880 45 "seq( [(i-1)*50], i=1..numrows ); " } {TEXT 1871 34 " # A sequence of row (time) values" }{TEXT 1867 0 "" }} {PARA 256 "> " 0 "" {MPLTEXT 1 1882 1 " " }}{PARA 243 "" 0 "" {TEXT 1850 72 " Now we insert the values of the data matrix to form a list o f triplets " }{TEXT 1872 8 "[x,y,z]:" }{TEXT 1850 1 " " }}{PARA 293 " " 0 "" {TEXT 1944 1 " " }}{PARA 258 "> " 0 "" {MPLTEXT 1 1895 104 "dat a_list := [ seq( [ seq([(i-1)*50,(j-1)*10, data_matrix[ i, j] ], j=1 ..numcol ) ], i=1..numrows ) ];" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 252 "" 0 "" {TEXT 1850 32 "Each point to plot is a s et of " }{TEXT 1872 7 "(x,y,z)" }{TEXT 1850 78 " values that we are \+ representing in Maple as a sublist. For example, for row " }{TEXT 1872 1 "1" }{TEXT 1850 12 " and column " }{TEXT 1872 1 "2" }{TEXT 1850 18 " there is the list" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }} {PARA 255 "> " 0 "" {MPLTEXT 1 1879 15 "data_list[1,2];" }}{PARA 302 " > " 0 "" {MPLTEXT 1 1963 1 " " }}{PARA 252 "" 0 "" {TEXT 1850 31 "We n ow plot the graph with the " }{TEXT 1872 8 "surfdata" }{TEXT 1850 66 " command, first in simple form, and then with labels and contours:" }} {PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1895 22 "surfdata( data_list );" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 131 "surfdata( data_list, labels = [Time, Position, Temperature], axes=boxed, style=patchcontour, orientation=[-20,50], shading=zhue ) ;" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 303 "" 0 "" {TEXT 1964 30 "Reading in (x, y, z ) Data Sets" }}{PARA 278 "" 0 "" {TEXT 1919 0 "" }}{PARA 252 "" 0 "" {TEXT 1850 172 "In a more realistic case our data might be the result s of a numerical simulation or a measurement and would reside in an ex ternal file. For this purpose we supply the file" }{TEXT 1872 15 "\nEq Heat_xyz.dat" }{TEXT 1850 53 " on the CD. We read in these data as bef ore with the " }{TEXT 1872 8 "readdata" }{TEXT 1850 51 " command, but \+ now with 3 columns for the values of " }{TEXT 1872 8 "(x,y,z):" } {TEXT 1850 0 "" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 12 "with(plots):" }}{PARA 243 "> " 0 "" {MPLTEXT 1 1881 57 "#data_list_ex := readdata( `EqHeat_xyz.dat`, 3 ); " }{TEXT 1850 49 " # Unix read, file in present working directory. " }} {PARA 243 "> " 0 "" {MPLTEXT 1 1881 74 "#data_list_ex := readdata( `/h ome/mcintyre/cpug/heat/EqHeat_xyz.dat`, 3 );" }{TEXT 1850 49 " \+ # Unix read, complete specification." }}{PARA 302 "> " 0 "" {MPLTEXT 1 1963 77 "#data_list_ex := readdata( \"Mac G3 HD:DMc:OSU wor k:CPUG:EqHeat_xyz.dat\", 3 ):" }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 95 "data_list_ex := readdata( \"C:\\\\My Documents\\\\Rubin\\\\Books\\ \\Intro\\\\Maple\\\\EqHeat_xyz.dat\", 3 ): " }}{PARA 255 "> " 0 "" {MPLTEXT 1 1879 27 "listplot3d( data_list_ex );" }}{PARA 258 "> " 0 " " {MPLTEXT 1 1879 0 "" }}{PARA 298 "" 0 "" {TEXT 1965 1 " " }{TEXT 1966 28 "Converting Lists to Matrices" }{TEXT 1959 0 "" }}{PARA 241 " " 0 "" {TEXT 1848 0 "" }}{PARA 252 "" 0 "" {TEXT 1850 79 "We have show n how to make surface plots from numerical data using the commands " } {TEXT 1872 10 "listplot3d" }{TEXT 1850 5 " and " }{TEXT 1872 10 "matri xplot" }{TEXT 1850 52 ". If the data are in the form of a Maple list, \+ then " }{TEXT 1872 10 "listplot3d" }{TEXT 1850 59 " is simpler. If the data are in explicit matrix form, then " }{TEXT 1872 10 "matrixplot" }{TEXT 1850 76 " is simpler. If you are given data, but are not sure \+ of its form, then the " }{TEXT 1872 4 "type" }{TEXT 1850 98 " command \+ will tell you. Once you know the data type, you can decide how to plot them. For example," }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 254 " > " 0 "" {MPLTEXT 1 1880 37 "data:=readdata( \"EqHeat_z.dat\",11 ); " }{TEXT 1871 42 " # Read in data list" }{TEXT 1867 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 1880 68 "data_matrix := con vert( data,matrix ); # " }{TEXT 1871 27 "Co nvert data list to matrix" }{TEXT 1867 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 1880 23 "type( data,listlist ); " }{TEXT 1871 89 " \+ # Is file data a list of lists?" }{TEXT 1867 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 1880 19 "type( data,list ); " }{TEXT 1871 92 " \+ # Is file data a list?" }{TEXT 1867 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 1880 26 "type( data_matrix, list ); " }{TEXT 1871 78 " \+ # Is file data_matrix a list?" }{TEXT 1867 0 "" }}{PARA 254 "" 0 "" {TEXT 1871 9 "> " }{TEXT 1891 40 " # Test if file data_matri x is an array?" }{TEXT 1867 0 "" }}{PARA 252 "> " 0 "" {MPLTEXT 1 1881 2 " " }{TEXT 1850 35 " # Test if file data is an array?" }} {PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}{PARA 252 "" 0 "" {TEXT 1850 71 "It is rather easy to convert from a list of lists to a matrix with the " }{TEXT 1872 7 "convert" }{TEXT 1850 9 " command:" }}{PARA 241 "" 0 "" {TEXT 1848 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 1880 25 " convert( data, matrix ); " }{TEXT 1871 71 " \+ # Convert variable data to matrix" }{TEXT 1867 0 "" }} {PARA 254 "> " 0 "" {MPLTEXT 1 1880 24 "matrixplot( data ); " } {TEXT 1871 69 " # Will work only if \+ data is matrix" }{TEXT 1867 0 "" }}{PARA 241 "> " 0 "" {MPLTEXT 1 1886 37 "convert( data, listlist ); " }{TEXT 1887 39 "# Conv ert variable data to list of list" }{TEXT 1848 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 1880 20 "listplot3d( data ); " }{TEXT 1871 88 " \+ # Will work only if dat a is a list" }{TEXT 1867 0 "" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1879 0 "" }}}{SECT 0 {PARA 292 "" 0 "" {TEXT 1935 5 " " }{TEXT 1936 5 " \+ " }{TEXT 1937 5 "4.12 " }{TEXT 1938 20 " Animations of Data*" } {TEXT 1935 0 "" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 257 "" 0 " " {TEXT 1883 304 "Below we present an animated plot of the wave mot ion resulting from plucking a string. The string is hanging under its \+ own weight and is affected by