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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 43 "Vi b\366rjar med att bil da funktionen i 1.34c:" }}{PARA 266 "" 0 "" {TEXT -1 1 " " }{TEXT 256 0 "" }}}{EXCHG {PARA 257 "> " 0 "" {MPLTEXT 1 0 30 "f:=(x,y)->(x+y)/sq rt(x^2+y^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGj+6$%\"xG%\"yG6 \"6$%)operatorG%&arrowGF)*&,&9$\"\"\"9%F0F0-%%sqrtG6#,&*$)F/\"\"#F0F0* $)F1F8F0F0!\"\"F)F)F)6$\"\"!F=" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 66 "Vi f\366rs\366ker rita den med plot3d. 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Det \344r dock lite sv\345rt at t se vad som h\344nder." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 158 "Vi skall nu d\344rf\366r studera hur funktionen f \+ beter sig l\344ngs linjer (x,y)=t*(cos(theta),sin(theta)) d\344r (co s(theta), sin(theta)) \344r en fix (riktnings)vektor." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT -1 51 "Vi b\366rjar med att uttrycka f i pol\344ra koordinater: " }}}{EXCHG {PARA 258 "> " 0 "" {MPLTEXT 1 0 43 "g:=(r,theta)->f(r*cos (theta),r*sin(theta));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGj+6$% \"rG%&thetaG6\"6$%)operatorG%&arrowGF)-%\"fG6$*&9$\"\"\"-%$cosG6#9%F2* &F1F2-%$sinGF5F2F)F)F)6$\"\"!F;" }}}{EXCHG {PARA 269 "" 0 "" {TEXT -1 134 "Sedan ritar vi g som funktion av r f\366r n\345gra olika v\344rde n p\345 theta . Detta svarar mot att best\344mma sk\344rningen mell an funktionsytan " }}{PARA 270 "" 0 "" {TEXT -1 138 "och planet genom origo som \344r parallellt med z-axeln och vektorn (cos(theta), sin( theta),0). (Observera att jag \344ven till\345ter negativa r" }}{PARA 271 "" 0 "" {TEXT -1 50 "vilket egentligen svarar mot att \366ka theta med Pi." }}}{EXCHG {PARA 268 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 259 "> " 0 "" {MPLTEXT 1 0 70 "plot([g(r,0),g(r,Pi/8),g(r,Pi/4)],r=-5..5,t hickness=4,legend=[\"theta=0" }{TEXT -1 0 "" }}{PARA 260 "> " 0 "" {MPLTEXT 1 0 30 "\",\"theta=Pi/8\",\"theta=Pi/4\"]);" }}{PARA 13 "" 1 "" {GLPLOT2D 486 486 486 {PLOTDATA 2 "6(-%'CURVESG6%7hn7$$!