Question
Use the Divergence Theorem to evaluate S Ss F.dS where F= = (5x8, 6yz4, -40z?) and...
Use the Divergence Theorem to evaluate S Ss F.dS where F= = (5x8, 6yz4, -40z?) and S is the boundary of the sphere 22 + y2 + z2 = 9 oriented by the outward normal. The surface integral equals
Answers
Solution: S aF F + + Given F = (5x8,6yz*, 40x’z) S: boundary of sphere x2 + y2 + z = 9 Using Divergence Theorem: Flux= F ds = [[[ div(F)dv We have div(F)= 40x? +624 - 40x? In spherical coordinates,x = Osin() cos(O), y = Øsin (w) sin(O), z=; = cos() The given sphere is : O=3=0S0S3, OSOS, Osos 27 and dV = o- sin()dod code :: Converting to spherical coordinates: div(F)= 62* = 6( pcos (©))* = 60* cos (6) Flux = [] F.nds = [[| div(F)dV S 2. 2 3 000 0 = III (6,0* cos* ()) (o* sin cod od gode) = 6ſ 40.cos(a)sin(modo }(odp) +6(e)){(+3.60 (cm (a)_](30). -0) +36-1-1 -1)(413 - 0) = 6(27-0) 524887 35
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