Vector Calculus Quizzes: Test Your Understanding

Question 1

Which physical quantity is a vector?

Accepted Answer

Displacement

Answer

Mass

Answer

Volume

Answer

Temperature

Question 2

Find the surface integral of the vector field F(x, y, z) = <x, y, z> over the surface of the unit sphere centered at the origin.

Accepted Answer

4π

Answer

0

Answer

π

Answer

2π

Question 3

Identify the theorem that relates line integrals to surface integrals.

Accepted Answer

Stokes' Theorem

Answer

Green's Theorem

Answer

Gauss's Theorem

Answer

Cauchy's Theorem

Question 4

Find the potential function for the vector field F(x, y, z) = <2x, 3y, z>.

Accepted Answer

x^2 + y^3 + z^2 + C

Answer

x^2 + y^3 + z^2

Answer

x^2 + 3y^2 + z^2

Answer

2x^2 + 3y^2 + z^2

Question 5

Express the vector line integral representing the work done by a force field F(x, y, z) along a curve C given by r(t) using precise notation.

Accepted Answer

∫C F(r(t)) · r'(t) dt

Answer

∫C F(x, y, z) · dr

Answer

∫C F(x, y, z) · r(t) dt

Answer

∫C r(t) · F(x, y, z) dt

Question 6

Which of the following is a defining property of a vector?

Accepted Answer

Has both magnitude and direction

Answer

Has magnitude only

Answer

Has direction only

Answer

Has neither magnitude nor direction

Question 7

What is the rate of change of the vector function r(t) = (2t, t^2)?

Accepted Answer

(2, 2t)

Answer

(2, t)

Answer

(2t, 2)

Answer

(t, 2t)

Question 8

Which integral formula calculates the circulation of a vector field F around a closed curve C?

Accepted Answer

∮[C] F ⋅ dr

Answer

∫[C] F ⋅ dr

Answer

∫[C] F ⋅ dC

Answer

∮[C] F ⋅ dC

Question 9

Which condition ensures that a vector field is conservative?

Accepted Answer

Curl of F is zero

Answer

Divergence of F is zero

Answer

F is continuous

Answer

F is differentiable

Question 10

What is the relationship between a conservative vector field F and its potential function φ?

Accepted Answer

∇φ = F

Answer

∇⋅ φ = F

Answer

∇× φ = F

Answer

∇² φ = F

Question 11

Which of the following is a scalar quantity?

Accepted Answer

Speed

Answer

Displacement

Answer

Velocity

Answer

Acceleration

Question 12

The dot product of two vectors is a/an:

Accepted Answer

scalar quantity

Answer

vector quantity

Answer

tensor quantity

Answer

matrix quantity

Question 13

The gradient of a scalar field represents the:

Accepted Answer

direction of maximum change

Answer

rate of change

Answer

curvature

Answer

divergence

Question 14

The surface integral of a vector field over a surface is:

Accepted Answer

the flux of the vector field through the surface

Answer

the volume enclosed by the surface

Answer

the curl of the vector field

Answer

the divergence of the vector field

Question 15

The curl of a vector field measures its:

Accepted Answer

circulation

Answer

divergence

Answer

gradient

Answer

potential

Question 16

Which application of vector calculus is commonly used in fluid dynamics?

Accepted Answer

Calculating velocity fields

Answer

Modeling heat transfer

Answer

Analyzing electromagnetic fields

Answer

Designing aircraft wings

Question 17

The vector equation of a plane in three-dimensional space is:

Accepted Answer

Ax + By + Cz + D = 0

Answer

r = r0 + tv

Answer

F = ma

Answer

∇ф = 0

Question 18

Given a vector field F(x, y, z) and a smooth curve C, the line integral of F along C is defined as:

Accepted Answer

The work done by F on a particle moving along C

Answer

The flux of F through a surface bounded by C

Answer

The divergence of F at each point on C

Answer

The curl of F at each point on C

Question 19

The divergence of a vector field F(x, y, z) at a point measures:

Accepted Answer

The outward flux of F from a small region around the point

Answer

The circulation of F around a small region around the point

Answer

The rate of change of the magnitude of F along a streamline

Answer

The rate of change of the direction of F along a streamline

Question 20

The curl of a vector field F(x, y, z) at a point measures:

Accepted Answer

The circulation of F around a small loop around the point

Answer

The outward flux of F from a small region around the point

Answer

The rate of change of the magnitude of F along a streamline

Answer

The rate of change of the direction of F along a streamline

Question 21

Stokes' Theorem can be used to calculate:

Accepted Answer

The circulation of a vector field around a closed curve

Answer

The flux of a vector field through a closed surface

Answer

The divergence of a vector field at a point

Answer

The curl of a vector field at a point

Question 22

In the context of electromagnetism, the curl of an electric field vector field represents:

Accepted Answer

The magnetic field

Answer

The electric potential

Answer

The electric charge density

Answer

The electric current density

Question 23

Calculate the gradient of the vector field F(x, y, z) = (x^2 + y^2)i + (y^2 + z^2)j + (z^2 + x^2)k.