friction. (We thanks Juan Vanegas for th is example.) This animation comes from a numerical simulation [CP 05] \+ that outputs its results to a file in the " }{TEXT 1941 7 "gnuplot" } {TEXT 1883 25 " format used for surface " }{TEXT 1941 7 "z(x,y) " } {TEXT 1883 77 "plots. As described in Sec. 4.10.5, the data are in th e form of a matrix of " }{TEXT 1941 1 "z" }{TEXT 1883 153 " values, wi th the rows of the matrix separated by blank lines. The first row corr esponds to time 1, with the place in the row corresponding to differen t " }{TEXT 1941 1 "x" }{TEXT 1883 34 " values. Row two contains all th e " }{TEXT 1941 1 "z" }{TEXT 1883 220 " values for time 2, and so\nfor th. As we see in the figures, the surface plot shows many ripples cor responding to oscillations of the string, but is not nearly as effecti ve a visualization as playing the animation below." }}{PARA 239 "" 0 " " {TEXT 1843 0 "" }}{PARA 248 "" 0 "" {TEXT 1862 23 " \+ " }}{PARA 304 "" 0 "" {TEXT 1967 35 "Animation of waves on a str ing. " }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 257 "" 0 "" {TEXT 1883 178 "In this section we indicate the steps needed to input \+ numerical data in surface plot format, and convert it into an animatio n. We start by forming a very long list of data named " }{TEXT 1941 4 "data" }{TEXT 1883 15 " from the file " }{TEXT 1941 12 "function.dat" }{TEXT 1883 70 " (Unix reads from present directory, Windows requires \+ full path name):" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{EXCHG {PARA 257 "> " 0 "" {MPLTEXT 1 1900 41 "data:=readdata('function.dat',1,floa t): " }{TEXT 1883 6 "# Unix" }}{PARA 257 "> " 0 "" {MPLTEXT 1 1900 97 "data:=readdata(\"C:/Documents and Settings/rubin/My Documents/Rubi n/Books/function.dat\",1,float): " }{TEXT 1883 27 "# Windows (note rev ersed /)" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1895 69 "data[1]; data[4]; \+ # examine some individual elements in list ``data''" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1895 60 "DATA:= array(data): # covert list ``data'' to array ``DATA''" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1895 18 "DATA[1]; D ATA[4];" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1895 0 "" }}}{PARA 257 "" 0 "" {TEXT 1883 27 "We next break up the array " }{TEXT 1941 4 "DATA" } {TEXT 1883 44 " into a list of sublists. The first sublist " }{TEXT 1941 8 "funct[1]" }{TEXT 1883 67 " corresponds to row 1 of the origina l matrix, the second sublist, " }{TEXT 1941 8 "funct[2]" }{TEXT 1883 55 " corresponds to row 2 of original matrix, and so forth:" }}{PARA 257 "" 0 "" {TEXT 1883 1 " " }}{EXCHG {PARA 258 "> " 0 "" {MPLTEXT 1 1895 116 "for t from 0 by 1 to 199\ndo func[t]:= [seq(DATA[i], i=101*t +1..101*(t+1))]: # for loop repeats for multiple t values" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1895 7 "end do:" }}}{EXCHG {PARA 258 "> " 0 " " {MPLTEXT 1 1895 0 "" }}}{PARA 257 "" 0 "" {TEXT 1883 38 "As a check, we print out the sublist " }{TEXT 1941 7 "func[1]" }{TEXT 1883 64 ". It is first row of input and will be first frame of the movie:" }} {PARA 257 "" 0 "" {TEXT 1883 0 "" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1895 35 "func[1]; # Print out individual row" }}{PARA 258 "> " 0 "" {MPLTEXT 1 1895 0 "" }}{PARA 257 "" 0 "" {TEXT 1883 116 "Now that we k now that the data look good, we plot several of the frames that we wil l put together to form the movie:" }}{PARA 257 "" 0 "" {TEXT 1883 0 " " }}{PARA 258 "> " 0 "" {MPLTEXT 1 1895 87 "listplot(func[0]); listplo t(func[10]); listplot(func[20]); # plot individ frames (rows)" }} {PARA 258 "> " 0 "" {MPLTEXT 1 1895 0 "" }}{PARA 257 "" 0 "" {TEXT 1883 32 "To create the movie, we use the " }{TEXT 1941 7 "display" } {TEXT 1883 113 " command . Yet, if we try to display several frames to gether, we just get multiple frames on top of each other: " }}{PARA 305 "" 0 "" {TEXT 1968 0 "" }}{PARA 257 "" 0 "" {TEXT 1883 0 "" }} {PARA 258 "> " 0 "" {MPLTEXT 1 1895 76 "display(listplot(func[0]), lis tplot(func[10]), listplot(func[20])); " }}{PARA 258 "> " 0 "" {MPLTEXT 1 1895 0 "" }}{PARA 257 "" 0 "" {TEXT 1883 75 "In order not t o display all of the frames on top of each other, we use the " }{TEXT 1941 3 "seq" }{TEXT 1883 73 " command to form a time ordered sequence \+ of plots. This is our animation." }}{PARA 257 "" 0 "" {TEXT 1883 0 "" }}{PARA 257 "> " 0 "" {MPLTEXT 1 1900 60 "display(seq(listplot(func[t] ), t=1..199), insequence=true); " }{TEXT 1883 15 "# Animated plot" }} {PARA 258 "> " 0 "" {MPLTEXT 1 1895 0 "" }}}{SECT 0 {PARA 292 "" 0 "" {TEXT 1936 3 " " }{TEXT 1937 5 "4.10 " }{TEXT 1938 23 " Key Words an d Concepts" }{TEXT 1935 1 " " }}{PARA 257 "" 0 "" {TEXT 1883 61 "abstr act data type abscissa animation contour plots" }{TEXT 1883 70 " dependent variable implicit plot independent variable \+ " }{TEXT 1883 17 "list matrix" }{TEXT 1883 40 " nonline ar functions ordinate " }{TEXT 1883 112 " set sequenc e surface plot 2D plot parametric plot polar plot \+ potential energy" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 306 " " 0 "" {TEXT 1969 41 "1. Does potential energy occur in nature?" }} {PARA 306 "" 0 "" {TEXT 1969 76 "2. Is there a reason for electric cha rge to occur always in integer values?" }}{PARA 306 "" 0 "" {TEXT 1969 66 "3. How does an abstract data type differ from an algebraic sy mbol?" }}{PARA 306 "" 0 "" {TEXT 1969 33 "4. How do you decide which i s an " }{TEXT 1970 11 "independent" }{TEXT 1969 16 " and which is a " }{TEXT 1970 9 "dependent" }{TEXT 1969 10 " variable?" }}{PARA 306 "" 0 "" {TEXT 1969 61 "5. When might it be a bad idea to use color in you r plotting?" }}{PARA 306 "" 0 "" {TEXT 1969 117 "6. List three ways in which you may change the apparent meaning of data by changing the way in which it is presented." }}{PARA 306 "" 0 "" {TEXT 1969 19 "7. What might be a " }{TEXT 1970 10 "dishonest " }{TEXT 1969 28 "way of prese nting your data?" }}{PARA 306 "" 0 "" {TEXT 1969 99 "8. Give examples \+ of the type of data that may be appropriate for 1D, 2D, 3D, and 4D vis ualizations." }}{PARA 306 "" 0 "" {TEXT 1969 52 "9. When are animation s a useful way to display data?" }}{PARA 306 "" 0 "" {TEXT 1969 103 "1 0. How, in a mathematical sense, does a phase-space (parametric) plot \+ differ from an ordinary 2D plot?" }}{PARA 306 "" 0 "" {TEXT 1969 52 "1 1. What is the difference between a list and a set?" }}{PARA 306 "" 0 "" {TEXT 1969 93 "12. How is a table of numerical data similar to, and different from, a mathematical function?" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}}{SECT 0 {PARA 307 "" 0 "" {TEXT 1971 1 " " }{TEXT 1972 2 " " }{TEXT 1973 5 "4.11 " }{TEXT 1974 18 " Further Exercises" }{TEXT 1971 0 "" }}{PARA 257 "" 0 "" {TEXT 1883 44 "1. On a single graph, dis play the function " }{XPPEDIT 18 0 "x^3*sin(x);" "6#*&%\"xG\"\"$-%$si nG6#F$\"\"\"" }{TEXT 1883 3 ", " }{XPPEDIT 18 0 "x^3*cos(x);" "6#*&% \"xG\"\"$-%$cosG6#F$\"\"\"" }{TEXT 1883 6 ", and " }{XPPEDIT 18 0 "x*l og(x);" "6#*&%\"xG\"\"\"-%$logG6#F$F%" }{TEXT 1883 64 ", each in a dif ferent color. Use an equal negative and positive " }{TEXT 1941 1 "x" } {TEXT 1883 93 " range, and pick that range to obtain the most interest ing comparison of the three functions." }}{PARA 257 "" 0 "" {TEXT 1883 0 "" }}{PARA 257 "" 0 "" {TEXT 1883 264 "2. A graphical approach \+ to solving equations plots the right and left hand sides of an equatio n as two separate functions, and then shows the solution to the equati on as the value of the abscissa at which the two functions intersect. \+ In other words, a solution of " }{TEXT 1941 11 " f(x)=g(x) " }{TEXT 1883 26 "occurs when the graphs of " }{TEXT 1941 13 "f(x) versus x" } {TEXT 1883 25 " intercepts the graph of " }{TEXT 1941 15 "g(x) versus \+ x. " }{TEXT 1883 129 "Use Maple's ability to plot several functions in the same graph to determine the approximate solutions of the followin g equations" }}{PARA 257 "" 0 "" {TEXT 1883 37 " \+ a) " }{XPPEDIT 18 0 "sin(x) = x^2;" "6#/-%$sinG6#%\"xG*$F' \"\"#" }{TEXT 1883 1 "," }}{PARA 257 "" 0 "" {TEXT 1883 37 " \+ b) " }{XPPEDIT 18 0 "x^2+6*x+1 = 0;" "6#/,(*$% \"xG\"\"#\"\"\"*&\"\"'F(F&F(F(F(F(\"\"!" }{TEXT 1883 1 "," }}{PARA 257 "" 0 "" {TEXT 1883 36 " c) " } {XPPEDIT 18 0 "H^3-9*H^2+4 = 0;" "6#/,(*$%\"HG\"\"$\"\"\"*&\"\"*F(*$F& \"\"#F(!\"\"\"\"%F(\"\"!" }{TEXT 1883 1 "." }}{PARA 257 "" 0 "" {TEXT 1883 0 "" }}{PARA 257 "" 0 "" {TEXT 1883 97 "3. Do a graphical experim ent in which you prove to yourself these very useful mathematical fact s:" }}{PARA 257 "" 0 "" {TEXT 1883 45 " a) the exponent " }{XPPEDIT 18 0 "exp(x);" "6#-%$expG6#%\"xG" }{TEXT 1883 29 " grows faster than any power " }{XPPEDIT 18 0 "x^n;" "6#)%\"x G%\"nG" }{TEXT 1883 1 "," }}{PARA 257 "" 0 "" {TEXT 1883 46 " \+ b) the logarithm " }{XPPEDIT 18 0 "ln(x);" "6#-%#l nG6#%\"xG" }{TEXT 1883 29 " grows slower than any power " }{XPPEDIT 18 0 "x^n;" "6#)%\"xG%\"nG" }{TEXT 1883 1 "." }}{PARA 257 "" 0 "" {TEXT 1883 0 "" }}{PARA 257 "" 0 "" {TEXT 1883 1 " " }{TEXT 1941 6 "Hi nt: " }{TEXT 1883 43 "To avoid overflow problems with very large " } {TEXT 1941 1 "x" }{TEXT 1883 44 " values, you may want to make semilog plots." }}{PARA 257 "" 0 "" {TEXT 1883 0 "" }}{PARA 257 "" 0 "" {TEXT 1883 50 "4. Do a graphical experiment to find the value of " } {TEXT 1941 1 "n" }{TEXT 1883 36 " for which these equations are true: " }}{PARA 257 "" 0 "" {TEXT 1883 30 " a) " } {XPPEDIT 18 0 "n*sin(2*x) = sin(x)*cos(x);" "6#/*&%\"nG\"\"\"-%$sinG6# *&\"\"#F&%\"xGF&F&*&-F(6#F,F&-%$cosG6#F,F&" }{TEXT 1883 1 "," }}{PARA 257 "" 0 "" {TEXT 1883 30 " b) " }{XPPEDIT 18 0 "n*cos(x)^2 = 1+cos(2*x);" "6#/*&%\"nG\"\"\"*$-%$cosG6#%\"xG\"\"# F&,&F&F&-F)6#*&F,F&F+F&F&" }{TEXT 1883 1 " " }}{PARA 257 "" 0 "" {TEXT 1883 209 "5. If two tones very close in frequency are played tog ether, your ear hears them as a single tone with an oscillating amplit ude. Make plots as a function of time of the results of adding the two sine functions" }}{PARA 308 "" 0 "" {XPPEDIT 18 0 "sin(100*t)+sin(b*t );" "6#,&-%$sinG6#*&\"$+\"\"\"\"%\"tGF)F)-F%6#*&%\"bGF)F*F)F)" }{TEXT 1975 0 "" }}{PARA 257 "" 0 "" {TEXT 1883 26 "Make a series of plot for " }{TEXT 1941 1 "b" }{TEXT 1883 13 " in the range" }}{PARA 309 "" 0 " " {TEXT 1976 14 " 90 < b < 100." }}{PARA 257 "" 0 "" {TEXT 1883 45 "Ma ke sure to plot for a long enough range of " }{TEXT 1941 1 "t" }{TEXT 1883 62 " values to see at least three cycles of any periodic behavior ." }}{PARA 257 "" 0 "" {TEXT 1883 0 "" }}{PARA 257 "" 0 "" {TEXT 1883 50 "6. Here are nine measurements given in the form (" }{TEXT 1941 6 "x, y):" }{TEXT 1883 0 "" }}{PARA 257 "" 0 "" {TEXT 1883 1 " " }} {PARA 309 "" 0 "" {TEXT 1976 116 "(0, 10.6), (25, 16.0), (50, 45.0), (75, 83.5), (100, 52.8), (125, 19.9), (150, 10.8), (175, 8.25), \+ (200, 4.7)" }}{PARA 310 "" 0 "" {TEXT 1977 0 "" }}{PARA 257 "" 0 "" {TEXT 1941 1 " " }{TEXT 1883 33 "Make a plot of these data points." }} {PARA 257 "" 0 "" {TEXT 1883 0 "" }}{PARA 257 "" 0 "" {TEXT 1883 203 " 7. The orbits of planets and comets are known to be conic sections. Co nic sections are the 2-D curves formed when a cone is cut (sectioned) \+ by a plane, and are given by the parametric equations in which " } {TEXT 1941 1 "s" }{TEXT 1883 18 " is the parameter:" }}{PARA 257 "" 0 "" {TEXT 1883 0 "" }}{PARA 257 "" 0 "" {TEXT 1883 44 " \+ a) hyperbola: " }{XPPEDIT 18 0 "(x(s), y(s)) = (4*cosh(s ), 1.4*sinh(s));" "6#/6$-%\"xG6#%\"sG-%\"yG6#F(6$*&\"\"%\"\"\"-%%coshG 6#F(F/*&-%&FloatG6$\"#9!\"\"F/-%%sinhG6#F(F/" }{TEXT 1883 0 "" }} {PARA 257 "" 0 "" {TEXT 1883 41 " b) ellip se: " }{XPPEDIT 18 0 "(x(s), y(s)) = (4*cos(s), 1.4*sin(s));" "6#/6$-% \"xG6#%\"sG-%\"yG6#F(6$*&\"\"%\"\"\"-%$cosG6#F(F/*&-%&FloatG6$\"#9!\" \"F/-%$sinG6#F(F/" }{TEXT 1883 0 "" }}{PARA 257 "" 0 "" {TEXT 1883 42 " c) parabola: " }{XPPEDIT 18 0 "(x(s), y( s)) = (s*cos(theta)-s^2*sin(theta), s^2*cos(theta)+s*sin(theta));" "6# /6$-%\"xG6#%\"sG-%\"yG6#F(6$,&*&F(\"\"\"-%$cosG6#%&thetaGF/F/*&F(\"\"# -%$sinG6#F3F/!