{VERSION 4 0 "SUN SPARC SOLARIS" "4.0" } {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 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 275 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple O utput" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 11 "Math 2280-2" }}{PARA 258 "" 0 "" {TEXT -1 20 "Friday April 6, 2001" }}{PARA 257 "" 0 "" {TEXT 257 41 "Forced oscillations: resonance mysteries " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "restart:\nwith(plots):\nwith(DEtool s):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "f1:=x->evalf(2*Pi*(frac((x+P i)/(2*Pi)))-Pi); #sawtooth!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1G R6#%\"xG6\"6$%)operatorG%&arrowGF(-%&evalfG6#,&*&%#PiG\"\"\"-%%fracG6# ,$*&,&9$F2F1F2F2F1!\"\"#F2\"\"#F2F " 0 "" {MPLTEXT 1 0 63 "plot(f1(t),t=-Pi..5*Pi,color=black);\n#has minim um period = 2*Pi" }}{PARA 13 "" 1 "" {GLPLOT2D 371 153 153 {PLOTDATA 2 "6&-%'CURVESG6#7co7$$!3)****4tk#fTJ!#<$!3_li!fk#fTJF*7$$!3i%RP<(fsIF F*$!3'[C<0(fsIFF*7$$!3VZQ@uIBtBF*$!3XLM:tIBtBF*7$$!35VO$>#\\>r>F*$!3vo G0@\\>r>F*7$$!3vJw!)*z\"\\m:F*$!3(yo2\"*z\"\\m:F*7$$!3;Io(4;7P;\"F*$!3 ?doXg@rj6F*7$$!3:`/)RwQG!z!#=$!3o'G\\/wQG!zFI7$$!3mB6Fv^?OSFI$!3hfwYt^ ?OSFI7$$!3-`B9b'=Ft$!#?$!3%RNuMl=Ft$FT7$$\"3A!4V(\\js[RFI$\"3-1(yzME([ RFI7$$\"3Q493gB()[!)FI$\"3)G*\\[cB()[!)FI7$$\"3tRVx>$Gg;\"F*$\"30KLD>$ Gg;\"F*7$$\"3mF?1Wwes:F*$\"3'*f$fLk(es:F*7$$\"3'pl'**Rj\"3)>F*$\"31%e6 \"Rj\"3)>F*7$$\"3%pcn9L?UP#F*$\"346nSI.AuBF*7$$\"3pBL)zeWGb#F*$\"3![lU oeWGb#F*7$$\"3X!3*\\W)o9t#F*$\"3_)fyK%)o9t#F*7$$\"3%**o^Mgpw$GF*$\"3Ib P=-'pw$GF*7$$\"3W*H/COqQ%HF*$\"337*)3h.(Q%HF*7$$\"3?/1)=uqp*HF*$\"3[! \\T0uqp*HF*7$$\"3%*3pN@62]IF*$\"3*z1%**>62]IF*7$$\"34h]468iwIF*$\"32e. s48iwIF*7$$\"3p8K$3]rJ5$F*$\"3RYmW*\\rJ5$F*7$$\"3***G-dfYk6$F*$\"3a!z4 VfYk6$F*7$$\"3Gm8d!p@(HJF*$\"3eNH<*o@(HJF*7$$\"3eU/W&y'*H9$F*$!3c>R'R_ )=SJF*7$$\"3W=&4.)=FcJF*$!3Iw25HM\"p7$F*7$$\"395#Gn#Q?OLF*$!31#[iF[\") p%HF*7$$\"3%=!p9td8;NF*$!3$y=Ck`\\qw#F*7$$\"3(f;C1.VZ$RF*$!3>lR8zAW[BF *7$$\"3f_;w$Q\"G0VF*$!38W?;ER!z(>F*7$$\"3b0KK%o?=r%F*$!3'=9#yDYOr:F*7$ $\"3YlxiOA%*)4&F*$!3!yb]O2VU=\"F*7$$\"3K?*RZ'H'G]&F*$!3fK))=dMA.yFI7$$ \"34OW(F*$\"3)4!p>8iwe6F*7$$\"3W4Ofz?gXyF*$\"3*p,)3onTi:F*7$$\"3hVo. )R&\\S#)F*$\"3u/[N'35t&>F*7$$\"3i'zm'))\\dA')F*$\"3kNS\"on*QRBF*7$$\"3 fG1(*4@pM))F*$\"3@*3B!)z1:b#F*7$$\"3zeWFJ#4o/*F*$\"37S@B>RijFF*7$$\"3e __;lv5U\"*F*$\"3&GN!3`A#*eGF*7$$\"3OYg0**eSP#*F*$\"3fl&Gpe?U&HF*7$$\"3 PU9+m]0&G*F*$\"3IpE&Qvp=+$F*7$$\"3PQo%HB/FL*F*$\"3mvnx?*=&\\IF*7$$\"3E P&>k\")GlN*F*$\"3tH)QU]VL2$F*7$$\"3:OA*)*R`.Q*F*$\"3-#)3q(3or4$F*7$$\" 3g&eG;plAR*F*$\"3i3>Vz.34JF*7$$\"3/N\\O$)z0:j*F*$!3%p#\\cGa'[$HF*7$$\"3sh'Rm\"y+N)*F*$!3(H%[v.GOJFF*7$$ \"3eoMe1:Q?5!#;$!3?5YsabbiBF*7$$\"3D\"\\'R!)=qg5F_]l$!3tUXx;=Nf>F*7$$ \"3!pL9*4-k)4\"F_]l$!3(Qgl<_o*z:F*7$$\"3VZKp&p'HQ6F_]l$!3Y#p`Tm.M=\"F* 7$$\"3K'z(3f\"oq<\"F_]l$!3&*QW\"QI!*o&zFI7$$\"3&Qn4\")Gew@\"F_]l$!3;b. 