Question

iii) Evaluate SD da over a spherical surface, a) if D = (2 + 16r2)2 and r = 2 in the xy plane. b) If D = 10cos Of for r = 2 and 0 sr="/2. iv) Verify the following identities: a) V (FA) = fVX A+ Vf A b) (f9) = fVg+gof c) V. (VX A) = 0 d) x (vf) = 0

Answer 1

以 a) x (HA) = {7xh + f xi LH: :- g e 最++下: e ze Let Ā= A x P + Ay } + Az #tre : Now fà = fax + fty } + f Az A co> 9 x(n) = 9 3x [s ze fax f Az f Ay ze ps (FA) |品 (feng) 是(s) (16) 2016 + Flas doi: 1]- الم 会, 1 Now f FX 二 品 员 Ax Ay Az 1 2 AZ_of dan 9 az ** 1 EXAJ 二 ** [f - + 中

38 籠+ 9 = (BV) be 86*+ Since (46) a + s) (46) ke ay B ӘA xe 十大 会+ Ке Xye + fe zu le pot five to 张力)-(+ Me, + Ax f5xã + If yĀ = be Zve Bev - Бие fel be Az so yo texte 十 ? Ax fe | w-第十張A4 f - Eve Az +R e. ə Az Lue f = X2 + EXA 十 he t Eve Zery-Zv 把 台+ from +R [器置 [*-*-[w - 1 x74 (3) Aa Ay +全 Ar ze 6e de Je Fe If XÃ = 後一定要一: 4 MON

e ze As from 6 & 6 = + R DAY _ füxtgfxa 금 (4Hz) - (4) - (fr) - (5] (4) - J (RA) LHS = RHS xfi) = fDxA + FfxA c) ⑦.(x)=0 NON FIX 읎 룷 Ax Ay Az JAZ A 영도, EXÃE y 또 22 ә9 01 K e e e Ge A ax 20 C 2 AR 5 + "5. (asy-f + 7 +3f(-1 - (-) + ( - 5.(x7) = (- - + - A Az 22 C Fue Eve SeeG Ze re

gay 2² Az Ə AX + * hexe = (yxA):4 dy az az az əyəx. gray + q 2 Aa az ay azax are Perfect differendiable 2 If Axe, Ay, Az operators: AZ they aaay a'y huge 2² ge AZ azaa dxaz ayda an AR 22 Ax ayaz azay 2 = (4x2/4 d) Ex (PF) = 0 T* of & af af + say 19:47 fe Z 6 of a NOW 2x <* 19- oleme Ze Exluf) af ә9 af əy af IZ хXC 27 the PÅ 22 bxaz ƏMƏZ bez/e Zele Ft (68) = 9) me +F FX (ff) =0 of arf Әң әу axay -)

= (6+)д fogtg of . 2. 0-5 = R+). ке fАС+ Бар 1 +(6+) ер + (6+) е (6) fog+97f [#6 +ttle- (ct it is-[Ev-z/k # # #s gies = 240 tefa+c+vet:= 60 Р 29 Now - : 5 1t-d) Difg) + (6) C+(52) с (45) –: 1:1 fa6t6af (6+) Д ( 4

j dady da = cov) da=&dodo f Given (1) D = (2 +1684)2 (2 +1684) ; =2 for Sphericat coordinates X = y sino cos Ø yersino sing = Xcos Metric elements (hirheg ha) -(1, 8, 8sino) & da=8 do do Let 5 = (2+1682) Z Mesxyz OQ ↑ ↑ 2 t +1689). @ xdrdo =) Now llo.ca -) for ood SS(2+16*) * do de [1.8.B-1] - Sjzvirde + f[re? de do - 3 bed ja t u besar di = [101*+[*101.5 4 2 2 fo.da [4] [] + 4[16] I z 2n + 327 = 347. D.dā = 347. -)