Question 1

A vector field p and a scalar field r are given by
p = (2x² - 3xy + z²) î + (2y² - 3yz + x²) ĵ + (2z² - 3xz + x²) k̂
r = 6x² + 4y² - z² - 9xyz - 2xy + 3xz - yz
Consider the statements P and Q.
P: Curl of the gradient of the scalar field r is a null vector.
Q: Divergence of curl of the ve

Accepted Answer

Both P and Q are TRUE. The curl of the gradient of any scalar field is always zero, confirming statement P as true. Additionally, the divergence of the curl of any vector field is also zero, validating statement Q as true.

Question 2

What is curl of the vector field 2x²y i + 5z² j − 4yzk ?

Accepted Answer

−14z i − 2x² k. The curl of a vector field measures the rotation at a point in the field. In this case, calculating the curl of the given vector field results in the components that match the correct option, indicating that the field has a specific rotational behavior characterized by the terms involving z and x.

Question 3

Let φ be a scalar field, and u be a vector field. Which of the following identities is true for div(φu)?

Accepted Answer

div(φu) = φ div(u) + u · grad(φ). The correct option is true because it follows the product rule for divergence, which states that the divergence of a product of a scalar field and a vector field can be expressed as the scalar field times the divergence of the vector field plus the vector field dotted with the gradient of the scalar field.

Question 4

A delivery agent is at a location R. To deliver the order, she is instructed to travel to location P along straight-line paths of RC, CA, AB and BP of 5 km each. The direction of each path is given in the table below as whole circle bearings. Assume that the latitude (L) and departure (D)

Accepted Answer

L = 0.0; D = 0.0. The correct option indicates that after following the paths RC, CA, AB, and BP, the delivery agent returns to the original latitude and departure of R, which is (0, 0) km. This means that the movements in different directions ultimately cancel each other out, resulting in no net change in position.

Question 5

Three vectors p, q and r are given as p = î + ĵ + k̂, q = î + 2ĵ + 3k̂, r = 2î + 3ĵ + 4k̂. Which of the following is/are CORRECT?

Accepted Answer

p × (q × r) + q × (r × p) + r × (p × q) = 0. The equation represents the vector triple product identity, which states that the sum of the cyclic permutations of the vector triple products equals the zero vector. This is a fundamental property of vector algebra, confirming that the vectors are arranged in such a way that their cross products cancel each other out.

Question 6

If a vector field V is related to another vector field A through V = ∇ × A, which of the following is true? Note: C and S_c refer to any closed contour and any surface whose boundary is C.

Accepted Answer

∮_C A · dl = ∬_S_c V · dS. This option is correct because it reflects Stokes' theorem, which states that the line integral of a vector field around a closed contour is equal to the surface integral of the curl of that vector field over the surface bounded by the contour. Since V is defined as the curl of A, the relationship holds true.

Question 7

The direction of vector A is radially outward from the origin, with |A|=krⁿ where r²=x²+y²+z² and k is a constant. The value of n for which ∇· A=0 is

Accepted Answer

−2. The divergence of a vector field in spherical coordinates can be computed, and for a radially outward vector with magnitude proportional to r raised to the power of n, the condition for divergence to be zero leads to the requirement that n must equal -2. This ensures that the vector field behaves properly at infinity and does not contribute to divergence.

Question 8

Consider a vector field A(r). The closed loop line integral ∮ A·dl can be expressed as

Accepted Answer

∬ (∇×A)·ds over the open surface bounded by the loop. The correct option relates to Stokes' theorem, which states that the line integral of a vector field around a closed loop is equal to the surface integral of the curl of that field over any surface bounded by the loop. Therefore, the integral of (∇×A)·ds over the open surface is the appropriate representation of the closed loop line integral.

Question 9

The divergence of the vector field A = x aₓ + y a_y + z a_z is

Accepted Answer

3. The divergence of a vector field measures the rate at which 'stuff' expands from a point. For the vector field A = x aₓ + y a_y + z a_z, the divergence is calculated as the sum of the partial derivatives of its components, which results in 3, indicating that the field is expanding uniformly in three-dimensional space.

Question 10

Match column A with column B.
Column A
1. Point electromagnetic source
2. Dish antenna
3. Yagi-Uda antenna
Column B
P. Highly directional
Q. End fire
R. Isotropic

Accepted Answer

1 → R
2 → P
3 → Q. A point EM source radiates isotropically (1->R), a dish antenna is highly directional (2->P), and a Yagi-Uda is an end-fire array (3->Q). This matching is option 1, not the stored option 3.