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" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 275 31 "Part 2: Fourier approximations " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "restart:with(linalg):with(plots):\n #we'll us e linear algebra and plots below" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warni ng, the protected names norm and trace have been redefined and unprote cted\n" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords \+ has been redefined\n" }}}{PARA 0 "" 0 "" {TEXT -1 146 "On the interval [-Pi..Pi] the functions \{1,cos(x),cos(2x),...sin(x),sin(2x),sin(3x), ...\} are mutually orthogonal with respect to the inner product" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Dot:=(f,g)->int(f(x)*g(x),x= -Pi..Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$DotGR6$%\"fG%\"gG6\"6$ %)operatorG%&arrowGF)-%$intG6$*&-9$6#%\"xG\"\"\"-9%F3F5/F4;,$%#PiG!\" \"F;F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 242 "Mag:=f->sqrt (Dot(f,f)); #computes the magnitude of\n #a vector\nDist:=(f,g)->M ag(f-g); #computes the ``distance'' between\n #two vectors. \nan gle:=(f,g)->arccos(Dot(f,g)/(Mag(f)*Mag(g)));\n #computes the ``an gle'' between two vectors" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$MagGR6 #%\"fG6\"6$%)operatorG%&arrowGF(-%%sqrtG6#-%$DotG6$9$F2F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%DistGR6$%\"fG%\"gG6\"6$%)operatorG% &arrowGF)-%$MagG6#,&9$\"\"\"9%!\"\"F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&angleGR6$%\"fG%\"gG6\"6$%)operatorG%&arrowGF)-%'arcc osG6#*&-%$DotG6$9$9%\"\"\"*&-%$MagG6#F4F6-F96#F5F6!\"\"F)F)F)" }}} {PARA 0 "" 0 "" {TEXT -1 80 " Verify that sin(3*x) and sin(4*x) are or thogonal, and compute their magnitudes:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "Dot(x->sin(3*x),x->sin(4*x));\nMag(x->sin(3*x));Mag(x ->sin(4*x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%sqrtG6#%#PiG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%sqrtG6#%#PiG\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 271 "Whe n we do the projection of a function f onto Vn, the coefficients for e ach basis vector are just the inner product of that basis vector with f, divided by the square of the magnitude of the vector Let's illust rate by projecting the absolute value function onto W10: " }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 212 " f:=x->abs(x); #we'll do Fourier for the absolute value function\nn:=1 0; #order 10 expansion\na:=vector(n); #cos coefficients\nb:=vector(n ); #sin coefficients\na0:=(1/Pi)*Dot(f,1); #order zero Fourier coeff icient" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG%$absG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG \"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG-%&arrayG6$;\"\"\"\"#57 \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG-%&arrayG6$;\"\"\"\"#57\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a0G%#PiG" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 129 "for i from 1 to n do #compute the projection coefficients\nb[i]:=(1/Pi)*Dot(f,x->sin(i*x));\na[i]:=(1/Pi)*Dot(f,x- >cos(i*x));\nod:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "evalm (b);#why will an even function have Fourier sine\n #coefficients all \+ equal to zero?