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Module V

Answer any three questions, each carries 5 marks.
Prove that ‏مل‎ (x? —yz)t+ (y? - ‏رصع‎ + (22 —xy)k . 67 is independent of the

path and evaluate the integral along any curve from (0,0,0) to (1,2,3).

Evaluate ‏عل‎ xy? dx +xy dy where C is a triangle with vertices at (0,0), (0,1)

and (2,1)

Evaluate ‏مل‎ 229 dx + (x? + y?)dy along the curve C:x = cost,y = sint,
7

0 > 2 > 3

Determine whether F(x, y) = 6൦ i+ ‏نم12‎ 15 a conservative vector field. If

so find the potential function for it.

107 = (sinz + ycosx)i + (sinx + 2cosy)j +(sin y + x cos z)k ,find

Div F and Curl F.

Module VI

Answer any three questions, each carries 5 marks.
Using Stoke’s theorem, evaluate | நீ F .d7 where C is the boundary of the

projection of the spherex? + y? + z? = 1 on the XY plane with

ह = நேர - y*zk
Using Green’s theorem evaluate ८ (y? — 7y)dx + (2xy + 2x) dy where C is the
circle x? +y? = 1
Evaluate using divergence theorem for F = x?i + 2j + yzk taken over the cube
bounded by x=0,x=1,y=0, y=1, z=Oandz=1

Evaluate the surface integral ff, 22 ds, where o is the portion of the curve

2 y?
Use Green’ theorem to find the area enclosed by the ellipse > 451
a b?

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