{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 Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 1 16 0 0 0 0 0 1 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 "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 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 0 "" 0 "" {TEXT 260 67 "Eigenvalue Problems in Li near Algebra and in Differential Equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 14 "Linear Algebra" }}{PARA 0 "" 0 "" {TEXT -1 20 "1) load the package:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "2) define a real, symmetric 2-by-2 matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "A:=Matrix([[1,3],[3,4]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"*%)3m\\\"-%'MATRIXG6#7$7 $\"\"\"\"\"$7$F/\"\"%%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Determinant(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"&" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "The matrix is not singular." }} {PARA 0 "" 0 "" {TEXT -1 147 "Now define a quadratic form. We need (x, y) both as a column, and as a row vector. This is best done using matr ices, rather than the Vector command." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "P:=Matrix([[x],[y]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG-%'RTABLEG6%\"*kSm\\\"-%'MATRIXG6#7$7#%\"xG7#%\"yG%'Matrix G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Prow:=Transpose(P);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ProwG-%'RTABLEG6%\"*/(o'\\\"-%'MAT RIXG6#7#7$%\"xG%\"yG%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "QF:=Prow . A . P;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#QFG-%'RT ABLEG6%\"*+9n\\\"-%'MATRIXG6#7#7#,&*&,&%\"xG\"\"\"*&\"\"$F2%\"yGF2F2F2 F1F2F2*&,&*&F4F2F1F2F2*&\"\"%F2F5F2F2F2F5F2F2%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 119 "Technically, this is a one-by-one matrix . To get our equation for the conic section we have to do the cumberso me thing:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "conic:=QF[1,1] =1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&conicG/,&*&,&%\"xG\"\"\"*&\" \"$F*%\"yGF*F*F*F)F*F**&,&*&F,F*F)F*F**&\"\"%F*F-F*F*F*F-F*F*F*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "How would we graph this object?" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 142 "We coul d try isolating y, and then using the regular plot command. However, t here is a command in Maple that should save us from this trouble:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "#?implicitplot" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 70 "implicitplot(conic, x=-2..2, y=-2..2,scaling=c onstrained,thickness=3);" }}{PARA 13 "" 1 "" {GLPLOT2D 864 400 400 {PLOTDATA 2 "6&-%'CURVESG6gp7$7$$!33++++++S=!#<$\"37.=\\qPdlB!#=7$$!3w )**********R&=F*$\"3'o************R#F-7$7$F/$\"3e'************R#F-7$$! \"#\"\"!$\"3EmznSk=rFF-7$F'7$$!3T++++++N=F*$\"3a,+++++]BF-7$7$$!3;++++ ++!o\"F*$\"33kjjjjj$)>F-F>7$FD7$$!3\"QE0@%ot>;F*$\"3)Qj_5Uotz\"F-7$7$$ !3C++++++?:F*$\"39.^v(QpMd\"F-FJ7$FP7$$!3Ox6%HN#)3S\"F*$\"3unF*7$$!3\"ef QaF*7$7$$!3Q+++++++7F*$\"3]xmmmmmmk!#>7$$!3 )=#p2Bp2`7F*$\"3[l************zFjo7$F[p7$Fgn$\"3c*[`R\"eDB6F-7$7$Ffo$ \"3#))[`R\"eDLM([zr[zr\"Fjo7$$!3Uymmmm;z$*F-$!3u#HLLLL $3AFjo7$7$$!3O.++++++))F-$!3'ywyyyyyy$FjoF[s7$7$$!3[/++++++))F-$\"3g^? G^?Gh8F*7$$!3Kbbbbbb<5F*$\"3d************>:F*7$F\\tF[r7$7$$!31.++++++s F-$!3))e&G9dG9(**Fjo7$$!3=A#p2Bp2t(F-$!3)[.++++++)Fjo7$Fht7$Fhs$!3kq(y yyyyy$Fjo7$7$$!3=/++++++sF-$\"3!Qz8C\"F*7$7$$!3WD9dG9dotF-Fju7$$!3)\\$*********\\n)F-$\"3g#******* **\\Z8F*7$7$$!3Iy([!y[!yy)F-$\"3m************f8F*F`v7$7$$!3=x([!y[!yy) F-$\"3U************f8F*Fgs7$7$FcuFft7$$!3vzmmmmm;oF-$!3!zKLLLLL=\"F-7$ 7$$!3'Q++++++g&F-$!3^%*o?'eF[k\"F-Fcw7$7$Fjw$\"3p'H'H'H'H15F*7$$!33tn4 (Q[N*eF-$\"3\")************R5F*7$FbxFbu7$7$$!3c.++++++SF-$!3]pQ[N>unCF -7$$!38$***********RTF-$!3_.++++++CF-7$F^y7$$!3w-++++++cF-F\\x7$7$Fjx$ \"3\\.+++++O%)F-7$$!3#f.Pq.PqL%F-$\"3#*)************z)F-7$F[zF_x7$Fix7 $$!3;;+++++DQF-$!3=\"**********\\d#F-7$7$Fay$!3%y**********fJ$F-Fbz7$7 $$!3C.++++++CF-$\"3))o&p3EyM%pF-7$$!3)=VI\"R<_cEF-$\"3_(************>( F-7$Fa[l7$Fjx$\"3Q-+++++O%)F-7$7$F[u$!3WsS2uS2uVF-7$$!3Y=)eqkw/>w(FjoFjw7$F`]lF[\\l7$7$F^p$ \"3yvS2uS2uVF-7$$!3yLZ!>w/>w(Fjo$\"3?(************f&F-7$Fh]lFi\\l7$F]] l7$$\"3aS&*********\\#*Fjo$!3Bh*********\\s&F-7$7$F5$!3+g&p3EyM%pF-F_^ l7$7$F5$\"3G-+++++;LF-7$$\"3=G)eqkunCF-F[`l7$7$F [^l$!3-'H'H'H'H15F*7$$\"3\"4/Pq.PqL%F-Fhs7$Fh`lFc_l7$7$$\"3K)********* ***f&F-$\"37)*o?'eF[k\"F-7$$\"3!f++++++9%F-F57$FbalFa`l7$7$Fd[l$!3q#z8 C(F-$\"3Iy&G9dG9(**Fjo7$7$Fex$!3Bj9MYTjW:F*7$$\"3*fbb bbbv,\"F*FQ7$FeelF[cl7$7$$\"3q'************z)F-$\"3i&zyyyyyy$Fjo7$$\"3 Wemmmm;z$*F-$\"3'zNLLLL$3AFjo7$7$Fex$!3o7([zr[zr\"FjoF_fl7$7$Fju$!3%z[ `R\"eDLF*7$$\"3'e! \\etPg'G\"F*$!3I++++++S=F*7$7$F_hlF(Fifl7$7$Fiv$!31([`R\"eDB6F-7$$\"3K Cp2Bp2`7F*F[u7$Fihl7$Fju$!3%3mmmmmmY'Fjo7$7$$\"3E'fQaw6%HN#)37F-7$7$F_t$!3#45bxQpMd\"F-Ffi l7$7$F_t$!3k+^v(QpMd\"F-7$$\"3.i_5Uot>;F*$!3gGE0@%otz\"F-7$7$Far$!3qij jjjj$)>F-Fcjl7$Fijl7$$\"3&))**********\\$=F*$!3#o***********\\BF-7$7$F [q$!3c-=\\qPdlBF-F][m7$7$Fbo$!3/kznSk=rFF-7$$\"3U,+++++a=F*Fay7$7$F[\\ m$!3!Q++++++S#F-7$F[q$!3I-=\\qPdlBF--%'COLOURG6&%$RGBG\"\"\"F:F:-%*THI CKNESSG6#\"\"$-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6$%\"xG%\"yG" 1 2 0 1 10 3 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "Our section happens to be a hyper bola rather than an ellipse. This was not obvious from the expression: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "simplify(conic);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*$)%\"xG\"\"#\"\"\"F)*(\"\"'F)F'F)% \"yGF)F)*&\"\"%F))F,F(F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 " The eigenvalue-eigenvector problem for matrix A (which defines the con ic section) will make it obvious." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Eigenvalues(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-% 'RTABLEG6%\"*cUm\\\"-%'MATRIXG6#7$7#,&#\"\"&\"\"#\"\"\"*(\"\"$F0F/!\" \"F.#F0F/F07#,&F-F0*(F2F0F/F3F.F4F3&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"*OVm\\\"-%'MATRIXG6#7$7#$\"+m>5ae!\"*7#$! *m>5a)F.&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 169 "One of the eigenvalues is negative. In an ellipse the eigenvalues are related to the inverse of the semimajor axes. A negative value implie s that it can't be an ellipse." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 192 "The eigenvectors should give us the prin cipal axes. Those are the symmetry axes for the conic section. They de fine some new coordinate system which is rotated with respect to the o riginal one." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "Before we continue let us switch to floating point numbers:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "A:=evalf(A);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"*O8o\\\"-%'MATRIXG6#7$7$$\" \"\"\"\"!$\"\"$F07$F1$\"\"%F0%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Eigenvalues(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-% 'RTABLEG6%\"*cWm\\\"-%'MATRIXG6#7$7#^$$!32Zo\\i'>5a)!#=$\"\"!F17#^$$\" 3q%o\\i'>5ae!# " 0 "" {MPLTEXT 1 0 20 "EV:=Eigenvectors( A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#EVG6$-%'RTABLEG6%\"*OXm\\\"- %'MATRIXG6#7$7#^$$!32Zo\\i'>5a)!#=$\"\"!F47#^$$\"3q%o\\i'>5ae!#@6Jd_F2F37$^$$\"3![L\">@6Jd_F2F3FE%'MatrixG" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 97 "The eigenvectors appear as columns in a matrix. Let us \+ verify the eigenvalue-eigenvector problem:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 31 "V1:=SubMatrix(EV[2],1..2,1..1);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#V1G-%'RTABLEG6%\"*#z$p\\\"-%'MATRIXG6#7$7#^$$!3y)R ?N33l])!#=$\"\"!F37#^$$\"3![L\">@6Jd_F1F2%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "V2:=SubMatrix(EV[2],1..2,2..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#V2G-%'RTABLEG6%\"*glp\\\"-%'MATRIXG6#7$7# ^$$!3![L\">@6Jd_!