📝 Divergence as flux density (15 MCQs)

📖 From Calculus • 16. Topics in vector Calculus • 15 questions available

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What is Divergence as flux density?

Divergence as flux density:
Divergence $\nabla \cdot \mathbf{F}$ at a point is the net flux per unit volume, representing the strength of a source or sink at that point.

Example:
For $\mathbf{F} = \langle x, 0, 0 \rangle$ , $\nabla \cdot \mathbf{F} = 1$ , indicating uniform outflow density.

Reason:
This interpretation links local field behavior to global flux via the Divergence Theorem, essential for continuity equations.

Easy

Medium

Hard

📝 All Divergence as flux density MCQs

Q1. A vector field $\mathbf{F}$ has divergence $\nabla\cdot\mathbf{F}=6$ at a point. Which interpretation best describes this value?

A.The field has magnitude 6 at that point.

B.The field produces a net outward flux of approximately 6 units per unit volume near that point. ✅

C.The field points in the positive

x

-direction with magnitude 6.

D.The field has zero flux through every sufficiently small surface around the point.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Divergence measures local net outward flux per unit volume. Thus $\nabla\cdot\mathbf{F}=6$ means that, for a sufficiently small closed region surrounding the point, the outward flux is approximately six times the region's volume. It does not directly specify the field's magnitude or direction.

Q2. For a smooth vector field, why is divergence more appropriately described as a flux density than simply as flux?

A.Because divergence depends only on the length of the boundary.

B.Because divergence measures total flux through an entire surface.

C.Because divergence measures net outward flux normalized by the volume of a very small region. ✅

D.Because divergence is always equal to the field magnitude.

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: Total flux depends on the size and shape of the surface enclosing a region. Divergence removes this size dependence by considering net outward flux per unit volume in the limiting sense around a point. Therefore, divergence provides a local density of flux production or depletion.

Q3. Two small cubic regions have equal volume and surround nearby points. Region A has net outward flux $0.004$ , while Region B has net outward flux $0.012$ . Which conclusion is most justified if the fields are approximately uniform in divergence within each cube?

A.The divergence in B is approximately three times the divergence in A. ✅

B.The divergence in A is approximately three times the divergence in B.

C.Both regions necessarily have the same divergence because their volumes are equal.

D.The divergence cannot be compared without knowing the field magnitude.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: When the two regions have equal volume and their divergence is approximately uniform, divergence is proportional to net outward flux divided by volume. Since B has three times the outward flux of A for the same volume, its divergence is approximately three times as large. Field magnitude alone is insufficient for comparison.

Q4. A fluid velocity field has positive divergence in a small region. A student claims that this necessarily means the fluid particles inside that region are moving faster. What is the best evaluation?

A.The claim is correct because positive divergence directly measures speed.

B.The claim is correct only when the region is spherical.

C.The claim is incorrect because positive divergence indicates local net expansion or outward flow, not necessarily increasing particle speed. ✅

D.The claim is incorrect because divergence always measures rotation.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Positive divergence indicates that more flow is leaving a sufficiently small region than entering it, corresponding to local expansion in a flow interpretation. Particles can have nearly constant speed while their trajectories spread apart. Therefore, divergence should not be confused with speed, acceleration, or rotational motion.

Q5. A field is $\mathbf{F}(x,y,z)=\langle 2x,-3y,z\rangle$ . At a point, the divergence is positive. Which modeling interpretation is most appropriate for a tiny control volume centered there?

A.The control volume tends to have greater outward flux than inward flux. ✅

B.The control volume must have zero total flux because the field is linear.

C.The field must circulate around the control volume without crossing its boundary.

D.The field magnitude must increase in every direction away from the point.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: For this field, $\nabla\cdot\mathbf{F}=2-3+1=0$ , so the premise that the divergence is positive is actually false. However, among the choices, the defining interpretation of positive divergence would be greater outward than inward flux. The question tests whether students connect divergence with control-volume behavior rather than field magnitude.

Q6. A researcher observes that the net outward flux through successively smaller closed surfaces centered at a point decreases toward zero. However, the ratio of flux to enclosed volume approaches $4$ . What does this suggest about the divergence at the point?

A.The divergence approaches zero because the flux approaches zero.

B.The divergence approaches

4

because flux density is obtained by normalizing flux by volume. ✅

C.The divergence becomes infinite because both flux and volume become small.

D.The divergence cannot be inferred from any limiting ratio.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Flux through a shrinking closed surface normally tends to zero because the surface becomes smaller. What matters for divergence is the limiting ratio of net outward flux to enclosed volume. If that ratio approaches $4$ , the divergence at the point is $4$ , assuming the required smoothness conditions hold.

Q7. A ventilation engineer models air movement through a small room as a vector field. Measurements show that the room has nearly equal inflow and outflow through its walls, but air density is changing because of compression. Which statement most carefully describes what divergence alone tells the engineer?

A.It completely determines the change in air density.

B.It directly gives the local net outward volumetric flux density of the velocity field. ✅

C.It always equals the pressure at the center of the room.

D.It determines the direction of every individual air particle.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: For a velocity field, divergence describes local net volumetric outflow per unit volume. It does not by itself determine density changes, pressure, or individual particle trajectories. In compressible flow, additional conservation laws and material properties are needed to connect velocity divergence to density evolution.

Q8. A student computes $\nabla\cdot\mathbf{F}=5$ and concludes that the flux through any closed surface enclosing the point must be positive. Why is this reasoning incomplete?

