{VERSION 5 0 "IBM INTEL LINUX" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 1 12 0 0 0 0 0 2 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "lucidatypewriter" 1 12 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 14 0 0 0 0 0 2 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "lucidatypewriter" 1 14 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 267 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times " 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 } {PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }1 1 0 0 12 12 1 0 1 0 2 2 19 1 } {PSTYLE "Heading 1" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 256 0 "" }{TEXT 257 11 "Worksheet: " }{TEXT 258 12 "Lecture3.mws" }{TEXT 262 0 "" }} {PARA 18 "" 0 "" {TEXT -1 10 "Lecture 3:" }{TEXT 260 1 " " }}{PARA 18 "" 0 "" {TEXT 261 39 "Laplace's Equation & Harmonic Functions" }} {PARA 19 "" 0 "" {TEXT -1 12 "Jon Jacobsen" }}{PARA 18 "" 0 "" {TEXT 259 18 "PCMI, Summer 2003" }}{PARA 0 "" 0 "" {TEXT -1 62 "This worksh eet contains some examples from the third lecture." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart:w ith(plots):" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 16 "Complex Analysis " }}{PARA 257 "" 0 "" {TEXT -1 76 "The real and imaginary parts of an \+ analytic function are harmonic functions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f:=(x,y)->(x+I*y)^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "plot3d(Re(f(x,y)),x=-3..3,y=-3..3,axes=framed,shading =zhue);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "plot3d(Im(f(x,y) ),x=-3..3,y=-3..3,axes=framed,shading=zhue);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f:=(x,y)->(x+I*y)^5;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 75 "plot3d(Re(f(x,y)),x=-3..3,y=-3..3,axes=framed,shadi ng=zhue,numpoints=1500);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "plot3d(Im(f(x,y)),x=-4..4,y=-4..4,axes=framed,shading=zhue,numpoints= 1500);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "f:=(x,y)->3*(x+I* y)^7-2*(x+I*y)^4;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "plot3d (Im(f(x,y)),x=-2..2,y=-2..2,axes=framed,shading=zhue,view=-1000..1500) ;" }}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "f:=(x,y)->exp(-(x+I*y)^2)/((x+I*y)^2)+(x+I*y)^4;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "plot3d(Im(f(x,y)),x=-2..2,y= -2..2,axes=framed,shading=zhue,view=-24..29,numpoints=2000);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "plot3d(Re(f(x,y)),x=-2..2,y= -2..2,axes=framed,shading=zhue,view=-26..16,numpoints=2000);" }}} {PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 25 "Visualizing Perturbations" }} {PARA 0 "" 0 "" {TEXT -1 58 "Here we use Maple to plot a surface and s ome perturbations" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "addcoor ds(z_cylindrical,[z,r,theta],[r*cos(theta),r*sin(theta),z]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "g:=(r,t)->5+3/4*r^2*sin(2*t) ;" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 266 18 "Here is a surface:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "plot3d(g(r,t),r=0..2,t=0..2* Pi,shading=zhue,grid=[35,35],coords=z_cylindrical,style=patchnogrid); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 267 44 "Here is an ex ample of a \"bump function\" phi:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "p:=(x,y)->piecewise(x^2+y^2 >=1,0,exp(1/(x^2+y^2-1))) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "plot3d(.1*p(x,y),x=-1. .1,y=-1..1,numpoints=500,style=patchnogrid,shading=zhue);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 268 35 "Let's plot u plus the perturbation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "animate3d([r*cos(t),r*sin(t),g(r,t )+7.5*cos(eps)*p(r*cos(t)-.9,r*sin(t)+.5)],r=0..2.3,t=0..2*Pi,eps=0..2 *Pi,axes=none,shading=zhue,grid=[35,35],style=patch);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "plot3d(g(r,t)-8.5*p(r*cos(t)-.8,r* sin(t)+.5),r=0..2.3,t=0..2*Pi,axes=none,shading=zhue,grid=[35,35],coor ds=z_cylindrical,style=patchnogrid);" }}}}}{MARK "3" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }