Page 65 - Engineering Mathematics Workbook_Final

Vector Calculus

63. The  yzdx + (xz + ) 1 dy + xydz , (c) x + 1 2 x + 2 2 2x x
1 3
C
where C is a simple closed curve, equals (d) (2 , x x + 3 , x 1 ) x [IISC 2002]
1
1
____________
67. A unit normal vector to the curve
(a) 0
( 
, , 
2
C : x x x R ) in the plane R at
2
(b) 3xyz + y
the point (0, 0) is given by _________
(c) length of C
)
−
(a) (0, 1 − ) (b) ( 1,0
(d) area enclosed by C [IISC 2005]
 1 1 
2
64. Let D be the square in R with vertices (c)   ,     (d) (1, 0)
(0, 0) (1, 0) (0, 1) (1, 1). The integral  2 2 

∫ where D is the boundary of
[IISC 2002]
the square, is equal to ________
3
F
(a) 0 (b) 0.5 68. Let : R − 3   0 → R be the vector
x
F
(c) 1 (d) 1.5 field defined by ( ) x = where
x
[IISC 2005]
x = ( , , x x  3   0 and
) R −
x
1
3
2
65. Let
2
2
2
) ( x y +
V = 2xyzi + ( x z y j + 2 3z 2 ) k . x = x + x + x . Then the
+
2
3
2
1
divergence of F (x) is ____
̅
Then the magnitude of curl at (1, 1, 1)
is ____ 1
(a) x (b)
(a) not defined x
(b) 1 2
(c) (d) 2  x
(c) 0 x
(d) strictly greater than 1 [IISC 2005]
[IISC 2003]
3
3
66. Let :V R → R be the vector field 69. Let G be the tetrahedron in R with
3
defined by vertices (0, 0, 0) (1, 0, 0) (0, 1, 0) and (0,
: x +
2
V ( , ,x x x 3 ) ( 1 2 x 2 2 , x x + x 2 3 , x x + x x ) 0, 1). The outward flux of the vector
2
1
1 2
1 3
2
the divergence of V is ________ field
(a) 4x + x V ( , ,x y z ) 2 cosz= ( ) ( z− 2 + ) 1 j yz+ 2 sin xy
xy i
( )k
1
3
across the boundary of G is _________
(b) 0

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