Satyabama niversity questions in vector

1. Satyabama niversity questions in Vector.
1. Find ∇ φ if φ = x3
y2
z4
.
2. Find the directional derivative of φ = x2
yz + 4xz2
+ xyz at (1,2,3) in
the direction of 2 i + j − k
3.State Gauss Divergence theorem
4.State Gauss-Divergence theorem.
5.Show that kyxzjzxizxyf ˆ)3(ˆ)3(ˆ)6( 223
−+−++= is irrotational.
6.Prove that grade 2
g
fgradgfgradf
g
f −
=





7. If u = x2
– y2
, prove that ∇2
u = 0.
8.Find grad ϕ at the point (1, -2, -1) when ϕ = 3x2
y – y3
z2
.
9. Show that F = (y2
– z2
+ 3yz – 2x) i + (3xy + 2xy) J +
(3xy – 2xz + 2z) K is solenoidal.
10.Show that F = (x + 2y) i + (y + 3z) j + (x – 2z) k is solenoidal .
11. State stoke’s theorem.
12.Find the directional derivative of x2
+ 2xy at (1, –1, 3) in the direction of x axis.
13. ∫ rdF.

is independent of the path when?
Prove that div grad f = ∇2
f.
14.. State Stoke’s theorem.
15.Find the directional derivative of φ = x2
y2
z2
at the point (1, 1, 1) in the direction
.kji
∧∧∧
++
16. Find the value of ‘a’ so that the vector
kji zxzayyxF
∧∧∧
=+−++= )3()3()3( is
solenoidal
17.Define solenoidal vector function and irrotational motion.
18. If ( ) ∫−+−=
4
2
32
)(,7635)( dttFfindthenktjtitttF
Find grad ∅ if ∅ = xyz at (1,1,1).
19. Show that kzjyixF
 222
++= is a conservative vector field
20.Find a unit normal vector ‘n’ of the cone of revolution
z2
= 4(x2
+ y2
) at the point (1, 0, 2).
21.. Is the flow of a fluid whose velocity vector v = [secx, cosecx, 0] is irrotational?
22.Determine the constant a so that the vector kzxjayxizxF

)5()3()( −++++= is
solenoidal.
23. State Green’s theorem.
24.If r is the position vector of the point (x, y, z), a is a constant vector and φ = x2
+ y2
+ z2
, then find (a) grad ( r ο a ) and (b) r °gradφ.

2. 25.. If F is a solenoid vector, find the value of curl (curl(curl(curl F )))
Show that F = (y2
- z2
+ 3yz - 2x)i + (3xz + 2xy) j + (3xy – 2xz + 2z) k is irrotational.
26.. State Green’s theorem in a plane.
27. Define irrotational vector
28.State Guass divergence theorem
29.If FcurldivfindkZjYiXF
 333
++=
30.Prove that 0. =∫
c
rdr

31.Prove that (3x + 2y + 4z)
→
i + (2x + 5y + 4z)
→
j + (4x + 4y – 8z)
→
k is irrotational.
32. State Gauss-Divergence theorem.
33.Find the directional derivative of φ = x2
y2
z2
at the point (1, 1, 1) in the direction
.kji
∧∧∧
++
34. Find the value of ‘a’ so that the vector
kji zxzayyxF
∧∧∧
=+−++= )3()3()3( is
solenoidal.
35.Determine the constant a so that the vector kzxjayxizxF

)5()3()( −++++= is
Solenoidal.
36.State Gauss Divergence theorem.
36.If uand v
 
are vector point functions, then prove that
( ) ( ) ( )u v v u u v∇ × = ∇× − ∇×
     
g g g .
38.State Green’s theorem in a plane.
39.Find a and b such that
→
F = 3x2 →
i + (ax3
-2yz2
)
→
j +(3z2
-by2
z)
→
k is irrotational.
40.Prove that div
→
r = 3 , if
→
r = x
→
i + y
→
j + z
→
k .
Using Green’s theorem prove that the area enclosed by the simple closed curve ‘C’ is
( )∫ − .
2
1
ydxxdy
41.. Prove
r
nn
nrr
→
−
=∇ 2
I. Verify Green’s theorem in the plane for
∫ (3x2
– 8y2
) dx + (4y – 6xy) dy where C is the boundary of the C
region defined by x = 0, y = 0, x + y =1.
II. (a) Show that the vector
F = (3x2
+ 2y2
+ 1) i + (4xy – 3y2
z – 3) j + (2-y3
) k is
irrotational and find its scalar potential.
(b) If F = xy i + (x2
+ y2
) j, find ∫ F. dr where C is the arc
of the parabola y = x2
– 4 from (2, 0) to (4, 12).

