Question

Please answer a,b and d(most important) . TQ

(BEKG 2443) PART B: ANSWER ONE QUESTION ONLY QUESTION 4 Given S is a surface for a part of the sphere x2 + y2 + x2 = 4 that lies above the cone z = √ x² + y² (a) Find a parametric representation for the surface S in terms of 6 and 0. Then, determine the domain of O and e. (5 marks) (b) Based on the expression written in (a), show that ||r4 x re|| = 4 sin o. (5 marks) (c) Hence, compute the surface integral of xyz ds where the surface S is given para- metrically by r(0,0). (8 marks) (d) Let C is the boundary curve along the bottom of the upper half of the sphere. Thus, use F. dr, where Stokes' Theorem to compute the line integral $ с F(x, y, z) = xzi+yzj+xy k and the upwards orientation. Hint: Write the parameterization of C in term of t. (7 marks) [25 marks

Answer 1

@ x²4Y²+ z2=4 and above the cone z= 12 + yl X= p sind uso where P = 2 & x=2 sing Corso Y= zsind sino Ze 2 cette Hence +($0) = ( 2 singl Glo) it ( 2-sänd gine) j+ fasolſ '. I- Jurtyr ..2060 = using cost u fin 24 Sinza 2 Cog® – 14 Bin 24 (Cote + Sin ) - zsind Cos&= sind 9 tand = 1 fo d = 1 Hence domain: ος φε 4 < 21 ( 1184 x 70)1 = ? Htc etc to (sino Curso) it is a Sindi sina) i+ (32) R 89 = (2 cos& caso ê +(2 Gosd Sing I - (sind) Ê

to= Jo Jo. Jo (2 sing Coso) êt ta ( 2 sinf sine ſ + d do ( 2 Corted) IC do = 628 info siha). ê +62 8ing. Colos) į tok dox do= I Lg  2 Colo Colo 2. Coff fino - sind - 2 sind sina asingolo 84 X to 2 Coff sino - 2sind 2686 colo -zfing 23ing Gesto Es 1-28ingsina of 2 Coff auto 2 Coso fino 4 -2singsing 2 sind so tøxto = (4 Bin 246 688) + (4.312 24 8 in) Î + 4C Spinel Cortif Gas 20 t shocesso Sin 20) 8qXdo= (ysin & colo) êt (ysinplino)j + 4 sind Golf Ê 1164xtoll & magnitude of vector de to

Boxdoll = /16 Sing & Casket 16 sin la finzt is fine es4 11d&xtoll = 16 8in1€ (sin Pot Cut?o ) +1631in24 Guf 24 1188 Xdolla vio Bin 48 + 16 sin ?$ Cep?ef ( 26 X toll – J16 Bin 24 (6924 48 in245) VIG 8 iu24 = 4find. Honce 1186xtoll = using SS xyzds hol' ds = 1128xroll do do where & Elly E21 SS vyzdse SS (2 sinf Gite) ( 24inf sins) (2 Gased) 1.4 Xpo lidé de - s Talsi 88 in 26 celo Bino Cerse) (43 indo) do do

2 = try 328in34 caso sino. Caso de do こ My 32 32 82"(cuss. Sins de). S('fin y as4 de) =168. (25 in. (se) dox s "it sinad )dø Els Pogin2o) do 1974 ["4 8 in 84. G84d4 25 Sin (20) dos 211 IN [Coup (20)] -- fus yoz Geo] 2 介 { "Sin (20)d0 = rų (14)=0 Therefore SS nyzds- ox My sin 34 oso do co S ISS nyzdsco affe & F. do where Fny, z)= nzity z5t nye using Stokes theorem SSCExP). ds P. di s

dr e 0 + Mae not + (a 2외 du 20 (Re) 지 Re (2) F(x) 킷= g(8) Re zh ZI Jz R de ~ c P _ = AXE

7x7=(-) - (4 – u) ſ tok Si upper half furface of the Sphere including its bettem Sz 다 ys, oc X & F. do SSCxF).alse SS EXF).ds + SS COXF) ds c S اک 2 (cider fulface Siin Z=0 for Sii So u?y? 4 a=foto, Y=2since & S(20) = (auto) it (2 finals

So=ds- S = who êt Sinos dr So-ds c Ersing ? + (wko) do Soxo = ra L. Celo sinca に =ri tsina Herso SSZx Flods e S (-y) it (ny) jt of} .sê dido SI O

fosf(587). ds to Si Now for Surface su x= 2sing coso, y = 2sihx shoo V=2sing shoes 2= 23614 we already found that rø Xto = (48 in24 (ata) î+ (usinep Sing)j + (48 lenf Cent) É Stéxa.ds - S San-y)i + (49) 5+of] Sz ((4.sin 24 (240) + (48in24 3ihee) 5 +48 nf (94)] de do SS Causines Congo (19)+ (ky) (MS inte fine) Jd¢dos SSEXF). ds y &in 24 (sino+ Cogo) (24 Zit S. mly Sind Caso 2 Sino sino) dado را ( 2 h sirsch Sin34 (dihes + Corgo) (Colo- Sins) d® do 58 sem (99 sin34 (cas's-sinta) def de

Cosze - sinza = cifra :SSEXF) .ds = og de masinse. (Goa) diy do Sz =88820) dox gh4681n34) df 0 e O as 20 Godzodoro 7-9

-otoco S Inally of Faton SSEF) de >ISSEXF). ds.co