n0 crAo DUc VA DAo TAo DAI HOC I{Ufi Hq vd t€n thi sinh: Sd bdo danh:

KV rHI ruyfx srNH sAU DAI Hoc NAM 2006 MOn thi: To6n cho vilt ly (ddnh cho: Cao hgc) Thdi gian ldm bdi: 180 phtit

CAU 1. Tim hbm u(r,y) th6a mdn phuong fiinh

.0'u

^

02u ^ 0'u a * r - " a r a a - o 6 r z : v

vi th6a m6n cdc didu kiOn

: O.

u(r, a)lr:o : 3r2 , Hlr:o

Ctllu 2. Tim phAn b0 nhi0t dE u(r,t) trong mQt thanh htru han c6 chidu ddi (, tat thdi didm bat ky t > 0. BiCt rang phAn bd nhiet d0 ban ddu trong thanh c6 dang u(r,,O) : Ar((, - r), (A li hang s0). Tiong thanh khOng c6 ngu6n nhiOt, hai ddu mrit ctra thanh luOn dugc gifr 6 nhiet dQ bang 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 gOc toa d0. Gia srl (r, p) ld cdc toa d0 c{c, (r,il ld cdc toa dO pC Cac hai chidu. Tim nghiOm cira phuong ffinh Laplace ddi vdi midn trong hinh trbn th6a mdn didu kiqn biOn Dirichlet:

u(r, p) - u(R,p) : A + B sin29, l":"

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.

Ghi chit: Cdn bd coi thi kh1ng gidi thich gi th€m