Question

(1 point) This problem will illustrate the divergence theorem by computing the outward flux of the vector field F(x, y, z) - 2ri + 5y + 3-k across the boundary of the right rectangular prism: -3 <<6, -15y<3,-425 oriented outwards using a surface integral and a triple integral over the solid bounded by rectangular prism. Note: The vectors in this field point outwards from the origin, so we would expect the flux across each face of the prism to be positive. Part 1 - Using a Surface Integral First we parameterize the six faces using 0 < x < 1 and 0 <t < 1: The face with z = 4:0,- (+1(3),8(t), 21(s, t)) 11(3) vi() = 2,(s, t) The face with z-5: 09 (+2(), yu(t), 29(s, t)) 32(s) 32(t) 22(s, t) 5 The face with x = -3: 93 = 13(s, t), y(s), zz(t)) 23(s, t) -3 93() z3(t) The face with x = 6: 04- (,(s, t), y(s),- (t)) 2.(s, t) -6 Y(s) za(t) - (15(s), ys(s, t), zs(t)) The face with y-.1: os Is(s) ys(s, t) -1 zo(t) The face with y = 3:00 = (z.(), y(s, t), <s(t)) To(s) ys(s, t) 26(t) 3

Answer 1

F(x, y, z) = 2 x1 + 5 y j + 32 k. -3<x< 6 -Iky <3 -45255 div E 2 (2x)+ a (54) + +2 (32) 2 22 ar 2.1+ 5.1+ 3.1 = 10 gy dio f By divergence theorem, we have fonds HI - divf dv At di 6 3 5 To dz dy dre 13-3 y=-| 2=-4 y = -1 = (loz dy dx -4 6 G 3 6 4 dx = 360 (da = 1 1 9 dy do X=-3 y = -1 6of de =-3 X=-3

Schedule your appointmel 360 (2) USJdive du= 360 (9) 3940 Now we want to calculate Affie fonds 10 11

Schedule your Appointments, Events and Important Dates SINCE 199 Saturday H M دله 11 N 12 ds outwaad wrat Nbo mal 1 X=-3 소 2 ܝܝ ZE-4 -K 2= 6 - Sp> OBLC dydz ? Sz→ QAMC y = -1 dada ng Sz OANB dady SyANPM dydz y=3 dada 17uhdM PLC dody fonds = {{P. nas + ff .n indst T. mdst SSF inds FTS Finds + SSAñds D Cash S6 S NBLP Ž-5 Week 50 - Day 351 SL 33 December 2018 F T S S 2017 January * M T W T F S S wk M T W -8 1 2 3 1 1 2 3 4 5 7

on x=-3 3 5 8 fimos (ar î+ 5y9+3zk).(-a) dzdy 9 si y=-12=-4 10 = S- dx dz dy s 3 11 ff €2)(-3).dedya dzdy 12 y=-124 3 1 9 = 216 ff I dy = 54 (4) 2 3 5. 2x azdy 4 Sy 3 on az 6 Sinds Y = -1 2-4 ja dy y=-1 108 (3+1) 5 = 120 Ten=6 6 108 xy = 432

your Appointments, Events and Important Dates Schedule on y = -1 ê ñ ds = $ dada 65 9 f ff (ac+s4j+326) 69 H-Sy d2 dx = - -56-8ffd2d2 X=> 2=4*y=-1) 54 9 obe = 45_(643) 10 6 11 = 5 12 -3 45692 405 1 an y = 3 2 P. ñ ds = ff sy dzdx = 15 ff de der 6 5 3 4 6 X=-32=-4 = 15 9 dx = 135 [9] = 1215 un - 363 Z=-4 S3 Em ds = Islanit Syſt3zK): LR dyda = (-3)-4) ff dydx X=-3 y=143 December 2017 January 2018 wk MTWT F S S wk M T W T F S S 2-3 y=-1

Schedule your Appointments, Events and Important Dates =12 (42 dae 8 Jh3 9 48 +6+32 = 48 X 9 = 432 6 B F.nds 32 dyda g= 3 y=-1 on 2=5 SS HT S6 11 = 15 (42 dae 12 3 1 60 (6+3) 2 540 all valves in 3 letting à ñ ds 3840 ff 4