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I A S

VECTOR ANALYSIS

Previous year Questions from 1992 To 2017

Syllabus

Scalar and vector fields, differentiation of vector field of a scalarvariable; Gradient, divergence and curl in cartesian and cylindri-cal coordinates; Higher order derivatives; Vector identities andvector equations. Application to geometry: Curves in space, Cur-vature and torsion; Serret-Frenet’s formulae. Gauss and Stokes’theorems, Green’s identities.

** Note: Syllabus was revised in 1990’s and 2001 & 2008 **

MATHEMATICS

ANALOG IASI N S T I T U T E

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Branches:

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20171. Suppose U and W are distinct four dimensional subspaces of a vector space V, where

dim V=6. Find the possible dimensions of subspace U W. (10 Marks)

2. Evaluate the integral : .

S

F nds where 2 2 3 2 ˆ3 3F xy i yx y j zx k and S is a surface

of the cylinder 2 2 4, 3 3y z x , using divergence theorem. (9 Marks)

3. Using Green’s theorem, evaluate the .C

F r dr counterclockwise where

2 2 2 2F r x y i x y j and d r dxi dy j and the curve C is the boundary of the

region 2, 1 2 .R x y y x (8 Marks)

20164. Prove that the vector ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ3 2 , 3 4 , 4 2 6a i j k b i j k c i j k

can form the sides

of a triangle find the length of the medians of the triangle (10 Marks)

5. Find f(r) such that 5

rf

r

and f(1)=0 (10 Marks)

6. Prove that C S

fd r d S f

(10 Marks)

7. For the of cardioid r = a(1+cos) show that the square of the radius of curvature at any

point (r,) is proportion to r. Also find the radius of curvature if = 0, ,4 2

.(15 Marks)

20158. Find the angle between the surfaces x2+y2+z2–9=0 and z=x2+y2–3 at (2,–1,2)

(10 Marks)

9. A vector field is given by 2 2 2 2F x xy i y x y j

. Verify that the field is irrotational

or not. Find the scalar potential. (12 Marks)

10. Evaluate sin cosx

C

e ydx ydy , Where C is the rectangle with vertices (0,0) (,0),

(,2

), (0,

2

) (12 Marks)

201411. Find the curvature vector at any point of the curve cos sin , 0 2r t t ti t t j t .

Give its magnitude also. (10 Marks)

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12. Evaluate by Stoke’s theorem r

ydx zdy xdz , where is the curve given by

x2+y2+z2–2ax–2ay=0, x+y=2a starting from (2a,0,0) and then going below the z-plane(20 Marks)

2013

13. Show the curve 21 1t t

x t ti j kt t

lies in a plane. (10 Marks)

14. Calculate 2 nr and find its expression in terms of r and n, r being the distance of

any point (x,y,z) from the origin, n being a constant and being the Laplanceoperator. (10 Marks)

15. A curve in space is defined by the vector equation 2 32r t i t j t k

. Determine the

angle between the tangents to this curve at the points t = +1 and t = –1(10 Marks)

16. By using Divergence Theorem of Gauss, evaluate the surface integral

1

2 2 2 2 2 2 2a x b y c z

dS. where S is the surface of the ellipsoid ax2+by2+cz2=1,

a,b and c being all positive constants. (15 Marks)

17. Use Stroke’s theorem to evaluate the line integral 2 2 3

c

y dx x dy z dz , where C is

the intersection of the cylinder x2+y2=1 and the plane x+y+z=1 (15 Marks)

201218. If 2 3 2 22 , 2A x yzi xz j xz k B zi y j x k

find the value of

2

A Bx y

at (1,0,–2) (12 Marks)

19. Derive the Frenet-Serret formulae. Define the curvature and torsion for a space

curve. Compute them for the space curve x = t, y = t2, z =32

3t . Show that the curvature

and torsion are equal for this curve. (20 Marks)

20. Verify Green’s theorem in the plane for 2 2

C

xy y dx x dy where C is the closed curve

of the region bounded by y = x and y = x2 (20 Marks)

21. If 2F yi x xz j xyk

, evaluate .S

F nds

where S is the surface of the sphere

x2+y2+z2=a2 above the xy - plane. (20 Marks)

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201122. For two vectors a

and b

give respectively by 2 35a t i t j t k

and

sin 5 cosb ti t j

determine:(i) .d

a bdt

and (ii) d

a bdt

(10 Marks)

23. If u and v are two scalar fields and f

is a vector field, such that u f

=gradv, find the

value of f

curl f

(10 Marks)

24. Examine whether the vectors , ,u v w are copalanar, where u,v and w are thescalar functions defined by: u=x+y+z, v=x2+y2+z2

and w=yz+zx+xy (15 Marks)