Question 11

Consider the two-dimensional vector field F(x,y)=x î+y ĵ, where î and ĵ denote the unit vectors along the x-axis and the y-axis, respectively. A contour C in the x-y plane, as shown in the figure, is composed of two horizontal lines connected at the ends by two semicircular arcs of uni

Accepted Answer

0. The vector field ( ext{F}(x,y) = x ext{hat{i}} + y ext{hat{j}}) is conservative, meaning that the line integral over any closed path is zero. This is because the field can be expressed as the gradient of a scalar potential function, which implies that the integral around a closed contour yields no net work done.

Question 12

The value of the line integral ∫_P^Q (z² dx + 3y² dy + 2xz dz) along the straight line joining the points P (1,1,2) and Q (2,3,1) is

Accepted Answer

24. The field (z^2,3y^2,2xz) is conservative with potential phi = x z^2 + y^3. The integral equals phi(Q)-phi(P) = (2*1+27) - (1*4+1) = 29 - 5 = 24, not -5.

Question 13

Divergence of the vector field V(x,y,z) = -(x cos xy + y)i + (y cos xy)j + (sin z² + x² + y²)k is

Accepted Answer

(A) 2z cos z². div V = d/dx[-(x cos xy + y)] + d/dy[y cos xy] + d/dz[sin z^2 + x^2 + y^2]. This gives (-cos xy + xy sin xy) + (cos xy - xy sin xy) + 2z cos z^2; the cos xy and xy sin xy terms cancel, leaving 2z cos z^2, which is option A.

Question 14

F(x,y) = (x² + xy) aₓ + (y² + xy) a_y. Its line integral over the straight line from (x,y) = (0,2) to (x,y) = (2,0) evaluates to

Accepted Answer

0. The line integral evaluates to zero because the vector field F is conservative, meaning it has a potential function and the integral over a path between two points depends only on the endpoints, not the path taken. Since the starting and ending points are different but the field is conservative, the integral results in zero.

Question 15

Divergence of the three-dimensional radial vector field r is

Accepted Answer

3. The divergence of a radial vector field in three dimensions is calculated using the formula for divergence in spherical coordinates, which results in a constant value of 3 for the radial vector field, indicating that the field is expanding uniformly in all directions.

Question 16

The direction of vector A is radially outward from the origin, with |A| = k rⁿ where r² = x² + y² + z² and k is a constant. The value of n for which ∇·A = 0 is

Accepted Answer

-2. The divergence of a radial vector field is calculated using the formula ∇·A = (1/r²)(∂/∂r)(r² A_r), where A_r is the radial component. For the given magnitude |A| = k rⁿ, setting n to -2 results in the divergence being zero, satisfying the condition ∇·A = 0.

Question 17

The flux density at a point in space is given by B = 4x aₓ − 2ky a_y − 8z a_z Wb/m². The value of constant k must be equal to

Accepted Answer

-2. Since B is a magnetic flux density, div B = 0. Computing: d/dx(4x) + d/dy(-2ky) + d/dz(-8z) = 4 - 2k - 8 = 0, which gives k = -2.

Question 18

The curl of the gradient of the scalar field defined by V = 2x² y³ z² + 4z² x is

Accepted Answer

0. The curl of the gradient of any scalar field is always zero, as it represents the rotation of a conservative vector field, which has no curl.

Question 19

Given a vector field y² aₓ + z y a_y + x² a_z, the line integral ∫ F · dl evaluated along a segment on the x-axis is from x = 1 to x = 2 is

Accepted Answer

0. The line integral of a vector field along the x-axis evaluates to zero because the vector field has no component in the direction of the x-axis when y and z are both zero, resulting in no work done along that path.

Question 20

The curl of the gradient of the scalar field defined by V = 2x²y³z² + 4z²x is

Accepted Answer

0. The curl of the gradient of any scalar field is always zero, as it represents the rotational component of a conservative vector field, which has no rotation.

Question 21

Given a vector field F = y² x aₓ − z y a_y − x² a_z, the line integral ∫ F · dl evaluated along a segment on the x-axis from x = 1 to x = 2 is

Accepted Answer

0. The line integral of the vector field along the x-axis evaluates to zero because the vector field has no component in the direction of the x-axis when y and z are both zero, resulting in no net work done along that path.

Question 22

In cylindrical coordinate system, the potential produced by a uniform ring charge is given by φ = f(r,z), where f is a continuous function of r and z. Let E be the resulting electric field. Then the magnitude of ∇ × E is ________.

Accepted Answer

(B) is 0.. The electric field produced by a static charge distribution, such as a uniform ring charge, is conservative, meaning that the curl of the electric field (∇ × E) is zero. This is a fundamental property of electrostatics, indicating that the electric field can be derived from a scalar potential function.

Question 23

The value of the line integral ∫_C (2xy² dx + 2x²y dy + dz) along a path joining the origin (0, 0, 0) and the point (1, 1, 1) is

Accepted Answer

2. The line integral evaluates the work done along the specified path, and in this case, the contributions from the terms in the integral simplify to yield a total value of 2 when calculated from the origin to the point (1, 1, 1).

Question 24

The value of the directional derivative of the function ϕ(x,y,z) = xy² + yz² + zx² at the point (2, -1, 1) in the direction of the vector p = i + 2j + 2k is

Accepted Answer

1. The directional derivative measures the rate of change of the function in the specified direction. By calculating the gradient of the function at the given point and taking the dot product with the normalized direction vector, we find that the result is 1, indicating the maximum rate of increase in that direction.

Question 25

Let R be a region in the first quadrant of the xy plane enclosed by a closed curve C considered in counter-clockwise direction. Which of the following expressions does not represent the area of the region R?

Accepted Answer

∮_C ydx. The expression ∮_C ydx represents the circulation of the vector field (0, y) around the curve C, which does not calculate the area of region R. In contrast, the other options either directly compute the area or relate to it through Green's Theorem.

Question 26

Let E(x,y,z)=2x²î+5yĵ+3zk̂. The value of ∭_V (∇·E) dV, where V is the volume enclosed by the unit cube defined by 0≤ x≤ 1, 0≤ y≤ 1, and 0≤ z≤ 1, is

Accepted Answer

10. div E = d(2x^2)/dx + d(5y)/dy + d(3z)/dz = 4x + 5 + 3 = 4x + 8. Integrating 4x+8 over the unit cube gives integral of (4x+8) dx from 0 to 1 = 2 + 8 = 10. Stored answer 8 is wrong.

Question 27

Spheres of unit diameter are centered at (l,m,n) where l, m, and n take every possible integer values. The distance between two spheres is computed from the center of one sphere to the center of the other sphere. For a given sphere, x is the distance to its nearest sphere and y is the dist

Accepted Answer

√2. The distance between the centers of two adjacent spheres is 2 units, while the distance to the next nearest sphere, which is diagonally adjacent, is √2 units. Therefore, the ratio y/x, where y is the distance to the next nearest sphere and x is the distance to the nearest sphere, simplifies to √2.

Question 28

The value of the integral
∯_S r · n dS
over the closed surface S bounding a volume V, where r = x î + y ĵ + z k̂ is the position vector and n is the normal to the surface S, is

Accepted Answer

3V. The integral of the position vector dotted with the outward normal over a closed surface can be evaluated using the divergence theorem, which relates the surface integral to a volume integral. In this case, the divergence of the position vector r is constant and equal to 3, leading to the result being 3 times the volume V.

Question 29

Given a function φ = 1/2 (x² + y² + z²) in three-dimensional Cartesian space, the value of the surface integral ∯_S n̂ · ∇φ dS, where S is the surface of a sphere of unit radius and n̂ is the outward unit normal vector on S, is

Accepted Answer

4π. The surface integral evaluates the flux of the gradient of the function φ through the surface of the sphere. Since the gradient ∇φ points radially outward and has a constant magnitude on the surface of the unit sphere, the integral simplifies to the product of the magnitude of the gradient at the surface and the surface area of the sphere, resulting in a value of 4π.

Question 30

The divergence of the vector field u = e^x(cos y î + sin y ĵ) is

Accepted Answer

2e^x cos y. div u = d/dx(e^x cos y) + d/dy(e^x sin y) = e^x cos y + e^x cos y = 2 e^x cos y. The stored single-term expression is incorrect.

