zeil`ivpxtic zeipaz :ixehwe aizka F = dx i x i ,dzr 1.R n -l ihxcphqd qiqaa e i xehwel mgkezn oeniqn xzei did `l dx i xy`k :mipalnl oixb htyna xkfp

Transkrypt

1 zeilìvpxtic zeipaz xfr ilkk xwira yeniyl eqpked xy`,zeipaz-1 ly byena epynzyd mcewd wxta mfilnxetd on xake,mipalnl oixb htyn zgkeda epnzg wxtd z`,z`f mr.ipeniq htynd ly dllkd oixb htyna ze`xl mivex epgp` m`y xexa htynd geqip ly zilnxet dyib rivp ep`.zeipaz-1-l dxifbd byen z` lilkdl epilr,itpi`d ly iceqid ziaihi`hpiè dxexa dbvd xytìy ilkk,zilìvpxtic zipaz ly zxfbpd zxcbdl zcnerd zihnznd divièhpi`d.(jynda qe`be qwehq ihtyne) oixb htyn ly icnin-n agxna zeicnin-k zerixil gtpd byen zxcbd ìd bivpy mfilnxetd ixeg`n mfilnxetd lre,dhppinxhcd ìd (dcigid) gtpd zivwpet n = k xear.(n > k-l),z`f mr. R n -a mixehwe zei-k-l dhppinxhcd byen zllkd lrk aeygl xyt` bivpy.dagxda ef dyib gztp `l F ly lìvpxticd z` epazke F zipaz-0-n eplgzd.zeipaz-1 eplaiw cvik xkfp F = n i=1 F x i dx i :ixehwe aizka,dzr 1.R n -l ihxcphqd qiqaa e i xehwel mgkezn oeniqn xzei did `l dx i xy`k ˆ ( F2. F ) ˆ 1 dd = F 1 dx 1 + F 2 dx 2 D x 1 D :mipalnl oixb htyna xkfp l`ny sbày oeeik,zgp i` ly znieqn dcin zxxern daezk ìdy itk,efd dgqepd -hpi` riten oini sbày cera (R 2 -a ixlwq dcy ly) ihxcphq onix lxbhpi` riten aezkl xyt` l`ny sb` z`,oaenk.zipaz-1 ly divxbhpi` ly deeqna onix lxb ˆ C (F 1, F 2 )dc heyt mivex epgp`y `l`.beqd eze`n mihwiae` miriten dèeynd itb` ipyay `ceel jke xy`,zeilìvpxtic zeipaz ly aizkd z` gztl mivex epgp` - xg` oeeika zkll èvnl,ok m`,ìd dpey`xd epzxhn.iyeniye c`n liri aizkk jynda xxazi azkpy itk,oixb htyn oini sbà cpxbhpi`d z` silgz xy`,zilìvpxtic zipaz ite` ilra mihwiae` el`,xac ly eceqia,ik cnln zeipaz-1 ly aizkd xak.lirl la`,(f 1 dx 1 + F 2 dx 2 lynl) "mipey mipeeika" zeipaz-1 mekql ozip mpn`.icnin-1 divw`xhpi` yiy e`,zilnxetn xzei ìd dnikqdy d`xp `l - dfd alya zegtl - zcigi lrk dx i lr aeygl xyt`,iaihièhpi` oteà.mipeyd mixaegnd oia idylk.e i oeeika zilniqhipitpi` :oken oekzn epl yi R 3 -a?lynl,ghy zecigil jxe` zecigin xearl ozip,cvik dltkny `l`.u, v i"r rawpd oeliawnd ly ghyd èd v u f`,v, u R 3 m` ly ghy lr xacl,lynl,dvxp m`,da ynzydl lkep `le,xicp ic wqr ìd zinipt dlhdd - e i -l ilècd ix`pild lpeivwpetd - e i lrk `l` e i lrk `l dx i lr aeygl xzei oekp,dyrnl 1 `l - xen`k - j`,zehppinxhck zeilìvpxtic zeipaz zpadl gztnd ìd ef dyib.i-d dhpicxe`-ewd lr.df `yepa xeaicd z` aigxp 1