\"\",&*&F(F5-F16#F3F/F/*&F(F/-F76#F3F/F/" }{TEXT 1883 3 ", " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT 1883 22 " = arbitrar y parameter" }}{PARA 257 "" 0 "" {TEXT 1883 0 "" }}{PARA 257 "" 0 "" {TEXT 1883 98 "Make parametric plots of these conic sections. Cover as much range as is needed for the parameter " }{TEXT 1941 1 "s" }{TEXT 1883 40 " in order to obtain the familiar shapes." }}{PARA 257 "" 0 " " {TEXT 1883 0 "" }}{PARA 257 "" 0 "" {TEXT 1883 71 "8. In polar coor dinates the conic section is described by the equation" }}{PARA 308 " " 0 "" {XPPEDIT 18 0 "alpha/r = 1+epsilon*cos(theta);" "6#/*&%&alphaG \"\"\"%\"rG!\"\",&F&F&*&%(epsilonGF&-%$cosG6#%&thetaGF&F&" }{TEXT 1975 0 "" }}{PARA 257 "" 0 "" {TEXT 1883 6 "where " }{XPPEDIT 18 0 "ep silon;" "6#%(epsilonG" }{TEXT 1883 25 " is the eccentricity and " } {XPPEDIT 18 0 "2*alpha;" "6#*&\"\"#\"\"\"%&alphaGF%" }{TEXT 1883 61 " \+ is the ltaus rectum of the orbit. An ellipse occurs when 0< " } {XPPEDIT 18 0 "epsilon" "6#%(epsilonG" }{TEXT 1883 21 " <1, a hyperbol a for " }{XPPEDIT 18 0 "epsilon" "6#%(epsilonG" }{TEXT 1883 24 " >1, a nd a parabola for " }{XPPEDIT 18 0 "epsilon" "6#%(epsilonG" }{TEXT 1883 140 " =1. Make polar plots of these three kinds of orbits based o n the polar equation of the conic section. Try various values for the \+ parameter " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT 1883 63 " (it \+ is inversely proportional to the energy of the planet). " }}{PARA 257 "" 0 "" {TEXT 1883 0 "" }}{PARA 257 "" 0 "" {TEXT 1883 47 "9. Make a surface plot of the Yukawa potential:" }}{PARA 309 "" 0 "" {XPPEDIT 18 0 "V(x,y) = exp(-r)/r;" "6#/-%\"VG6$%\"xG%\"yG*&-%$expG6#, $%\"rG!\"\"\"\"\"F.F/" }{XPPEDIT 18 0 "cos(x/r);" "6#-%$cosG6#*&%\"xG \"\"\"%\"rG!\"\"" }{TEXT 1976 5 ", " }{XPPEDIT 18 0 "r = sqrt(x^2+y ^2);" "6#/%\"rG-%%sqrtG6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF+F," }{TEXT 1976 1 "." }}{PARA 257 "" 0 "" {TEXT 1883 54 "10. Make a contour plot of th is same Yukawa potential." }}{PARA 257 "" 0 "" {TEXT 1883 0 "" }} {PARA 257 "" 0 "" {TEXT 1883 48 "11. A standing wave is described by t he equation" }}{PARA 257 "" 0 "" {TEXT 1883 0 "" }}{PARA 308 "" 0 "" {XPPEDIT 18 0 "y(x,t) = sin(10*x)*cos(12*t);" "6#/-%\"yG6$%\"xG%\"tG*& -%$sinG6#*&\"#5\"\"\"F'F/F/-%$cosG6#*&\"#7F/F(F/F/" }{TEXT 1975 0 "" } }{PARA 257 "" 0 "" {TEXT 1883 0 "" }}{PARA 257 "" 0 "" {TEXT 1883 6 "w here " }{TEXT 1941 1 "x" }{TEXT 1883 35 " is the distance along the st ring, " }{TEXT 1941 1 "y" }{TEXT 1883 39 " is the height of the distur bance, and " }{TEXT 1941 1 "t" }{TEXT 1883 14 " is the time. " }} {PARA 257 "" 0 "" {TEXT 1883 66 " a) Create an d animation of this function." }}{PARA 257 "" 0 "" {TEXT 1883 142 " \+ b) Create a surface plot of this function, and s ee if you agree with us that it is not as revealing as the animation. " }}{PARA 257 "" 0 "" {TEXT 1883 0 "" }}{PARA 257 "" 0 "" {TEXT 1883 49 "12. A traveling wave is described by the equation" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 311 "" 0 "" {XPPEDIT 531 0 "y(x,t) = sin( 10*x-12*t);" "6#/-%\"yG6$%\"xG%\"tG-%$sinG6#,&*&\"#5\"\"\"F'F/F/*&\"#7 F/F(F/!