25. If 4 2u yi x j z k

calculate double integral u d s

over the hemisphere given

by x2+y2+z2=a2, z 0 (15 Marks)

26. If r

be the position vector of a point, find the value(s) of n for which the vector. nr r

(i) irrotational, (ii) solenoidal (15 Marks)

27. Verify Gauss’ Divergence Theorem for the vector 2 2 2v x i y j z k

taken over the

cube 0 x, y, z 1. (15 Marks)

201028. Find the directional derivative of f(x,y)=x2y3+xy at the point (2,1) in the direction of a

unit vector which makes an angle or 3

with the x-axis. (12 Marks)

29. Show that the vector field defined by the vector function v xyz yzi xy j xyk

is

conservative. (12 Marks)

30. Prove that ( ) .div f V f divV grad f V

where f is a scalar function. (20 Marks)

31. Use the divergence theorem to evaluate s

V nd A

where 2 2V x zi y j xz k

and S is

the boundary of the region bounded by the paraboloid z=x2+y2 and the plane z=4y.(20 Marks)

32. Verify Green’s theorem for e–xsinydx+e–xcosydy the path of integration being the

boundary of the sqaure whose vertices are (0,0), ( ,0)2

, ( , )

2 2

, and (0, )

2

(20 Marks)

200933. Show that div(gradrn)=n(n+1)rn–2 where 2 2 2r x y z . (12 Marks)

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34. Find the directional derivative of (i) 4xz3–3x2y2z2 (i) at (2,–1,1) along z-axis

(ii) –x2yz+4xz2 at (1,–2,1) in the direction of 2 2i j k

. (6+6=12 Marks)

35. Find the work done in moving the particle once round the ellipse 2 2

125 16

x y , z=0

under the field of force given by 22 3 2 4F x y z i x y z j x y z k

.

(20 Marks)

36. Using divergence theorem, evaluate .s

A d S where 3 3 3A x i y j z k

and S is the

surface of the sphere x2+y2+z2=a2 (20 Marks)

37. Find the value of .s

f d s

taken over the upper portion of the surface

x2+y2–2ax+az=0 and the bounding curve lies in the plane z=0, when

2 2 2 2 2 2 2 2 2F y z x i z x y j x y z k

(20 Marks)

200838. Find the constants a and b so that the surface ax2–byz=(a+2)x will be orthogonal to

the surface 4x2y+z3=4 at the point (1,–1,2). (12 Marks)

39. Show that 3 2 22 3F xy z i x j xz k

is a conservative force field. Find the scalar

potential for F

and the work done in moving an object in this field (1,–2,1) to (3,1,4).(12 Marks)

40. Prove that 2

22

2d f dff x

dr r dr where

12 2 2 2r x y z . Hence find f(x) such that

2 0f r . (15 Marks)

41. Show that for the space curve x=t, y=t2, z=32

3t the curvature and torsion are same at

every point. (15 Marks)

42. Evaluate c

Adr

along the curve x2+y2=1, z=1 from (0,1,1) to (1,0,1) if

2 2A yz x i xz j xy z k

. (15 Marks)

43. Evaluate

s

F nds

where 2 2A yz x i xz j xy z k

,

s

F nds

and S is the surface

of the cylinder bounded by x2+y2=4, z=0 and z=3 (15 Marks)

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200744. If r

denotes the position vector of a point and if r be the unit vector in the direction of

,r r r

determined grad (r–1) in terms of r and r.. (12 Marks)

45. Find the curvature and torsion at any point of the curve x=acos2t. y=asin2t, z=2asint.(12 Marks)

46. For any constant vector, show that the vector a

represented by curl a r

is always

parallel to the vector a

, r

being the position vector of a point (x,y,z,z) measured from

the origin. (15 Marks)

47. If r xi y j x k

find the value(s) of n in order that nr r

may be

(i) solenoidal (ii) irrotational (15 Marks)

48. Determine c

ydx zdy xdz by using Stoke’s theorem, where C is the curve defined

by (x–a)2+(y–a)2+z2=2a2, x+y=2a that starts from the point (2a,0,0) goes at firstbelow the z-plane (15 Marks)

200649. Find the values of constants a,b and c so that the directional derivative of the function

2 2 2f axy byz cz x at the point (1,2,–1) has maximum magnitude 64 in the direction

parallel to z-axis. (12 Marks)

50. If 2 , , 4 3 7A i K B i j k C i j K determine a vector R satisfying the vector

equation & . 0R B C B R A (15 Marks)

51. Prove that nr r is an irrotational vector for any value of n but is solenoidal only ifn+3=0 (15 Marks)

52. If the unit tangent vector t and binormal b make angles and respectively with a

constant unit vector a prove that sin

.sin

d k

d

. (15 Marks)