2 zglven didz `l mpn`y,ziniptd dltknd ly dllkd gztp okl.r 4 -a mighyn zltkn (ly hlgend jxr), u v xicbp u, v R n ozpda.epikxv lr dprz j`,denk epgp` R 3 -a la` 2.u, v i"r zrawpd ziliawnd ly ghyd zeidl,v-e u ly fixhd :xagl xyt` f` R 3 -a mixehwe u, v, x, y m`y `"f) xehwe èd u v :xzei milawn lrk eilr aeygl xyt`y) oeeik yi u v-l aeyg zegt `ly dne,(u v + x y ly dneiwl zetvl ozip `l zinipt dltkn xcrda ixd?z`f lilkp cvik.("oniq" èd ekxe`y,(cigi) u v xehwe u, v mixehwe bef lkl dni`znd zix`pil divwpet i"r zxvepd ziliawnd heyt ìd u v - oihelgl ilnxet oteà - xicbp okl. u v,u oeeika mcwzp ziy`x :divhpixe` ozip u v ziliawnl?oniqd iabl dne.u, v.dlgzdd zcewpl v-e u xefgp fè v mcwzp k"g` zeiliawnd ly "mekqd" èd u v+x y :mixagn heyt?zeiliawn mixagn cvik zeivièhpi`d lr dpriy ick j`,ilnxet mekq edf,xen`k.x y-e u v zepeeknd ilnxetd mekqly ick,ok lr xzi.miqiqa miweg xtqn miiwl eilr eplgzd odn alya.xiaq ixabl` dpan el didiy c`n i`ck,ipeniq leaxql xarn miyeniy eidi?z`f dyrp cvik.ixehwe agxnl "zeiliawnd agxn" z` jetdl dvxp,oey`x.v v = 0-y heyt fixkp okl.( v v = 0 epyxce) zpeepn ìd v v ziliawnd okl.dketd divhpixeà j`,v u ziliawnl ddf u v ziliawndy xexa cer.u v + v u = 0 xicbp miyp.λ(u v) = λu v :icnl irah oteà,xicbdl lkep,xlwqa ltkl xyà okl,λ(v u) = λv u = u λv-y al.λu v = u λv ltkdy gihadl wx xzepy jk,ilnxet oteà - xekfk - epxcbd,xeaigd z` :xicbp okl,xeaigl qgia iaiheaixhqic èd xlwqa.λ(u v + w z) = λ(u v) + λ(w z),ixehwe agxn didi zeipazd sqe`y dyixcd on wx aiegn epi` xacd ik s` -y icnl zirahd dyixcd z` siqep.u v + u v = (u + u ) v ( (u v) = ( u) v mr sexva u v = v u-n) ihnehe` oteà,lawpy oaenk.u v + u v = u (v + v ) mb ipy ly dhppinxhcd ixdy,zehppinxhcl zeilìvpxtic zeipaz oia xywl ztqep divwicpi` idef 2.miqxet mdy ziliawnd ghy,oniq ick cr,ìd R 2 -a mixehwe 2

3 xzeia zirahd dl`yd.ixehwe agxnl zepeeknd zeiliawnd sqe` z` epktd jk èd dx 1,..., dx n,xekfk.qiqad ly cnind edn ìd ixehwe agxn eplaiwy rbxa agxna e`) R n -a xehwe lky oeeik.(ilècd agxnl weic xzil e`) R n -l qiqa -e`y (oexg`d i`pzdn) lawp,dx 1,..., dx n mixehwed ly ix`pil sexiv èd (ilècd.(?recn) epipay ixehwed agxnl qiqa mieedn 1 i < j n mr dx i dx j zebefd sq ( ) n zeipaz-2-d agxn `xwp eplaiwy ixehwed agxnl. èd agxnd cniny `"f 2.zeiqiqad ly ixehwed agxnd èd zeipaz-1-dy myk.zeipaz-2 xicbdl èd àd alyd dxevd on mihwiae`d lk n F i dx i i=1 zeidl zillk zipaz-2 xicbp,zetivx zeivwpet F i xear F i,j dx i dx j 1 i<j n,lirl excbedy millkd z` eniiwiy oteà excbei xlwqa ltkde xeaigd,oaenk,xy`k lynl.f 1 dx 1 dx 2 + F 2 dx 2 dx 1 = (F 1 F 2 )dx 1 dx 2 xehxte` xicbdl epilr - l"pd oeicd z` eplgzd epnn - oixb htynl xefgp m`,dzr -a) zipaz-2 didz zlawznd d`vezdy oteà (R 2 -a) zeipaz 1-d sqe` lr dxifb zipaz-2 wx R 2 -a yi xlwqa ltk ick cry,al miyp.htyna driten efy itk.(r 2 :gihadl dvxp oixb htyna okl.dx 1 dx 2 :zg` ziqiqa ( F2.d(F 1 x 1 + F 2 x 2 ) = F ) 1 dx 1 dx 2 x 1 ogap,ziy`x?z`f lawp cvik.miytgn epgp`y dxifbd xehxte` z` oiivn d xy`k :dvxp f`.f 2 = 0-y dxwind z`.d(f 1 dx 1 ) = F 1 dx 1 dx 2 = F 1 dx 2 dx 1 :zeipaz-0 xefbl mircei xak epgp`y xkfp,dzr df = F x 1 dx 1 + F dx 2 df 1 dx 1 = F 1 dx 2 dx 1 -y al miype :xicbp okl.mikixv epidy enk weica 3

4 .dω := df dx i = j i f`.r n -a zipaz-1 ω := F dx i idz 0.1 dxcbd F x j dx j dx i f` ω = n i=1 F idx i m` :ix`pil oteà dxcbdd z` aigxpe.dω = n df i dx i i=1 :heyt èd (oalnl) oixb htyn ly geqipd,z`fd dxcbdd mry `ceel lw ˆ D ˆ dω = lr zexiyi riavnd,gepe ihpbl` geqip edf.r 2 -a zipaz-1 ω-e oaln èd D xy`k D ω.itpi`d ly iceqid htynd ly dllkd htynd ly ezeid.dxebq okle,zwiecn zipaz (epxcbdy itk) ìd df f` idylk zipaz-0 F idz m` dxebq ìd F = F 1 dx 1 + F 2 dx 2 zipazy epxcbd,xekfk. F 1 = F 2 x 1 dxebq zipaz ly dxcbdd epgzity ycgde llkeynd dxifbd xehxte` mry,al miyp :xzei daxd dheyt zkted.dω = 0 m` wxe m` dxebq ìd ω f`.r 2 -a zipaz-1 ω idz 0.2 dprh mipeniql lbxzdl ick hexta dirl xearp la`,zilìaixh ìd dgkedd dω = :dgked f` ω = F 1 dx 1 + F 2 dx 2 idz.miycgd ( F2 F ) 1 dx 2 dx 1 x 1.dxebq zipaz ω xy`k weica 0-l zizedf deey efd zipazde :illk oteà xicbdl lkep,okl m` zwiecn ìd ω zipaz-1.dω = 0 m` dxebq ìd (R n -a) ω zipaz dxcbd.α zipaz-0 efi`l ω = dα :ziciin dzr ìd dàd dprhde.dxebq ìd zwiecn zipaz dprh 4

5 oteà.zeipaz-1 ly dxifbae zeipaz-2-a oeicd z` wiqtdl daiq lk oi`y oaenk f` ω = F dx i dx j m`y xicbdl ozip,irah dω = df dx i dx j -iecnd zeipaz 3-d sqe` z` lawp jk.ix`pil oteà dxifbd xehxte` z` aigxdle agxnl zeipaz-3-d zwlgn z` aigxp zeipaz-2 lye zeipaz-1 ly dxwna enk.zew.d`ld jke,i < j < k xear F dx i dx j dx k dxevd on miwiae`d lk i"r xvepy ixehwed :xicbp,illk oteà.dgezt dveaw U R n idz 0.5 dxcbd :dxevd on hwiae` ìd U lr zilìvpxtic zipaz-k.1 F I dx I I.I = (i 1,..., i k ) xear,x I = x i1... x ik -e U lr dwlg divwpet F I xy`k zeiqiqad zeipaz-k-d ze`xwp I = (i 1,..., i k ) xear dx I dxevd on zeipaz-k-d.2.u lr -eàe.f I F J dx I dx J ìd F J dx J zipaz-l mr F I dx I zipaz-k ly fixhd zltkn.3 ( ) ( ). FI dx I FJ dx J = (F I dx I F J dx J ) I,J :xzei illk ot :àd oteà lret dxifbd xehxte` F I dx I zipaz-k ozpda.4.d(f I dx I ) := df I dx I = n i=1 F I x i dx i dx I agxnl U lr zeipaz-k-d agxnn zix`pil divwpet èd d dxifbd xehxte`.u lr zeipaz-(k + 1)-d on wlg,evxz m`,edf) zeilìvpxtic zipaz k xear miniiwzn miàd miqgid :(dxcbdd on zeraepy zepekz `l,dxcbdd j l miniiw,r"gg ìd σ-y `"f) selig ìd σ : {1,..., k} {1,..., k} m` f` (h j, l lkl σ(h) = h sqepae,σ(l) = j-e σ(j) = l-y jk dx I = dx σ(i) -d ly oniqd z` dpyn I-a miqwcpi` bef ly xcqd ztlgd,zexg` milina.zipaz-k.f I dx I + G I dx I = (F I + G I )dx I 5

6 :zeilìvpxtic zeipaz-k ly zeàd zepekzd z` `ceel lw,dzr :if`,u R n lr zipaz-l α-e zipaz-k ω dpidz 0.6 dprh.j l dfi`l i j = i l -e I = (i,1,..., i k ) m` dx I = 0.1.ω = 0 f` k > n m`.ω = ±α f` zeiqiqa zeipaz ω, α-e k = n = l m`.2.ω α = ( 1) kl α ω.3.df ω = (df ) ω + F dω.4.d(ω α) = dω α + ( 1) k ω dα.5.d(dω) = 0.6 :dlw ìd dgkedd cg` cvn.σ(l) = j-e σ(j) = l seligd σ : {1,..., k} {1,..., k} idz.1 dx I = dx σ(i),(1) dpekz itl,ipy cvn.(x ij = x ik ik) dx I = dx σ(i).dx I = 0 gxkda okl σ : {1,..., n} yie I = {i 1,..., i n } f` k = n m`.ω = dx I -y ziy`x gipp.2 -elig ly dakxdk aezkl ozip σ lky oeeik.j lkl i j = σ(j)-y jk {1,..., n} -ritend miteligd xtqn èd t xy`k,dx I = ( 1) t dx 1... dx n -y lawp mit dx J =-y d`xi oerihd eze` α = dx J m`.l"pd oteà σ ly daizka mi,i j = i l -y lk j l yi gxkda,k > n m`,dzr.yxcpk,( 1) t dx 1... dx n.mcewd sirqdn zraep dprhde ztlgzn fixhd zltkny llba) zeqiqa zeipazl dprhd z` gikedl witqi.3.l lr divwecpià lynl,xexa df dfd dxwna la`.(minekq mr.àd sirqd ly ihxt dxwn edf.4 -e ω = F I dx I xear dprhd z` gikedl witqi,zix`pil divwpet d-y oeeik.5 d(ω α) = d(f I G J dx I dx J ) = d(f I G J ) dx I dx J dfd dxwna.α = G J dx J ipy cvn dω α + ( 1) k ω dα = (df I )G J dx I dx J + ( 1) k F I dx I ((dg J ) dx J ) 6

7 la` F I dx I ((dg) dx J ) = ( 1) k F I dg J dx I dx J weica miyxcp jkle,dx I zipaz-k-d oial dg J zipaz-1-d oia silgdl epilk ik -y ziciin raep uipaiil zgqepny oeeik.(lirl (3) e`xe) mitelig k d(f I G J ) = G J df I + F I dg J.zraep dprhd dxevd on zeipazl gikedl witqi,zxfbpd zeix`pil llba,libxk :heyt oeaygd.6 d(f dx I ) = F x i dx i dx I f`.f dx I d(d(f dx I )) = 2 F x j x i dx j dx i dx I -e,zeaxernd zexfbpd oeieeyn 2 F x i x j = 2 F x j x i -e dx i dx j dx I = dx j dx i dx I.d(d(F dx I ) = 1 2 ( 2 F x i x j 2 F x j x i ) dx i dx j dx I = 0 :lawp okl ix`pil lpeivwpet lrk aeygl xyt` dx i lr ik xikfp df wxt meiql.dx i (v 1,..., v n ) = v i zix`pil-ihlen divwpet lrk dx I lr aeygl xyt` dnec oteà dx i1 ( v 1 )... dx ik ( v 1 ).dx I ( v 1,..., v k ) = det.. dx i1 ( v k )... dx ik ( v k ) zegtl e`) divxbhpi`l zeilìvpxtic zeipaz ly xywd xzei ile` xexa efd dbvda zilìvpxtic zipaz ly oniqd z` zexicbnd zepekzd zexdazn oke,(gtp iaeyigl.'eke 7

8 divwpet ozpda :zeilìvpxtic zeipazl zrbepd zg` zifkxn dl`y dxzep dzr f g dakxdd,v megz dfi`l g : V U (dwlg) divwpete U megz lr (dwlg) f :xzei zipkh oeyla e`,v megzl f ly "g zervnà mebxzd" icnl irah oteà - ìd jeynl ozip (- dn myle - cvike) m`d.g (jxc e`) zervnà f ly xeg`l dkiynd ly divxbhpi` epxcbdyk epiyry dn df,yxetna z`f xnel ilan?zeipaz mb xeg`l z` xeg`l epkyne (g divwpetd ef) mewrd ly divfixhnxt epxga :mewr lr zipaz-1 ick cr - yi R lry oeeikne) R lr zilìvpxtic zipaz-1 lawl ick g jxc zipazd eprci mixikn ep`y onix lxbhpi` mr zcklzny,zg` ziqiqa zipaz-1 wx - oniq micnina mb eply dibhxhq`d didz ef.(efd zipaz 1-d ly lxbhpi`d z` aygl cvik zeicnin-k (zeixhnxt) zerixi lr zeipaz-k ly milxbhpi` aygl ick :xzei mideab -l divfixhnxtd jxc "zipazd z` xeg`l jeynp",drixid ly divfixhnxt `vnp rcp,mixikn xak epgp` dze`y,zg` ziqiqa zipaz-k wx yi R k -ay oeeikne,r k xeg`l dkiynd z` xicbdl epilr :xdfidl epilr la`.lxbhpi`d z` aygl cvik on xake) divfixhnxtd zxigaa dielz didz `l lxbhpi`d aeyiga d`vezdy jk z` okl.(zilìaixh `l dibeqa xaecny xexa dpzynd iepiy htyn ly geqipd.àd wxtl dgcp ef dibeqa letihd 8