Luận văn Lý thuyết thế vị ứng dụng trong trọng lực học và từ trường

cbi gay,nUt,dUtdo~n,ne'p16i...thuongla duongtheo huangn~mngang.Khi do,ne'udQdi thuongla tuye'ntinh,tacoth~xemxetcac ngu6nha'pdftnho~cngu6ntunhumQtduongcohuangkhangthayd6ivav~th~ WcdoduQcxemlanhulahaichi~uvaiphanb6m~tdQd~ngrex,y,z)=rex,z). "£'uj.JLmUL 1luJ-euj 58 g uin 7/(Joid(f{J,mL Cacngu6nhaichi~uIad~hlnhdungvad~thie'tI~pmohlnhhdncacngu6n 3 chi~utu'dngungm6ikhi di~uki~ndiachfftchophep.MQtchumcachlnhtn,l songsongse t:;wnenmQtd:;mgddngiancuamohlnhhaichi~u.Khi do,dQdi thu'ongcoth€ du'Qcxffpxi b~ng(2.2),vdi \I1mnIa h!chffpd~nt(;lidi€m mgayra boihlnhtn,ln vdim~tdQddnvi. B~ngthuc(1.50)chotathe'hffpd~n~cuav~th€ haichi~uvdim~tdQkh6i p(x,z) u =2yJp(S)Iog!dS, . s r vditichphandu'QcIffytrenm~thie'tdi~nngangS vavdi rIa khoangcachvuong gocWithanhph§ncuav~th€ vadu'Qchoboi r =)(X-X,)2+(Z-Z,)2. B€ ddngian,tachuy€ndi€m quailsattOig6ctQadQvaxemphanb6m~tdQ Iah~ng.Khi do,phu'dngth~ngdungcuatru'onghffpd~ndu'Qchoboi (P) = au = 2 JJ z'dX'dZ'.g a YP ,2 ,2Z x +z (2.7) Tich phanIffytheox', tadu'Qc f[ x' x' ]g =2yp arctan--2 - arctan-L dzI ,Z' z' vdi XI' va X2'Ia hai hamcua z', va nhu'da:chi ra tronghlnh2.4, bi€u di~nhai du'ongriengbi~tbaaquanhchuvi cuam~tthie'tdi~nngang.Tich phanIffytheo chi~ukimd6ngh6dQctheochuvi,tadu'Qc x' g =2ypr1iarctan~z'. LJJ z' (2.8) Thaythe'chuvi boi cacc(;lnhcua dagiacbien,(2.8)trathanh J2ttt/-1l tJall Inf!£ uj 59 g ~L 7IfJO-idr1lhiuL N Zn+l X' g =2ypI farctan~dz', n=1Z Zn (2.9) trongd6ZnvaZn+!d€ chIcaode>cuahaidgumutcuac<;lnht iln cuadagiac. p >~\::~)"( ! / \ \,,-~ ._-,---' /'""""'-..--z. x ';~-e,,+ f ~~D x .-...---...-....-----..- ;:;: Hinh 2.4.Xapxl vQtthi haichi~ubangdagidcN dink. Dungduongth~ngbi€u di6n x' theo z', taduQc , I n.x = aDz+ fJn, (2.10) voi an xn+!- xn Zn+l- Zn ~n = xn - anzn The'(2.10) vao(2.9), taduQc g =2ypi zJ arctan( anz'~~nJ dZIn=1Zn Z g - 2ypi { 7t(Zn+l - Zn )+ ( Znarctan ~ - zn+larctan Zn+l ) + n=1 2. xn Xn+l + ~n2 [ log~x;+;- Z;+1 - Gt, ( arctan zn+l- arctan~ J]} ' l+an ~x -z Xn+l xnn n Hai s6 h<;lngd:1ulien trongda"ungo~cdon cuat6ngtie'nWi khongtrenba"t ..l2tlijn1001fhLp! ul 60 q ~l 7ItJoid(JlJull'L ky dagiacnaG.Do d6,d£ngthuctrenc6d~ngdongianhall, g =2ypI ~n2 [ 10grn+l - an(en+l-en) ] , n=l1+an rn (2.11) vai rnva endu'Qcxacdinhnhu'tronghlnh2.4. Lvc hapd~ncuav~tth€ haichi€u dod6phVthuQcVaGvi tri ~uaN dinh cuadagiac.N€u tatu'dngtu'Qngc6 N du'ongth£ngvetudi€m quailsatWim6i dinhcuadagiacthllvc hapd~nphvthuQcVaGchi€u daicuam6idu'ongnayva g6ct~obdigiuachungvaim~tph£ngchuami€n dagiac(hlnh2.4). 2.1.2.Caemohlnhtli trtiCing Tu tru'ongcuamQtngu6ntUdu'Qchobdi B = -Cm\7p fM.\7 Q!dv , R r (2.12) vai M la dQtuh6a, r lakhoangcachtudi€m quailsatP taithanhphfindvcua ngu6n.Gia tricuah5ngs6CmphvthuQcVaGh~th6ngdonvi du'QchQn.Hfiuh€t cacnghienCUlltUtru'ongdo dQdi thu'ongcuatru'ongtoanphfinho~cthanhphfin doncuaB. DQdi thu'ongcuatru'ongtoanphfindu'Qchoxapxi bdi - f 1 ~T = -CmF.\7p M.\7Q-dv , R r (2.13) vai F lavectodonvi tronghu'angcuami€n tru'ong. Tu'ongtvnhu'ly lu~nv€ cacngu6nhapd~n3chi€u,mQts6thu~ttoandu'Qc khaosatnh5mtinhtoanmQthanhphfincuaB tu(2.12)ho~cdQdi thu'ongtru'ong tO~lllphfinchomQtngu6ntuc6hlnhd~ngvaphanb6tuchotru'acb5ngphu'ong phapthu~n. J2LLijPLomL lnf!£ uj 61 q ~L ']{)oOirnJuin B~ngcachxemme)tngu6ntunhu'la me)tt~pcaclu'ongqI'ctuhaytutich,ta nh~ndu'Qcnhi~ucachkhacnhaud~nh~ndu'Qcacmahlnhxa'pXlchongu6ntu. Volume of l'-tagnctizalion Vohtll'\C and Sur'facc Charge ~-Q$ L/ \/OIUlllC and Surface C:::lttTr~nt$ Poisson's Relation ('--?\ (~l~=~~~)-'f ~-~.:-/ $ D Hinh 2.5: Bondt;mgphanb6'cuasf!tuh6a. f)£ngthlic(2.12)ho~c(2.13)coth~tinhtoantu'ongminhchonhITngv~th~ cohlnhdangdongianvacoth~dungchophu'ongphapthu~n.Khi do,me)tngu6n tucoth~du'QchiathanhN ph~ndongianvakhidocactinhtoandu'Qcthvchi~n tu'ongtv nhu'd6ivoi(2.2).Bathanhph~ncuatutru'ongtrdthanh B,J = N IMjbjj, j=l (2.14) trongdo, Bj la tutru'ongt<;=tidi~mquailsatthlij, Mj la de)IOncuade)tuhoacua ph~nthlii, vabijla tUtru'ongt<;=tidi~mquailsatthli j gayrabdiph~nthli ivai de) tuhoadonvi bo. 1J f - 1 = - CmV'P M.V'Q-dv , 0 r 1 (2.15) NSucacph~nchiadunho,chungco th~du'Qcxemnhu'code)tuhoad~u. TrangthvctS,caca phaiconhITnghlnhdangdongian.Ch~ngh<;=tnnhu'hlnhhQp chITnh~tho~ccacc~plu'ongcvc,d~d8danghontrongvi~ctinhtoanbi~uthlic (2.14). "!!.llt/-I'LlJillL /},f!R uj 62 q v4n 7IJoa1(],lJUYL Nhu'dad~c~ptrongcacph~ntru'ac,tichphankh6icua (2.12)cothedu'QC chuyenthanht6ngcuatichphanffi~tva tichphankh6ib~ngcachapdl,mgh<% thlicvecto V.(~A)=V~.A+~V.A vaapdl,mgdinhly Divergence.The'tuco d~ng v = CmfM,VQ!dv = CmfM.nds - CmfV.M dv R r sr Rr =Cm fQs ds + CmfQv dv. s r R r (2.16) Cactichphantrong(2.16)coclingd~ngnhu'the'ha"pd~n,vaid~ilu'Qngvo hu'angQs va Qv tu'dnglingla bieudi6nchotutichtrenb~ffi~tvaphiatrong cuav~tthe.Ne'udQtuhoad~u,thltichphanthlihaicua(2.16)tri<%ttieu,vathe' vidu'Qchobdi v = C fM.nm -ds. s r (2.17) Ne'udQtuhoad~u, v~thecothehoantoandu'Qcbieudi6nbdis1,1'phanb6 cuacactutichtrenb~ffi~tcuav~the.S1,1'bi€u di6nnayd~ntaiffiQtso'cacthu~t toan hi<%uq a,ffiQtso'trongcacthu~toandosedu'Qcd~c~pd ph~nsau. M~tkhac,Dinhly Biot - Savartchotha"ytutru'ongcuaffiQtvongdi<%nnho xua'thi<%nt~inhii'ngkhoangcachnhu'tru'ongcuaffiQtc~plu'ongc1,1'c.Tu do,ffiQt phanb6cuacacc~plu'ongc1,1'ccothedu'QCxeffinhu'la cacthanhph~nkh6icua dong di<%n Is=Mxn Iv =Y'xM Ne'utlr'hoa la d~u,thl dongdi<%nkh6i tri<%ttieu va ffiQtv~ttu co thedu'QC ,I2,'tPLmin fhf!l! uj 63 g,d,L 7J()oii1(llJuin thaythe-bdi mQtkh6i r6ngeo cimghinhd.~lllgmangdongdi~ntich trenb€ m~t euano (hinh2.7). Is Hinh 2.6.. DongLuanchuyingayra d(J tith6acilavi,ltthi titd8ngnhatLatUdngdudng wJi dongdi~ntrenb~m(itcilavi,ltthi. 1\ n Is Hinh 2.7.. DongLuanchuyingayra d(Jtith6acilavi,ltthi titd8ngnhatLatUdngdudng wJi dongdi~ntrenb~m(itcilavi,ltthi. 