Page 81 - Engineering Mathematics Workbook_Final

Vector Calculus
 ydx zdy +
+
182. The value of xdz y $ j + z k
$
C ( x + 2 y + 2 z 2 ) 3/ 2 ( x + 2 y + 2 z 2 ) 3/ 2
where C is the intersection of
2
2
2
x + y + z = a and x z a . $ $ $
2
+
=
where i , j , k are unit vectors along
183. The value of the axes of a right-handed rectangular /
)
( (  2x − ) y i − yz j − y zk dr . Cartesian coordinate system. The surface
2
2
u r
u r
u r
C integral   f  dS (where dS is an
Where C is the boundary of upper half of elemental surface area vector) evaluated
the surface of the sphere over the inner and outer surfaces of a
2
2
2
x + y + z = 1 above xy plane is spherical shell formed by two concentric
spheres with origin as the center, and
−
(a)  (b)  internal and external radii of 1 and 2,
respectively, is
(c) 2 (d) 0
(a) 4 (b) 0
184. Given vector
)
1 (c) 2 (d) 8
$
$
3 $
$
3
3
u = ( − y i + x j + z k and n as
3 [GATE-20-ME-SET 1]
the unit normal vector to the surface of
the hemisphere 186. The value of
)
( x + y + z = 1;z  0 ) , the value of (  4xi − 2y j + z k  nds where S
$
2
2
2
2
2
S
)
integral (   u  n dS evaluated is bounded by x + 2 y = 2 4 , z = and
0
on the curved surface of the hemisphere z = 2 is
S is
(a) 16 (b) 48
 
(a) − (b) (c) 32 (d) 84
2 3
$
 187. The value of  F  nds where S is the
(c) (d)  S
2 surface of the sphere x + y + z = 4
2
2
2
$
[GATE-19-ME-SET 2] where n is the unit normal and
185. A vector field is defined as F = xi + y j + zk is ________
ur x
f ( , ,x y z = ) 3/ 2 $ i + (a) 32 (b) 16
( x + 2 y + 2 z 2 )
(c) 8 (d) 64

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