Question

9) State the Divergence Theorem and "verify" it for me re, y, z) =< 3x, xy, 2xz> and the unit cube F defined by verify" it for the vector field

Answer 1

Statement of Gauss Divergence theorem Let S be the closed surface enclosing a volume v. It Ē is a continuously . differentiable vector point Function Then div Fdy - Finds, where ñ is the out ward drawn normal vector at any point of s. Givens Ē (9,9,2)= <3n, ay, 222> and unit F = 371 +xy j +222 k cube E defined by Osm,y,zel div F dv - { die de aj s farsa) dan di de (3+34) da dy dzi 7-0 4-0 Z=0 divf = o.fi Hivf sine B 9 12 + lon 2 +k? Bai + Beyj+242 k ay 02 din F = + x +2x) = 3+32 0 Schiet doo (33) mosha -> ] [139 *3c09)- oJ distrz y=0 20 ( ( y . 4-0 2-0 Scanned with CamScanner

I dive du = 2 jtesia divp dv : 1-0) dz z=0 (1-0) . div F dr = 2 O YA > verification we will cal erlate the value of Pfinds over six faces , o of the cube (1) For si = PQAS: unit out ward normal ñzi = 1: ds = dydz 04yL1, OEZI '. Fiño 3x 31 is a 3 231 Sl Ends=1 l 3 dydz ZO = 3 j y l de = 3 j' (1-0) dz CS Scanned with CamScanner ZS-D

=3 tt-o] JS er ds as tas-- - for 52 - OCRB unit outward drawn normal ñ=-1 =0, ds = dydz : 0242 047 41 F.ñ = 3% -0 250 d o dydz = 0 ..soos - foto *. $$ Funds=-— @ ds = dady : ñ=k. for Sg = RBQP: 2=1, OSX41) 0444 Fn = 2x2 Fin = 2x2 =1 : Jf finds = , , 2x du dy 4-0 =0 =. J dy = yl y=0 : 1-0 =1 fin dy est d'with CamScanner

For s4 - OASC 2-O -K s - O4*10cy1 fin = -272 EN=0 Z0 En ds = 1 To dondy =0 u sasa x = y= D Ends = 0 - (V) ds a da dz Foy 5 = PSCR! PSCR U EL Y-, ña) ñ=j Ocael, Oszel Fin F = ny = y=1 ds = i l'a de de no 2-0 lo *} [231 ç (10) = { frids - 4 - 0 cs CSS Scanner

For SG= OBQA, Y=0, n=-j ds = dx dz Ocnel, Oczel En = -xy Fń - y=0 Finds = o dadz = 0 X=0 220 s Ende =0 @ Foods = $58 dat setias fetade u sfonds = 3 + 0 + 0 + 1 2 +0 - 4+1 Js ands - - From Equ o s Į div k dr = sfinds Scanned with CamScanner