Accepted Answer

2xi + 2yj + 2zk

Answer

x^2i + y^2j + z^2k

Answer

2xi - 2yj + 2zk

Answer

x^2i - y^2j + z^2k

Question 24

Which of the following is NOT a vector quantity?

Accepted Answer

Temperature

Answer

Force

Answer

Displacement

Answer

Velocity

Question 25

If a vector field F(x, y, z) is conservative, then:

Accepted Answer

F can be expressed as the gradient of a scalar function

Answer

Curl of F is zero

Answer

Divergence of F is zero

Answer

F is a constant vector field

Question 26

Which of the following is a potential function for the vector field F(x, y) = 2xyi + (x^2 - y)j?

Accepted Answer

x^2y - xy^2

Answer

y^2x - xy^2

Answer

x^2y + xy^2

Answer

y^2x + xy^2

Question 27

Calculate the value of the line integral ∫C (x + y) ds, where C is the curve defined by the vector function r(t) = (t, t^2) for t ranging from 0 to 1.

Accepted Answer

∫[0,1] (t + t^2) dt

Answer

1

Answer

(1 + 1) ds

Answer

∫[0,1] (x^2 + y^2) ds

Question 28

Which of the following is NOT a fundamental characteristic of a vector?

Accepted Answer

Polarization

Answer

Magnitude

Answer

Direction

Answer

Length

Question 29

What is the formula to determine the gradient of a scalar field f(x, y, z)?

Accepted Answer

∇f = (df/dx, df/dy, df/dz)

Answer

∇f = (dx/df, dy/df, dz/df)

Answer

∇f = (f/dx, f/dy, f/dz)

Answer

∇f = (df/dx, df/dy, df/dz)^2

Question 30

Which of the following is NOT one of the fundamental theorems of line integrals?

Accepted Answer

Cauchy's Integral Theorem

Answer

Conservative Vector Field Theorem

Answer

Green's Theorem

Answer

Stokes' Theorem

Question 31

Which of the following is the formula to calculate the line integral ∫C F · dr?

Accepted Answer

∫C F(x, y, z) · (dx, dy, dz)

Answer

∫C F(x, y, z) dx + F(x, y, z) dy + F(x, y, z) dz

Answer

∫C F(x, y) dx + F(y, z) dy + F(z, x) dz

Answer

∫C F(x, y, z) / (dx, dy, dz)

Question 32

Which of the following is the correct Gauss's Theorem?

Accepted Answer

∫∫S F · dS = ∫∫∫V ∇ · F dV

Answer

∫S F · dS = ∫V ∇ · F dV

Answer

∫S F · dS = ∫∫V F · dS

Answer

∫∫S F · dS = ∫∫∫V ∇ · F dS

Question 33

Find the curl of the vector field F = (x^2, y^2, z^2)

Accepted Answer

(-2y, 2x, 0)

Answer

(2y, -2x, 0)

Answer

(0, 0, 2z)

Answer

(0, 0, 0)

Question 34

Which of the following is NOT a type of surface integral?

Accepted Answer

Volume integral

Answer

Flux integral

Answer

Circulation integral

Answer

Area integral

Question 35

Which physical quantity is a vector?

Accepted Answer

Speed

Answer

Distance

Answer

Temperature

Answer

Mass

Question 36

The dot product of vectors A and B calculates:

Accepted Answer

The component of vector A in the direction of vector B

Answer

The magnitude of the cross product of the vectors

Answer

The angle between the vectors

Answer

The sum of the magnitudes of the vectors

Question 37

Which is NOT a condition for a surface to be orientable?

Accepted Answer

The surface is homeomorphic to a sphere

Answer

The surface is connected

Answer

The surface is smooth

Answer

The surface has a unit normal vector

Question 38

Which vector field is not solenoidal?

Accepted Answer

F(x, y, z) = (x^2, y^2, z^2)

Answer

F(x, y, z) = (x, y, z)

Answer

F(x, y, z) = (y, -x, 0)

Answer

F(x, y, z) = (1, 1, 1)

Question 39

The curl of a conservative vector field is:

Accepted Answer

Zero

Answer

Non-zero

Answer

Parallel to the vector field

Answer

Perpendicular to the vector field

Question 40

A vector function r(t) = (t^2, t^3, t^4) represents the position of a particle's motion. What is its acceleration vector?

Accepted Answer

(2t, 3t^2, 4t^3)

Answer

(0, 2, 6)

Answer

(2, 6, 12)

Answer

(0, 0, 0)

Question 41

Identify the scalar quantity from the following options:

Accepted Answer

Electric potential

Answer

Displacement

Answer

Velocity

Answer

Acceleration

Question 42

Which theorem establishes a connection between line integrals and double integrals?

