Page 61 - Engineering Mathematics Workbook_Final

Vector Calculus

39. Let ∫ (2 − ) − − where C is

C = (  , x ) y  R 2 :max  , x y = 1 the circle x + 2 y 2 1 = , z = , oriented
0
  .
The value of the line integral counter clockwise. [JAM 2005]
( ))
( ))
(  xy + 2y + sin e x dx + ( x y + cos e y dy
2
2
C 43. The vector field
) ( x −
4
is F = ( 2xy − y + 3 i + 2 4xy 3 ) j
(a) 0 (b) -2 is conservative then its potential and also
(c) -4 (d) -8 work done in moving particle from (1, 0)
)

+
+
=
x
40. Let r xi y j zk . If ( , , y z is to (2, 1) along some curve.
a solution of the Laplace equation then [JAM 2005]
the vector field (   r +
) is _________
44. Let C be the circle x + 2 y 2 1 = taken in
the anticlock wise sence. Then the value
(a) neither solenoidal nor irrotational
of the integral
(b) solenoidal but not irrotational
 (2xy + ) y dx + (3x y + 2x ) dy =
3
2
2
(c) both solenoidal and irrotational C
_________
(d) irrotational but not solenoidal

[JAM 2005] (a) 1 (b)
2

(c)  (d) 0

41. Let F = xi + 2y j + 3zk , S be the [JAM 2006]

surface of the sphere x + y + z = 1 45. Let r be the distance of a point P(x, y, z)
2
2
2

and be the inward unit normal vector from the origin O. Then r is a vector
⋀
to S then   F  $ ________.
n ds is equal to
S
_______ (a) orthogonal to OP

(a) 4 (b) 4− (b) Normal to the level surface of r at p.
(c) normal to the surface of revolution
(c) 8 (d) 8−
(d) normal to the surface of revolution
[JAM 2005] generated by OP about y axis.

2
2
2
42. Let S be the surface x + y + z = 1, [JAM 2006]
z = 0 use stoke’s theorem to evaluate

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