Page 56 - Engineering Mathematics Workbook_Final

Vector Calculus
 ( r r r  ( r r ) and y + z = 2, then which of the


) 
(II) div a r r = a r
  ur r

following is (are) equal to C  F dr
then,
?
(a) both statements are true 2
(b) only statement I is true (a) 0  0  1 (1 2 sin+ r )r dr d

(c) only statement II is true  1 2 
(d) both statements are false (b) 0  2 0 1  2 + 3 sin      d

[JAM GP 2010] 2 1  

3
3
8. Let : f R → R be a scalar fields (c) 0  0  (1 2 sin + r )dr d
r
: v R → R be a vector field and let (d) 0  2 (1 sin + )d
3
3
r r
a  R be a constant vector. If r
3
represent the position vector [JAM MA 2015]
$
$
+
x i + $ y j zk , then which one of the 10. Consider the vector field
ur
)
$
$
−
following is FALSE? F = r  ( y i x j , WHERE   ,
R
r
r
r
( )
$
(a) curl f v = grad ( ) f  + f curl v r x i + $ y j and r = r
v
( )
r =
r . If the
(b) absolute value of the line integral
ur
r
 F  dr along the closed curve
  2  2  2 
div ( grad f   + + 2    f C
( )) = 
2
  x 2  y 2  z  : c x + 2 y = 2 a (oriented counter
(c) ( r r ) r r clockwise) is 2 , then  is

curl a r = 2 a r
(a) -2 (b) -1
   r  r r
0
(d) div   r 2 r   = 0 , for r  (c) 1 (d) 2
  r  
  [JAM MA 2016]
11. The values of the line integral
[JAM MA 2018]  
3
2
ur  (3x + 2xy ) dx + ( x + x 2 ) dy  
9. Let F be a vector field given by  
ur from M (0, 0) to N (1, 1) along the paths
)
$
$
$
3
F ( , ,x y z = − y i + 2xy j + z k , for 2
: =
C 1 : y = x and C y x are,
2
( , ,x y z ) R 3 . If C is the curve of respectively
intersection of the surfaces x + 2 y 2 1 = (a) 2 and -1 (b) 3 and 3

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