3. III.(a) Show that F = (6xy + z3
) i + (3x2
– z) j + (3xz2
–y) k is solenoidal and hence find
the scalar potential.
(b) Find the directional derivative of f(x,y,z) = x2
yz + 4xz2
at (1,–2, –1) in the
direction of 2 i – j – k
IV. Verify Gauss divergence theorem
F = (x2
– yz) i + (y2
– zx) j + (z2
– yx) k and s is the surface of the rectangular
parallelepiped bounded by x=0, x=a, y=0, y=b, z=0, z=c.
V. (a) Prove that
22
)1( −
+=∇ nn
rnnr where kzjyixr ˆˆˆ ++=
b) Prove kxzjxizxyF ˆ3ˆˆ)2( 223
+++= is a conservative force. Find φ so that
F=∇φ .
VI. Given the vector kxyyjxyiyxF ˆ)(ˆ2ˆ)( 222
−++−= verify Gauss-Divergence
theorem over the cube with centre at the origin and of side length a.
VII. Verify the Gauss divergince theorem for kyzjyixzF +−= 2
4 over the cube
bounded by x = 0, x = 1, y = 0, y = 1, z = 0 and z = 1.
VIII. (a) Find the directional derivative of f = xyz at (1, 1, 1) in the directions of i + j +
k, i, -i.
(b) Evaluate kzyjxizFwheredsnF
s
22
,. −+=∫∫ and S is the surface of the
cylinder x2
+ y2
= 1, included in the first octant between the planes z = 0 and z =
2.
IX. Verify Stoke’s theorem for KyxJxzizyF 222
++= where S is the open surface of the
cube formed by the planes x = ± a, y = ± a, and z = ± a in which the plane z = -a
is cut.
X. Verify Gauss divergence theorem for KzJyIxF 222
++= where S is the surface
of the cubold formed by the planes x = 0, x = a, y = 0, y = b, z = 0 and z = c.
XI. Show that F = (y2
+ 2xz2
) i + (2xy – z) j + ( 2x2
z – y + 2z) k is irrotational and
hence find its scalar potential.
XII. Verify Gauss divergence theorem F = x2
i + y2
j + z2
k, where S is the
surface of the Cuboid formed by the planes x = 0, x = a, y = 0, y = b, z = 0, z = c.
XIII. (a) Prove that
22
)1( −
+=∇ nn
rnnr .
(b) Find ∫
c
rdF.

, kzjyixF
 22
24 +−= where S is the upperhalf of the surface of
the sphere 1222
=++ zyx , C is its boundary.
XIV. a) Find the workdone in moving a particle in the force field
F = 3x2
i + (2xz-y)j + zk along (i) the straight line from
(0,0,0) to (2,1,3). (ii) the curve defined by x2
= 4y, 3x3
= 8z
from x=0 to x = 2.
(b) Prove that ∇r n
= nr n-2
r

, where .zkyjxir ++=

XV. (a) Prove that ∇ x (∇ x V) = ∇ (∇.V) - ∇2
V.
(b) Verify Gauss divergence theorem, for f = 4xzi – y2
j + yzk
taken over the cube bounded by x = y = z = 0 of x =y=z= 1.
XVI. Show that 





=∇+=∇ −
.0
1
)1( 222
r
duceandhencedernnr nn

4. XVII.a.) Verify Gauss divergence theorem for
kji yzyxzF
∧∧∧
+−= 2
4 over the
cube bounded by x = 0, x = 1, y = 0, y = 1. z = 0 and z = 1.
b.) Using Stoke’s theorem evaluate ∫c
[(x + y) dx + (2x – z) dy +
(y + z) dz] where c is the boundary of the triangle with vertices
(2, 0, 0), (0, 3, 0) and (0, 0, 6).
XVIII.a,) Verify Divergence theorem for ( ) ( ) ( ) kxyzjxyiyzxF −+−+−= 222
3 taken
over the rectangular parallelopiped 0 ≤ x ≤ a, 0 ≤ y ≤ b, 0 ≤ z ≤ c.
b.)Verify Stokes theorem for kzxjyzixyF

