Question

GIVEN: Ω isthe portion of the surface of the sphere centered at the origin of radius 3 above 1.2 1(xy, z) the plane, z-2: Ω: the field F = (x, x,x). a) FIND the flux of VrF through Ω in the given direction: n has positive 2-component. HINT: (radius a)on Q:(spherical coordinates) b) Parameterize the path,c-a2, (r,g,z)asin g dode with orientation to agree with the given n for Ω ANS: (a) 5 c) With positive orientation,an -e DETERMINE: F.ds ANS: (c) Ω: be the circular disk so that its boundary is the same as the boundary of Ω. (note: on Ω:-2) (d) LET: a Ω,-20, (With carried orientaion) FIND: dSon Ω2 ANS: (d) ds drdy e) FIND the flux of V x F through Ω2 with orientation induced by a.-ao. ANS: (e) 5x Egl Batchelor Math 03 HOME-WORK (for FE.) Page 17 of (19)

Answer 1

lhe parametric representation of the surFace 2 is given aS such that Cos90- 2

αφ 9 -3 sinQ 3 cos9 sirn 3 cos θ cos 3 Sing cos 3 sine Sin 3 sine 3 Cose sin 3 sin9 Cos | 3 cos θ cos 3 Sin -3 sins Sinф 3 coss sin 3 cose Cosd 3 sin9 COs 3 sin sin 9 sinscoss cos 49 sins cose sin')

Thus, 2T 6o 2π 2 2TT Sin

60 S0 2 But cos θ,- L2 he radius of the circle on the plane 2 2, centered at the origin is given as 32- 22 5. So, the parametric equaticn of the circle is where φ [0, 2n) is a dummy variable it can be replaced by Since Y5 cost, J5 sint, 2 ), ete[o, 2m)

dv - dr dt , since dt is parametrized by 't' where, F- cost, S sint, 2 sint) dt, (5 cost) dt, o Also, the field F on the curve is givena F- 15 cost , s cost, s cost since, X V5 cost on the curve c Thererore dr 5 cost sintt 5 cos2 t dt OW,F.di- (-5 cost sint5 co) dt 2TT -5 Cost sint dt + 5cos't dt

coSt sint dt -COS 0 2TT cos* t dt = | (1 + cos 2 t 2 + COS 2t dt 2 2 2 CoS2t dt -T Clearlu (d) dS on a since the disk is on the plane 22 so the normal vectr ( unit normal vechor) So to the disk surgace is given as unit normal vechor)

ơxe.ds. 1 〉. 〈o, o, ds) 〈。,-1, dS 2 lds is the area of the circular disc of radus s