Question 31

For a position vector r = x î + y ĵ + z k̂ the norm of the vector can be defined as |r| = √(x² + y² + z²). Given a function ϕ = ln|r|, its gradient ∇ϕ is

Accepted Answer

r/(r·r). The gradient of the function ϕ = ln|r| involves the derivative of the logarithm of the norm of the vector, which leads to a result that is proportional to the vector itself divided by the square of its magnitude. This is consistent with the formula for the gradient of a logarithmic function, resulting in the correct option being r/(r·r).

Question 32

A two dimensional flow has velocities in x and y directions given by u = 2xyt and v = -y²t, where t denotes time. The equation for streamline passing through x = 1, y = 1 is

Accepted Answer

xy² = 1. Streamline: dx/(2xyt)=dy/(-y^2 t) gives -dx/x = 2 dy/y, so -ln x = 2 ln y + C, i.e. x*y^2 = constant. At (1,1) the constant is 1, so the streamline is x*y^2 = 1.

Question 33

The velocity field in a fluid is given to be V = (4xy)î + 2(x² - y²)ĵ. Which of the following statement(s) is/are correct?

Accepted Answer

The flow is incompressible.. The flow is incompressible because the divergence of the velocity field is zero, indicating that the fluid density remains constant throughout the flow.

Question 34

The area of a triangle formed by the tips of vectors a, b and c is

Accepted Answer

1/2 |(a − b)×(a − c)|. The area of a triangle formed by two vectors can be calculated using the cross product, which gives the magnitude of the parallelogram spanned by the vectors. Dividing by two yields the area of the triangle, hence the correct option is 1/2 |(a − b)×(a − c)|.

Question 35

The directional derivative of the scalar function f(x,y,z)=x²+2y²+z at the point P=(1,1,2) in the direction of the vector a =3î−4ĵ is

Accepted Answer

-2. The directional derivative is calculated by taking the dot product of the gradient of the function at the given point and the unit vector in the direction of the specified vector. At point P, the gradient of the function is evaluated, and when the unit vector of the direction is used, the resulting value is -2.

Question 36

The divergence of the vector field 3xz i^ + 2xy j^ - yz² k^ at a point (1,1,1) is equal to

Accepted Answer

3. Divergence = 3z + 2x - 2yz. At (1,1,1) this equals 3 + 2 - 2 = 3. The stored value 0 is incorrect.

Question 37

For the spherical surface x² + y² + z² = 1, the unit outward normal vector at the point (1/√2, 1/√2, 0) is given by

Accepted Answer

1/√2 î + 1/√2 ĵ. The unit outward normal vector at a point on a sphere is simply the normalized position vector from the origin to that point. At (1/√2, 1/√2, 0), the vector is (1/√2, 1/√2, 0), which, when normalized, remains the same since its magnitude is already 1 in the x-y plane.

Question 38

The following surface integral is to be evaluated over a sphere for the given steady velocity vector field F = x i + y j + z k defined with respect to a Cartesian coordinate system having i, j and k as unit base vectors.
∬_S (F · n) dA
where S is the sphere, x² + y² + z² = 1 and n is the o

Accepted Answer

4. The surface integral evaluates the flux of the vector field F through the sphere's surface. Since F is a linear function and the sphere is symmetric, the total outward flux can be computed using the divergence theorem, which gives a result of 4 for this specific vector field over the unit sphere.

Question 39

The integral ∮_C (y dx − x dy) is evaluated along the circle x² + y² = 1/4 traversed in counter clockwise direction. The integral is equal to

Accepted Answer

−π/2. The integral represents the area enclosed by the curve, and since the curve is a circle of radius 1/2, the area is π times the square of the radius, which gives us π/4. However, the orientation of the curve and the specific form of the integral leads to a negative sign, resulting in the final value of -π/2.

Question 40

For an incompressible flow field, V which one of the following conditions must be satisfied?

Accepted Answer

∇ · V = 0. For an incompressible flow, the divergence of the velocity field must be zero, indicating that the fluid density remains constant and there is no net flow of fluid into or out of any infinitesimal volume.

Question 41

The volumetric flow rate (per unit depth) between two streamlines having stream functions 1 and 2 is

Accepted Answer

1 - 2. The volumetric flow rate per unit depth between two streamlines is determined by the difference in their stream functions, which represents the flow potential. Therefore, the correct option is the difference between the two stream functions, 1 - 2.

GATE Engineering Mathematics: Vector Calculus questions with solutions

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