2.1.2.1. Xffpxi moblobbaebi~u Caee~ph.tonget.fe MQtngu6ntubaehi€ueoth€ duQexa'pXlbdimQt~peaethanhphftnnho hall,eodC;lngdongian.CaeluongevetuIa mQtvi d\ldongiannha't.Tu truong euamQte~pluongevetu,duQe hobdi (1.69) B m [( -- ) - - J= Cm? 3 m.r r-m r:;t:0, (2.18) vai m = m.m la mamenluongeve,r =r.r la veetoduQedinhhuangtu .l2uqlL luilL tj,1f£ ilj 64 I'd,,&t 7IfJom (/(JUUL luBngclfctOidi~rnquailsat. M~cdli tinhtoantrongthlfct€ co daichutphuct.W,tutru'ongcuarnQtv~t th~coth~du'<;1ctinhb~ngcachchiav~tth~thanhnhungthanhphftnnho,vaighl sala cacthanhphftnnhonaycachnhaurnQtkhmlngnhu'rnQtIu'Bngclfc.sa dl;lng (2.18),tasuyratutruongcuarn6ic~pIu'Bngclfc.Tu tru'ongcuangu6n.tubandftu b~ngt6ngcactutruongcuata'tCelcacluBngclfc.MarnenIu'Bngclfcrncuarn6i thanhphftndu'<;1cchobditichcuadQtuhoavath~tichcuathanhphftndo. xa'pXlb~nghinhhQpchii'nh~t Chungtaclingcoth~xa'pXlrnQtngu6ntubachi~ubdirnQt~pcachlnhhQp chITnh~t.Tu truongcuarn6ihlnhhQpchITnh~tdu'<;1cchobdiBhattacharyya(xern [1]).M6i hlnhhQpchITnh~tdu'<;1cdinhhu'angsongsongvai cac trl;lcx, y, z giOi hqn bdi Xl ~ X ~ X2,Yl ~ Y ~ Y2, Zl ~ Z <00,vacodQtuhoa M =M(iMx +]My +kMz). N€u dQ di thu'onggay ra bdi hlnh hQpdu'<;1cquail sat tqi rnQttru'ongdia phuongdinhhuangsongsongvai F = (Fx,Fy,Fz)' thldQdi thu'ongtru'ongtoan phftnquailsattqig6ctQadQdu<;1cchobdi ~T = CmM [ a2310g ( r-X' J2 r+x' + 0,13I ( r - y, J -og- 2 r+y' - al2log(r +Zl) M,F, arctan(X'2+X;{+z; ) ( x'y' J M F arctan 2 X ,2 y y r +rzl - + M F ( X' ' ]J 1X'=X2IY'=Y2 Z z arctan ~ rzl I I ,Ix'=x I IY'=YI (2.19) ,J21l1!lLIfflt'L litf!£ uj 65 g ~L ';J600irJlhiu'L vOi r2 = X.2 + y.2 + Z~. D£ngthuc(2.19)chotagiatridithu'ongtru'ongto~mph~ncuamQtkh6ihlnh hQpchITnh~tvoi Zl::;;Z<00.Ne'utinhtoannaydu'c)cthljchi~nhail~n,mQtl~n cho Zl =ZtvaM =Mo vamQtl~nchoZ\=ZbvaM =-Mothl theonguyen1'1ch6ng chat,t6ngcuahail~ntinhnaysechotatutru'ongcuakh6ihlnhhQpchITnh~tvoi dQtuhoaMo,cohaidaylaZtva Zb. B~ngcachchiangu6ntuthanhmQt~pcachlnhhQpchITnh~t,(2.19)coth~ du'c)csadl,mgl~pdi l~pl~inh~mdi€u chlnhmohlnhchocacngu6ntucohlnh d~ngtuy y. Cac kh6i hlnhhQpchITnh~tclingdu'c)cdungtrongphu'dngphap nghichd~tinhtrljctie'pvectd M tudQdi thu'ongtru'ongtoanph~ndod~cdu'c)c. Ch6ngcaebanmong Talwani(xem[1])dffsadl,mgtichphanbQi(2.12)d~du'arathu~ttoantinh toantUtru'ongsinhbdi ngu6ntu co hlnhd~ngtuy '1.Phu'dngphap nay tu'dngtlj nhu'phu'dngphapcuaTalwanivaEwingtrongtinhtoandQdi thu'ongtru'onghap d§ncuangu6nbachi€u dffd€ c~pd ph~ntren.Ngu6ntudu'c)cxapXlbdich6ng cacbanmong,m6ibanmongl~idu'c)cxapXlbdimQtdagiac.TudotichphanbQi (2.12)du'c)cchuy~nthanhtichphantheox,y vat6ngtheoz. Plouff md rQngphu'dngphap cua Talwani (xem [1]) b~ngcach thay the' ch6ngvo h~ncacbanmongbdi mQtt~nghU'uh~ncacbanmongvoi dQdaynho. Phu'dngphapnaybi~udi~nmQtcachchinhxachdnmohlnhcuangu6ntu.Ngoai fa, ngu'oita condungphu'dngphapxapXl cuaTalwaniva Plouff chobai toan al2 = MxFy + MyFx' al3 = MxFz + MzFx' a23 = MyFz + MzFy, J2I1ijl'L mi1L fJ,f!£ ill 66 g,4,L '7{)oairmullL nguejctinhbathanhphc1ncuadQtllhoatn!ctie'ptlldQdithuongtruongloanphc1n dod~cduejc. Caekh6idadi~n Ne'udQtll hoala d~u,mQtngu6ntll co th~duejcmahlnhbdicactUtich lIen b~m~tcuangu6ntll. Dt!avaod~cdi~mnay,Bott,Barnett,Hanssenva Wang (xem [1]), duara phuongphapxa'"pXl ngu6ntll bdi mQtkh6i .dadi~nma m6im~tcuano la mQtdagiacph~ng.(hlnh2.8) y z Hinh 2.8. MQtngu6ntitv6'ihlnhd(;mgtuyydu(femohlnh nhumQtkhatdadi?nv6'ieaemijtLaeaedagiaephdng. Phuongtrlnh (2.17)chota the'tll sinhbdi mQtngu6ntll co dQtll hoa d6ng nha't,voi tll truongduejcchobdi - fM.n-B = - VpV = Cm zrds. s r (2.20) £)~dongian,xettruonghejpg6ctQadQduejcd~tt~idi~mquailsatP. Khi b~ m~tcuangu6ntll duejcthaythe'bdiN m~tdagiacph~ng,(2.20)trdthanh N - ; B = CmL(M.n) f2ds, i=\ s r, (2.21) J2JltP' ImlL tltf!£ £lj 67 g nil, 7{;oiiir1lJuU, vai Sj lami[ttthli i, n)avectophaptuy€nhu'angrang oaitu'ongling. (a) A :;;;X 1\ . -/~ Vr - n;j I..P I IJ p/ P , con1cr j+ I z (0) ~ c I \ I A \ I" I" \ ",\1--- - D '-pl Alca AlJCD '" Arca ABP' +Area BCP' +ArcoilCOP' - Area ADP' Hinh 2.9. (a) M{it thz1i cila kh5i da di?n trong h? tr1;lctf,JadQx, y, z. Gia sumi[tthli i la mQtdagiac Kj dlnh(hlnh2.9).Ta quyu'acdug cac dlnhcuaba'tkymi[ttdagiacnaod~udu'Qcxettheochi~ukimd6ngh6khinhlntli ngoaingu6ntU.Tli P h(;ldu'ongvuonggoc vai mi[ttph~ngchlia da giac,ca:tmi[tt ph~ngnay t(;liPi', gQi Iij la vec to donvi tli dlnhj tai dlnhj+1, Pij la vec to donvi tUdiem P/ vuonggocvaic(;lnhj. Xet Kj tamgiacsaochomQitamgiac clingcomQtdlnhla P/ vac(;lnhdO'idi~ndlnhnayla mQtc(;lnhcuadagiacbien. Khi do, di~ntichcuaKj mi[ttcuakhO'ida di~nb~ngvai t6ngdi~ntichcuaKj tam giacvai trQngsO'di[ttb~ng+1 hoi[tc-1 tliy theovec td Iij co hu'angtraihayphai khi quailsatt(;lidiemP/. Tli do, tichphantrong(2.21)co thedu'QCthayth€ b~ng t6ngcuacactichphanla'ytrenKj tamgiac, f~dsr2 Sj = K- I Ilij~ij , j=1 (2.22) trongdo Ilij=1 n€u Ijj xPjj la clingchi~uvai nt 11..=-1n€u f xP. la ngu'ocr'IJ 1J 1J . chi~uvai nt Ilij=0 n€u Ijj xPjj =0, va ~jj la tichphantrenmi~ntamgiact(;lO bdidlnhPi' vac(;lnhthlij cuadagiacbien. J!.tUpL IJmL thf!£ nj 68 g ~L ,,;/6oai(J{J;,mt " PH IJ It.(-.r-- c:rncr .J Iv corncr j+ I f~' Hinh 2.10: Mr;ittamgidctranghf trl:lct(JadQmaiu,v,w. ChQng6ctQadQt<;tidi~mPi, tn;lctQadQmoiu,v va w songsongvoi Pij, Ijj va n( hinh2.10).Trangh<%trt;lctQadQmoinay,cacdlnhcuami€n tamgiacco tQadQla (0, 0, di), (uij,Vij,dj)va (Ujj,Vj,j+l,dj) voi dj 1akhmlngcachtli'P tOiPj'. Tich phanla'ytrenmi€n tamgiactrdthanh ~ij - fr- -dsr2 sij U f ijV fb U-p .. +vf +d,~- IJ IJ 1 lJ d d- 3 U v, OVa (u2+v2+dnZ Doc lu'Qngrichphankeptren,tadu'Qc - [ a [ r+p J 1 r +V ] - [ 1 ( r+p J]~ij = Pij -J1+a2 log ldJ - 2"logr-v - Ijj -J1+a2 log ldJ [ ]J IV=V' , I - dj a(r-Idjl) 1.]+ - n; jdJarClan( r+a2Id;1 - IV-Vii (2.23) vOi va ViP , Vi.j+lU= va Vb = - Uij Ujj J2114nmln lhLfl!uj 69 q ~t 'j6oai {J(JuHl , . v ~ 2 2 J 2 2 d2VOl a = - P = u.. +V r = u.. +V + ., 1J' 1J I . Uij Ne'uP n~mtrongm~tph~ngchuami€n tamgiac, (2.23)trdnensuybie'nva truonghQpnayduQcxU'ly b~ngcachbodi mQtph~nhlnhqU(;ltnhobaaquanh di~mPi' rakhoimi€n tamgiacdangxet,ke'tquathuduQcla - [ a ( ( r+p ) 1 r +V J] ~ij = Pjj l+log - - -log- ~1+a2 28 2 r - V - Ijj [ 1 ( 1+log r +p J] IV=Vi,.i+1 ~1+ a2 28 I , Iv=vij (2.24) trongdoE lamQts6nhotuyy. 2.1.2.2. Xa'pXlmohinhhaichi~u Nguoi ta sU'd\lngkhai ni~mb€ m~ttiI'tich d~xay d1!ngmQts6 thu?ttoan choma hlnhV?tth~hai chi€u va thaythe'cac d(;lngthie'tdi~nc~tngangcuaV?t th~bdicacdagiacN-dlnh.Ne'uV?tth~la tiI'hoad€u, dQtiI'hoacoth~duQcthay the'bdicactUtichtrenb€ m~tcuano(hlnh2.11). x x CB -~~~\~ z z Hinh 2.11.Xdpxl cilamQtvt;'itthi hai chiiu v6'icae ddih?p vahc;mtittich. Khi do,bai toanduQcchuy~nthanhvi~ctinhtoanl1!ctiI'cuaN daiph~ngtiI' tich,mdrQngva h(;lntheohuang+yva -yo TiI' truongcuamQtV?ttiI'd€u vai th~ tichR vab€ m~tS duQcchobdi (2.20), J2tttJn olin tJlf!£ u] 70 g r&L 'Jf5oO1 (j{JUUL - f M.n- B = Cm~rds. s r Ding thlicnayWongtl!nhuchotrUonghfipd~ncuamQtvor6ngclInghlnhd(;lng S a(S) - g = -y ~rds, s r vai a(S) la m~tdQm~trendonvi di<%ntich,trongdotittruongd~uc;;uamQtv~t th~titvaith~tichR coth~du<;jctinhb~ngcachthaythe'-M.~ choa(S) va Cm choy. (a) (b) ,. ~x (c) \0 ~ Hinh 2.12. (a)Ddi ndmngangmiJr(3ngvahc;mrheatrl!cy. (b) thanhv~tthi dili vahc;mrheatrl!cy cdtm(itphdngx,z t(li (x' , z') X6tmQtdaihypphing,n~mngangvaim~tdQb~m~tla a, mdrQngvoh(;ln theohuang+yva-y, codQrQngcuadaitit(Xl,z') tai (X2,z') (hlnh2.12a).MQt thanhph~ndx cuadaihypc~tm~tphingX, z t(;li(x', z') va Wongduongvai thanhv~tth~dai vo h(;lndQctheotrl;lcy co kh6i lu<;jngtrendonvi chi~udai A=adx(hlnh2.12b).Nhudffd~c~pd ph~ntruac,thanhv~tth~ co truonghfip d~nquailsatt(;lig6ctQadQla g = -2YA~r J!.llijn 'Jan tnLfl!uJ 71 g P4.n7!6oiL1rnJuiI" " - = 2YA x l+z'k X'2 '2'+z (2.25) Truongha'pd§n cuadai h(fpn~mngangdu<;5ctinhb~ngcachd~tA=crdx trong(2,25)vala'ytichphantheox, Xz , gx = 2ycrf '2X '2 dx'x +z XI r ~ = 2ycrlog~, rl Xz dx' g = 2ycrz'f---;z- '2z X +z XI = 2ycr(81-82), g [ - r - ]= 2ycrilog r~+k(81- 82) , (2.26) trongd6ri var2la khoangcach tUdi~mP taic(;tnh1va2,81va82 la cacg6c giuatn:lcx va cacduongn6ic(;tnh1,c(;tnh2 tuongung(hlnh2.12b).£)~ngthuc (2.26)la truongha'pd§ncuamQtdaih(fpn~mngang.T6ngquat,vaiba'tky dai h(fpnao,chungtaquaydaih(fpmQtg6ctuyyvavaihaivectodonvi Ii va ; lin IU<;5tlaphapvectovavectochIphuongcuadaih(fptuongung(hlnh2.12c). - - Vec to s luonc6huangsongsongvai daih(fptITc(;tnh1tai c(;tnh2.Vec to n vuongg6cvaidiEh(fpvadu<;5cdinhhuang nX = sz, nz = -sx' Thanhphincuatruongha'pd§ntheohuang~ van du<;5cchoboi (2.26) r gs = 2ycrlog~, rl gn = - 2ycr(81- 82), £llIpl (WI flu.a uj 72 g ~I '7()tJitirnluitrl Thanhph~ntheohuangxvaz duQc hoboi gx = l.g = sxgs+nxgn = sxgs+szgn [ - r - ]= 2ycrSxlog r~-sz(8t -82) . gz = k.g, = szgs+nzgn = szgs- sxgn [ - r - ]= 2ycrSzlog~-sx(8t-82) . D~ngthuctrencho ta h/c ha'pdfincuaffiQtdeliv~tth€ hyp dai vo h(;ln.D€ chuy€n v€ truonghQptUtruong,chungta chi c~nd<1ity =Cmvacr=-M.n B, = -2Cm(M.~{S,lOg~ - S,(8,-82)} (2.27) - [ - r - ]Bz = -2Cm(M.n) szlogr~ - sx(8\-82) . (2.28) D~ngthuc(2.27) va (2.28)bi€u di€n choh;tctu cuaffiQtdelihyptu tich, chungcoth€ duQCsadlJngN l~nd€ tinhtminh;tctucuaffiQtv~th€ hlnhlangtrlJ N-dlnh N B = I(rn\x+kB\z)'i=\ (2.29) J2tLlpL tJiUL thLJ-eu] 73 q P4n '7{JoOi(](J,iYL vai B1xva BIz la cacthanhph~ncuaB gayra bdi m~t1.Tru'ongtoanph~ndu'Qc tinhbdi N ~T = I(FxBlx +FzBlz)'1=1 (2.30) vai Fx va Fz la thanhph~ntheohu'angx, z cuatru'ongxungquanb,khangbi nhi~ulo<;ln.Ma hlnhhaichi~ud~dangxaydvnghdnnhi~usovai mahlnhba chi~u,chungthu'ongdu'QCsadl;lngm6ikhidi~uki~ndiav~tly chophep. 2.2. Phuongphap nghjch Trongph~ntru'ac,taco cacquailh~giuangu6nva tru'ongthe'sinhrabdi ngu6n.Vdi R la vungchuangu6n,rex, y, z) la di~mquailsatd~tngoaiR, Q (x', y', z') la di~mla"ytichphann~mtrongR, r la vec td du'Qcdinhhu'angtu Q Wi P. Ta co tru'ongha"pd~ntheophu'dngth~ngdung g(P) = J z-z' - y p(Q)~dv, R r (2.31) tutru'ongtheophu'dngth~ngdung Bz(P) = 8 J 1 - Cm- M(Q).V -dv8z Q rR = CmfMr~Q)[3;(Z-ZI)-rkJdv, R (2.32) dQdi thu'ongtoanph~nxa"p Xl - J 1 ~T(P) = -CmF.VpM(Q)VQ-dv R r = C fM(Q) [3(F.;);- FJdv,m r3R (2.33) .l2tltPl lfflfil tJllJP ilj 74 q ~l '7fJLJmf'J1Jtml trongd6,p(Q)vaM(Q) l~nluQtla phanb6m~tde)vade)tli'h6acuangu6n.Vec todonvi F lahuangcuatli'truongkhongbinhi~u,vectodonvi k duQcdinh huangth~ngdung.H~so"y vaCmlacach~ngso"dfid~c~ptrongchuang1. Phuongtrlnh(2.32)va(2.33)c6dc.lUgchungla fer)=fs(Q).G(P,Q)dv. R (2.34) Chungta sexemxetdangdi~udinhhuangcuade)tli'h6atrongme)twi truonghQp. Gia sU'de)tli'h6aduQchoantoancamungboi truongxungquanh. Lucd6,de)tli'h6aseg~nnhutheome)thuangduynha"tne-ude)camlad~nghuang va khonggiantli'dangxetkhongquaIOn.Trongnhfi'ngtruonghQpnay,trong (2.32)va (2.33),tac6th€ d~tM(Q)=M(Q)M vachuy€nvectodonvi vaotrong da"ungo~c.Cacphuongtrlnhtli'(2.31)de-n(2.33)c6dc.lUgt6ngquat fer) =fs(Q).\V(P,Q)dv, R (2.35) trongd6fer) la giatri truongthe-tC;liP, s(Q)motame)tdC;liuQngv~tly (phanb6 m~tde)ho~cde)tli'h6a)tC;liQ. Phuongtrlnh(2.35)la me)tphuongtrlnhFredholm, trongd6\V(P,Q)vaG(P,Q)duQCgQilacachamGreen. Phuongtrlnh(2.35)th€ hi~nra slj khacnhaucuaphuongphapthu~nva phuongphapnghich.Pheptinhloanthu~n h~mml;lcdichtinhloanfer) tli'cac hams(Q),\V(P,Q)vavungngu6nR chotruac.Gia tri fer)duQchoanloanduQC xacdinhtUcacthongtinv~hams(Q),\V(P,Q)va vungngu6nR. N6i khacdi, phuongphapthu~nc6 duynha"tnghi~m.Tuy nhien,c~nchuy r~ngm~cdu phudngphapthu~nluauchoduynha"tnghi~mv~m~tloanhQC,nhungmohlnh cacngu6ntli'hayngu6nha"pd~nxaydljngtheophuongphapthu~nthlkhongduy nha"t.NguQcIC;li,trongphuongphapnghich,hamfer) chotruacboi cacphepdo J2tl4rl luin flute uj 75 q v&l 7f5omrn.Juirl d(,lctrongtht!cte',tudo tatlms(Q)hayR. BfLitmlnxacdinhs(Q)la bili loan ngl1(Jctuytntinh,trongkhibaitoanxacdinhR labililoanngl1(Jcphituytn. Xet bai toantuye'ntinh.D~ngthuc (2.35)coth€ vie'tl(,lidu'aid(,lngma tr~n N fj =:L>j\Vjj, j=l i =1,2,...,L (2.36) Khi L>N, ngu'oitadungphu'ongphapblnhphu'ongt6i thi€u d€ tlm N gia tri cuaSj' Di~unaykhongdonghlndokhokhand:1utieng~pphaila nghi~mcua bai toanthu'ongkhongduynhc1t.Ngayca khi ta bie'tf(P) chinhxac,ta clIng khongth€ xacdinhnghi~ms(Q)mQtcachduynhc1t.St!duynhc1tcuabaitoanco th€ du'Qcxacdinhbdi st!t6nt(,lihaykhongnghi~mkhongt:1mthu'onga(Q) trong Ja(Q)\V(P,Q)dv= O. R Khi phu'ongtrlnhnayconghi~mkhongt:1mthu'ong,thlnghi~ms(Q),ne'u co, la khongduynhc1t.Lop tc1tca a(Q)du'QcgQila nhancua\V(P,Q) va vung ngu6nR. D€ kh3:cphl;!cst!khongduynhc1tnghi~m,ngu'oitathu'ongtlmcachdongian hoacac gia thuye'tv~ngu6n.Ch~ngh(,lnta co th€ gia sa dQtu hoacuangu6nla d~uho~cngu6ndu'QcmdrQngvoh(,lntheomQthu'ang.Chuydingnhil'nggiadinh nay thu'ongcho phepgiambot s61u'Qngcuacac nghi~md€ It!achQn.MQtcach tie'pc~nkhacla HmcachxacdinhmQtsO'd~cdi€m, khiac(,lnhcuangu6nchung chomQinghi~m.Ch~ngh(,lnhu'HmcachxacdinhdQsaut6idacuangu6n. 2.2.1. Bai tminngu'Qctuye'ntinh Theo (2.35),tru'onghc1pd~nhay ttttru'ongla phl;!thuQctuye'ntinhvaom~t ""21tiJ-fILmYL thf!