{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 1 12 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" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 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 0 }3 0 0 -1 8 8 0 0 0 0 0 0 -1 0 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Nor mal" -1 257 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 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 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 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 40 "Partial Differential Equa tions PowerTool" }}{PARA 19 "" 0 "" {TEXT -1 16 "by Dr. Jim Herod" }}} {PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 256 56 "Sect ion 6.5: Laplace's Equation on a Ring or a Half Disk" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 258 30 "Maple Packages for Section 6.4" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 " Suppose that u is the solu tion for " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT -1 27 " u = 0 o n any region. Take " }{XPPEDIT 18 0 "p[0];" "6#&%\"pG6#\"\"!" }{TEXT -1 77 " to be any point on the interior of the region. Take C to be a circle about " }{XPPEDIT 18 0 "p[0];" "6#&%\"pG6#\"\"!" }{TEXT -1 116 " which is small enough that the circle lies in the region. We dr aw the picture as though the region were a square, " }{XPPEDIT 18 0 "p [0];" "6#&%\"pG6#\"\"!" }{TEXT -1 57 " were in the 2nd quadrant and \+ C is a small circle with " }{XPPEDIT 18 0 "p[0];" "6#&%\"pG6#\"\"!" } {TEXT -1 39 " as its center and lying in the square." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "plot([[t,-1,t=-1..1],[t,1,t=-1..1],[-1,t ,t=-1..1],[1,t,t=-1..1],\n [-1/2+cos(t)/4,1/2+sin(t)/4,t=-Pi..Pi] ],\n color=black,scaling=constrained);" }}}{EXCHG }{PARA 0 "" 0 "" {TEXT -1 52 " Agree that the integral of u about that circle " }}{PARA 0 "" 0 "" {TEXT -1 8 " " }{XPPEDIT 18 0 "1/(2*Pi);" "6# *&\"\"\"F$*&\"\"#F$%#PiGF$!\"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "int( u(x(t),y(t)),t = -Pi .. Pi);" "6#-%$intG6$-%\"uG6$-%\"xG6#%\"tG-%\"yG6 #F,/F,;,$%#PiG!\"\"F3" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "w ould give the average value of u on that circle. We show below that t he value of u at " }{XPPEDIT 18 0 "p[0];" "6#&%\"pG6#\"\"!" }{TEXT -1 19 " is this integral. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "You ask what is the significance of this result? It \+ would imply that the value of u at " }{XPPEDIT 18 0 "p[0];" "6#&%\"pG6 #\"\"!" }{TEXT -1 55 " is the average of the values of u on any circle about " }{XPPEDIT 18 0 "p[0];" "6#&%\"pG6#\"\"!" }{TEXT -1 133 " whic h lies in the region on which u is defined. This gives an understandin g of the Maximum Principle. How could u have a maximum at " }{XPPEDIT 18 0 "p[0];" "6#&%\"pG6#\"\"!" }{TEXT -1 46 " if it is the average of \+ values all around it?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "The idea generalizes to three dimensions." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 479 "Think of a cube w ith specified, unchanging temperature, perhaps different on each face. The temperature at each point on the interior will be exactly the ave rage of the surrounding temperatures in the sense described above. Thi s seems to be an interesting way to conceive the construction of the h eat distribution. One might think it would not be possible to make suc h distributions -- each point has the average value property -- knowin g only the temperatures on the boundaries." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "That it is possible is surely \+ a tribute to our predecessors: Fourier, Cauchy, Laplace, Poisson, etc. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "Here \+ is verification of this property: Think of " }{XPPEDIT 18 0 "p[0];" "6 #&%\"pG6#\"\"!" }{TEXT -1 14 " as being the " }{TEXT 259 6 "center" } {TEXT -1 45 " of a circle with radius r. Then u satisfies " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT -1 74 "u = 0 in this circle and defi nes boundary conditions on the circle. Then, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 "1/( 2*Pi);" "6#*&\"\"\"F$*&\"\"#F$%#PiGF$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "int(u(r,theta),theta = -Pi .. Pi);" "6#-%$intG6$-%\"uG6$%\"rG%&t hetaG/F*;,$%#PiG!\"\"F." }{TEXT -1 3 " = " }{XPPEDIT 18 0 "1/(2*Pi);" "6#*&\"\"\"F$*&\"\"#F$%#PiGF$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "in t(a[0],theta = -Pi .. Pi);" "6#-%$intG6$&%\"aG6#\"\"!/%&thetaG;,$%#PiG !\"\"F." }{TEXT -1 3 " + " }{XPPEDIT 18 0 "sum(a[n]*r^n*int(cos(n*thet a),theta = -Pi .. Pi),n);" "6#-%$sumG6$*(&%\"aG6#%\"nG\"\"\")%\"rGF*F+ -%$intG6$-%$cosG6#*&F*F+%&thetaGF+/F5;,$%#PiG!\"\"F9F+F*" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "sum(b[n]*r^n*int(sin(n*theta),theta = -Pi .. Pi) ,n);" "6#-%$sumG6$*(&%\"bG6#%\"nG\"\"\")%\"rGF*F+-%$intG6$-%$sinG6#*&F *F+%&thetaGF+/F5;,$%#PiG!\"\"F9F+F*" }{TEXT -1 2 " ." }}{PARA 0 "" 0 " " {TEXT -1 39 " = " }{XPPEDIT 18 0 "a[0];" "6#&%\"aG6#\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "To see that all the sum terms \+ actually are zero, one has only to perform the integration and remembe r that n is an integer." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "a ssume(n,integer):\nint(cos(n*theta),theta=-Pi..Pi);\nint(sin(n*theta), theta=-Pi..Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 260 28 "Laplace's Equation on a Ring" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "We find a function u which satisfies " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 " " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT -1 14 " u = 0, for 1" }{XPPEDIT 18 0 "` \+ ` <= ` `;" "6#1%\"~GF$" }{TEXT -1 1 "r" }{XPPEDIT 18 0 "` ` <= ` `;" " 6#1%\"~GF$" }{TEXT -1 6 "2, - " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" } {XPPEDIT 18 0 "` ` <= ` `;" "6#1%\"~GF$" }{XPPEDIT 18 0 "theta;" "6#%& thetaG" }{XPPEDIT 18 0 "` ` <= ` `;" "6#1%\"~GF$" }{XPPEDIT 18 0 "Pi; " "6#%#PiG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "with" }} {PARA 0 "" 0 "" {TEXT -1 37 " u(1," } {XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 8 ") = u(2," }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 8 ") = sin(" }{XPPEDIT 18 0 "theta " "6#%&thetaG" }{TEXT -1 9 ") for - " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{XPPEDIT 18 0 "` ` <= ` `;" "6#1%\"~GF$" }{XPPEDIT 18 0 "theta;" "6#% &thetaG" }{XPPEDIT 18 0 "` ` <= ` `;" "6#1%\"~GF$" }{XPPEDIT 18 0 "Pi; " "6#%#PiG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "We have seen that the two differential equations \+ associated with this problem are" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 "theta;" "6#%&thetaG " }{TEXT -1 6 " '' + " }{XPPEDIT 18 0 "lambda^2*theta;" "6#*&%'lambdaG \"\"#%&thetaG\"\"\"" }{TEXT -1 6 " = 0, " }{XPPEDIT 18 0 "theta;" "6#% &thetaG" }{TEXT -1 4 " (- " }{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 3 " ( " } {XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 7 "), and " }{XPPEDIT 18 0 "th eta;" "6#%&thetaG" }{TEXT -1 5 " '(- " }{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 4 " \+ '( " }{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 2 ")." }}{PARA 0 "" 0 " " {TEXT -1 3 "and" }}{PARA 0 "" 0 "" {TEXT -1 18 " r( rR ') ' - " }{XPPEDIT 18 0 "lambda^2;" "6#*$%'lambdaG\"\"#" }{TEXT -1 7 " R = 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "The sol utions for the first equation, with eigenvalues and eigenfunctions are " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " \+ " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 9 " = 0, " } {XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 5 " = 1." }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 "-lambda^2;" "6#,$*$%'lambdaG\" \"#!