Question

(a) [4]Use the Divergence Theorem to prove that ſſF. n dS = 8, S where S is the surface of the cube -15 x 51, -15y51, -15 z 51 oriented outward, and F(x, y, z) = (4y2, 32-cosx, z’ - x). (b) [1] Would it have been easier to just calculate the given surface integral directly? Explain.

Answer 1

Hope the solution will be helpful. Thank you.

solution: (9 Here we have, F(2,4, 2) = < 443, 32-C3, 22->) Here we have also, - exsle :-). 3 Y 21, -142=1. Now , 1 VF = (3x+ 39173 RX 4821+(37-6055) 5+ (228) => VF = 0 + 0 + 322 => V = = 322 the divergence Now according to theorem ISFn SIS OF av Fonds V where dr dr = dady dzi

(parues? = 8 spul ISF. 4-x 2 درا درا 12% 12 [+1] + . 12 = T. PE [[+1].cz), ZP ZZ مج a - Z P 2 1- ZP 6' 22.707, hi] 9 a are il = ZP hp ze -1 Zphp [i+1] EE zphp , Ix ZE ✓ ni |- |- |- S Zphpkp z 5 = {{S, = spurt is O

(6) It been easier would have not to just calculate the giren surface integral directly since of here is the surface it has 6 faces. S СА cube 7

NOW the of 'n' and the Coordinate value anes face, will be changed in cach And we need to calculate the value of the integral s sfonds .( S six will for each faces. which be long process compare using divergence theorem.