53. Verify Stoke’s theorem for the function 2F x i xy j integrated round the sqaure in the

plane z=0 and bounded by the lines x=0, y=0, x=a and y=a, a>0. (15 Marks)

2005

54. Show that the volume of the tetrahedron ABCD is 1

6AB AC AD

Hence find the

volume of the tetrahedron with vertices (2,2,2), (2,0,0), (0,2,0) and (0,0,2) (12 Marks)55. Prove that the curl of a vector field is independent of the choice of coordinates

(12 Marks)

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56. The parametric equation of a circular helix is cos sinr a ui a u j cu k

where c is a

constant and u is a parameter. Find the unit trangent vector i at the point u and the

arc length measured form u=0 Also find di

ds

where S is the arc length. (15 Marks)

57. Show that curl 1 1

. 0k grad grad k gradr r

where r is the distance from the origin

and K is the cunit vector in the direction OZ (15 Marks)58. Find the curvature and the torsion of the space curve (15 Marks)

59. Evaluate 3 2 2

s

x dydz x ydzdx x zdxdy by Gauss divergence theorem, where S is the

surface of the cylinder x2+y2=a2 bounded by z=0 and x=b (15 Marks)

200460. Show that if A and B are irrotational, then A B is solenodial. (12 Marks)61. Show that the Frenet-Serret formuale can be written in the form

, &dT d N d B

T N Bds ds dx

, where T k B . (12 Marks)

62. Prove the identity . . .A B B A A B B A A B (15 Marks)

63. Derive the identity 2 2

v s

D dV ndS Where V is the volume

bounded by the closed surface S. (15 Marks)

64. Verify Stoke’s theorem for 22f x y i yz j z k

where S is the upper half surface of

the sphere x2+y2+z2=1 and C is its boundary. (15 Marks)

200365. Show that if a' b' and c' are the reciprocals of the non-coplanar vectors, a, b and c ,

then any vector r may be expressed as .a' .b' .c'r r a r b r c . (12 Marks)

66. Prove that the divergence of a vector field is invariant w.r. to co-ordinate transformations. (12 Marks)

67. Let the position vector of a particle moving on a plane curve be r(t), where t is thetime. Find the components of its acceleration along the radial and transversedirections. (15 Marks)

68. Prove the identity 2 2 . 2A A A A A where i j k

x y z

(15 Marks)

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69. Find the radii of curvature and torsion at a point of intersection of the surface

x2–y2=c2, tanhz

y xc

. (15 Marks)

70. Evaluate .scurlA ds Where S is the open surface x2+y2–4x+4z=0, z 0 and

2 2 2 2 2 2 2 2 22 3A y z x i z x y j x y z k

. (15 Marks)

2002

71. Let R be the unit vector along the vector r t . Show that 2

dR r drR

dt r dt where

r r (12 Marks)

72. Find the curvature k for the space curve x=acos, y=asin, z=atan (15 Marks)

73. Show that 2curlv grad divv v (15 Marks)

74. Let D be a closed and bounded region having boundary S. Further, let f is a scalarfunction having second partial derivatives defined on it. Show that

2 2.s v

fgradf nds gradf f f dv Hence .s

fgradf nds or otherwise evaluate

for f=2x+y+2z over sx2+y2+z2=4 (15 Marks)75. Find the values of constants a,b and c such that the maximum value of directionial

derivative of 2 2 2f axy byz cx z at (1,–1,1) is in the direction parallel to y-axis and

has magnitude 6. (15 Marks)

200176. Find the length of the arc of the twisted curve r = (3t.3t2,2t3) from the point t=0 to the

point t=1. Find also the unit tangent t, unit normal n and the unit binormal b at t=1(12 Marks)

77. Show that 3 3 5

3.

a r a rcurl a r

r r r

where a is constant vector.. (12 Marks)

78. Find the directional derivative of 2 3f x yz along x=e–t, y=1+2sint, z=t–cost at t=0

(15 Marks)79. Show that the vector field defined by F=2xyz3i+x2z3j+3x2yz2k is irrotataional. Find also

the scalar u such that F= grad u (15 Marks)80. Verify Gauss’ divergence theorem of A=(4x,–2y2,z2) taken over the region bounded by

x2+y2=4, z=0 and z=3 (15 Marks)

200081. In What direction from the point (–1,1,1) is the directional derivative f=x2yz3 a

maximum? Compute its magnitude (12 Marks)

Page.No.9ANALOG IAS INSTITUTE - The Right Choice of Achieverswww.analogeducation.in

82. (i)Show that the covariant derivatives of the fundamental metric tensors gij, gij, i

j,

Vanish(ii) Show that simultaneity is relative in special relativity theory. (6+6=12 Marks)

83. Show that(i) (A+B).(B+C)(C+A)=2A.BC(ii) (AB)=(B.)A–B(.A)–(A.)B+A(.B) (7+8=15 Marks)

84. Evaluate .s

F Nds Where F=2xyi+yz2j+xzk and S is the surface of the parallelepiped

bounded by x=0, y=0, z=0, x=2, y=1 and z=3 (15 Marks)

85. If gij and

ij are tow metric tensors and defined at a point and

l

ij

and l

ij are the corre

sponding Christoffel symbols of the second kind, then prove that l

ij

–l

ij is a mixed

tensor of the type lijA (15 Marks)

86. Establish the formula E=mc2 the symbols have their usual meaning. (15 Marks)

199987. If , ,a b c are the position vectors of A,B,C prove that a b b c c a is vector

perpendicular to the plane ABC (20 Marks)

88. If 3 3 3 3 find .f x y z xyz F (20 Marks)

89. Evaluate sin cosx x

c

e ydx e ydy (by Green’s theorem), where C is the rectangle

whose vertices are (0,0), (,0), ,2

and 0,2

(20 Marks)

199890. If r

1 and r

2 are the vectors joining the fixed points A(x

1,y

1,z

1) and B(x

2,y

2,z

2) respectively

to a variable point P(x,y,z) then the values of grad (r1.r

2) and curl(r

1r

2). (20 Marks)

91. Show that (ab)c=a(bc) if either b=0 (or any other vector is 0) or c is collinear with aor b is orthogonal to a and c (both) (20 Marks)

92. Prove that logk

ig

ik x

. (20 Marks)

199793. Prove that if ,A B and C are there given non-coplanar vectors F then any vector can

be put in the form F B C C A A B

for given determine ,,.(20 Marks)

94. Verify Gauss theorem for 2 24 2F xi y j z k

taken over the region bounded by

x2+y2=4, z=0 and z=3 (20 Marks)

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95. Prove that the decomposition of a tensor into a symmetric and an anti-symmetric partis unique. Further show that the contracted product S

ijT

ij of a tensor T

ij with a

symmetric tensor Sij is independent of the anti-symmetric part of T

ij. (20 Marks)

199696. State and prove ‘Quotient law’ of tensors (20 Marks)

97. If xi y j z k

and r r

show that

(i) 0r grad f r

(ii) 3n ndiv r r n r

(20 Marks)

98. Verify Gauss’s divergence theorem for 2 2F xyi z j yz k

on the tetrahedron

x=y=z=0 , x+y+z=1 (20 Marks)

199599. Consider a physical entity that is specified by twenty -seven numbers A

ijk in given

coordinate system. In the transition to another coordinates system of this kind. LetA

ijkB

jk transform as a vector for any choice of the anti-symmetric tensor. Prove that the

quantities Aijk

–Aijk are the components of a tensor of third order. Is A

ijk the component

of tensor? Give reasons for your answer (20 Marks)100. Let the reason V be bounded by the smooth surfaces S and Let n denote outward

drawn unit normal vector at a point on S. If is harmonic in V, Show that 0s

dsn

(20 Marks)

101. In the vector field u(x) let there exists a surface curlv on which v=0. Show that, at anarbitary point of this surface curlv is tangential to the surface or vanishes.

(20 Marks)

1994102. Show that nr r

is an irrotational vector for any value of n, but is solenoidal only if n=–3.

(20 Marks)

103. If 2F yi x xz j xy k

evaluate s

F nds

Where S is the surface of the sphere

x2+y2+z2=a2 above the xy plane. (20 Marks)

104. Prove that logi

gik x

. (20 Marks)

1993105. Prove that the angular velocity or rotation at any point is equal to one half or the curl of

the velocity vector V. (20 Marks)

106. Evaluate s

Fnds

where S is the upper half surface of the unit sphere x2+y2+z2=1

and F zi x j y k

(20 Marks)

Page.No.11ANALOG IAS INSTITUTE - The Right Choice of Achieverswww.analogeducation.in

107. Show that p

q

A

x

is not a tensor even though A

p is a covariant tensor or rank one

(20 Marks)

1992

108. If 2 2 2 2 2 2, ,F x y z y z i z x j x y k

then calculate c

f dx

where C consist

of(i) The line segment from (0,0,0) to (1,1,1)(ii) the three line segments AB,BC and CD where A,B,C and D are respectively thepoints (0,0,0), (1,0,0), (1,1,0) and (1,1,1)

(iii) the curve 2 2 ,x ui u j u k u

from 0 to 1. (20 Marks)

109. If a

and b

are constant vectors, show that

(i) 2div x a x xa

(ii) 2 2div x a x b x a b x b a x

(20 Marks)

110. Obtain the formula 1/ 2

1

1

ij

gdiv A A i

x gg

where A(i) are physical components

of A

and use it to derive expression of divA

in cylindrical polar coordinates

(20 Marks)