Vector Calculus: Dive into the World of Vector Fields, Integrals, and Theorems

Question 1

Which of the following best defines a vector field?

Accepted Answer

A function that assigns a unique vector to each point in a region of space

Answer

A function that assigns a unique scalar to each point in a region of space

Answer

A function that assigns a unique curve to each point in a region of space

Answer

A function that assigns a unique plane to each point in a region of space

Question 2

What is the divergence of a vector field?

Accepted Answer

A scalar field that measures the outward flux of the vector field

Answer

A scalar field that measures the inward flux of the vector field

Answer

A vector field that points in the direction of greatest increase

Answer

A vector field that points in the direction of greatest decrease

Question 3

What is the curl of a vector field?

Accepted Answer

A vector field that measures the rotation of the vector field

Answer

A vector field that measures the circulation of the vector field

Answer

A scalar field that measures the circulation of the vector field

Answer

A scalar field that measures the rotation of the vector field

Question 4

What is the gradient of a scalar field?

Accepted Answer

A vector field that points in the direction of greatest increase

Answer

A vector field that points in the direction of greatest decrease

Answer

A scalar field that measures the rate of change

Answer

A scalar field that measures the magnitude of change

Question 5

Which of the following is the Laplacian of a scalar field?

Accepted Answer

The divergence of the gradient

Answer

The curl of the gradient

Answer

The dot product of the gradient with itself

Answer

The cross product of the gradient with itself

Question 6

Which of the following vector fields is conservative?

Accepted Answer

F = (e^x + y) i + (x + e^y) j

Answer

F = (y) i + (x) j

Answer

F = (x + y) i + (y - x) j

Answer

F = (x^2 + y) i + (y^2 - x) j

Question 7

Which of the following is the definition of a vector field?

Accepted Answer

A function that assigns a vector to each point in a region

Answer

A function that assigns a scalar to each point in a region

Answer

A function that assigns a matrix to each point in a region

Answer

A function that assigns a tensor to each point in a region

Question 8

What is the line integral of a vector field F(x, y) along a curve C?

Accepted Answer

The integral of the dot product of F(x, y) and the unit tangent vector to C

Answer

The integral of F(x, y) over the length of C

Answer

The integral of the cross product of F(x, y) and the unit tangent vector to C

Answer

The integral of the divergence of F(x, y) over the region enclosed by C

Question 9

What is the unit normal vector to a surface?

Accepted Answer

A vector that is perpendicular to the surface at every point

Answer

A vector that is tangent to the surface at every point

Answer

A vector that is parallel to the surface at every point

Answer

A vector that is zero at every point

Question 10

What is the definition of a vector-valued function?

Accepted Answer

A function that assigns a vector to each point in a region

Answer

A function that assigns a scalar to each point in a region

Answer

A function that assigns a matrix to each point in a region

Answer

A function that assigns a tensor to each point in a region

Question 11

Which of the following is NOT a fundamental theorem of vector calculus?

Accepted Answer

Cauchy-Schwarz inequality

Answer

Green's theorem

Answer

Stokes' theorem

Answer

Divergence theorem

Question 12

Express the correct formula for the line integral of a vector field F(x, y, z) along a smooth curve C.

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(t)) · r(t) dt

Question 13

Given a surface S defined by z = f(x, y) and a vector field F(x, y, z) = (x, y, z), calculate the surface integral of F over S.

Accepted Answer

∫∫[S] F(x, y, f(x, y)) · ||∇f(x, y)|| dA

Answer

∫∫[S] F(x, y, z) · dS

Answer

∫∫[S] F(r(u, v)) · ||∂r/∂u x ∂r/∂v|| du dv

Answer

∫∫[S] F(x, y, z) · dA

Question 14

State the divergence theorem.

Accepted Answer

∫∫[S] F(x, y, z) · dS = ∫∫∫[D] ∇ · F dV

Answer

∫[C] F(x, y, z) · dr = ∫∫[S] F(x, y, z) · dS

Answer

∫∫[S] F(x, y, z) · dS = ∫[C] F(r(t)) · r'(t) dt

Answer

∫∫[S] F(x, y, z) · ||∇f(x, y)|| dA = ∫∫[S] F(x, y, f(x, y)) · dS

Question 15

Evaluate the line integral of the vector field F = (xy, xz, yz) around the circle parametrized by r(t) = (cos(t), sin(t)).

Accepted Answer

2π

Answer

0

Answer

π

Answer

4π

Question 16

Which of the following is NOT a property of vector fields?

Accepted Answer

They are solely defined in terms of Cartesian coordinates.

Answer

They can describe physical quantities.

Answer

They can be used to model fluid flow.

Answer

They can be utilized to calculate work done by forces.

Question 17

Evaluate the following integral using Green's Theorem: ∮[C] (3x + y) dx + (x - 2y) dy, where C is the boundary of the region enclosed by the curves x^2 + y^2 = 1 and x^2 + (y - 1)^2 = 1.

Accepted Answer

0

Answer

1

Answer

2

Question 18

Calculate the flux of the vector field F(x, y, z) = (x, y, z) through the surface S, which is the cylinder x^2 + y^2 = 1, oriented outward, and bounded by the planes z = 0 and z = 3.

Accepted Answer

3π

Answer

9π

Answer

6π

Answer

12π

Question 19

Find the gradient of the scalar field f(x, y) = x^2 + y^2.

Accepted Answer

(2x, 2y)

Answer

(y, x)

Answer

(x + y, x - y)

Answer

(x, y)

Question 20

Which of the following is NOT a fundamental theorem of vector calculus?

Accepted Answer

Chain rule for vector fields

Answer

Divergence theorem

Answer

Stokes' theorem

Answer

Green's theorem

Question 21

Let F = (x, y, z) and S be the surface of the sphere x^2 + y^2 + z^2 = 4. What is the flux of F through the surface?

Accepted Answer

4π

Answer

8π

Answer

16π

Answer

0

Question 22

The divergence theorem can be expressed mathematically as:

Accepted Answer

∫∫_(S) F·dS = ∭_(V) (∇·F) dV

Answer

∫_(S) f dS = ∭_(V) f dV

Answer

∮_(C) F·dr = 0 if F is conservative

Answer

∫_(C) F·dr = ∭_(D) (∇×F) dV

Question 23

State the divergence theorem.

Accepted Answer

The flux of a vector field through a closed surface is equal to the divergence of the vector field integrated over the enclosed region.

Answer

The divergence of a vector field is equal to the negative curl of the field.

Question 24

State Stokes' theorem.

Accepted Answer

The line integral of a vector field around the boundary of a surface is equal to the surface integral of the curl of the vector field over the surface.

Answer

The surface integral of a vector field over a closed surface is equal to the divergence of the vector field integrated over the enclosed region.

Question 25

The divergence theorem is a mathematical result that relates a closed surface's outward flux of a vector field to the divergence of the vector field in the enclosed region. Which statement accurately describes the divergence theorem?

Accepted Answer

The flux of the vector field through the closed surface is equal to the divergence of the vector field integrated over the enclosed volume.

Answer

The line integral of the vector field around a closed curve is equal to the circulation of the vector field along the curve.

Question 26

Which topic is not directly related to vector calculus?

Accepted Answer

Vector spaces

Answer

Surface integrals

Answer

Divergence theorem

Answer

Line integrals

Question 27

Find the divergence 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

2(x + y + z)

Answer

0

Answer

x + y + z

Question 28

Which of the following is the Jacobian of the transformation x = u + v, y = u - v?

Accepted Answer

2

Answer

1

Answer

-1

Answer

-2

Question 29

Calculate the work done by the vector field F(x, y, z) = (y, -x, z) along the path C parameterized by r(t) = (t, t^2, t^3) for 0 ≤ t ≤ 1.

Accepted Answer

2/3

Answer

2

Answer

1

Answer

1/3

Question 30

Find a potential function for the vector field F(x, y) = (2x + y) i + (x - 2y) j.

Accepted Answer

x^2 + xy - y^2

Answer

x^2 - y^2

Answer

2x^2 + y^2

Answer

2xy - y^2

Question 31

Determine whether the vector field F(x, y, z) = (e^x, e^y, e^z) is irrotational.

Accepted Answer

Yes

Answer

No

Answer

Cannot be determined

Answer

None of the above

Question 32

Evaluate the line integral of the vector field F(x, y) = (x^2 + y, xy) along the curve C given by y = x^2 from (0, 0) to (1, 1).

Accepted Answer

1/3

Answer

2/3

Answer

1

Answer

1/2

Question 33

Find the surface integral of the vector field F(x, y, z) = (x, y, z) over the surface S given by the part of the plane z = 2 that lies within the cylinder x^2 + y^2 = 1.

Accepted Answer

4π

Answer

8π

Answer

6π

Answer

2π

Question 34

Which of the following theorems relates the flux of a vector field through a closed surface to the divergence of the vector field within the enclosed region?

Accepted Answer

Divergence Theorem

Answer

Green's Theorem

Answer

Fundamental Theorem of Calculus

Answer

Stokes' Theorem

Question 35

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

Accepted Answer

2(x + y + z)

Answer

1

Answer

0

Answer

x^2 + y^2 + z^2

Question 36

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

Accepted Answer

Scalar field

Answer

Conservative vector field

Answer

Solenoidal vector field

Question 37

Find the gradient of the scalar field f(x, y, z) = ln(x^2 + y^2 + z^2).

Accepted Answer

(2x/(x^2 + y^2 + z^2), 2y/(x^2 + y^2 + z^2), 2z/(x^2 + y^2 + z^2))

Answer

(-x, -y, -z)

Answer

(x, y, z)

Question 38

Which of the following is a property of a potential function?

Accepted Answer

Its gradient is the vector field.

Answer

Its divergence is zero.

Answer

Its curl is zero.

Answer

Its Laplacian is zero.