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7,\"\"!F 'F'F'F'F'F'F'F'F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 268 267 "Th e answer to the rhetorical question above is that the product of an ev en function with an odd function is an odd function. We already asser ted that the integral of an odd function over an interval [-L,L] is al ways zero. By the way, could you prove that assertion?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "evalm(a ); #Could you predict higher order coefficients\n #from the pattern \+ you see here?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7,,$*&\" \"\"F)%#PiG!\"\"!\"%\"\"!,$F(#F,\"\"*F-,$F(#F,\"#DF-,$F(#F,\"#\\F-,$F( #F,\"#\")F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 269 146 "The answ er to this rhetorical question is yes: the i=even cos coefficients ar e zero, the i=odd ones are -4 divided by the product of Pi with i^2." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "approx:=x->sum(b[k]*sin(k* x),k=1..n)\n +sum(a[m]*cos(m*x),m=1..n)\n +a0/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'approxGR6#%\"xG6\"6$%)operatorG%& arrowGF(,(-%$sumG6$*&&%\"bG6#%\"kG\"\"\"-%$sinG6#*&F4F59$F5F5/F4;F5%\" nGF5-F.6$*&&%\"aG6#%\"mGF5-%$cosG6#*&FDF5F:F5F5/FDF " 0 "" {MPLTEXT 1 0 50 "approx(x); # order 10 Fourier expansion for abs(x):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&-%$cosG6#%\"xG\"\"\"%#PiG!\"\"!\"%*&#\"\"%\"\"*F)*&-F&6#,$F( \"\"$F)F*F+F)F+*&#F/\"#DF)*&-F&6#,$F(\"\"&F)F*F+F)F+*&#F/\"#\\F)*&-F&6 #,$F(\"\"(F)F*F+F)F+*&#F/\"#\")F)*&-F&6#,$F(F0F)F*F+F)F+*&#F)\"\"#F)F* F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "almost:=plot(appro x(x),x=-Pi..Pi,color=black):\nexact:=plot(abs(x),x=-Pi..Pi,color=black ):\ndisplay(\{almost,exact\});" }}{PARA 13 "" 1 "" {GLPLOT2D 240 182 182 {PLOTDATA 2 "6&-%'CURVESG6$7eo7$$!3)****4tk#fTJ!#<$\"3a>b2,+9yIF*7 $$!3Mue[-KZCJF*$\"3_;g)Q'y?xIF*7$$!3E[2 g9v#F*$\"3'*R/eB-9YFF*7$$!3)4577.flh#F*$\"33&z$4*3uag#F*7$$!3u\"QM::*H #[#F*$\"3\"poW<42J[#F*7$$!3x\\AhcI#yN#F*$\"3*H^$*GknhO#F*7$$!3&e=d-FN* GAF*$\"3As%=ao-AB#F*7$$!3u%*GBP$Rc4#F*$\"3`@w*>7x**3#F*7$$!3lCj&f)3xi> F*$\"3g0)=FVWx&>F*7$$!3'or4AN*4E=F*$\"3a=A2N'y$H=F*7$$!3QQQ*3**=dq\"F* $\"3\\vC$>OT=r\"F*7$$!3d!>jg@*>q:F*$\"3Qau3m:;q:F*7$$!3_;m,%)H7M9F*$\" 3m3&GF_yzU\"F*7$$!3v(4F,K))HI\"F*$\"3SRKU?TU+8F*7$$!3R*4!R#[0R=\"F*$\" 33f/YM\"*o)=\"F*7$$!3oY@QqWIU5F*$\"3#eg)=]\\\"y/\"F*7$$!3Wp3w$R)\\B#*! 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So in this problem what they are really approx imating is the 2*Pi-periodic extension of the absolute value function \+ restricted to the interval [-Pi..Pi]. This is called a saw-tooth func tion, by the way." }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "38 1" 1 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }