Đề thi tuyển sinh sau đại học môn Toán cho Vật lý năm 2006 - 2009

n0 crAo DUc VA DAo TAo<br /> <br /> Hq vd t€n thi sinh:<br /> Sd bdo danh:<br /> <br /> DAI HOC I{Ufi<br /> <br /> KV rHI ruyfx srNHsAUDAI Hoc NAM 2006<br /> MOn thi: To6n cho vilt ly (ddnhcho: Cao hgc) Thdi gian ldm bdi: 180phtit CAU1. Tim hbm u(r,y) th6a mdn phuongfiinh<br /> 02u ^ 0'u .0'u ^<br /> <br /> a*r-"araa-o6rz:v vi th6a m6n cdc didu kiOn<br /> <br /> : u(r, , a)lr:o 3r2 Hlr:o : O.<br /> Ctllu2. Tim phAnb0 nhi0t dE u(r,t) trong mQtthanhhtru han c6 chidu ddi (, tat thdi didm bat ky t > 0. BiCtrang phAnbd nhiet d0 ban ddu trong thanhc6 dang : u(r,,O) Ar((,- r), (A li hang s0). Tiong thanhkhOngc6 ngu6n nhiOt,hai ddu mrit ctra thanh luOn dugc gifr 6 nhiet dQbang khOng. 86 qua su ffao Cdi ntriet qua mdt b0n, vdn tdc truydn nhigt ffong thanh bdng a. Ciu 3. X€t hinh trbn bdn kinh R c6 tam nam tai gOctoa d0. Gia srl (r, p) ld cdc cira phuong toa d0 c{c, (r,il ld cdc toa dO pC Cac hai chidu. Tim nghiOm ffinh Laplace ddi vdi midn trong hinh trbn th6a mdn didu kiqn biOnDirichlet: u(r,p) l":" - u(R,p): A + B sin29,<br /> <br /> trong d6 A vh s ld c6c hang sd CAu 4. Cho b6n kinh vecto i_ ri+aV + rE, r: ld. Hdy tfnh: div [r.grad (r-')] , trong d6 n ld sd nguyOn.<br /> <br /> Ghi chit: Cdn bd coi thi kh1nggidi thich gi th€m<br /> <br /> nQ cmo DUc vA DAOTAO HU6 DAIHQC<br /> Ky THI TUyfN<br /> <br /> Ho ud,t€n th{ s'inh: danh: 56 b6,o<br /> I^'<br /> <br /> SINH SAU DAI HQC ivAvt 2ooz<br /> {,/<br /> r\<br /> <br /> M6n thi: Todn cho Vat li (dd'nhcho Cao hP") ld,mbd'i:180 Phrit Thdi g'i,an<br /> <br /> .-<br /> <br /> t<br /> <br /> L:-r-6<br /> <br /> CAu I. dS, trons d6 7 ra vecto kh6ng ddi, 7: vects (r. Tfnh tich phan m6t i - $ ; (;;) Js \ / vj tri, d: vects dcrn vi ph6p tuydn cria m{t ,S. vects b. Sr! dpng dinh tf Oxtr6gratxki-Gaoxo,hey tfnh t!6ng lugng crLa , i , - z . 3 , i+ a 3j * r z y k grli qua mQt md,tkfn ,9 gi6i h?n bdi hinh n6n c6 bdr- kinh ddy.R, chibu caoH:<br /> *2 )- o,2 72<br /> <br /> Tsp,<br /> <br /> o1z1H!,,<br /> <br /> CAu II. - 0,n: tai X6c dlnh dao d6ng tg do crlamQtd6y huu hq,t, gXnch5,t c6c mri.tr bi6t d9 lQchban db,udugc cho bdi:<br /> <br /> ,<br /> <br /> u(",0):T,0(<br /> <br /> Am(0--)<br /> <br /> rS(,<br /> <br /> cbn v6,nt6c ban dbu bXng 0. C5.u III. MQt thanh dbng chdt c6 chiEudei / v6i c6c mdt b€n cd trao ddi nhi6t v6i m6i trulng xung quanh, nhiet d6 m6i trulng bXng 0, cbn c6,cmrit n : 0, n : {, dugc gifr & nhiQtag f.nO"! ddi bHng 0. Tlm ph6,nbd nhiQttr6n thanh iric t > 0? Bidt rH,ngnhi€t d6 ban db,ucrla thanh c6 dang: u(r,O) : An (A: hXngs6). Ciu IV. trong hinh chfr nh6t A I r I a, 0 < g ( b vb thod,md,n u(r,A) diEu hoA, fim hh,m c6c dibu kiQobi6n: 0Sn3a u(r,0) :0, u(r,b)-n, 0 < g < bu r ( O , g ): 0 , u r ( a , A ): 0 , C6.u V. a. Cho F - r^l trong toq,dQ ch,u, vecto dcrnvi theo phuong xuy6n t6,rn. Chftng {:<br /> <br /> minhrxng:<br /> <br /> v* F-',vF -(n+z)rn-r.<br /> b. Chfrng minh rXng, hh,m<br /> <br /> t)dt (, + ir cost i'ysinf, * J_*f tuc tdj' v6i f (r, t) le hA,m i, gi&itfch theor vh 1i6n theot, lb hhmdibu hohgidi th{ch'gi' th,€m. Ghi chfr: Cd,nb6 coi ttti, kh'6ng<br /> <br /> u(r,A,z) -<br /> <br /> f1r<br /> <br /> e0 ctAo DVCvA DAo T40<br /> DAr HgC HUE<br /> ) v<br /> <br /> /<br /> <br /> Hg ud,t€n tht s'inh: Sd b6,odanh:<br /> <br /> KV THI TUYEN SINH SAU D4I HQC NAM 2OO8<br /> M6n thi: TOAN CHO VAr r,Y<br /> (d,anh cho Cao hp") Thdi g'ian ld,mbd,i: 180 phft CAU I. Vecto cubng d6 di6n trulng do di6n tich didm q dqt tai gdc toa d6 O Sa^y tai ra didm M d,uoc cho b&i c6ng thirc:<br /> <br /> i -nn4,<br /> T"<br /> <br /> trong 7 -ffi,, d6<br /> <br /> : l?1, k lA, hXng sd.<br /> <br /> a. Tfnh divE tai didm M c6 , + \ b. Tfnh th6ng lugng cria trulng E qua mdt cbu t6,m O, b5"nkfnh R. C6 thd d,p dpng dinh lf Ostr6gratxki-Ga.oxo tld tintr th6ng lucrng n6i tr6n iluoc kh6ng? Vi sao? CAU II. Tim hbm u(r,t) th6a m5n phuong trinh vb c6c dibu kion sau: : 1trtt A2Urr, 0 ( t I (,,,t > 0, u*(0,t) : u((.,t) - 0, t ) 0, u(r,O) : r, ut(r,0) : 1, 0 S r < (..<br /> <br /> CAU III. MOt thanh mAnh, ilbng chdt c6 chibu dei / v6i c6c m6,tb6n cach nhi6t, cbn c6c db,umrit r : 0 vd,n, : / lu6n cluoc giu d nhi6t d6 kh6ns eldi bXng 0. Tbongthanh kh6ng.c6ngubnnhi6t. Tim ph6,nb6 nhi0t tt6 trong thanh tai thli didm bdt ky t > 0. Bi6t rXng phdn bd nhi6t dO ban dh,utrong thanh c6 dang u ( r , O ): T . r ( ( . - r ) , trong d,6T" lb hXngs6. Cho vdn t6c truybn nhi6t trong thanh bHnga. CAu fV. Tim hbm u(r,y) dibu hba trong hinh chir nhd,t0 < r I a, 0 ( g < b vb thda m6,ncd,c dibu ki6n bi6n : u ( 0 , a ): 0 , u ( a , A ) 0 , A < y I b , u(r,0) :0, u(r,b): f("),0S rSa. Ap dungchotrubng hep a - b: Tr f @): sinr.<br /> Ghi chf: Cd,n b6 coi,thi khdng gid,i,thfuh gi th€m.<br /> <br /> BO GIAO DL,C\TA DAO T4O /<br /> <br /> H o vd t6n thi si nh:<br /> <br /> DAr HgC HUtr<br /> <br /> S5 b 6 o d a n h : . . . . . .<br /> <br /> KY rlrr ruydw srr{rr sAu DAr Hec xArn 200e(Dot r) M6n thi: roAx cHo vAT t..i<br /> (dd"nhcho Cao hq") Thdt gian ldm bdi: 180 phrit he CAu I. 1. Xd.cdinh c6,c sd Lame trong he toa do chu (r,0,'g). Tu do viSt ra ttong he toa do ndy. bidu thf'c cria dir,,AphAn 2. Tinh tfch<br /> I -<br /> <br /> r-<br /> <br /> Ill<br /> \;'<br /> <br /> z" d.rdad,z<br /> <br /> 9rrt<br /> <br /> vor V Ia hinh chu 12 + y2 * z2 < R2. CAu II. Tim nghiem u - u(r,t)<br /> utt : urr<br /> <br /> cua bai to6n h5n hcrp sau<br /> <br /> ,(0,/) : 0. u,({,/) : 0<br /> u(r,0) : t. ut(r, 0) :<br /> qln vrrr rry ll.L (\ t, JI qtn vrrr<br /> <br /> 0< r 1{, t > 0 t> 0<br /> 3rr<br /> !(<br /> <br /> _ :(<br /> <br /> u<<br /> <br /> r{[<br /> <br /> vot {. ) 0 cho tru6c. CAu III. Tim phAn bo nhiet cvthcvidi6m / > 0 tren mQt thanh dbng chdt c6 chibudei (. vot c5,cmdt ben c6ch nhiet khi bi6t nhiet do ban dbu b5ng 0. mrit r: - n +^ ducycgiu cr nhiet do khong doi bing 7, cbn mrit r - ( duoc giu cr nhiet o o g khongdoi bXn U. Tim hdm u(r,g) @ho trong toa d6 cuc) diEu h6a trong hinh trdn c6 tAm d e6c toa d6 O vd brin kinh E. th6a man dibu ki6n tr6n bi6n cria hinh trdn u(R, P) : '4 (sin(P+ 2 cos2P)' tt trinh CAu V. Tim ne.hiArn - u(r,y) cua phucrng<br /> ur, - 2 sin g,'r-t,ra .or2 fr.uaa - cos r.uo - Q<br /> <br /> Cdu IV.<br /> <br /> th6a min cac dlEu ki6n sau u ( r , c o sr ) - s i nr r u r ( r ) c o sr ) : r * c o sr .<br /> <br /> Ghi chri: Cdn bo coi thi khdng giAi thfch gi them.<br /> <br />