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Problem 18 Use the Divergence Theorem to calculate the surface integral || FdS , F(x,y,z) =< x²yz,xy-z, xyz? > S is the surface of the box enclosed by the planes x = 0, x = a, y = 0, y = b, z = 0, and z = C, where a, b, c are positive numbers.

Answer 1

Problem 18:- Given Vector E= <ayz, ny, ry zry.. . Given S is the surface of the boni enclosed by the planas . 1-0, 4-a, yao, yab, F-0 and 70' which is Shown belonio Now SF-nds = S + S + S + + $ 0 **** Notas s S halbere S, 8, ....So are the 6 faces of all p l the box bounded by given plany. Do S;: OBCE, Sy: ABCD 9:0GAB Sg: ADFG S: OEFG S6 : COEF Now on the faces: OBCE We have $70, dss dady, ñ=- Now [Fonds = s. * Senigayir Qay'357 Cya”)) • (-E)dandy Soyed a go go mayat dady Y=0 ya = fsb 2, ).dnay (130) N:ota Now on the face uş: OGLAB We have y co, dse dads, na I' 3:00. Now Foods e f (Bugyi Hajj tayzu) ). j) daudz O 00 - je l'énys) dads -jj-100)=dadx

10 b - on the face : OFF We have ñ= -2, x=0, ds=dydz. . . 9:07b, 3:0-e... .: S Funds = gb sp Cangasit (kg)]+(9 357) ).0)dy dos yoo 320 . c = [ -.93dy dit ]-Xy3 dydt = . yao 720 - yao 320 20 (1:*=0). on the face : Sui ABCD: 1. We have nza, n=ē, ds. dydis :07, 3:0+. [/F15 ds = L / Caja) dy ds - 54 yoo 0 za sody (Sžds) că. CED makes a big

Aloed on the face sv. Antes We have 306, d9= drdy. ñ= 2:07a, 4:0-56. Noro Decade a S Sonygos dady. no yed so share y dady = cüde) Clien) .. .): (?) 3: 00 Now on the face. Se: CDEF w We have ob do dedz, raj. n.osa ... 5db = s fcut= egy nyo? a)..duds "56 - so scegah dads - La fuchy z dodo = $. Cladu(køds) = b. a - bm. 96 1. From od onoga ons Finds = 070704 vit

By By Game Fido como Gauss Divergence theorem. I Founds = { divendu Casanoſt rutit (again) Sina F = Guya) + (499)ſt (1997) € . divF= 0.F = 2 (vý ) (xyz)+ ( 242") = 2ny3 + 2xy 3 + 2xyz c. Gogg CO ya Now Selin Edo - ano en el rango dude = of Játu) (sin) (jáds) <0.680 aby". = S'Fiodo Thus LF-5d - Sdnos do = { Caley