Timedora
Free Practice Quiz Question List

07 Vector Fields Online Quiz Questions

Use this free practice quiz with 20 questions to review 07 Vector Fields, test your knowledge, and prepare for your next test or exam.

20 questions
01
True or false
1 point

A vector field assigns a vector to every point in a region of space.

  1. A

    True

  2. B

    False

02
Written response
1 point

For the conservative field F(x,y)=⟨2xy+3,x2+4y⟩\mathbf{F}(x,y)=\langle 2xy+3,x^2+4y\rangle, a potential function is f(x,y)=x2y+3x+2y2f(x,y)=x^2y+3x+2y^2. What is the work along any path from A=(0,1)A=(0,1) to B=(2,3)B=(2,3)?

03
Fill in the blank
1 point

For a two-dimensional velocity field v=⟨P(x,y),Q(x,y)⟩\mathbf{v}=\langle P(x,y),Q(x,y)\rangle, a streamline satisfies dydx=\frac{dy}{dx}= when .

04
Choose one
1 point

If F=∇f\mathbf{F}=\nabla f, what does the Fundamental Theorem for Line Integrals imply about ∫CF⋅dr\int_C\mathbf{F}\cdot d\mathbf{r}?

  1. A

    The value depends on the length of the path.

  2. B

    The value depends only on the initial and terminal points.

  3. C

    The value is always zero, even when the endpoints differ.

  4. D

    The value depends only on the speed used to parametrize the path.

05
True or false
1 point

A streamline of a time-dependent velocity field necessarily traces the path followed by one specific fluid particle over time.

  1. A

    True

  2. B

    False

06
Choose all
1 point

Select all conditions that must hold for a three-dimensional field F=⟨P,Q,R⟩\mathbf{F}=\langle P,Q,R\rangle to pass the cross-partial test for conservativeness.

  1. A

    Py=QxP_y=Q_x

  2. B

    Px=QyP_x=Q_y

  3. C

    Pz=RxP_z=R_x

  4. D

    Qz=RyQ_z=R_y

  5. E

    Px=RzP_x=R_z

07
Written response
1 point

What adjective describes a vector field whose curl is zero?

08
Fill in the blank
1 point

For a fluid velocity field, positive divergence indicates local , whereas negative divergence indicates local .

09
Choose one
1 point

Which function is a potential function for F(x,y)=⟨2xy+3,x2+4y⟩\mathbf{F}(x,y)=\langle 2xy+3,x^2+4y\rangle?

  1. A

    f(x,y)=x2y+3x+2y2f(x,y)=x^2y+3x+2y^2

  2. B

    f(x,y)=2x2y+3x+4y2f(x,y)=2x^2y+3x+4y^2

  3. C

    f(x,y)=x2y+3y+2x2f(x,y)=x^2y+3y+2x^2

  4. D

    f(x,y)=x2+3x+2y2f(x,y)=x^2+3x+2y^2

10
Choose all
1 point

Select all statements supported by the vector-calculus interpretation of a fluid velocity field.

  1. A

    A flow with zero divergence is modeled as incompressible.

  2. B

    Positive divergence indicates local expansion or source-like behavior.

  3. C

    Zero divergence guarantees that the flow has zero curl.

  4. D

    Zero curl indicates no local rotational tendency.

11
Open ended
1 point

Explain why checking that the curl is zero is not, by itself, enough to conclude that a vector field is conservative. State the domain condition that makes the test sufficient and describe what can go wrong when the domain has a hole.

12
Choose one
1 point

Which statement correctly relates a conservative force F\mathbf{F}, potential energy UU, and the work WW done as a particle moves from AA to BB?

  1. A

    F=∇U\mathbf{F}=\nabla U and W=U(B)−U(A)W=U(B)-U(A)

  2. B

    F=∇U\mathbf{F}=\nabla U and W=U(A)−U(B)W=U(A)-U(B)

  3. C

    F=−∇U\mathbf{F}=-\nabla U and W=U(A)−U(B)W=U(A)-U(B)

  4. D

    F=−∇U\mathbf{F}=-\nabla U and W=U(B)−U(A)W=U(B)-U(A)

13
Choose one
1 point

Let F(x,y)=⟨x2+3y, 3x+4y2⟩\mathbf{F}(x,y)=\langle x^2+3y,\,3x+4y^2\rangle on all of R2\mathbb{R}^2. Which conclusion is justified?

  1. A

    It is not conservative because Py≠QxP_y\neq Q_x.

  2. B

    It is conservative because Py=QxP_y=Q_x and the domain is simply connected.

  3. C

    It is not conservative because the field has two components.

  4. D

    It is conservative only if P=QP=Q.

14
Choose one
1 point

A conservative vector field has potential function f(x,y)=x2+y2f(x,y)=x^2+y^2. What is the line integral from A=(1,0)A=(1,0) to B=(0,2)B=(0,2), regardless of the path?

  1. A

    −3-3

  2. B

    11

  3. C

    33

  4. D

    55

15
True or false
1 point

True or false: In a time-dependent fluid velocity field, a streamline necessarily gives the path traced by one particular fluid particle.

  1. A

    True

  2. B

    False

16
Choose one
1 point

Compute the divergence of F(x,y,z)=⟨x2y, yz, z2x⟩\mathbf{F}(x,y,z)=\langle x^2y,\,yz,\,z^2x\rangle.

  1. A

    x2+y2+z2x^2+y^2+z^2

  2. B

    2xy+yz+2zx2xy+yz+2zx

  3. C

    x2y+yz2+zx2x^2y+yz^2+zx^2

  4. D

    2xy+z+2zx2xy+z+2zx

17
Written response
1 point

What term describes a vector field whose curl is zero?

18
Choose one
1 point

Find one potential function for F(x,y)=⟨2xy+3, x2+4y⟩\mathbf{F}(x,y)=\langle 2xy+3,\,x^2+4y\rangle.

  1. A

    x2y+3x+2y2x^2y+3x+2y^2

  2. B

    2x2y+3x+4y22x^2y+3x+4y^2

  3. C

    x2+3xy+2y2x^2+3xy+2y^2

  4. D

    x2y+3y+2x2x^2y+3y+2x^2

19
Written response
1 point

For the planar velocity field v(x,y)=⟨−y,x⟩\mathbf{v}(x,y)=\langle -y,x\rangle, what is the exact scalar curl Qx−PyQ_x-P_y?

20
Choose one
1 point

A conservative force has potential energy U(x,y)=x2+y2U(x,y)=x^2+y^2. What work does the force do when an object moves from A=(0,3)A=(0,3) to B=(0,2)B=(0,2)?

  1. A

    −5-5

  2. B

    55

  3. C

    1313

  4. D

    3636