{VERSION 4 0 "IBM INTEL NT" "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 Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "Courier" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "Courier" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 1 0 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 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{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 "Normal" -1 256 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 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }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 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 34 "K\344yr\344parven kohtisu orat leikkaajat" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "K\344yr\344parven " }{TEXT 256 26 "kohtisuoriksi leikkaaj iksi" }{TEXT -1 504 " kutsutaan toista k\344yr\344parvea, jonka k\344y r\344t leikkaavat ensinmainitun parven k\344yr\344t kohtisuorasti joka isessa leikkauspisteess\344. Kahden k\344yr\344n kohtisuora leikkaamin en tarkoittaa leikkauspisteeseen asetettujen tangenttien kohtisuoruutt a. \n\nOngelma on luonteeltaan geometrinen, mutta sill\344 on merkitys my\366s sovelluksissa: esimerkiksi s\344hk\366kent\344n kentt\344viiv at ja tasapotentiaalipinnat (tai tasokuvassa tasapotentiaalik\344yr \344t) ovat toisiaan vastaan kohtisuorat. \n\nPoistetaan mahdolliset a ikaisemmat muuttujat: \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 " restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 190 "Yksinkertainen esimerkki k\344yr\344parven kohtisuorista leikkaajista on origokeskinen ympyr\344parvi, jonka kohtisuorat leikk aajat ovat origon kautta kulkevat suorat, kuten seuraava kuvio osoitta a: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "with(plots): with(plottools):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 65 "ympyraparvi:= seq(plot([r*cos(t), r*sin(t), t= 0..2*Pi]), r=1..7):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "suor aparvi:= seq(line([0,0], [7.5*cos(2*t*Pi/20), 7.5*sin(2*t*Pi/20)]), t= 0..20):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "display(suorapar vi, ympyraparvi, scaling=constrained);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 192 "Annetun k\344yr\344parve n kohtisuorat leikkaajat voidaan l\366yt\344\344 differentiaaliyht\344 l\366iden avulla. T\344ll\366in muodostetaan ensin annetulle parvelle \+ sen differentiaaliyht\344l\366. T\344m\344 on periaatteessa muotoa " } {TEXT 257 1 "y" }{TEXT -1 4 "' = " }{TEXT 282 1 "f" }{TEXT -1 1 "(" } {TEXT 258 1 "x" }{TEXT -1 2 ", " }{TEXT 259 1 "y" }{TEXT -1 25 ") ja a ntaa siis pisteen (" }{TEXT 260 1 "x" }{TEXT -1 2 ", " }{TEXT 261 1 "y " }{TEXT -1 144 ") kautta kulkevan k\344yr\344n tangentin kulmakertoim en t\344ss\344 pisteess\344. Saman pisteen kautta kulkevan kohtisuoran leikkaajan kulmakerroin on t\344ll\366in " }{TEXT 264 1 "y" }{TEXT -1 7 "' = -1/" }{TEXT 283 1 "f" }{TEXT -1 1 "(" }{TEXT 262 1 "x" } {TEXT -1 2 ", " }{TEXT 263 1 "y" }{TEXT -1 96 "). T\344m\344 on kohtis uorien leikkaajien differentiaaliyht\344l\366. Se voidaan kirjoittaa m y\366s muotoon -1/" }{TEXT 265 1 "y" }{TEXT -1 4 "' = " }{TEXT 284 1 " f" }{TEXT -1 1 "(" }{TEXT 266 1 "x" }{TEXT -1 2 ", " }{TEXT 267 1 "y" }{TEXT -1 127 "), mist\344 n\344kyy, ett\344 alkuper\344isen parven di fferentiaaliyht\344l\366st\344 p\344\344st\344\344n kohtisuorien leikk aajien yht\344l\366\366n tekem\344ll\344 korvaus \n" }}{PARA 258 "" 0 "" {BITMAP 78 42 42 1 "?TMGr;B:nH>?nCN?F:C:;::JB;\\:::::::_R:B::::::JCZ::::::Z^V\\;?>CC:>:::::ZM Fb[^@_:;:::::R:]CB_jC<>;J;B:<::::JC:_R:B:::::j: L=;Z:bB_:;:Z<>B<:::=J:DJLR>>:;:C[:>Z;>><:::R:_ZLV<>:N\\:> Z:>Z;^@>:::R:_ZLF:>:N\\:B:;jC<>;:::ZLF:>:f\\\\N^\\N>]BGb:?cB?[:K::JLH; >:F\\KF\\KZLZ;F\\KF<<<;::aRJZ:JC<:;jB<<;:ZKF<]:;Z:FZk:<:_:;b:_:;:::ZLb >L<:;R:;[:::::b>B:Z;Z:::::r>;\\:::::::@KJB::::::Z;Z:::::::@JTJ:::: NZ\\:;:Z;N<>::::=R><<;:Z;N<>::::@JJB::::::Z;Z:::::::@:<::::::R:B:::::: Z;Z:::::::@:<::::::R:B::::::Z;Z:::::::?RB;[:::::::@JJB:5: " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "Korvausta teht\344ess\344 ei yht\344l\366n tarvitse olla \+ normaalimuodossa " }{TEXT 268 1 "y" }{TEXT -1 4 "' = " }{TEXT 285 1 "f " }{TEXT -1 1 "(" }{TEXT 269 1 "x" }{TEXT -1 2 ", " }{TEXT 270 1 "y" } {TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 333 "Lopuksi ratkaistaan saatu kohtisuorien leikkaajien diffe rentiaaliyht\344l\366. Yleisess\344 ratkaisussa esiintyv\344 m\344\344 r\344\344m\344t\366n vakio, integroimisvakio, on parven parametri. Kut akin t\344m\344n (sallittua) arvoa vastaa jokin parven k\344yr\344. \n \nSeuraava esimerkki valaisee menettely\344. \n\nOlkoon annettuna k \344yr\344parvi, miss\344 parven parametriksi on merkitty " }{TEXT 271 1 "C" }{TEXT -1 3 ": \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "yhtalo:= y=C*x*exp(x^2+y^2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "Koska yht\344l\366\344 ei voida ra tkaista muuttujan " }{TEXT 281 1 "y" }{TEXT -1 70 " suhteen, on kuva p arvesta piirrett\344v\344 kirjoittamalla yht\344l\366 muotoon \n" }} {PARA 257 "" 0 "" {XPPEDIT 18 0 "C = y/(x*exp(x^2+y^2));" "6#/%\"CG*&% \"yG\"\"\"*&%\"xGF'-%$expG6#,&*$F)\"\"#F'*$F&F/F'F'!