−−= 2 where S is the open
surface of the rectangular parallelepiped formed by the planes x = 0, x=1, y=0,
y=2 and z = 3 above the XOY plane.
XIX.. Find the values of the constants a,b,c so that
kyxzjczxibzaxyF

)3()3()( 223
−+−++= may be irrotational. For these
values of a,b,c find also the scalar potential of F

.
XX.Verify Green’s theorem for ∫C
[(xy + y2
) dx + x2
dy], where C is bounded by y = x;
and y = x2
.
XXI..a.) Using Stoke’s theorem evaluate ∫C
[(x + y) dx + (2x – z) dy + (y + z) dz]
where C is the boundary of the triangle with vertices (2, 0, 0), (0, 3, 0) and (0, 0, 6).
b.)Prove that kxzjxyizxyF

)23()4sin2()cos( 232
++−++= is irrotational and find
its scalar potential.
XXII. Verify Stroke’s theorem for a vector field defined by jxyiyxF

2)( 22
+−= in
the rectangular region in the XOY plane bounded by the lines x=0, x=a, y=0 and
y=b.
XXIII. (a) Prove the relation: curl(curl F ) = ∇(∇° F ) - ∇2
F .
(b) Verify Gauss divergence theorem for F = x2
i + y2
j + z2
k , where S is the
surface of the cuboid formed by the planes x=0, x=a, y=0, y=b, z=0 and z=c.
XXIV.. (a) Verify Stoke’s theorem for F = - y i + 2yz j + y2
k, where S is the upper half
of the sphere x2
+ y2
+ z2
= a2
and C is the circular boundary on the xoy plane.
XXV. Verify Stokes theorem when F = (2xy – x2
)i – (x2
– y2
)j and C is the boundary of
the region enclosed by the parabolas y2
= x and x2
= y.
XXVI. Verify Gauss divergence theorem for F = x2
i + y2
j + z2
k where S is the surface of
the cuboid formed by the planes x = 0, x = a, y = 0, y = b, z = 0 and z = c.
XXVII.(a) Find the angle between the surfaces x2
+y2
=2 and x+2y- z=2 at (1,1,1)
(b) Find the value of m, if ( ) ( ) ( )kyxz3jzmxizxy6F 223
−+−++=
is irrotational. Find also φ such that F = grade φ
XXVIII. Verify Divergence theorem for ( ) ( ) ( )kxyzjzxyiyzxF 222
−+−+−=
taken over the rectangular parallelopiped bounded by
x=o,x=a,y=o,y=b,z=o,z=c

5. XXIX. (a) Find directional derivative of Φ = 3x2
+2y-3z at (1,1,1) in the direction 2
kji

−+ 2
(b) Show that kyxzjzxizxyF

)3()3()6( 223
−+−++= is irrotational vector and
find the scalar potential function Φ∇=F

XXX. verify Gauss divergence Theorem for the function kzjxiyF 2
++=

over the
cylindrical region bounded by x2
+y2
=9, z=0 and z=2.
XXXI. (a) Find the directional derivative of φ = xy + yz + zx at (1, 2, 3) in the direction of
3
→
i +4y
→
j + 5
→
k .
(b) Find the work done in moving a particle by the force
→
F = 3x2 →
i + (2xz-y)
→
j + z
→
k along the line joining (0, 0, 0) to (2, 1, 3).
XXXII. Verify Green’s theorem in the plane for dyxyydxxyx
C
)2()( 232
−+−∫
where C is a sqare with the vertices (0, 0), (2, 0), (2, 2), (0, 2).
XXXIII, A light horizontal strut AB is freely pinned at A and B. It is under the action of
equal and opposite compressive forces P at its ends and it carries a load ‘w’ at its
center. Then for
.,
2
cos
sin
2
Pr.
2
000.
2
1
,
2
0
2
2
2
EI
P
n
x
nl
n
nx
p
w
ythat
ove
l
xat
dx
dy
andxatyAlsox
w
Py
dx
yd
EI
L
x
=










−
=
====−=+<<
Show that 





=∇+=∇ −
.0
1
)1( 222
r
duceandhencedernnr nn
XXXIV.a.) Verify Gauss divergence theorem for
kji yzyxzF
∧∧∧
+−= 2
4 over the
cube bounded by x = 0, x = 1, y = 0, y = 1. z = 0 and z = 1.
b.)Prove that kxyzjzxyiyzxA

)6()4()2( −−+++= is solenoidal as well as
irrotational. Also find the scalar potential of A.
XXXV.Verify Green theorem in the plane for ∫ +−
C
xydydxyx ],2)[( 22
where C is the
closed curve of the region bounded by y = x2
and y2
= x.
XXXVI.(a) Verify Green’s theorem in a plane for the integral ( 2 )
C
x y dx x dy− +∫
taken around the circle C:
2 2
1x y+ = .
(b) Determine ( )f r so that ( )f r r

is Solenoidal.

6. xxxviii.(a) Evaluate
µ
S
F n dS∫∫
u
g where
F yz i zx j xy k= + +
u   u
and S is the part of the surface of the sphere
2 2 2
1x y z+ + = which lies in the first octant.
(b) Verify Stoke’s theorem 2 2
( ) 2F x y i xy j= + −
u  
taken around the rectangle
bounded by the lines , 0,x a y y b= ± = = .
XXXIX. (a) What is the angle between the surfaces x2
+ y2
+ z2
= 9 and x2
+ y2
+ z2
– 3 =
0 at (2, -1, 2).
(b) Show that
→
F = (2xy+z3
)
→
i + x2 →
j + 3xz2 →
k is a conservative field. Find the
scalar potential φ .
XL.. Verify Stoke’s theorem for
→
F = xy
→
i + xy2 →
j taken round the square bounded by
the lines x = 1, x = -1, y = 1, y = -1.
XLI.a)Prove that the given ( ) ( ) kzxjxyizxyF 232
34sin2cos +−++= is irrotational
and find the scalar potential
(b) Using Green’s theorem evaluate ( ) ( )∫ −+−
c
dyxyydxyx 6483 22
where C is the
boundary of the region enclosed by 2
xyandxy ==
XLII. (a) If kzjyixr ++= then prove that
div (grad (rn
)) = n(n + 1) rn–1
.
(b) Verify Gauss Divergence theorem for the given function kzjxiyF 2
++= over
the cylindrical region bounded by x2
+ y2
= 9, z = 0 and z = 2.

7. xxxviii.(a) Evaluate
µ
S
F n dS∫∫
ur
g where
F yz i zx j xy k= + +
ur r r ur
and S is the part of the surface of the sphere
2 2 2
1x y z+ + = which lies in the first octant.
(b) Verify Stoke’s theorem 2 2
( ) 2F x y i xy j= + −
ur r r
taken around the rectangle
bounded by the lines , 0,x a y y b= ± = = .
XXXIX. (a) What is the angle between the surfaces x2
+ y2
+ z2
= 9 and x2
+ y2
+ z2
– 3 =
0 at (2, -1, 2).
(b) Show that
→
F = (2xy+z3
)
→
i + x2 →
j + 3xz2 →
k is a conservative field. Find the
scalar potential φ .
XL.. Verify Stoke’s theorem for
→
F = xy
→
i + xy2 →
j taken round the square bounded by
the lines x = 1, x = -1, y = 1, y = -1.
XLI.a)Prove that the given ( ) ( ) kzxjxyizxyF 232
34sin2cos +−++= is irrotational
and find the scalar potential
(b) Using Green’s theorem evaluate ( ) ( )∫ −+−
c
dyxyydxyx 6483 22
where C is the
boundary of the region enclosed by 2
xyandxy ==
XLII. (a) If kzjyixr ++= then prove that
div (grad (rn
)) = n(n + 1) rn–1
.
(b) Verify Gauss Divergence theorem for the given function kzjxiyF 2
++= over
the cylindrical region bounded by x2
+ y2
= 9, z = 0 and z = 2.