\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "-n^2;" "6#,$*$%\"nG\"\"#! \"\"" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "Theta;" "6#%&ThetaG" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 10 ") = cos(n \+ " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 13 ") or sin(n " } {XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 2 ")." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "The second equation is \+ now " }{XPPEDIT 18 0 "r^2;" "6#*$%\"rG\"\"#" }{TEXT -1 16 " R '' + r \+ R ' - " }{XPPEDIT 18 0 "n^2;" "6#*$%\"nG\"\"#" }{TEXT -1 6 "R = 0." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "This has \+ led to two familiar ordinary differential equations which have solutio ns" }}{PARA 0 "" 0 "" {TEXT -1 37 " R(r) = 1 and ln(r), \+ " }{XPPEDIT 18 0 "Theta(theta);" "6#-%&ThetaG6#%&thetaG" }{TEXT -1 20 " = 1, in case n = 0," }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 0 "" 0 "" {TEXT -1 21 " R(r) = " }{XPPEDIT 18 0 "r^n" "6# )%\"rG%\"nG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "r^(-n)" "6#)%\"rG,$% \"nG!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "Theta(theta)" "6#-%&ThetaG 6#%&thetaG" }{TEXT -1 9 " = sin(n " }{XPPEDIT 18 0 "theta" "6#%&thetaG " }{TEXT -1 11 ") and cos(n" }{XPPEDIT 18 0 "theta" "6#%&thetaG" } {TEXT -1 10 ") in case " }{XPPEDIT 18 0 "n <> 0;" "6#0%\"nG\"\"!" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 20 "Combine these to get" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 " u(r ," }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "a[0];" "6#&%\"aG6#\"\"!" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "b[0]; " "6#&%\"bG6#\"\"!" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(r);" "6#-%#lnG6 #%\"rG" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "sum((a[p]*r^p+a[-p]*r^(-p))* cos(p*theta)+(b[p]*r^p+b[-p]*r^(-p))*sin(p*theta),p)" "6#-%$sumG6$,&*& ,&*&&%\"aG6#%\"pG\"\"\")%\"rGF-F.F.*&&F+6#,$F-!\"\"F.)F0,$F-F5F.F.F.-% $cosG6#*&F-F.%&thetaGF.F.F.*&,&*&&%\"bG6#F-F.)F0F-F.F.*&&FA6#,$F-F5F.) F0,$F-F5F.F.F.-%$sinG6#*&F-F.F " 0 "" {MPLTEXT 1 0 54 "solve(\{b[1]+b[ -1]=1,2*b[1]+1/2*b[-1]=1\},\{b[1],b[-1]\});" }}}{PARA 0 "" 0 "" {TEXT -1 18 "We can now make u." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "u:=(r,theta)->(r+2/r)/3*sin(theta);" }}}{PARA 0 "" 0 "" {TEXT -1 42 " We check the boundary conditions and that " }{TEXT 257 1 "u" }{TEXT -1 27 " solves Laplace's Equation." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "u(1,theta); u(2,theta);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "diff(r*diff(u(r,theta),r),r)/r+diff(u(r,theta),theta, theta)/r^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%); " }}}{PARA 0 "" 0 "" {TEXT -1 125 "It is interesting to view the graph of this solution for Laplace's Equation on a ring, keeping in mind th e maximum principal." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "cyl inderplot([r,theta,u(r,theta)],r=1..2,theta=0..2*Pi,\n orient ation=[-5,70],axes=NORMAL, shading=zhue, lightmodel=light1, style=patc hnogrid);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 261 33 "Laplace's Equation on a Half Disk" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 28 "We suppose that u satisfies " }{XPPEDIT 18 0 "Delt a;" "6#%&DeltaG" }{TEXT -1 48 " u = 0 for 0 < r < 1, with u( r, 0) = 0 = u(r, " }{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 10 "), u( 1, " } {XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 8 " ) = . " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "We know what is th e PDE: " }}{PARA 0 "" 0 "" {TEXT -1 19 " " } {XPPEDIT 18 0 "diff(r*diff(v(r,theta),r),r)/r+diff(v(r,theta),`$`(thet a,2))/(r^2);" "6#,&*&-%%diffG6$*&%\"rG\"\"\"-F&6$-%\"vG6$F)%&thetaGF)F *F)F*F)!\"\"F**&-F&6$-F.6$F)F0-%\"$G6$F0\"\"#F**$F)F:F1F*" }{TEXT -1 8 " = 0, " }}{PARA 0 "" 0 "" {TEXT -1 83 "and that this leads to the ordinary differential equations with boundary conditions" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 "Theta;" "6#%&ThetaG" }{TEXT -1 6 " '' + " }{XPPEDIT 18 0 "lambda ^2*Theta;" "6#*&%'lambdaG\"\"#%&ThetaG\"\"\"" }{TEXT -1 7 " = 0, " } {XPPEDIT 18 0 "Theta;" "6#%&ThetaG" }{TEXT -1 11 " (0) = 0, " } {XPPEDIT 18 0 "Theta;" "6#%&ThetaG" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "Pi ;" "6#%#PiG" }{TEXT -1 5 ") = 0" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 17 " r( rR ') ' - " }{XPPEDIT 18 0 "lambda ^2;" "6#*$%'lambdaG\"\"#" }{TEXT -1 7 " R = 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "It follows that " } {XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 7 " = n, " }{XPPEDIT 18 0 "Theta;" "6#%&ThetaG" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "theta;" "6# %&thetaG" }{TEXT -1 10 ") = sin(n " }{XPPEDIT 18 0 "theta;" "6#%&theta G" }{TEXT -1 14 "), and R(r) = " }{XPPEDIT 18 0 "r^n;" "6#)%\"rG%\"nG " }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 24 " General solutio n is" }}{PARA 0 "" 0 "" {TEXT -1 20 " u(r, " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 5 " ) = " }{XPPEDIT 18 0 "sum(a[n ]*r^n*sin(n*theta),n);" "6#-%$sumG6$*(&%\"aG6#%\"nG\"\"\")%\"rGF*F+-%$ sinG6#*&F*F+%&thetaGF+F+F*" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 31 "The requirement that 1 = u( 1, " }{XPPEDIT 18 0 "theta;" "6#%&t hetaG" }{TEXT -1 5 " ) = " }{XPPEDIT 18 0 "sum(a[n]*sin(n*theta),n);" "6#-%$sumG6$*&&%\"aG6#%\"nG\"\"\"-%$sinG6#*&F*F+%&thetaGF+F+F*" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 21 "We computer the a 's." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "f:=theta->theta^4*(Pi-theta);" }}}{PARA 0 "" 0 "" {TEXT -1 86 "W e compute 10 coefficients, but printing them does not seem to serve a \+ useful purpose." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "for n fr om 1 to 10 do\n a[n]:=int(f(theta)*sin(n*theta),theta=0..Pi)/\n \+ int(sin(n*theta)^2,theta=0..Pi):\nend do:\nn:='n':" }}} {PARA 0 "" 0 "" {TEXT -1 28 "Here is the definition of u." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "u:=(r,theta)->sum(a[n]*r^n*sin(n*th eta),n=1..10);" }}}{PARA 0 "" 0 "" {TEXT -1 23 "We draw the graph of u ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "cylinderplot([r,theta, u(r,theta)],r=0..1,theta=0..Pi,\n orientation=[-18,75],axes=N ORMAL, shading=zhue, numpoints=1000);" }}}{EXCHG }}{EXCHG }{EXCHG } {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 264 "In this \+ Section, we have illustrated that Laplace's equation on half disks or \+ on rings can be solved with similar techniques to those that have come before. It is not a surprise. The main point of interest is how to ex press Laplace's Equation in polar coordinates." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "On the other hand, the av erage value property is conceptually pretty." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "EMAIL: herod@math.gatech.edu \+ or jherod@tds.net" }}{PARA 0 "" 0 "" {TEXT -1 38 "URL: http://www.ma th.gatech.edu/~herod" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 36 "Copyright \251 2003 by James V. Herod" }}{PARA 258 "" 0 "" {TEXT -1 19 "All rights reserved" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 1 1 2 33 1 1 }