\"&\"\"!$! \"\"F*7$$!3YLLLe%G?y%!#f_PF0F+7$$!3K++vo1YZNF0F+7$$ !3;LL3-OJNLF0$!2))***************F07$$!3p***\\P*o%Q7$F0F+7$$!3Kmmm\"RF j!HF0F+7$$!33LL$e4OZr#F0F+7$$!3w*****\\n\\!*\\#F0F+7$$!3&)*****\\ixCG# F0F+7$$!3\"******\\KqP2#F0F+7$$!3:LL3-TC%)=F0F+7$$!3Zmmm\"4z)e;F0FF7$$ !3Lmmmm`'zY\"F0F+7$$!3#****\\(=t)eC\"F0FF7$$!3\"ommmh5$\\5F0F+7$$!3S$* **\\(=[jL)!#=F+7$$!3*f***\\iXg#G'FfoF+7$$!3ndmmT&Q(RTFfoFF7$$!3$\\mmTg =><#FfoF+7$$!3g***\\7yQ16\"FfoF+7$$!3uDMLLe*e$\\!#?F+7$$\"3;$4aj%e#R&> Ffp$\"\"\"F*7$$\"327;/EvuV))Ffp$\"2))***************F07$$\"358Hd?pNt:! #>Fjp7$$\"3*\\mT&)3RBE#FdqFjp7$$\"3xo\"zWU..k$FdqF_q7$$\"3bsmTgxE=]Fdq Fjp7$$\"37!o\"HKk>uxFdqFjp7$$\"3womT5D,`5FfoFjp7$$\"3Gq;zW#)>/;FfoFjp7 $$\"3zrm;zRQb@FfoF_q7$$\"3lOLL$e,]6$FfoFjp7$$\"3_,+](=>Y2%FfoFjp7$$\"3 rumm\"zXu9'FfoFjp7$$\"3$4+++]y))G)FfoFjp7$$\"3H++]i_QQ5F0Fjp7$$\"3b++D \"y%3T7F0F_q7$$\"3,++]P![hY\"F0Fjp7$$\"3jKLL$Qx$o;F0Fjp7$$\"3Z+++v.I%) =F0Fjp7$$\"3@mm\"zpe*z?F0Fjp7$$\"3;,++D\\'QH#F0Fjp7$$\"3'HL$e9S8&\\#F0 Fjp7$$\"3s++D1#=bq#F0F_q7$$\"3\"HLL$3s?6HF0Fjp7$$\"3a***\\7`Wl7$F0Fjp7 $$\"3enmmm*RRL$F0$\"3C+++++++5F07$$\"3%zmmTvJga$F0Fjp7$$\"3^MLe9tOcPF0 F_q7$$\"31,++]Qk\\RF0Fjp7$$\"3![LL3dg6<%F0Fjp7$$\"3&ymmmw(GpVF0Fjp7$$ \"3C++D\"oK0e%F0Fjp7$$\"36,+v=5s#y%F0Fjp7$$\"\"&F*Fjp-%'COLOURG6&%$RGB G$\"#5F,$F*F*Fjw-%'LEGENDG6#Q)theta=0|+6\"-F$6%7hn7$F($!3ewj(['Hc18F07 $F.Fdx7$F2$!3!oPw['Hc18F07$F5Fdx7$F8Fdx7$F;Fdx7$F>Fdx7$FAFdx7$FDFhx7$F IFdx7$FL$!3Nwj(['Hc18F07$FOFby7$FRFdx7$FUFdx7$FXFdx7$FenFby7$FhnFby7$F [oFdx7$F^oFdx7$FaoFdx7$FdoFby7$FhoFhx7$F[pFdx7$F^pFby7$FapFdx7$FdpFby7 $Fhp$\"3Nwj(['Hc18F07$F]qFdz7$Fbq$\"3ewj(['Hc18F07$FfqFhz7$FiqFdz7$F\\ rFhz7$F_rFhz7$FbrFhz7$FerFhz7$FhrFdz7$F[sFhz7$F^sFhz7$FasFhz7$FdsFdz7$ FgsFhz7$FjsFhz7$F]tFhz7$F`tFhz7$FctFdz7$FftFhz7$FitFhz7$F\\uFdz7$F_uFh z7$FbuFhz7$FeuFdz7$Fhu$\"3!oPw['Hc18F07$F]vFdz7$F`vFdz7$FcvFdz7$FfvFdz 7$FivFhz7$F\\wFhz7$F_wFhz7$FbwFhz-Few6&FgwFjwFhwFjw-F\\x6#Q+theta=Pi/8 F_x-F$6%7hn7$F($!3:&4tBc8UT\"F07$F.Fd]l7$F2Fd]l7$F5Fd]l7$F8$!3#\\4tBc8 UT\"F07$F;Fd]l7$F>Fd]l7$FAFd]l7$FDFj]l7$FIFd]l7$FLFd]l7$FOFd]l7$FRFd]l 