Vector Calculus Learning Quizzes: Master the Fundamentals of Vector Analysis

Question 1

What is the mathematical expression for the line integral of a vector field F along a curve C given by the vector function r(t)?

Accepted Answer

∫C F(r(t)) · r'(t) dt

Answer

∫C F(x, y, z) · dr

Answer

∫C F(x, y, z) · ds

Answer

∫C F(r(u, v)) · |r_u × r_v| du dv

Question 2

Which of the following statements correctly describes a consequence of the Divergence Theorem?

Accepted Answer

The flux of a vector field through a closed surface is zero if the field is divergence-free

Answer

The divergence of a vector field represents the net outward flux per unit volume

Answer

The curl of a vector field represents the circulation per unit area

Answer

The gradient of a vector field represents the rate of change in the direction of maximum decrease

Question 3

What is the surface integral of a vector field F over a surface S given by the vector function r(u, v)?

Accepted Answer

∫∫S F(r(u, v)) · |r_u × r_v| du dv

Answer

∫∫S F(x, y, z) · dA

Answer

∫∫S F(x, y, z) · ds

Answer

∫∫S F(r(t)) · ds

Question 4

Which of the following is an application of Stokes' Theorem in Vector Calculus?

Accepted Answer

Calculating the circulation of a vector field around a closed curve

Answer

Evaluating the flux of a vector field through a surface

Answer

Finding the area enclosed by a closed curve

Answer

Computing the volume of a solid region

Question 5

What is the physical interpretation of the divergence of a vector field in Vector Calculus?

Accepted Answer

The net outward flux per unit volume

Answer

The net circulation per unit area

Answer

The rate of change in the direction of maximum increase

Answer

The amount of work done by the vector field per unit volume

Question 6

Which of the following integral formulas is used to evaluate the line integral of a vector field F along a curve C given by a vector function r(t)?

Accepted Answer

∫C F(r(t)) · r'(t) dt

Answer

∫C F(x, y, z) · ds

Answer

∫C F(r(u, v)) · |r_u × r_v| du dv

Answer

∫C F(x, y, z) · dr

Question 7

Which of the following is **not** a fundamental theorem of vector calculus?

Accepted Answer

The Chain Rule for Vector Fields

Answer

Gauss's Theorem

Answer

Green's Theorem

Answer

Stokes' Theorem

Question 8

Which of the following is the divergence of the vector field $\mathbf{F}(x,y,z) = x^2 \mathbf{i} + y^2 \mathbf{j} + z^2 \mathbf{k}$?

Accepted Answer

2

Answer

0

Answer

4

Answer

6

Question 9

Determine if the following vector field is solenoidal: $\mathbf{F}(x,y,z) = x \mathbf{i} - y \mathbf{j} + z \mathbf{k}$.

Accepted Answer

Yes, it is solenoidal.

Answer

No, it is not solenoidal.

Answer

Cannot be determined from the given information.

Question 10

Which of the following is a potential function for the vector field $\mathbf{F}(x,y,z) = y \mathbf{i} + x \mathbf{j} + z^2 \mathbf{k}$?

Accepted Answer

$\, f(x,y,z) = xy + z^2$

Answer

$\, f(x,y,z) = x^2 y + z^2$

Answer

$\, f(x,y,z) = y^2 x + z^2$

Answer

$\, f(x,y,z) = xy^2 + z^2$

Question 11

In vector calculus, which of the following is not a fundamental theorem?

Accepted Answer

Chain Rule

Answer

Divergence Theorem

Answer

Stokes' Theorem

Answer

Green's Theorem

Question 12

Stokes' Theorem is used to calculate the circulation of a vector field around a:

Accepted Answer

Closed curve

Answer

Open curve

Answer

Parametric curve

Answer

Helix

Question 13

The curl of a vector field F represents its:

Accepted Answer

Circulation

Answer

Divergence

Answer

Gradient

Answer

Flux

Question 14

The gradient of a scalar field φ is a:

Accepted Answer

Vector field

Answer

Scalar field

Answer

Tensor field

Answer

Matrix field

Question 15

A vector field that describes the velocity of a fluid is called a:

Accepted Answer

Flow field

Answer

Force field

Answer

Electric field

Answer

Magnetic field

Question 16

Use the Divergence Theorem to find the outward flux of the vector field F(x, y, z) = (x, y, z) through the surface of the unit sphere x^2 + y^2 + z^2 = 1.

Accepted Answer

4π

Answer

0

Answer

8π

Answer

12π

Question 17

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

Accepted Answer

4

Answer

0

Answer

2

Answer

6

Question 18

Which of the following is a vector field?

Accepted Answer

Velocity of a fluid at each point

Answer

Temperature at each point in a room

Answer

Height of a mountain at each point

Question 19

What is the line integral of a vector field F along a curve C?