£ .Ill 76 g ~L 76o.itifllJuirL dQhaydQtuh6avadod6,baitmlnxacdinhphanb6m~tdQhaydQtuh6atudfi' ki~ncuatnI'ongha'pd~nhaytu tru'ongsinhra boi ngu6ndu'<jcgQiIa bai toan ngu'<jctuye'ntinh. Phu'dngtrlnh(2.4)matathanhph~ntheophu'dngth~ngdungcuatru'ongha'p d~ncuamQtv~tth€ voim~tdQd€u. Ne'uhlnhd(;lngcuav~th€ du'<jcbie'thltich phantrong(2.4)c6 th€ du'<jctinhboi phu'dngphaptie'n0 ph~ntru'oc.DQ di thu'ongha'pd~ndu'<jcdot(;liN di€mroinhaula gi =P\Vj i =I,2,...,N , trongd6 h~ngsO'P c6 th€ du'<jcxac dinhboi pheph6i guytuye'ntinhddngian. (xem[1]). Chungta c6 th€ xetbai toanphuct(;lphdnb~ngcachchiav~tth€ thanhcac ph~nnhohdnva sadl:mgphu'dngphapblnhphu'dngt6i thi€u d€ tinhtoanphanb6 m~tdQcuatungph~n.Cachtie'pc~nnayclingdu'<jcxemxettrongph~nsaucho tru'ongtutru'ong. 2.2.1.1. D{)tit'hoacuam{)tfing TrongmQts6mahlnhtlmdQtuh6acuadaybi€n b~ngphu'dngphapngu'<jc, Bot,BottvaHutton,EmiliavaBodvarsson(xem[1])giadinhr~ngdQtuh6ala haichi€u, trongd6mahlnhngu6ntunaybaag6mmQtdaycaca dagiac(hlnh 2.13),voicaca du'QCs~pxe'pl(;lisaDchodinhcuam6ia tu'dngungvoidinhcua , tangtu. Trong (2.36),dQdi thu'ongtru'ongtoanph~nt(;lidi€m i du'Qcvie'tl(;lithanh N ~Tj = LMj\Vjj j=l i =I,2,...,L, (2.37) trongd6Mj la dQIOncuadQtuh6aaj, \Vjjla dQdi thu'ongtru'ongtoanph~nt(;li J2tt4rt tJiit't Ihf!£ uJ 77 g PAn ';/{;tffli(}{J,.tin di~mi gayraboi0j voidOtithoadonvi.vt tniicua (2.37)baag6mto~mbOcac d<;lilU<;5ngdfidodU<;5c.Ma tr~n\!Iijcoth~du<;5cHnhtoanvoi (2.30).Tit do,chIco duynha'tmOtd<;lilu<;5ngchuaxacdinhtrong(2.37)la N giatri cuadOtithoa,va n€u N <L, nhii'ngd<;lilU<;5ngayco th~du<;5ctinhb~ngphuongphapblnhphuong t6i thi~u(xem[1]). .6Ti ...... Hinh 2.13: Mo hinhnguf/caatangtadaybiin.Dr)dj thuimgtruimg formphdn,11;duf/cdophia trenhangcaacac0, miJrr)ngvoht;mvuonggac wJi m{itgilly.MtJi 0 La tahaadiu vaidr)tahaaMj. Voi mOtdOSailchotruoccuat~ngtit,BottvaHutton(xem[1])khaosattinh 6ndinhcuaphuongphapnghich.Nhuminhhatronghlnh2.14,hd~~Ti=O0 kh~pnoi,tritt<;linhii'ngdi~mdon,noi~Ti=1nTvadung(2.37)d~tinhdOtithoa. ... «1) 1- 0 -. --- 0 [IIIITIII1TTT I I I I IU.l.lITilIIIII]IITIIJ = 2 "V II "" J (b) s= ~ c£ .!2 .~ 0 a:;c:: OJ) ~ 0/\-\/=3 " -'~i~----- -3 Hinh 2.14: nghifmkhongtJndjnhciladr)tahaacilatangta (a) dr)dj thuimgtruimg loan phdn Lakhong tra tc;zicac diim d(/fl. Tdng Latgp h(fpcac 0 vai dr)rr)ng W va d{ittc;zi dr)sauD. Dr)daycilatangLa2W. (b) Ke'tquaciladr)tahaakhiD/W=3. R6rang,mOtnghi~mth\fcchoMj gayradOdi thuongd~cbi~tnaylaxa'pXl J21u!-1'LtJiiI't Ihq.£ ui 78 grAn 7!fJoiIirmuHt khongvoi mQij. D~thaythe'hQtimnghi~mdQtuhoakhongth\ickhacnhau (hinh2.14)valu'Qngbie'nd6iph\!thuQcvao dQsaucuat~ngvadQrQngcuacac 0 (hinh2.15). BottvaHuttonducke'theokinhnghi~mr~ngne'udQrQngcua0nhohondQ mQthayhail~nsovoidQsaucuat~ng,thinghi~mngu'Qccua (2.37)1akhong6n dinh. J§- .~3 - ::;;N '::::: 2 .. :!::! ~ ~ i 1- J2o~. :! :." D/W Rink 2.15:Duongcongbiiu diJn chocaegiatrj thudu(1cilacaethinghi~m otaiJ hinh2.15.Th1!chi~nbiJi BottvaHutton[38J, Ly do cua tinh khong6n dinh nay co th~tha'ydu'oidc,mgd<;lis6. Vie't l<;li (2.38) M6i cQtcuamatr~n\Vij bi~udiencho dQdi thu'ongtru'ongtoanph~ndQc theomQtprofintrenmQt0 don,la 0j. Ne'udQrQngcua0 la nhosovoi dQsau,thi profintren0donj sera'tgi6ngvoiprofintren0j+1ho~c0j-l. Noi cachkhac,dQ rQng0 nhothi caccN g~nnhaucuamatr~n\Vijla g~ngi6ngnhau.Haytheo (2.37) d d<;lngma trn, ta co - T1 \VII \VII ... \VII M1 T2 \VII \V12 ... \VIN M2 T3 = \V21 \V22 ... \V2N M3 TL \VLJ \Vu ... \VLN MN j;!1lJ/1LOii1L Ihf!£ uj 79 q,d,L 760-m{J{J,mL ngonngud£.lis6thlmatr~nla xffu.:£)6hi6udu'<;5csv anhhu'angcuatru'ongh<;5p nayde'nnghi~mMj, taxetmQttru'ongh<;5pdongiancua (2.38), Zl =al1X + a12Y, Zz=aZlx+allY, ~ trongd6Zl,zzlad£.liu'<;5ngdodu'<;5c,al1,a12,azJ,azzlacacd£.liu'<;5ngdffbie'tvax,y lacacgiatric~nxacdinh.Nhu'chiratronghlnh2.16,haid~ngthlicnayxacdinh haidu'ongth~ngtrongm~tph~ngx,y,voi nghi~m(xo,Yo)chinhla tQadQgiao di6mcuachung. ( _\. \.'"I (0) 'Y x x xo Hinh 2.16: Saisf)'cuaZlhoi;icZ2tamchocaeduimgthllngdickchuyfntenhoi;icxuf)'ng (a)Ntu caeduimgldn nettflOvoinhaum(Jtgacl6'n,thisai sf;'nhocuaZl vaZ2khong lamanhhuangnhi~udennghi?m.(b)Ntuhaiduimgtagdnsongsongvoinhau,thisaiso' nhocuaZlvaZ2seanhhUClngrfitl6n dennghi?m. Cacsais6trongphepdoZlva Zzgayrasvdichchuy6nsongsongcuacac du'ongth~ngbi6udi€n. Ne'uhai du'ongth~ngt£.lOvoi nhaumQtg6cIOn(hlnh 2.16(a» thl sv dichchuy6ntrenkh6nganhhu'angnhi~ude'nvi~cxac dinh nghi~m(xo,Yo).Tuynhien,ne'uhaidu'ongth~ngla g~nsongsongvoinhau(hlnh 2.16(b»sais6dlinhocuaZlhayZzsegayrasail~chdangk6trongvi~cxacdinh nghi~m(xo,Yo),vanghi~mlakhong6ndinh. Hai du'ongse g~nsongsongvoi nhaune'uall ~ a21 . Du'oid£.lngt~ngtli' a12 a22 J2tUP'Loiit'Lfhf!£ ul 80 goPAn7txfflirmw.n (hlnh2.13)tlnhtr~ngnayseKayrane'ugiatri tru'ongt~idi€m i gayrabdi0j la g~ngi6ngvoigiatrjtru'ongt~idi€m i gayrabdi0 0+1)vane'ugiatrj tru'ongt~i di€m i gayrabdi0j lag~ngi6ngvoigiatritru'ongt~idi€m (i+l) gayrabdi0j. Phu'ongtrlnh(2.38)chila tru'ongh<JpmdrQngN chi~ucuatru'ongh<Jpkhao satneutren.Hangva CQtcua\!fijt~othanhcachamdutrOll.Tu do, cacpheptinh thu~nchocac~Tj tuMj choke'tquala mN hamtrolltrongkhi cacpheptinh ngu'<Jcl~ithlkhong.Honfilla,dQsauhoncuat~ngtucom6ilienh~voidQrQng cuaO.Ne'udQrQngcua0 la ra'tnhosovoidQsaucuat~ng,matr~n\!fijtrdthanh xa'uvamQtslj thayd6inhocua~TisegayramQtsais61Ontrongcactinhroan Mj. 2.2.1.2. Xacdinhhu'ongcuade)tithoa De)tITboa d~u Trongnhi~utru'ongh<Jpdiacha'tquantn;mg,ngu'oitacoth€ giasudingv~t th€ la tuhoad~umalingdl;lngthu'ong~pcuabairoanngu'<Jcd~ngnay13bai roanxacdinhdQtuhoacuamli bi€n. Khi do,cacphepdo d~cv~dQsauva dQdi thu'ongtru'ongroan ph~nthu'ongdu'<Jcthljc hi~ntren m~tbi€n. Nui bi€n nay thu'ongdu'<Jcgiasula tuhoad~uvadu'<JCgiOih~nbdihait~ng: t~ngtrenvat~ng du'oi. Phu'ongtrlnh(2.33)du'<Jcvie'tl~id~ng - R a 1 a 1 a 1 ] ~T(P)=-CmF.Vp Mx(Q)--+M y(Q)--+Mz(Q)-- dv. ax'r ay'r az'r Khi M lad~u,taco ~T(P) = M,[-CJVP 1:'1V]... .I211fjrLtJiiI'L tJu,£ uj 81 q ni.n 7J{;oQ1(J{J,jyL + M,[ -CJ\7p I~'1v] + Mz [ -CmF.Vpf~~v ]8z'rR ilT(P) = Mx~x(P) + MA/P) + Mz~z(P) ~ (2.39) voi ~X, ~Y' ~z la s6h,;mgtichphantuongling,la'ytheod~nghinhhQccuanui bi~n.Gia sa dQdi thuongcua truongto~mphftnduQCdo t~iN vi tri khacnhau, (2.39)co th~duQcvie'tl~i(j d~ngmatr~n ilT, ilT2 ~lx ~2x ~lY ~2Y ~lz [ Mx ] ~2z M. y I Mz~Nz (2.40)= ilTN ~Nx ~NY Ba CQtcuamatr~n ~jjtrong (2.40)bi~udi€n cho dQdi thuongtruongtoaD phftnt~icacvi tri khacnhau,giasadQtuhoadonvi theocachuang x, y, z tuong ling.M6i thanhphftncuamatr~nco th~duQctinhb~ngcachsad\lngphuong phapthu~n,nhungd~lamduQcdi€u naydoihoiphaicoslfxa'pxi hinhd~ngv~t th~theocacmahinhdongian,ch~ngh~nnhula mQtt~pcackh6ihQpchii'nh~t, mQtch6ngcacbanmong,haymQtch6ngcactftng.Lucdo, (2.40)choN phuong trinhtheo3 bie'nmatagQila bathanhphftncuadQtuhoavachungcoth~duQc tinhb~ngky thu~tbinhphuongt6i thi~u(xem[1]);tlic la, timMx,My,Mz saD cho d?i IU<;1ng N 2 "'" ' 2 E = L.i(ilTj -ilTj) i=l la nhonha't,voi ilT' j, i=1,2, ...,N la cacgiatridQdi thuongdoduQc.Honnii'a, ~I2Il1P'LmYL fhf!£ vj 82 q nin 7It5oidrmum b~ngcachtrudi truongdiaphuongF(P), (2.39)trdthanh ~T(P) = Mx~x(P)+ My~/P) + Mz~z(P)- F(P) Cac d<:;tiluQngkhacnhauco th€ duQcdnh d€ daubgia kha nangtrungkhit giua ma hlnh don gian nay voi du li~u do duQc. Truong ph~n du, e = ~T - ~T' vathamsf)I I I N LI~Til r - i=l- N Lied i=l duQcdungd€ daubgiasvsaikhaccuamahlnh. D()tit boakhongd~u Trongth1!ct€, conhi~uIi dod€ dftnd€n d('>tuhoala khangd~u.Nui bi€n coth€ duQchlnhthanhlientl,lcclingnhucacthanhph~nhoahQCvakhoangcha't khangd6ngnha'tcuanuibi€n gayranhungthayd6idangk€ cuatruongtu. M('>tsf)nghiencUudia ly chophepta phanngu6ntu thanhcacph~nroi nhau,m6iph~nla d('>tuhoad~u.Khi do,nguoitavftncoth€ dungphuongphap blnhphuongtf)ithi€u d€ Hmd('>tuhoatrongm6iph~n. M('>tcachti€p c~nkhacduQcd~xua'tbdi Parkeret al. (xem[1])b~ngcach thayvi~cxemxetd('>tuhoathanhhaithanhph~nd6ngnha'tvakhangd6ngnha't, saDcho thanh ph~nkhang d6ng nha'tnh6 d€n milc co th€. Y tudngnay duQctom Hitnhusail D('>dithuongtruongtoanph~ncuanuibi€n duQc hobdi (2.33) - f 1 ~T(P) = -CmF.Vp M(Q),VQ-dv R r Duoi d<:;tngdongianhon J2um'LtJiULlime uj. " . 83 g uin 7HJo-ai(llJULI'L ~Ti = - JM(Q).Oj(Q)dv R (i =1,2,..., L), (2.41) trongdochIsfSi d~chIcacdi~mquailsatroi nhau.HamM(Q) trongdAngthuc trenla mQthamvecto.T~pta'tCelcachamtithoanayt~onenmQtkhonggian Hilbertvoh~nchi€u.HamGreenGj(Q)trong(2.41)gifSngvoivectotQadQtrong khonggian3 chi€u, va duQCgQila hamtQadQ.Tich trongcuahaithanhph~n A(Q)vaB(Q)cuakhonggianHilbertnaydu<Jchobdi (A,B) = JA(Q).B(Q)dv R tuongtt,I'nhutichvohuangcuahaivecto.Tit do, (2.41)du<Jcvie"tduoid~ngtich vohuang ~Tj = (M,OJ, (2.42) va ~Tj trdthanhhamtuye"ntinhcuaM(Q). "DQIon"cuamQtthanhph~nduQc dobdichuffn 1 IIAII = (A,A)2 1 = [pAl'dvr vadQsaikhaccuahaithanhph~nduQc hobdi IIA- BII. TrongphuongphapcuaParkeret aI.,dQtit hoaM(Q) trong (2.41)duQc phantichthanhhaithanhph~nd6ngnha'tvakhongd6ngnha't, M(Q) = Mo + MN(Q), voiMola mQtvectoh~ngsfS.Phuongtrlnh(2.42)trdthanh ~Tj = (Mo,GJ + (MN,GJ (i=I,2,...,L) (2.43) Thanhph~nM(Q) cogiatrjnhonha'tla MN(Q)la dQtithoag~nd€u voiba't kyMochotruoc. J2Wp'Lo£ULl~ uj 84 g ,,4,'L7I{joOit'IlhtiIL Xettruongh<jpModffbie't.Lucdo,thanhph~nMN(Q)cochufinnhonha'tvai di~uki~nthoa (2.43)vaimii=I,..,L.,khita'tcacacd(,\ilu<jngd~ubie'tngo(,\itru MN(Q).Parkeretal. chIrar~ngdQtuhoanaycoth~du<jcbi~udi€n b~ngkhai tri~nhamGreen, L MN(Q) = IajG/Q). j=1 (2.44) Ta't ca cach~sO'khaitri~naj 0=1,2,...,L) d~udu<jcxacdinh.The' (2.44) VaG(2.43),tadu<jc L ~Ti = (Mo,G)+ Ia/Gj,G) j=1 L = (Mo G. ) + "aT.., I L.J J 1J j=1 (2.45) tich trong lij duila matr~nGram.Ta'tca cacd(,\ilu<jngtrong (2.45)d~u dffxacdinh, L h~sO'khaitri~nIa xacdinhmQtcachduynha'tbdi Cj lakhongsuy bie'n.Vec toh~ngMola chuaxacdinhtronglingdl;lngdQtuhoanuibi~n,vala d(,\ilu<jngcgntinh. Bai toand~trala : TIm Mo vaMN(Q)d~ MN(Q)cochufin nhonha't.MothuQckhonggiancon3 chi~ucuakhonggianHilbert,va co th~ du<jcvie'tduaid,;mgtudngtvnhu(2.44) 3 Mo = IPkXk k=1 (2.46) vai Xk, k = 1,2,3 ladQtuhoaddnvi theo3huangtrvcgiao.The't6ngtrenVaG (2.45)tadu<jc 3 L ~Ti = IPk(Xk,G) + Ialij k=1 j=1 (2.47) Tu do,chungtasetlmcacthamsO'PI. P2,P3,aI. a2,..., aL d~ ,.e,'.tP'LomL 1Ju,.el{j 85 q uin ';J()OO1fllhtin I ( L L J 2 11M-Moll = ~ajGj' ~akGk (2.48) la nhonha't,vaigiathuye'trangbuQcla (2.47).Ta'tcacacd<;lilu<;1ngtrong(2.47) va(2.48)d~udffxacdinh,tnIL+3thamsf)~J,~2,~3,aJ, a2,..., aL.Parkeretal. [210] dua ra cach xac dinh nhungtham sf) tren b~ngphuongphap nhan tU' Lagrange.Vai (L+3)thamsf)nay,huangtuh6ad~u,du<;1cchobdi (2.46).ChuY dog, ne'udQtu h6a thvcsv d~u, (2.48)tri<$ttieu, (2.47) gian lu<;1cthanhd<;lng (2.39). 2.2.2. Bili toaDngtiqcphituye'n. Truongthe'd ve'traicua(2.35)la mQthamtuye'ntinhcuasvphanbf)khf)i lu<;1nghaycha'tu.Ch~ngh<;ln,Svga'pdoidQIOncuadQtuh6a,lamga'pdoidQ IOn cuadQdi thuongtruongtoanphgn,trongkhiSvga'pbadQtuh6aselamga'p badQIOncuadQdi thuongtru'ongtoanphgn.MQtcacht6ngquat,mQth<$dU<;1c gQila tuye'ntinhne'un6thoaman:Ne'ufl(P) la truonggayrabdisvphanphf)i ngu6nSI(Q), fir) la tru'ongayrabdisvphanphf)ingu6nS2(Q),thltruonggay rabdisvphanphf)iasl(Q)+bs2(Q)chidongianla truongafl(P)+bf2(P).(a,b la h~ngsf)) Di~unayse khongdungchocacthongsf)khaccuangu6n.Truongthe' khonglahamtuye'ntinhcuadQsail,dQday,hayhlnhd<;lngcuangu6n.Ta'tcacac thamsf)trenduCJchuatrong\V(P,Q)vatronggiOih<;lncuatichphantrenth€ tich R d (2.35).PhuongphapngU<;1cduaracachdnhtoancacthams6tren,vadU<;1c gQila cacphuongphapkhongtuye'ndoh,nhungtrongthvcte'cacphuongphap khongtuye'ntinhd§nde'nsvdongianh6acacgiathuye't,madi~ud6l<;liamcho baitoantrdnentuye'ntinh. J2tltpL om'LIhLf-R-uj 86 g ~L ';J(Joai{J{hmL 2.2.2.1. ffinh d~ngngu6n Caephu'dngphapl~p Nhudad~c~pd ph~ntruac,mohinhphuongphapthu~nduQcxaydl!ng theo3 buacxU'ly. DQdi thuongduQctinhtumohinhvaduQCsosanhvOidQdi thuongquansatduQC.Mo hinhseduQcdi~uchinhvahoanthi~ntheo~sl!sosanh. BabuacxU'ly seli[Lpdi li[Lpl~ichode'nkhimohinhmaithoamanyellc~udi[Ltfa. ChungtasexemxetchungtheophuongphapnguQc. c; ---r--r, CO]J . . -, . L_J .: " II - '.', ',' ',_11\ Hinh 2.17..Ma hinhthittdi~nngangcuaLuuvifCtramtich,trongphu(jflgphapcuaBatt. Luu vifcdu(Jcgia thuyttLamiJrQngvah(;lnvadu(Jcchia thanhcackhoihQpchilnh(jt, mJi khathQptrenmQttru(mgdilm. MQtvi dl,1duQcduarabdiBottd~danhgiad~ngthie'tdi~nngangcuahtu vl!ctr~mtich.Trangphuongphapnay,htuvl!cduQcgiathuye'tlamdrQngvoh~n theomQthuangvacom~tdQd~u.Ltfuvl!cduQcchiathanhN kh6ihQpchii'nh~t, mdrQngvoh~ntheohuangsongsongvai Itfuvl!Cva mdrQngtheochi~uSailtj, j=1,2, ...,N (hinh2.17).Chi co N truongdi~mgi (i=1,2, ...,N) dctheocacprafin vuonggocvai Itfuvl!cduQcxet,va m6i truongdi~mco tamphiatrenm6ikh6i. Gia thuye'tm6ikh6ila mQtphie'nvoh~ntheochi~ungang.Truongha'pdfincua phie'nvoh~nduQcchobdi g = 21typtk, vai p,t la m~tdQvadQdaycuaphie'nvoh~n,k lavectodonvi theohuang J2tltPL miLLlltf!£ £{j 87 g 1'&11';J{)oiLi(/(j,nLL th£ngdung.Luc d6,dQdaycuam6iphie'nvoh(;lndu<;5cdoboi gj (I) - - , tj - 27ty~p j =1,2, ..., N. ChI s6trenchIs61:1nl~p.Nen phuongphap3buocsed~nWi Yi<%cchlnhsaa dQdaycuakh6i.Cacbuocdu<;5ctie'nhanhnhusan,yoi k d€ chI s61:1nJ~p 1. Truong gjk) du<;5ctinht(;lim6idi€m quailsatgayra boi tit.cacackh6i, giasar~ngdQdaydu<;5ctinhtheophepl~pke'truoc. 2. Ph:1ndu gj - gjk) du<;5ctinh t(;lim6i di€m quail sat. 3. Sv xip Xl phie'nYOh(;ln du<;5csa d\lllg cho toi khi uoc lU<;5ngdu<;5cdQ day maioSv hi<%uchlnhcho m6i kh6i du<;5ctinh duOigia thuye'tdng m6ikh6ila mOtphie'nYOh(;lnyoimOtdQdaytinhtheoph:1ndu,tucla dQdaymoicuaphie'nYOh(;lndu<;5ctinhtheocongthuc ( (k)) t~k+l)= gj - gj +t~k) J 27ty~p J Ba buoctrendu<;5cl~pdi l~pl(;lichode'nkhi mohlnhmoi la thoamanyell c:1u. Cordellva Hendersondahoanthi<%nphuongphapnayb~ngcachsad\lllg dli li<%ud <;5cdo(ho~cnQisuy) trenmOtm(;lngluoicackh6ichlinh~t,lucd6cac ngu6ndu<;5cxemxettrongkhonggian3 chi€u. Cac ngu6ndu<;5cmohlnhnhumOt b6 cackh6i chli nh~t,m6i kh6i c6 truonghip d~nrieng(hlnh2.18).DQ daycua kh6i tj, j =1,2, ...,N du<;5cdinhnghIac6 lien quailWi mOtb€ m~tthamchie'u,d6 la b€ m~tc6 th€ du<;5cxemch£ngh(;lnnhum~tdlnh,ho~cm~tday cuatoanbQ cac kh6i. Tu'ongtv nhuphuongphapBott,dQday kh6i band:1udu<;5cuoc lu<;5ng b~ngcachgiathuye'tm6ikh6i la mOtphie'nYOh(;ln.Do d6,ty s6 J2tltP'L (JaIL /hf!£ uj 88 g uin 360mrmuitt t~k+l) ~ --L- = (k) t~k) gjJ du'Qcsu d\lngd€ di€u chInhl<;LidQday cuaphie'nvo h<;Lnxa'pxl. Phu'ongphap3 bu'ocsetinhtoan,sosanh,di€u chInht<;Lim6ibu'ocl~p. R.:.:::nccsurfacc; Hinh 2.18: mahinh3chducilaphU{fflgphapZ{IpdoCordellva Henderson d~xuat. DI) day cila khaT-phf:lthul)cvao b~m{Itthamchitu chung. TruOngh5pdJn quansatdu(Jcdo trenml)tZullichilnh{jt. M9t cachtie'pc~nhoikhacdu'Qcdu'arabdi JachensvaMoring.Gi6ngnhu' haiphu'ongphaptren,hQu'oclu'Qngcacd<;Lnghtuvf!cvoim~tdQtr~mtichthffp, nhu'nghQcotinhde'nkhanangm~td9thayd6icuat~ngn€n. Phu'ongphapnay chiaphepdotru'ongha'pd~nthanhhaiph~n: thanhph~ngayrabdibanthanhill vf!cvathanhph~ngayra sf!bie'nd6im~tdQcuat~ngn€n. Xet g bi€u di~ncho tru'onghffpd~ndu'Qcquailsat,saukhidalo<;Libocactru'ongdiaphu'ong.D~tg= gb+gd,vOigbla d9di thu'onghffpd~ngayrabdit~ngn€n vagdla dQdi thu'ong gayrabdicaclOptrffmtichm~td9tha'p.Phu'ongphapdu'Qctie'nhanhtheocac bu'ocsau: 1. Bu'ocl~pthunha'tgiasur~nggbdu'QcxacdinhChId nhungvi tridinh xutrent~ngcosdtr6ilenvatinhtoanmQtb€ m~tnhanvoinhungdu J.!.tU!1'LtJiit'L fJu,£ uj 89 grAn '3tJoa1rmum li<%unay,nhuchIrabaiduongnetg~ch(hlnh2.19).Dieunayl~pnen xa'pXlthlinha'tgb(l)chotruongha'pd~ncuat~ngcosagb. 2. gdduQCxa'pxIl~n thlinha'tbai pheptrUcuatruongha'pd~nquailsatg cho gb(l).Ph~ndu mOi gd(l)duQCsa d\lngd~Omxa'pXl thli nha'tcho chieusaucuat~ngcosasad\lngphepxa'pXlphi€n banvQh~ntuong ttfnhuphuongphapBott. 3. Hi<%ulingha'pd~ncuahillvtfccoth~duQctinhbaimQts6phuongphap khacnhau.La'ytruongha'pd~nt~ngcdsatrltchok€t quaOmduQccho taphepxa'pXlk€ ti€p cuatruongha'pd~nt~ngcdsa gb(2). Iteration1 \ /\ CdGravityV Observ Iteration1 , / I / , Hinh 2.19..ChiaphdnducuatntilnghapdJn thanhhaiphdn,thanhphdngayra biJisf!thayd6imtJtdQtrongtangcosiJvathanhphdngayrabiJihtuvf!C.Cacnot chambiiu thichophepdotrenphdntr8ileucuatangcosiJ,cachlnhtrimlanlim trenbi m(ittramrich. Ba buGCnayduQcl~pdi l~pl~ichod€n khi nghi<%mla thoamanyell c~ud~t fa. H<%qua la : OmduQChlnhd~ngcuaIttuvtfctr~mtichm~tdQtha'pva truong ha'pd~ncuat~ngcosakhongchillanhhuangcuaIttuvtfc. .1211f/l'LomL tluJ-e .uj 90 g"&L dVomrnJuuL Tuye'ntinhhoahili tminphituye'n. M~cdli tntongthe'phl;!thuOcphi tuye'nvao cacthongsO'tv nhiencua ngu6n,nhungSvphl;!thuOcnayla "gffntuye'ntinh"khichiconhii'ngthayd6inho trencacthongsO'.Ch~ngh(;ln,truongthe'cuamOthlnhkhO'idagiacphil thuOcvao tQadOcuacacdinhcuadagiactheohamarctangva logarit(xem (2.23),(2.27), (2.28)),tucla truongthe'lamQthamphituye'ncuatQadOcacdinhdagiac,mO'i quailht%phituye'n aycoth€ lamchotuye'ntinhb~ngcachthayd6ichutit cac thongsO'.Vi dl;!truongha'pd~nva tITtruonggay ra bdi mOtt~pcac hlnhkhO'ida giac co th€ du<;1ckhai tri€n theochu6iTaylor dva tren thay d6i tQadOcua da giac.Ne'usVthayb6i tQadOla nho,thlchu6iTaylorco th€ du<;1crutgQn,ham phl;!thuOcvaosvthayd6inaydodotrdnentuye'ntinh.MOtthu~toandu<;1cd€ xua'tnh~mxacdinht~pcachlnhkhO'idagiacto'tnha'tlingvdimOtdOdi thuong chotrudc,vdi cacd(;lngthie'tdit%ngangcuacachlnhkhO'idu<;1cchinhl~pbdi IU<;1nghotrongphuongphapblnhphuongto'ithi€u tuye'ntinh. Cachtie'pc~nnaydU<;1cmotatheocacd(;lngkhacnhauchocadOdi thuong ha'pd~n(xemCorbato,AI-Chalabiva Colestrong[1])va dOdi thuongtIT(xem Johnson,McGrathva Hood,Rao va Babutrong[1]).Cac d€ c~psail tomt~t phuongphapd€ xua'tbdiJohnson. Phuongtrlnh (2.2) va (2.30)cling ca'pthanhphffnha'pd~ntheo phuong th~ngdungvadOdi thuongtrtiongloanphffntuongling cuamOthlnhkhO'ivo h(;ln mdrOngtheomOthuangvdim~tdOhaydOtIThoad€u vacothie'tdit%nladagiac N-dinh(hlnh2.20).D~tAi la mOttrongL phepdoroi r(;lCcuadOdi thuongha'p d~nhaydOdithuongtIT. Ai = A(x~,ZI' X2'Z2'..., XN' ZN'Xi' z) Jl.tuJ-rLoibL I~ uj 91 g nhL JfJoai (J{JuUL = A(Xi,Zi'W) i =1,2,...,L vai (Xi,Zj)la vi tricuaphepdothui. (x\' Z'i) la tQadQcuaN dlnhcuadagiac, vaduQckyhi~ubdimQtmang2Nchi€u w.Honnlia,coth~xemcacdlnhcuada giaccoth~dichchuy~ntheohuangtuyy. ~B ~B .\" x .: Rink 2.20.. Xapxl cilamQtngu6nhaichdu blingmQthinhkhatmiJrQngvah{lnvcJithief difn nganghinhdagiac. Vecf{JM va BfahinhchituciladQtah6avatatnt(rngxung quanhfenm{itph/lngx, z. GQiAi va Ai tuongungIa dQdithuongquailsatvadQdithuongdoduQct~i mQtdi~mquailsat.ChungtaseHmvectoWsaDchoblnhphuongsais6 L 2 E2 = 2:[Ai -Ai(W)] i=l (2.49) la nhonha't.Bdi Ai la mQthamphi tuye'ncuaw, chungtakhongth~sadvng phuongphapblnhphuongt6ithi~u.Thaythe',chungtased6icacthanhph§ncua WmQtluQngnho.Ne'uthayd6itrongWla nho,Ai sela hamg§ntuye'ntinhd6i vai nhlingsl,thayd6inay. Chu6iTaylorla congcvthichhQpd~thl,tchi~ncongvi~ctren.Vi dV,giatri cuahamf(x,y)coth~duQcngo~isuytheo(x+~x,y+~y)bdichu6i .£tapLoillL thf!£-uj 92 g PAn7/f;oidfJlJuin f(x +~x,y+~y) = f(x,y) + 8f(x,y)~ + 8f(x,y)~y + ... ax ay voicacs6h~ngb~ccaDhondtiQclo~ibobdi~x va~y dtiQCla'ydunho. (k) 9 - ? ( "" " ) ? Ttiongtl!,chungta xem w Ia biendiencua xI' zl' X2'Z2'..., xN' ZN 0 phepl~pthilk.Lucdo,khaitri€n chu6iTaylorcuadQdi thtiongt~idiemi la 2N 8 Ai(W(k+l») ~ Ai(W(k») + L-Ai(w(k»)~W~) m=lawm (i =1,2, ...,L) (2.50) voi ~W(k)= W(k+l)- W(k) ChuY/ r~ng A. (W(k+l») la mothamtuye'ntinhcuam m m . I . ~w~)voi m=1,2,...,2N.Tilc lachungtadiituye'ntinhhoabairoanphituye'n. Thay the'(2.50)vao (2.49)tadtiQc L [ 2N ] 2 E2 =L Ai -Ai(w(k») - L~Ai(W(k»)~W~) i=1 m=l8wm Bi€u thilc Ai(W) dii dtiQcsuyra cl;1th€ tu (2.22)va (2.30), chungtaco th€ Hm bi€u thilcd~ohamriengcuachung.Do do,chotrtiocmQtmangkhdidfiucua thams6 W(k)thl cac iin s6cfinHmtrong trenla ~Wffi(k) , m=1,2, ...,2N. B€ Hm caciin s6naysaDcho E2la nhonha't,chungtatinhcacd~ohamriengcuaE2 rheaWj,j=I,2,...,2N, vaxeth~ t [ Ai - Ai(W(k») - I:~Ai(W(k»)~W~) ] . [ ~Ai(W(k») ] =0 (j =1,2, ..., i=l m=lawm awj 2N.) ? trongdochungtal~itie'ptl;1Clo~ibocacs6h~ngb~ccaD.d d~ngmatr~ntaco 2N a. ="G .~W(k)J ~ ffiJ m m=l (2.51) J2tUP'LtnUL lluJ-e uj 93 f1,d,L 7HJoa1t1lJuin voi aj =I[Ai -Ai(W(k»)J~Ai(W(k») i=\ 8wj (2.52) Gmj =I [ ~Ai(W(k»)~Ai(W(k») ]i=\ 8wm 8wj (2.53) cacbu'ocdu'<Jctie"nhflllhnhu'sau 1. ChQn mQt t~p cac gia tri ban dffu cua tQa dQ cac dlnh W = (x\, ZIpx'z,z'z,...).Gia sar~ngtfitcacacdlnhcuadagiac1a co th6dichchuy6ntuyy. Trongthl;tcte"thl tased6hffuhe"tcacdlnh cuadagiac1ac6 dinh,chidichchuy6nmQtvai dlnhtrongquatrlnh tinhloan. 2. T' h A .( (k») , 8 A .( (k») ,.. - 1 2 L ' - 1 2 2NIn I W va - I W VOl 1- , ,..., va m- , ,..., . 8wm 3. Tinhaj va Gmjtheo(2.52)va(2.53). 4. TImmatr~n ghichdaotrong(2.51)d6tinh~W(k). 5. Bi€u chlnhWchophuh<Jpvoi yell cffud~tfa. Bu'oc2 du'<Jcl~pdi l~pl~ichode"nkhi nghi~mOmdu'<Jc1ahQitv, tucla, cho de"nkhi E2du'<Jcnit gQntOimQtmuctheoly thuye"tchIra hay~Wm,m =1,2,...,2N aunho. v~uth~compact Bfitky sl;tphanph6ikh6i1u'<Jnghaychfittii'cuatru'onghfipd~nhaytru'ong tii'tu'ongling,du'<JcOmd6thoaman(2.35).Phu'ongtrlnh(2.35)la rangbuQcchu ye"u.Chungtacoth6du'athemvaomQtvairangbuQckhac.Vi~cthemvaocac ./211ijPLtJ-ii#L1Ju,e uJ 94 g ~L ']{)oa;r1lJuYL rangbuQcdalamql'cti~uhoahaycvcd(;lihoamQtvai thuQctinhvahu'angcua .ngu6n.Ch~ngh(;ln,LasvaKubikdamatavi~clamsaod~tlmmQtv~th~vai th~tichnhonha'thoamancacrangbuQc.H9 chia ngu6nthanhN kh6ichlinh~t, haiho:)cbachi~u.llinh 2.21chotru'ongh<jp2chi~u,giclsadingm~tdQcuam6i kh6i la d6ngnha't.Cho bi~ttru'ongha'pdin dodu'<jct(;liL vi tri roi r(;lc\rangbuQc (2.35)du'Qcvi~tl(;li(j d(;lng N gj =L\VjjPj + ej, j=l (i =1,2, ...,L) (2.54) g; 7 /' --...." '" ,,'" -. ." ""-~ -. "'.-'" ""Oo-- _..- -..r_- - =+-++=--+3:- ,-J - ==£:1::~-:._- . ,. ~.:-->-+-g=t-._' ::-'-~~--::t:--,:::1:- - =i=i:::.::1=:J::j.:: p~./ ",/ Hinh 2.21..Mi~nngu8ndu(lcchiathanhcaekhatchflnh(tt,veJiso'Lu(lngcaekho'ivu(lt quaso'Lu(lngcaephepdo.CJ day,xetngu8nLahai chiiu, vai ej, i =1,2, .."L, la cacsaiso'cuam6iphepdo.N~uL >N, chungtacoth€ sad1,mgphu'dngphapblnhphu'dngt6i thi~ud~tlmm~tdQchom6ikh6i,di~u naydadu'Qcd~c~ptrongcacphgntru'achotutru'ong.Vai L <N va doihoi phgnkhackhangcuami~nngu6nnhod~nmilccoth~,mi~nla svdanhgiahlnh d~ngcuav~th~compactla thoamanL phepdotru'ongha'pdin noitren. N~ud:)t 2 { o I' Pj1m = E~O pf+E 1 (Pj=0) (pj :;t:0) - 1:::, -'-- .-4 £'UpL IJiUL IhLfl! uj 95 gnin 7tJoairnltan thlth~tich(haydi~ntich)cuav~th~duQcchobdi N p~ V = ~VlimL ~ 'E~O . 1P. +EJ= J vai~V la th~tichcuamQtkh6irieng(ho~cdi~ntichcuamQthlnhchii'nh~trieng trongtruonghQphaichi€u).Thu~tminhIedodbihoiphaiQfcti~uhaad<;liluQng vahuang N p2 L q =L 2 j + Lwie~, j=l pj +E i=l (2.55) trongdoL bi~uthucrangbuQccua (2.55),vaiEduQcchQndunhovaWi, i=1,2, ...,L lahamnhi€u. Dodo,nghi~mOmduQclac1;l'cti~uhoacath~tichcuav~th~vat6ngtrQng cuablobphuongphgndu. N giatq cuaPj coth~b5ngkhongho~ckhackhong. Caca khackhongtrongmi€n ngu6nclingcffpmQtuacluQngcuahlnhd<;lngcua v~th~compactthoamancacphepdotruonghffpdftn. 2.2.2.2. Dt)saungu6n CacphuongphapdaubgiadQsitucuangu6ntuhayngu6nhffpdftncoth~ duQcchiathanhhailOp:MQtlopphuongphapphilotichmQtdQdi thuongdon,co l~p. Lop phuongphapcon l<;liphilo dch profin cuanhi€u ngu6n.MQt vai vi d\l cuam6ilo<;liseduQcnghiencUud phgnduaiday. PhuOngphapPeters Ne'udQdi thuongtUduQCgayra bdi mQtv~tth~hai chi€u theochi€u dQc, co dQtu hoad€u theochi€u dQCva dQsitulOn,hie do dQsitucuav~tth~co th~ duQctinhxa'pXlb5ngphuongphapd6thi (hlnh2.22). J211fj.lL tJiUL thf!£- £Ii 96 grAn ';j{joQir1lJuUL Ve haidu'ongth~ngsongsong voih~s6gocb~ngYzgradientclfcd~icua dQdi thu'ong,mQtdu'ongtie'pxucvoidi€m clfc cuadQdi thu'ong,du'ongconl~i tie'pxucvoi di€m clfc ti€u cuadQdi thu'ong.Khmlngcachtheophu'ongn~m nganggiuahaidu'ongth~ngla ty l~thu~nvoidQsancuav~tth€. H~s6tyl~la 1.2chonhungv~tth€ mongva la 2.0chonhungv~tth€ day;giatq 1.6thu'ong, duQCsadl;lngchungchocacv~th€ khac. " ,,'I "x " /~I" "...r-.. " //J / / / / / / ,- / ,- //~ // / J ' / . / J / b // b / ' a // 20" ;' x =1.2 d if body very thin x =2.0 d if body very thick x =J.6 d for intermediate thickness Rinh 2.22: Minh hQachophu(JflgphdpPeters Phu'ongphapPeterdoihoinhi6uslfdongianhoacacgiathuye'tv6v~th€. Tuynhienphu'ongphapthlfchi~nd€ dang,chicc1ngia'"yvevabutchIchungtaco th€ daubgiasoluQcvanhanhchongdQsancuangu6ntrongnhi6utr~ngthaidia ch§tkhacnhau. DQsaucticd~i ChungtaxetmQtvaiphu'ongphapthu~nti~n,HnhloannhanhchongdQsan clfcd~icuacacngu6n,du'Qcphattri€n bdiSmith,Bottva Smith(xem[1]).Cac phu'ongphapnaydlfatrend~ohamb~cnh§t,b~chai,b~cbacuadQdi thu'ong h§pd§nvadQdi thu'ongtu,du'QcdodQctheoprofin(hlnh2.23).Chungd~cbi~t r§t hi~uquaVI chungkh6ngdoi hoi gia thuye'tnaov6 hlnhd~ngcuav~tth€ ngu6n. J21lf/l'L milL f~ ui 97 g Nin ';/{Jo.QirnJuuL .A(:\")",.....