Page 68 - Engineering Mathematics Workbook_Final

Vector Calculus

84. If A & B are irrotational vectors then 88. Evaluate  f  dr where

the divergence of A B is ______ C
f = ( x − yz i + 2 xz ) j + ( z − xy )k
) ( y −
2
2
−
(a) 1 (b) 0 from (1, 1, 0) to (2, 0, 1).
−
(c) 2 (d) 3− (a) 7/3 (b) 6 −
)

x
85. Let r xi y j zk= + + . If ( , , y z is (c) 0 (d) 8
a solution of the laplace equation then Greens Theorem
r
the vector field V + is

89. The value of  A dr where
(a) neither solenoidal nor irrotational C
A = (2y − y i + ) y j and C is
) (2x +
2
(b) solenoidal but NOT irrotational
the closed curve of the region bounded
(c) both solenoidal and irrotational by y = x and x = y , is
2
2
(d) irrotational but NOT solenoidal
(a) 1/2 (b) 3/10
Line Integral (c) 1/6 (d) 4/5



86. The value of the line integral F dr 90. The value of the line integral
2
2
+
, where F = 3x i + (2xz − ) y j zk  c ( y dx + x dy ) where C is the
2
and C is the straight line from (0,1, 1 − ) boundary of the square bounded by
=
x = 0 , x a , y = , y a is
=
0
to (1, 2, 0) is
)
+
−
(a) 1 (b) 3 (a) 0 (b) ( 2 x y
(c) 0 (d) None (c) 4 (d) 4/3
87. The work done in moving a particle in  − )
the force field 91. Find the value of (x dy y dx
) (2xy +
f = ( x − y + x i − ) y j around the circle x + 2 y = 2 1
2
2
x
2
along the parabola y = from (0, 0) to
(a)  (b) 2
(1, 1) is
−
(c) 0 (d) 
(a) 24 (b) 18
(c) 3/2 (d) 2/ 3− Surface Integral

92. Evaluate  f  nds or   f  n ds
S S
over the surface of the cylinder

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