Accepted Answer

Green's Theorem

Answer

Stokes' Theorem

Answer

Divergence Theorem

Answer

Cauchy-Riemann Equations

Question 43

Determine the mathematical expression for the curl of a vector field F:

Accepted Answer

The cross product of the del operator and F

Answer

The gradient of the divergence of F

Answer

The divergence of the gradient of F

Answer

The dot product of the del operator and F

Question 44

Which application of vector calculus is relevant to fluid dynamics?

Accepted Answer

Calculating the circulation of a fluid

Answer

Finding the flux of heat through a surface

Answer

Determining the work done by a force field

Answer

Solving electrostatic problems

Question 45

Identify the condition for a vector field F to possess a potential function:

Accepted Answer

Curl of F is zero

Answer

Divergence of F is zero

Answer

Line integral of F over any closed curve is zero

Answer

Surface integral of F over any closed surface is zero

Question 46

Which of the following is not a characteristic of a conservative vector field?

Accepted Answer

Its divergence is always non-zero.

Answer

Its line integral is path independent.

Answer

Its curl is zero.

Answer

Its gradient is a vector field.

Question 47

Given the vector field F(x, y, z) = (x, y, z), determine its divergence.

Accepted Answer

3

Answer

0

Answer

1

Answer

x + y + z

Question 48

Identify an example of a solenoidal vector field.

Accepted Answer

F(x, y, z) = (y, -x, 0)

Answer

F(x, y, z) = (x, y, z)

Answer

F(x, y, z) = (x^2, y^2, z^2)

Answer

F(x, y, z) = (1, 0, 0)

Question 49

What is the physical interpretation of the curl of a vector field?

Accepted Answer

Circulation density

Answer

Divergence

Answer

Gradient

Answer

Flux

Question 50

Which of the following is NOT a vector quantity?

Accepted Answer

Temperature

Answer

Displacement

Answer

Velocity

Answer

Acceleration

Question 51

Which of the following is the divergence theorem?

Accepted Answer

∫∫S F · dA = ∫∫∫V ∇ · F dV

Answer

∮C F · dr = ∫∫S (∇ × F) · dA

Answer

∫∫S F · dA = ∮C F · dr

Answer

∇ × (∇ × F) = ∇(∇ · F) - ∇^2 F

Question 52

Which of the following is the surface integral of the vector field F(x, y, z) = xi + yj + zk over the surface of the unit sphere?

Accepted Answer

4π

Answer

0

Answer

8π

Answer

16π

Question 53

Which of the following is the Gauss's theorem?

Accepted Answer

∫∫S F · dA = ∫∫∫V ∇ · F dV

Answer

∮C F · dr = ∫∫S (∇ × F) · dA

Answer

∫∫S F · dA = ∮C F · dr

Answer

∇ × (∇ × F) = ∇(∇ · F) - ∇^2 F

Question 54

Which of the following is the curl of a gradient?

Accepted Answer

Zero

Answer

Non-zero

Answer

Depends on the gradient

Answer

Depends on the function

Question 55

Which of the following accurately defines a vector field?

Accepted Answer

A function that assigns a vector quantity to each point.

Answer

A function that assigns a scalar quantity to each point.

Answer

A set of vectors that form a closed loop.

Answer

A set of vectors that are parallel to each other.

Question 56

What is the mathematical expression for the line integral of a vector field along a curve?

Accepted Answer

The sum of the dot products of the vector field with the unit tangent vector along the curve.

Answer

The total variation of the vector field over the curve.

Answer

The average value of the vector field along the curve.

Answer

The area under the curve defined by the vector field.

Question 57

State the Divergence Theorem.

Accepted Answer

The integral of a vector field over a closed surface is equal to the net outward flux of the vector field through the surface.

Answer

The gradient of a scalar field is perpendicular to the curl of a vector field.

Answer

The divergence of a vector field is proportional to the surface area of a closed surface.

Answer

The curl of a vector field is equal to the line integral of the vector field around a closed loop.

Question 58

Explain the physical interpretation of the divergence of a vector field.

Accepted Answer

The net outward flux of the vector field through a closed surface.

Answer

The net inward flux of the vector field through a closed surface.

Answer

The total amount of the vector field in a region.

Answer

The rate of change of the vector field with respect to time.

Question 59

What does Stokes' Theorem state?

Accepted Answer

The line integral of a vector field around a closed loop equals the surface integral of the curl of the vector field over any surface bounded by the loop.

Answer

The surface integral of a vector field over a closed surface equals the volume integral of the divergence of the vector field over the enclosed region.

Answer

The gradient of a scalar field is equal to the curl of a vector field.

Answer

The divergence of a vector field is equal to the surface area of a closed surface.