Accepted Answer

The total work done by F along C

Answer

The total flux of F across C

Answer

The total change in F along C

Question 20

What is the surface integral of a vector field F over a surface S?

Accepted Answer

The total flux of F across S

Answer

The total work done by F on S

Answer

The total amount of F flowing through S

Question 21

What is the Divergence Theorem?

Accepted Answer

A theorem that relates a volume integral to a surface integral

Answer

A theorem that relates a surface integral to a line integral

Answer

A theorem that relates a line integral to a surface integral

Question 22

What is the Stokes' Theorem?

Accepted Answer

A theorem that relates a surface integral to a line integral

Answer

A theorem that relates a line integral to a surface integral

Answer

A theorem that relates a volume integral to a surface integral

Question 23

What is the curl of a vector field F?

Accepted Answer

The divergence of F

Answer

The gradient of F

Answer

The Laplacian of F

Question 24

What is the potential function of a vector field F?

Accepted Answer

A function whose gradient is F

Answer

A function whose divergence is F

Answer

A function whose curl is F

Question 25

Find the gradient of the vector field F(x, y) = x^2y + xy^2.

Accepted Answer

(2xy, x^2 + 2xy)

Answer

(2y, 2x)

Answer

(x, y)

Answer

(x^2, y^2)

Question 26

Which of the following is the divergence of the vector field F(x, y, z) = (x^2)i + (y^2)j + (z^2)k?

Accepted Answer

2

Answer

0

Answer

1

Answer

x^2 + y^2 + z^2

Question 27

Use the Divergence Theorem to calculate the flux of the vector field F(x, y, z) = (x + y)i + (y - x)j + (z + x)k across the boundary of the sphere S: x^2 + y^2 + z^2 = 4.

Accepted Answer

32π

Answer

0

Answer

16π

Answer

64π

Question 28

Which of the following is the curl of the vector field F(x, y, z) = (x^2y)i + (y^2z)j + (z^2x)k?

Accepted Answer

(y-z)i + (z-x)j + (x-y)k

Answer

(x+y)i + (y+z)j + (z+x)k

Answer

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

Answer

0

Question 29

Use Stokes' Theorem to calculate the line integral of F(x, y, z) = (x + y)i + (y - x)j + (z + x)k around the boundary of the surface S: z = x^2 + y^2, 0 ≤ z ≤ 1.

Accepted Answer

4π

Answer

0

Answer

2π

Answer

8π

Question 30

Find the potential function f for the vector field F(x, y) = (2xy + y^2)i + (x^2 + 2xy)j.

Accepted Answer

f(x, y) = x^2y + xy^2

Answer

f(x, y) = x^2y^2

Answer

f(x, y) = y^2 + x^2

Answer

f(x, y) does not exist

Question 31

Suppose C is a closed curve that encloses a region D. If F is a conservative vector field and C is positively oriented with respect to D, then the circulation of F around C is equal to:

Accepted Answer

0

Answer

The flux of F through D

Answer

The divergence of F at any point in D

Answer

The curl of F at any point in D

Question 32

Which of the following is NOT a basic vector operation?

Accepted Answer

Scalar triple product

Answer

Dot product

Answer

Cross product

Answer

Vector addition

Question 33

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

Accepted Answer

The net outward flux of the field per unit volume

Answer

The net circulation of the field per unit area

Answer

The rate at which the field is changing at a given point

Answer

The magnitude of the field at a given point

Question 34

Which of the following is a closed surface?

Accepted Answer

A sphere

Answer

A hemisphere

Answer

A cone

Answer

A torus

Question 35

What is the relationship between the line integral and the potential function of a vector field?

Accepted Answer

The line integral is the path integral of the potential function

Answer

The line integral is the gradient of the potential function

Answer

The potential function is the curl of the line integral

Answer

The line integral is zero if and only if the potential function is constant

Question 36

Which theorem relates the surface integral of a vector field over a closed surface to the divergence of the field within the surface?

Accepted Answer

The Divergence Theorem

Answer

The Stokes' Theorem

Answer

The Green's Theorem

Answer

The Gauss' Theorem

Question 37

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

Accepted Answer

The circulation of the field per unit area

Answer

The net outward flux of the field per unit volume

Answer

The magnitude of the field at a given point

Answer

The direction of the field at a given point

Question 38

Which theorem relates the line integral of a vector field around a closed curve to the surface integral of the curl of the field over any surface bounded by the curve?

Accepted Answer

The Stokes' Theorem

Answer

The Divergence Theorem

Answer

The Green's Theorem

Answer

The Gauss' Theorem

Question 39

A vector field F(x, y, z) is called irrotational if:

Accepted Answer

Its curl is zero

Answer

Its divergence is zero

Answer

Its gradient is zero

Answer

Its Laplacian is zero

Question 40

Calculate the curl of the vector field  **F** =  ⟨y, x, z⟩.

