{VERSION 4 0 "IBM INTEL LINUX22" "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 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 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 "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 "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 19 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 8 8 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 12 "L/int23d.mws" }}{PARA 19 "" 0 "" {TEXT -1 10 "7.4.02 HA" }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 11 "Alustukset " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(plots): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "wi th(plottools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "setoption s3d(axes=boxed,orientation=[-30,50]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "#read(\"/home/apiola/opetus/peruskurssi/v2-3/maple/v2 02.mpl\");" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 17 " Luento ti 9.4.02 " }}{PARA 0 "" 0 "" {TEXT -1 69 "Avaruusintegraalin laskeminen [LP] s. 173, [VA] s. 132, [Ad] s. 851 ." }}{PARA 0 "" 0 "" {TEXT 256 6 "Esim1 )" }{TEXT -1 12 " Laskettava:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Int(Int(Int(x*y*z,z=0..x*y),y=0..1-x),x=0..1);" }}}{PARA 0 "" 0 "" {TEXT -1 110 "K\344sin laskettaessa riitt\344\344 useimmiten piirt \344\344 kappaleen projektio xy-tasoon (xy-projisioituvassa tapauksess a)." }}{PARA 0 "" 0 "" {TEXT -1 212 "Kuvan piirto on tietysti asiaa ha vainnollistava. Maplella se onnistuu mainiosti. Kuvan piirto Maplella \+ on aivan samanlainen teht\344v\344 kuin integroimisrajojen m\344\344r \344\344minen. Siten n\344m\344 teht\344v\344t tukevat hyvin toisiaan. " }}{PARA 0 "" 0 "" {TEXT -1 25 "(Erityinen kiitos Maplen " }{TEXT 0 7 "plot3d:" }{TEXT -1 20 "n suunnitelijoille!)" }}{PARA 0 "" 0 "" {TEXT -1 43 "Projektio saadaan t\344ss\344 ihan perusplotilla:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "plot(1-x,x=0..1);" }}}{PARA 0 "" 0 "" {TEXT -1 15 "Nyt se 3d-kuva:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "suoraseina:=plot3d([x,1-x,z],x=0..1,z=0..x*(1-x),labe ls=[x,y,z]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "display(suo raseina,plot3d(x*y,x=0..1,y=0..1-x),style=patchcontour);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "K\344sin laskien luennolla muotoon:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "Int(x*Int(y*(x*y)^2/2,y=0..1 -x),x=0..1): %=value(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "Int(Int(Int(x*y*z,z=0..x*y),y=0..1-x),x=0..1): %=value(%);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 7 "Esim. 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "plot( [2*cos(Theta),2*sin(Theta)-1,Theta=0..2*Pi],scaling=constrained);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "solve(r^2+2*r*sin(Theta)-3=0 ,r); R:=%[1]; # Voi vaihdella eri ajokerroilla." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "x:=r*co s(Theta): y:=r*sin(Theta):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "taso:=plot3d([x,y,3-2*y],Theta=0..2*Pi,r=0..R,labels=[\"x\",\"y\", \"z\"]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "taso;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "paraboloidi:=plot3d([x,y,r^2 ],Theta=0..2*Pi,r=0..R,labels=[\"x\",\"y\",\"z\"]):paraboloidi;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "display(taso,paraboloidi,sty le=patchcontour);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "Int((3 -2*r*sin(Theta)-r^2)*r,r=0..-sin(Theta)+sqrt(sin(Theta)^2+3));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "rint:=%:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "vV:=int(rint,Theta=0..2*Pi);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Komeasti Maple sen osasi! Katsotaan, milt\344 n\344ytt \344\344 integraalifunktio:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "int(rint,Theta);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "sim plify(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Eip\344 tuosta k\344 sin laskien selvi\344isi!" }}{PARA 0 "" 0 "" {TEXT 258 66 "Oliskohan m uuten mennyt helpommin ilman napakoordinaattimuunosta? " }}{PARA 0 "" 0 "" {TEXT -1 31 "Kannattaa kokeilla! No tehd\344\344n." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "x:='x': y:='y': z:='z':" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Tarkistetaan ensin, ett\344 rajat tulivat oikein:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "plot3d([x,y,0] ,x=-2..2,y=-1-sqrt(4-x^2)..-1+sqrt(4-x^2),view=[-2..2,-3..1,-1..1],gri d=[50,50],color=red,style=patchnogrid);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "T\344h\344n saakka tehtiin k\344sin taululla:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "Int(Int(3-2*y-x^2-y^2,y=-1-sqrt(4-x ^2)..-1+sqrt(4-x^2)),x=-2..2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "Int(int(3-2*y-x^2-y^2,y=-1-sqrt(4-x^2)..-1+sqrt(4-x^2)),x=-2.. 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "Int(simplify(int(3-2*y-x^2-y^2,y=-1 -sqrt(4-x^2)..-1+sqrt(4-x^2))),x=-2..2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "infolevel[int]:=3:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "value(%%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "infolevel[int]:=0: # Takaisin normaalitilaan!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 41 "Eik\366 nyt parempaa muuttujanvaihtoa l\366ydy? " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 129 "Katsotaanpa, miten p iirsimme aivan ensimm\344isen kuvan. Projektiokiekko piirrettiin ottam alla origoksi piste (0,-1) ja sitten vain " }{TEXT 260 13 "siirrettiin . " }}{PARA 0 "" 0 "" {TEXT -1 33 "No jospa siirret\344\344n takaisinp \344in." }}{PARA 0 "" 0 "" {TEXT -1 34 "Ts. muunnos: u = x \+ x=u" }}{PARA 0 "" 0 "" {TEXT -1 46 " v=y+1 y=v-1" }}{PARA 0 "" 0 "" {TEXT -1 24 "T\344ll\366in Jakobin de t = 1 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "c1:=(u,v)->u^2+(v -1)^2; c2:=(u,v)->3-2*(v-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c1G R6$%\"uG%\"vG6\"6$%)operatorG%&arrowGF),&*$)9$\"\"#\"\"\"F2*$),&9%F2F2 !\"\"F1F2F2F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c2GR6$%\"uG%\" vG6\"6$%)operatorG%&arrowGF),&\"\"&\"\"\"*&\"\"#F/9%F/!\"\"F)F)F)" }}} {EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "Int(Int(c2(r*cos(Theta),r*sin(Theta))-c1(r*cos(Theta) ,r*sin(Theta))*r,r=0..2),Theta=0..2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$,(\"\"&\"\"\"*(\"\"#F*%\"rGF*-%$sinG6#%&ThetaGF*! \"\"*&,&*&)F-F,F*)-%$cosGF0F,F*F**$),&*&F-F*F.F*F*F*F2F,F*F*F*F-F*F2/F -;\"\"!F,/F1;F@,$%#PiGF," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 " value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG\"\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "N\344inp\344 t\344m\344 muuttui puhtaaksi k\344sinlaskuksi." }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 6 "M\366bius " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "plot3d([4+x*cos(1.5*y), y,x*sin(1.5*y)],\nx=-Pi..Pi,y=-0..2*Pi,coords=cylindrical,style=patchn ogrid,grid=[60,60],\norientation=[42,138],scaling=constrained);" }}}}} {MARK "2 16 32 0 0" 44 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }