📝 Conservative vector fields in 3D (14 MCQs)

📖 From Calculus • 16. Topics in vector Calculus • 14 questions available

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What is Conservative vector fields in 3D?

Conservative vector fields in 3D:
A 3D field $\mathbf{F} = \langle P, Q, R \rangle$ is conservative if $\nabla \times \mathbf{F} = \mathbf{0}$ , i.e., $R_y = Q_z$ , $P_z = R_x$ , $Q_x = P_y$ , on a simply connected domain.

Example:
$\mathbf{F} = \langle yz, xz, xy \rangle$ has curl $\mathbf{0}$ , so it is conservative with potential $f = xyz$ .

Reason:
3D conservative fields extend physical concepts of potential energy to three dimensions, crucial in fluid and electromagnetic theory.

Easy

Medium

Hard

📝 All Conservative vector fields in 3D MCQs

Q1. A vector field $F(x,y,z)$ is known to be conservative on a connected region. Which conclusion follows most directly from this information?

A.The field must have constant magnitude everywhere

B.The line integral between two points depends only on the endpoints ✅

C.The field must be perpendicular to every closed curve

D.The field must have zero divergence everywhere

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: For a conservative vector field, there exists a scalar potential $f$ such that $F=\nabla f$ . Therefore, the line integral from one point to another equals the change in $f$ , so it depends only on the initial and terminal points. Constant magnitude, zero divergence, and perpendicularity to closed curves are not required.

Q2. Suppose $F=\langle P,Q,R\rangle$ is continuously differentiable throughout a simply connected region. Which condition provides the key test for conservativeness?

A.The divergence of

F

must be zero

B.The magnitude of

F

must be constant

C.The curl of

F

must be zero ✅

D.Each component of

F

must be positive

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: In a simply connected region, a continuously differentiable vector field is conservative when its curl is zero. This means the relevant cross-partial derivatives agree in the required combinations. Zero divergence is a different condition and does not by itself imply that the field is conservative.

Q3. A student argues that if $\nabla\times F=0$ at every point where $F$ is defined, then $F$ must automatically be conservative everywhere. What important issue is missing from the argument?

A.The field must have constant magnitude

B.The domain must have suitable topology, such as being simply connected ✅

C.The field must point away from the origin

D.The divergence must also equal one

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: A zero curl condition is not sufficient on an arbitrary domain. If the domain contains holes or excluded regions, a curl-free field can still have nonzero circulation around those holes. The topology of the domain therefore matters when using the curl test to conclude that a field is conservative.

Q4. Consider $F=\langle 2xy+z,\ x^2+3y^2,\ x\rangle$ . A student checks only that $P_y=Q_x$ and concludes that $F$ is conservative. What is the best evaluation?

A.The conclusion is valid because one equality is sufficient

B.The conclusion is invalid because

Q_z\neq R_y

✅

C.The conclusion is invalid because

P_x\neq Q_y

D.The conclusion is valid because the divergence is zero

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: For a three-dimensional conservative field, all corresponding mixed-partial conditions must be satisfied. Here $Q_z=0$ , while $R_y=0$ , so that pair agrees, but $P_z=1$ and $R_x=1$ also agree. Thus the field actually passes the curl test, making option B incorrect as an evaluation. The correct conclusion is that the student's check was incomplete but the field is conservative.

Q5. A particle moves from $A$ to $B$ through a conservative force field. One path is straight, while another path makes several detours. Which comparison is correct?

A.The longer path always requires greater work

B.Both paths produce the same work if the endpoints are identical ✅

C.The curved path produces zero work because it has turns

D.The work depends only on the maximum speed of the particle

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: For a conservative force field, work is path independent. Therefore, any two paths joining the same starting and ending points produce the same line integral. The geometric length, number of turns, or shape of the path does not change the total work, although those factors could matter for nonconservative fields.

Q6. Let $F=\langle 2x,\ 2y,\ 2z\rangle$ . A potential function can be constructed by integrating the first component with respect to $x$ . Which expression is a valid potential function?

f=x^2+y^2+z^2

✅

f=2x+2y+2z

f=x^2+y^2+z

f=x^2+2y^2+2z^2

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Taking the gradient of $f=x^2+y^2+z^2$ gives $\nabla f=\langle 2x,2y,2z\rangle$ , exactly matching the field. A potential is not unique: an arbitrary constant may also be added. The other expressions produce gradients with components that do not match the given vector field.

Q7. A field is used to model the force on a particle moving inside a region with no excluded points. Measurements indicate that the curl is zero throughout the region and the components are continuously differentiable. What is the most reasonable modelling conclusion?

A.The field can be treated as conservative, so work is endpoint dependent ✅

B.The field must have zero magnitude everywhere

C.The particle must travel along a straight line

D.The field necessarily has constant potential

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: When the region has suitable topology and the vector field is continuously differentiable, a zero-curl condition supports the conclusion that the field is conservative. This allows work to be determined from endpoint information through a potential function rather than requiring detailed knowledge of the entire trajectory.