Accepted Answer

⟨0, 0, 0⟩

Answer

⟨-1, -1, -1⟩

Answer

⟨1, 1, 1⟩

Question 41

Evaluate the line integral of the vector field **F** = ⟨x², y²⟩ along the line segment C from (0, 0) to (1, 1).

Accepted Answer

2/3

Answer

1/3

Answer

1

Answer

2

Question 42

Calculate the flux of the vector field **F** = ⟨x, y, z⟩ across the surface S, where S is the unit sphere centered at the origin.

Accepted Answer

4π

Answer

0

Answer

2π

Answer

π

Question 43

Determine the divergence of the vector field **F** = ⟨x², y², z²⟩.

Accepted Answer

2x + 2y + 2z

Answer

x² + y² + z²

Answer

x + y + z

Answer

0

Question 44

The Divergence Theorem connects a surface integral over a closed surface to a volume integral over the enclosed region. Which of the following accurately represents this theorem mathematically?

Accepted Answer

∫∫_S **F** · d**S** = ∫∫∫_V div(**F**) dV

Answer

∫∫_S **F** · d**S** = ∫_C **F** · d**r**

Answer

∫∫_S **F** · d**S** = ∫∫∫_V **F** dV

Answer

∫∫_S **F** · d**S** = ∫∫∫_V curl(**F**) dV

Question 45

Stokes' Theorem establishes a connection between a line integral around a closed curve and a surface integral over a surface bounded by that curve. Which of the following accurately represents this theorem mathematically?

Accepted Answer

∫_C **F** · d**r** = ∫∫_S curl(**F**) · d**S**

Answer

∫_C **F** · d**r** = ∫∫∫_V curl(**F**) dV

Answer

∫_C **F** · d**r** = ∫∫∫_V div(**F**) dV

Answer

∫_C **F** · d**r** = ∫∫_S div(**F**) d**S**

Question 46

If **F** represents the velocity of a fluid, the divergence of **F** at a specific point provides information about:

Accepted Answer

The rate of expansion or compression of the fluid at that point.

Answer

The curl of the fluid at that point.

Answer

The direction of the fluid flow at that point.

Question 47

Which scenario is best explained by Stokes' Theorem?

Accepted Answer

Calculating the work done by a force field as a particle traverses a closed loop.

Answer

Finding the volume enclosed by a closed surface.

Answer

Determining the amount of fluid flowing across a surface.

Question 48

Given a vector field F = <P, Q, R>, which of the following represents its curl?

Accepted Answer

∇ × F = (∂R/∂y - ∂Q/∂z) i + (∂P/∂z - ∂R/∂x) j + (∂Q/∂x - ∂P/∂y) k

Answer

∇F = (∂P/∂x, ∂Q/∂y, ∂R/∂z)

Answer

∇ ⋅ F = ∂P/∂x + ∂Q/∂y + ∂R/∂z

Question 49

A vector field is considered conservative if:

Accepted Answer

Its curl is zero.

Answer

Its divergence is zero.

Answer

It is a gradient field.

Question 50

Evaluate the line integral ∫C F ⋅ dr, where F = <x, y> and C is the circle x² + y² = 4 traversed counterclockwise.

Accepted Answer

8π

Answer

0

Answer

2π

Answer

4π

Question 51

Which theorem relates the outward flux of a vector field across a closed surface to the divergence of the field within the enclosed volume?

Accepted Answer

Divergence Theorem

Answer

Green's Theorem

Answer

Stokes' Theorem

Question 52

The primary purpose of a surface integral is to calculate:

Accepted Answer

The flux of a vector field across a surface.

Answer

The volume of a solid.

Answer

The work done by a force along a path.

Answer

The length of a curve.

Question 53

Stokes' Theorem relates a line integral around a closed curve to:

Accepted Answer

The surface integral of the curl of the vector field over the surface enclosed by the curve.

Answer

The line integral along another closed curve.

Answer

The divergence of the vector field over the enclosed volume.

Question 54

Find the flux of the vector field F = <x, y, z> across the sphere x² + y² + z² = 4.

Accepted Answer

32π/3

Answer

0

Answer

4π

Answer

8π

Question 55

A vector field with zero divergence is said to be:

Accepted Answer

Solenoidal

Answer

Conservative

Answer

Irrotational

Question 56

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

Accepted Answer

It can be represented by a constant.

Answer

It has both magnitude and direction.

Answer

It can be visualized as a collection of arrows.

Answer

It can be differentiated and integrated.

Question 57

Calculate the work done by the force field F = <x, y> in moving an object along the straight line segment from (0, 0) to (2, 2).

Accepted Answer

2

Answer

-1

Answer

1

Answer

0