A.Divergence is never related to closed-surface flux.

B.A pointwise positive divergence does not guarantee positive total flux unless the divergence is appropriately considered throughout the enclosed volume. ✅

C.Flux is determined only by the field magnitude at the center.

D.A closed surface cannot have outward flux.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The divergence theorem connects total outward flux through a closed surface to the volume integral of divergence throughout the entire enclosed region. A positive divergence at one point does not determine the sign of that integral. Other parts of the volume may have negative divergence large enough to reverse the total flux.

Q9. A student argues: 'If the divergence at $P$ is zero, then the flux through every small closed surface centered at $P$ must be exactly zero.' Which correction is most accurate?

A.The statement is always true for every surface size.

B.Zero divergence means the field has zero magnitude at

P

C.For sufficiently small surfaces, the flux-to-volume ratio approaches zero, although a finite surface may still have nonzero total flux. ✅

D.Zero divergence means the field has no direction at

P

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Divergence is a local limiting quantity. If divergence is zero at $P$ , the net outward flux divided by the enclosed volume approaches zero as the region shrinks toward $P$ . A finite closed surface may still enclose locations where divergence is nonzero, producing nonzero total flux.

Q10. Consider a graph showing net outward flux $\Phi$ versus volume $V$ for increasingly small regions around the same point. The plotted points lie close to a straight line through the origin with slope $7$ . What quantity is best estimated by this slope?

A.The magnitude of the vector field

B.The divergence at the point ✅

C.The surface area of the region

D.The circulation density at the point

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: When small regions shrink around a point, divergence is represented by the limiting ratio $\Phi/V$ . On a graph of flux versus volume, that ratio is the slope. Therefore, a slope near $7$ indicates that the local divergence is approximately $7$ , provided the plotted data are sufficiently close to the limiting regime.

Q11. A rectangular control volume is enlarged while remaining inside a region where $\nabla\cdot\mathbf{F}=3$ everywhere. If its volume changes from $2$ to $10$ cubic units, how should its total outward flux change?

A.It remains

3

because divergence is constant.

B.It changes from

6

30

, assuming the divergence remains

3

throughout both volumes. ✅

C.It changes from

2

10

because flux equals volume.

D.It becomes zero because the control volume is larger.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: If divergence is uniformly $3$ throughout the region, the total outward flux equals the volume integral of divergence, which is $3V$ . Thus a volume of $2$ gives flux $6$ , while a volume of $10$ gives flux $30$ . This illustrates the distinction between local flux density and accumulated total flux.

Q12. Two analysts estimate the divergence at the same point. Analyst A uses a tiny cube and obtains $2.01$ . Analyst B uses a much larger cube and obtains $2.80$ . The field varies significantly across the larger cube. Which estimate is generally more appropriate for the pointwise divergence?

A.Analyst A, because divergence is defined through a local limiting process. ✅

B.Analyst B, because larger volumes always reduce measurement error.

C.Both estimates must be identical for any vector field.

D.Neither estimate can represent divergence because cubes cannot be used.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Pointwise divergence is a local quantity obtained by examining the flux-to-volume ratio as the region shrinks toward the point. A larger region averages behavior over locations where the divergence may differ substantially. Therefore, the tiny cube's estimate is generally closer to the pointwise value, assuming adequate numerical accuracy.

Q13. Suppose $\nabla\cdot\mathbf{F}=0$ throughout a solid region. A second field $\mathbf{G}$ has positive divergence in one part and negative divergence elsewhere, with the volume integral of divergence also equal to zero. Which comparison is correct for a closed surface enclosing the entire region?

A.Both fields must have positive outward flux.

B.Both fields must have negative outward flux.

C.Both fields can have zero total outward flux, even though the second field has local sources and sinks. ✅

D.Only the first field can have zero flux because zero divergence is required everywhere.

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: For the first field, zero divergence everywhere immediately implies zero net outward flux through any closed surface contained in the region. For the second field, positive and negative divergence can cancel in the volume integral. Thus its total flux can also be zero despite substantial local source-like and sink-like behavior.

Q14. A simulation displays a tiny spherical region around a point. The arrows crossing the sphere appear slightly denser outward on one side and slightly denser inward on the opposite side, while the field magnitude near the center is almost unchanged. The computed net outward flux is positive. What is the strongest conclusion?

A.The field necessarily has positive divergence because its magnitude is increasing.

B.The positive net flux is evidence of positive local divergence if the sphere is sufficiently small and the calculation represents the full boundary flux. ✅

C.The field must have nonzero curl and therefore cannot have divergence.

D.The field has zero divergence because its magnitude is nearly unchanged.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Divergence does not require the field magnitude to increase. It concerns the balance of outward and inward flux through a small closed surface. If the full boundary calculation gives positive net outward flux and the region is sufficiently small, the flux density provides evidence that the local divergence is positive.

Q15. A numerical model gives $\Phi(V)\approx 5V-0.2V^{3/2}$ for small control volumes around a point. As $V\to0$ , what does this model predict for the divergence at that point?

A.0

B.-4.8

C.-5 ✅

D.It diverges to infinity.

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Divergence is obtained from the limiting ratio $\Phi(V)/V$ as the volume shrinks to zero. Here $\Phi(V)/V=5-0.2\sqrt{V}$ . Since $\sqrt{V}\to0$ , the ratio approaches $5$ . Therefore, the model predicts a divergence of $5$ , even though the finite-volume flux includes a correction term.

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