Vector Differential and Integral Calculus

Prepared By: Satyaban Panigrahi, NIST, Berhampur

+ + =
r
. Find curl V
r
[APRIL-2004]
3. Let k z x j y x i x F

2 2 3
+ + =
r
. Find F div
r
[APRIL-2004]
4. Prove that . . . ) ( v curl u u curl v v u div
r r r r r r
= [APRIL-2004]
5. Find the directional derivative of
2 2 2
1
) , , (
z y x
z y x f
+ +
= at a point P(3,0,4) in the direction of the
vector k j i a

+ + =
r
.[APRIL-2004] [APRIL-2007]
6. Find a unit vector perpendicular to both j i a

2

3 =
r
and k j i b

2

+ =
r
.[JULY-2004(S)]
7. Find the unit vector normal to the surface 6
2
= + yz y x at the point (2,3,-2). [JULY-2004(S)]
8. Determine the vector k z j y i x V

+ + =
r
is solenoidal and/or irrotational. [JULY-2004(S)]
9. Find the projection of a
r
on b
r
if k j i a

6

12 + =
r
and k j i b

4

2 + + =
r
.[JULY-2004(S)]
10. Find )

(
2
k xz j y i xy curl + + at (-2, 4, 1). [JULY-2004(S)]
11. Determine the scalar point function if it exists for k z j y i x u

2 2 2
+ + =
r
.[JULY-2004(S)]
12. Find a unit vector perpendicular to k j i a

3

2 + =
r
and j i b

2

4 =
r
.[APRIL-2005]
13. Find the directional derivative of function
2 2 2
2 ) , , ( z y x z y x + = at the point (2, -2, 2) in the direction
of k j i

4

2 .[APRIL-2005]
14. If
2 2 2
) , , ( z y x z y x + = , find the value of ) ( grad curl at any point (x,y,z). [APRIL-2005]
15. Find the volume of the tetrahedron whose vertices are (1,3,6), (3,7,12), (8,8,9), (2,2,8). [APRIL-2005 ]
16. Find F div
r
and F curl
r
at the point P(1,2,3) if k xyz j z xy i yz x F

2 2 2
+ + =
r
.[APRIL-2005]
17. Find the divergence of
3
r
r
u
r
r
= at (-2,4,1). [JUNE-2005(S)]
18. Find the curl of the vector field k xz j y i xy u

2
+ + =
r
at the point (-2,4,1). [JUNE-2005(S)]
19. Find a vector of length 10, which is perpendicular to j i a

3

2 + =
r
and k j i b

2

+ =
r
.[JUNE-2005(S)]
20. Find the directional derivative of function
2 2 2
4 3 ) , , ( z y x z y x f + + = at the point (1, 0, 1) in the direction
of k j i

+ .[JUNE-2005(S)]
21. Find a scalar point function such that k z j y i x grad

+ + = .[JUNE-2005(S)]
22. Verify the identity 0 . = u
r
when .

cos

sin k e j y e i y e u
x x x
+ + =
r
[JUNE-2005(S)]
23. Find the vector projection of ) 7 , 4 , 3 ( = a
r
in the direction of ) 2 , 5 , 2 ( = b
r
. (APRIL-2006)
24. Test whether the vector v = (x, y, -z) is irrotational and incompressible. (APRIL-2006)
25. Find the unit normal vector at (2, 2, 3) to the surface . 26 2
2 2 2
= + + z y x (APRIL-2006)
26. Find the angle between the two planes x+y+z = 8, 2x+y-z = 3. (APRIL-2006)
27. If , ) , , (
2 2
z y x z y x f + = calculate Curl (grad f). (APRIL-2006)
28. If
2 2 2
z y x f + + = find the value of Curl (grad f). [JUNE-2006(S)]
29. Find the unit vector normal to the surface
2 2
y x z + = at the point (1, 1, 2). [JUNE-2006(S)]
30. Show that g f g f g f div + = . ) (
2
.[JUNE-2006(S)]
31. Show that v v div grad v Curl Curl
r r r
2
) ( ) ( = .[JUNE-2006(S)]
32. Find the components of ] 4 , 0 , 4 [ = a
r
in the direction of ] 1 , 1 , 1 [ = b
r
. [APRIL-2007]
33. If ] 3 , 0 , 4 [ = a
r
, ] 1 , 1 , 1 [ = b
r
, ] 1 , 2 , 1 [ = c
r
find ] [ c b a
r
r
r
. [APRIL-2007]
34. Prove that curl(grad f) = 0. APRIL-2007
Vector Differential and Integral Calculus
Prepared By: Satyaban Panigrahi, NIST, Berhampur

