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-1 384 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 385 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 386 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Norm al" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 } {PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "Author" 0 19 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 8 8 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 72 "Maple Illustrations of Ke y Topics from Vector and Multivariable Calculus" }}{PARA 19 "" 0 "" {TEXT -1 20 "J. A. Ziegler, Ph.D." }}{PARA 262 "" 0 "" {TEXT -1 37 "So uthern Polytechnic State University" }}{PARA 264 "" 0 "" {TEXT -1 17 " jziegler@spsu.edu" }}{PARA 263 "" 0 "" {TEXT -1 44 "http://www2.SPSU.e du/math/ziegler/index.html" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 266 "" 0 "" {TEXT -1 36 "With corrections of November 5, 2001" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 385 615 "Learning vector and multivariable calculus has until r ecently been impeded by the difficulty of visualizing the structures a nd processes being studied. Maple's graphics capabilities can largely remove this impediment. This is a revolutionary development, but its use is spreading more slowly than might have been expected. One of t he reasons for this is that, in spite of an extensive Maple literature , standard procedures have been lacking for creating those illustratio ns which everyone teaching the subject would find most useful. The pu rpose of this paper is to provide standard methods for some of these. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 386 226 "The first part is the easiest and is presented in detai l. The second part has somewhat more complicated procedures. In each case the Maple code is given fully. It is available for academic use from the web site given above." }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 24 "PART I: VECTOR CALCULUS" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "restart:with(plots):with(plottools):with(linalg):" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 37 "Vector Calculus and Parametric Curves" }} {EXCHG {PARA 4 "" 0 "" {TEXT -1 14 "Two Dimensions" }}{PARA 5 "" 0 "" {TEXT -1 33 "A Particle Moving Along the Curve" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "To plot a parametric curve with parameter t in the \+ interval [a,b], the parabola x = t, y = t^2 for t in [" }{XPPEDIT 18 0 "-2,4;" "6$,$\"\"#!\"\"\"\"%" }{TEXT -1 60 "] will be used as an exa mple. First, define the end-points." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "a:=-2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "b: =4:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Next, in Maple, the domain of t is written thus:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "T _int:=a..b:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Then draw the curv e. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "c:=plot([t,t^2,t=T_i nt],color=black):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "Now define t he position of a point on the curve via the vector-valued function, " }{TEXT 256 1 "R" }{TEXT -1 59 ", and draw a suitable line segment to r epresent the vector." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "R:= t->[t,t^2];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 385 "The effects of (1 ) changing the orientation of the curve, and (2) changing its paramete rization can easily be seen by using a small colored disk to mark the \+ point and act as a \"particle\" which will move along the curve. But \+ first, the behavior for the present parameterization should be observe d. The easiest way to do this is to define a function, p, from t to t he disk centered at " }{TEXT 266 1 "R" }{TEXT -1 84 "(t). This is an i nteresting extension of \"function\" for most students at this point. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "p:=t->disk(R(t),0.3,col or=magenta):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Choose n, the num ber of pictures that will constitute the animation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "n:=30:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "Then calculate the values of t from a to b at which these pictu res will be taken." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "S:=se q(a+i*(b-a)/n,i=0..n):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Each pi cture will show the curve and the vector " }{TEXT 257 1 "R" }{TEXT -1 40 "(t). Make a sequence of these pictures." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "pic:=[seq(display(c,p(s)),s=S)]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Now display them in sequence. This is th e animation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "display(pic ,insequence=true,scaling=constrained);" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 25 "Changing the Orientation " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 232 "Changing the orientation of the curve is easy. Since th e sequence S begins with a and ends with b, to reverse this, t should \+ be replaced, in R, with b + a - t. Then the sequence of values \"seen \" by R begins with b and ends with a. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "R_rev:=t->[b+a-t,(b+a-t)^2];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "p_rev:=t->disk(R_rev(t),0.3,color=cyan):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "pic1:=seq(display(c,p_rev(s) ),s=S):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "display(pic1,ins equence=true,scaling=constrained);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "If desired, these may be viewed in succession by displaying bot h pic aand pic1, in that order." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "display(pic,pic1,insequence=true,scaling=constrained);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "To view them simultaneously, each \+ individual picture should contain the curve and both points." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "pic2:=seq(display(c,p(s),p_r ev(s)),s=S):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "display(pic 2,insequence=true,scaling=constrained);" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 29 "Changing the Parameterization" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 175 "The interest here is to compare the motion produced by d ifferent parameterizations. The curve and the sequence of times at wh ich the pictures are made remains the same. Let " }{TEXT 258 2 "R2" } {TEXT -1 66 " be the second parameterization of the curve. As an exam ple, let " }{XPPEDIT 18 0 "x = t^3;" "6#/%\"xG*$%\"tG\"\"$" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = t^6;" "6#/%\"yG*$%\"tG\"\"'" }{TEXT -1 89 ". So that the (x, y)-end points will be the same as before, the i nterval for t must be " }{XPPEDIT 18 0 "[-2^(1/3), 4^(1/3)];" "6#7$,$ )\"\"#*&\"\"\"\"\"\"\"\"$!\"\"F+)\"\"%*&\"\"\"F)\"\"$F+" }{TEXT -1 0 " " }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 264 9 "Rema rk: " }{TEXT -1 192 "In Maple, writing roots other than square roots \+ is awkward. In writing text, rational exponents must be used, but in \+ writing commands requiring the nth root of the real number x, the comm and " }{TEXT 265 9 "surd(x,n)" }{TEXT -1 176 " is necessary, when x is negative, if real values are to be obtained. Americans unfamiliar wi th British terminology (Maple, of course, is Canadian) are unlikely to guess this." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "a2:=surd(-2 ,3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "b2:=surd(4,3):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "R2:=t->[t^3,t^6];" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "p2:=t->disk(R2(t),0.3,color= blue):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 201 "This time there will a gain be n pictures taken of the two motions, but these will find the p articles at different positions. In making the sequence of pictures, \+ this difficulty is handled as follows: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "pic3:=[seq(display(c,p2(a2+i*(b2-a2)/n),p(a+i*(b-a)/n )),i=0..n)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "display(pic 3,insequence=true,scaling=constrained);" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 16 "Three Dimensions" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 " To apply these procedures to curves in three dimensions is simple: jus t replace " }{TEXT 259 4 "plot" }{TEXT -1 4 " by " }{TEXT 260 10 "spac ecurve" }{TEXT -1 5 " and " }{TEXT 261 4 "disk" }{TEXT -1 4 " by " } {TEXT 262 6 "sphere" }{TEXT -1 48 ". As an example, consider the elli ptical helix " }}{PARA 0 "" 0 "" {TEXT -1 33 "x = 2 cos(t), y = sin(t) , z = t/(" }{XPPEDIT 18 0 "2*Pi;" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 15 ") for t in [0, " }{XPPEDIT 18 0 "4*Pi;" "6#*&\"\"%\"\"\"%#PiGF%" } {TEXT -1 2 "]." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "a:=0:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "b:=4*Pi:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 12 "T_int:=a..b:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "c:=spacecurve([2*cos(t),sin(t),t/(2*Pi)],t=T_int,colo r=black,thickness=2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "R: =t->[2*cos(t),sin(t),t/(2*Pi)];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "p:=t->sphere(R(t),0.2,color=red,style=patchnogrid):" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "The inclusion of the option " } {TEXT 263 19 "style = patchnogrid" }{TEXT -1 133 " is important. Othe rwise, Maple's default plotting style will cover the sphere with a net work of gridlines which will turn it black." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "n:=40:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "S:=seq(a+i*(b-a)/n,i=0..n):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "pic:=seq(display(c,p(s)),s=S):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "display(pic,insequence=true,scaling=constrained,axes= normal);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 171 "The procedures given above for illustrating the effects of changing the orientation of a c urve or its parameterization may be used unchanged in the three dimens ional case." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 4 "" 0 " " {TEXT -1 75 "The Use of Line Segments to Represent the Velocity and \+ Acceleration Vectors" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 136 "In discu ssing parametric curves in terms of vector valued functions, it is obv iously desireable to be able to show the position vector, " }{TEXT 267 1 "R" }{TEXT -1 50 ", and also the velocity and acceleration vecto rs, " }{TEXT 268 1 "V" }{TEXT -1 5 " and " }{TEXT 269 1 "A" }{TEXT -1 162 ". It is easiest to use colored line segments for this purpose an d, as with the \"particle\", above, to produce them via \"graphical ob ject valued\" functions of t. " }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 16 "Three Dimensions" }}{PARA 5 "" 0 "" {TEXT -1 27 "Showing the Posit ion Vector" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Continuing with the previous example, define the \"R-vector\", via" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 52 "Rvec:=t->line([0,0,0],R(t),color=black,thickne ss=2):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "Now simply include it i n the previous sequence of pictures and display the result." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "pic:=seq(display(c,p(s),Rvec (s)),s=S):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "display(pic,i nsequence=true,scaling=constrained,axes=normal);" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 98 "The time required to compute the animation will be noticeably less if the \"particle\" is omitted. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "To add the velocity vector to the pictures, the \+ simplest way is this:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dR :=map(diff,R(t),t):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "V:=u napply(dR,t):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "Vvec:=t->l ine(R(t),R(t)+V(t),color=blue,thickness=3):" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 37 "The need to use the peculiarly named " }{TEXT 270 7 "un apply" }{TEXT -1 172 " command to turn dR into a function is one of th e quirks of Maple. Now include this in the sequence of pictures, this time omitting the particle, and display the result. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "pic:=seq(display(c,Rvec(s),Vvec(s)) ,s=S):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "display(pic,inseq uence=true,scaling=constrained,axes=normal);" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 64 "This process may be repeated to produce the acceleratio n vector." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dV:=map(diff,V (t),t):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "A:=unapply(dV,t) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "Avec:=t->line(R(t),R(t )+A(t),color=red,thickness=3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "pic:=seq(display(c,Rvec(s),Vvec(s),Avec(s)),s=S):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "display(pic,insequence=true,scaling =constrained,axes=normal);" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 14 "Two Dimensions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Except in the definitions of " }{TEXT 271 1 "R" }{TEXT -1 97 ", c, and Rvec, where the origin must now be written as [0,0], there are no other changes to make." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 45 "The Acceleration and the Unit Vectors T a nd N" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "T o see that, in three dimensions, " }{TEXT 272 1 "A" }{TEXT -1 30 " lie s in the plane defined by " }{TEXT 273 1 "T" }{TEXT -1 31 ", the unit \+ tangent vector, and " }{TEXT 274 1 "N" }{TEXT -1 64 ", the principal u nit normal vector, the process for calculating " }{TEXT 275 1 "V" } {TEXT -1 54 " need only be slightly extended. To find the norm of " } {TEXT 276 2 "V," }{TEXT -1 45 " it is best not to use the build-in com mand, " }{TEXT 277 4 "norm" }{TEXT -1 49 ", but to calculate it from t he definition, using " }{TEXT 278 12 "innerproduct" }{TEXT -1 12 " ins tead of " }{TEXT 279 10 "dotproduct" }{TEXT -1 82 " to avoid the incon venience of having to declare that t is a real variable. Thus," }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "ndR:=simplify(sqrt(innerprod(dR,dR) )):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "T:=unapply(map(x->x/ ndR,dR),t):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "Tvec:=t->lin e(R(t),R(t)+T(t),color=cyan,thickness=4):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Having turned the derivative of " }{TEXT 280 1 "T" } {TEXT -1 53 " into a function, the process is repeated to produce " } {TEXT 281 1 "N" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dT:=map(diff,T(t),t):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "ndT:=simplify(sqrt(innerprod(dT,dT))):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "N:=unapply(map(x->x/ndT,dT),t):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "Nvec:=t->line(R(t),R(t)+N(t),color= maroon,thickness=4):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Include t he curve, c, and the vectors " }{TEXT 282 1 "R" }{TEXT -1 2 ", " } {TEXT 283 1 "A" }{TEXT -1 2 ", " }{TEXT 284 1 "T" }{TEXT -1 6 ", and \+ " }{TEXT 285 1 "N" }{TEXT -1 42 " in each picture, then display the re sult." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "pic:=seq(display(c ,Rvec(s),Avec(s),Tvec(s),Nvec(s)),s=S):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "display(pic,insequence=true,scaling=constrained,axes= normal);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "In order to show more clearly that " }{TEXT 293 1 "A" }{TEXT -1 30 " lies in the plane cont aining " }{TEXT 294 1 "T" }{TEXT -1 5 " and " }{TEXT 295 1 "N" }{TEXT -1 122 ", a picture of a portion of this plane may be included in the \+ following way provided it is never vertical for t in [a, b]." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "The following equation defines the plane containing " }{TEXT 286 1 "T" }{TEXT -1 5 " and " }{TEXT 287 1 "N" }{TEXT -1 4 " at " }{TEXT 292 1 "R" }{TEXT -1 4 "(t)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "eq:=innerprod(crossprod(T(t),N(t)), ([x,y,z]-R(t)))=0:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 129 "This is so lved for z to obtain an expression for this plane in x and y at t. Th is, in turn, is transformed into a function of t." }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 32 "Pln:=simplify(solve(eq,z),trig):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Plnfunc:=unapply(Pln,t):" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "The \"object valued\" function, TN , has as its value at t the picture of the plane containing " }{TEXT 288 1 "T" }{TEXT -1 5 " and " }{TEXT 289 2 "N " }{TEXT -1 2 "at" } {TEXT 290 2 " R" }{TEXT -1 125 "(t). This is then included in each pi cture in the sequence of pictures of the animation. (R(t)[1] means th e x-component of " }{TEXT 291 1 "R" }{TEXT -1 10 "(t), etc.)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "TN:=t->plot3d(Plnfunc(t),x= R(t)[1]-1..R(t)[1]+1,y=R(t)[2]-1..R(t)[2]+1,color=yellow,style=patchno grid):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "pic1:=seq(display (c,Rvec(s),Avec(s),Tvec(s),Nvec(s),TN(s)),s=S):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 62 "display(pic1,insequence=true,scaling=constrain ed,axes=normal);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {EXCHG {PARA 3 "" 0 "" {TEXT -1 32 "PART II: MULTIVARIABLE CALCULUS" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 37 "Does f(x, y) have a Limit at ( a, b)? " }}{EXCHG {PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 301 0 "" } {TEXT 296 11 "Definition:" }{TEXT 297 3 " " }{TEXT 298 78 "Let f be \+ a function defined in at least some deleted neighborhood of (a,b). \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 303 31 " W e say that" }{TEXT 302 3 " " }{XPPEDIT 18 0 "Limit(f(x,y) = L,(x, y) = (a, b));" "6#-%&LimitG6$/-%\"fG6$%\"xG%\"yG%\"LG/6$F*F+6$%\"aG%\"bG " }{TEXT -1 3 " " }{TEXT 299 0 "" }{TEXT 300 23 "if and only if for \+ each" }{TEXT -1 6 " " }{TEXT 304 0 "" }{XPPEDIT 18 0 "epsilon;" " 6#%(epsilonG" }{TEXT -1 1 " " }{TEXT 312 6 "> 0 " }}{PARA 0 "" 0 "" {TEXT 305 32 " there is a " }{TEXT 314 1 " " } {XPPEDIT 18 0 "delta;" "6#%&deltaG" }{TEXT -1 1 " " }{TEXT 313 6 "> 0 \+ " }{TEXT 306 12 "such that if" }{TEXT 311 7 " 0 <" }{TEXT 310 1 " " }{TEXT -1 1 " " }{TEXT 309 0 "" }{XPPEDIT 18 0 "abs(abs((x, y)-(a, \+ b)));" "6#-%$absG6#-F$6#,&6$%\"xG%\"yG\"\"\"6$%\"aG%\"bG!\"\"" }{TEXT -1 3 " < " }{XPPEDIT 18 0 "delta;" "6#%&deltaG" }{TEXT -1 3 ", " } {TEXT 307 0 "" }{TEXT 308 4 "then" }{TEXT 315 3 " " }{XPPEDIT 18 0 " abs(f(x,y)-L);" "6#-%$absG6#,&-%\"fG6$%\"xG%\"yG\"\"\"%\"LG!\"\"" } {TEXT -1 3 " < " }{XPPEDIT 18 0 "epsilon;" "6#%(epsilonG" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 265 "" 0 "" {TEXT -1 178 "Interpreting this graphically, Maple may be used to see if it is plau sible that a given function has a limit at (a, b). The idea is to d raw the surface and the planes z = L + " }{TEXT 316 0 "" }{XPPEDIT 18 0 "epsilon;" "6#%(epsilonG" }{TEXT -1 13 " and z = L - " }{TEXT 317 0 "" }{XPPEDIT 18 0 "epsilon;" "6#%(epsilonG" }{TEXT -1 25 " over the di sk of radius " }{XPPEDIT 18 0 "delta;" "6#%&deltaG" }{TEXT -1 75 " cen tered at (a, b). Geometrically, the definition says that if, for each " }{TEXT 320 0 "" }{XPPEDIT 18 0 "epsilon;" "6#%(epsilonG" }{TEXT -1 36 " > 0, there is a sufficiently small " }{XPPEDIT 18 0 "delta" "6#%& deltaG" }{TEXT -1 53 " > 0 such that the surface lies entirely between the " }{TEXT 322 0 "" }{TEXT -1 1 " " }{XPPEDIT 18 0 "epsilon;" "6#%( epsilonG" }{TEXT 321 0 "" }{TEXT -1 171 "-planes, except perhaps exact ly at (a,b), then the limit exists and is equal to L. For simplicity, in the illustrations the point (0, 0) and the value L = 0 will be use d." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "restart:with(plots):w ith(plottools):with(linalg):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 323 10 " Example 1." }{TEXT -1 5 " " }{XPPEDIT 18 0 "f1(x,y) = (xy+y^3)/(x^ 2+y^2);" "6#/-%#f1G6$%\"xG%\"yG*&,&%#xyG\"\"\"*$F(\"\"$F,F,,&*$F'\"\"# F,*$F(\"\"#F,!\"\"" }{TEXT -1 3 " ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f1:=(x*y+y^3)/(x^2+y^2):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "To begin with, choose " }{XPPEDIT 18 0 "epsilon = 1/10;" "6#/%(epsilonG*&\"\"\"\"\"\"\"#5!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "delta = 1/10;" "6#/%&deltaG*&\"\"\"\"\"\"\"#5!\"\"" }{TEXT -1 66 " . These may subsequently be changed and a new picture produced. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "eps:=10^(-1);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "del:=10^(-1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "Make a picture of the surface over the di sk of radius " }{XPPEDIT 18 0 "delta;" "6#%&deltaG" }{TEXT -1 43 " cen tered at (0, 0) and another of the two " }{TEXT 319 0 "" }{TEXT -1 1 " " }{XPPEDIT 18 0 "epsilon;" "6#%(epsilonG" }{TEXT 318 0 "" }{TEXT -1 28 "-planes. Then display them." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "p1:=plot3d(f1,x=-del..del,y=-sqrt(del^2-x^2)..sqrt(de l^2-x^2),numpoints=50^2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "Epsilon:=plot3d(\{eps,-eps\},x=-del..del,y=-sqrt(del^2-x^2)..sqrt (del^2-x^2),color=yellow,style=patchnogrid):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "display(p1,Epsilon,style=patchcontour,axes=frame ,orientation=[90,1]);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Rotate this. \+ Clearly, for the chosen value of " }{XPPEDIT 18 0 "delta;" "6#%&delta G" }{TEXT -1 49 ", the surface does not lie entirely between the " } {XPPEDIT 18 0 "epsilon;" "6#%(epsilonG" }{TEXT -1 37 "-planes. Choose a smaller value for " }{XPPEDIT 18 0 "delta;" "6#%&deltaG" }{TEXT -1 51 ". A few repetitions, each time choosing a smaller " }{XPPEDIT 18 0 "delta;" "6#%&deltaG" }{TEXT -1 41 ", make it plausible that no limi t exists." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 324 10 "Example 2." }{TEXT -1 15 " f2(x,y) = " }{XPPEDIT 18 0 "x^2*y/(x^2+y^2);" "6#*(%\"xG\" \"#%\"yG\"\"\",&*$F$\"\"#F'*$F&\"\"#F'!\"\"" }{TEXT -1 2 " ." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f2:=x^2*y/(x^2+y^2):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "eps:=10^(-1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "del:=10^(-1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "p2:=plot3d(f2,x=-del..del,y=-del..del,numpoin ts=50^2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "display(p2,sty le=patchcontour,axes=frame);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "H ere, inspection of the graph suggests that there may be a limit; namel y, L = 0. Again, draw the " }{XPPEDIT 18 0 "epsilon;" "6#%(epsilonG" }{TEXT -1 32 "-planes and consider the result." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 71 "eu:=plot3d(eps,x=-del..del,y=-del..del,color=y ellow,style=patchnogrid):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "el:=plot3d(-eps,x=-del..del,y=-del..del,color=yellow,style=patchno grid):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "display(p2,eu,el, style=patchcontour,axes=frame);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "This continues to be e ncouraging. Change eps to " }{XPPEDIT 18 0 "10^(-2);" "6#)\"#5,$\"\"# !\"\"" }{TEXT -1 69 ", then, after looking at the final display, try \+ to find a value for " }{XPPEDIT 18 0 "delta;" "6#%&deltaG" }{TEXT -1 68 " which will, once again, place the graph of f2 entirely between th e " }{XPPEDIT 18 0 "epsilon;" "6#%(epsilonG" }{TEXT -1 53 "-planes. R epeating this several times, first making " }{XPPEDIT 18 0 "epsilon;" "6#%(epsilonG" }{TEXT -1 15 " smaller, then " }{XPPEDIT 18 0 "delta;" "6#%&deltaG" }{TEXT -1 33 ", suggests strongly that L = 0. " }}{PARA 0 "" 0 "" {TEXT -1 75 "The Maple \"exploration\" has again suggested w hat one should try to prove. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 37 " The geometry of par tial derivatives." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart :with(plots):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "To show the geom etric meaning of " }{XPPEDIT 18 0 "diff(f(x,y),x)" "6#-%%diffG6$-%\"fG 6$%\"xG%\"yGF)" }{TEXT -1 154 " at (X,Y), first cut the surface z = f( x,y) with the vertical plane y = Y and find the curve of intersection \+ between this and the surface. The value of " }{XPPEDIT 18 0 "diff(f( x,y),x);" "6#-%%diffG6$-%\"fG6$%\"xG%\"yGF)" }{TEXT -1 108 " at (X,Y) \+ is the slope of the line tangent to this curve at this point. Use thi s to draw the tangent line. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 " \+ As an example, let " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "f:=( x,y)->15-5*(x-1)^2-3*y^2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 152 "Con sider that portion of this surface lying above the region [0,2] x [-1, 2] and choose (X,Y) in this region. As an illustration, let (1.5,1) b e chosen." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "X:=1.5:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "Y:=1:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Draw this portion of the surface." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "S:=plot3d(f(x,y),x=0..2,y=-1..2,color=cya n):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "The plane y = Y may be def ined parametrically." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "PY: =plot3d([x,Y,z],x=0..2,z=0..15,color=yellow):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "The curve of intersection is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "CY:=spacecurve([x,Y,f(x,Y)],x=0..2,color=red,thi ckness=4):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "So far, the picture is " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "display(S,PY,CY,sty le=patchcontour,axes=frame);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "S ince f has been defined as a function in x and y, then as a function o f x and y, " }{XPPEDIT 18 0 "diff(f(x,y),x)" "6#-%%diffG6$-%\"fG6$%\"x G%\"yGF)" }{TEXT -1 24 " is given most simply by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "fx:=D[1](f);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "The line tangent to CY at (1.5,1) may be defined by" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "TLx:=spacecurve([x,Y,fx(X,Y) *(x-X)+f(X,Y)],x=0..2,color=red,thickness=3):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 155 "To mark the point (X,Y), a short \"pin\" will be used . This is often convenient when, as in this case, using the same scal e on each axes is not a good idea." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "Pt:=spacecurve([X,Y,t],t=f(X,Y)-2..f(X,Y)+2,color=yel low, thickness=5):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "Lgd1:=textplot3d([2,-1,25,c at(\"Diff(f(x,y),x) = \",convert(evalf(fx(X,Y),3),string))],font=[HELV ETICA,BOLD,14],color=red):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Omi tting the plane y = Y, the picture is now" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 76 "display(TLx,S,CY,Pt,Lgd1,style=patchcontour,axes=fr ame,orientation=[50,60]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Let \+ the curve and tangent line associated with " }{XPPEDIT 18 0 "diff(f(x, y),y);" "6#-%%diffG6$-%\"fG6$%\"xG%\"yGF*" }{TEXT -1 109 " at (X,Y) be added to the picture. The curve of intersection between x = X and th e surface may be defined by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "CX:=spacecurve([X,y,f(X,y)],y=-1..2,color=blue,thickness=4):" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{XPPEDIT 18 0 "diff(f(x,y),y )" "6#-%%diffG6$-%\"fG6$%\"xG%\"yGF*" }{TEXT -1 12 " is given by" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "fy:=D[2](f);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "So the line tangent to CX at (1.5,1) may \+ be defined by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "TLy:=space curve([X,y,fy(X,Y)*(y-Y)+f(X,Y)],y=-1..2,color=blue,thickness=3):" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Including a legend in the picture \+ by using " }{TEXT 326 10 "textplot3d" }{TEXT -1 17 " can be useful. \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "Lgd2:=textplot3d([2,-1 ,25,cat(\"Diff(f(x,y),x) = \",convert(evalf(fx(X,Y),3),string),\", D iff(f(x,y),y) = \",convert(evalf(fy(X,Y),3),string))],font=[HELVETICA, BOLD,14],color=red):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "Now displ ay the complete picture. To see the full legend, it is necessary to m agnify the picture." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "disp lay(S,CY,CX,TLx,TLy,Pt,Lgd2,style=patchcontour,axes=frame,orientation= [50,60]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 378 "From this picture, two important facts m ay easily be seen: first, there will be a plane tangent to the surfac e at (X,Y) containing these tangent lines, and, second, at a locally h igh or low point on the surface where there is also a tangent plane, t his plane must be horizontal. So both partial derivatives must be zero there. This is easily seen if (X,Y) is changed to (1,0)." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 26 "The Geometric Meaning of " }{XPPEDIT 18 0 "diff(f(x,y),y,x);" "6#-%%diffG6%-%\"fG6$%\"xG%\"yGF*F)" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "The mixed partial " } {XPPEDIT 18 0 "diff(f(x,y),y,x);" "6#-%%diffG6%-%\"fG6$%\"xG%\"yGF*F) " }{TEXT -1 43 " gives the instantaneous rate of change of " } {XPPEDIT 18 0 "Diff(f(x,y),y);" "6#-%%DiffG6$-%\"fG6$%\"xG%\"yGF*" } {TEXT -1 273 " as one moves in the x-direction. This may be suggested in the following way. First, draw the curve of intersection between \+ the surface z = f(x,y) and a plane x = X. Then draw a portion of the \+ line tanget to this curve at (X,Y). The slope of this line is the val ue of " }{XPPEDIT 18 0 "Diff(f(x,y),y);" "6#-%%DiffG6$-%\"fG6$%\"xG% \"yGF*" }{TEXT -1 234 " at (X,Y). Holding Y fixed, now consider a seq uence of such curves and tangent lines as X is increased. The visably changing slope of these tangent lines suggests the instantaneous rate of change given by the mixed partial at (X,Y)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "As an illustration, consider the surface, z = " } {XPPEDIT 18 0 "f(x,y) = -2*sin(x)*sin(y);" "6#/-%\"fG6$%\"xG%\"yG,$*( \"\"#\"\"\"-%$sinG6#F'F,-F.6#F(F,!\"\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "restart:with(plots):with(plottools) :with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "f:=(x,y)- >-2*sin(x)*sin(y):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "surf: =plot3d(f(x,y),x=0..3,y=-3..0):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "Choose Y. This will be used to show the curve along which (X,Y) w ill move." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "Y:=-2.4;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Now find both partials as function s of x and y." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "fx:=unappl y(diff(f(x,y),x),(x,y)); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "fy:=unapply(diff(f(x,y),y),(x,y));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "Draw C, the curve with y = Y along which (X,Y) will move and the curve, Cx, which has slope " }{XPPEDIT 18 0 "diff(f(x,y),y); " "6#-%%diffG6$-%\"fG6$%\"xG%\"yGF*" }{TEXT -1 10 " at (X,Y)." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "C:=spacecurve([t,Y,f(t,Y)],t =0..3,color=yellow,thickness=5):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "Cx:=s->spacecurve([s,y,f(s,y)],y=-3..0,color=cyan,thi ckness=5):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "Now draw the line t angent to Cx at (X,Y) and define a \"pin\" to mark the location of (X, Y)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "T:=s->spacecurve([s, t,fy(s,Y)*(t-Y)+f(s,Y)],t=-3..-1.2,color=red,thickness=4):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "Pt:=s->spacecurve([s,Y,t],t=f(s,Y)- 0.3..f(s,Y)+0.3,thickness=4,color=green):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "To show the changing values of x and of the slope during the animation, a legend can be added by using " }{TEXT 325 10 "textpl ot3d" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "tx t:=s->textplot3d([0,-3.1,3,cat(\"Diff(f(x,y),y) = \",convert(evalf(fy( s,Y),3),string),\", x = \",convert(evalf(s,2),string))],font=[HELVET ICA,BOLD,14],color=red):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Final ly, make a sequence of pictures and display it. This is the animation ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "S:=[seq(1+i/10,i=0..19 )]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "pic:=seq(display(sur f,C,T(s),Cx(s),txt(s),Pt(s)),s=S):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "display(pic,insequence=true,axes=frame,orientation=[- 20,80]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Stepping through the \+ animation one picture at a time, it can be seen that " }{XPPEDIT 18 0 "diff(f(x,y),y);" "6#-%%diffG6$-%\"fG6$%\"xG%\"yGF*" }{TEXT -1 58 " is decreasing as the pin passes through (2,Y). Now find " }{XPPEDIT 18 0 "diff(f(x,y),y,x)" "6#-%%diffG6%-%\"fG6$%\"xG%\"yGF*F)" }{TEXT -1 26 " and evaluate it at (2,Y)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "A:=" }{XPPEDIT 19 1 "diff(diff(f(x,y),y),x);" "6#-%%diffG6$-F$6 $-%\"fG6$%\"xG%\"yGF,F+" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "evalf(subs(x=2,y=Y,A),3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 159 "That this is negative is now expected from viewing the a nimation, which showed that the slopes of the red lines decrease as x \+ increases along the yellow curve." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 22 "Surfaces an d Gradients" }}{EXCHG {PARA 0 "" 0 "" {TEXT 327 310 "The surface z = f (x,y) is the same as the level surface F(x,y,z) = 0, where F(x,y,z) = \+ f(x,y) - z. However, the gradient of f(x,y) and the gradient of F(x ,y,z) are not the same. It is important to understand the relationshi p between them. The gradient of F at (X,Y,f(X,Y)) is normal to the su rface there. " }{TEXT -1 1 " " }{TEXT 335 176 "The gradient of f at (X ,Y) lies in the x,y-plane and is normal, there, to the level curve f(x ,y) = f(X,Y). To illustate this relationship, the following procedure may be used." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "restart:wi th(plots):with(plottools):with(linalg):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "First, choose f. As an illustration, let f(x,y) = " } {XPPEDIT 18 0 "x^2/2+y^2/4+1;" "6#,(*&%\"xG\"\"#\"\"#!\"\"\"\"\"*&%\"y G\"\"#\"\"%F(F)\"\"\"F)" }{TEXT -1 27 " and let (X,Y) be (1.5, 1)." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f:=(x,y)->x^2/2+y^2/4+1;" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "X:=1.5:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "Y:=1:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "Next, define F as a function. Because the previously defined ex pression f(x,y) appears in the definition of F, the " }{TEXT 328 7 "un apply" }{TEXT -1 34 " command must be used to do this. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "F:=unapply(f(x,y)-z,(x,y,z));" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "Now find the gradient of f and ev aluate it at (X,Y). The somewhat elaborate procedure is required beca use " }{TEXT 329 4 "grad" }{TEXT -1 128 " produces, quite properly, a \+ vector, but for present purposes it is much more convenient to have gf as a \"list-valued\" function." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "gf:=unapply(convert(grad(f(x,y),[x,y]),list),(x,y)):" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Next, find the gradient of F at (X ,Y,f(X,Y)). " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "gF:=unappl y(convert(grad(F(x,y,z),[x,y,z]),list),(x,y,z)):" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 8 "Whether " }{TEXT 330 4 "line" }{TEXT -1 3 " or" } {TEXT 331 11 " spacecurve" }{TEXT -1 140 " is used to produce the line segments representing the two gradients is rather a matter of taste, \+ but both expect lists as arguments. Since " }{TEXT 332 6 "evalm " } {TEXT -1 51 "is needed to do the \"arithmetic\", this produces an " } {TEXT 333 5 "array" }{TEXT -1 41 " which must again be converted to a \+ list." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "gf_vec:=t->convert (evalm([X,Y,0]+t*[op(gf(X,Y)),0]),list):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "gF_vec:=t->convert(evalm([X,Y,f(X,Y)]+t*gF(X,Y,f(X,Y) )),list):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 132 "Now draw line segme nts, Vgf and VgF, which have the magnitude and direction of the gradie nt of f(x,y) and of F(x,y,z), respectively." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "Vgf:=spacecurve(gf_vec(t),t=0..1,color=blue,thickn ess=5):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "VgF:=spacecurve( gF_vec(t),t=0..1,color=red,thickness=5):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "Mark the point (X,Y,f(X,Y)) with a small, red sphere and \+ the point (X,Y,0) with a small, blue one." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 56 "p:=sphere([X,Y,f(X,Y)],.15,color=red,style=patchnog rid):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "p1:=sphere([X,Y,0] ,.15,color=blue,style=patchnogrid):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 132 "To help to show that Vgf is the projection of VgF onto the x,y -plane, their corresponding ends may be connected with thin red lines. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "L1:=line([X,Y,0],[X,Y,f (X,Y)],color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "L2:=l ine(gf_vec(1),gF_vec(1),color=red):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "Next draw the level curve f(x,y) =f(X,Y). Because this is the \+ ellipse " }{XPPEDIT 18 0 "x^2/2+y^2/4 = f(x,y);" "6#/,&*&%\"xG\"\"#\" \"#!\"\"\"\"\"*&%\"yG\"\"#\"\"%F)F*-%\"fG6$F&F," }{TEXT -1 35 " + 1 , which may be rewritten as " }{XPPEDIT 18 0 "x^2/(2*(f(x,y)-1))+y^2/( 4*(f(x,y)-1)) = 1;" "6#/,&*&%\"xG\"\"#*&\"\"#\"\"\",&-%\"fG6$F&%\"yGF* \"\"\"!\"\"F*F1F**&F/\"\"#*&\"\"%F*,&-F-6$F&F/F*\"\"\"F1F*F1F*\"\"\"" }{TEXT -1 70 " , it is easy to find a parametric representation for it and then use " }{TEXT 334 10 "spacecurve" }{TEXT -1 23 " to draw it. \+ This is " }{XPPEDIT 18 0 "x = sqrt(2*(f(x,y)-1))*cos(t);" "6#/%\"xG* &-%%sqrtG6#*&\"\"#\"\"\",&-%\"fG6$F$%\"yGF+\"\"\"!\"\"F+F+-%$cosG6#%\" tGF+" }{TEXT -1 9 " and y = " }{XPPEDIT 18 0 "sqrt(4*(f(x,y)-1))*sin(t );" "6#*&-%%sqrtG6#*&\"\"%\"\"\",&-%\"fG6$%\"xG%\"yGF)\"\"\"!\"\"F)F)- %$sinG6#%\"tGF)" }{TEXT -1 15 " for t in [0, " }{XPPEDIT 18 0 "2*pi; " "6#*&\"\"#\"\"\"%#piGF%" }{TEXT -1 2 "]." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 107 "c:=spacecurve([sqrt((2*(f(X,Y)-1)))*cos(t),sqrt((4 *(f(X,Y)-1)))*sin(t),0],t=0..2*Pi,color=red,thickness=3):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "To make the top of the surface smooth, le t it extend to, say, z = 4. Then, " }{XPPEDIT 18 0 "x^2/2+y^2/4 = 3 " "6#/,&*&%\"xG\"\"#\"\"#!\"\"\"\"\"*&%\"yG\"\"#\"\"%F)F*\"\"$" } {TEXT -1 22 " and, if y = 0, x is " }{XPPEDIT 18 0 "sqrt(6);" "6#-%%s qrtG6#\"\"'" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "-sqrt(6);" "6#,$-%%sqr tG6#\"\"'!\"\"" }{TEXT -1 41 ". Using this as the x-interval, solving " }{XPPEDIT 18 0 "x^2/2+y^2/4 = 3;" "6#/,&*&%\"xG\"\"#\"\"#!\"\"\"\" \"*&%\"yG\"\"#\"\"%F)F*\"\"$" }{TEXT -1 40 " for y yields the necessar y y-interval. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "surf:=pl ot3d(f(x,y),x=-sqrt(6)..sqrt(6),y=-sqrt(12-2*x^2)..sqrt(12-2*x^2),styl e=patchcontour,shading=zhue):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 268 "Finally, display all of these things together. Vgf should be normal \+ to c at (X,Y) and VgF should be normal to the surface at (X,Y,f(X,Y)). If the angles seem wrong, press the \"1:1\" button. Be sure to enlar ge the picture and rotate it to inspect its various features." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "display(p,p1,Vgf,VgF,L1,L2,s urf,c,axes=normal,orientation=[-50,70]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 337 45 "Surfaces, Critical Points, and Tangent Planes" }{TEXT 338 0 " " }}{EXCHG {PARA 256 "" 0 "" {TEXT -1 384 "Illustrating the connection between local extrema and a horizontal tangent plane is quite easy. R elative extrema of several functions f(x,y) and a saddle point will be exhibited to illustrate that if a tangent plane exists at such a poin t, then it is horizontal. The gradient of F(x,y,z) = z - f(x,y) at su ch a point is clearly < 0,0,1 > and is a normal vector for the tangent plane." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "restart:with(plo ts):with(plottools):with(linalg):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Example 1. f(x,y) = " }{XPPEDIT 18 0 "20-2*x^2+y^2+8*x-6*y;" "6#, ,\"#?\"\"\"*&\"\"#F%*$%\"xG\"\"#F%!\"\"*$%\"yG\"\"#F%*&\"\")F%F)F%F%*& \"\"'F%F-F%F+" }{TEXT -1 47 ". Maximum at (-2,3,3). Tangent plane: z = 37." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "f1:=20-(2*x^2+y^2 +8*x-6*y):" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "s1:=plot3d(f1,x=-4..0,y=1..5):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "Tp1:=plot3d(37,x=-4..0,y=1..5,color=blue,style=wirefr ame):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "P1:=sphere([-2,3,3 7],.1,color=red,style=patchnogrid):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "display(s1,Tp1,P1,style=patchcontour);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Example 2. f(x,y) = " }{XPPEDIT 18 0 "1- (x^2+y^2)^(1/3);" "6#,&\"\"\"\"\"\"),&*$%\"xG\"\"#F%*$%\"yG\"\"#F%*&\" \"\"F%\"\"$!\"\"F1" }{TEXT -1 85 ". Maximum at (0,0,1). Function is \+ not differentiable at (0,0), so no tangent plane." }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 22 "f2:=1-(x^2+y^2)^(1/3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "s2:=plot3d(f2,x=-2..2,y=-2..2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "P2:=sphere([0,0,1],.1,color=red,sty le=patchnogrid):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "display (s2,P2,style=patchcontour);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Ex ample 3. f(x,y) = " }{XPPEDIT 18 0 "x^2*y^2;" "6#*&%\"xG\"\"#%\"yG\" \"#" }{TEXT -1 45 ". Minimum at (0,0,0). Tangent plane: z = 0." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f3:=x^2*y^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "s3:=plot3d(f3,x=-2..2,y=-2..2):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "Tp3:=plot3d(0,x=-2..2,y=-2. .2,color=yellow,style=patchnogrid):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "P3:=sphere([0,0,0],.1,color=red,style=patchnogrid):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "display(s3,Tp3,P3,style=p atchcontour);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 257 "This is a parti cularly nice picture as it shows that the surface is in contact with t he tangent plane exactly along the x and y axes. To confirm this, cli ck on the picture, then use the button on the context bar to select a \+ \"normal\" set of coordinate axes." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Example 4. f(x,y) = " }{XPPEDIT 18 0 "-x^3+4*x*y-2*y^2+1;" "6# ,**$%\"xG\"\"$!\"\"*(\"\"%\"\"\"F%F*%\"yGF*F**&\"\"#F**$F+\"\"#F*F'\" \"\"F*" }{TEXT -1 131 ". Local maximum at (4/3,4/3, 59/27); saddle po int at (0,0,1). Tangent planes at these points are z = 59/27 and z=1, respectively." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f4:=-x^3+ 4*x*y-2*y^2+1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "s4:=plot3 d(f4,x=-2..3,y=-2..4):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "P 41:=sphere([0,0,1],.1,color=yellow,style=patchnogrid):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "P42:=sphere([4/3,4/3,59/27],.1,colo r=red,style=patchnogrid):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "Draw n simply, the nature of the points is difficult to see. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "display(s4,P41,P42);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 431 "Pressing the 1:1 button just makes thing s worse, although enlarging the picture does show a hump. However, it \+ is impossible to see if is associated with one of the colored points. \+ Introducing a \"box\" coordinate system shows that the extent of the \+ surface in the z-direction is much greater than that in the x- or y-di rections. But how can this be decreased without changing the interval s on the x and y axes? Using the optional " }{TEXT 336 4 "view" } {TEXT -1 100 " command, as shown, solves this problem. This is anothe r Maple \"trick\" that is well worth knowing. " }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 45 "display(s4,P41,P42,view=[-2..3,-2..4,-2..4]); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "Finally, the tangent planes a re defined and they and the surface are displayed together. " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "TpMax:=plot3d(59/27,x=-2..3, y=-2..4,style=wireframe,color=yellow):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "TpS:=plot3d(1,x=-2..3,y=-2..4,style=wireframe,color=p ink):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "display(s4,P41,P42 ,TpMax,TpS,view=[-2..3,-2..4,-2..4]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 79 "Volumes by \+ Double Integration: Drawing the Object and Animating the Integration" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "restart:with(plots):with(p lottools):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 573 "In order to find t he volume between the x,y-plane and a surface z = f(x,y) over a closed , bounded region, R, of the x,y-plane it usually necessary to visualiz e it. To see how to use use Maple to help do this, it is well to begi n with a simple case in which R is vertically simple or horizontally s imple. Lines and spacecurves to draw R, then only that portion of the surface above (or below) R will be drawn, and, finally the vertices o f these will be connected with line segments. It will be convenient t o define f as a Maple function. Consider the following example. " }}} {EXCHG {PARA 4 "" 0 "" {TEXT 349 37 "Example 1: A Vertically Simple Re gion" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Find the volume enclosed \+ between the graph of the function f(x,y) = " }{XPPEDIT 18 0 "4-(x/2)^ 2-(y/3)^2;" "6#,(\"\"%\"\"\"*$*&%\"xGF%\"\"#!\"\"\"\"#F**$*&%\"yGF%\" \"$F*\"\"#F*" }{TEXT -1 83 " and the x,y-plane, over the region R encl osed by the curves x = 0, y = x + 3, and " }}{PARA 0 "" 0 "" {TEXT -1 4 "y = " }{XPPEDIT 18 0 "x^2+1;" "6#,&*$%\"xG\"\"#\"\"\"\"\"\"F'" } {TEXT -1 58 ". Here R is vertically simple. The \"lower curve\" is y = " }{XPPEDIT 18 0 "x^2+1;" "6#,&*$%\"xG\"\"#\"\"\"\"\"\"F'" }{TEXT -1 36 " and the \"upper curve\" is y = x + 3." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "First, define the function. The parentheses in (x,y) \+ are important." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "f:=(x,y)- >4-(x/2)^2-(y/3)^2:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Next, defi ne the upper and lower curves as functions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Uc:=x->x+3:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Lc:=x->x^2+1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "so lve(Uc(x)=Lc(x),x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 129 "So, in R, x will be in [0,2]. Now use this fact and the upper and lower curve \+ functions to draw the graph of the surface over R." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "s:=plot3d(f(x,y),x=0..2,y=Lc(x)..Uc(x),grid =[15,15]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 153 "Now, draw the regi on R. If the lower and upper curves did not meet, as they do in this e xample, there would be a second line at the right-hand edge of R." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "L1:=line([0,1,0],[0,3,0],col or=blue,thickness=2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "L2 :=spacecurve([x,Uc(x),0],x=0..2,color=blue,thickness=2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "L3:=spacecurve([x,Lc(x),0],x=0..2,c olor=blue,thickness=2):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 125 "Colle ct these together and call the result R. In order to show R in its ex pected relationship to the coordinate system, the " }{TEXT 339 4 "view " }{TEXT -1 38 " command has been used in the display." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "R:=\{L1,L2,L3\}:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "display(s,R,axes=normal,view=[0..2,0..5,0 ..4]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "Now define the vertical connecting lines between the vertices of R and those of the surface. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "C1:=line([0,1,0],[0,1,f (0,1)],color=green,thickness=3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "C2:=line([0,3,0],[0,3,f(0,3)],color=green,thickness=3 ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "C3:=line([2,5,0],[2,5 ,f(2,5)],color=green,thickness=3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "C:=\{C1,C2,C3\}:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "display(s,R,C,axes=normal,view=[0..2,0..5,0..4]);" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 150 "Rotating the pic ture helps to reveal conpletely the volume will be given by the iterat ed integral, first with respect to y and then with respect to x:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "V:=Int(Int(f(x,y),y=Lc(x)..U c(x)),x=0..2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "The value of th is is given by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(V); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Of course, its value may be f ound directly via" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "int(in t(f(x,y),y=Lc(x)..Uc(x)),x=0..2);" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 50 "Animating the Process by \"Sweeping Out\" the Volume" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 651 "To show the outer integral \"sweeping out the volume\", one may use a parametric surfac e to represent the \"very thin\" slice of the volume at x. The Maple \+ function producing this slice is is an \"object-valued\" function of x and the parameters are y and z. Its definition assumes that the surf ace is above the x,y-plane over all of R. However, as will be seen be low, it succeeds equally well when the situation is reversed. To show the idea, first a single slice is drawn at x = 1. Note that using x = 2 would result in a plotting error because the slice has zero exten sion in the y-direction. A way to overcome this difficulty will be sh own below." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "A:=x->plot3d( [x,y,z],y=Lc(x)..Uc(x),z=0..f(x,y),color=red,style=patchnogrid):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "display(A(1),s,R,C,axes=norm al,view=[0..2,0..5,0..4],style=wireframe);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 123 "To make an \"animation\" of the sweeping out of this vol ume, a sequence of pictures is made, then displayed using the option \+ " }{TEXT 340 15 "insequence=true" }{TEXT -1 197 ". By using x = i/20 \+ +0.001 as the argument of A, a sequence of 40 pictures as x varies fro m 0 to 20 can be made and, because of the 0.001 term, the plotting err or for i = 40 can be avoided. The " }{TEXT 341 11 "orientation" } {TEXT -1 212 " option can initially be omitted as the point of view th e animation can best be seen is, of course, not yet known. To run the \+ animation, click on the picture and use the buttons which appear on th e \"context bar\"." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "SV:=s eq(display(A(i/20+0.001),s,C,R),i=0..40):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 83 "display(SV,insequence=true,axes=normal,view=[0..2,0 ..5,0..4],orientation=[-23,56]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 348 9 "Example 2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 169 "This procedure c an also be applied to a case in which the surface z = f(x,y) over R ex tends both above and below the x,y-plane. Consider the following inte gral problem." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 344 10 "Evaluate " }{XPPEDIT 18 0 "Int(Int(x^4-2*y,y = -x^2 .. x^2),x = -1 . . 1);" "6#-%$IntG6$-F$6$,&*$%\"xG\"\"%\"\"\"*&\"\"#F,%\"yGF,!\"\"/F/;, $*$F*\"\"#F0*$F*\"\"#/F*;,$\"\"\"F0\"\"\"" }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 " Naturally, it would be interesting to s ee the object over which the integration is to be performed." } {MPLTEXT 1 0 1 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "restart:with(p lots):with(plottools):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Define \+ the graph of " }{XPPEDIT 18 0 "z = x^4-2*y;" "6#/%\"zG,&*$%\"xG\"\"%\" \"\"*&\"\"#F)%\"yGF)!\"\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 18 "f:=(x,y)->x^4-2*y:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "The upper and lower curves are y = " }{XPPEDIT 18 0 "x^2; " "6#*$%\"xG\"\"#" }{TEXT -1 9 " and y = " }{XPPEDIT 18 0 "-x^2;" "6#, $*$%\"xG\"\"#!\"\"" }{TEXT -1 15 ", respectively," }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 11 "Uc:=x->x^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Lc:=x->-x^2:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 " The surface is then defined as in the previous example." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "s:=plot3d(f(x,y),x=-1..1,y=Lc(x)..U c(x),style=wireframe, color=navy,grid=[15,15]):" }}{PARA 0 "" 0 "" {TEXT -1 483 "Drawing the curved, vertical sides of the object can be \+ quite involved and can be avoided by simply connecting the vertices of the surface and R with vertical lines. However, if the sides are des ired, there are several things to notice. First, these surfaces are d efined parametrically, using x and z as the parameters. Second, the x and y components are defined just as in the two spacecurves. Third, \+ even though the surface is being defined parametrically, because the c ommand " }{TEXT 345 7 "plot3d " }{TEXT -1 283 "is used, the end-points of the y-range can be specified as functions of x. Fourth, the heig ht of the sides of the solid is z, where, in the first case, z runs fr om the curve c1 up to the x,y-plane (z = 0), while, in the second case , z runs from the x,y-plane down to the curve c2. " }{TEXT 346 1 " " } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 141 "Because this ext ends both above and below the x,y-plane, the relevant portion of the x ,y-plane may be usefully included in the final picture." }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 70 "p:=plot3d(0,x=-1..1,y=-1..1,style=wireframe,co lor=black,grid=[10,10]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "The r egion R is bounded by the curves " }{XPPEDIT 18 0 "y = x^2;" "6#/%\"y G*$%\"xG\"\"#" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "y = -x^2;" "6#/% \"yG,$*$%\"xG\"\"#!\"\"" }{TEXT -1 129 " in the x,y-plane and the line s x = -1 and x = 1. The vertical walls of the volume will extend from these curves to the surface " }{XPPEDIT 18 0 "z = x^4-2*y;" "6#/%\"zG ,&*$%\"xG\"\"%\"\"\"*&\"\"#F)%\"yGF)!\"\"" }{TEXT -1 28 " . The curv es are given by" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "c1:=spacecurve([ t,t^2,0],t=-1..1,color=red,thickness=2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "c2:=spacecurve([t,-t^2,0],t=-1..1,color=red,thickness =2):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "and the lines are given b y" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "L1:=line([-1,-1,0],[-1 ,1,0],color=red,thickness=2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "L2:=line([1,-1,0],[1,1,0],color=red,thickness=2):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 483 "Drawing the curved, vertical sides of th e object can be quite involved and can be avoided by simply connecting the vertices of the surface and R with vertical lines. However, if t he sides are desired, there are several things to notice. First, thes e surfaces are defined parametrically, using x and z as the parameters . Second, the x and y components are defined just as in the two space curves. Third, even though the surface is being defined parametricall y, because the command " }{TEXT 342 7 "plot3d " }{TEXT -1 283 "is used , the end-points of the y-range can be specified as functions of x. F ourth, the height of the sides of the solid is z, where, in the first case, z runs from the curve c1 up to the x,y-plane (z = 0), while, in the second case, z runs from the x,y-plane down to the curve c2. " } {TEXT 343 1 " " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "S1:=plot3d([x,x^2,z],x=-1..1,z=x^4-2*x^2..0,color=yellow,style=p atchnogrid):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "S2:=plot3d( [x,-x^2,z],x=-1..1,z=0..x^4+2*x^2,color=cyan,style=patchnogrid):" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 34 " Finally, we display the solid. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "display(s,p,c1,c2,L1,L2,S1,S2,a xes=frame);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 345 "As can now be see n directly, the 3-dimensional region over which the integration takes \+ place is that bounded by the surface z = f(x,y), the x,y-plane, and th e cyan and yellow surfaces. The value of the integral is positive wher e the surface is above the plane and negative where it is not. The fi gure should be rotatied and inspected carefully." }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 40 "The iterated integral for this volume is" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Int(Int(x^4-2*y,y=-x^2..x^2),x=-1..1);" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "Replacing I by i puts the comman d in active form. Maple then finds" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "V1:=int(int(x^4-2*y,y=-x^2..x^2),x=-1..1);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 50 "Animat ing the Process by \"Sweeping Out\" the Volume" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 173 "To create an animation, \+ the same \"object-valued\" function as before may be used to draw the \+ \"slice\". To check that it is going to work, a single slice is first displayed at" }}{PARA 0 "" 0 "" {TEXT -1 141 "x = 0.8. Then a suitab le sequence of pictures is made, this time avoiding a slice of zero ar ea (and a plotting error) at x = 0. Again, the " }{TEXT 347 11 "orien tation" }{TEXT -1 71 " option can initally be omitted. Finally, the a nimation is displayed. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 " A:=x->plot3d([x,y,z],y=Lc(x)..Uc(x),z=0..f(x,y),color=red,style=patchn ogrid):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "display(A(0.8),s ,p,c1,c2,S1,S2,L1,L2,axes=frame);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "SV:=seq(display(A(i/20+0.001),s,p,c1,c2,S1,S2,L1,L2), i=-20..20):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "display(SV,i nsequence=true,axes=frame,style=patchcontour,orientation=[126,57]);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 17 "Crossed Cylinders" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "Two Cylinders" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 350 292 "Finding the volume enclosed by a pair of cylinders of the same ra dius whose axes are perpendicular and in the same plane is a well-know n exercise in multiple integration. The difficulty of visualizing thi s object when considered for the first time is also well-known. Maple is a great help." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "restar t:with(plots):with(plottools):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }{TEXT 351 60 "We describe the cylinders parametrically and display \+ them. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "A:=plot3d([x,cos (t),sin(t)],x=-1..1,t=0..2*Pi,color=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "B:=plot3d([cos(t),y,sin(t)],y=-1..1,t=0..2*Pi,colo r=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "display(A,B,scal ing=constrained);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 352 129 "Now the region common to both is described and displayed. Notice that a parametric description of the surfaces is not necessary." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "a:=plot3d(\{sqrt(1-x^2),-sqr t(1-x^2)\},x=-1..1,y=-x..x,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "b:=plot3d(\{sqrt(1-y^2),-sqrt(1-y^2)\},x=-y..y,y=-1.. 1,color=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "display( a,b,scaling=constrained,style=patchcontour);" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }{TEXT 353 38 "The volume of this region is given by \+ " }{XPPEDIT 18 0 "Int(Int(sqrt(1-x^2),y = -x .. x),x = 0 .. 1);" "6#-% $IntG6$-F$6$-%%sqrtG6#,&\"\"\"\"\"\"*$%\"xG\"\"#!\"\"/%\"yG;,$F/F1F//F /;\"\"!\"\"\"" }{TEXT -1 3 " . " }{TEXT 354 25 "Calculating this, we f ind" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "v:=int(int(sqrt(1-x^ 2),y=-x..x),x=0..1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 355 23 "So the t otal volume is " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "V:=8*v;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "Three Cylinders" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "restart:with(plots):with(plottools):" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 356 13 "The Problem " }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 160 "To use Maple to draw the surface enclo sing the points contained by the intersection of the three unit cylind ers whose axes are the x,y,and z axes, respectively." }{TEXT 357 0 "" }{TEXT -1 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 19 "Drawing the Sur face" }}{EXCHG {PARA 258 "" 0 "" {TEXT -1 20 "The three cylinders." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "A:=implicitplot3d(x^2+z^2=1 ,x=-1..1,y=-1.5..1.5,z=-1..1,color=red,style=patchcontour):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "B:=implicitplot3d(z^2+y^2=1, x=-1.5..1.5,y=-1..1,z=-1..1,color=green,style=patchcontour):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "C:=implicitplot3d(x^2+y^2=1, x=-1..1,y=-1..1,z=-1.5..1.5,color=yellow,style=patchcontour):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "display(A,B,C,scaling=constr ained,orientation=[40,60]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 358 0 "" }{TEXT -1 0 "" }{TEXT 359 20 "Using the result of " } {TEXT 360 13 "Two Cylinders" }{TEXT 361 0 "" }{TEXT -1 0 "" }{TEXT 362 88 ", the volume defined by the intersection of the red and green \+ cylinders, alone, is this." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "A1:=plot3d(\{sqrt(1-x^2),-sqrt(1-x^2)\},x=-1..1,y=-abs(x)..abs(x), color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "B1:=plot3d( \{sqrt(1-y^2),-sqrt(1-y^2)\},x=-abs(y)..abs(y),y=-1..1,color=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "display(A1,B1,scaling=con strained,style=patchcontour);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }{TEXT 363 125 "Adding the yellow cylinder to this, just what must b e removed to define the volume common to all three cylinders may be se en." }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "displ ay(A1,B1,C,scaling=constrained,axes=frame,style=patchcontour);" }}} {EXCHG {PARA 259 "" 0 "" {TEXT -1 117 "Those pieces of the red cylinde r which are to be saved may be defined as follows. These are collecte d as the set La." }{TEXT 364 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "a1:=plot3d(\{sqrt(1-x^2),-sqrt(1-x^2)\},x=-1/sqrt(2). .1/sqrt(2),y=-abs(x)..abs(x),color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "a2:=plot3d(\{sqrt(1-x^2),-sqrt(1-x^2)\},x=1/sqrt(2).. 1,y=-sqrt(1-x^2)..sqrt(1-x^2),color=red):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 94 "a3:=plot3d(\{sqrt(1-x^2),-sqrt(1-x^2)\},x=-1..-1/sq rt(2),y=-sqrt(1-x^2)..sqrt(1-x^2),color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "La:=\{a1,a2,a3\}:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 365 152 "In a similar fashion, those pieces of the green cylinde r which are to be saved may be defined in the following way. These a re collected as the set Lb." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "b1:=plot3d(\{sqrt(1-y^2),-sqrt(1-y^2)\},x=-abs(y)..abs(y),y=-1/sqr t(2)..1/sqrt(2),color=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "b2:=plot3d(\{sqrt(1-y^2),-sqrt(1-y^2)\},x=-sqrt(1-y^2)..sqrt(1-y ^2),y=1/sqrt(2)..1,color=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "b3:=plot3d(\{sqrt(1-y^2),-sqrt(1-y^2)\},x=-sqrt(1-y^2 )..sqrt(1-y^2),y=-1..-1/sqrt(2),color=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Lb:=\{b1,b2,b3\}:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 384 107 "After careful investigation, the necessa ry parts of the yellow cylinder are defined as parametric surfaces." } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "s1:=plot3d([x,sqrt(1-x^2), z],x=-1/sqrt(2)..1/sqrt(2),z=-abs(x)..abs(x),color=yellow):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "s2:=plot3d([x,-sqrt(1-x^2),z ],x=-1/sqrt(2)..1/sqrt(2),z=-abs(x)..abs(x),color=yellow):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "s3:=plot3d([sqrt(1-y^2),y,z],y=-1/s qrt(2)..1/sqrt(2),z=-abs(y)..abs(y),color=yellow):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "s4:=plot3d([-sqrt(1-y^2),y,z],y=-1/sqrt(2). .1/sqrt(2),z=-abs(y)..abs(y),color=yellow):" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 366 45 "Now assemble all of the parts of the surface." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "L:=\{a1,a2,a3,b1,b2,b3,s1,s2 ,s3,s4\}:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 367 20 "Finally, display it ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "display(L,scaling=cons trained, style=patchcontour,orientation=[50,50]);" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 176 "Now use the buttons on the tool bars to put \+ a coordinate box around it and enlarge it, then rotate it, checking th at it has a circular cross-section in each coordinate plane. " }}} {EXCHG {PARA 261 "" 0 "" {TEXT -1 237 "As a last picture, draw the cyl inders A, B, and C in wire-frame style and display them together with \+ the surface. This looks best when it is enlarged. If the result is t oo cluttered, try combining one cylinder at a time with the solid." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "a:=implicitplot3d(x^2+z^2=1 ,x=-1..1,y=-1.5..1.5,z=-1..1,color=wheat,style=wireframe,grid=[6,6,6]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "b:=implicitplot3d(z^2+ y^2=1,x=-1.5..1.5,y=-1..1,z=-1..1,color=wheat,style=wireframe,grid=[6, 6,6]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "c:=implicitplot3d (x^2+y^2=1,x=-1..1,y=-1..1,z=-1.5..1.5,color=wheat,style=wireframe,gri d=[6,6,6]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "display(L,a, b,c,scaling=constrained,style=patchnogrid);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 "Indirec t Calculation of the Volume" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 368 151 "Method: Calculate the volume of the corners that are cu t off, then subtract this from the volume enclosed by the two crossed \+ cylinders. (See above and " }{TEXT 369 19 "Crossed Cylinders.)" } {TEXT 370 0 "" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 371 68 "Again, we look at the yellow cylinder together with the volume from " }{TEXT 373 17 "Crossed Cylinders" }{TEXT -1 2 ". " } {TEXT 377 37 "This shows just what must be removed." }{TEXT -1 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "display(A1,B1,C,scaling=con strained,axes=frame,style=patchcontour);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 372 24 "Looking down the z-axis:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "display(A1,B1,C,scaling=constraine d,axes=normal,view=[-1.5..1.5,-1.5..1.5,-1.5..1.5],style=patchnogrid,o rientation=[-90,0]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 374 132 "There are four \"corners\", each consisting of four symmetric parts of equal volume. Make a seperate picture of one of these parts . " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "part:=plot3d(sqrt(1- x^2),x=1/sqrt(2)..1,y=sqrt(1-x^2)..x,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "LL:=line([1/sqrt(2),1/sqrt(2),0],[1/sqrt(2),1 /sqrt(2),1/sqrt(2)],color=yellow,thickness=4):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 67 "LL1:=line([1/sqrt(2),1/sqrt(2),0],[1,1,0],colo r=green,thickness=4):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "LL 2:=line([0,0,0],[1/sqrt(2),1/sqrt(2),0],color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "CC:=spacecurve([x,sqrt(1-x^2),0],x= 0..1,color=yellow,thickness=4):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "display(CC,LL1,LL2,LL,part,style=patchcontour,scaling=constra ined,axes=normal,view=[0..1,0..1,0..1],orientation=[-125,50]);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 379 52 "Careful inspection shows that the volume is given by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "f:=Int(Int(sqrt(1-x^2),y=sqrt(1-x^2)..x),x=1/sqrt(2). .1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(f);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 375 54 "The total volume o f the corners is sixteen times this." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "16*%;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 376 127 "Subtracting this from the volume of the two crossed cylinders , we have the volume enclosed by the three intersecting cylinders." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "16/3-%;" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 378 20 "The solid once more." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 71 "display(L,scaling=constrained, style=patchcontour,o rientation=[50,50]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "An Elegant Method of Drawing th e Volumes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 380 65 "This method \+ of drawing the volumes depends on the use of Maple's " }{TEXT 381 3 "m in" }{TEXT 382 131 " command and is due to Dr. John Gosselin, Associat e Professor of Mathematics at the University of Georgia (personal comm unication)." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "Two Cylinders" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "plot3d(\{min(sqrt(1-x^2),sqrt(1-y^ 2)),-min(sqrt(1-x^2),sqrt(1-y^2))\},x=-1..1,y=-1..1,scaling=constraine d,grid=[35,35]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 " Three Cylinders" }}{PARA 0 "" 0 "" {TEXT 383 199 "Dr. Gosselin also su ggested that perhaps the three-cylinders volume might be generated in \+ a similar way if the cylinders were expressed in spherical coordinates . This is correct, as we will now see." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 344 "plot3d(\{min(1/sqrt(sin(phi)^2*cos(theta)^2+cos(phi) ^2),1/sqrt(sin(phi)^2*sin(theta)^2+cos(phi)^2),1/sin(phi)),-min(1/sqrt (sin(phi)^2*cos(theta)^2+cos(phi)^2),1/sqrt(sin(phi)^2*sin(theta)^2+co s(phi)^2),1/sin(phi))\},theta=0..2*Pi,phi=0..Pi,coords=spherical,scali ng=constrained,grid=[70,70],shading=zgreyscale,lightmodel=light1,style =patchnogrid);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}}} {MARK "0 7 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }