Question

2.) Show that the fundamental theorem of divergences (aka Gauss's theorem, aka Green's theorem), shown below, holds for the (vector) function v from the previous problem. (Use the cube shown below as the basis for your work; the cube has sides of length 3.) fundamental theorem of divergences (V.v)dr v-da 24 A(v) (ii) 47 (iv) (ii) (vi)

Answer 1

| Given the following vector function 0 =(x)% + 2 +1) + 1 ) 7.ū= ( Oxt $ 2y +233)-) +1242) g déanja? - Da (23) + 2y (242) +32 (388) - 1+22 131 1 2 3 4 | 3x + y + 2z Cartesian unit vectors.

regral of the normal lenon ū taken around Gauss's theorem of Divergence : Statement: are surface integral of component of a vector function & fak a closed surface is equal to the integral of the divergence of o tarken over the volume ? enclosed by the surface og. Mathematically slü ñda = JIJ Gudo proof: let go in 2282(19) Let 5 = vititur 29 Putting the values of o, and n in the statement of the divergence theorem 4 72.fr(wy) we have J (vitu, itupk) n ds. SS56 a to by ok 31) (0,1+0.3+h; x dadyda us ou ons opdx dyd?

we have to evaluate and prove equation . so let us evaluate zadalary) - S au dedy dz = J L dos de indy pe you get a zadilary Of w = 1 { uz (X, Y, dz) - U₂ (x, YS,) andy for the upper part of the surface ire se we have dady dia cost, a M. hdse !! Again for the lower part of the surface ies, we have dady=-ds, coss, 2-ni, kids, so, If Us (x, y, f2) dady = // Vg nz r dsa id JJ Uz (r, y, fi) dady = -1/8 nikds, Pulling the values from equation and inequation ③ we get and

ou die dydz - Jl Bus in . k ds, JJ ogni ads, E or, JJ d dr = uz non da similarly, it can be shown that solg t is avaient = do - Il audy als viñeida Now adding equation Ô © and we vu get Wome 02 I where af (hituitogů) . ñda or iss dr = 11 Buda a entido JJ F-dy = S/, Fida Proved) the This is the Gauss's Divergence fundamental law of theorem.
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