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11 Theorems of Vector Calculus Online Quiz Questions

Use this free practice quiz with 20 questions to review 11 Theorems of Vector Calculus, test your knowledge, and prepare for your next test or exam.

20 questions
01
True or false
1 point

The divergence theorem can be applied directly only when the surface is closed and oriented outward.

  1. A

    True

  2. B

    False

02
Written response
1 point

What is the name of the quantity measured by the closed-curve line integral ∮CF⋅dr\oint_C\mathbf F\cdot d\mathbf r?

03
Fill in the blank
1 point

Flux through a surface depends on the of the vector field.

04
Choose one
1 point

Which component of a vector field contributes to its flux through an oriented surface?

  1. A

    The component tangent to the surface

  2. B

    The component parallel to the boundary curve

  3. C

    The component normal to the surface

  4. D

    The component perpendicular to the boundary curve

05
Choose all
1 point

Which conditions are required for the direct application of Stokes’ theorem? Select all that apply.

  1. A

    The curve is closed.

  2. B

    The spanning surface is piecewise smooth and oriented, and the curve is its positively oriented boundary.

  3. C

    The vector field has continuous first partial derivatives on a region containing the surface.

  4. D

    The spanning surface is planar.

06
True or false
1 point

If ∇⋅F=0\nabla\cdot\mathbf F=0 throughout a solid, then the outward flux of F\mathbf F through every suitable closed surface bounding that solid is zero.

  1. A

    True

  2. B

    False

07
Written response
1 point

Let F(x,y,z)=⟨x2,yz,z3⟩\mathbf F(x,y,z)=\langle x^2,yz,z^3\rangle. What is ∇⋅F\nabla\cdot\mathbf F at (1,2,−1)(1,2,-1)?

08
Fill in the blank
1 point

Complete both statements: Stokes’ theorem relates circulation to the surface integral of , while the divergence theorem relates outward flux to the volume integral of .

09
Choose one
1 point

A circle in the xyxy-plane is traversed counterclockwise when viewed from above. Which normal direction is compatible with Stokes’ theorem?

  1. A

    The downward normal, because the curve is counterclockwise

  2. B

    The upward normal, because the curve is counterclockwise when viewed from above

  3. C

    Either normal, because orientation does not affect the integral

  4. D

    The outward radial direction in the plane of the circle

10
Choose all
1 point

Which statements correctly describe curl and divergence? Select all that apply.

  1. A

    The curl is a vector describing local rotational tendency.

  2. B

    The normal component of curl gives limiting circulation per unit area.

  3. C

    Divergence is a vector describing the axis of local rotation.

  4. D

    For suitable smooth fields, ∇⋅(∇×F)=0\nabla\cdot(\nabla\times\mathbf F)=0.

11
Open ended
1 point

Explain how to evaluate the circulation around the counterclockwise circle x2+y2=a2x^2+y^2=a^2 in the xyxy-plane for F=⟨−y/2,x/2,0⟩\mathbf F=\langle-y/2,x/2,0\rangle without parameterizing the circle. State the surface, orientation, theorem, and resulting value.

12
Choose one
1 point

What is the outward flux of F(x,y,z)=⟨x,y,z⟩\mathbf F(x,y,z)=\langle x,y,z\rangle through the sphere x2+y2+z2=R2x^2+y^2+z^2=R^2?

  1. A

    3πR23\pi R^2

  2. B

    4πR24\pi R^2

  3. C

    4πR34\pi R^3

  4. D

    43πR3\frac{4}{3}\pi R^3

13
Written response
1 point

What term describes a region where a vector field has positive divergence and therefore behaves locally like a source?

14
Choose one
1 point

Which component of a vector field contributes to its circulation along an oriented curve?

  1. A

    The normal component of the field across the curve

  2. B

    The tangential component of the field along the curve

  3. C

    The divergence of the field inside the curve

  4. D

    The curl magnitude at every point on the curve

15
Choose one
1 point

Which geometric condition is required before applying the divergence theorem directly to a surface?

  1. A

    A curve that is open and has no orientation

  2. B

    Any surface, whether or not it has a boundary

  3. C

    A solid bounded by a closed, outward-oriented surface

  4. D

    A planar region whose boundary may be omitted

16
True or false
1 point

True or false: If the relevant second partial derivatives are continuous, then ∇×(∇f)=0\nabla\times(\nabla f)=\mathbf 0.

  1. A

    True

  2. B

    False

17
Choose one
1 point

A boundary curve is traversed in a specified direction. According to Stokes’ theorem, how should the compatible surface normal be chosen?

  1. A

    Curl the fingers of your right hand in the direction of traversal; your thumb gives the normal direction.

  2. B

    Point your left thumb along the traversal direction; your fingers give the normal direction.

  3. C

    Point the normal toward the interior of the surface regardless of traversal.

  4. D

    Choose the normal so that the curl vector always points away from the surface.

18
Written response
1 point

What mathematical quantity describes the local net rate at which a vector field spreads outward?

19
Choose one
1 point

Let F(x,y,z)=⟨−y2,x2,0⟩\mathbf F(x,y,z)=\left\langle-\frac{y}{2},\frac{x}{2},0\right\rangle. What is ∇×F\nabla\times\mathbf F?

  1. A

    ⟨0,0,−1⟩\langle 0,0,-1\rangle

  2. B

    ⟨1,0,0⟩\langle 1,0,0\rangle

  3. C

    ⟨0,0,1⟩\langle 0,0,1\rangle

  4. D

    ⟨0,1,0⟩\langle 0,1,0\rangle

20
Choose one
1 point

Which local differential equation expresses conservation of a density ρ\rho with flux J\mathbf J?

  1. A

    ∂ρ∂t−∇⋅J=1\frac{\partial\rho}{\partial t}-\nabla\cdot\mathbf J=1

  2. B

    ∂ρ∂t+∇×J=0\frac{\partial\rho}{\partial t}+\nabla\times\mathbf J=0

  3. C

    ∂ρ∂t=∇⋅J\frac{\partial\rho}{\partial t}=\nabla\cdot\mathbf J

  4. D

    ∂ρ∂t+∇⋅J=0\frac{\partial\rho}{\partial t}+\nabla\cdot\mathbf J=0