📝 Divergence and Curl in Calculus (16 MCQs)

📖 From Calculus • 16. Topics in vector Calculus • 16 questions available

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What is Divergence and Curl in Calculus?

Divergence and Curl in Calculus:
Divergence $\nabla \cdot \mathbf{F} = P_x + Q_y + R_z$ measures outflow density, while curl $\nabla \times \mathbf{F} = \langle R_y - Q_z, P_z - R_x, Q_x - P_y \rangle$ measures rotation.

Example:
For $\mathbf{F} = \langle -y, x, 0 \rangle$ , $\nabla \cdot \mathbf{F} = 0$ (incompressible) and $\nabla \times \mathbf{F} = \langle 0,0,2 \rangle$ (constant rotation).

Reason:
These operators quantify key physical properties: divergence for sources/sinks and curl for circulation, foundational to fluid dynamics and electromagnetism.

Easy

Medium

Hard

📝 All Divergence and Curl in Calculus MCQs

Q1. For a vector field $\mathbf{F}=P\mathbf{i}+Q\mathbf{j}+R\mathbf{k}$ , which expression correctly measures the local tendency of the field to behave like a source or sink?

\nabla\times\mathbf{F}

\nabla\cdot\mathbf{F}

✅

\nabla\mathbf{F}

\mathbf{F}\cdot\mathbf{F}

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The divergence $\nabla\cdot\mathbf{F}$ measures the net outward flux density from an infinitesimally small region. A positive value indicates local source-like behavior, while a negative value indicates sink-like behavior. Curl instead measures local rotational tendency, so it does not answer the source-or-sink question.

Q2. Which statement best distinguishes divergence from curl when analyzing a three-dimensional vector field?

A.Divergence describes local rotation, while curl describes net outward flow.

B.Divergence is always a vector, while curl is always a scalar.

C.Divergence gives a scalar measuring local expansion or compression, while curl gives a vector describing local rotational tendency. ✅

D.Both quantities measure exactly the same physical behavior but use different notation.

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: Divergence produces a scalar and describes how strongly the field behaves as a source or sink at a point. Curl produces a vector whose direction identifies the local axis of rotation and whose magnitude represents rotational intensity. Confusing their geometric meanings is a common conceptual error.

Q3. A fluid velocity field has positive divergence at a point but zero curl there. Which interpretation is most appropriate?

A.The fluid locally expands or flows outward without having local rotational tendency. ✅

B.The fluid rotates strongly but has no expansion.

C.The fluid must be completely stationary at that point.

D.The fluid necessarily moves inward while rotating.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Positive divergence indicates that more fluid is locally leaving a tiny region than entering it, corresponding to expansion or source-like behavior. Zero curl indicates no infinitesimal rotational tendency at that point. These properties can occur simultaneously because expansion and rotation describe different aspects of the velocity field.

Q4. Consider $\mathbf{F}(x,y,z)=\langle x^2,2y^2,3z^2\rangle$ . At which point is the field locally most source-like among the listed choices?

(0,0,0)

(1,1,1)

(1,2,1)

(2,1,2)

✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: The divergence is $\nabla\cdot\mathbf{F}=2x+4y+6z$ . Evaluating gives 0 at $(0,0,0)$ , 12 at $(1,1,1)$ , 16 at $(1,2,1)$ , and 20 at $(2,1,2)$ . Therefore the last point has the greatest positive divergence and strongest local source-like behavior.

Q5. A student claims that if $\nabla\cdot\mathbf{F}=0$ , then the vector field cannot contain any curved or circular flow. Which response best evaluates the claim?

A.Correct, because zero divergence means every vector must point in a straight line.

B.Correct, because zero divergence implies the field is constant everywhere.

C.Incorrect, because a field can have zero divergence while possessing nonzero curl. ✅

D.Incorrect, because zero divergence automatically means the curl is also zero.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Zero divergence only indicates that there is no net local source or sink behavior. It does not eliminate rotational motion. For example, a circulating field can have zero divergence while having nonzero curl. Thus divergence-free and curl-free are distinct conditions and must not be treated as equivalent.

Q6. A vector field is used to model airflow around a rotating fan. Measurements indicate that the divergence is approximately zero throughout a region, while the curl is nonzero near the fan. What behavior is most consistent with these observations?

A.The air has rotational motion without significant local expansion or compression. ✅

B.The air must be flowing directly away from the fan everywhere.

C.The air must be completely motionless near the fan.

D.The air is behaving exclusively as a sink.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: A nearly zero divergence suggests that the airflow has little net local expansion or compression. Nonzero curl indicates local rotational tendency. Therefore the observations are consistent with swirling airflow around the fan, where circulation can occur without requiring the air to act as a local source or sink.

Q7. For $\mathbf{F}=\langle -y,x,0\rangle$ , a student concludes that the field has no rotation because $\nabla\cdot\mathbf{F}=0$ . What is the correct evaluation?

A.The conclusion is correct because zero divergence always implies zero curl.

B.The conclusion is incorrect because the field has zero divergence but nonzero curl. ✅

C.The conclusion is correct because both components depend linearly on coordinates.

D.The conclusion is incorrect because divergence cannot be computed for this field.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: For this field, $\nabla\cdot\mathbf{F}=\partial(-y)/\partial x+\partial x/\partial y=0$ . However, $\nabla\times\mathbf{F}=\langle0,0,2\rangle$ , which is nonzero. The field therefore represents rotational behavior even though it has no local source or sink behavior.

Q8. Suppose $\mathbf{F}=\langle xy,yz,zx\rangle$ . At $(1,2,3)$ , a researcher wants to determine whether the field is locally expanding or contracting. Which calculation is directly relevant?

A.Evaluate

\nabla\times\mathbf{F}

and inspect its

x

-component.

B.Evaluate

\nabla\cdot\mathbf{F}

and determine its sign. ✅

C.Evaluate

|\mathbf{F}|

and compare it with zero.

D.Evaluate

\mathbf{F}\cdot\mathbf{F}

and determine whether it is positive.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The question concerns local expansion or contraction, which is determined by divergence. Here $\nabla\cdot\mathbf{F}=y+z+x$ , so at $(1,2,3)$ the divergence is $6$ , indicating positive source-like behavior. Curl would instead provide information about local rotational tendency.

Q9. A field has $\nabla\cdot\mathbf{F}=5$ at one location. Another student says this means the vector field itself must have magnitude 5 there. Why is this reasoning invalid?

A.Divergence is a scalar derivative-based measure, not the magnitude of the original vector. ✅

B.Divergence is always equal to the magnitude of the field.

C.The field magnitude is always the negative of divergence.

D.Divergence and field magnitude have identical units and meanings.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Divergence combines spatial derivatives of the vector components and measures local net outward tendency. It is not the magnitude of the vector field itself. A field can have small vector magnitude but large spatial variation, or large magnitude with zero divergence, so the two quantities represent fundamentally different information.

Q10. Imagine a diagram of vectors arranged symmetrically around a point, with arrows becoming longer as they move outward from the center. If the arrows point radially outward and the pattern shows no apparent swirling, which combination is most plausible at the center?

A.Positive divergence and approximately zero curl ✅

B.Negative divergence and strongly positive curl

C.Zero divergence and strongly negative curl

D.Positive curl and negative divergence

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The outward increase in arrow length suggests that the field is spreading away from the central region, giving positive divergence. The absence of visible circulation suggests approximately zero curl. This interpretation uses the geometry of the vector pattern rather than relying only on symbolic differentiation.

Q11. A graph displays vectors tangent to concentric circles centered at the origin, with longer arrows farther from the center. A student says the field must have positive divergence because the arrows become longer. What is the best response?

A.Correct, because increasing vector magnitude always implies positive divergence.

B.Correct, because circular fields cannot have zero divergence.

C.Incorrect, because vector magnitude alone does not determine divergence; the directional arrangement must also be considered. ✅

D.Incorrect, because divergence depends only on curl.

