Question

13 Points .. Imagine that the filter of a dishwasher machine, that looks like the boundary of V = {(x, y, z) € R3 : Vx2 + y2 <z 51}, is immersed by water with a flow given by: F(x, y, z) = (2yz cos(y2) + x3,2x2 cos cos(x2) + 43,0) Q1.1 3 Points Draw a sketch of the filter, and mark any points you think will be relevant. Q1.2 8 Points Compute the flux through the filter (Sſs F. ñdS). Q1.3 2 Points Is there more water entering the filter or leaving it? Entering Same O Leaving

Answer 1

Anwar Celian (2y2 cos (42) + x², 2x 2 cos (2) 443, o) Svelgence the dem Equation N-{(x, y, z) E1R3: Jaby? (221) mate flow compute flow through the filten. "The surface inleged of the normal component of a vector point function 'f' over a chored Surface s'u equal to the volume integral of the divergence of P taken over the volume Venclorad by the surface is save av lands. au (ryzcos (42)+²) + Sy (@X2.C03(x2)+y)+ $210) 3x²+352 .3 (x7ye) applying Divergence I. nads:)) du Pav), 3(x²ye) dady dz Jxhyr 2221 = mindkerye 2221 So, Thearam. 2) osas Sxeyer, 12nsa =) - Sinne eine :31) Sex?ty e) de dydz dn (x27 y ²) dz lawyer (20 40 ) Varme

Bjor Wi-xa (a bry) dy Sina + 1/ (43) 1.x2 3 -3 jdx (xP. 2din +112 (1-x2y3/2) "x"Jaydet's jumuylene) put x =sind dx = cosodo प|2 sino.cosocosodott cos o cosodo 5/2 -1/2 π/2 sineo.cos²odott cosu Copy -IT/2 12 sino.costo do the east odo) -1/2 92 -t/2 -1/2 6 costodo omla Co-cošolca odots } costo do) 2) costado) (i costo de la costado) -12 (436) - 1/2 = 12x Talu 12 X 7ZIT 16 = 35-371/2 - 3T (1-1) = Flux, )Sands=311 Rince hare flex cu puitive, so more water is leaving it - 3T 2. a XX =)