1. n0 crAo DUc VA DAo TAo Hq vd t€n thi sinh:
DAI HOC I{Ufi Sd bdo danh:
KV rHI ruyfx srNHsAUDAIHoc NAM 2006
MOn thi: To6n cho vilt ly
(ddnhcho: Cao hgc)
Thdi gian ldm bdi: 180phtit
CAU1. Tim hbmu(r,y) th6amdnphuongfiinh
02u ^ 0'u .0'u ^
a*r-"araa-o6rz:v
vi th6am6ncdcdidukiOn
u(r,a)lr:o: 3r2, Hlr:o
: O.
Ctllu2. Tim phAnb0 nhi0tdEu(r,t) trongmQtthanhhtruhanc6 chiduddi (,tat
thdi didmbatky t > 0. BiCtrangphAnbd nhietd0 banddutrongthanhc6 dang
u(r,,O): Ar((,- r), (A li hangs0). TiongthanhkhOngc6 ngu6nnhiOt,hai ddu
mrit ctrathanhluOndugc gifr 6 nhiet dQbangkhOng.86 qua su ffao Cdintriet
quamdt b0n,vdn tdc truydnnhigt ffong thanhbdnga.
Ciu 3. X€t hinh trbn bdnkinh R c6 tamnamtai gOctoa d0. Gia srl(r,p) ld cdc
toad0 c{c, (r,il ld cdctoa dO pCCac hai chidu.Tim nghiOmciraphuong
ffinh Laplaceddi vdi midn trong hinh trbn th6amdn didu kiqn biOnDirichlet:
u(r,p)
l":"
- u(R,p): A + B sin29,
trongd6 A vh s ld c6changsd
CAu4. Chob6nkinh vectoi_ ri+aV + rE,r: ld. Hdy tfnh:
div[r.grad(r-')] ,
trongd6 n ld sdnguyOn.
Ghi chit: Cdn bd coi thi kh1nggidi thichgi th€m

2. Ky THI TUyfN SINH SAU DAI HQC ivAvt 2ooz
M6n thi: Todn cho Vat li
(dd'nhchoCaohP")
Thdi g'i,anld,mbd'i:180 Phrit
r
I -
t
^ '
{,/ .- L:-r-6
CAu I.
(r. Tfnh tich phanm6t i - $ ; (;;) dS,trons d6 7 ravectokh6ngddi, 7: vects
Js /
vj tri, d: vects dcrnvi ph6p tuydn criam{t ,S.
b. Sr! dpngdinh tf Oxtr6gratxki-Gaoxo,hey tfnh t!6ng lugng crLavects
i, -z.3,i,+a3j *rzyk
grli qua mQtmd,tkfn ,9gi6i h?n bdi hinh n6n c6 bdr-kinh ddy.R, chibu caoH:
*2 )- o,2 72
Tsp,
o1z1H-
CAu II.
X6c dlnh daod6ngtg do crlamQtd6yhuu hq,t,gXnch5,ttai c6cmri.tr - 0,n: !,,
bi6t d9 lQchban db,udugc chobdi:
A m ( 0 - - )
, u(",0):T,0( rS(,
cbnv6,nt6c ban dbu bXng0.
C5.uIII.
MQt thanh dbng chdt c6 chiEudei / v6i c6c mdt b€n cd trao ddi nhi6t v6i m6i
trulng xung quanh,nhiet d6 m6i trulng bXng0, cbn c6,cmrit n : 0,n : {,dugc gifr &
nhiQtag f.nO"! ddi bHng0. Tlm ph6,nbd nhiQttr6n thanh iric t > 0? Bidt rH,ngnhi€t
d6 ban db,ucrlathanh c6 dang: u(r,O) : An (A: hXngs6).
Ciu IV.
nQcmo DUcvA DAOTAO
DAIHQCHU6
Ho ud,t€n th{ s'inh:
56 b6,odanh:
fim hh,mu(r,A)
c6cdibu kiQobi6n:
diEuhoA,trong hinh chfrnh6t A I r I a, 0 < g ( b vb thod,md,n
u(r,0) :0, u(r,b)-n, 0Sn3a
ur(O,g): 0, ur(a,A): 0, 0 < g < b-
C6.uV.
a. Cho F - r^l trong toq,dQch,u,{: vecto dcrnvi theo phuongxuy6nt6,rn.Chftng
minhrxng:
v* F-',vF -(n+z)rn-r.
b. Chfrngminh rXng,hh,m
f1r
u(r,A,z) -
J_*f
(, + ir cost* i'ysinf,t)dt
v6i f (r,t) le hA,mtdj' i, gi&itfchtheor vh 1i6ntuc theot, lb hhmdibuhoh-
Ghi chfr: Cd,nb6 coi ttti,kh'6nggidi th{ch'gi'th,€m.

