Page 59 - Engineering Mathematics Workbook_Final

Vector Calculus

1   ur $ $ F ( , x y = ) y i − x j . Let
F 
 S n ds , where n is the unit x + 2 y 2 x + 2 y 2

outward drawn normal to the surface S,  , : 0,1 → 2

  R be defined by
is _____.
t =
 ( ) (8cos ,17sin2 t  ) and
t 
[JAM MA 2016]
 ( ) (26cos2 , 10sin2 t  ). If
t =
t  −
26. The line integral of the vector field
ur 3   F  dr − 4  F  dr = 2m ,
$
$
$
+
F = zx i xy j + yzk along the
then m =
boundary of the triangle with vertices (1,
0, 0), (0,1,0) and (0,0,1), oriented 29. Consider the unit sphere
)
anticlockwise, when viewed from the S = (  , , y z  R 3 : x + y + z =  1
2
2
2
x
point (2,2,2) is
)
and the unit normal vector n = ( , , y z
x
1 −
(a) (b) -2 at each point (x,y,z) on S. The value of
2 the surface integral
1   2x   y   2z   
2 

z


(c) (d) 2   S   + sin y 2  x +    e −    y +  + sin y z d =
 
 
 
2             

[JAM MA 2017]
30. Let
27. Let S be the surface of he cone
)
 = (  , , y z  R 3 : 1 x y z   1
− 
x
, ,
z = x + 2 y bounded by the planes
2
z = 0 and z = 3. Further, let C be the and :   → R be a function whose all
closed curve forming the boundary of second order partial derivatives exist and
ur are continuous. If  satisfied the
the surface S. A vector field F is such

ur Laplace equation  = 2 0 for all
$
$
that  F = − x i − y j . The absolute ( , ,x y z   , then which one of the
)
ur
r
value of the line integral  F  dr , following statements TRUE in  ?
C
where
(a)   is solenoidal but not irrotational
(a) 0 (b) 9
(b)   is irrotational but not solenoidal
(c) 15 (d)
(c)   is both solenoidal and
[JAM MA 2016] irrotational
28. Let F be a vector field defined on (d)  is neither solenoidal nor
R 2 / 0,0 irrotational
  by

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