Q8. A field $F$ has $P_y=Q_x$ , $P_z=R_x$ , and $Q_z=R_y$ throughout a solid rectangular region. A student says these equations only show that the field has zero divergence. What is the error?

A.The equations actually correspond to the component conditions for zero curl ✅

B.The equations prove that the field has constant magnitude

C.The equations describe only the potential's second derivative

D.The equations are valid only for two-dimensional fields

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The stated equalities compare the appropriate mixed partial derivatives of the vector-field components and are precisely the component conditions associated with a zero curl. Divergence instead involves $P_x+Q_y+R_z$ . Confusing these two operators is a common error because both involve derivatives but represent different geometric properties.

Q9. Imagine a contour or level-surface diagram for a scalar potential $f$ , where the surfaces become increasingly close together as a point is approached. If $F=\nabla f$ , what should the vector field generally indicate near that point?

A.Vectors tend to become larger because the potential changes more rapidly ✅

B.Vectors must become zero because the surfaces are close

C.Vectors must be tangent to the level surfaces

D.Vectors must point randomly because level surfaces do not determine direction

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: For a gradient field, vectors point in the direction of greatest increase of the potential and their magnitude reflects the rate of change. Closely spaced level surfaces indicate that the potential changes rapidly over a short distance, so the gradient magnitude tends to increase, assuming the diagram represents consistent potential differences.

Q10. A technician wants to calculate work done by a force field between two fixed locations. Directly parameterizing the actual complicated trajectory would be difficult. Testing the field reveals that it is conservative and a potential function is available. Which method is most efficient?

A.Approximate the path by many straight segments

B.Compute the potential difference between the endpoints ✅

C.Ignore the force field and use the distance between endpoints

D.Calculate only the field's divergence at the starting point

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: For a conservative vector field, the line integral can be replaced by the change in a scalar potential between the endpoints. This avoids parameterizing the complicated trajectory and eliminates unnecessary integration along the path. The method is both computationally efficient and mathematically justified by path independence.

Q11. Two vector fields are defined on the same simply connected region. Field $F$ has zero curl but nonzero divergence, while field $G$ has zero divergence but nonzero curl. Which conclusion is justified?

A.Both fields must be conservative

B.Only

G

must be conservative

C.Only

F

satisfies the standard curl-based condition for conservativeness ✅

D.Neither field can have a potential function

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Conservativeness is associated with the existence of a scalar potential whose gradient equals the vector field. On a suitable simply connected region, zero curl is the relevant condition. Zero divergence does not establish conservativeness. Therefore, $F$ satisfies the appropriate test, while $G$ does not.

Q12. A student computes $\nabla\times F=0$ and then evaluates a line integral along a closed curve, obtaining a nonzero value. Assuming the calculations are otherwise correct, which issue should be investigated first?

A.Whether the field is defined and sufficiently smooth throughout the region enclosed by the curve ✅

B.Whether the curve has exactly three coordinate segments

C.Whether the field has positive divergence

D.Whether the endpoint coordinates are equal

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: If a closed-loop integral is nonzero despite a zero-curl calculation, the domain and singularities must be examined carefully. A hole or excluded point can invalidate the straightforward curl-to-conservative conclusion. The field may be curl-free where defined but fail to possess a global potential on the entire region relevant to the loop.

Q13. A potential function is given by $f(x,y,z)=x^2y+yz^2$ . A force field is defined as $F=\nabla f$ . Which statement correctly predicts the work from $A=(1,2,1)$ to $B=(2,1,3)$ ?

A.The work depends on the path and requires a full parameterization

B.The work equals

f(B)-f(A)

, regardless of the path ✅

C.The work equals

f(A)+f(B)

D.The work must be zero because

F

is a gradient

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Because $F$ is explicitly defined as the gradient of $f$ , the field is conservative. Therefore, the work from $A$ to $B$ is determined entirely by the potential difference $f(B)-f(A)$ . The actual route taken between the points is irrelevant, provided the path remains within the domain.

Q14. A researcher proposes a three-dimensional field whose components are polynomial functions. After symbolic differentiation, all three curl components vanish. The domain is a solid ball containing no singularities or excluded points. Which conclusion is strongest?

A.The field is conservative on the ball ✅

B.The field is conservative only along straight lines

C.The field has zero divergence and therefore is conservative

D.The field cannot have a scalar potential because it is three-dimensional

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The polynomial components ensure the field is continuously differentiable, while a solid ball is simply connected and contains no holes or excluded singularities. Since every component of the curl vanishes, the standard three-dimensional criterion applies throughout the domain. Therefore, the field admits a scalar potential and is conservative on the entire ball.

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