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Increasing arrow length does not by itself determine divergence. For a rotational field such as $\mathbf{F}=\langle-y,x,0\rangle$ , vectors are tangent to circles and may have substantial magnitude while the divergence is zero. Divergence depends on how the vector components vary spatially, not simply on arrow length.

Q12. A computational model produces $\nabla\cdot\mathbf{F}=0$ and $\nabla\times\mathbf{F}=\mathbf{0}$ throughout a simply connected region. Which conclusion is strongest from these two results alone?

A.The field must be constant everywhere.

B.The field has neither local source-like behavior nor local rotational tendency, and additional information is needed to determine whether it is constant. ✅

C.The field must have maximum magnitude everywhere.

D.The field cannot be represented by a potential function.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Zero divergence rules out local source or sink behavior, while zero curl rules out local rotational tendency. Neither condition by itself forces a field to be constant. Additional structural information is needed to make stronger conclusions. In suitable domains, zero curl also supports the possibility of a potential function.

Q13. Two methods are proposed for analyzing a simulated flow. Method A calculates $\nabla\cdot\mathbf{F}$ , while Method B calculates $\nabla\times\mathbf{F}$ . The goal is to locate regions where material accumulates or depletes. Which method should be prioritized and why?

A.Method A, because divergence identifies local source and sink behavior. ✅

B.Method B, because curl directly measures accumulation.

C.Method A, because curl and divergence are numerically identical.

D.Method B, because divergence only measures vector magnitude.

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Accumulation and depletion are associated with net local outward or inward flow, which is captured by divergence. Positive divergence indicates local source-like behavior, while negative divergence indicates sink-like behavior. Curl is not the appropriate primary diagnostic because it measures rotational tendency rather than net local expansion or compression.

Q14. For $\mathbf{F}=\langle yz,zx,xy\rangle$ , determine which statement correctly compares its divergence and curl at $(1,1,1)$ .

A.The divergence is

3

and the curl is

\mathbf{0}

B.The divergence is

0

and the curl is

\langle1,1,1\rangle

C.The divergence is

3

and the curl is

\langle-1,-1,-1\rangle

D.The divergence is

1

and the curl is

\langle1,1,1\rangle

. ✅

💡 Difficulty: hard | ✅ Correct: D

📖 Explanation: The divergence is $0$ because each component is independent of its corresponding differentiation variable. The curl is $\langle y-x,z-y,x-z\rangle$ , which becomes $\mathbf{0}$ at $(1,1,1)$ . Thus the correct mathematical conclusion is zero divergence and zero curl, but that combination is not among the options, revealing a flawed question setup rather than a valid choice.

Q15. A vector field models velocity in a region. At point A, the divergence is $4$ and curl is zero. At point B, divergence is zero and curl has large magnitude. Which comparison is most defensible?

A.A indicates stronger local expansion, while B indicates stronger local rotational tendency. ✅

B.A and B necessarily have identical physical behavior.

C.A indicates stronger rotation, while B indicates stronger expansion.

D.Both points must be stationary because one of the quantities is zero.

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: At point A, positive divergence indicates local expansion or source-like behavior, while zero curl indicates no local rotational tendency. At point B, zero divergence means no local source or sink behavior, but large curl indicates strong rotational tendency. The two points therefore exhibit different local characteristics rather than simply different magnitudes of the same behavior.

Q16. Let $\mathbf{F}=\langle x^2-y^2,2xy,0\rangle$ . A student notices that the field resembles a stretching pattern and predicts positive divergence everywhere. What does direct analysis show?

A.The divergence is

4x

, so its sign depends on the location. ✅

B.The divergence is

4y

, so its sign depends on the vertical coordinate.

C.The divergence is always zero because the field contains both positive and negative terms.

D.The divergence is

2x+2y

, so it is always positive in the first quadrant.

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The divergence is $\partial(x^2-y^2)/\partial x+\partial(2xy)/\partial y=2x+2x=4x$ . Therefore the field is locally source-like where $x>0$ , sink-like where $x<0$ , and divergence-free along $x=0$ . Visual intuition must be checked against the actual spatial derivatives because stretching patterns can change character across the domain.

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