\"\"" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "ja piirt\344m\344l l\344 oikeana puolena olevan kahden muuttujan funktion korkeusk\344yri \344. T\344m\344 voidaan tehd\344 " }{TEXT 272 5 "Maple" }{TEXT -1 13 "n komennolla " }{TEXT 275 11 "contourplot" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "ca rvot:= -0.64, -0.32, -0.16, -0.08, -0.04, -0.02, -0.01, 0, 0.01, 0.02, 0.04, 0.08, 0.16, 0.32, 0.64:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "kuva1:= contourplot(y/(x*exp(x^2+y^2)), x=-2..2, y=-2..2, con tours=[carvot], grid=[50, 50], color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "display(kuva1, scaling=constrained);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "Parven di fferentiaaliyht\344l\366n johtamiseksi ajatellaan, ett\344 muuttuja " }{TEXT 273 1 "y" }{TEXT -1 4 " on " }{TEXT 274 1 "x" }{TEXT -1 13 ":n \+ funktio: \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "parvi:= subs (y=y(x), yhtalo);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 129 "Differentiaaliyht\344l\366 saadaan derivoimalla p arven yht\344l\366 ja eliminoimalla alkuper\344isest\344 ja derivoidus ta yht\344l\366st\344 parviparametri " }{TEXT 276 1 "C" }{TEXT -1 1 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "derivparvi:= diff(parvi, x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "eliminoi:= eliminate(\{parvi, derivparvi\}, C); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 277 9 "Eliminate" }{TEXT -1 157 " -komento antaa listan, jonka viimeisen \344 alkiona on eliminoinnin tulos polynomin muodossa. Ratkaisu muunne taan yht\344l\366muotoon merkitsem\344ll\344 polynomi nollaksi." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "diffyht:= eliminoi[-1][]=0;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "T\344st\344 saadaan kohtisuorien l eikkaajien differentiaaliyht\344l\366 tekem\344ll\344 edell\344 esitet tu korvaus:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "diffyht2:= subs(diff(y(x), x)=-1/diff(y(x), x), di ffyht);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "Differentiaaliyht\344l\366 ratkaistaan ja saadun ratkais un avulla muodostetaan kohtisuorien leikkaajien yht\344l\366:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "ratk:= dsolve(diffyht2, y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "parvi2:= y(x)^2=rhs(ratk[2])^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "yhtalo2:= subs(\{y(x)=y, _C1=C\}, parvi2);" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 " Parviparametrina on integroimisvakio " }{TEXT 278 3 "_C1" }{TEXT -1 29 ", jota on merkitty lyhyemmin " }{TEXT 279 1 "C" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "Kohtisuor ien leikkaajien parvi voidaan piirt\344\344 samalla menettelyll\344 ku in alkuper\344inen parvi: \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "lauseke2:= solve(yhtalo2, C);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "carvot:= -0.8, -0.7, -0.6, -0.5, -0.4, -0.3, -0.2, - 0.1, 0, 0.1, 0.2, 0.4, 0.7, 1.1, 1.6, 2.2, 2.9, 3.7;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "kuva2:= contourplot(lauseke2, x=-2..2, y= -2..2, contours=[carvot], grid=[50, 50], color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "display(kuva2, scaling=constrained) ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Molemmat parvet samassa kuvassa: \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "display(kuva1, kuva2, scaling=constrained);" }}} {EXCHG {PARA 5 "" 0 "" {TEXT -1 8 "Teht\344v\344 " }}{PARA 0 "" 0 "" {TEXT -1 25 "Muodosta hyperbeliparven " }{XPPEDIT 18 0 "x^2-2*y^2 = C; " "6#/,&*$%\"xG\"\"#\"\"\"*&F'F(*$%\"yGF'F(!\"\"%\"CG" }{TEXT -1 26 " \+ kohtisuorat leikkaajat. \n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 280 19 "SK K & MS 12.07.2001" }{TEXT -1 0 "" }}}}{MARK "0 2 2" 59 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }