Question

(point) Compute the outward flux of the vector field F(x..:) - 2 + 4xj + zlecross the boundary of the right cylinder with radius 3 with bottom edge at height 2-2 and upper edge at 27. Note: The vectors in this field point outwards from the origin, so we would expect the flux across each face of the cylinder to be positive. Part 1 - Using a Surface Integral First we parameterize the three faces of the cylinder Note: In the following type as theta. The bottom face (6.0) = (x), y16.0). 216.where o SOS205rs and *16.8) = 71.8) = 21.)-2 The top face: 63,0) = 0.0.2.0). 26.owhero 0 505 250575 3 and X. yr.) - .) -7 The lateral tace: 090.2) = (x88.21.790.2).2960, zwhere 0 S 0S 24-25=5 Tand 0) = 2) = Then mind the orientation) II. ..d = 0,"Iro) Com ho pedra "Pole ) dra ("room d: do Part 2 - Using the Divergence Theorem I nos- // VFav dz dr

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Answer 1

F (219, 2) = 2x1 + 4y = + 327 The equation 2=7 of eylinders is 2²+ y² = 9 -23217 : Calle the Z=-2 3 three surfaus eve bottom Sunface as 8,, top se Serface as 2 and lateral surface as ro, for the bottom Surface of the parametraie Surface follows: Tere x,(2,0) = pesso Z(0,0)=- 2 27 (P. o) on Sino The top Surface Zz (0,0) = 7 X₂ (0,0) = p eoso 7 CP,0) = plino For laternal face is 29 Co) = 3 caso Y₃ (0) = 3 Sino -2 ? Z $7. Zz • (2) = .z i 20 now I Findsa Cari 20 F001) ) dn deo do dout ออ 3 !" / 03( 5 » 9-3 bis 2! View (26o ang az do de do ao ( erso, sino, o Op 36 - (ino po sino, peaso, o) 20 votos a, K - -psino plos o 0. E-PR eso sino o

a * 362 = no [; the dibution is upwand] ise 203 (0,0,1) a 83 ao (38.40, 8 Coso0) 2 k ao a z ars OO 1 + 8650 7° + some 3) - 381no scorso a it [orieutiction is tve] JJ Firids - j"JCeme Cencoso 7+2p8in0 3–51.6-7 &°) dndo 201 3 4 (ners of + 2p Str 0 +21+") -(ne) dodo +5*J* Cous +12 sing J +37ū): (30907't s sino 7) -2 & zdo = 6 !" ! *mdir pdrdo + 21 /" !donde 4 !"(irca 1960) dzio 2 T 3 7 = 27 | ** ( 2 ) dot 18+9 (146520)] d zdo 2 IT 2 21x9 do + (27 é o casozo) x9 do = 27 x 9 9 X To 9 x 27.2 = 729 1.

F = 2xPaay? +32% J'A32) dir 7 Sa+ 7 ms rū Ba). ( 227 Aug = (2+4+3) = g. theonem hoge S Finds 7.3 25 ep up ap upp ao - 2 pao 00 using divergent Il dire du ST 1" L" 81 5. ** [m2: do 7 Ap up z p al 6 pco 7--2 3 gp xg drdo- こ OP 214 3 81 AP 6 — 0 Dec 81 81 x9 *[0]." 81% 2M 729 T . 2