Vector Calculus
Chapter
01. If the linear velocity V
—
V=X
is given by
2
yi+xyz j—yz 2 k then the angular
05. The expression curl (grad fi where . f is a
scalar function is
(GATE-96[MED
velocity W at the point (1, 1, — 1) is
(GATE-93)
02. The directional derivative of
f(x, y) = 2x2 + 3y2 + z2 at point P(2, 1, 3) in
the direction of the vector a = i - 21c is
(GATE — 94)
(a) 4 /15-
(b) — 4 /
(c) & /4
(d) — .15 /4
03. If V is a differentiable vector function andf
(a) Equal to V2f
(b) Equal to div(gradf )
(c) A scalar of zero magnitude
(d) A vector of zero magnitude
06. The directional derivative of the function
f(x,y,z) = x + y at the point P(1,1,0) along
the direction i + j is
(GATE — 96)
(a) 1 /112
(b)Vi
(c) —
(d) 2
is sufficiently differentiable scalar function
07. For the function (I) = axe y — y3 to represent
then curl (f V) =
(GATE-95[MED
(a) (grad f) x V + (f curl V)
(b) o
(c) f curl V
the velocity potential of an ideal fluid, V2 (I)
should be equal to zero. In that case, the
value of 'a' has to be
(GATE — 99)
(a) —1
(b) 1
(c) —3
(d) 3
(d) (grad f)xV
04. The derivative of f(x, y) at point (1, 2) in the
08. Given a vector field F , the divergence
theorem states that
(GATE-02 DEED
direction of vector i +j is 242 and in the
(a) f F.ds = fV.Fdv
direction of the vector —2 j is —3. Then the
(b)SP. ds=i-VxFdv
derivative of f(x, y) in direction —i — 2 j is
(GATE — 95)
(a) 2 -‘,/2 + 3/2
(b) — 7 /J
(c) — 2 i — 3/2
(d) 1 /
I.1(1 I mit ci
(c) f Fxds=1V.Fdv
(d) iFxds= SV.Fdv
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: 63 :
The directional derivative of the following
function at (1, 2) in the direction of (4i+3j)
is: f(x, y) = x2 + y2
(GATE — 02)
(a) 4 /5
(b) 4
(c)2 /5
(d) 1
10. The vector field F = xi — y j (where i and
j are unit vectors) is
(GATE — 03)
(a) divergence free, but not irrotational
(b) irrotational, but not divergence free
(c) divergence free and irrotational
(d) neither divergence free nor irrotational
11. Value of the integral
Vector Calculus
(c) a line integral and a volume integral
(d) gradient of a function and its surface
integral.
x 2 3/ 2
14. For the scalar field u = —
2 + — , the
3
magnitude of the gradient at the point (1, 3)
is
(GATE-05[EE])
(b)
9
2
xydy — y2 dx , where,
15. If a vector R(t) has a constant magnitude
c is the square cut from the first quadrant by
the line x = 1 and y = 1 will be (Use Green's
theorem to change the line
integral into
double integral)
(GATE — 05)
(a) 1/2
(b) 1
(c) 3/2
(d) 5/3
then
(GATE-0511[111)
(a) R .
dk
=0
dt
(c) R • R =
dR
dt
(b) K x
dR
=0
dt
(d) R x R =
dk
dt
12. The line integral I V.dr of the vector
function V(r) = 2xyz i + x2 z j + x2 y k from
the origin to the point P (1, 1, 1)
(GATE'05)
(a)
(b)
(c)
(d)
is 1
is zero
is —1
cannot be determined without specifying
the path
16. A scalar field is given by f = X23 + y2/3 ,
where x and y are the Cartesian coordinates.
The derivative of 1' along the line y = x
directed away from the origin at the point
(8, 8) is
(GATE-05[IN] )
-sh
13. Stokes theorem connects (GATE-05[ME])
(a) a line integral and a surface integral
(b) a surface integral and a volume integral
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17. Which one of the following is Not
associated with vector calculus?
(GATE-05 [PI])
(a) Stoke's theorem
(b) Gauss Divergence theorem
(c) Green's theorem
(d) Kennedy's theorem
22. The angle (in degrees) between two planar
P (b) V2 /3 + V(V.P)
(d) V (V.P)- V2 /3
23. Divergence of the vector field
(GATE-05 [EC]
(c) V 2 /3 + (V x P)
19. The
directional
derivative
the
f (x, y, z) =2x2 + 3y2 + z2 at
of
point
p(2,1,3) in the direction of the vector
a=
1 - - -1 a x bxc
(c) 2
1
(d) -(a xb) •c
.
.
vectors a = —1+
1 j- and b =
1+ 1- j
2 2
2 2
(GATE-07 [PI])
is
(a) 30
(b) 60
(c) 90
(d) 120
18. V x x P) where P is a vector is equal to
(a) Px V xP-
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v(x, y, z) = -(x cosxy + y) i + (y cosxy) j +
[(sinz2) + x2 + y2] is
(GATE-07[EN)
(a) 2z cosz2
(c) x sinxy - cosz
(b) sin xy + 2z cosz2
(d) none of these
i - 2k is
(a) - 2.785
(c) - 1.789
(GATE-06[CE')
(b) - 2.145
(d) 1.000
24. Consider points P and Q in xy - plane with
P = (1, 0) and Q = (0, 1). The line integral
21 (x dx + y dy) along the semicircle with the
20. The velocity vector is given as
v=
5xyi + 2y2 j + 3yz2 k. The divergence of
this velocity vector at (1, 1, 1) is
(GATE-07 ICE])
(a) 9
(b) 10
(c) 14
(d) 15
21. The area of a triangle formed by the tips of
vectors a, b and c is
(GATE-07[ME])
line segment PQ as its diameter
(GATE-08 [EC])
(a) is -1
(b) is 0
(c) 1
(d) depends on the direction (clockwise (or)
anti-clockwise) of the semi circle
25. The divergence of the vector field
(a) (a - b) • (a - c)
2
(x - y)i + (y - x) j + (x + y + z)k is
(b) I (a - b) x (a 2
(a) 0
(c) 2
(GATE-08 [ME])
(b) 1
(d) 3
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65 :
Vector Calculus
e
26. The directional derivative of the scalar
(a) fV.di=fi
function f(x, y, z) = x2 + 2y2 + z at the point
P = (1, 1, 2) in the direction of the vector
a =3i-4j is
(b)
= ff cTs'
s,
(GATE-08 [ME])
(a) — 4
(b) —2
(c) —1
(d) 1
(ofvxV.di=fivxXZ
sc
(d) fvxA.di=f1
27. If r is the position vector of any point on a
closed surface S that encloses the volume V
then if (r • ds) is equal to
(GATE-08[PI])
31. A sphere of unit radius is centered at the
origin. The unit normal at a point (x, y, z) on
the surface of the sphere is the vector.
(a) V
(b) V
(c) 2V
(d) 3V
2
(GATE-09[IN])
(a) (x, y, z)
( 1 1 1
(b)
„
28. For a scalar function f(x, y, z) = x2+3y2+2z2,
the gradient at the point P(1, 2, —1) is
(GATE-09 [CE]
(a) 2i + 6j + 4k
(b) 2i +12j — 4k
(c) 2i +12 j + 4k
(d) 156.
29. For a scalar function f(x, y, z) = x2+3y2+2z2,
(c)
(d)
x
y
z
x
y
z
Ah.