7$FUFd]l7$FXFj]l7$FenFd]l7$FhnFj]l7$F[oFd]l7$F^oFj]l7$FaoFd]l7$FdoFd]l 7$FhoFd]l7$F[pFj]l7$F^pFj]l7$FapFj]l7$FdpFj]l7$Fhp$\"3:&4tBc8UT\"F07$F ]q$\"3#\\4tBc8UT\"F07$FbqFe_l7$FfqFe_l7$FiqFe_l7$F\\rFe_l7$F_rFb_l7$Fb rFe_l7$FerFb_l7$Fhr$\"3q%4tBc8UT\"F07$F[sFe_l7$F^sFb_l7$FasFb_l7$FdsFe _l7$FgsFe_l7$FjsFe_l7$F]tFe_l7$F`tFb_l7$FctFb_l7$FftFb_l7$FitFb_l7$F\\ uFb_l7$F_uFe_l7$FbuFb_l7$FeuFe_l7$FhuFb_l7$F]vFe_l7$F`vFe_l7$FcvFb_l7$ FfvFb_l7$FivFb_l7$F\\wFe_l7$F_wFe_l7$FbwFb_l-Few6&FgwFhwFhwFjw-F\\x6#Q +theta=Pi/4F_x-%*THICKNESSG6#\"\"%-%+AXESLABELSG6$Q\"rF_xQ!F_x-%%VIEWG 6$;F(Fbw%(DEFAULTG" 1 2 0 1 10 4 2 9 1 4 2 1.000000 47.000000 46.000000 0 1 "theta=0\n" "theta=Pi/8" "theta=Pi/4" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 258 390 "Vi ser att d\345 (x,y) varier ar l\344ngs en str\345le (x,y)=r*(cos(theta),sin(theta)) (med r>0) s \345 h\345ller sig funktionen konstant --- och om vi r\344knar efter \+ \344r konstanten sin(theta)+cos(theta). D\345 (x,y) g\345r mot noll \+ l\344ngs en s\345dan str\345le g\345r funktionsv\344rdet allts\345 mot denna konstant, men f\366r olika str\345lar har vi olika konstanter. \+ Det finns allts\345 inget gemensamt gr\344nsv\344rde f\366r alla str \345larna. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 272 "" 0 "" {TEXT -1 44 "Vi g\366r nu samma sak f\366r funktionen i 1.34a):" }}}{EXCHG {PARA 261 "> " 0 "" {MPLTEXT 1 0 29 "f1:=(x,y)->x*y/sqrt(x^2+y^2);" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1Gj+6$%\"xG%\"yG6\"6$%)o peratorG%&arrowGF)*(9$\"\"\"9%F/-%%sqrtG6#,&*$)F.\"\"#F/F/*$)F0F7F/F/! \"\"F)F)F)6$\"\"!F<" }}}{EXCHG {PARA 262 "> " 0 "" {MPLTEXT 1 0 45 "g1 :=(r,theta)->f1(r*cos(theta),r*sin(theta));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#g1Gj+6$%\"rG%&thetaG6\"6$%)operatorG%&arrowGF)-%#f1G 6$*&9$\"\"\"-%$cosG6#9%F2*&F1F2-%$sinGF5F2F)F)F)6$\"\"!F;" }}}{EXCHG {PARA 263 "> " 0 "" {MPLTEXT 1 0 70 "plot([g1(r,0),g1(r,Pi/8),g1(r,Pi/ 4)],r=-5..5,thickness=4,legend=[\"thet" }}{PARA 264 "> " 0 "" {MPLTEXT 1 0 33 "a=0\",\"theta=Pi/8\",\"theta=Pi/4\"]);" }}{PARA 13 " " 1 "" {GLPLOT2D 486 486 486 {PLOTDATA 2 "6(-%'CURVESG6%7S7$$!\"&\"\"! $F*F*7$$!3YLLLe%G?y%!#f_PF/F+7$$!3K++vo1YZNF/F+7$$! 3;LL3-OJNLF/F+7$$!3p***\\P*o%Q7$F/F+7$$!3Kmmm\"RFj!HF/F+7$$!33LL$e4OZr #F/F+7$$!3w*****\\n\\!*\\#F/F+7$$!3&)*****\\ixCG#F/F+7$$!3\"******\\Kq P2#F/F+7$$!3:LL3-TC%)=F/F+7$$!3Zmmm\"4z)e;F/F+7$$!3Lmmmm`'zY\"F/F+7$$! 3#****\\(=t)eC\"F/F+7$$!3\"ommmh5$\\5F/F+7$$!3S$***\\(=[jL)!#=F+7$$!3* f***\\iXg#G'FcoF+7$$!3ndmmT&Q(RTFcoF+7$$!3$\\mmTg=><#FcoF+7$$!3uDMLLe* e$\\!#?F+7$$\"3zrm;zRQb@FcoF+7$$\"3_,+](=>Y2%FcoF+7$$\"3rumm\"zXu9'Fco F+7$$\"3$4+++]y))G)FcoF+7$$\"3H++]i_QQ5F/F+7$$\"3b++D\"y%3T7F/F+7$$\"3 ,++]P![hY\"F/F+7$$\"3jKLL$Qx$o;F/F+7$$\"3Z+++v.I%)=F/F+7$$\"3@mm\"zpe* z?F/F+7$$\"3;,++D\\'QH#F/F+7$$\"3'HL$e9S8&\\#F/F+7$$\"3s++D1#=bq#F/F+7 $$\"3\"HLL$3s?6HF/F+7$$\"3a***\\7`Wl7$F/F+7$$\"3enmmm*RRL$F/F+7$$\"3%z mmTvJga$F/F+7$$\"3^MLe9tOcPF/F+7$$\"31,++]Qk\\RF/F+7$$\"3![LL3dg6<%F/F +7$$\"3&ymmmw(GpVF/F+7$$\"3C++D\"oK0e%F/F+7$$\"36,+v=5s#y%F/F+7$$\"\"& F*F+-%'COLOURG6&%$RGBG$\"#5!\"\"F+F+-%'LEGENDG6#Q)thet|+a=06\"-F$6%7U7 $F($\"3))oj'H&pwnNeXaF5F/7$FL$\"3o8#yC^T!)f*Fco7$FO$\"3L)os&e[ZN))Fco7$FR$\"3wF. sKqxp!)Fco7$FU$\"3?_Y:(H&)=L(Fco7$FX$\"3+5%p*4*3=m'Fco7$Fen$\"3&4SIWFB ]'eFco7$Fhn$\"32#)\\eE8/!>&Fco7$F[o$\"3u_ATeo([S%Fco7$F^o$\"3h!>!3jK() 4PFco7$Fao$\"3>:n&o;Wt%HFco7$Feo$\"3oW'G[9O7A#Fco7$Fho$\"3S,fd(f=OY\"F co7$F[p$\"33\\t&f'=*)yw!#>7$$!3g***\\7yQ16\"Fco$\"3!o=`$o5qERF`z7$F^p$ \"3KX-\\2F5X:8N^ac*Fco 7$Fir$\"3+HeAzrEH5F/7$F\\s$\"3F&Hk)>/S06F/7$F_s$\"3cE%\\#zdsy6F/7$Fbs$ \"3qRq$)\\:r`7F/7$Fes$\"3b)o%Q+k2G8F/7$Fhs$\"3!=P.[(*4kR\"F/7$F[t$\"3/ ,r_izsu9F/7$F^t$\"3I8FQ/lxW:F/7$Fat$\"3YO#z,'GY>;F/7$Fdt$\"3Ez2WKs%4p \"F/7$FgtFiu-Fjt6&F\\uF+F]uF+-Fau6#Q+theta=Pi/8Fdu-F$6%7U7$F($\"3+++++ +++DF/7$F-$\"3smm;HU,\"R#F/7$F1$\"3?L$3FH'='H#F/7$F4$\"3hmmTgBa*=#F/7$ F7$\"3wmm\"H_\">#3#F/7$F:$\"3dL$3_!4Nv>F/7$F=$\"3km;/wfHw=F/7$F@$\"3;+ ]PM.ttc\"F/7$FI$\"3;LL$epjJX \"F/7$FL$\"3amm\"z/otN\"F/7$FO$\"35++]P[_\\7F/7$FR$\"3#*****\\7)Q79\"F /7$FU$\"3'*****\\i^)o.\"F/7$FX$\"3kkmT50A@%*Fco7$Fen$\"3OKLLeaR%H)Fco7 $Fhn$\"3kJLLLo#)RtFco7$F[o$\"3[)**\\PfO%HiFco7$F^o$\"3.MLL$3`lC&Fco7$F ao$\"3q'**\\P4u\"oTFco7$Feo$\"3*z**\\7G-89$Fco7$Fho$\"3%)GL$3Fp)p?Fco7 $F[p$\"3YKL3-$ff3\"Fco7$Fbz$\"3)z**\\i!R>`bF`z7$F^p$\"3'Grmm\"z%zY#F`p 7$Fjz$\"3%QM$3_D1l_F`z7$Fbp$\"3vNLe*)>px5Fco7$Fep$\"3/,+v$f4t.#Fco7$Fh p$\"3NPL$e*GstIFco7$F[q$\"3\"*******\\#RW9%Fco7$F^q$\"3[,+]7j#>>&Fco7$ Faq$\"3t-+D1RU0iFco7$Fdq$\"3+++](=S2L(Fco7$Fgq$\"3Fkmm;p)=M)Fco7$Fjq$ \"3O-++v=]@%*Fco7$F]r$\"36L$e*[$z*R5F/7$F`r$\"3e++]iC$p9\"F/7$Fcr$\"3q m;H2qcZ7F/7$Ffr$\"3O+]7.\"fFN\"F/7$Fir$\"3Xmm;/Ogb9F/7$F\\s$\"3****\\i lAFj:F/7$F_s$\"3,MLL$)*ppm\"F/7$Fbs$\"3(RL$3xe,t#[(>F/7$F[t$\"3SnmT&G!e&3#F/7$F^t$\"3#RLLL)Qk%=#F/7$Fa t$\"3o**\\iSjE!H#F/7$Fdt$\"3c+]P40O\"R#F/7$FgtF]`l-Fjt6&F\\uF]uF]uF+-F au6#Q+theta=Pi/4Fdu-%*THICKNESSG6#\"\"%-%+AXESLABELSG6$Q\"rFduQ!Fdu-%% VIEWG6$;F(Fgt%(DEFAULTG" 1 2 0 1 10 4 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "thet\na=0" "theta=Pi/8" "theta=Pi/4" }}}}{EXCHG {PARA 265 "" 0 "" {TEXT -1 134 "I detta fall ser vi att d\345 (x,y) g\345r m ot (0,0) l\344ngs de str\345lar som vi betraktar s\345 g\345r funktion sv\344rdet mot noll. Detta troligg\366r att " }}{PARA 273 "" 0 "" {TEXT -1 177 "gr\344nsv\344rdet existerar och \344r lika med noll. Eft ersom vi bara unders\366kt n\345gra s\344tt att g\345 mot (0,0) har vi \344nnu inte visat att s\345 \344r fallet, men det \344r l\344tt att \+ g\366ra med hj\344lp av" }}{PARA 274 "" 0 "" {TEXT -1 19 "pol\344ra ko ordinater." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 261 112 "Att funktionen har gr\344nsv\344rdet 0 d\345 (x,y) g\345r mot (0,0) verkar \344ven t roligt efter att vi ritat funktionsytan med" }}{PARA 0 "" 0 "" {TEXT 262 7 "plot3d:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "plot3d([f1(x,y),0], x=-1..1,y=-1..1,grid=[50,50],color=[blue,red],transparency=.1,axes=nor mal);" }}{PARA 13 "" 1 "" {GLPLOT3D 486 486 486 {PLOTDATA 3 "6&-%%GRID G6';$!\"\"\"\"!$\"\"\"F)F&X,%)anythingG6\"6\"[gl'!%\"!!#_gt\"S\"S3FE6A 09E667F3BCD3FE626B56DB2BA183FE5A51C19010C203FE51B5C42027CFA3FE488FDE70 DF7C33FE3ED889D910ACD3FE34885630C9ACC3FE29980D726B54C3FE1E00DE72CAA263 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