3. T40 Hg ud,t€n tht s'inh:
Sd b6,odanh:
) v
TUYEN SINH SAU D4I HQC NAM 2OO8
M6n thi: TOAN CHO VAr r,Y
(d,anhcho 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^yra tai
didm M d,uoc cho b&i c6ng thirc:
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 < (..
CAUIII.
MOt thanh mAnh,ilbng chdt c6 chibudei / v6i c6cm6,tb6n cachnhi6t, cbnc6c
db,umrit r : 0 vd,n,: / lu6n cluocgiu d nhi6t d6 kh6ns eldibXng0. Tbongthanh
kh6ng.c6ngubnnhi6t. Tim ph6,nb6 nhi0t tt6 trong thanh tai thli didm bdt ky t > 0.
Bi6t rXngphdn bd nhi6t dOban 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,cdibu 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 dungchotrubnghep a - b: Tr f @): sinr.
e0 ctAo DVCvA DAo/
DAr HgC HUE
KV THI
i -nn4,T"
trongd6 7 -ffi,, : l?1, k lA,hXngsd.
Ghi chf: Cd,nb6 coi,thi khdng gid,i,thfuh gi th€m.

4. BO GIAO DL,CTA DAO T4O
/
DAr HgC HUtr
KY rlrr ruydw srr{rr sAu DAr Hec xArn 200e(Dotr)
M6nthi: roAx cHo vAT t..i
(dd"nhcho Cao hq")
Thdt gian ldm bdi: 180 phrit
CAu I. 1. Xd.cdinh c6,che sd Lame trong he toa do chu (r,0,'g). Tu do viSt ra
bidu thf'c cria dir,,A-ttong he toa do ndy.
2. Tinh tfch phAn
r-I -
9 r r t
z" d.rdad,z
vor V Ia hinh chu 12 + y2 * z2 < R2.
CAu II. Tim nghiem u - u(r,t) cua bai to6n h5n hcrpsau
utt : urr
,(0,/) : 0. u,({,/) : 0
u(r,0) : t. ut(r, 0) :
r r y
- l l . L
q l n - J - q t n
v r r r
( t ,
I v r r r
! (
thi sinh:
b6o danh:......
r 1{, t > 0
0
r{[
Ho vd t6n
S5
Ill;'
0<
t>
3rr
_ u<
:(
vot {.) 0 cho tru6c.
CAu III. Tim phAnbo nhiet cvthcvidi6m / > 0 tren mQtthanh dbng chdt c6 chibu
(. vot c5,cmdt ben c6chnhiet khi bi6t nhiet do ban dbu b5ng 0. mrit r:
ducycgiu crnhiet do khong doi bing 7, cbn mrit r - ( duoc giu crnhiet
khongdoi bXng U.
Cdu IV. Tim hdm u(r,g) @hotrong 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)'
CAu V. Tim ne.hiArntt - u(r,y) cuaphucrngtrinh
ur, - 2 sin g,'r-t,ra- .or2 fr.uaa- cosr.uo - Q
th6a min dlEu ki6n sau
u(r, cosr) - sinrr ur(r) cosr) : r * cosr.
dei
-n
+ ^
oo
cac
Ghi chri: Cdn bo coi thi khdng giAi thfch gi them.