32. The
divergence
the directional derivative at the point
3xz i + 2xy — yz2
P(1, 2, —1) in the direction of a vector
equal to
- + 2k is
(GATE-09ICED
(a) —18
(b) —3 .4
(c) 3 Ai&
(d) 18
30. If a vector field V is related to another field
of
the
(GATE-09IECD
Note: C and Sc refer to any closed contour
field
at a point (1, 1, 1) is
(GATE-091111E1)
(a) 7
(b) 4
(c) 3
(d) 0
33. F(x, y) = (x2 + xy) fix + (y2+xy) a y . Its line
integral over the straight line from
(x, y) = (0, 2) to (x, y) = (2, 0) evaluates to
A through V=VxA, which of the
following is true?
vector
(GATE-09 [EE])
(a) — 8
(b) 4
(c) 8
(d) 0
and any surface whose boundary is C.
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ACE
34. The line integral of the vector function
F = 2x i + x 2 31 along the x — axis from x = 1
to x = 2 is
(a) 0
(c) 3
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(GATE-09 [PI])
(b) 2.33
(d) 5.33
vectors is perpendicular to both the
(GATE-11 [PI])
vectors AB and BC .
(a) 16i +9 j —12K
(b) 16i-9 j+12k
(c) 161-9j-12K
(d) 16i+9 j+12K
P2
35. Divergence of the 3 — dimensional radial
vector field r is
39. The line integral
f (ydx+ xdy) from
(GATE-10[EE])
1 (x1 y1 ) to P2 (x2, y2 ) along the semi-circle
1
(a) 3
(b)
(c) 1 + 3+ k
(d) +3+0
—
PiP2 shown in the figure is (GATE-11[PI])
Y
(a) x2 y2 — xi yi
36. If A = xy a•••• x + x 2 a y then A •cl I- over the
(b)
x2
2
(c)
path shown in the figure is
(y2
2, ) 4 (x2
2 .311
.
(x 2
x i)
(y2
i
y1)
(d) (Y2 — Y1)2 + (x 2
Y
(x2 y2)
—
1)
P1 (x1 3,
2
►
40. If T(x, y, z) = x2 + y2 + 2z2 defines the
3
temperature at any location (x, y, z) then the
2
0
x
magnitude of the temperature gradient at
point P(1, 1, 1) is
Itsh 2/.1S
(GATE-10[EC])
(GATE-11[PI])
(a) 2J
(b) 4
(c) 24
(d)
37. If a and b are two arbitrary vectors with
magnitudes a and b respectively,
41. Consider a closed surface 'S' surrounding a
a x i)12 will be equal to (GATE-1110ED
volume V. If r is the position vector of a
(a) a2 b2 — (a . b)2
2
(c) a2 b2 +(a.b)
(b) ab—a.b
(d) ab+a.b
38. If A (0, 4, 3), B(0, 0, 0) and c(3, 0, 4) are
there points defined in x, y, z coordinate
system, then which one of the following
ACE Engineering Publications
point inside S with n the unit normal on
S', the value of the integral ff5r.nds is
(a) 3V
(c) 10 V
(GATE-11 [EC])
(b) 5V
(d) 15 V
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Vector Calculus
42. The two vectors [1, 1, 1] and [1, a, a2] where
46. Consider a vector field A-(f). The closed
(GATE-111EED
loop line integral A • dl can be expressed
a = 21+j—
2 are
(a) Orthonormal
(c) Parallel
as
(b) orthogonal
(d) Collinear
(GATE - 20131ECI)
(a) if (V x A) • ds over the closed surface
bounded by the loop
43. The direction of vector A is radially outward
from the origin, with lAl = K rn
where r2 = x2 + y2 + z2 and K is constant.
The value of n for which V. A = 0 is
bounded by the loop
(c)
(GATE-12[EC, EE, IN])
(a) -2
(c) 1
(b) ficf (V • A)dv over the closed volume
(b) 2
f f f (V • A)dv over the open volume
bounded by the loop
(d) Li (V x A) • ds over the open surface
(d) 0
bounded by the loop
44. For the spherical surface x2 + y2 + z2 = 1, the
unit outward normal vector at the point
i
Ln2
A = xa x + }Ta y + za, is (GATE - 20131ECI)
1_ oj is given by
2
(GATE-12[ME, PI])
(a)
1
1
j
N/ 2 +2
.‘,/
(b)
47. The divergence of the vector field
1 ••
i- 1j
N/2
(a) 0
(c) 1
(b) 1/3
(d) 3
48. For a vector E, which one of the following
statements is NOT TRUE?
