Question

Imagine that the filter of a dishwasher machine, that looks like the boundary of V = {(x, y, z) e R3 : Vx2 + y2 <z<1}, is immersed by water with a flow given by: F(x, y, z) = (2yz cos(yº) + x3,2xz cos(x²)+y3,0)

Answer 1

4 {(x, y, z) e 173: Sxxy? (7513 water flow F(x,y,z) 2 ( 282603(y-) +x3, 2x2 609 (x2)+420) compute flow Hirough the filter. x Divergence Theorem "The sunface internal of the normal component of a vecton point function f over a closed sunface s is equal to the volume integral of the divergence of & taken over the volume v enclosed by the sunface". SSS divf.dv a SS Fonds divez (217c0nlr) +x)) + (2x7ics (x2)+73) +20 3x + 3y = 3(x²+x2) so applying Divergence Theorem. SSF ñds - SSS div F'dv - SSS 3 C**+y") dady dz- V s 2x S ✓ Xr+y2 <Z sll =) min x+y=sts o <z sl – xray si risasi = -√1-x2 sy<di-xe :.3555 (x+y=) dedydz - 3 Sox S dy Į GEtY2) dz -S1-XL = 3 s (2x + y 2) dy s'ax S

1 VI-2 = 3 dx f (x + y ) dy -1 11-xt + 3 $(9371e** T-X ✓i-xi 2 - 1 - T/ • Th -T/ -T/ + 3 (dx. (x².251-x² + yep 1 - x03%) 6 [ fx/1-x2 dx + s'e_2-3% dea] (put x = sind, dx, 2 casodo) 6[ sin?e.cose.coso do +} sceso.corodo 6[ "Sino.ce*ode + 4 5 "testode] 6[ 591-60340) 60320d8 + Ź sco So cesto do - / - T/ T/ [ { "cato do - Sthcosto do - TV T/2 212 [ { coro do -2% scosso do 12 [ 1/4 - 2/2 x 3%] 12811/4 f2 x 7 x 31/16 = 37 - 37/ = 30 (1-²) 3/ Ans. TV T/ O : 12 3 or s - flux, frnds 31/ Since here flux is positive, rearing So more is Water