Page 63 - Engineering Mathematics Workbook_Final

Vector Calculus
−
−
(a) 8 (b) 4 (c)  r  dr = 0 for every piece wise
C
(c) 4 (d) 8 smooth closed curve C in D.

[JAM 2008] (d) The differential equation
+
pdx qdy is exact in D.
) xy +
−
2 2
T x
52. Let ( , , y z = 2 2z x z be
[JAM 2009]
the temperature at the point (x, y, z). The
unit vector in the direction in which the 54. The value of C for which there exists a
temperature decreases most rapidly at twice differentiable vector field F with
)
(1,0, 1 − is ________
+
curl F = 2xi − 7y j czk is _______
1 $ 2 $
(a) − i + k (a) 0 (b) 2
5 5
(c) 5 (d) 7
1 $ 2 $
(b) i − k [JAM 2009]
5 5
x $
$
$
2
2
2
x
x
2 $ 3 $ 1 $ 55. Let F = 2xyze i + ze j + ye k be
(c) i + j + k the gradient of a scalar function. The
14 14 14

 2 $ 3 $ 1 $  value of  F dr along the oriented

 
(d) −   i + j + k C
 14 14 14  path L from (0, 0, 0) to (1, 0, 2) and then
to (1, 1, 2) is ______
[JAM 2009]
(a) 0 (b) 2e
$
)
53. Suppose V = P ( , x y )i + $ q ( , x y j is 2
(c) e (d) e
a continuously differentiable vector field
2
defined in a domain in R . Which one [JAM 2010]
of the following statements is not Let F = + −
equivalent to the remaining ones? 56. xyi y j yzk denote the
force field on a particle traversing the
(a) There exists a function ( ,x y ) such path L from (0, 0, 0) to C (1, 1, 1) along
the curve of intersection of the cylinder
 
)
=
2
that = p ( ,x y and y x and the plane z x. The work
=
 x
  = q ( ,x y for all ( , x y  done by F is __________
)
) R
 y
1
(a) 0 (b)
 p  q 4
(b) = holds at all points of D.
 y  x

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