Page- 2 -
35. If z z e y e z y x f
y x
cos sin cos ) , , ( + + =

find f
2
. [APRIL-2007]
36. If a
r
and b
r
are any two vectors prove that
|

\
|
+ = + +
2
2
2 2
2 b a b a b a
r
r
r
r
r
r
. [APRIL-2007]
37. Find the volume of the tetrahedron of its vertices are (1, 3, 6), (3, 7, 12), (8, 8, 9) and (2, 2, 8). [APRIL-
2007]
38. Find the unit vector perpendicular to k j i

+ + and k j i

2 . [JUNE-2007(S)]
39. Find the directional derivatives of
2 2 2
) , , ( z y x z y x + = at the point ) 2 , 1 , 1 ( J[JUNE-2007(S)]
40. Determine whether the vector is k z j y i x

+ + solenoidal; irrotational. [JUNE-2007(S)]
41. Find a scalar point function such that k z j y i x grad

2 ) ( + = . [JUNE-2007(S)]
42. Find the directional derivative of the function y y x y x 4 ) , (
3 2
= at the point (2, -1) in the direction of
the vector ) 5 , 2 ( = v
r
. [APRIL-2008]
VECTOR INTEGRAL CALCULUS
1. Evaluate

+ +
1
0
1
0
1
0
dxdydz e
z y x
[APRIL-2004]
2. Evaluate

+
+
5
1 0
2
) 2 1 (
x
y x
dydx e x .[APRIL-2004] [APRIL-2005] JUNE-2007
3. Using Greens theorem evaluate

+ +
c
dy xy x dx y ) 3 (
2 3 3
where C is the boundary of the region between
3
x y = and y = x, 1 0 x .[APRIL-2004]
4. Evaluate

s
dA n F .
r
for the data k z j y i x z y x F

) , , ( + + =
r
and S: k u j v u i v u r

sin

cos
2
+ + =
r

v u , 0 .[APRIL-2004]
5. Using Stokes theorem evaluate

c
r d F
r
r
. where k y z j y x i y z F

) 3 (

) 4 3 (

) 2 ( + + + =
r
and C is the unit
circle in the plane z = 2. [APRIL-2004] [SUPPL-2006]
6. Show that the integral

+
+
) 0 , 4 , 2 (
) 1 , 1 , 0 (
) 2 (
2
dz dy dx e
z y x
is exact; evaluate it. [JULY-2004(SUPPL)]
7. Evaluate the double integral

= =
4 /
0
cos
0
2
sin

y
y
x
ydxdy x .[JULY-2004(SUPPL)] APRIL-2007
8. State Gauss divergence theorem. [APRIL-2005] [SUPPL-2006] JUNE-2007
9. Evaluate the integral

C
r d r F
r r
). ( where ] , , 2 [ y x z F =
r
and C is given by ] 2 , sin , [cos t t t r =
r
from (0,0,0)
to (1,0,4 ).[APRIL-2005] (APRIL-2006) JUNE-2007
10. Evaluate the integral

C
r d r F
r r
). ( using Stokes theorem where k
y
j
z
i y F

2
3

+ + =
r
and C is the circle:
3 , 6
2 2
+ = = + + x z z z y x .[APRIL-2005] [JUNE-2005(SUPPL)]
11. Use Greens theorem to evaluate the line integral

C
r d r F
r r
). ( where ] 2 ln ,
1
, [ x x e
x
e F
y y
+ =
r
and
2 1 :
4
+ y x C .[JUNE-2005(SUPPL)]
12. Use Gauss divergence theorem to evaluate the surface integral

+
S
ds k z x j y x i x )

4 (
2 2
where S is the
surface of tetrahedron whose vertices are (0,0,0), (1,0,0), (0,1,0) and (0,0,1); thus surfaces are x = 0, y
= 0, z = 0, and x+y+z = 1. [JUNE-2005(SUPPL)]
Vector Differential and Integral Calculus
Prepared By: Satyaban Panigrahi, NIST, Berhampur

Page- 3 -
13. Use Greens theorem to evaluate

C
r d F
r
r
. where
)
`

= + = =
4
1
: ) , ( ), , (
2 2
y x y x C x y F
r
. (APRIL-06)
14. Find the volume of the region in space in the first octant section cut from the region inside the cylinder
2 2 2
a y x = + by the planes y =0, z = 0, x = y. (APRIL-2006)
15. Evaluate the following integral using Gauss divergence theorem

s
dA n F .
r
where
) cos , sin , (cos z x y F =
r
and S is the surface . 2 , 4
2 2
+ z y x (APRIL-2006)
16. Verify Stokes theorem for ) 0 , 5 , (
2
x z F =
r
and S: . 1 , 1 0 , 1 0 = z y x (APRIL-2006)
17. Evaluate the following:

c
xdy ydx ) ( where C is the rectangle (in the cartesian plane) whose vertices
are (0, 0), (2, 0), (2, 3) and (0, 3). [SUPPL-2006]
18. Evaluate

S
ds n A , .
r
where k yz j x i y x A

2

) (
2
+ + =
r
, and S is the surface bounded by the planes x = y
= z = 2x + y + 2z 6 = 0. [SUPPL-2006]
19. Use Stokes theorem to evaluate

S
ds n F , ). (
r
where k xy j xz x i y F

) 2 (

+ =
r
and S is the surface
of the sphere
2 2 2 2
a z y x = + + above the xy-plane. [SUPPL-2006]
20. Evaluate the integral

c
dr r F ). (
r
if ] [
2 2
y x xy F = and C, the quarter circle from (2, 0) to (0, 2) with
center at (0, 0). APRIL-2007
21. Evaluate the following integral using Gauss divergence theorem

s
dA n F .
r
if ] , , [
3 3 3
z y x F = and S is
the sphere 9
2 2 2
= + + z y x . APRIL-2007 APRIL-2008
22. Use Gauss Divergence theorem to evaluate the surface integral

s
dS n F .
r
for the data
] , , [
z y x
e e e F =
r
and S is the surface of the cube 1 , 1 , 1 z y x . JUNE-2007[APRIL-2005]
23. Change the order of the integration

0 0
x
xy
ydydx e . [APRIL-2008]
24. Evaluate the double integral

1
0
1
2
) sin(
x
dydx y . APRIL-2008
25. Find the volume of the sphere
2 2 2 2
a z y x = + + using multiple integral. APRIL-2008
26. Evaluate the line integral

+
C
xydy dx x
4
by using Greens theorem where C is the triangular curve
consisting of the line segments from (0, 0) to (1, 0), from (1, 0) to (0, 1) and from (0, 1) to (0, 0).
APRIL-2008
27. Show that ) 3 , 2 , ( ) , , (
2 2 3 3 2
z xy xyz z y z y x F =
r
is a conservative vector field, and evaluate

+ +
) 1 , 0 , 1 (
) 0 , 0 , 0 (
2 2 3 3 2
) 3 2 ( dz z xy dy xyz dx z y
. APRIL-2008
28. Evaluate

S
dS F.
r
, where )) sin( , , ( ) , , (
2
2
xy e y xy z y x F
xz
+ =
r
where S is the surface of the region
bounded by the parabolic cylinder
2
1 x z = and the plane z = 0, y = 0 and y + z = 2. APRIL-
2008