- \ d .::-\ (:.:, I ~ \ d:t::: I""<%.. L I dA(_:) IrL>.: '..a... --0---- Hinh 2.23.. DQsauqtc d(licilangu8ndlfatrend(lohambljcmQt,bljchai,bljcbaciladQ di thuCln.Profile A(x) bilu ddn dQdi thuClngtithaydQdi thuClnghapJan. BQ saugiai h(;lntheoSmith, Bott va Smithdu'<Jctomta:tnhu'du'Oidaycho ca dQdi thu'ongtUva dQdi thu'ongha"pdftn.Trong cac ba"td&ngthucnay A(x) bi~udi€n la profincuadQdi thu'ongha"pdftnho~cdQdi thu'ongtu.Trongtru'ong h<Jptru'ongha"pdftn,A(x)bi~udi€n thanhphftnha"pdftnth&ngdung.Trongtru'ong h<Jptu tru'ong,A(x) bi~udi€n cho thanhphftncua tu tru'ongsongsongvai ; = (;x,;y,;z)(ch&ngh(;lnA(x)la dQdi thu'ongtru'ongtoanphftn,; la vecto theohu'angcuatru'ongKungquanh),thams6dladQsaucuangu6n.Ky hi~u A =A(x), A' = dA(x) A" = d2A(x) Alii - d3A(x) dx ' dX2' - dX3 chIs6"max"d~chIgiatri IOnnha"tthudu'<JcdQctheotr\lCx. Thams6/)'wla ml'a dQIOncuaffiQtdQdi thuongd6iKung,tucla,khoangcachgiuavi tridQdi thuong d(;ltgiatri qtcd(;livavi tridQdi thu'ongd(;lt'12giatriqtcd(;li.Gia triqtcd(;licua ffi~tdQhaydQtuhoaduQckyhi~uboiPmaxhayMmax.Cach~thucsauduQcsuy ratrongh~EMU Vi) di thliunghill'dlin3chilu ,.8ujn oiULIhf!P uj 98 g,d,L 'dtJoaifllIuin TruiJngh(1pchung d :s;5.40 YPmax I A" I max d2 :s;6.26 YPmax I A"' I max TruiJngh(lpm(itd()hoimtoimduang(ho(ichoimtoimamntudauthayd6ithich h(lp) A d :s;1.5jA1' '\Ix d2 < A- -3- A" ' '\Ix E {A"<O} A d :s;0.86~ A' max d :s;2.70 YPmax I A" I max d2 :s; 3.13 YPmax I A"I I max EJi)dj thlldnghapdan2 chiiu M(itd()hoimtoimduangho(ichoimtoimam. d < A - jA1' '\Ix d2 < A- -2- A" ' '\Ix E {A"<O} d :s;0.65 Amax IA'lmax '£'upL oillL 1Ju,.euj 99 g ~L 71&a1t1lJuin d ~ f'j,w (dQdtthuangdoLxang) 81)dj thlilingtit3 chi€u Khongcosf!rangbuQcnaochodQtithoa 1-2 -2 -2- M d ~ 6.28(4rx +3ry + 3rz)21Afaxmax 1 2 -2 -2 -2 - M d ~ 9.73(3rx+2ry + 2rz)2, rxA" max DQtahoasongsongvacoClAngchdu tc;zimQidilm 1-2 -2 -2 - M d ~ 3.14(4rx+3ry + 3rz)21A'C:x 1 2 -2 -2 -2 - M d ~ 4.87(3rx+2ry + 2rz)21 rxA" max r vaM cohuangthangdang M d ~ 5.18 max IA'lmax d2 ~ 6.28 Mmax I A" I max r vaM cohuangthangdang,tc;zimQidilm M coClAngchiiu M d ~ 2.59~ IA'lmax d2 M~ 3.14 max IA"lmax 81)dj thlilingtit2 chi€u JlJltpt luU'L Ihlf1!-uj 100 gNin 7IJoairmHut Kh6ngco sZ;trangbuQcnaochodQtithoa I d ~ 8(;~+ ;~)2Mmax IA'lmax I 2 -2 -2 - M d ~ 9.42(rx + rz)2 1 m l ax A" max M songsongvacungchiiu tt;limQidilm I -2 -2 - M d ~ 4(rx + rz)2~ IA'lmax I 2 -2 -2 - M d ~ 4.7l(rx + rz)2 1 m l ax A" max PhuongtrinhEULER Cac phu'ongphapu'oc1u'<;1ngdQsaud~c~pd trenla thichh<;1pchodQdi thu'ongayrabdicacngu6ndon,col~p.MQtlOpcackythu~tkhacxemxetdQdi thu'ongha'pd§:nhaydodi thu'ongtitgayrabdinhi~ungu6ntu'ongd6idongian. Ch~ngh<;tn,phu'ongphapu'oclu'Qngvi tri cuamQtv~tth€ dongian(donqfc, lu'ongclfc,...)titmQtvaiphepdocuatru'ongtitho~ctru'onghffpd§:n,coth€ ap dl;lngVaGmQtprofindaib~ngcachchiaprofinthanhcac0 cuacacphepdolien tl;lc,m6i0 la mQtu'oc1u'<;1ngdonchovi tri ngu6n. Phu'ongtrlnhEulerd d<;tngt6ngquatdu'<;1cchobdi LVf =-nf Cachamf thoaphu'ongtrlnhEulerdu'<;1cgQila thufinnhfftb~cn,ne'uchung clingthoaLaplace,chungcoth€ du'<;1cbi€u di~ntrongtQadQcfiunhu'lat6ngcua cachamdi~uboam~tcfiu.£)<;tohamcuabfftkyhamthufinnhfftnaGclinglaham J2,ItPL tJdIL Ihq.e uj 101 g nin 7J{}oQ1rnJuln thu~nha't.Ch£ngh(;ln,la'yd(;lohamriengtheox cahaivficua Euler a a a a -[r.Vf] = -f +r.V-f = -n-fax ax ax ax r.V [ ~f ] = -(n+1)~fax ax vad§:ntoi :x f la thu~nha'tb~c(n+1) D€ tha'y,ham! =Vr thoaphu'dngtrlnhEulervoin=1.Do dothficuamQt cha'tdi6m(haymQthinhc~ud6ngnha't)la thoaphu'dngtrinhEuler.Bdivi tru'ong thfi cua cac lo(;lingu6ndi6mkhac(c~plu'ongqic ...)chuad(;lohamcua Vr nen chungcGngthoa phu'dngtrlnh Euler. DQ di thu'ongtru'ongtoan ph~ncua c~p lu'ongqI'ctu(ho~chinhc~uvoi dQtuhoad6ngnha't)du'<;1cchobdi - 1 ~T =Cmb'v(m.-)r voi b la vectdddnvi songsongvoitru'ongxungquanh,vam la momenlu'ong cl;l'c.D€ tha'yding~T thoamanphu'dngtrlnhEulervoi n=3.Thams6n trong phu'dngtrlnhEulerdu'<;1cgQila chIso'e[{utruehay tYl~suygiam.Bang2.1du'a rachIso'diutruechotunglo(;lingu6nkhacnhau n Ki~ungu6n 1 Du'ongkh6i 2 Du'onglu'ongcl;l'c 2 Cha'tdi6m(hinhc~uvoi m~tdQd€u) 3 Lu'ongcl;l'cdi6m(hinhc~uvoi dQtuhoad€u) Bang2.1.."chislYdiu true"cilacaelo(lingu8nhapdJn vangu8ntitkhacnhau. Phu'ongtrinhEulerdfidu'<JCsad\lngbdimQts6tacgiadungd€ phantich caedQdi thu'ongtuvadQdi thu'ongha'pd~n.Ch~ngh(;ln,xetdQdi thu'ongtru'ong toaDphelntrenmQtv~tth€ dongiannhu'hlnhceluhayhlnhtr\l.B~t~Tila di€m thlii cuamQtnghiencUututrenmQtv~th€ dongian,voidi€m dot(;li(x,y,z) va tamcuav~tth€ t(;li(xo,Yo,zo).The'vao Euler tadu'<Jc r x- x ] [ ~~Ti' ~~Ti' ~~Ti ] y-y~ =n~Ti ax Oy oz z-zo Gia sa chungta dfi do du'<Jcho~ctinhdu'<JcgradientcuadQdi thu'ong tru'ongtoaDpheln.Phu'ongtrinhtrenchicob6nffnlaXo,Yo,Zovan, baffndelucho tavi tri cuav~tth€. Chungtaco th€ thudu'<Jcnhi€u theoyell Celli,b~ngcachvie't nayt(;licacvi tridokhacnhau a -~Tiax a -~T2ax a -~T1 Oy a -~T2 Oy a -~T1oz a -~T2oz r x- xo l [ Sf, ]~=:: =n !1~: vasad\lngphu'ongphapblnhphu'ongt6ithi€u d€ tlmnghi~m.Ne'uvi tricuav~t th€ du'<Jcbie't,chungtacoth€ tlm nvabie'tdu'<Jcd(;lngngu6n(xemBarongo[1]). M~t khac,ne'uchungta nghi ngo d(;lngv~tth€ ngu6n,chungta co th€ chnn thichh<Jpvagiai tlmvi tri v~tth€. M~cduphu'ongtrinhEulerclingca'pmQtcachhfi'uhi~ud€ xacdinhvi tri cuav~tth€ ngu6n19tu'dng,nhu'v~tth€ ngu6nd(;lngCelli,d(;lnghlnhtr\l,nhu'ng phu'ongphapnayco mQtvai h(;lnche'khi apd\lngchocacd(;lngv~tth€ ngu6n khac.Trongcactru'ongh<Jpnay,n coth€ khongphaila h~ngs6d6ivoidQsauva J2lLi):n tJiUL fJu,.e uj 103 g'nilL '7l5oidrmuYL vi tri cuav~tth~ngu6n,bCiivi f khongchila dt;lohamcua1/rmala tichphan trentoanbQphanb6 ngu6n.Ravat(xem[1])chi ra r~ngphuongphapnayap dlJngdu<;1ckhi dQdi thuongco h~so'suygiamty l~voi khoangcachtinhtu ngu6n.