Question

Use the divergence theorem to find the outward flux of F across the boundary of the region D. F=3./x2 + y2 + 2? (xi + yj + zk) D: The region 35x2 + y2 +z+s4 The outward flux is- (Type an exact answer, using a as needed.)

Answer 1

F = 3 x² + y²42² ( x2 + y J + ZF) F = (3x5x²+442² C + (3y 3x² + y2 + 22/9 + (32J x² + y² + 22 ) & Di The region 34 x² y² + 2² < 4. Flux off across the boundary of region D is - Flux= 11 Fonds. we are using Divergence theorem to find flex- Flux = // Ends = Il divf dv Hence, Flux div F du V e dev F = ( 2 1 + 2 5 + 2 F ) . (3x / 27y²4 2²a + (3 45x4942–) 9° 4325x*y*22)] div F = (3xx 2 - x 2x + 35x²+4²4 27 ) + 2x² + y²7 22 139 x 2y + 3 Je²+ y²+ z² | 1 8J x2+42422 [3 2 1 XX2 + 35x244²22 & J x² + y² + z² div F = 13 (x2 + y² + ²) + 9x² + y² +22 1 x² + y² + 2² dove = 12 ſaty ²+2² 2020-5-11 06:08

using spherical coordinates - x= rsuio Cost y = rsu o sue o 2= roso x+y+z² = r2 ardydx=22su o dr dodo OSOL 211, OLOST 35r² < 4. 73 4842 Now, 2 210 II Flux = Ill div F du = l l l 12 r. rtsu o dr do do 0 0 0 0 8=3 21T IT 2 12.83 suo drdodo 0 0 0 0 8=53 ZIT, IT i su o dodo 3 6=0 Ozo = "/" 3.[46-9] && o dodø D=0 0=0 -3.7 12h [-coso de 20 2IT = 21 / 2 - [-2-1] do div F dv = 21x2 x 21 = 841 Hence Net outward flux is (B4 I/ Ang 2020-5-11 06:10