{VERSION 6 0 "IBM INTEL NT" "6.0" } {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 "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 "Normal" -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 "T imes" 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 {PARA 257 "" 0 "" {TEXT 256 56 "Section 6.5: Laplace's Equatio n on a Ring or a Half Disk" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT 258 30 "Maple Packages for Section 6.5" }} {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 solution for " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT -1 27 " u = 0 on any region. Take " } {XPPEDIT 18 0 "p[0];" "6#&%\"pG6#\"\"!" }{TEXT -1 77 " to be any poin t 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 smal l enough that the circle lies in the region. We draw the picture as th ough 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=blac k,scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{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 "would give the ave rage value of u on that circle. We show below that the value of u at \+ " }{XPPEDIT 18 0 "p[0];" "6#&%\"pG6#\"\"!" }{TEXT -1 19 " is this inte gral. " }}{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 " which lies in \+ the region on which u is defined. This gives an understanding of the M aximum 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 with spe cified, unchanging temperature, perhaps different on each face. The te mperature at each point on the interior will be exactly the average of the surrounding temperatures in the sense described above. This seems to be an interesting way to conceive the construction of the heat dis tribution. One might think it would not be possible to make such distr ibutions -- each point has the average value property -- knowing 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 v erification 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;,$%#PiG!\"\"F9F+F*" }{TEXT -1 2 " ." }}{PARA 0 "" 0 " " {TEXT -1 39 " = " }{XPPEDIT 18 0 "a[0];" "6#&%\"aG6#\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "To see that all the sum terms \+ actually are zero, one has only to perform the integration and remembe r that n is an integer." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "i nt(cos(n*theta),theta=-Pi..Pi);\nint(sin(n*theta),theta=-Pi..Pi);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 260 28 "Laplace's Equation on a Ring" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "We find a function u which sati sfies " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 " " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT -1 14 " u = 0, for 1" }{XPPEDIT 18 0 "` ` <= ` `;" "6#1%\"~GF$ " }{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$" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "with" }}{PARA 0 "" 0 "" {TEXT -1 37 " \+ u(1," }{XPPEDIT 18 0 "theta" "6#%&thet aG" }{TEXT -1 8 ") = u(2," }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 8 ") = sin(" }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 9 ") for - " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{XPPEDIT 18 0 "` ` <= ` `;" "6#1 %\"~GF$" }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{XPPEDIT 18 0 "` ` <= ` `;" "6#1%\"~GF$" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "We have s een that the two differential equations associated with this problem a re" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " \+ " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 6 " '' + " } {XPPEDIT 18 0 "lambda^2*theta;" "6#*&%'lambdaG\"\"#%&thetaG\"\"\"" } {TEXT -1 6 " = 0, " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 4 " (- " }{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 3 " ( " }{XPPEDIT 18 0 "pi;" "6#%#pi G" }{TEXT -1 7 "), and " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 5 " '(- " }{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 4 ") = " } {XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 4 " '( " }{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }} {PARA 0 "" 0 "" {TEXT -1 18 " r( rR ') ' - " }{XPPEDIT 18 0 "lambd a^2;" "6#*$%'lambdaG\"\"#" }{TEXT -1 7 " R = 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "The solutions for the fir st equation, with eigenvalues and eigenfunctions are" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 "l ambda;" "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 ordin ary differential equations which have solutions" }}{PARA 0 "" 0 "" {TEXT -1 37 " R(r) = 1 and ln(r), " }{XPPEDIT 18 0 "Th eta(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,$%\"nG!\"\"" }{TEXT -1 2 " , " }{XPPEDIT 18 0 "Theta(theta)" "6#-%&ThetaG6#%&thetaG" }{TEXT -1 9 " = sin(n " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 11 ") and co s(n" }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 10 ") in case " } {XPPEDIT 18 0 "n <> 0;" "6#0%\"nG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 20 "Combine these to get" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 10 " u(r," }{XPPEDIT 18 0 "theta" "6 #%&thetaG" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "a[0];" "6#&%\"aG6#\"\"! " }{TEXT -1 3 " + " }{XPPEDIT 18 0 "b[0];" "6#&%\"bG6#\"\"!" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(r);" "6#-%#lnG6#%\"rG" }{TEXT -1 3 " + " } {XPPEDIT 18 0 "sum((a[p]*r^p+a[-p]*r^(-p))*cos(p*theta)+(b[p]*r^p+b[-p ]*r^(-p))*sin(p*theta),p)" "6#-%$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.-%$sinG6#*&F- F.F " 0 "" {MPLTEXT 1 0 54 "solve(\{b[1]+b[-1]=1,2*b[1]+1/2*b[-1]=1\}, \{b[1],b[-1]\});" }}}{PARA 0 "" 0 "" {TEXT -1 18 "We can now make u." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "u:=(r,theta)->(r+2/r)/3*si n(theta);" }}}{PARA 0 "" 0 "" {TEXT -1 42 "We check the boundary condi tions and that " }{TEXT 257 1 "u" }{TEXT -1 27 " solves Laplace's Equa tion." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "u(1,theta); u(2,the ta);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "diff(r*diff(u(r,the ta),r),r)/r+diff(u(r,theta),theta,theta)/r^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}}{PARA 0 "" 0 "" {TEXT -1 124 "I t is interesting to view the graph of this solution for Laplace's Equa tion on a ring keeping in mind the maximum principal." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "cylinderplot([r,theta,u(r,theta)],r=1..2, theta=0..2*Pi,\n orientation=[-5,70],axes=NORMAL);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT 261 33 "Laplace's Equation on \+ a Half Disk" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "We suppose that u satisfies " }{XPPEDIT 18 0 "Delta;" "6#%&Delt aG" }{TEXT -1 48 " u = 0 for 0 < r < 1, with u( r, 0) = 0 = u(r, " } {XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 10 "), u( 1, " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 8 " ) = . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "We know what is the PDE: \+ " }}{PARA 0 "" 0 "" {TEXT -1 19 " " }{XPPEDIT 18 0 "diff(r*diff(v(r,theta),r),r)/r+diff(v(r,theta),`$`(theta,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 diffe rential 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 "T heta;" "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#*$%'lambdaG\" \"#" }{TEXT -1 7 " R = 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 18 "It follows that " }{XPPEDIT 18 0 "lambda;" "6#%' lambdaG" }{TEXT -1 7 " = n, " }{XPPEDIT 18 0 "Theta;" "6#%&ThetaG" } {TEXT -1 1 "(" }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 10 ") = \+ sin(n " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 14 "), and R(r) = " }{XPPEDIT 18 0 "r^n;" "6#)%\"rG%\"nG" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 24 " General solution is" }}{PARA 0 "" 0 "" {TEXT -1 20 " u(r, " }{XPPEDIT 18 0 "theta;" "6#%&thetaG " }{TEXT -1 5 " ) = " }{XPPEDIT 18 0 "sum(a[n]*r^n*sin(n*theta),n);" " 6#-%$sumG6$*(&%\"aG6#%\"nG\"\"\")%\"rGF*F+-%$sinG6#*&F*F+%&thetaGF+F+F *" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 31 "The requirement tha t 1 = u( 1, " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{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-th eta);" }}}{PARA 0 "" 0 "" {TEXT -1 86 "We compute 10 coefficients, but printing them does not seem to serve a useful purpose." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "for n from 1 to 10 do\n a[n]:= int(f(theta)*sin(n*theta),theta=0..Pi)/\n int(sin(n*the ta)^2,theta=0..Pi):\nod:\nn:='n':" }}}{PARA 0 "" 0 "" {TEXT -1 28 "Her e is the definition of u." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "u:=(r,theta)->sum(a[n]*r^n*sin(n*theta),n=1..10);" }}}{PARA 0 "" 0 " " {TEXT -1 23 "We draw the graph of u." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "cylinderplot([r,theta,u(r,theta)],r=0..1,theta=0..Pi, \n orientation=[-18,75],axes=NORMAL);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 264 "In \+ this Section, we have illustrated that Laplace's equation on half disk s 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 express Laplace's Equation in polar coordinates." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "On the other hand, the \+ average 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.math.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" 56 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 1 1 2 33 1 1 }