\"\"" }{TEXT 1978 0 "" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }} {PARA 257 "" 0 "" {TEXT 1883 6 "where " }{TEXT 1941 1 "x" }{TEXT 1883 35 " is the distance along the string, " }{TEXT 1941 1 "y" }{TEXT 1883 39 " is the height of the disturbance, and " }{TEXT 1941 1 "t" } {TEXT 1883 14 " is the time. " }}{PARA 257 "" 0 "" {TEXT 1883 65 " \+ a) Create an animation of this function." }} {PARA 257 "" 0 "" {TEXT 1883 142 " b) Create a surface plot of this function, and see if you agree with us that it i s not as revealing as the animation." }}{PARA 257 "" 0 "" {TEXT 1883 0 "" }}{PARA 257 "" 0 "" {TEXT 1883 202 "13. Give an example or two of the type of function(s) that would be visualized best with each of th e following plots:\n a) 2-D plot b) 3-D plot c) multifunction plo t d) parametric plot e) animation " }}{PARA 257 "" 0 "" {TEXT 1883 21 " f) 3-D animation" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 239 "" 0 "" {TEXT 1948 3 "14." }{TEXT 1843 1 " " }{TEXT 1948 44 "Expla in in just a few words what is meant by" }{TEXT 1843 0 "" }}{PARA 239 "" 0 "" {TEXT 1843 5 " " }{TEXT 1948 24 "a) an abstract data type " }{TEXT 1843 0 "" }}{PARA 239 "" 0 "" {TEXT 1843 5 " " }{TEXT 1948 35 "b) a parametric or phase space plot" }{TEXT 1843 0 "" }} {PARA 257 "" 0 "" {TEXT 1883 38 " c) a function of three variable s" }}{PARA 257 "" 0 "" {TEXT 1883 33 " d) a set of three variable s" }}{PARA 257 "" 0 "" {TEXT 1883 34 " e) a list of three variabl es" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 239 "" 0 "" {TEXT 1948 21 "15. Plot the function" }{TEXT 1843 2 " " }{XPPEDIT 530 0 "f( x)=sec(x) +4" "6#/-%\"fG6#%\"xG,&-%$secG6#F'\"\"\"\"\"%F," }{TEXT 1948 19 " over the interval " }{XPPEDIT 528 0 "[0, 4*Pi];" "6#7$\"\"!* &\"\"%\"\"\"%#PiGF'" }{TEXT 1843 3 " . " }{TEXT 1948 142 "Since Maple' s automatic scaling does not work well here, you will need to specify \+ a range for the ordinates to obtain a useful visualization. " }{TEXT 1843 0 "" }}{PARA 239 "" 0 "" {TEXT 1843 0 "" }}{PARA 312 "" 0 "" {TEXT 1979 139 "16.Given the points (1, 0.53) , (1.5, 0.65) , (2, 0. 91) , (2.5 , 0.95) and (3, 1.10 ). Create a plot containing these po ints as well as" }}{PARA 313 "" 0 "" {TEXT 1980 14 "the functions " }} {PARA 239 "" 0 "" {TEXT 1843 10 " " }{XPPEDIT 527 0 "a(x) = s in(x/2);" "6#/-%\"aG6#%\"xG-%$sinG6#*&F'\"\"\"\"\"#!\"\"" }{TEXT 1948 19 ", " }{XPPEDIT 525 0 "b(x) = x^2/5;" "6#/-%\"bG6#% \"xG*&F'\"\"#\"\"&!\"\"" }{TEXT 1843 1 "," }}{PARA 314 "" 0 "" {TEXT 1981 58 "and thereby determine which function fits the data better?" } }}{PARA 315 "" 0 "" {TEXT 1982 0 "" }}{PARA 315 "" 0 "" {TEXT 1982 0 " " }}{PARA 315 "" 0 "" {TEXT 1982 0 "" }}{PARA 316 "" 0 "" {TEXT -1 0 " " }}}{MARK "1 2 0" 1 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 1 0 2 33 52 1 }