Question

7. Find (a) the curl and (b) the divergence of the vector field F(x, y, z)= e' sin yi+e' cos yj+zk F.de where is the curve of intersection of the plane : = 5 - x and the cylinder rº + y2 = 9. 8. Use Stokes Theorem to evaluate F(x, y, - ) = xyi +2=j+3yk

Answer 1

The answer is given as

7 (a) - The curl of vector field Frary, z) = e sing it e by jt zk È Curl li k leasing en Losy z File - e *103)-3 Coresiny) - i (0) – 3 (0) + E [ et cosy - e a cosy) = ico) – 3 (0) + R (0) ² zol Turl

7(b) : 2x Divergence of the vector field ²(x, y, z) = e singite cosyj+Zk div. (P) = V Ê = (1 + 50 by + kg) • F =(18x+ 5 mg +K 2): (emsing i +ePogo Letsing) + 3 ay l ex cosy) + Sz (2) - "siny – ex siny + 1 t = 1 - [ air (P) = 2 div ( -

& we have to evaluate SF. der I by using stokes theorem where È cm 9; 2) = nyi+azĝ+ 3yr and c is the curve of intersection of the plane z = 5- and the cylinder x² + y2 = 9. Stoke's theorem state that - I SF. ds - Scurt F. ñ ds 0 Now first . . curl F - colă wcle E con az sy az 30 | il 3-2] -3 (0-0) +R [6->) ] = i–x Ŕ curl Now we para me trizes. take a=u, y=v & z=5-x=5-4

10. is- Henco the parametrization i luid) = 24, 8, 5-u> Now riu Llog - 1> <0,1 0 > = a eru x OIA = 11 To ! i(0+1) 3 (0)4k (1) = i + = [1,0, 17) - 1 ru x hence we have -

Ssuarl F. n as Glu, wo) (ru xh) da - n taking card f = G} =SS<1,0 ,-43.<1,0,1) da = 5S [1+0-4] DA R

و 273م ماه وهو (ی 0 9 - 1 " = ا ر د مه. [ope) 9 - =

- Sora do get a caso do - 9. 20 – 9 [sin o] 2* = an – a coro] q r hena S curl P. ds – at hence from equ. (1) - P des – at I