(c)
(GATE - 20131IN1)
(a) If V.E = 0, E is called solenoidal
45. For the parallelogram OPQR shown in the
sketch.
OP = a I+ b j and OR = ci+ d j .
The area of the parallelogram is
(GATE-12 [CE])
(b) If V x E = 0, E is called conservative
(c) If V x E = 0, E is called irrotational
(d) If V.E = 0, E is called irrotational
Q
(a) ad - bc
(b) ac + bd
(c) ad + bc
(d) ab - cd
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4.
49. The following surface integral is to be
53. The magnitude of the gradient for the
evaluated over a sphere for the given steady
function f (x , y, z) = x 2 + 3y2 +z3
velocity vector field F = xi + yj + zk defined
point (1, 1, 1) is
with respect to a Cartesian coordinate
at the
(GATE — 14 — EC — Set 4)
system having i, j and k as unit base vectors.
54. The
ff —
1 (F.n )dA
4
directional
derivative
of
Where S is the sphere, x2+y2+z2 = 1 and n is
Y + y) at (1, 1) in the direction
f(x, y) = xr__
A/2
the outward unit normal vector to the sphere.
of the unit vector at an angle of
The value of the surface integral is
(GATE — 2013[ME])
(a) TC
(b) 27c
(c) 3 m/4
(d) 4 n
y — axis, is given by
(GATE — 14 — EC — Set 4)
55. Given F = za x + xa y
50. The curl of the gradient of the scalar field
—
rc with
4
ya z .
If S represents
the portion of the sphere x2 + y 2 + z2 =1
defined by V = 2x2y + 3y2z + 4z2x is
(GATE — 2013IEED
for z 0, then f (V x F.Jds is
(a) 4xyax + 6yzay + 8zxa7
(GATE — 14 — EC — Set 4)
(b) 4a„ + 6ay +
(c) (4xy+4z2)a„+(2x2 + 6yz)ay+ (3y2 + 8zx)a,
(d) 0
56. The line integral of function F = yzi, in the
counterclockwise direction, along the circle
51. Given a vector field
F = y 2 xa x — yza y — x 2 a, , the line integral
.I. F.d1 evaluated along a segment on the x-
x2 + y2 = 1 at z= 1 is
(GATE — 14 — EE — Set 1)
(a) —27c
(b) —IT
(c) TC
(d) 27c
axis from x = 1 to x = 2 is
(GATE — 2013 [EN)
57. Let V.(/' = x2 y + y2 z + z2 x, where f and
(a) 2.33
(b) 0
V are scalar and vector fields respectively.
(c) 2.33
(d) 7
If V = yi + zj + xk, then V.(Vf) is
(GATE —14 — EE — Set 3)
(a) x2 y + y2 z + z2 x
52. If r = xa x + ya y + za , and r = r, then div
(b) 2xy + 2yz + 2zx
(c) x + y+ z
(r 2 V(1nr))=
(d) 0
(GATE — 14 — EC — Set 2)
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58. A vector is defined as f = y 1+ x j+ z k
Vector Calculus
(a) 0
(c) 5
(b) 3
(d) 6
A A
Where i , j , and k are unit vectors in
cartesian (x,y,z) coordinate system.. The
41
surface integral of.ds over the closed
surface S of a cube with vertices having the
following coordinates:
(0,0,0), (1,0,0), (0,1,0), (0,0,1), (1,0,1),
(1,1,1), (0,1,1), (1,1,0) is
(GATE —14 — IN — Set 1)
59. The integral
(ydx — xdy) is evaluated
along the circle x 2
y2
1
=—
traversed in
4
counter clockwise direction. The integral is
equal to
(GATE — 14 — ME — Set 1)
(a) 0
(c)—
(b) —
(d)
2
4
4
60. Curl of vector f = x2z2i-2xy2ij+ 2y2z3k
is
(GATE — 14 — ME — Set 2)
(a) (4yz3 + 2 xy2 + 2 x2z3 2y2A
(b)
yzi + 2 xy2
2x2;j 2y2zi
(c) 2xz21 —4xyzj+ 6y2z2k
62. A particle moves along a curve whose
parametric equations are: x=t3+2t, y = —3e-21
and z = 2 sin (5t), where x, y and z show
variations of the distance covered by the
particle (in cm) with time t (in s). The
magnitude of the acceleration of the particle
(in cm/s2) at t = 0 is
(GATE-14—CE—Set 1)
63. Directional derivative of (I) = 2xz—y2' at the
point (1, 3, 2) becomes maximum in the
direction of
(GATE —14 —PI— Set 1)
(a) 4i+2j-3k
(b) 4i-6j+2k
(c) 2i-6j+2k
(d) 4i-6j-2k
64. If 4 = 2x3y2z4 then V24'is
(GATE 14 — PI — Set 1)
2z4+4x2z4+
(a) 12xy
20x3y2z3
(b) 2x2y2z+4x3z4+24x3y2z2
(c) 12xy2z4+4x3z4+24x3y2z2
(d) 4xy2z+4x2z4+24x3y2z2
65. The directional derivative of the field
u(x,y,z) = x-3yz in the direction of the
1 vector ( + j 2 k) at point (2,-1,4) is
(d) 2xz2i + 4xyzj + 6y2z2k
(GATE-15—CE—Set 1)
61. Divergenece of the vector field
x2zi + xy3 yz2k at (1, —1, 1) is
(GATE — 14 — ME — Set 3)
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4.)
ay _
(b) 515
x2yz a7 . Which one of the following
(d) 9k
66. A vector P is given by P = x3y a, —
x2y2
statements is TRUE?
(GATE — 15 — EC — Set 1)
(a) P is solenoidal, but not irrotational
(b) P is irrotational, but not solenoidal
(c) P is neither solenoidal nor irrotational
(d) P is both solenoidal and irrotational
(c) Div Curl P = 0 (d) Div (0P)= 0DivP
Q. Gauss's Theorem
R. Divergence Theorem
S. Cauchy's Integral Theorem
List-II
2. f(z)dz = 0
[(3x-8y2)dx + (4y-6xy)dy], (where C is
71. The velocity field on an incompressible flow
3. fff ( v .A)dv = ff A.ds
4. ff (v xA).ds = A . d /
(GATE —15 — EE — Set 2)
1
3
4
70. The value of
the region bounded by x=0, y=0 and x+y=1)
. (GATE — 15 — ME — Set 3)
is
1.ff D.ds = Q
1
vector valued function in a threedimensional space. Which one of the
following is an identity?
(GATE — 15 — ME — Set 3)
(b) DivP = 0
List-I
P. Stoke's Theorem
Q
scalar function and V be an arbitrary smooth
(a) Curl(ili = V(0DivP)
67. Match the following.
Codes:
P
(a) 2
(b) 4
(c) 4
(d) 3
69. Let 0 be an arbitrary smooth real valued
R
4
3
1
2
S
3
2
2
1
68. The magnitude of the directional derivative
of the function f(x,y)= x2+3y2 in a direction
normal to the circle x2 +y2 =2, at the point
(GATE — 15 — IN — Set 2)
(1,1), is
is given by
V = (aix + a2y +a3z) i + (bix + b2y b3z)
+ (cix + c2y + c3z) k,
where al = 2 and c3 = —4 . The value of b2
is
(GATE —15 — ME — Set 1)
72. Curl of vector V(x,y,x) = 2x2i + 3z2 j +y3k at
x= y = z = 1 is
(GATE — 15 —ME — Set 2)
(b) 3i
(a) —3i
(c) 3i — 4j
(d) 3i — 6k
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73. The surface integral if — (9xi — 33/j) • n dS
TC
over the sphere given by x2 + y2 + z2 9 is
(GATE — 15 — M E — Set 2)
74. The
value
of
line
integral
Vector Calculus
The value of the integral is
(GATE-16-ME-SET2)
78. The value of the line integral f F.rds, where
C is a circle of radius
units is
f (2xy2dx + 2x2ydy + dz) along a path
Here, F(x, y) = yl + 2x j and r is the UNIT
joining the origin (0, 0, 0) and the point
(1, 1, 1) is
(GATE — 16 — EE — Set 2)
(a) 0
(b) 2
(c) 4
(d) 6
75. The line integral of the vector field
tangent vector on the curve C at an arc
length s from a reference point on the curve.
and j are the basis vectors in the x-y
Cartesian reference. In evaluating the line
integral, the curve has to be traversed in the
counter-clockwise direction.
(GATE-16-ME-SET3)
F = 5xz i + (3x2 +2y) j +x2z k along a path
from (0, 0, 0) to (1, 1, 1) parameterized by
(t, t2, t) is
(GATE — 16 — EE — Set 2)
76. The vector that is NOT perpendicular to the
vectors (i + j + k) and (i + 2j + 3k)
(GATE —16—IN)
is
(a) (i — 2j + k)
(1)) (—i + 2j — k)
(c) (Oi + Oj + Ok)
(d) (4i + 3j + 5k)
77. A scalar potential 9 has the following
gradient: VT = yzi + xzj + xyk . Consider the
integral LV(p.cif
r=
on
,t
the
curve
+ yj + zk . The curve C is
79. Suppose C is the closed curve defined as the
circle x2 + y2 = 1 with C oriented anticlockwise.
The
value
of
2
2
(xy dx + x ydy) over the curve
f
equals
. (GATE — 16 — EC — Set 2)
80. The smaller angle (in degrees) between the
planes x + y + z = 1 and 2x — y + 2z = 0
is
(GATE-17—EC)
81. If
the
function
P.a.(3y—kiz)+ay(k2x-2z)— az(k3y + z)
is irrotatiotial, then the values of the
constants kl, k2 and k3 respectively, are
parameterized as follows:
x=t
y = t2 and 1 t
z = 3t2
vector
(a) 0.3, —2.5, 0.5
(c) 0.3, 0.33, 0.5
(GATE-17—EC)
(b) 0.0, 3.0, 2.0
(d) 4.0, 3.0, 2.0
I Delhi I Bhopal I Pune I Bhubaneswar LucknowiPatrialBengaluruiChennailVijayawadalVing airupati I Kukarpalty I KolkataACE Engineering Publications 45Hyderabad
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: 72 :
Fagineaing Publications
lo
82.
Let I =
(2z dx+2y dy + 2x dz) where
x, y , z are real , and let C be the straight
line segment from point A: (0,2, 1 ) to
point B: (4,1,-1) . The value of I
is
(GATE-17—EC)
83. For the vector V = 2yzi + 3xzj + 4xyk, the
Engineering Mathematics
where F = (x+y)I + (x+z)j + (y+z)k and n is
the unit outward surface normal,
yields
(GATE-17—ME)
85. The divergence of the vector field V = x2i +
2y3 j+z4 katx= 1,y= 2,z= 3 is
(GATE-17—CE)
value of v.(v x ) is
(GATE-17—ME)
84. The surface integral LI.F.ndS over the
surface S of the sphere x2 + y-2 +
ACE Engineering Publications
z2 =
9,
yderabadiDelhilBhopallPurtelBhubaneswarl LucknowiPatnalBengalurulChennailVijayawadaiVizag ITirupati Kukatpally1 Kolkata