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:.. Ghi chil: Cdn bo coi thi kh6ng giai thfch gi th€m.
Trang 1BQ GIAO DUC VA DAO TAO
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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: orv r ' 1 i \ \
\,,/ va grad [ai" (;)]
Cflu 2 GiAi bdi todn
bdne cach ddI u :
{ * " - - 2 , 0 1 x 1 n 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,
l u ( x , 0 ) : s i n x ,
u ( r , ! ) : 0 , 0 3 y S T t ,
u ( x , t t ) = 0 , 0 5 ; x 1 r c
( u r t : u x x + 2 , 0 < x 1 T t , f > 0
1 u @ , 0 ) : 0 , u t ( x , o ) = 0 , o < x S n 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 ) : l h - , k h i o < l x l l c
[ o k h i c < l x l < * o o
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.