BQ GIAO DUC VA DAO TAO t DAI HQC HUE aPq6 Hp vd tln thi sinh: Sii Uao danh; xV rul TUYEN sINH sAu DAI Hgc xau 2012 (Dqt l) Mdn thi: TOAN CHO VAT LV (Danh cho cao hqc) Tl4di gian ldm bdi.' 180 phut Ciu l. Tinh V' (i f (r)),trong do F - xi + yj + ti ld b6n kinh vecto, r : lil, f (r) la ham v6 hucrng chi phU thudc vdo r vd V ld to6n tu Nabla. Ap dung kOt qua tr6n dti tintr r ' 1 i \ \ r: orv \,,/ va grad [ai" (;)] Cflu 2. GiAi bdi todn bdne cach ddI u : {*" 2, 01x1n tw(0) - 0, w(n) - 0. CAu 3, Cho mQt thanh m6ng, ddng chht, chi6u diri l; dAu x - 0 cua thanh dugc git o nhiQt d0 kh6ng d6i bing ?nr, dAu x = t dugc giti o nhiQt d0 khdng ddi bdng ?"r. Tim ph6n bd nhiqt tr€n thanh hic f > 0? Bi6t ring nhiet dQ ban dAu cua thanh bing 0. CAu 4. Tim nghi€m u(x,y) cua phuong trinh Laplace u*, * ,X, = 0 trong hinh cht nh4t D - {(x,y) € IRz[0 < x I Tr,0 < y <n] thoa mdn c6c di€u kiQn bi6n sau: {u(0,!) : o, lu(x,0) : sin x, u(r,!):0, 03ySTt, u(x,tt)=0, 05;x1rc. (urt:uxx+2, 0<x1Tt,f >0 1u@,0) :0, ut(x,o)=0, o<xSn fr(0, t) : 0, u(n,f) : 0, f > 0, v * w trong do w - w(x) la nghiQrn cua bdi todn CAU 5, 1. Tinh luu s6 cua hirm vecto: F @y)i+ (bx)i, (a,b la hing sO; Ogc theo ducrng trdn b6n kinh ^R nim trong mflt phing xq, co tAm trung vdi g6c tqa d0. 2. MQt sqi dAy vd hpn (-m I x 1+oo) dugc kich thfch dao dQng tU do boi mQt d0 lQch ban dAu co dang: ( . hlxl u(x,o):lh- , khi o<lxllc [o khi c<lxl<*oo. Hdy vE dpng cua sgi ddy tai c6c thdi iliOm tr, = ki voi k = 1vi k = 2 n6u vdn tdc truydn song tr6n ddy a, - 2, con vfln t6c ban dAu cua dAy bing 0. Ghi chil: Cdn bo coi thi kh6ng giai thfch gi th€m. . TAO t DAI HQC HUE aPq6 Hp vd tln thi sinh: Sii Uao danh; xV rul TUYEN sINH sAu DAI Hgc xau 2012 (Dqt l) Mdn thi: TOAN CHO VAT LV (Danh cho cao hqc) Tl4di gian ldm bdi.'. bdi todn bdne cach ddI u : {*" 2, 01x1n tw(0) - 0, w(n) - 0. CAu 3, Cho mQt thanh m6ng, ddng chht, chi6u diri l; dAu x - 0 cua thanh dugc git o nhiQt d0 kh6ng d6i bing ?nr,. vdi g6c tqa d0. 2. MQt sqi dAy vd hpn (-m I x 1+oo) dugc kich thfch dao dQng tU do boi mQt d0 lQch ban dAu co dang: ( . hlxl u(x,o):lh- , khi o<lxllc [o khi c<lxl<*oo. Hdy