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Antalet v \344ljer du med \"" }{TEXT 299 13 "contours = n " }{TEXT -1 14 "\", de fault \344r " }{TEXT 300 6 " n= 10" }{TEXT -1 12 "). Gl\366m ej " } {TEXT 290 16 "axis=constrained" }{TEXT -1 63 " (g\366r vi med musen f \366rst\345s) f\366r att se att de \344r ortogonala. 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0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 163 "Men helst vill vi ju se funktionsytan (ritas med plot3d), kanske med niv\345kurvorna i \+ ytan (det klickar vi fram med musen: klicka med h\366germusen i figure n och v\344lj s" }{TEXT 291 26 "tyle = patch and contour, " }{TEXT -1 79 "g\366r det direkt f\366r att kolla; pr\366va \344ven de andra inst \344llningarna, ba de olika " }{TEXT 297 7 "color =" }{TEXT -1 1 " " } {TEXT 296 12 "light schema" }{TEXT -1 5 "...)," }{TEXT 292 1 " " } {TEXT -1 101 "men allrahelst f\366rst\345s vill vi se allt: F\366r att kunna rita en 2d-plot och en 3d-plot tillsammans med " }{TEXT 294 7 " display" }{TEXT -1 83 ", \"transformerar\" vi 2d-plotten s\345 att den ligger i rummet med z-koordinaten t.ex. " }{TEXT 293 7 "z=-0.3 " } {TEXT -1 120 "s\345 att din definitionsm\344ngd med niv\345kurvorna li gger synligt under ytan, vidare kan du fylla ut definitionsomr\345det med " }{TEXT 295 11 "filled=true" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "p2ny:=contourplot(f(x,y),x=-3..3,y=-2.6..2. 3,coloring=[yellow,red],filled=true):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "pr:=transform((x,y)->[x,y,-0.3]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "pxy:=pr(p2ny):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 39 "p3:=plot3d(f(x,y),x=-3..3,y=-2.6..2.3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "display([p3,pxy]);" }}{PARA 13 "" 1 "" {GLPLOT3D 343 301 301 {PLOTDATA 3 "63-%%GRIDG6%;$!\"$\"\"!$\"\"$F );$!#E!\"\"$\"#BF/W(\\bm\":\":3FC8550B20F71D1F3FC769811B44E35B3FC62DCB 143F5A403FC4AB32DDADDA103FC2FE5FDA06BF423FC15905EC62DD243FBFFCD8491B59 6D3FBE73E63B00B6CF3FBEA10E238D2B563FC06FE3B295FCC23FC2A003956979B43FC5 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rita v\345ran yta s\345 h\344r (nu ritar vi bara definitionsomr \345det med). 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C2D3FCC4C476DAD9D7B4007FFFFFFFFFFFE3E5EE0DFE182BFD93FCC4C476C45F6D0400 7FFFFFFFFFFFE3E61D081297CA9C43FCC4C476ADE50254007FFFFFFFFFFFE3E6430926 237F39A3FCC4C476976A97A4007FFFFFFFFFFFE3E6690A39AF33D703FCC4C47680F02C F4007FFFFFFFFFFFE3E68F0B4D3AE87473FCC4C4766A75C244007FFFFFFFFFFFE3E6B5 0C60C69D11C3FCC4C47653FB5744007FFFFFFFFFFFE3E6DB0D745251AF43FCC4C4763D 80EC94007FFFFFFFFFFFE3E7008743EF032643FCC4C476270681E4007FFFFFFFFFFFE3 E71387CDB4DD7503FCC4C476108C1734007FFFFFFFFFFFE3E72688577AB7C3C3FCC4C4 75FA11AC84007FFFFFFFFFFFE3E73988E140921263FCC4C475E39741D-%)CONTOURSG6 #\"#V-%&TITLEG6#%+niv|`ykurvorG-%+AXESLABELSG6%%\"xG%\"yG%!G" 1 4 0 1 0 2 1 1 3 2 1.000000 64.000000 -40.000000 0 }}}}{EXCHG {PARA 13 "" 1 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 49 "EXEMPEL2: funk tionalmatris, funktionaldeterminant" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "Vi forts\344tter ned samma funktion: Vi skall ber\344kna statio n\344ra punkter och deras typ:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "solve(\{v[1],v[2]\},\{x,y\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"yG\"\"!/%\"xGF&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "OBS: \+ Vi f\345r naturligtvis ej alla l\366sningar, t.ex f\345s l\366sningen \+ (" }{XPPEDIT 18 0 "0,Pi;" "6$\"\"!%#PiG" }{TEXT -1 6 ") med" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "fsolve(\{v[1],v[2]\},\{x,y\} );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"xG\"\"!/%\"yG$\"+aEfTJ!\"* " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 124 "F\366r att best\344mma av vi lken typ den station\344ra punkten (0,0) \344r ber\344knar vi Hessemat risen i punkten (0,0) (se anvisningarna):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "H1:=subs(x=0,y=0,hessian(f(x,y),[x,y]));" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#H1G-%' matrixG6#7$7$,$*&-%$cosG6#\"\"!\"\"\"-%$expGF.F0\"\"#F+7$F+F/" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "H:=simplify(H);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"HG-%'matrixG6#7$7$\"\"#\"\"\"7$F+\"\"!" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "Anm: Att ber\344kna Hesse-matrisen g\345r s\344kert snabbare \344n att ber\344kna alla derivator " }} {PARA 0 "" 0 "" {TEXT -1 5 "(med " }{TEXT 264 1 "D" }{TEXT -1 1 "[" } {TEXT 265 1 "i" }{TEXT -1 1 "," }{TEXT 266 1 "k" }{TEXT -1 2 "](" } {TEXT 261 1 "f" }{TEXT -1 2 ")(" }{TEXT 262 1 "x" }{TEXT -1 1 "," } {TEXT 263 1 "y" }{TEXT -1 4 ") , " }{TEXT 267 1 "i" }{TEXT -1 1 "," } {TEXT 268 1 "k" }{TEXT -1 66 " = 1,2). Du kan f\366rst ber\344kna Hess ematrisen i en godtycklig punkt:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Hess:=hessian(f(x,y),[x,y]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Och sedan plocka fram " }{TEXT 272 1 "D" }{TEXT -1 1 " [" }{TEXT 273 1 "i" }{TEXT -1 1 "," }{TEXT 274 1 "k" }{TEXT -1 2 "](" }{TEXT 269 1 "f" }{TEXT -1 2 ")(" }{TEXT 270 1 "x" }{TEXT -1 1 "," } {TEXT 271 1 "y" }{TEXT -1 6 ") med " }{MPLTEXT 1 0 9 "Hess[i,k]" } {TEXT -1 24 " ! I punkten (0,0) f\345s " }{TEXT 275 2 " H" }{TEXT -1 17 " s.o. d\345 s\345 h\344r:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "H:=subs(x=0,y=0,eval(Hess)): " }{TEXT -1 19 "(pr\366va !! gl \366m ej " }{TEXT 280 4 "eval" }{TEXT -1 1 ")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "D\345 kan vi nu avg\366ra om " }{TEXT 276 1 "H" } {TEXT -1 71 " \344r positivt definit, negativt definit eller indefinit : Maples svar \344r " }}{PARA 0 "" 0 "" {TEXT -1 1 "'" }{TEXT 277 5 "f alse" }{TEXT -1 9 "' eller '" }{TEXT 278 4 "true" }{TEXT -1 12 "' elle r, om " }{TEXT 279 1 "H" }{TEXT -1 84 " inneh\345ller parametrar, vill kor som m\345ste vara uppfylda f\366r att ... skall vara sant:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "definite(H,positive_def);definite( H,negative_def);definite(H,positive_def);definite(H,negative_def);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Eftersom det inte st\344mmer , s\345 \344r " }{TEXT 281 1 "H" } {TEXT -1 132 " indefinit och (0,0) allts\345 sadelpunkt. Man kan f \345 det enklare genom att ber\344kna egenv\344rdena eller genom att b er\344kna determinanten:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "eigenvalues(H);det(H);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&*$-%%sqrt G6#\"\"#\"\"\"\"\"\"F*F*,&F*F*F$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Eftersom " }{TEXT 302 1 "H" }{TEXT -1 53 " har ett positivt och ett negativt egenv\344 rde s\345 \344r " }{TEXT 304 2 " H" }{TEXT -1 19 " indefinit. Eller: " }}{PARA 0 "" 0 "" {TEXT -1 42 "Eftersom determinanten \344t negativ, s\345 \344r " }{TEXT 303 2 "H " }{TEXT -1 31 " indefinit. 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