#=$\"\"!F37#^$$!3y)R?N33l])F1F2%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Evals:=EV[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&EvalsG-%'RTABLEG6%\"*OXm\\\"-%'MATRIXG6#7$7#^$$!32Zo \\i'>5a)!#=$\"\"!F37#^$$\"3q%o\\i'>5ae!# " 0 "" {MPLTEXT 1 0 17 "A.V1,Evals[1]*V1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\"*K!)p\\\"-%'MATRIXG6#7$7#^$$\" 3m0O0!GDaE(!#=$\"\"!F17#^$$!31dezl(z-\\%F/F0%'MatrixG-F$6%\"*G2q\\\"-F (6#7$7#^$$\"3'f'4jz_UlsF/$!\"!F17#^$$!3YXY`l(z-\\%F/F0F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "A.V2,Evals[2]*V2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\"*?Zq\\\"-%'MATRIXG6#7$7#^$$!37`_ " 0 "" {MPLTEXT 1 0 47 "V1:=convert(V1,Vector): V2:=convert(V2,Vector ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "PL0:=implicitplot(con ic, x=-2..2, y=-2..2,scaling=constrained,thickness=3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "PL1:=implicitplot(V1[1]*x+V1[2]*y=0 ,x=-2..2,y=-2..2,color=blue,scaling=constrained,thickness=3):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "PL2:=implicitplot(V2[1]*x+V2 [2]*y=0,x=-2..2,y=-2..2,color=green,scaling=constrained,thickness=3): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "display(PL0,PL1,PL2);" }}{PARA 13 "" 1 "" {GLPLOT2D 503 268 268 {PLOTDATA 2 "6'-%'CURVESG6hp7 $7$$!33++++++S=!#<$\"37.=\\qPdlB!#=7$$!3w)**********R&=F*$\"3'o******* *****R#F-7$7$F/$\"3e'************R#F-7$$!\"#\"\"!$\"3EmznSk=rFF-7$F'7$ $!3T++++++N=F*$\"3a,+++++]BF-7$7$$!3;++++++!o\"F*$\"33kjjjjj$)>F-F>7$F D7$$!3\"QE0@%ot>;F*$\"3)Qj_5Uotz\"F-7$7$$!3C++++++?:F*$\"39.^v(QpMd\"F -FJ7$FP7$$!3Ox6%HN#)3S\"F*$\"3unF*7$$!3\"efQaF*7$7$$!3Q+++++++7F*$\"3]xmmmmmmk!#>7$$!3)=#p2Bp2`7F*$\"3[l********** **zFjo7$F[p7$Fgn$\"3c*[`R\"eDB6F-7$7$Ffo$\"3#))[`R\"eDLM([zr[zr\"Fjo7$$!3Uymmmm;z$*F-$!3u#HLLLL$3AFjo7$7$$!3O.++++++))F-$!3' ywyyyyyy$FjoF[s7$7$$!3[/++++++))F-$\"3g^?G^?Gh8F*7$$!3Kbbbbbb<5F*$\"3d ************>:F*7$F\\tF[r7$7$$!31.++++++sF-$!3))e&G9dG9(**Fjo7$$!3=A#p 2Bp2t(F-$!3)[.++++++)Fjo7$Fht7$Fhs$!3kq(yyyyyy$Fjo7$7$$!3=/++++++sF-$ \"3!Qz8C\"F*7$7$$!3WD9dG9dot F-Fju7$$!3)\\$*********\\n)F-$\"3g#*********\\Z8F*7$7$$!3Iy([!y[!yy)F- $\"3m************f8F*F`v7$7$$!3=x([!y[!yy)F-$\"3U************f8F*Fgs7$ 7$FcuFft7$$!3vzmmmmm;oF-$!3!zKLLLLL=\"F-7$7$$!3'Q++++++g&F-$!3^%*o?'eF [k\"F-Fcw7$7$Fjw$\"3p'H'H'H'H15F*7$$!33tn4(Q[N*eF-$\"3\")************R 5F*7$FbxFbu7$7$$!3c.++++++SF-$!3]pQ[N>unCF-7$$!38$***********RTF-$!3_. ++++++CF-7$F^y7$$!3w-++++++cF-F\\x7$7$Fjx$\"3\\.+++++O%)F-7$$!3#f.Pq.P qL%F-$\"3#*)************z)F-7$F[zF_x7$Fix7$$!3;;+++++DQF-$!3=\"******* ***\\d#F-7$7$Fay$!3%y**********fJ$F-Fbz7$7$$!3C.++++++CF-$\"3))o&p3EyM %pF-7$$!3)=VI\"R<_cEF-$\"3_(************>(F-7$Fa[l7$Fjx$\"3Q-+++++O%)F -7$7$F[u$!3WsS2uS2uVF-7$$!3Y=)eqkw/>w(FjoFjw7$F`]lF[\\l7$7$F^p$\"3yvS2uS2uVF-7$$!3yLZ!>w/>w (Fjo$\"3?(************f&F-7$Fh]lFi\\l7$F]]l7$$\"3aS&*********\\#*Fjo$! 3Bh*********\\s&F-7$7$F5$!3+g&p3EyM%pF-F_^l7$7$F5$\"3G-+++++;LF-7$$\"3 =G)eqkunCF-F[`l7$7$F[^l$!3-'H'H'H'H15F*7$$\"3\"4 /Pq.PqL%F-Fhs7$Fh`lFc_l7$7$$\"3K)************f&F-$\"37)*o?'eF[k\"F-7$$ \"3!f++++++9%F-F57$FbalFa`l7$7$Fd[l$!3q#z8C(F-$\"3Iy&G 9dG9(**Fjo7$7$Fex$!3Bj9MYTjW:F*7$$\"3*fbbbbbv,\"F*FQ7$FeelF[cl7$7$$\"3 q'************z)F-$\"3i&zyyyyyy$Fjo7$$\"3Wemmmm;z$*F-$\"3'zNLLLL$3AFjo 7$7$Fex$!3o7([zr[zr\"FjoF_fl7$7$Fju$!3%z[`R\"eDLF*7$$\"3'e!\\etPg'G\"F*$!3I++++++S=F*7$7$ F_hlF(Fifl7$7$Fiv$!31([`R\"eDB6F-7$$\"3KCp2Bp2`7F*F[u7$Fihl7$Fju$!3%3m mmmmmY'Fjo7$7$$\"3E'fQaw6%HN#)37F-7$7$F_t$!3#45bxQpMd\"F-Ffil7$7$F_t$!3k+^v(QpMd\"F-7$$\"3 .i_5Uot>;F*$!3gGE0@%otz\"F-7$7$Far$!3qijjjjj$)>F-Fcjl7$Fijl7$$\"3&))** ********\\$=F*$!3#o***********\\BF-7$7$F[q$!3c-=\\qPdlBF-F][m7$7$Fbo$! 3/kznSk=rFF-7$$\"3U,+++++a=F*Fay7$7$F[\\m$!3!Q++++++S#F-7$F[q$!3I-=\\q PdlBF--%'COLOURG6&%$RGBG\"\"\"F:F:-%*THICKNESSG6#\"\"$-F$6`p7$7$$!3w8^ Qxz1O7F*F87$$!3\"HDIfB\"HA7F*$!3GZ(pSw3x(>F*7$7$Ffo$!3h4+o'yS;%>F*Fc]m 7$7$FerFj]m7$$!3QSP8umRD=P6F*$!3&)*** *********R=F*F^^m7$Fd^m7$$!3iFsL7@1+6F*$!3ssFm()y$*z:e:uuLtF-$!3%)[=We_i'= \"F*7$7$Fcu$!3E1!3?Z%)\\;\"F*F\\cm7$Fbcm7$$!3J%R;wz,Es'F-$!3Yh$Q-#)Rx3 \"F*7$7$$!3$[f-C[`vU'F-FfqFfcm7$F\\dm7$$!3'yE^'zhX6hF-$!3!3u[.#Qa)))*F -7$7$Fjw$!3%p/SyL!*41*F-F`dm7$Ffdm7$$!3TThoh0J+bF-$!3%p'QJQ%*o**))F-7$ 7$$!3+/X\\+\"*pQaF-FhsFjdm7$7$Faem$!3e0++++++))F-7$$!3'\\,@P%\\;*)[F-$ !33$**yi0N3\"zF-7$7$$!3i7ke=Z%)\\WF-FcuFgem7$7$$!3=8ke=Z%)\\WF-Fcu7$$! 3%z)evD$>!yUF-$!3L?TCu1)>#pF-7$7$Fjx$!3uN+gbf8skF-Fdfm7$Fjfm7$$!3%4w!z 2P(om$F-$!3[Y#4AHEJ$fF-7$7$$!3C@$ymL!*4Y$F-FjwF^gm7$Fdgm7$$!3/Nc#)*3Gd 0$F-$!3isV<5>FW\\F-7$7$$!37I-xaf8sCF-$!35/++++++SF-Fhgm7$7$F_hmFjx7$$! 3f30'=Z#eWCF-$!3w)\\R\"GvTbRF-7$7$Fay$!3WB+Ot:G$)QF-Fehm7$7$F][lF\\im7 $$!3'=Q&*Q&oVL=F-$!3!\\i/h9jl'HF-7$7$$!3uQ@'Gd\"G$[\"F-F_\\mF`im7$7$$! 3-R@'Gd\"G$[\"F-F_\\m7$$!3ob-$fB\"HA7F-$!3K^(pSw3x(>F-7$7$F[u$!3o6+7\" >FWH\"F-F]jm7$Fcjm7$$!3f\"H^'zhX6hFjo$!3A!y[.#Qa)))*Fjo7$7$$!3(\\ZS&4> FW\\FjoF[uFgjm7$7$$!3Ow/a4>FW\\FjoF[u7$$!32>Q_A'ohq#!#L$!3J4MQ\"GDFB%F g[n7$7$$\"3LL/a4>FW\\Fjo$\"35k************zFjoFd[n7$7$$\"3UN/a4>FW\\Fj oF^p7$$\"3ZP7lzhX6hFjo$\"3S\"p[.#Qa)))*Fjo7$7$F^p$\"3-++7\">FWH\"F-Fd \\n7$7$F^p$\"3I++7\">FWH\"F-7$$\"3T]-$fB\"HA7F-$\"3sU(pSw3x(>F-7$7$$\" 3eM@'Gd\"G$[\"F-F5Fa]n7$Fg]n7$$\"3'oP&*Q&oVL=F-$\"3-;Y5YJcmHF-7$7$$\"3 I'************R#F-$\"3!=,gLd\"G$)QF-F[^n7$7$F5Fd^n7$$\"3K.0'=Z#eWCF-$ \"3W!\\R\"GvTbRF-7$7$$\"3CE-xaf8sCF-F__lFh^n7$7$$\"3'fAqZ&f8sCF-$\"3M' *************RF-7$$\"3/Ic#)*3Gd0$F-$\"3ujV<5>FW\\F-7$7$$\"3!oJymL!*4Y$ F-F[^lFg_n7$F]`n7$$\"3]c2z2P(om$F-$\"3fP#4AHEJ$fF-7$7$F__l$\"3kC+gbf8s kF-Fa`n7$Fg`n7$$\"3%H)evD$>!yUF-$\"3X6TCu1)>#pF-7$7$$\"3u3ke=Z%)\\WF-F d[lF[an7$7$FbanF]el7$$\"3S45sV\\;*)[F-$\"3?%)*yi0N3\"zF-7$7$$\"3c*\\% \\+\"*pQaF-F\\clFfan7$F\\bn7$$\"3'e8'oh0J+bF-$\"3;fQJQ%*o**))F-7$7$F[^ l$\"3/Q+%yL!*41*F-F`bn7$Ffbn7$$\"3Ji7lzhX6hF-$\"3-L([.#Qa)))*F-7$7$$\" 3]\"f-C[`vU'F-$\"3/++++++S5F*Fjbn7$7$$\"3R!f-C[`vU'F-Fex7$$\"3l(Q;wz,E s'F-$\"3eg$Q-#)Rx3\"F*7$7$Fd[l$\"390!3?Z%)\\;\"F*Ficn7$F_dn7$$\"3A::e: uuLtF-$\"3'z%=We_i'=\"F*7$7$$\"3A\"o5V'yS;uF-FjuFcdn7$Fidn7$$\"3nTmaLI *[%zF-$\"3ON`k'p5bG\"F*7$7$$\"3;t(=iCi_S)F-$\"3))************f8F*F]en7 $7$$\"30s(=iCi_S)F-Fiv7$$\"37o<^^'Qgb)F-$\"3]A)[[8'R%Q\"F*7$7$F\\cl$\" 3/1?B5*pQU\"F*F\\fn7$Fbfn7$$\"3Y$*oZpU=n\"*F-$\"3!*4B0t:G$[\"F*7$7$$\" 3'G'o7Gm6%R*F-F_tFffn7$F\\gn7$$\"3/@?W())H$y(*F-$\"3G(zb7,n@e\"F*7$7$$ \"3:&\\.55(HQ5F*FarF`gn7$Ffgn7$$\"3k92a]v%*Q5F*$\"3m%Gf%\\C0\"o\"F*7$7 $Fex$\"3Q2gX[`v#o\"F*Fjgn7$F`hn7$$\"3RFsL7@1+6F*$\"31sFm()y$*zRD=P6F*F[qFdhn7$Fjhn7$$\"3#*RP8umF*F^in7$Fdin7$$\"3Y_-$fB\"HA7F*$\"3QY(pSw3x(>F*7$7$$\"3a8 ^Qxz1O7F*FboFhin-Fe\\m6&Fg\\m$F:F:Fcjn$\"*++++\"!\")Fi\\m-F$6V7$7$Fg^m $\"3!\\I%>RD=P6F*7$$!3G3+o'yS;%>F*Fju7$F][o7$F8$\"3)R6&Qxz1O7F*7$7$$!3 %*************z;F*$\"3\"e\\.55(HQ5F*7$$!3Q2gX[`v#o\"F*Fex7$Fj[o7$F(F[[ o7$7$FEFh[o7$$!3_8%[DvTbn\"F*$\"3>8%[DvTb.\"F*7$7$FQ$\"3Ino7Gm6%R*F-Fa \\o7$7$Fgn$\"3\\w(=iCi_S)F-7$$!3/1?B5*pQU\"F*F^z7$F^]oFg\\o7$7$Fgn$\"3 gx(=iCi_S)F-7$$!3?68T9jlc7F*$\"3c0J6WJcmxF-7$7$Ffo$\"3b%o5V'yS;uF-Ff]o 7$7$Ffq$\"3$[f-C[`vU'F-7$$!390!3?Z%)\\;\"F*Fd[l7$7$Fd^oF]elF\\^o7$7$Fh s$\"3+/X\\+\"*pQaF-7$$!3/Q+%yL!*41*F-F[^l7$7$$!3%p.SyL!*41*F-F[^lF`^o7 $Fi^o7$$!3K%3UFw3xP)F-$\"3-x?ui(3x<&F-7$7$Fcu$\"317ke=Z%)\\WF-Fd_o7$7$ Fjw$\"3C@$ymL!*4Y$F-7$$!3kC+gbf8skF-F__l7$Fa`o7$Fcu$\"3i7ke=Z%)\\WF-7$ F^`o7$$!3/^5P\"Qa))=%F-$\"31W5P\"Qa))e#F-7$7$Fjx$\"3SI-xaf8sCF-Fi`o7$7 $Fay$\"3-R@'Gd\"G$[\"F-7$$!3!=,gLd\"G$)QF-F57$Ffao7$FahmF`ao7$7$F[u$\" 3ux/a4>FW\\Fjo7$$!3I++7\">FWH\"F-F^p7$F_boFcao7$7$F[u$\"3Ow/a4>FW\\Fjo 7$$!3wA$R%yduX=!#K$\"3#**f[!)Qc=:\"Fjbo7$7$F^p$!3.M/a4>FW\\FjoFgbo7$7$ F5$!3eM@'Gd\"G$[\"F-7$$\"3S6+7\">FWH\"F-F[u7$7$$\"3o6+7\">FWH\"F-F[uF^ co7$7$F__l$!3'fAqZ&f8sCF-7$$\"3WB+Ot:G$)QF-Fay7$F`do7$F5$!3'[8iGd\"G$[ \"F-7$F]do7$$\"3'\\,r8Qa))=%F-$!3I@5P\"Qa))e#F-7$7$F[^l$!3M<$ymL!*4Y$F -Fhdo7$7$Fd[l$!3u3ke=Z%)\\WF-7$$\"3uN+gbf8skF-Fjx7$FeeoF^eo7$Fbeo7$$\" 3y[?ui(3xP)F-$!3#[0UFw3x<&F-7$7$F\\cl$!3c*\\%\\+\"*pQaF-Fjeo7$7$Fex$!3 R!f-C[`vU'F-7$$\"3/[+%yL!*41*F-Fjw7$7$$\"3;\\+%yL!*41*F-Fjw7$F[flFafo7 $7$$\"3%**************>\"F*$!3A\"o5V'yS;uF-7$$\"3E1!3?Z%)\\;\"F*Fcu7$F ego7$$\"3f************R5F*Fefo7$7$FjuFcgo7$$\"3m28T9jlc7F*$!3M$38T9jlw (F-7$7$F_w$!3;t(=iCi_S)F-F^ho7$7$F_t$!3'G'o7Gm6%R*F-7$$\"3:2?B5*pQU\"F *Feem7$F[io7$FivFeho7$Fhho7$$\"3?5%[DvTbn\"F*$!3(4T[DvTb.\"F*7$7$$\"3q ************z;F*$!3:&\\.55(HQ5F*Faio7$7$F[q$!3X/V>RD=P6F*7$$\"3q3gX[`v #o\"F*Ffq7$F`joFgio7$7$Fbo$!3K8^Qxz1O7F*7$$\"3$)4+o'yS;%>F*Ffo7$7$$\"3 h4+o'yS;%>F*FfoF]jo-Fe\\m6&Fg\\mFcjnFdjnFcjnFi\\m-%(SCALINGG6#%,CONSTR AINEDG-%+AXESLABELSG6%%\"xG%\"yG-%%FONTG6#%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "We have shown that the eigenvectors define the symmetry axes for t he hyperbola." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 11 "Exercise 1:" }}{PARA 0 "" 0 "" {TEXT -1 196 "Change the \+ entries in A such that two positive eigenvalues are found, and demonst rate that an ellipse emerges. Measure the semimajor axes and determine their relationship with the two eigenvalues." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 140 "One last step that is us eful to do with our example is to transform the matrix A with a transf ormation matrix made up from the eigenvectors." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 33 "A_diag:=Transpose(EV[2]).A.EV[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'A_diagG-%'RTABLEG6%\"*)))3(\\\"-%'MATRIXG6#7 $7$^$$!3uVo\\i'>5a)!#=$\"\"!F3^$$\"3+^<)fJ2#)>&!#LF27$^$$\"3\"[GHWpOI` (F7F2^$$\"3#Ho\\i'>5ae!# " 0 " " {MPLTEXT 1 0 11 "k:=n->n*Pi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" kGf*6#%\"nG6\"6$%)operatorG%&arrowGF(*&9$\"\"\"%#PiGF.F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "fB:=n->sin(n*Pi*x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fBGf*6#%\"nG6\"6$%)operatorG%&arrow GF(-%$sinG6#*(9$\"\"\"%#PiGF1%\"xGF1F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "For normalization, and for projection we define an inner product for real functions on the interval [0,1]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "IP:=(f,g)->int(f*g,x=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#IPGf*6$%\"fG%\"gG6\"6$%)operatorG%&arrowGF)-%$i ntG6$*&9%\"\"\"9$F2/%\"xG;\"\"!F2F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "IP(fB(2),fB(3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# \"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "We find that our basis f unctions are orthogonal under this inner product." }}{PARA 0 "" 0 "" {TEXT -1 32 "(check this out for other (m,n))" }}{PARA 0 "" 0 "" {TEXT -1 31 "Now we normalize our functions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "IP(fB(1),fB(1)),IP(fB(2),fB(2)),IP(fB(3),fB(3)), IP(fB(4),fB(4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&#\"\"\"\"\"#F#F#F# " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "A simple factor for all funct ions will suffice:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "fn:=n ->sqrt(2)*sin(n*Pi*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fnGf*6#% \"nG6\"6$%)operatorG%&arrowGF(*&-%%sqrtG6#\"\"#\"\"\"-%$sinG6#*(9$F1%# PiGF1%\"xGF1F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "IP( fn(5),fn(5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 137 "Now take some function defined on the in terval [0,1] with boundary conditions that it vanishes at 0 and at 1, \+ and decompose in our basis:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "g:=x*(1-x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG*&%\"xG\"\"\" ,&F'F'F&!\"\"F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "cn:=seq( IP(fn(n),g),n=1..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#cnG6,,$*( \"\"%\"\"\"\"\"##F)F*%#PiG!\"$F)\"\"!,$**F(F)\"#F!\"\"F*F+F,F-F)F.,$** F(F)\"$D\"F2F*F+F,F-F)F.,$**F(F)\"$V$F2F*F+F,F-F)F.,$**F(F)\"$H(F2F*F+ F,F-F)F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "The simplest approxim ant:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "gA1:=cn[1]*fn(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$gA1G,$*(\"\")\"\"\"%#PiG!\"$-%$si nG6#*&F)F(%\"xGF(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "gA 3:=add(cn[n]*fn(n),n=1..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$gA3G ,&*(\"\")\"\"\"%#PiG!\"$-%$sinG6#*&F)F(%\"xGF(F(F(*&#F'\"#FF(*&F)F*-F, 6#,$*(\"\"$F(F)F(F/F(F(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "plot([g,gA1,gA3],x=0..1,color=[red,blue,green],thickness=2);" }} {PARA 13 "" 1 "" {GLPLOT2D 536 240 240 {PLOTDATA 2 "6(-%'CURVESG6$7S7$ $\"\"!F)F(7$$\"3emmm;arz@!#>$\"3GB,pBQ?K@F-7$$\"3[LL$e9ui2%F-$\"3dSMrO S65RF-7$$\"3nmmm\"z_\"4iF-$\"3tYSw2qhBeF-7$$\"3[mmmT&phN)F-$\"3c*)edZQ \"zl(F-7$$\"3CLLe*=)H\\5!#=$\"3Mr/<0_&>R*F-7$$\"3gmm\"z/3uC\"FB$\"3gRf 6k`!=4\"FB7$$\"3%)***\\7LRDX\"FB$\"35!=n.G_:C\"FB7$$\"3]mm\"zR'ok;FB$ \"3)**yvvJovQ\"FB7$$\"3w***\\i5`h(=FB$\"3MZ3We!eT_\"FB7$$\"3WLLL3En$4# FB$\"3%H![C4hKb;FB7$$\"3qmm;/RE&G#FB$\"35fz*Hz?Iw\"FB7$$\"3\")*****\\K ]4]#FB$\"3MR#)=s]Zv=FB7$$\"3$******\\PAvr#FB$\"3)fVP\"*eH!z>FB7$$\"3)* *****\\nHi#HFB$\"3=R\\\"RmZ*p?FB7$$\"3jmm\"z*ev:JFB$\"3@nAox]%G#FB7$$\" 3w***\\7o7Tv$FB$\"3t$z(*)ykxWBFB7$$\"3'GLLLQ*o]RFB$\"3Ai](Hs%*)*Q#FB7$ $\"3A++D\"=lj;%FB$\"3Uln*))H00V#FB7$$\"31++vV&RVN& \\#FB7$$\"3\\***\\(=>Y2aFB$\"3+doZyuR$[#FB7$$\"39mm;zXu9cFB$\"3u>&Q-\" *3AY#FB7$$\"3l******\\y))GeFB$\"3oPA@$\\%HJCFB7$$\"3'*)***\\i_QQgFB$\" 3!3GiYgv@R#FB7$$\"3@***\\7y%3TiFB$\"3Ik\\dc3(fM#FB7$$\"35****\\P![hY'F B$\"3?!\\8K*4/&G#FB7$$\"3kKLL$Qx$omFB$\"3>#=y1p^;A#FB7$$\"3!)*****\\P+ V)oFB$\"39'[x'47%\\9#FB7$$\"3?mm\"zpe*zqFB$\"3k!3'\\\"=xt1#FB7$$\"3%)* ****\\#\\'QH(FB$\"3]uaeq$=Q(>FB7$$\"3GKLe9S8&\\(FB$\"3CEp#\\iIu(=FB7$$ \"3R***\\i?=bq(FB$\"3ey\\cBr,o26%H>ZA:FB7$$\"3#pmmm'*RRL)FB$\"3Imi')HW[)Q\"FB 7$$\"3Qmm;a<.Y&)FB$\"3WnT%)zecU7FB7$$\"3=LLe9tOc()FB$\"3')**Hzf/(*)3\" FB7$$\"3u******\\Qk\\*)FB$\"3Gzr:eMJ+%*F-7$$\"3CLL$3dg6<*FB$\"39T`K#\\ >9g(F-7$$\"3ImmmmxGp$*FB$\"3U:q07WK4fF-7$$\"3A++D\"oK0e*FB$\"3t$o+gN?( =SF-7$$\"3A++v=5s#y*FB$\"3Agp!olzb7#F-7$$\"\"\"F)F(-%'COLOURG6&%$RGBG$ \"*++++\"!\")F(F(-F$6$7SF'7$F+$\"3EFt^o-VlvV^:^%p8\"FB7$FP$\"35ia3J5n)G\"FB7$FU$\"3t?]\"zX;UV \"FB7$FZ$\"3M'GaWx>td\"FB7$Fin$\"3q)\\R0u(G(p\"FB7$F^o$\"3lo-G-p'\\#=F B7$Fco$\"3_jh-[>uW>FB7$Fho$\"3%f8^Xce;0#FB7$F]p$\"3N4WEX<7T@FB7$Fbp$\" 3\"Rq@`M,wB#FB7$Fgp$\"3WLkQS!y0J#FB7$F\\q$\"3%R$p=Lj*\\Q#FB7$Faq$\"3*H w?xI(>TCFB7$Ffq$\"3]I#oO%R9#\\#FB7$F[r$\"3u*yQ@]H+`#FB7$F`r$\"3w(=fCVL $eDFB7$Fer$\"3QB&H-#*=Td#FB7$Fjr$\"3?hWl_'>,e#FB7$F_s$\"3_!y&z]*4Ud#FB 7$Fds$\"3Sk;HFB7$Ffv$\"3k'*pUZ\"4s#=FB7$F[w$\"3(pRDa13Hq\"FB7$F`w$\"3;+X61x= u:FB7$Few$\"3xRs.t#)RK9FB7$Fjw$\"31DdA8bj*G\"FB7$F_x$\"3'QQX1s%)z8\"FB 7$Fdx$\"3peQKy(**f#)*F-7$Fix$\"3e\"zc4;)>g$)F-7$F^y$\"34Ywq5ulUmF-7$Fc y$\"3O8@y2O(*y]F-7$Fhy$\"37>\\N)pT-R$F-7$F]z$\"3)RC*>+(H)fF-7$F1$\"3wbg wp!QKl$F-7$F6$\"3OHLWhI$*GbF-7$F;$\"39X#z%p;*GP(F-7$F@$\"3!>\\'4@gV]\" *F-7$FF$\"3s!p8()HDO2\"FB7$FK$\"3hHoscecI7FB7$FP$\"3I!pxm'4B%Q\"FB7$FU $\"3\\yyS$3?z_\"FB7$FZ$\"3+0.Gg_Cl;FB7$Fin$\"3'\\yqX)>1xUr*>FB7$Fho$\"3i4!RFkIu3#FB7$F]p$\"3Y>!y_V!eg @FB7$Fbp$\"3H@^6x***oB#FB7$Fgp$\"3e(GJ'pny#H#FB7$F\\q$\"3GP\\^M^3[BFB7 $Faq$\"3Mn()e[2o)Q#FB7$Ffq$\"3B$)o9]>fCCFB7$F[r$\"31PUx(pP2X#FB7$F`r$ \"3mQ+x%)[&*pCFB7$Fer$\"3?(ez>*Qb![#FB7$Fjr$\"3O3!HYegX[#FB7$F_s$\"3yb (ocw91[#FB7$Fds$\"3MY2zsETqCFB7$Fis$\"39fVZp->_CFB7$F^t$\"3Ik,a%o$GDCF B7$Fct$\"3oj)H5#Qr!R#FB7$Fht$\"3'***33%*)o\"\\BFB7$F]u$\"3[U$*)R7\"G$H #FB7$Fbu$\"32/o2tA\"RB#FB7$Fgu$\"3GqU`X*f0;#FB7$F\\v$\"3ml#eZL,\\3#FB7 $Fav$\"3kxZcL&=>*>FB7$Ffv$\"3I[W&pupW*=FB7$F[w$\"3iqz(yCAAy\"FB7$F`w$ \"3'e8`kR%Gi;FB7$Few$\"3To-L`#\\h_\"FB7$Fjw$\"3_X9j*f&>&Q\"FB7$F_x$\"3 N&GJDF- 7$Fbz$\"3z6u5#>@3^$F^dl-Fez6&FgzF(FhzF(-%*THICKNESSG6#\"\"#-%+AXESLABE LSG6$Q\"x6\"Q!F^^m-%%VIEWG6$;F(Fbz%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "What happens when we try this with a function that does not vanish at the boundaries?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "g:=cos(x*Pi/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG-%$cosG6#,$*(\"\"#!\"\"%#PiG\"\"\"%\"xGF-F-" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "cn:=seq(IP(fn(n),g),n=1..10) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#cnG6,,$**\"\"%\"\"\"\"\"$!\"\" \"\"##F)F,%#PiGF+F),$**\"\")F)\"#:F+F,F-F.F+F),$**\"#7F)\"#NF+F,F-F.F+ F),$**\"#;F)\"#jF+F,F-F.F+F),$**\"#?F)\"#**F+F,F-F.F+F),$**\"#CF)\"$V \"F+F,F-F.F+F),$**\"#GF)\"$&>F+F,F-F.F+F),$**\"#KF)\"$b#F+F,F-F.F+F),$ **\"#OF)\"$B$F+F,F-F.F+F),$**\"#SF)\"$*RF+F,F-F.F+F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "The simplest approximant:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "gA1:=cn[1]*fn(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$gA1G,$*&#\"\")\"\"$\"\"\"*&%#PiG!\"\"-%$sinG6#*&F,F* %\"xGF*F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "gA3:=add(c n[n]*fn(n),n=1..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$gA3G,(*&#\" \")\"\"$\"\"\"*&%#PiG!\"\"-%$sinG6#*&F,F*%\"xGF*F*F*F**&#\"#;\"#:F**&F ,F--F/6#,$*(\"\"#F*F,F*F2F*F*F*F*F**&#\"#C\"#NF**&F,F--F/6#,$*(F)F*F,F *F2F*F*F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "gA5:=add(c n[n]*fn(n),n=1..5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$gA5G,,*&#\" \")\"\"$\"\"\"*&%#PiG!\"\"-%$sinG6#*&F,F*%\"xGF*F*F*F**&#\"#;\"#:F**&F ,F--F/6#,$*(\"\"#F*F,F*F2F*F*F*F*F**&#\"#C\"#NF**&F,F--F/6#,$*(F)F*F,F *F2F*F*F*F*F**&#\"#K\"#jF**&F,F--F/6#,$*(\"\"%F*F,F*F2F*F*F*F*F**&#\"# S\"#**F**&F,F--F/6#,$*(\"\"&F*F,F*F2F*F*F*F*F*" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 70 "plot([g,gA1,gA3,gA5],x=0..1,color=[red,blue,gr een,black],thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 536 240 240 {PLOTDATA 2 "6)-%'CURVESG6$7S7$$\"\"!F)$\"\"\"F)7$$\"3emmm;arz@!#>$\"3 gf&fZ1RT***!#=7$$\"3[LL$e9ui2%F/$\"3MNt%3#y]z**F27$$\"3nmmm\"z_\"4iF/$ \"3G%Gk?7uC&**F27$$\"3[mmmT&phN)F/$\"3MqF'\\wzR\"**F27$$\"3CLLe*=)H\\5 F2$\"3-1bg6NZk)*F27$$\"3gmm\"z/3uC\"F2$\"3ygtY(GY'3)*F27$$\"3%)***\\7L RDX\"F2$\"3p]m$\\UK3u*F27$$\"3]mm\"zR'ok;F2$\"3#)pE!y(G1g'*F27$$\"3w** *\\i5`h(=F2$\"3K-l94u()o&*F27$$\"3WLLL3En$4#F2$\"3WS;:W$oSY*F27$$\"3qm m;/RE&G#F2$\"3QVhylsfi$*F27$$\"3\")*****\\K]4]#F2$\"3SL=(\\'RAQ#*F27$$ \"3$******\\PAvr#F2$\"3I 06J)F27$$\"3'GLLLQ*o]RF2$\"3'zIFh%[XN\")F27$$\"3A++D\"=lj;%F2$\"3)33' \\'o@Q$zF27$$\"31++vV&RY2aF2$\"3 AQ:tW>K/mF27$$\"39mm;zXu9cF2$\"3-yxKnrPcjF27$$\"3l******\\y))GeF2$\"3- ^jD^F:$4'F27$$\"3'*)***\\i_QQgF2$\"3WJ_*f%f'*GeF27$$\"3@***\\7y%3TiF2$ \"3g6\\@t2MnbF27$$\"35****\\P![hY'F2$\"3-uBt6.Dq_F27$$\"3kKLL$Qx$omF2$ \"375=!pksw*\\F27$$\"3!)*****\\P+V)oF2$\"3IyCEf$y5q%F27$$\"3?mm\"zpe*z qF2$\"3[\\0#=[SwU%F27$$\"3%)*****\\#\\'QH(F2$\"3>5#)fC[#R7%F27$$\"3GKL e9S8&\\(F2$\"3o)\\\\C&[*Q$QF27$$\"3R***\\i?=bq(F2$\"3]H8`-(Qm_$F27$$\" 3\"HLL$3s?6zF2$\"3x]Q-cS^AKF27$$\"3a***\\7`Wl7)F2$\"3dxm17]_+HF27$$\"3 #pmmm'*RRL)F2$\"3Imi06+F(e#F27$$\"3Qmm;a<.Y&)F2$\"3KX*H/b%3kAF27$$\"3= LLe9tOc()F2$\"3:0%\\Dn#4T>F27$$\"3u******\\Qk\\*)F2$\"3\"R_E-J?Ck\"F27 $$\"3CLL$3dg6<*F2$\"3WvP@!)HE)H\"F27$$\"3ImmmmxGp$*F2$\"3ql4(Rc05*)*F/ 7$$\"3A++D\"oK0e*F2$\"3q]oNP1@%e'F/7$$\"3A++v=5s#y*F2$\"3sUsbzwM7MF/7$ F*$\"3Mint&*RBBh!#M-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-F$6$7S7$F(F(7$ F-$\"3#oKeEyK!3eF/7$F4$\"3s0*Q<%y.%3\"F27$F9$\"3//R%QV$HX;F27$F>$\"3<# [+u(e!G?#F27$FC$\"31x`@!yDxu#F27$FH$\"3]w%fy/J>C$F27$FM$\"3EuDH0&*RSPF 27$FR$\"3_7e)*fxbRUF27$FW$\"3ODwr@IQ=ZF27$Ffn$\"3aJkiiS<*=&F27$F[o$\"3 7\\XJ^G&Qe&F27$F`o$\"36$eioX+R+'F27$Feo$\"3EPS2jX%zR'F27$Fjo$\"3k)=UF8 'o\\nF27$F_p$\"3e8*4,?3S/(F27$Fdp$\"3e&=!*=O89O(F27$Fip$\"3ZAh[lq\\,wF 27$F^q$\"37!eMbVBj%yF27$Fcq$\"3g4htIs@J!)F27$Fhq$\"3/=(e[%\\#))>)F27$F ]r$\"3+/$)ROMYB$)F27$Fbr$\"3&ou<[jzlT)F27$Fgr$\"3+I#3!3=^o%)F27$F\\s$ \"3'=(=REMD)[)F27$Fas$\"3Q]tc'G6)o%)F27$Ffs$\"3Q-]NjS\")=%)F27$F[t$\"3 !)='Qs4c/L)F27$F`t$\"3e'oLrE$4-#)F27$Fet$\"3^Y^$e.,1/)F27$Fjt$\"3'e^A# \\,?^yF27$F_u$\"3'GtzKR_Og(F27$Fdu$\"3G[g(=#*p([tF27$Fiu$\"36VS_nW#R/( F27$F^v$\"3zll&3ym'RnF27$Fcv$\"3a(QFK(y%zP'F27$Fhv$\"3yQ(=x9x7,'F27$F] w$\"3I(od!zHM-cF27$Fbw$\"3u5G!3(*p)y^F27$Fgw$\"3IXqAL9S7ZF27$F\\x$\"3H #eW!z2tUUF27$Fax$\"3a2Z3p'>Qu$F27$Ffx$\"3u1;CbViKKF27$F[y$\"31V>6_\\R] FF27$F`y$\"3Mn\\&QpY`=#F27$Fey$\"3!zw7CM:4n\"F27$Fjy$\"3ryY*G\"[M:6F27 $F_z$\"3/t!=9l2'*y&F/7$F*$\"3$H$=n)[7&R5!#L-Fhz6&FjzF(F(F[[l-F$6$7gnFa [l7$$\"3ILLL3x&)*3\"F/$\"3gG\\6p^)pY(F/7$F-$\"3m()fFA#3'*[\"F27$$\"3.+ +D\"y%*z7$F/$\"3*43u]u++8#F27$F4$\"37tw>&))QAw#F27$$\"33++voMrU^F/$\"3 O-=mrJagMF27$F9$\"3pp='e*4@UTF27$$\"3emmmm6m#G(F/$\"3#RZ))\\l\"Q3[F27$ F>$\"3C&G*\\8NR^aF27$$\"36++v=ddC%*F/$\"3s,#4t^ua1'F27$FC$\"3w(*)=L()p 5l'F27$$\"3')***\\(=JN[6F2$\"3/tYx$o-k;(F27$FH$\"3_zD4OMO`wF27$FM$\"3E ii&RBL\\c)F27$FR$\"3/_6Vq2\"Fdhl 7$$\"3CLLe9r5$R#F2$\"3'\\WNRLlv3\"Fdhl7$F`o$\"35SIIs@7%4\"Fdhl7$$\"3() ******\\jB4EF2$\"33*)Rm0M%o4\"Fdhl7$Feo$\"3%z\"\\gIm%e4\"Fdhl7$$\"3n** ***\\-w=#GF2$\"3_4(Rh3K:4\"Fdhl7$Fjo$\"3q1Os0L5%3\"Fdhl7$F_p$\"3U:[s%G \"Gj5Fdhl7$Fdp$\"36,([8AZx-\"Fdhl7$Fip$\"3`@0s)ph7!**F27$F^q$\"3)=y\\F nL#[YE:\"*)F27$Fhq$\"3)G;&4^Q2a$)F27$F]r$\"3dj@-AK5=y F27$Fbr$\"3aJHj*HP6F(F27$Fgr$\"3'y^&GR4H$z'F27$F\\s$\"36H](ed5hJ'F27$F as$\"3]>67`ZeseF27$Ffs$\"3_7F)4\"yINbF27$F[t$\"3,XS:Gp/%F27$ Fbw$\"3fu9lnb[3RF27$Fgw$\"32(fyyp%==PF27$F\\x$\"3Co[D)H`c[$F27$Fax$\"3 #40L!*G3[>$F27$Ffx$\"3A,c*pt/H&GF27$F[y$\"3CNhl?Q]$\\#F27$F`y$\"3uLmtf >&G.#F27$Fey$\"37*>(=5@t#e\"F27$Fjy$\"3)>_Uf#pSr5F27$F_z$\"355;#Gcssg& F/7$F*$\"3aPE&=87)45Fbdl-Fhz6&FjzF(F[[lF(-F$6$7aoFa[l7$$\"3WmmmT&)G\\a !#?$\"3?q#Q5my;%fF/7$Fidl$\"3#\\\">JTx`'=\"F27$$\"3$*****\\ilyM;F/$\"3 Odn@74Kv$ \"3%*y(4Uvrt4)F27$Fifl$\"3Xzho(>!*Q%))F27$FC$\"3W)>0L/y*)\\*F27$Fagl$ \"3.*f&p$4Q@+\"Fdhl7$FH$\"3YoeS$HOg/\"Fdhl7$$\"33LLe*ot*\\8F2$\"35^.U0 mm#3\"Fdhl7$FM$\"3oj4OJyU56Fdhl7$$\"3/LLekGhe:F2$\"3Zsv:H`2I6Fdhl7$FR$ \"3!\\)[!)*fF49\"Fdhl7$$\"3#)*****\\2`vr\"F2$\"3()R;^B>AV6Fdhl7$$\"39L L3_(>/x\"F2$\"3!32*p\"R]N9\"Fdhl7$$\"3Xmm;HkGB=F2$\"3ZsgfN7+U6Fdhl7$FW $\"33!GPG3p'Q6Fdhl7$$\"3gm;HdG\"\\)>F2$\"3EE^Q[(Rm7\"Fdhl7$Ffn$\"3=_N- uIV36Fdhl7$F[o$\"3'e%[D^YUk5Fdhl7$F`o$\"3/SY*fR_G+\"Fdhl7$Feo$\"3'Rcpr eS/O*F27$Fjo$\"3ZF^T^PQQ()F27$F_p$\"3BX`-IG,Q#)F27$Fdp$\"3_m'RdX\"ejxF 27$Fip$\"3s%4:\"\\K?#[(F27$$\"3mm;HdO2VOF2$\"3U?:\"yVK4P(F27$F^q$\"3Av _c#\\*e'H(F27$$\"3Im;HK5S_QF2$\"35&=6(ojRfsF27$Fcq$\"3E?R.f0MYsF27$$\" 3Em;H#GF&eSF2$\"3SNR'*4;&fD(F27$Fhq$\"3yDt>#GYeG(F27$F]r$\"35;P8tH?ztF 27$Fbr$\"3ANDZ(o36\\(F27$Fgr$\"3m3*pdhZ%pvF27$$\"3%***\\(=7O*))[F2$\"3 K@\\lz$[2f(F27$F\\s$\"3i)Hgy/W@f(F27$$\"3@m;/^7I0^F2$\"3]CpWDK0pvF27$F as$\"3US1UB*f#=vF27$$\"3ALLLe,]6`F2$\"3oY?kZf3]uF27$Ffs$\"3WQ&H/J4!ftF 27$F[t$\"3QqS\\b*p\\3(F27$F`t$\"3[/C#[M/+q'F27$Fet$\"3DO&\\3#HYViF27$F jt$\"3y%3%*fZvGv&F27$F_u$\"3)3q*=Z-c(=&F27$Fdu$\"3#>DK1\"[)Hp%F27$Fiu$ \"3yq=\\RL+4UF27$F^v$\"3R@KcZ\\:HQF27$Fcv$\"3_p@RPz7!\\$F27$Fhv$\"3#4# yhVT%eC$F27$F]w$\"3%z5Kh%42gIF27$Fbw$\"3+zB**ei!4$HF27$Fgw$\"3g.$y#4Wn EGF27$F\\x$\"3q;&)\\*Q[*GFF27$Fax$\"3MB\\&Q(*[Ng#F27$Ffx$\"3=)GHBR4#HC F27$F[y$\"3(*>\")Q4jQ4AF27$F`y$\"39C9/@i8x=F27$Fey$\"3*>`'=uy,2:F27$Fj y$\"3QCgPJv[X5F27$F_z$\"3-))Q)f`8@b&F/7$F*$\"32B\"*yx?J05Fbdl-Fhz6&Fjz F)F)F)-%*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q!Fc`n-%%VIEWG6$;F(F* %(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "C urve 1" "Curve 2" "Curve 3" "Curve 4" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 164 "While this seems like a crazy way to approximate somethi ng smooth, it is nevertheless something useful, especially if the boun dary condition problem can be avoided." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 292 "The method as developed so far, corr esponds to the Fourier series approximation technique. It is particula rly suited for approximating periodic functions. Complicated periodic \+ functions are represented as a sine series using sine functions that i nvolve multiples of the fundamental frequency." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 247 "Our examples worked abov e because the integrals needed for the projection of the function g on to the basis functions f(n) could be obtained in closed form. In gener al, one needs to use numerical integration to calculate the expansion \+ coefficients." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 126 "In the remainder of this worksheet we explore the conver sion of a differential equation problem into a linear algebra problem. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "In t he notes there is a derivation of the Poisson equation that yields the electrostatic potential of a charge distribution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "rho:=r->4*exp(-2*r)/(4*Pi);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$rhoGf*6#%\"rG6\"6$%)operatorG%&arro wGF(*&-%$expG6#,$*&\"\"#\"\"\"9$F3!\"\"F3%#PiGF5F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "This charge distribution encloses a charg e of 1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Q:=4*Pi*int(r^2* rho(r),r=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"QG\"\"\" " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "This charge density can desc ribe the charge cloud in an atom. The length unit would be a Bohr radi us." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "Th e Poisson equation is expressed in units in which 4Pi times epsilon0 i s set to 1." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "PE:=diff(Phi (r),r$2)=-r*rho(r)*4*Pi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#PEG/-%% diffG6$-%$PhiG6#%\"rG-%\"$G6$F,\"\"#,$*(\"\"%\"\"\"F,F4-%$expG6#,$*&F0 F4F,F4!\"\"F4F:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "The boundary c onditions:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "at r=0 the function Phi vanishes [assuming that the potential \+ is finite as in the case of a charged sphere]" }}{PARA 0 "" 0 "" {TEXT -1 132 "at r=infinity the function Phi should correspond to the \+ total value of charge accumulated, so that V(r)=Q/r as r becomes very \+ large." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "sol:=dsolve(\{PE, Phi(0)=0,Phi(infinity)=Q\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$sol G/-%$PhiG6#%\"rG*&,(\"\"\"!\"\"F)F-*$)-%$expGF(\"\"#F,F,F,F0!\"#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "expand(rhs(sol));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&\"\"\"F%*$)-%$expG6#%\"rG\"\"#F%!\"\"F-* &F(!\"#F+F%F-F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "plot( [rhs(sol)/r,Q/r,4*Pi*r^2*rho(r)],r=0..10,color=[red,blue,green],thickn ess=2,view=[0..10,0..1]);" }}{PARA 13 "" 1 "" {GLPLOT2D 948 326 326 {PLOTDATA 2 "6(-%'CURVESG6$7fn7$$\"3`*****\\n5;\"o!#?$\"3Gcr6yFp****!# =7$$\"33+++N@Ki8!#>$\"3)3')RxVz()***F-7$$\"3<+++-K[V?F1$\"3S=+c9BF(*** F-7$$\"3;+++qUkCFF1$\"3g[wvVN=&***F-7$$\"3s*****\\Smp3%F1$\"3#pVXbh3$* )**F-7$$\"3i******R&)G\\aF1$\"3%R+Wz$yC\")**F-7$$\"3W******4G$R<)F1$\" 3wQkUmk#*e**F-7$$\"31+++3x&)*3\"F-$\"3g#Rc^71*G**F-7$$\"3*)*****>c'yM; F-$\"3iZ!)*yL(H[)*F-7$$\"39+++;arz@F-$\"3\"eRN:mySu*F-7$$\"3?+++!y%*z7 $F-$\"3LjYq')HB=&*F-7$$\"33+++XTFwSF-$\"3-H9i8')y]#*F-7$$\"3&******4z_ \"4iF-$\"3;j%\\CmBYc)F-7$$\"3C+++S&phN)F-$\"3Y@V?(4Vq$yF-7$$\"3%****** *)=)H\\5!#<$\"3\\!oog9C_8(F-7$$\"31+++[!3uC\"F]p$\"3WpB7Q@/IlF-7$$\"35 +++J$RDX\"F]p$\"3C.>t'=i,'fF-7$$\"3'******zR'ok;F]p$\"3#zmL*GG$QV&F-7$ $\"31+++1J:w=F]p$\"3%R**>Jbd.(\\F-7$$\"3))*****zgsO4#F]p$\"3u2#zq,&*=b %F-7$$\"3!)*****R!RE&G#F]p$\"3W>x2*>OqA%F-7$$\"3\")*****\\K]4]#F]p$\"3 +=41U\"QV!RF-7$$\"3;+++vB_(R#F-7$$\"3M+++V&RF-7 $$\"34+++=>Y2aF]p$\"3Kb63(Ge!\\=F-7$$\"3#)*****zdWZh&F]p$\"3Oo2*G]o3y \"F-7$$\"3,+++\\y))GeF]p$\"3TP$H5y\"\\:Qx$omF]p$\"3s<,nEmf*\\\"F-7$$\"3Q+++u.I% )oF]p$\"3Ib?uF%oDX\"F-7$$\"3O+++(pe*zqF]p$\"37=F.J&HCT\"F-7$$\"3I***** R#\\'QH(F]p$\"3ig5&4**45P\"F-7$$\"3[*****H,M^\\(F]p$\"3#RuS#ea>M8F-7$$ \"3t*****\\?=bq(F]p$\"3cI`sf*oxH\"F-7$$\"3N+++2s?6zF]p$\"3Mf1yv!GSE\"F -7$$\"3K+++IXaE\")F]p$\"3\\dDpHV`I7F-7$$\"3+*****\\'*RRL)F]p$\"3)QX!30 ?\"**>\"F-7$$\"3=,++`<.Y&)F]p$\"3-\"*4&oIL,<\"F-7$$\"3;+++8tOc()F]p$\" 3^n8KAe-U6F-7$$\"3?******[Qk\\*)F]p$\"3VOR<7FO<6F-7$$\"3a******o0;r\"* F]p$\"3![m2:Uu.4\"F-7$$\"3E*****\\w(Gp$*F]p$\"3g13y2pJn5F-7$$\"3C***** *zE`!e*F]p$\"3)*>ygXKyV5F-7$$\"33+++<5s#y*F]p$\"3`!z&Q]/@A5F-7$$\"#5\" \"!$\"235tKx*******F--%'COLOURG6&%$RGBG$\"*++++\"!\")$Fe]lFe]lF_^l-F$6 $7dp7$F($\"3ekD?c:3o9!#:7$F/$\"3V@G,\"y2/M(!#;7$F5$\"3W\\#R&)=0O*[Fj^l 7$F:$\"3s5k]!*Q?qOFj^l7$$\"3:+++Q`!eS$F1$\"30wS(>6jh$HFj^l7$F?$\"3'4%4 n$f-oW#Fj^l7$$\"3)******>ZF\"oZF1$\"3Q)eBiOfs4#Fj^l7$FD$\"3s0KDX>5N=Fj ^l7$$\"38+++5'\\/8'F1$\"3ib([%GYL\"Fj^l7$FI$\"3[qa$oH,MA\"Fj^l7$ $\"3k+++!)Q4b))F1$\"3JxBX/NHH6Fj^l7$$\"3/+++X\\DO&*F1$\"3mJ=+$oH'[5Fj^ l7$$\"3%******4gT<-\"F-$\"3%**\\yqP5sy*F]p7$FN$\"3yEgEE(4b<*F]p7$$\"3/ +++:Q(z:\"F-$\"36p1:eQxN')F]p7$$\"3.+++A**3E7F-$\"33yPCU'3g:)F]p7$$\"3 /+++Gg?%H\"F-$\"3ZSw8M]uExF]p7$$\"3/+++N@Ki8F-$Fi^lF]p7$$\"3.+++U#Q/V \"F-$\"3!)RP7_X'3*pF]p7$$\"3#******zMa&)\\\"F-$\"3/Ta'\\)z4tmF]p7$$\"3 #******\\Xqmc\"F-$\"3!>zb))GjHQ'F]p7$FS$\"3Q_t<%[1q6'F]p7$$\"3()****** oE!Hq\"F-$\"3i_\"[RAEB(eF]p7$$\"37+++w(=5x\"F-$\"3a')=EAvYYcF]p7$$\"3- +++#)[8R=F-$\"3F^cy(4RtV&F]p7$$\"3,+++*)4D2>F-$\"3He\"4]T[JC&F]p7$$\"3 *)*****>?$[V?F-$\"3'*\\#R&)=0O*[F]p7$FX$\"3S8I8j[v(e%F]p7$$\"31+++O.D) H#F-$\"3CSOM)4O6N%F]p7$$\"33+++d_y;CF-$\"3GHcY))zsPTF]p7$$\"34+++y,KND F-$\"3j3-%>+vU%RF]p7$$\"3u*****z4bQl#F-$\"3%\\2QDC.\"oPF]p7$$\"3C+++R \\#4*GF-$\"37JR\"R`+\"fMF]p7$Fgn$\"3-&GK8GOp>$F]p7$$\"34+++iW8-OF-$\"3 U*)GF#zJhx#F]p7$F\\o$\"3+pQns1A`CF]p7$$\"3I+++3Q\\4YF-$\"3iV6e2cVp@F]p 7$$\"3!*******pMrU^F-$\"3W&==nw)\\W>F]p7$$\"3K******HJ$fn&F-$\"3'R-oP% [#=w\"F]p7$Fao$\"3/#3EmnD0h\"F]p7$$\"3]*****\\;hEG(F-$\"37d]MNW7t8F]p7 $Ffo$\"3UC2+t0s'>\"F]p7$F[p$\"3\")=E/9%z,`*F-7$Fap$\"3@;Z`hHi;!)F-7$Ff p$\"37f]9k^\\%)oF-7$F[q$\"3+^$QwWPr+'F-7$F`q$\"3q\\:F.a0I`F-7$Feq$\"3O I>&3&fHwZF-7$Fjq$\"3)pDX'GA'eP%F-7$F_r$\"3)zjvx0![)*RF-7$Fdr$\"35w6W2D #)zOF-7$Fir$\"3!RBz>vmtT$F-7$F^s$\"38:B$f/%\\4KF-7$Fcs$\"3qS+9>v+$*HF- 7$Fhs$\"3_tWa;$H7$GF-7$F]t$\"3-*Q?uHXPm#F-7$Fbt$\"3WA'[`!Q?JDF-7$Fgt$ \"3kT+JoO<+CF-7$F\\u$\"3+2Zcy*=uG#F-7$Fau$\"3kV+8nq`!=#F-7$Ffu$\"3a$R \"*)**=#34#F-7$F[v$\"3N&Hs(4j(>+#F-7$F`v$\"3(e)Qf%fZt\">F-7$Fev$\"3KMD CEkH\\=F-7$Fjv$\"3Yp3r<\\-\"y\"F-7$F_w$\"3y^1qQJf:MX(=2 c;F-7$Fiw$\"3[p#=lb&G-;F-7$F^x$\"3+7)*4%y:la\"F-7$Fcx$\"3)4i&z(=:'*\\ \"F-7$Fhx$\"3!)*4h.V!e_9F-7$F]y$\"3A/!\\$=wV79F-7$Fby$\"3)4F3UC:5P\"F- 7$Fgy$\"3C>bIf*)>M8F-7$F\\z$\"3q;V$4DrxH\"F-7$Faz$\"3EfJr*eHSE\"F-7$Ff z$\"3Pk(G4JN0B\"F-7$F[[l$\"3[6')R^E\"**>\"F-7$F`[l$\"37\"*\\hGP8q6F-7$ Fe[l$\"3J$Hd&)4E?9\"F-7$Fj[l$\"3RJO4#F- 7$$\"3Lm;ajW8-OF-$\"3YFK8O&Q__#F-7$$\"3[LL$e9ui2%F-$\"3%=l9dN57%HF-7$$ \"3\"pm\"H2Q\\4YF-$\"3-zl&omx0Q$F-7$$\"3z***\\(oMrU^F-$\"3O#ps*G[G#y$F -7$$\"3oK$3-8Lfn&F-$\"3F7@Zx*G79%F-7$$\"3nmmm\"z_\"4iF-$\"3c\"3-z0dXX% F-7$$\"3Unmmm6m#G(F-$\"3]9tOc_(R%\\F-7$$\"39ommT&phN)F-$\"3wK)psz*H^_F -7$$\"37M$3-js.*))F-$\"3;QC@'R`=M&F-7$$\"3A,+v=ddC%*F-$\"3-%>Iv:+[R&F- 7$$\"3wM3-jsn\"p*F-$\"3$RdFhWf\"3aF-7$$\"3?n;H2)y(e**F-$\"38>,s#4>LT&F -7$$\"3'*\\i:N!)eA5F]p$\"3/R2`')3p5aF-7$$\"3KLLe*=)H\\5F]p$\"35PLBPqo+ aF-7$$\"3-++v=JN[6F]p$\"3\"4t4$45'fI&F-7$$\"3smm\"z/3uC\"F]p$\"37S\\&) p?hN^F-7$$\"3ULLe*ot*\\8F]p$\"3/XLZ:([$**[F-7$$\"3!****\\7LRDX\"F]p$\" 3sZ*[/!4:?YF-7$$\"3[LLekGhe:F]p$\"3#os!z3\\r-VF-7$$\"3%om;zR'ok;F]p$\" 3%)Q')=\")e0qRF-7$$\"3OLL3_(>/x\"F]p$\"35WP!*)H1Xj$F-7$$\"33++D1J:w=F] p$\"3r#eH)yVi.LF-7$$\"3+n;HdG\"\\)>F]p$\"3'3mqi=!)[(HF-7$$\"3oLLL3En$4 #F]p$\"3E'yl'H\")yiEF-7$$\"3_++Dc#o%*=#F]p$\"3\"[3Fvu.VS#F-7$$\"3#pmmT !RE&G#F]p$\"3%)3-x*\\2E;#F-7$$\"3D+++D.&4]#F]p$\"3Syrhj(F17$$\"3_LLL347TLF]p$\"3/nlP7zy%f&F17$$\"3nLLLLY.KNF]p$\"3!e)[P\" *>*zE%F17$$\"33++D\"o7Tv$F]p$\"3ag#okS'Q#4$F17$$\"3?LLL$Q*o]RF]p$\"3KI '=R.H9J#F17$$\"3m++D\"=lj;%F]p$\"3Uaca!e(**p;F17$$\"3S++vV&R7F17$$\"3CML$e9Ege%F]p$\"3Kh4[YX+T()F*7$$\"3]LLeR\"3Gy%F]p$\" 3$*e=.P#QST'F*7$$\"3emm;/T1&*\\F]p$\"3w\\4ldZ)fd%F*7$$\"3=nm\"zRQb@&F] p$\"3>$)HCQo&*4KF*7$$\"3:++v=>Y2aF]p$\"3=j5MK\\j]BF*7$$\"3Znm;zXu9cF]p $\"3=&)*\\9')GUn\"F*7$$\"34+++]y))GeF]p$\"3]CE9vOxv6F*7$$\"3H++]i_QQgF ]p$\"3Eb_Vi.-*H)!#@7$$\"3b++D\"y%3TiF]p$\"3b;slXFv5fFdbn7$$\"3+++]P![h Y'F]p$\"3a(\\')G%Q2XSFdbn7$$\"3iKLL$Qx$omF]p$\"33;U[x:#4(GFdbn7$$\"3Y+ ++v.I%)oF]p$\"37td>f9z')>Fdbn7$$\"3?mm\"zpe*zqF]p$\"3))Q!HVWZ3U\"Fdbn7 $$\"3;,++D\\'QH(F]p$\"3CH#=s'=;J)*!#A7$$\"3%HL$e9S8&\\(F]p$\"3ELXqFp4T pFcdn7$$\"3s++D1#=bq(F]p$\"3eT\\gi\\a;[Fcdn7$$\"3\"HLL$3s?6zF]p$\"3q$* oTD@zkLFcdn7$$\"3a***\\7`Wl7)F]p$\"3dH$*4K&[!3BFcdn7$$\"3enmmm*RRL)F]p $\"3!zrep]2) the differ ence between Q/r and the true V(r) is getting small." }}{PARA 0 "" 0 " " {TEXT -1 150 "In the case of a homogeneously charged sphere of radiu s R [rho(r)=const for rR] to a parabol a that yields V(r) for r " 0 "" {MPLTEXT 1 0 38 "plot(rhs(sol),r=0..10,title=\"Phi(r)\");" }}{PARA 13 "" 1 "" {GLPLOT2D 934 340 340 {PLOTDATA 2 "6&-%'CURVESG6$7Z7$$\"\"!F)F (7$$\"3WmmmT&)G\\a!#>$\"3'fGyu%*p!RaF-7$$\"3GLLL3x&)*3\"!#=$\"3M%[xv[4 @3\"F37$$\"3$*****\\ilyM;F3$\"3y`g8:k)*4;F37$$\"3emmm;arz@F3$\"3\\yb%z %=$R7#F37$$\"3v***\\7y%*z7$F3$\"3e,1\"3J)HxHF37$$\"3[LL$e9ui2%F3$\"3If fKX](3x$F37$$\"3z***\\(oMrU^F3$\"3uL!*)*p=1'e%F37$$\"3nmmm\"z_\"4iF3$ \"3$>)z=#p0zJ&F37$$\"3Unmmm6m#G(F3$\"3!)=52`lRsfF37$$\"39ommT&phN)F3$ \"3Ga'*3#3m([lF37$$\"3A,+v=ddC%*F3$\"33iv&)oY`]qF37$$\"3KLLe*=)H\\5!#< $\"3%[Y&zwx(p[(F37$$\"3smm\"z/3uC\"F`o$\"3#*pEj;riX\")F37$$\"3!****\\7 LRDX\"F`o$\"3Y5:!f**ptl)F37$$\"3%om;zR'ok;F`o$\"3O?D?*fFc/*F37$$\"33++ D1J:w=F`o$\"3f,a\\h<:D$*F37$$\"3oLLL3En$4#F`o$\"3nBI,7z**F37$$\"3_LLL347TLF`o$\"3I%eh_B2c%**F37$$ \"3nLLLLY.KNF`o$\"31Vg!R$zBh**F37$$\"33++D\"o7Tv$F`o$\"3#=5P&Q6#R(**F3 7$$\"3?LLL$Q*o]RF`o$\"3Q&eYy#4n\")**F37$$\"3m++D\"=lj;%F`o$\"3)o5%*[9u v)**F37$$\"3S++vV&R%***F37$$\"3]LLeR\"3Gy%F`o$\"3/Fh3nj%f***F37$$\"3emm;/T1&*\\F`o$\"3m Z@XV7D(***F37$$\"3=nm\"zRQb@&F`o$\"3'3@7kLm\")***F37$$\"3:++v=>Y2aF`o$ \"3'*RabrAr)***F37$$\"3Znm;zXu9cF`o$\"3O6)Q2x@\"****F37$$\"34+++]y))Ge F`o$\"3ylB](>4%****F37$$\"3H++]i_QQgF`o$\"3EO+&R]*f****F37$$\"3b++D\"y %3TiF`o$\"376m=&HD(****F37$$\"3+++]P![hY'F`o$\"3Qtm6>%>)****F37$$\"3iK LL$Qx$omF`o$\"3/@3TFi()****F37$$\"3Y+++v.I%)oF`o$\"3')z`Oqt\"*****F37$ $\"3?mm\"zpe*zqF`o$\"3/B[8UF%*****F37$$\"3;,++D\\'QH(F`o$\"3Kq:X$oh*** **F37$$\"3%HL$e9S8&\\(F`o$\"33+/0fP(*****F37$$\"3s++D1#=bq(F`o$\"2y[Z] M#)*****F`o7$$\"3\"HLL$3s?6zF`o$\"2=KmH-))*****F`o7$$\"3a***\\7`Wl7)F` o$\"2')GSf-#******F`o7$$\"3enmmm*RRL)F`o$\"2kHBOh%******F`o7$$\"3%zmmT vJga)F`o$\"21Z\"f&R'******F`o7$$\"3]MLe9tOc()F`o$\"2]xw6e(******F`o7$$ \"31,++]Qk\\*)F`o$\"2#H%4TK)******F`o7$$\"3![LL3dg6<*F`o$\"2gGx***))** ****F`o7$$\"3%ymmmw(Gp$*F`o$\"2)[$[aC*******F`o7$$\"3C++D\"oK0e*F`o$\" 2O=#Q&\\*******F`o7$$\"35,+v=5s#y*F`o$\"2%4hyc'*******F`o7$$\"#5F)$\"2 15tKx*******F`o-%'COLOURG6&%$RGBG$Fg\\l!\"\"F(F(-%&TITLEG6#Q'Phi(r)6\" -%+AXESLABELSG6$Q\"rFd]lQ!Fd]l-%%VIEWG6$;F(Ff\\l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 482 "The interesting information in Phi(r) is the found at distances r=0 to about r=5, i.e., the region where the c harge density times the Jacobian (r^2) is essentially different from z ero. That is where Phi(r) varies. We should obtain a reasonable soluti on over a finite range of r, as long as we keep in mind that the bound ary condition Phi(a)=Q makes only sense if the integral of the radial \+ density (r^2*rho(r)) from zero up to a does enclose the total charge ( practically speaking)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "Suppose we pick a=4. We would really need a functio n space with boundary condition f(0)=0, f(a)=const." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "a:=4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"aG\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "k:=n->n*Pi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kGf*6#%\"nG6\"6$%)operatorG%&arr owGF(*&9$\"\"\"%#PiGF.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "fB:=n->sin((2*n-1)*Pi*r/a/2)/sqrt(2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fBGf*6#%\"nG6\"6$%)operatorG%&arrowGF(*&-%$sinG6#,$* &#\"\"\"\"\"#F3**,&*&F4F39$F3F3F3!\"\"F3%#PiGF3%\"rGF3%\"aGF9F3F3F3-%% sqrtG6#F4F9F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "For normali zation, and for projection we define an inner product for real functio ns on the interval [0,1]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "IP:=(f,g)->int(f*g,r=0..a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# IPGf*6$%\"fG%\"gG6\"6$%)operatorG%&arrowGF)-%$intG6$*&9$\"\"\"9%F2/%\" rG;\"\"!%\"aGF)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "IP(f B(2),fB(3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "seq(IP(fB(j),fB(j)),j=1..5);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"\"F#F#F#F#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 121 "We have a basis that should work well for our probl em, because it is made of eigenfunctions of the differential operator. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "plot([seq(fB(j),j=1..5) ],r=0..a);" }}{PARA 13 "" 1 "" {GLPLOT2D 912 392 392 {PLOTDATA 2 "6)-% 'CURVESG6$7S7$$\"\"!F)F(7$$\"3Hmmmm;')=()!#>$\"38%[j>5#e?CF-7$$\"3RLLL e'40j\"!#=$\"3?HOa4\"4X_%F-7$$\"3mmmm;6m$[#F3$\"3%HoJLm0d)oF-7$$\"3fmm m;yYULF3$\"3%QX#y@Eua#*F-7$$\"3%HLL$eF>(>%F3$\"3y&*H2#Q2-;\"F37$$\"3Qm mm\">K'*)\\F3$\"31?<;lJnw8F37$$\"3P*****\\Kd,\"eF3$\"39D(pre.%*f\"F37$ $\"3-mmm\"fX(emF3$\"3Ub>(=N-!G=F37$$\"3.*****\\U7Y](F3$\"3KC./L#[Q0#F3 7$$\"3'QLLLV!pu$)F3$\"3qIVTW=z$G#F37$$\"3xmmm;c0T\"*F3$\"3!*=r]_o7%[#F 37$$\"3#*******H,Q+5!#<$\"3BP#Hwmbpq#F37$$\"3)*******\\*3q3\"F`o$\"3.$ *>JXIdFHF37$$\"3)*******p=\\q6F`o$\"3mj[9?i(p8$F37$$\"3mmm;fBIY7F`o$\" 3)fhSFb>UK$F37$$\"3GLLLj$[kL\"F`o$\"36lYaBA-VNF37$$\"3?LLL`Q\"GT\"F`o$ \"3:#Q7?79\\s$F37$$\"3!*****\\s]k,:F`o$\"3z)RY'e@FKRF37$$\"39LLL`dF!e \"F`o$\"3K$>!)*=]$=6%F37$$\"33++]sgam;F`o$\"34+-!QnFVI%F37$$\"3/++]z.$[%F37$$\"3QLLLe/TM=F`o$\"3#fR)pxs_kYF37$$\"3JLL$eDBJ \">F`o$\"3S\\G2e9aE[F37$$\"3immmTc-)*>F`o$\"3%>vHS&=7'*\\F37$$\"3Mmm;f `@'3#F`o$\"3?pR[2dQm^F37$$\"3y****\\nZ)H;#F`o$\"31O/4>Oc4`F37$$\"3Ymmm Jy*eC#F`o$\"3a)4W*HfxeaF37$$\"3')******R^bJBF`o$\"3$GF`o$\"3$*= pIi#*=SjF37$$\"3$*******pfaL&*\\F37$FU$\"3Qp)R23Y%oaF37$FZ$\"3=1IVN([%)*eF37$Fin$\"3\"**o@Y URgA'F37$F^o$\"3kn8z))f-MlF37$Fdo$\"3s[rMu[TvnF37$Fio$\"3\"GbjYVX8%pF3 7$F^p$\"3&3UN>sKR.(F37$Fcp$\"3wJ>9(>?52(F37$Fhp$\"3N(QCg4#4SqF37$F]q$ \"3p=))RiE^KpF37$Fbq$\"3/G\"fr*)RRx'F37$Fgq$\"3$z\")z:?*>LlF37$F\\r$\" 3(H3t-8R7C'F37$Far$\"3'GH#H5#)QueF37$Ffr$\"3]?lk1xo%[&F37$F[s$\"3\\Ovb %R;;,&F37$F`s$\"35kspKLDnWF37$Fes$\"3)\\`R@NSR&RF37$Fjs$\"3%4I&R.pPjLF 37$F_t$\"3s4'ym/g'>FF37$Fdt$\"3gcihGN3j?F37$Fit$\"3P#)ew!y?'39F37$F^u$ \"3K)o<8Kn;n'F-7$Fcu$!3lCCL+[Q+d!#@7$Fhu$!3O*)zq=A?RsF-7$F]v$!3!y)*f/R m%o8F37$Fbv$!3q>cKy5jf?F37$Fgv$!3%Q'4f$Q$*4p#F37$F\\w$!3\\b>cMD+DLF37$ Faw$!3=Gp=yeL8RF37$Ffw$!3It:\"49:)*[%F37$F[x$!3[o]zT\"H9+&F37$F`x$!3!H xP(G^AvaF37$Fex$!3)[l1*>!R6*eF37$Fjx$!3:LVpPOUAiF37$F_y$!3]UX*GjI&QlF3 7$Fdy$!3#GvWL>S5w'F37$Fiy$!3Cu\"p+csL$pF37$F^z$!3IA)*fSW.MqF37$Fcz$!3s va'=\"y1rqF3-Fhz6&FjzF(F[[lF(-F$6$7inF'7$F+$\"3jSwM#GBY?\"F37$F1$\"3^_ $pI3F`A#F37$F7$\"3X1sgf[D8LF37$F<$\"3M]\"[<)zk9VF37$FA$\"3!*QEB$z(z*=& F37$FF$\"33d\\V#eo8(eF37$FK$\"3htUc8LVFkF37$FP$\"3Q9bljBFFoF37$$\"3_KL L3!z;3(F3$\"3;P7`z-VcpF37$FU$\"3*e2fJ\\Vw.(F37$$\"3HLL3F>8AxF3$\"3I^Tu \\hagqF37$$\"3Wmm;H9lRzF3$\"3[6/D)Rr02(F37$$\"3e***\\7$40(F37$$\"3K+++DI(yv)F3$\"3uP(GZ#>#H*pF37$Fin$ \"3Mqb\"e_PV*oF37$$\"3+LLLe%GCd*F3$\"3-ss+&z:nt'F37$F^o$\"3ir)ocI$zIlF 37$Fdo$\"3mvYr\\:mxfF37$Fio$\"3wBw9SL9\"G&F37$F^p$\"35(RcoA/a_%F37$Fcp $\"3_ZmI%=8!)\\$F37$Fhp$\"3r&>+(phxSDF37$F]q$\"3#)*ej_S)3d8F37$Fbq$\"3 d!p&z)oJyt#F-7$Fgq$!3(37k;C&*H@*F-7$F\\r$!3*)zZ@Q$)GN?F37$Far$!3\"=R45 ))e39$F37$Ffr$!3%R%e&e^]'ySF37$F[s$!3g$4t+Ez0)\\F37$F`s$!3izRe!=44x&F3 7$Fes$!3,\\\"z(fG2>jF37$Fjs$!35\"[_(ew$)\\nF37$$\"3;LL$e[E()G#F`o$!3W% ))*e!*o'H!pF37$F_t$!3rnqbK'3t+(F37$$\"3s****\\AYXtBF`o$!3=sKj\"4l91(F3 7$Fdt$!3a=\"pDmay1(F37$$\"3k***\\(3S*eX#F`o$!3#[(fO\\m_GqF37$Fit$!3=:( 49P)oWpF37$$\"3m***\\PcY9a#F`o$!3LW^7F3$\"37+S98KO%Q#F37$F1$\"3#[e\"oNrEkIF37$$\"3/++](Q&3d? F3$\"3uILn%Gt()y$F37$F7$\"3K+0(e\"GChWF37$$\"3jmmmmW18HF3$\"33!zGa\\ni 2&F37$F<$\"3UN&\\gGY1i&F37$FA$\"3!>P7%y4/lkF37$$\"3Q*****\\Z7Mf%F3$\"3 D<3X!*p1QnF37$FF$\"3dZT)**\\R7$pF37$$\"3M*****\\ZjZ>&F3$\"3!>RKf*H3**p F37$$\"3IKLLeZ*)*R&F3$\"3u!zVqtxY/(F37$$\"3HlmmTg-0cF3$\"3!*R0I'yyy1(F 37$FK$\"3C\\e(>R7'oqF37$$\"3Jmmm\"R/B-'F3$\"3%*[^*[/Od/(F37$$\"39KLLe9 XMiF3$\"3Eh^.i[!*)*pF37$$\"33*****\\_)fYkF3$\"31Sd)\\1y#GpF37$FP$\"3y$ 3t\"yd4MoF37$F`fl$\"3)>qAQ'fRxlF37$FU$\"3w8>:oP*=B'F37$FZ$\"3+`m'Q9kHE &F37$Fjgl$\"3\\l\\'R\"))GPZF37$Fin$\"3+=HkK=5fTF37$Fbhl$\"3qPXHq!yLX$F 37$F^o$\"3%fm$HXE:*p#F37$$\"3&*******RXpV5F`o$\"3'ps:7(Gq.>F37$Fdo$\"3 Exot*R'H\"3\"F37$$\"3()******4/vG6F`o$\"3\\Kq)\\'\\KTFF-7$Fio$!3qw0vVl Om`F-7$$\"3@LLe9rR37F`o$!3I\\#QV2gqE\"F37$F^p$!3+Qsl=$RP)>F37$$\"33++D h`P\"H\"F`o$!3sU1y@/J2GF37$Fcp$!3-Pnhe^oF37$Fgq$!3wy&=(*\\n-,(F37$$\"3+++v3N3 (o\"F`o$!3Sl?uw#380(F37$$\"3&******\\%4i2xl@vmHIIF37$Fes$!3 1H1%e%>dSBF37$$\"3NLLe*HTW?#F`o$!3s%RL(fSkm:F37$Fjs$!3VkD0n]#Rs(F-7$Fg [m$\"3'4\")z02J\\&e!#?7$F_t$\"3ACGv9`!o)))F-7$F_\\m$\"3AW:+zc(*)o\"F37 $Fdt$\"3!*>;D&)**)oY#F37$Fg\\m$\"3uj+m!))\\&)=$F37$Fit$\"3]a)4hi_1(QF3 7$F_]m$\"3e;oX(o29d%F37$F^u$\"3sY\\'=;iA?&F37$Fcu$\"3n(Qs*oROIhF37$$\" 3Emmm^b`5FF`o$\"3kRZ(**ysX]'F37$Fhu$\"3;UAKblA(y'F37$$\"3)HL$e9=&Gz#F` o$\"3i&\\()4\")G4'pF37$F]v$\"3%RKJ#)\\jT0(F37$$\"30L3_!zyE%GF`o$\"3CdK '>Kza1(F37$$\"3i**\\(=5uL&GF`o$\"3QGFia$)oqqF37$$\"3?m\"HKTpS'GF`o$\"3 E$3rR4'ypqF37$$\"3?LLeCZwuGF`o$\"3?DY%)>LxiqF37$$\"3ym;HZ`:'*GF`o$\"3v %QyAvX/.(F37$Fbv$\"3EC8U#RZlF37$F\\w$\"3=NS(=f:/w&F37$Faw$\"3'e!3`?&3Tp%F37$$\"3 \\mm\"zM]v?$F`o$\"3f:_i%z]m.%F37$Ffw$\"3*QFpUn#oALF37$$\"31LLe**o4#H$F `o$\"3#R#H:Q3&4f#F37$F[x$\"33kL([sqb#=F37$$\"3mmm;WV*fP$F`o$\"3.]O)))z =%=5F37$F`x$\"3U913wtNu>F-7$$\"39++v8)z/Y$F`o$!3e*o(=-'yU>'F-7$Fex$!3* Q:;kQ\"pgAL%F37$$\"3\"*****\\n' *33PF`o$!3='*\\FBv69\\F37$Fdy$!3N*yhcZExV&F37$Fiy$!3QH#*p&*p=KjF37$Fd` m$!3:m#\\GZ*>UmF37$F^z$!3o#)H%4\\B-(oF37$$\"31+]i0j\"[$RF`o$!3==Ghit&y &pF37$F\\am$!3$zFz>=x1-(F37$$\"3-+](=5s#yRF`o$!3o7K]7*e%eqF3F_dl-Fhz6& FjzF(F(F[[l-F$6$7erF'7$Ffam$\"3y!R&*fQp^3\"F37$F+$\"3ro(=;uGY9#F37$F^b m$\"3r.!Qp3xf-$F37$F1$\"3y&=O/&Q-`QF37$Ffbm$\"3%)QO\"oCJ)*p%F37$F7$\"3 Qu>\"HD9+W&F37$F^cm$\"3YdB%e7C/1'F37$F<$\"3`y.ELn^TlF37$$\"3/++](GI)pP F3$\"3#y^=`*)H5(oF37$FA$\"3Kh#[*pl3WqF37$$\"3'*****\\(oZiH%F3$\"3O:TS+ 3PhqF37$$\"3Wmmm;EI&R%F3$\"3eRH([L,+2(F37$$\"3!HLLeadV\\%F3$\"35m<;&z5 BLF37$F]gl$\"3&\\9\\W34!GBF37$FZ$\"3WVH!Qo#)zF\"F37$Fjgl$\"3%zadj(e2tK F-7$Fin$!3=K2V_Rh$H'F-7$Fbhl$!3IMO=^2o\"p\"F37$F^o$!3,S3XH[v9FF37$Fjgm $!3u&Rq[ZI'yOF37$Fdo$!3-2>([skkb%F37$Fbhm$!3h(Q!olpz,`F37$Fio$!3b#yMdm \\>$fF37$Fjhm$!3CpS*3%4&GR'F37$F^p$!3'Qr.;<)=RnF37$$\"3OL$3-')Q)o7F`o$ !3'*zn$ylj\"))oF37$Fbim$!37R/(QEiM*pF37$$\"3KL3x6Ok-8F`o$!30NQVRJ^HqF3 7$$\"3dm;Hi=\"RJ\"F`o$!3#4;/gI;W0(F37$$\"3\")*\\7G6!=D8F`o$!3Kt?0rA8oq F37$Fcp$!3;r5?$GR12(F37$$\"37LLe\\S*fM\"F`o$!3[Hum?**)R1(F37$$\"3?LL$e tRbN\"F`o$!3_b\"ex2,$\\qF37$$\"3GLL3Aa3l8F`o$!3(*f&[F`o$\"3ik#QrN^m:%F37$F[s$\"3%\\\\ y@w))\\'\\F37$$\"3[mmT+07U?F`o$\"3s$z7)\\RG'o&F37$F`s$\"3YZ?oyMvpiF37$ Fc]n$\"3I_eE,\"[Vl'F37$Fes$\"3qo!**)*Ghm\"pF37$$\"3Mm;aLIr$=#F`o$\"3%[ -5Y!on0qF37$F[^n$\"3I8DQR(4r0(F37$$\"3&o;/EV0[@#F`o$\"3ul,'=fQ'oqF37$$ \"3!***\\il&p^A#F`o$\"3y;ojxToqqF37$$\"3'H$ek)pLbB#F`o$\"3)*R+O_PCjqF3 7$Fjs$\"3#[c%3)HFj/(F37$Fg[m$\"37e()p&)*Rm(oF37$F_t$\"3!ps_6&=p\\lF37$ F_\\m$\"3e2mP%p@[3'F37$Fdt$\"3-?fyR,w'[&F37$Fg\\m$\"3;D0\"z]JOz%F37$Fi t$\"3[O)p\\5iA+%F37$F_]m$\"3CS![ImU#GIF37$F^u$\"33V[N7Jux>F37$$\"3Nmm; %30pi#F`o$\"3;dz(*=\"HU!**F-7$Fcu$!3]F]DeR654\" F37$Fhu$!3#e*Q&oU599#F37$Fg`n$!3Em+0$F37$F]v$!3Q5O'>)4Q+RF37$F^bn $!3egF37$Fgv$!3Z'p)3? c/9lF37$$\"3cmm;W/8SIF`o$!3O>gd/ss\\oF37$F\\w$!3Xw2gGY?MqF37$$\"3amTgK <\\#4$F`o$!3OE+d3GwbqF37$$\"3LL$3F=wF5$F`o$!3Inq(e(***z1(F37$$\"37+D\" GjgI6$F`o$!33zohk***32(F37$$\"3Ymm\"H3XL7$F`o$!3c'[gP%*eW1(F37$$\"3f** \\7$)R\"R9$F`o$!3U>weP-gBqF37$Faw$!3M$Qj:pRc%pF37$Ficn$!3'*>eH6')GkmF3 7$Ffw$!36!fgD*=$)GiF37$Fadn$!3i7AatL7tcF37$F[x$!3#ovXM75d*\\F37$Fidn$! 3uf=U/YC#>%F37$F`x$!3i`hqf>t%H$F37$Faen$!3ik:?QbfJBF37$Fex$!3:(zVu59qJ \"F37$Fien$!3s'G:ZjJbe$F-7$Fjx$\"3kU6!H'p*e1'F-7$Fafn$\"3/l+0Q(zxp\"F3 7$F_y$\"3W%>!>'GHuu#F37$Fifn$\"3&=-**e(3.IOF37$Fdy$\"3]<+&p\\^:W%F37$$ \"33LLe*3k**y$F`o$\"3m$*=\\!\\Y1@&F37$Fiy$\"3w4CyYyxjeF37$Fd`m$\"3C5D: E+(34SnF37$Fjgn$\"35#3F+&4D%)oF37$F\\am$\"3MiSl)pL y)pF37$$\"3E+D\"G:3u'RF`o$\"35)G7\"H$3U-(F37$Fbhn$\"3y!\\qddG-0(F37$$ \"3!)*\\P40O\"*)RF`o$\"3ol$*=#3ce1(F3Fbz-Fhz6&FjzF[[lF(F[[l-%+AXESLABE LSG6$Q\"r6\"Q!Ffcp-%%VIEWG6$;F(Fcz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "This is a set of \+ cosine functions that incorporates the correct boundary conditions." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "Now watc h: we know the result of the second derivative operator on them:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "LHS:=seq(diff(fB(n),r$2)/fB( n),n=1..15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$LHSG61,$*&\"#k!\"\" %#PiG\"\"#F),$*(\"\"*\"\"\"F(F)F*F+F),$*(\"#DF/F(F)F*F+F),$*(\"#\\F/F( F)F*F+F),$*(\"#\")F/F(F)F*F+F),$*(\"$@\"F/F(F)F*F+F),$*(\"$p\"F/F(F)F* F+F),$*(\"$D#F/F(F)F*F+F),$*(\"$*GF/F(F)F*F+F),$*(\"$h$F/F(F)F*F+F),$* (\"$T%F/F(F)F*F+F),$*(\"$H&F/F(F)F*F+F),$*(\"$D'F/F(F)F*F+F),$*(\"$H(F /F(F)F*F+F),$*(\"$T)F/F(F)F*F+F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "RHS:=seq(-4*Pi*int(fB(n)*rho(r)*r,r=0..a),n=1..15);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$RHSG61,$**\"$G\"\"\"\"\"\"##F)F*, (*&\"#KF)%#PiGF)!\"\"*&\"%/BF)-%$expG6#!\")F)F)*(\"\"(F)F3F))F/F*F)F)F ),(\"&Ob'F)*&\"$7&F)F9F)F)*$)F/\"\"%F)F)F0F),$**\"$%QF)F*F+,(*&F.F)F/F )F)*(\"#@F)F3F)F9F)F)*&\"$o(F)F3F)F)F),(F;F)*&\"%3YF)F9F)F)*&\"#\")F)F ?F)F)F0F0,$**F(F)F*F+,(*&\"$g\"F)F/F)F0*(\"$v\"F)F3F)F9F)F)*&F2F)F3F)F )F),(F;F)*&\"&+G\"F)F9F)F)*&\"$D'F)F?F)F)F0F),$**F(F)F*F+,(*&\"$C#F)F/ F)F)*(\"$V$F)F3F)F9F)F)*&F2F)F3F)F)F),(F;F)*&\"&)3DF)F9F)F)*&\"%,CF)F? F)F)F0F0,$**\"%_6F)F*F+,(*&F.F)F/F)F0*(\"#jF)F3F)F9F)F)*&\"$c#F)F3F)F) F),(F;F)*&\"&s9%F)F9F)F)*&\"%hlF)F?F)F)F0F),$**F(F)F*F+,(*&\"$_$F)F/F) F)*(\"$Z)F)F3F)F9F)F)*&F2F)F3F)F)F),(F;F)*&\"&_>'F)F9F)F)*&\"&TY\"F)F? F)F)F0F0,$**F(F)F*F+,(*&\"$;%F)F/F)F0*(\"%$=\"F)F3F)F9F)F)*&F2F)F3F)F) F),(F;F)*&\"&Gl)F)F9F)F)*&\"&h&GF)F?F)F)F0F),$**FCF)F*F+,(*&FSF)F/F)F) *(\"$D&F)F3F)F9F)F)*&FIF)F3F)F)F),(F;F)*&\"'+_6F)F9F)F)*&\"&D1&F)F?F)F )F0F0,$**F(F)F*F+,(*&\"$W&F)F/F)F0*(\"%B?F)F3F)F9F)F)*&F2F)F3F)F)F),(F ;F)*&\"'oz9F)F9F)F)*&\"&@N)F)F?F)F)F0F),$**F(F)F*F+,(*&\"$3'F)F/F)F)*( \"%FDF)F3F)F9F)F)*&F2F)F3F)F)F),(F;F)*&\"'K[=F)F9F)F)*&\"'@.8F)F?F)F)F 0F0,$**FCF)F*F+,(*&FjnF)F/F)F0*(\"%H5F)F3F)F9F)F)*&FIF)F3F)F)F),(F;F)* &\"'#zD#F)F9F)F)*&\"'\"[%>F)F?F)F)F0F),$**F(F)F*F+,(*&\"$O(F)F/F)F)*( \"%.PF)F3F)F9F)F)*&F2F)F3F)F)F),(F;F)*&\"'[3FF)F9F)F)*&\"'T)z#F)F?F)F) F0F0,$**F(F)F*F+,(*&\"$+)F)F/F)F0*(\"%vVF)F3F)F9F)F)*&F2F)F3F)F)F),(F; F)*&\"'++KF)F9F)F)*&\"'D1RF)F?F)F)F0F),$**FeoF)F*F+,(*&\"#'*F)F/F)F)*( \"$n&F)F3F)F9F)F)*&F[pF)F3F)F)F),(F;F)*&\"'[KPF)F9F)F)*&\"'T9`F)F?F)F) F0F0,$**F(F)F*F+,(*&\"$G*F)F/F)F0*(\"%()eF)F3F)F9F)F)*&F2F)F3F)F)F),(F ;F)*&\"'#fI%F)F9F)F)*&\"'\"G2(F)F?F)F)F0F)" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 27 "c:=n->evalf(RHS[n]/LHS[n]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cGf*6#%\"nG6\"6$%)operatorG%&arrowGF(-%&evalfG6#*&& %$RHSG6#9$\"\"\"&%$LHSGF2!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "c(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+z!3il\"! \"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "phi5:=add(c(n)*fB(n) ,n=1..5);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%phi5G,@*($\"+&RS5G)!#5 \"\"\"-%$sinG6#,$*(\"\")!\"\"%#PiGF*%\"rGF*F*F*\"\"##F*F4F**($\"+O$[%f ;F)F*-F,6#,$**\"\"$F*F0F1F2F*F3F*F*F*F4F5F**($\"+ubPcY!#6F*-F,6#,$**\" \"&F*F0F1F2F*F3F*F*F*F4F5F**($\"+He1X:FAF*-F,6#,$**\"\"(F*F0F1F2F*F3F* F*F*F4F5F**($\"+?9/oe!#7F*-F,6#,$**\"\"*F*F0F1F2F*F3F*F*F*F4F5F**($\"+ e'z'eDFRF*-F,6#,$**\"#6F*F0F1F2F*F3F*F*F*F4F5F**($\"+**[t@7FRF*-F,6#,$ **\"#8F*F0F1F2F*F3F*F*F*F4F5F**($\"+!y1$Qk!#8F*-F,6#,$**\"#:F*F0F1F2F* F3F*F*F*F4F5F**($\"+)yBmd$FeoF*-F,6#,$**\"#F*F0F1F2F*F3F*F*F*F4F5F**($\"+OE/;8FeoF*-F ,6#,$**\"#@F*F0F1F2F*F3F*F*F*F4F5F**($\"+S9-+')!#9F*-F,6#,$**\"#BF*F0F 1F2F*F3F*F*F*F4F5F**($\"+!GTMo&FfqF*-F,6#,$**\"#DF*F0F1F2F*F3F*F*F*F4F 5F**($\"+C&H>'RFfqF*-F,6#,$**\"#FF*F0F1F2F*F3F*F*F*F4F5F**($\"+qZwfFFf qF*-F,6#,$**\"#HF*F0F1F2F*F3F*F*F*F4F5F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "phi2:=add(c(n)*fB(n),n=1..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%phi2G,&*($\"+&RS5G)!#5\"\"\"-%$sinG6#,$*(\"\")!\"\"% #PiGF*%\"rGF*F*F*\"\"##F*F4F**($\"+O$[%f;F)F*-F,6#,$**\"\"$F*F0F1F2F*F 3F*F*F*F4F5F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "phi3:=add( c(n)*fB(n),n=1..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%phi3G,(*($\" +&RS5G)!#5\"\"\"-%$sinG6#,$*(\"\")!\"\"%#PiGF*%\"rGF*F*F*\"\"##F*F4F** ($\"+O$[%f;F)F*-F,6#,$**\"\"$F*F0F1F2F*F3F*F*F*F4F5F**($\"+ubPcY!#6F*- F,6#,$**\"\"&F*F0F1F2F*F3F*F*F*F4F5F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "plot([phi2,phi3,phi5,rhs(sol)],r=0..a,color=[red,blue ,green,black],thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 1042 382 382 {PLOTDATA 2 "6)-%'CURVESG6$7S7$$\"\"!F)F(7$$\"3Hmmmm;')=()!#>$\"3> xx)*>0K:kF-7$$\"3RLLLe'40j\"!#=$\"3U<8ow\\Q(>\"F37$$\"3mmmm;6m$[#F3$\" 3l@&o+IOt\"=F37$$\"3fmmm;yYULF3$\"3sJ=>Ow>LCF37$$\"3%HLL$eF>(>%F3$\"3* HL\"R8:ENIF37$$\"3Qmmm\">K'*)\\F3$\"3_1!4!eI\\\"e$F37$$\"3P*****\\Kd, \"eF3$\"3`D_I!o*yKTF37$$\"3-mmm\"fX(emF3$\"3'[GUQJ^ao%F37$$\"3.*****\\ U7Y](F3$\"3Sf\"Rky?l@&F37$$\"3'QLLLV!pu$)F3$\"3NYCF\"po+u&F37$$\"3xmmm ;c0T\"*F3$\"3O/\\*\\6*e!='F37$$\"3#*******H,Q+5!#<$\"39+&=rbd=l'F37$$ \"3)*******\\*3q3\"F`o$\"3#4O(H<.O(4(F37$$\"3)*******p=\\q6F`o$\"3OKbm &=Y#*\\(F37$$\"3mmm;fBIY7F`o$\"3-7\"4)p))3SyF37$$\"3GLLLj$[kL\"F`o$\"3 YK^qq0y9#)F37$$\"3?LLL`Q\"GT\"F`o$\"3\"Q)\\E&omd])F37$$\"3!*****\\s]k, :F`o$\"3Nl-OQ-\\8))F37$$\"39LLL`dF!e\"F`o$\"3Uf50O]De!*F37$$\"33++]sga m;F`o$\"3a*zundjrH*F37$$\"3/++]F`o$\"3uQN9M\"oS\")*F37$$\"3imm mTc-)*>F`o$\"3,*HOc2@z$**F37$$\"3Mmm;f`@'3#F`o$\"3e1_)p\\CR+\"F`o7$$\" 3y****\\nZ)H;#F`o$\"3$*H-&4T,1,\"F`o7$$\"3YmmmJy*eC#F`o$\"3o=gx$)Qr:5F `o7$$\"3')******R^bJBF`o$\"3Z,VW1M()=5F`o7$$\"3f*****\\5a`T#F`o$\"3_Nf Y/w0?5F`o7$$\"3o****\\7RV'\\#F`o$\"35F`o7$$\"3k*****\\@fke#F` o$\"3%*Gj]'>:u,\"F`o7$$\"3/LLL`4NnEF`o$\"3[w6E*H%=95F`o7$$\"3#******* \\,s`FF`o$\"30k+O48h45F`o7$$\"3[mm;zM)>$GF`o$\"3)=wrBG\\Y+\"F`o7$$\"3$ *******pfa$=i (y*F37$$\"3#)****\\7yh]KF`o$\"3Io/&pB#e<(*F37$$\"3xmmm')fdLLF`o$\"3Ocz cuwZ_'*F37$$\"3bmmm,FT=MF`o$\"3q^t([&)y)*e*F37$$\"3FLL$e#pa-NF`o$\"3$o R3\"4U?L&*F37$$\"3!*******Rv&)zNF`o$\"3))Rz@.['p[*F37$$\"3ILLLGUYoOF`o $\"3I)fZNWu>W*F37$$\"3_mmm1^rZPF`o$\"3;2+\"Q:8)4%*F37$$\"34++]sI@KQF`o $\"3Q%e^PMKYQ*F37$$\"34++]2%)38RF`o$\"3cE@:'*f\")p$*F37$$\"\"%F)$\"3o) yRVHXVO*F3-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-F$6$7SF'7$F+$\"3eGoa0i: PvF-7$F1$\"3S8m:]Ti/9F37$F7$\"3_dF/![\"*e7#F37$F<$\"3=TUkl+,NGF37$FA$ \"3G%[fpYu&=NF37$FF$\"3JHI2.!z#GTF37$FK$\"3)o6,+bg8t%F37$FP$\"3V+dE*>e 7K&F37$FU$\"3[y/k6!>>(eF37$FZ$\"3')4'p%>$)z'R'F37$Fin$\"3E*Gx?hTE#oF37 $F^o$\"3y%olc3a+E(F37$Fdo$\"3ko>*=1XSl(F37$Fio$\"3!fl-F#f1\"*zF37$F^p$ \"3'=K(*Q]G:E)F37$Fcp$\"3BTQoL=aS&)F37$Fhp$\"3W\\o())eX\"=]*F37$Fes$\"3eg-@zX`<&*F37$Fjs$\"3Sylq`KaG&*F37$F _t$\"3*Gl?$\\3;O&*F37$Fdt$\"38y>VGVOU&*F37$Fit$\"3]TpxNu6\\&*F37$F^u$ \"3)f(y-eHGf&*F37$Fcu$\"3C-)pCo)*>d*F37$Fhu$\"3ozr0*GU,f*F37$F]v$\"34` ,!op&*4h*F37$Fbv$\"3e`[0EsfQ'*F37$Fgv$\"3gL2K7Y$)o'*F37$F\\w$\"3\\Ip<. ZB/(*F37$Faw$\"3HaR45gtT(*F37$Ffw$\"3Wh.+CD#Hy*F37$F[x$\"346l?')e@B)*F 37$F`x$\"3#f`w@r0P')*F37$Fex$\"3sk>xVoy,**F37$Fjx$\"3.TErZs#Q$**F37$F_ y$\"3%[5A6%z\"e'**F37$Fdy$\"33+$R:Yp\"*)**F37$Fiy$\"3kw;b4Hx+5F`o7$F^z $\"3^#)Hiuh(=+\"F`o7$Fcz$\"3-:lJCcG-5F`o-Fhz6&FjzF(F(F[[l-F$6$7SF'7$F+ $\"3Fs?9BqdS')F-7$F1$\"3_KD]ruF+;F37$F7$\"3/(\\JiJ\"R'R#F37$F<$\"3uXf; P\"pH:$F37$FA$\"3))=**pCb7bQF37$FF$\"3Oo(G!eQxeWF37$FK$\"3Cl#*=kImN]F3 7$FP$\"3A%Q&)**zu9e&F37$FU$\"37cuwQ`QvgF37$FZ$\"3!zKK@`CK`'F37$Fin$\"3 I![Y;'=m'*oF37$F^o$\"3K`&)y!>oWE(F37$Fdo$\"3%Qo\"Hs=Z$f(F37$Fio$\"3#p% Q$R*=svyF37$F^p$\"3S2Y@1I*[5)F37$Fcp$\"3)Ql.Ol#fY$)F37$Fhp$\"38kRcsnYF &)F37$F]q$\"3kv32`y?8()F37$Fbq$\"3Y8OF*RXz&))F37$Fgq$\"3w(Q$)RPrz**)F3 7$F\\r$\"3nC^G0$)*\\6*F37$Far$\"3YsnkYC'=A*F37$Ffr$\"3H%)=odLl2$*F37$F [s$\"3IV([]wQ%)Q*F37$F`s$\"3it5GXL6h%*F37$Fes$\"3?\"oC9%[J;&*F37$Fjs$ \"3'oOmz#obo&*F37$F_t$\"3(4H+(>rK:'*F37$Fdt$\"3#pNp7'=qa'*F37$Fit$\"3s **>WwsX(o*F37$F^u$\"37ekKL\\`=(*F37$Fcu$\"3qzYANXYU(*F37$Fhu$\"35UrK#p ![k(*F37$F]v$\"3Q%>I(*3G;y*F37$Fbv$\"3_aM+'>svz*F37$Fgv$\"3v+%Q&eP85)* F37$F\\w$\"3gv7_Xw2@)*F37$Faw$\"34wYf_h,I)*F37$Ffw$\"3#)*fAS&H)y$)*F37 $F[x$\"3(>geU%)*F37$F`x$\"3Q9_H(R)\\\\)*F37$Fex$\"3CPB\\,:c`)*F37$F jx$\"3?CR()f'oj&)*F37$F_y$\"3oK%))=vK(e)*F37$Fdy$\"3]zMf5BIg)*F37$Fiy$ \"3SfoiW7bh)*F37$F^z$\"3q/yTVILi)*F37$Fcz$\"3G&\\u7FOE')*F3-Fhz6&FjzF( F[[lF(-F$6$7SF'7$F+$\"3/hQj**QLy')F-7$F1$\"3j](=]Y$*eg\"F37$F7$\"3[#Qd fh$\\.CF37$F<$\"3))G@j*)=DiJF37$FA$\"3mS>_c.[nQF37$FF$\"33j'o4AwTZ%F37 $FK$\"3WvQl@Cz`]F37$FP$\"3%znYHAS=g&F37$FU$\"3K*e^KT%z(4'F37$FZ$\"3Y\\ f#34d!elF37$Fin$\"3].iqUt,CpF37$F^o$\"31R!QUHP[H(F37$Fdo$\"3Gby#*\\zlE wF37$Fio$\"3d*\\>(>AF6zF37$F^p$\"3>6h;-iVU\")F37$Fcp$\"3iJP.gxj'Q)F37$ Fhp$\"3arI(G[!*)p&)F37$F]q$\"3EE38&Q&fe()F37$Fbq$\"3+#eO#eC!f!*)F37$Fg q$\"3u:Q_\"3W./x#*F37$Ffr$ \"379+7nQ>l$*F37$F[s$\"3K=C'H*)>([%*F37$F`s$\"3iN#fBvnU_*F37$Fes$\"3EQ ;'GXI=e*F37$Fjs$\"3\"[ZO(yFWO'*F37$F_t$\"3]lhy#)*)f&o*F37$Fdt$\"3#R:?n 7Dus*F37$Fit$\"3ku;/wesi(*F37$F^u$\"3s*pLXQ>nz*F37$Fcu$\"3'4#e\"3G%=B) *F37$Fhu$\"3)[\"*)z(=Jx%)*F37$F]v$\"3_4Kln)yq')*F37$Fbv$\"3'**pj*yS[&) )*F37$Fgv$\"3wR9([i60!**F37$F\\w$\"3%*eM)oD`T\"**F37$Faw$\"3^[V(fr5d#* *F37$Ffw$\"3*ogv-Yth$**F37$F[x$\"3sOVi0i([%**F37$F`x$\"3UtH+x!oD&**F37 $Fex$\"3IG^u(p]\"f**F37$Fjx$\"3sd$zs'=Sk**F37$F_y$\"3e:(Qc41'p**F37$Fd y$\"33>:xd3it**F37$Fiy$\"3AOH^-gKx**F37$F^z$\"37qT5x%*Q!)**F37$Fcz$\"3 Iv[gooA$)**F3-Fhz6&FjzF)F)F)-%*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q\"r6 \"Q!Fegm-%%VIEWG6$;F(Fcz%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {MARK "76 0 0" 71 }{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 149660884 149664064 149668704 149671400 149664256 149664336 149681336 149664456 149664536 149692316 149693792 149696560 149698032 149700728 149704720 149707416 149708888 }{RTABLE M7R0 I6RTABLE_SAVE/149660884X,%)anythingG6"6"[gl!"%!!!#%"#"#"""""$F(""%6" } {RTABLE M7R0 I6RTABLE_SAVE/149664064X,%)anythingG6"6"[gl!"%!!!##"#""%"xG%"yG6" } {RTABLE M7R0 I6RTABLE_SAVE/149668704X,%)anythingG6"6"[gl!"%!!!##"""#%"xG%"yG6" } {RTABLE M7R0 I6RTABLE_SAVE/149671400X,%)anythingG6"6"[gl!"%!!!#""""",&*&,&%"xG"""%"yG""$F+F* F+F+*&,&F*F-F,""%F+F,F+F+6" } {RTABLE M7R0 I6RTABLE_SAVE/149664256X*%)anythingG6"6"[gl!#%!!!"#"#,&#""&""#"""*$F)#F+F*#""$F *,&F(F+F,#!"$F*6" } {RTABLE M7R0 I6RTABLE_SAVE/149664336X*%)anythingG6"6"[gl!#%!!!"#"#$"+m>5ae!"*$!*m>5a)F)6" } {RTABLE M7R0 I6RTABLE_SAVE/149681336X,%)anythingG6"6"[gl!"%!!!#%"#"#$"""""!$""$F)F*$""%F)6" } {RTABLE M7R0 I6RTABLE_SAVE/149664456X*%)anythingG6"6"[gl(#%!!!"#"#BFEB54CDA58FBBF00000000000 00000040176A99B4B1F77E00000000000000006" } {RTABLE M7R0 I6RTABLE_SAVE/149664536X*%)anythingG6"6"[gl(#%!!!"#"#BFEB54CDA58FBBF00000000000 00000040176A99B4B1F77E00000000000000006" } {RTABLE M7R0 I6RTABLE_SAVE/149692316X,%)anythingG6"6"[gl("%!!!#%"#"#BFEB38880B4603E400000000 000000003FE0D2CA0DA1530C0000000000000000BFE0D2CA0DA1530C0000000000000000BFEB388 80B4603E400000000000000006" } {RTABLE M7R0 I6RTABLE_SAVE/149693792X,%)anythingG6"6"[gl("%!!!##"#""BFEB38880B4603E400000000 000000003FE0D2CA0DA1530C00000000000000006" } {RTABLE M7R0 I6RTABLE_SAVE/149696560X,%)anythingG6"6"[gl("%!!!##"#""BFE0D2CA0DA1530C00000000 00000000BFEB38880B4603E400000000000000006" } {RTABLE M7R0 I6RTABLE_SAVE/149698032X,%)anythingG6"6"[gl("%!!!##"#""3FE73FD61D9DF54000000000 00000000BFDCBCDFD6997EF800000000000000006" } {RTABLE M7R0 I6RTABLE_SAVE/149700728X,%)anythingG6"6"[gl("%!!!##"#""3FE73FD61D98263880000000 00000000BFDCBCDFD69250DB00000000000000006" } {RTABLE M7R0 I6RTABLE_SAVE/149704720X,%)anythingG6"6"[gl("%!!!##"#""C0089F188BDCD7AE00000000 00000000C013EB4FCABF811600000000000000006" } {RTABLE M7R0 I6RTABLE_SAVE/149707416X,%)anythingG6"6"[gl("%!!!##"#""C0089F188BD8550A00000000 00000000C013EB4FCABBDAF800000000000000006" } {RTABLE M7R0 I6RTABLE_SAVE/149708888X,%)anythingG6"6"[gl("%!!!#%"#"#BFEB54CDA58FBBED00000000 000000003CCB24000000000000000000000000003CC2BA8000000000000000000000000040176A9 9B4B1F77C00000000000000006" }