Question

(1 point) This problem will illustrate the divergence theorem by computing the outward flux of the vector field F(x, y, z) = Aci+ 4y + tek across the boundary of the right rectangular prism: -ISXS 4.-2 Sys7.-2 Szs 7 criented 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 05:51 and Sis 1: The face with z=-2: 61 = (x+(3), y(0), 2(5.1)) x]() = yı(t) - zi(s. 1) = -2 The face with 2 = 7: 63 = (x2(8),y20), 22(5.1)) ya(t)- 22(3,1) 7 The face with x=-1: X3(8.1). 73(8), 2300) x3(3.1) = -1 y(s) z3(1) = The face with x = 4: 04 = (xas. 1), ya(s). z4(1) x4(3.1) 4 y(s) - 24(1) The face with y=-2: os = (x3(8), 79(3.1), 25(1) xS() = ys(s. 1) = -2 zg(t) = The face with y = 7:06 = (x6(8). 76, 7). 26(0) x6(s) yos. 1) 7 ze(t) = Then (mind the orientation) der: d84 do4 F.Ads Fai) dsdt + F(02) dsdt + Fay) dsdt + X dsdt dr ds de 1 dery dery des des Fay) ds de + ds de -%% +6L come out these + S'I +60 tehnika Flow X х Part 2-Using the Divergence Theorem ST" 5-SS TV Finds dividV dx dy dz

Answer 1

> Now Za=7 202 = Noel 1 Simillary ne 42 24-12+97 7 Given, FGG, 4, 7) = 4ni + 4y +42k with fandony alsus4, -2 sy 57, -237<7 Part-1: osast, and ous • Noce Z 3-2, 0,= (x,G), y, (+), 7 Cánt)) 3. X. (3)=T-1+55) 8.4)-1.2 + got 2,- -27 Gis (-1+55,-2+95,-2) 69 = ((126), yo(L), 72 (8,4)) (o)=1-1+68 49+).F2 + 9+ [-!+55,'2+ 97 , 7) X:-1, 63= (x3 (s, t), yz (5) - zlt)) Y = -1 89-3495] Zz--2+ 9+ 03 -6, -2+ 99,-2 +9+) for 4.4 (3) = [- 2+95 y=-2, x5(s)-1-1+38] 76 ya 72 no (o)= ago 76 - Then, Fa <40g. 47,42), 0,- (-1+55, -2 +97,-2) . i (61)= (-4+205., -8+ 367,-8) 2004 2003 = <0,5.0).(2.5,0,0)= (0,0545) s'<-4+208,-8+364,-1)(0,0:45)z 1360 Similarly, for others, then we get, sf Findia 98601 O dir F 2 Pa (4x) + Sy (44) +32 (42) - 4+4+4= 12 (4+1)(3+2)(2+2) SIF Fonds 12 dx dydz z gogo from 2 + 9+ 2+9 360]+(1960]+[324]+[296] +363] + 1260) 4 7 7 1२४ 21

Thank you so much sir