Page 62 - Engineering Mathematics Workbook_Final

Vector Calculus

46. Let S be a closed surface for and let ( , ) = + or ( , x y 
) D
2
which∬ . = 1. Then the volume
̅
̂
$
. If n is the outward unit normal to C,
enclosed by the surface is __________
nds , evaluated counter-
then     $
1 C
(a) 1 (b) clockwise over C, is equal to _______
3
2 (a) 0 (b)  − 2
(c) (d) 3
3 (c)  (d)  + 2

[JAM 2006] [JAM 2007]

2
R
47. Let : f R → be thrice differentiable 50. The work done by the force
+
2
and vanish on the boundary of the region F = 4yi − 3xy j z k in moving
)
( 1,1  −
 = − ) ( 1,1 . Then particle over the circular path
0
)
  1 1 div (grad f )( , x y dx dy is x + 2 y 2 1 = , z = from (1, 0, 0) to (0,
− 1 − 1 1, 0) is _______
_______
(a)  + 1 (b)  − 1
(a) never 0 (b) 1
(c)  − 1 + (d)  − 1 −
(c) 0 (d) depends on f
[JAM 2008]
[IISC 2007]
3
,
48. Let 51. Let V = ( { , , y z  x ) R
+
u = (ae x sin y − 4x i + + x cos ) y j azk 1  x + y + z  1} and
) (2y e
2
2
2
, where a is a constant. If the line integer 4
 u dr over every closed curve C is F = xi + y j + zk for

C ( x + 2 y + 2 z 2 ) 2
zero, then a is equal to ___________
$
( , ,x y z ) V Let n denoted the
−
(a) 2− (b) 1
outward unit normal to the boundary of
(c) 0 (d) 1 V and S denoted the part
  1  
) R x +
[JAM 2007]  ( , , y z  3 : 2 y + z =  2
2
x
  4 
49. Let C denote boundary of semi-circular
disc of the boundary of V. Then
$
)
D = (  ,x y  R 2 ;x + y  1, y   0   S F  nds _______
2
2

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