Page 80 - Engineering Mathematics Workbook_Final

Vector Calculus

168. Value of the integral [GATE-2014-EE-SET 1]
C  ( xy dy − y dx ) , where C is the $
2

178. The value of   curlv n ds where
square cut from the first quadrant by the S
2
lines x = 1 and y = 1 will be (use v = 2yi + 3x j − z k and S is the
Green’s theorem to change the line upper half surface of the sphere
integral into double integral)
$
2
x + y + z = 9 , n is the positive
2
2
1
(a) (b) 1 unit normal vector to s and c is its
2 boundary ____
3 5 (a) 3 (b) 9
(c) (d)
2 3
(c) 18 (d) 32

[GATE-2005] $
)
179. The value of  (   F  n ds
− yi + x j S
169. For a > 0, b > 0 let F = zi + x j +
b x + a y 2 where F = yk and S is a
2 2
2
be a planar vector field. Let hemisphere z = 1 x − y of unit
2
2
−
C = (  , x ) y  R 2 / x + y = a − b 2  radius above xy plane.
2
2
2
be the circle oriented anticlockwise. (a)  (b) 2
Then  F  dr = _____
C 
(c) (d) 4
2 2
(a) (b) 2
ab 180. The value of
 sin zdx − cos xdy + sin ydz

(c) 2 ab (d) 0 C

 xdy − ydx where c is the boundary of the rectangle


0 x  , 0 
y 
4
170. The value of 2 2 taken in 2 , z = .
C x + y
the positive direction over any closed (a) 1 (b) 2
continuous curve C with origin inside it.
(c) 3 (d) 4
171. The line integral of function F = yzi, in
the counter clockwise direction, along 181. The value of  F  dr where
the circle x + 2 y = 2 1 at z = 1 is C
2
F = y i + x j − (x + 2z )k where C
2
−
(a) 2 (b) − is the boundary of the triangle with
vertices at (0, 0, 0), (a, 0, 0), (a, a, 0).
(c)  (d) 2

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