Question

Verify the Divergence Theorem by evaluating I SF F. Nds as a surface Integral and as a triple Integral. F(x, y, z) = 2xi – 2yj + z2k S: cube bounded by the planes x = 0, x = a, y = 0, y = a, 2 = 0, z = a

Answer 1

= S # To verify . we first the divergence theorem III div F'dV FoWXS, calculate the volume interne and then surface integral, fo a fi î of Fa Î f F ĥ (sory) 2x 2 -2y7+pen 272 IFI 07 22 ha 28 1 & 26t 27 Bon dirigenzoys DENPADS Ils dividu E 3 11 2pkp XP 20 رز B Y=0 ZO a D a a oo t 5. da feady be de - G 0 E •277757 2 7 × ax a² lan 11

the we S с B and then A D S PRIS (627-28+) We shall now to calculate surface integral: Il Finds calculate // Ends over the six faces of the cube . seperately add these six results. GEAD . ry 11 G - trådatan GEAD = . 2x dy dz GEAD On the face GEAD, 11 ja laady 2=0 y= a x=2Const. and y anda both Verdes O to a a 2 20: s dy / dz 0 ñ , dx=0 2 Ñ ds = dyda 20. [9]."[z]." 2a. Q. a 2 203 on the face FBAE, ñ - Î , US Foods - Il corpo zaſız?k). (doh + dzdaj + hool) Y = a - dyzo FBAE FBAE Ñ as= 7 dzdx a 1 / 629) de du -20.a.a -1-2017 LED 720

üi) AÑOS Oven the face, Gold, Azeri Ñ = - y=0 dy - ds.' -ġdadx GOCD 2 S S (22 " - 20.0t zzi). Alisation dodaſ salona) GOCD PX021 ə over the face ABCD, ĥ Berganti iv) ss Funds and = constant. ABCD Z-a dE=0 .ñ ds = dedy a a 11 // Car ? aga +222) 0 (tedy ff) x=0 Y=0 ſ Seco a2 dx dy . 20 son jary 22 2 YO az, a.a a4 over the (V) 2=0, SI Èñds EFOG (2x2-22h). Eidemedy face EFGG, A =) NZ=0, and Ñ = -ĥ → N ds=-Ŕ dxdy FF.xmrftcovered = 3x - 247

(vi) Ovely the face OFBC, // 7hds and x=0 7 daro OF BC 11 - ñ ds = _î dydz and F= (-2y 9 +2²2) ; Fonds = (-2y +2212). Lideos = 0 Now the abore six. results, we have adding SS E. WAS = (203) + (1222) +(0) +(09) +(0)+ (0) S - a4 (u) 个 (0) زی یع Thus from Il Pads - as - Cantar у S is verified. Theorem le the divergence

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